Modern Physics

MODERN

PHYSICS S E C O N D

E D IT I O N

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Physical Consta nts and Useful Values Speed of light Gravitational constant

2.99792458 X 108 mls 6.67 X 10-11 N. 2/ 2 m kg

c

G

Planck's constant

6.6260690 X 10-34 J. s 1.054572 X 1O-:J4 J s

BOltzmann constant

.

1.380650 X 10-23 JIK 6.0221418 X 1023 mol-I 1.60217649 X 10-19

Avogadro's number Fund amental charge Perrnitivity of free space

C

"0

Permeability of free space

47T X 10-7 NIA2

Coulomb constant

8.987552 X 109 N. m2 /C2

Electron mass

9.1093822 X IO-:J\ kg 1.6726217 X 10 2 7 kg 1.6749273 X 10-27 kg

Proton mass

-

Neutron mass Atomic mass unit

8.854187817 X 10- 1 2 C2/N' 2 m

u

1.6605389 X 10-27 kg = 931 .4941 MeV/c2

Conte nts

SectiollS marked willi a Gl lire optional and at Ihe same 11'1'1'1 as the main uctionJ, SeeliG/IS marked witll {/ rJ are optional mid more advanced.

Prefa ce

1

ix

The Dawn of a New Age

1.1 Troubling Questions

[

1.2 A Glimpse of the New World

3

2

Special Relati vity

2.1 Basic Ideas 2.2 Consequences of Einstein's Postulates

5 8 13

2.4 The 1\vin Paradox t!l

24

2.5 The Doppler Effect

28

2.6 Velocity Transformation

31

2.1 Momentum and Energy

33

2.8 General Relativity and a First Look 2.9 The light Barrier [!J 2.10 The Fourth Dimension rJ

3

43 54 56

Progress and Applications

60

Summary

61

Exercises

62

Waves and Particles I: Electromagnetic Radiation Behaving as Particles

13

3.1 Blackbody Radiation: A New

Fundamental Constant 3.2 The Photoelectric Effect

18

3.4 The Compton Effect

80

3.5 Pair Production

85

3.6 Is It a Wave or a Particle?

81

Progress and Applications

90

Summary

92

Exercises

93

Waves and Particles II: Matter Behaving as Waves

91

5

2.3 The Lorentz Transformation Equations

at Cosmology

3.3 The Production of X-Rays

74 15

4

4.1 A Double-Slit Experiment 4.2 Properties of Matter Waves

98 102

4.3 The Free-Particle SchrOdinger Equation 109 4.4 The Uncertainty Principle

113

4.5 The Not-Unseen Observer

121

4.6 The Bohr Model of the Atom [!]

122

4.1 Mathematical Basis of the Uncertainty

5

Principle-The Fourier Transform rJ

12'

Progress and Applications

131

Summary

133

Exercises

13'

Bound States: Simple Cases

141

5.1 The Schrooinger Equation

142

5.2 Stationary States

143

5.3 Physical Conditions: Well�Behaved Functions

145

v

147

5.4 A_ole_Bounds..... 5.5 c.. I, _In • 80>-The InfuUte

149

� 5.6 eo. 2, 1be Finite WoJI

158

5.7 c..l:�SJmpIeHannonkOsciUaror

164

5.1

ExpecIadon Volua. Uncertainties, _Open_

5.10 The CGmputer Approach iii

169 176 177

5.1 I WoJI·Defined a.rv.bl.., EJaenvaJues D

180

5.9 Nonstadonary S ..... iii

Progress and AppUC4IItJons

183

7.6 TheHydrogen Alom 7.1 Radial Probabiliry

7.8

Hydrogenlike Atoms

7.9 A Solurion Examined rJ 7.10 Photon Emission: Rules and Rates rJ

Exercises

276 277 279

Spin and Atomic Physics

289

Progress and Applications Summary

8

Summary

185

8.1 Evidence of Angular Momentum Quantization: A New Property

Ex"",1xs

187

8.2 Identical Particles

9,3 The Exclusion Principle

6 Unbound States: Steps,

8.4 Multielectron Atoms and the

Tunneling, and Particle-Wave 195 Propagation

6.1 The Potential St.p 6.2 The Potential Barrier and Tunneling 6.3 AJpha Decay and Other Applications 6." Particle-Wave Propagation Progress and Applications Summary

Exercises

7 Quantum Mechanics i n Three Dimensions and the Hydrogen Atom

7.1 The Schnklinger Equation in Three

195 202 207 213 221 223 224

231

7.2 The 3D Infinite WeD 7.3 Energy Quantization and Spectral Lines In Hydrogen

238

7." 1be Schrtklinger Equation for a Central Force 7.5 Angular Behavior in a Central Force

8.5 Characteristic X·Rays The Spin·Orbit Interaction rJ

8.7 Adding Angular Momenta rJ 8.8 External Magnetic Fields and the Z·Axis rJ 8.9 Excitation Spectra rJ Progress and Applications Summary Exercises

9 Statistical Mechanics

309 316 318 320 325 331 335 337 339 347

9.8 The Laser 9.9 Specific Heats rJ

390

9.3 The Boltzmann Distribution 9.4 Classical Averages 9.S Quantum Distributions 9.6 The Quantum Gas 9.7 Massless Bosons: The Photon Gas

245 248

289 300 307

348 351 355 366 169 374 182 385

9.1 A Simple Thermodynamic System 9.2 Entropy and Temperature C!l

231 233

Dimensions

Periodic Table

8.6

255 262 266 269 271

c_

10

PrOgT�S5 and Applications

399

Summary

�Ol

Exercises

403

Bonding: Molecules and Solids

413

10.1 When Atoms Come Together

413

10.2 Molecules

414

10.3 Rotation and Vibration

422

10.4 Crystalline Solids

428

IO.S Energy Bands and Electrical Conduction

431

10.6 Conductors, Insulators. and Semiconductors

439

10.7 Semiconductor Theory

444

10.8 Semiconductor Devices

449

10.9 Superconduclivity

453

10.10 Fullcrenes

460

12

Fundamental Particles a nd Interactions

11

464

Summary

465

Exercises

467

Nuclear Physics

47 5

523

12.1 How Forces Act

523

12.2 Antiparticles

525

12.3 Forces and Particles: How Many?

52S

12.4 Particle Production and Detection

53 6

12.5 Decay Modes and Conservation Rules in the Standard Model

539

12.6 Parity, Charge Conjugation. and Time Reversal

545

12.7 Unified Theories and Cosmology

548

Progress and Applications

554

Summary

555

Exercises

555

Appendices A The Michelson-Morley Experiment

Progress and Applications

..-

A-I

8 The Lorentz Transformation: Plotting

Events

8-1

C Planck's Blackbody Radiation Law

C-I

0 Solving for the Fourier Transfonn

0-1

E The Momentum Operator

E-I

11.1 Basic Structure

475

F Time Evolution of a Gaussian Wave Packet

F-l

11.2 Binding

478

G The Operator for L2

G-I

11.3 Nuclear Models

486

H Energy Distributions

H-I

11.4 Nuclear Magnetic Resonance and MRI 9

491

11.5 Radioactivity

492

J Probability, Mean, Standard Deviation. and Numbers of Ways

11.6 The Radioactive Decay Law

498

K Some Important Math

11.7 Nuclear Reactions

503

Properties of Isotopes

1-1

J-I K-l

Answers to Selected Exercises

A:-I-l CR-\

Progress and Applications

515

Credits

Summary

517

Index

Exercises

518

1'<- 1

Prefa ce

This lex!. formerly known as Nonclassical Physics: Beyond Newton's View. is written for the student who has completed a calculus-based introductory course in classical physics. Thus, it is appropriate for a course taught either in the sophomore year or the early part of the junior year. It will introduce the reader to a wide range of fascinating physics principles that have emerged since the lurn of the twentieth century. As technology advances and we study the things around us ever more closely, physics and other disciplines­ engineering. chemistry, biology-will continue to draw nearer together. so a grasp of the principles of modern physics becomes more and more essential. This text will enable students to apply these principles quantitatively in all areas of natural science as weJl as to further study in physics. Having taught this subject for many years, I know the excitement students feel about it. After classical physics, it is quite a shock, but a delightful one. However, because it often seems counterintuitive, explanations and discussions must be much more thorough than in classical physics; otherwise. the excite­ ment may be overshadowed by confusion. Providing clear introductions that address common misconceptions is one of the main goals of the text. A related goal is 10 present a consistent approach to the various topics. [nconsistency can be extremely distracting to students, so care has been taken not only to avoid oversimplified explanations on fundamental points that should be treated with rigor at this level, but also to avoid doggedly preserving rigor when students at this level have not had the preparation to follow it. Accordingly, more than the usual amount of time is spent on some basic ideas. If these ideas are not grasped in this course, they often never are, for they are frequently taken for granted in later courses. For example, How can two observers each see the other aging slower? What is the true basis of the uncertainty principle? What really causes quantization? At the other end of the spectrum, topics too com­ plex for a quantitative treatment are clearly delineated and are discussed so as to give the studentlhe best grasp of the basic qualitative ideas.

Course Prerequisites To get the most out of this text, the student should be acquainted with the stan­ dard classical t opics of mechanics. waves, thermodynamics, and electricity and magnetism-in particular, the first two. Regarding the math prerequisites. familiarity with partial derivatives is assumed. and use is made of complex

ix

numbers. though not much beyond their definition and simple arithmetic. Allhough quantum mechanics cannot be understood without some use of dif· ferential equations. the text does not assume a prior understanding of them: the studenr is guided through the few necessary steps. Above all, the topic of the lext is physics. not math. The mathematics of differential equations. for in"tance. is merely a tool. and if there is a known mathematical solution to an equation, it will usually be applied without much concern about how it is obtained. Appendix K, which summarizes several math topics. discusses solu· rions to the differential equations used most oflen in the text.

Organization Each chapter is divided into sections. and mOSI chaplers have sections expressly designated as optional. of which there are two Iypes. Optional sec· lions at the same level as main sections are mnrked with 00 in the Contents and within the chapters. while advanced optional sections are marked with a. Occasionally, an optional section will be mentioned elsewhere in the text. but rarely would it need 10 be explicitly covered to understand the discussion where it is memioned. Quantitative application of material in either type of optional section is restricted almost exclusively to that section or to closely related optional sections. Nevertheless. even skipping all the optional sections may not satisfy time or curricwum constraints. Therefore. many sections not explicitly designated as optional have been made sufficiently self·contained that they can be sldpped without serious consequence. Indeed. most as self·contained as possible.

chaplers are wrinen to be

Each chapter closes with a Progress and Applications section that dis· cusses various topics related to the material in the chapter-recent advances, lingering mysteries, interesting and important applications-followed by a summary of the cbapter's important points and concluding with end-of-chapter exercises. In addition to conceptual and computational exercises (discussed below), each Chapter has a large number of quantitative exercises grouped according to chapter section, as well as comprehensive exercises that bring together ideas from other areas. Exercises are either standard ones. which rdIlge from easy to moderate difficulty, or challenging ones, involving deeper physical principles or more complex math. Challenging exercises are distin· guished wim asterisks. By making suitable choices, students at all levels can be appropriately challenged.

New in the Second Edition The second edition brings a number of new features: •

•

Chapter outlines and brief introductions give students a road map to the material covered in each chapter. One of the most important changes has been a signUicant reorganiza­

tion to make the text more flexible for users of various interest: or lev­ els of preparation. Advanced sections are now uniformly found

<.It

the

end of the chapter. Some sections not designated as advanced involve topics thai can be understood well on one level without sophisticated math but that may also be enhanced by a more mathematical treatment. Because it would be inappropriate to displace that treatment to its own section. these sections address the more-sophisticated math in feature boxes entitled A Closer Look. Covering them certainly gives a deeper understanding. but omitting them will not disturb the now of the sec­ tion. Some end-of-chapter exercises relate specifically to material in these subsections. •

•

•

Pedagogically sophisticated figures, complete with chalkboard-style annotations. provide great clarification of main principles. Updated in-text Examples walk students step-by-step through solving problems to better prepare them for the end-of-chapter exercises.

Real-World Examples have been added to almost all chapters. These examples contain expanded discussions and focus on applications stu­ dents hear about in other disciplines and in the news. In many cases, they discuss modem technology that students have actually used or will soon use in their academic careers.

•

All chaplers now have an array of Conceptual Questions. The main aim is to ensure a firm qualitative understanding of the chapter's ideas. Many address points where misconceptions arise. Some provide a brief introduction to ideas discussed in later chapters.

•

Most chapters now have dedicated Computational Exercises. Many applications in modem physics are particularly well suited to numeri­ cal solution, animation, or approximation on a computer, and pursuing these can provide insight impossible to gain otherwise. The computa­ tional exercises vary widely in depth and in required skills. Several would better qualiFy as projects rather than exercises.

•

End-oF-chapter exercises have been increased by more than 70 percent, with particular attention paid to the later chapters on atomic, statistical, molecular, and solid-state physics.

•

The Progress and Applications features at the end of each chapter have been completely rewritten and updated, with a somewhat greater emphasis on applications in engineering, computer science, and biology.

•

Users of the first edition will note that a short introductory chap­

ter has been added. It provides a perspective on modem physics, pointing out some of the most baffling problems facing classical physics at the dawn of the modem age. It then briefly charts the course ahead.

•

Chapter Summaries now incorporate the Basic Equations section From the first edition, emphasizing the relationship and i mportance of the equations to the chapter's key topics.

•

Appendices differ significantly From the first edition. A few have been shortened or dropped, with some ideas moved to inductive end-of­ chapter exercises. An important addition is Appendix J, which covers the basics of probability and averages. Students not already well acquainted with these ideas now have a convenient resource.

d_ All chaplers have bee n streamlined. Nonessential equations have been omit� red and explanations made more concise. Specific changes of note are as follows: Chapter 3. on panicle properties of light, adds a brief introduction to blackbody radiation. Chapter 5. on quanlum�mechanjcal bound states, now has the three o;imple cases in consecutive sections. The approach to solving the SchrOdinger equation is simplified in the infinite weU. but the more complete approach of the first edition is preserved in A Closer Look in the finite well section. Expectation values and operators are moved nearer the chapter's end and can be omitted if desired. Reference to a numerical technique for solving the SchrOdinger equation ha.o; bee n fonnalized and moved to its own optional section. In Chapter 6, on unbound states. the section dealing with phase and group velocity (Section 6.4) is no longer designated advanced. Much of the underlying math is moved to an appendix. In Chapter 7. the discussion of spectra now precedes that of the hydro� gen Schrodinger equation. Much of the math surrounding the latter is now con� centrated in A Closer Look. Ir can be omiued, but it is much easier to follow than before. Chapter 8, on atomic physics, discusses characteristic X-rays earlier than in the flIst edition. Much of the math in the section on identical particles is now in A Closer Look. and the entire section can be omitted or treated very quickJy. The topics in Chapter 9 on statistical mechanics are essentially unchanged, but the focus has been improved. The section on microscopic rime reversibility has been dropped. and noticeably fewer equations distract the reader. Chapter 10, on molecules and solids, refers less to statistical mechanics (losing a section, but not all of its content) and adds a section on fullerenes. Fundamenlal. particles in Chapter 12 discusses fewer conservation rules but expands its follow-up to the cosmology begun in the relativity chapter with a brief discussion of the Fried­ mann equation and the effect of a cosmological constant.

Ancillaries Offered in conjunction with the text are the following ancillaries. developed especially for the second edition by the textbook author: •

Instructor Solutions Manual (ISBN 0-8053-1709-0): These solutions

(Q all the end�of�chapter exercises provide not just (he bare mathemati­

cal steps but also words of explanation linking key ideas .

•

Companio" WebsiJe (www.wps.aw.comlHarris_mp_2):This open­

access website will provide a variety of resources (Q supplement the text. including appJets and animations, a classical physics review. and more.

The text is up to date with current research as of publication. but progress occurs at a dizzying pace in modem physics. An lntemet search on any of the key words in the text will provide a wealth of further information and is liable to reveal advances and discoveries no more than months or even days old.

Acknowledgments My warmest appreciation goes to the entire editorial and production team at Pearson Addison-Wesley. Acquisitions Editor Lothl6rien Hornet's vision and drive have contributed greatly to the improvements in the second edition.

Preface

It has been a pleasure working with Assistant Editor Ashley Taylor Anderson. who kept the project on track and managed countless elements of the process with great poise despite hectic and changing deadlines. Development Editor Brad Patlerson's unwavering aUention to consis tency and style has been won­ derful. Many thanks also go to copyeditor Tara Joffe for her excellent, meticu­ Jous work. And Emily Autumn at GGS Book Services and proofreader Denne Wesolowski deserve special recognition for their hard work and attention to detail.

Most sincere thanks go to all those who have reviewed the text: John Albright, Florida State University Bradley Ambrose, Grand Vafley State University Paul Avery, University of Florida Bereket Berhane, Embry-Riddle Aeronautical University Charles C. B!at chley, Pittsburgh State University

Arthur 1. Braundmeier, Southem ltJinois University Anthony J. Buffa, California Polytechnic State University Marshall Burns, Tuskegee University Louis H. Cadwell, Providence College Marcelo Cam peri, University of San Francisco Bernard Chasan, Boston University Randall T. Dillingham, Northern Arizona Universify Joseph DiRienzi. College of Notre Dame Theodore Einstein, University of Mary/and Nayer Er adat, Middle Tennessee State University

James Erskine, University of Texas Massimilliano Galeazzi, Miami University Heather Galloway, Texas State University

Cynthia Galovich, University of North Carolina Scot Gould, Claremont College Richard F. Haglund, Vanderbilt University L. Michael Hayden, University of Maryland

Tracy Hodge, Rice Unil'ersity Georg Hoffstaelter, Cornell University Yong Suk Joe, BalJ State University Chung Ngoc Leung, University of Delaware Michael Lieber, University ofArkansas Linda L. McDonald, North Park College Stephen Mecca, Providence University

Gary Mitchell, Duke University Enrique Moreno, West Virginia University Herbert M u eth er. SUNY Stony Brook Ernest B. Nischmidt, Idaho State Universiry Linda 1. Olafsen, Baylor University Feryal Ozel, University ofArizona L awrence Pinsky, Unil'ersity of HOl/ston Anthony Pitucco, Pima Community College

John A. Polo, Edinboro University

xiii

Chandra Prayaga, Ultil'er.sity of Western Florida Palric..'ia Rankin, UniJ'ersiry of Colorado Warren Rogers. l�estmoflt College Jeffrey Schmidr. UniJ'ersity of Wiscollsin-Parkside Raji", Singh. Uni�'ersity of California. Dads Darre' Smith. Embry-Riddle Aeronautical Unit'ersiry Shirvel Stanislaus. Valporaiso Ullit'ersity Dejan Slojkovic. Case Western Resenle University Andrew Suslich, Arkansas State Unh'ersity John Teming. University of California. DOI'is Mark Timko, ElmhursT Unh'ersity Sleven Tonsfeldt. Buena Vi.sta Unil'ersit)' John Truedson, Bemidji State Unfl'ersit)' Blair R. Thule. Pel/I/syli'ania Stale University-Erie James Whiteneon, Southern Polytechnic SllIfe UniversiTY Douglas Young. Mercer Ullillersity I am deeply indebted to all the users of the text who have provided So

many importan! comments and suggestions. Above ali, 1 am profoundly grate­ ful to my students! None could be belter al catching mistakes, questioning explanations, or discerning ambiguities. and no (ext dealing with ideas so challenging could be successful without this invaluable feedback. Randy Harris Davis, California July 2007

1 The D aw n

of a New Age

Chapler Outline I I

1_2

Troubling Quc"ion<; A Glimp<;e of the New World

I

ntroductory cla'isical physics covers a body of knowledge that can explain a vast amount of what goes on around us, from the every­ day to the ab�tracl. Why does a bicycle go forward when the rider pu�he.. downward on a pedal? What fonm can energy take. and if it I!. conserved. wby can'l we reu<,c indefinitely,! How do waves propagate energy and mfonnation without any matter moving from source to receiver'! What are electric and magnetic fields? However, early in the 20th century. revolutionary ideas arose lhal .,hook classical physics to liS foundallon. Even the most basic and "obvious" truth.;, about space and time and the nature of the matter around u .. came under assault. We will slUdy these ideas in detail in the chapters to come. In this chapter. we discuss c,everal telling signs that something w� wrong with the classical picture. then we survey the path ahead. (Cenain topics from cla..!oical physics are panicularly Imponant to modem physic.!.. A review of these topi" is available on the companion website.)

1.1

Troubling Questions

The core topics of cla.!.sical physics are mechanics. electromagnetism. and thennodynamics. Let us take a look al some of the problems that loomed In these areas al the dawn of the modem age. Classical mechanics allained a cohesive fOffil in the late 160Cb with the work of Sir Isaac Newton. Triumphant in ex.plaining the behaviors of macro­ scopic objects at ordinary ..peeds. Newlon's work reigned unchallenged for een­ tunes. As convincing as anything were its successes in celestial application'> When the orbit of the planet Uranus was found to devime <;lightly from what ... to attribute Newton's laws predicted it should be, the inslincti\'e respon..e wa. the deviation not to any failure of those law<; but rather to some

un..een

heavenly

body. Newton's laws predicted its location, and Neptune was later found right where It should be. After such <;3lisfying confinnation. it was natural to expct:t funher example". Mereu!)'·... Orbit I" somewhat elliptical. but the ellip� i� not retraced again and again. It prece!o\Cs-that is. it .. point .. of ma.ximum r.Jdiu.. advance slightly with each orbit about the Sun. Ne\\-1on's law" prooif.:ted one

1

2 QIptIr 1

The Dlwn of. New AJC'

rate of precession: actual ob.liervalion differed. The discrepancy was again lhought 10 be due 10 an un"een planet. bul in Ihis case. none could be found_ Maluring in rhe mid-19rh century with James Clerk Maxwell's comple_ tion and inlegration of the law" of Gau".., Ampere. and Faraday. electromag_ Delism has been extremely liuccessful. Ir superbly explains the physics behind telephonic communication and elecrrical power supply, which had become commonplace b}' the end of the 19th cenlUry. and it conlinues to prove its \'alidiry in counlless applications 10 this day. Early on, however, doubts were raised about the theory's prediction of waves of electromagnelic radiation. For one thing. me lack of explicit reference to a medium of propagation seemed to put ligh t in a special category among wave phenomena. Another problem had 10 do wilh the energy expected in electromagnetic radiation exch anged with maller. Charges in the maHer "hould jiggle around at rates dependent on Ihe temperature, producing and absorbmg electromagnetic energy as they do. A slandard wave calculation predicted that the electromagnetic intensity nearby should be infinite! Yet another perplexing question concerned the abil­ lly of lig ht to eject electrons from a metal, known as the photoelectric effect. In the classical view. a light wave is simply a pair of self-propagating electric and magnetic fields spread diffusely through some region of space. If light encounters an electron in a metal. these fields should be able to transfer energy to the electron and knock it OUI of the melaJ. Light of low intensity might require considerable time to deposit enough energy. but a high intensity should knock electrons out at a high rate and--owlOg 10 its stronger electric and magnetic fields-should eject them with greater kinetic energy. In fact, even very low-inrensity light can eject electrons immediately, and the kinetic energy of an ejected electron is completely independent of the light's intensity. The frequency of [he light seemed to be the deciding faclor, and classical elec­ tromagnetism could not explain why. By the end of Ihe 19th century, statistical thennodynamics had become one of the cornerstones of physics. Its laws had been established, and correct predictions were being made. A major step forward was (he formulation of the equipartition theorem. This says that each independent degree of freedom pos­ �essed by a particle in a thermodynamic system should manifest of energy on average, where is the Boltzmann constant and T is the tempera­ ture. There are three dimensions of translational freedom, leading to the famous fonnuJa for the average translational kinetic energy of a particle: In a solid, each atom has three additional degrees of freedom due to elastic potential energy in each dimension. With SIX 100ai degrees of freedom per atom, the equipartition theorem predict) that a solid should have a heat capaciry­ energy per degree per mole-of where is Avogadro's number This prediction supported an early emplflcal observation that many solid, !l:eem to have a heat capacity very near thiS value. However. even solids that adhered to the prediction at ordinary temperatures deviated noticeably at low temperatures, where the heat capacity seemed to drop off toward zero. To cla�· sical statistical physics. this was thoroughly baffling. Perhaps attracting more scrutmy than any other classically inexplicablt phenomenon at the tum of the 20th cemury was the subject of atomic 5.pectra. Atoms cnllt only certain wavelengths of light. Why? With the discovery of tho

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� kB T.

3kBNAy'

NAY

1 _2

ele<:tron by J. J. Thomson In 1898. hopes of explaining spectra rose-for a ,mall charged particle somehow jiggling around in an atom should emit electromag­ netic radiation. Neither the proton nor the momic nucleus was yet known, and Thomson's model of the mom assumed that its electron.. were embedded 10 a uniform sea of positive charge. No plausible oscillations of the electrons in thi, model could explain the observations. About a dozen years later, the work of Ernesl Rutherford and his students produced the now-familiar model of electron .. orbiting a positive nucleus. Although the quantum age had by then begun, Rutherford's nuclear model was still classical, and il actually compounded the mystery. An orbiting electron would be accelerating continuously, and any time a charged particle accelermes. it radiates electromagnetic energy. The nuclear atom should be unstable, with the electron spiralling into the nucleus! Perhaps not �ur­ prisingly, this model also failed to explain the spectral evidence. Some of its baliic elements survived, but it was 10 be profoundly altered by the new paradigm. We will discover the answers to these problems and many others that faced classical physics in the chapters to come.

1.2 A Glimpse o f the New World

Much of modem physics rests on two basic ideas: First. space and time are not the absolutes they might seem to be. Second, things we might think of as particles may behave as waves, and vice versa. Although these ideas are now accepted as fundamental to physics, they initially met with considerable opposition. The main reason they remained hidden for so long is that they involve behaviors not easily observed. And the reason they were not universally wel­ comed is thai they often seem counterintuitive. These two reasons are related. A behavior certainly would not be counterintuitive if it were subject to simple observation. By the same token, we cannOl justifiably claim to possess intu­ ition about a phenomenon we cannot observe. Loosely speaking, modem physics is the study of the slllall and the fast, but no one has ever actually seen an indi­ vidual atom nor has anyone ever traveled at a significant fraction of the speed of light relative to Earth. We must be careful not to apply preexisting notions based on experience to situations in which we have no experience. The branch of modem physics dealing with space and lime is special rela­ tivity. Although it is often said to be the physics that applies when objects or frames of reference are moving at very high speed, this is rather misleading. for it makes relativity sound like a special case. Relativity agrees with classical physics at low speeds, but it also succeeds at speeds comparable to that of light. where classical physics fails. Thus. classical mechanics is the special case. Of the many startling claims of special relativity, one is particularly helpful as an introduction and preparation for the challenges ahead: If passengers on a (vel)) high-speed train confirm that clocks at the front and back of their train ,trike noon simultaneously, observers on the ground will confirm that these clock.. do not strike noon simultaneously. This discrepancy is very small at ordinary speeds. but it is

1101 an optical illusion.

The other main branch of modem physics is quantum mechallics. which has its own challenging notions. In classical mechanics and electromagneli\m,

AGlim�oftheNcwWor

ld

3

we rrcal the electron as a panicle. bUI In 'imaJl confines. il behaves as a diffuse wave. It does nol have a �pecific location! All we can know are probabilities of finding the electron if an attempt were made 10 do 0;0. and the probabilities are related 10 the amplilude of the wave. These claims are mel with amuse­ men! and disbelief by some people. bUI upon them rest major areas of science. ..uch a'l chemisoy. modem electronics. materials science. modem optics (lasers). nuclear physic... and a host of others. We say that quantum mechanks applies 10 Ihe realm of "ery small confines. not usualJy open to casual scrutiny. but as 10 the ca�e of relativity. this is somewhat inaccurate. Quantum mechanics is correel for small and large and converges to the special case of classical mechanics in the Illnit of large things. It is nalural to ask how classical mechanics can be a special case of two dIfferent things. The figure schematically depicls the realms of applicability of the difef rent theories. The special relativity we study in this text is valid only for large things. It is nOI correct quantum mechanically. Similarly. the quan­ tum mechanics we study is, with n few noled exceptions. nol relalivislically correct; rather. it is valid only for slow-moving things. Classical physics is lhe region where these two overlap. The region conspicuously missed by both regions is the realm of the small and fast. This is the focus of high-energy physics-the search for the fundamenlal struclure of the universe-which we introduce in the text's final chapter. Slow

ClassicJI phym'�

Small

Fa�[

Special reJa!i\ II)'

QuanllJm mt:chanics

Our journey begins with relativity in Chapter 2. In Chaplers 3 through 8, we learn (he basics of quantum mechanics. Afterward are several chapters on specific topics in modem physics. These point the way to the frontiers of physics and the furure of technology. In many ways. they are applications of the earlier ma!erial, but each presents new challenges. We will learn important model.., each with its own set of fundamental assumptions. Classical physics is tremendously successful in its realm, but modem physics is truly an eye�opener. On to the new world!

2 Special Relativity Chapter Outline

2.1

:u 2.3 [!] 2.4 2.5 2.6 2.7 2.8

Basic Idt:as Consequences of Einstein's Posluiales The Lorentz Trnnsfomlalion Equations The l\vin Paradox The Doppler Errect Velocity Tnm!.formalion Momentum and Energy General Relntivity and a First Look at Cosmology The Light Barrier

III 2.9 r:J 2.10 The Fourth Dimension

S

pedal relalivity is the theory governing physical phenomena when one object or reference frame moves relative to another al a speed

comparable to that of light: c = 3 X lOS m/s. At such high speed<.., classical NewlOllilill mechanics has been found to be inadequate. and �eemingly obvious truths about space, time, and energy are profoundly altered. It is very important \0

bear in mind that essenlially all speeds with which we have experience are

"low" speeds. much less lhan c. Spacecraft. for instance. lravel no faster than a mere 25.000 mi/h. or about 10" mls. At such speeds. relativistic effects are imperceptibly small. and classical mechanics is satisfactory. Because we have no experience with high speeds. we really have no reliable intuition by which to judge the predictions of special relativity. Startling though these predictions are. it cannot be said that they are conlrary to observation and therefore incorrect. Given the high speed required, it is little wonder that special relativity emerged so late-there was no obvious evidence of its effects. Its discovery by Alben Einstein in 1905 was brought on by advances in the field of eleclfo· magnetism in the mid-19th century. Even so, Einstein's theory-a work of genius-was a tremendous leap forward and ran counter to many cherished beliefs. Not surprisingly, detractors abounded. With technological advances in the years since its discovery, however. the experimental evidence of its validity has mounted. until no serious opposition remains.

2.1

Basic Ideas

An imponant concept in special relativity is that of an inertial frame of refer· ence, or simply inertial frame. By definition, this is a frame of reference in which an object e'
velocity. In classical physics. this is any frame moving at constant velocity.

5

6

Chlpter 2

Special Rcllltl'dl)'

Consider. for example. an object noaling freely in space. Any observer moving al conslant velocity, perhaps gliding by in an unpowered spacecraft. would determine the object's velocity 10 be constant (ils aclual value dependent on the

spacecraft's velocity) and would thus be in an inertial frame. An observer in an accelerating rocket would see the object's velocity changing relative to h i m- Or hers.elf. so the rocket would not be an inertial frame. Einstein's fundamental postulates of special relativiJy can be stated as follows:

I . The form of each physical law is the same in all inertial frames.

2. Light moves at the same speed relative to all observers.

According to the first postulate, an experiment should proceed identically

in a spacecraft noaling at rest relative to Earth as in a spacecraft moving at a constant 1 ,000,000 mJs relative to Earth. No amount of experimentation could prove that a given inertial frame is actually moving. Consider a simple exam_ ple: If a man standing on the ground drops a ball from shoulder height, it lands at his feet a certain span of time later. If he is inside a high-speed train moving horizontally at a constant 100 mJs and drops the ball, it lands at his feet after the same span of time. In each case, his observation agrees with a law of the same form,

F

= ma,

without taking into account any motion relative to any­

thing else. The physical behaviors are identical. In fact. were it nol for give­ aways like bumps, noise, windows, and so forth, the man simply could not prove that his frame is in motion relative to Earth. Since he has no evidence that it is

he who moves. he can justifiably claim that he is stationary, and it is

Earth that moves. According to Einstein's first postulate, then, it is impossible to prove that anyone frame is truly in motion or thai any frame is truly at rest.

We say that there is no universal rest frame relative to which all motion might be measured. No motion is absolute-all is relative. Einstein'S first poslulate seems quite reasonable when applied to simple mechanical examples like the man on the train. Bul Einstein's second postulate can be very troubling. Imagine that Anna, a track star, runs to the right at 0.2c relative 10 Bob, who is standing still, as depicled in Figure 2. 1 . Anna carries a Figure 2.1

Anna and Bob walch a light

beam.

nashlight and shines it in the direction in which she is running. According to Einstein'S second postulate. Anna must determine that lhe beam moves at c rela­

tive to herself. and Bob must also detennine that the beam moves at c relative

to

himself A student of classical physics is often tempted to claim either that the

Anna

beam moves at I.Oe relative to Anna and 1 .2e relative to Bob or that it moves at I.De relative to Bob and O.Se relative to Anna. The underlying assumption is that we may add the velocity of the beam relative to Anna to the velocity of Anna \'

-

0.«

Bob

relative to Bob to obtain the velocity of the beam relative to Bob. We will see later that this classical assumption about adding velocities is invalid. In any case. the two claims are incompatible. Therefore, we face the question, Relative to what or whom does light move at c? Besides the obselVer-that is. besides Einstein's answer-there are only two alternatives: the source and a medium. As to the flISt alternative. Maxwell's equations. which govern all electro­ magnetic phenomena. predici thai light propagates al a speed independent of the speed of the source. All experiments in which light from high-speed sources has been timed agree. The second alternative calls for a bit more thought.

2. 1 Ba.sic Ide.

7

Mechanical waves require a medium whose deformation is communicaled from one point to another via internlolecular forces. resulting in a propagating wave. The �peed of the wave depends on properties of the medium. not on the

i

motion of source or observer. An explosion causes pressure varations III the air around it. This "deformation" propagates outward as a sound wave at a speed dependent only on properties of the medium-air. It is reasonable to suppose that the propagation of light also requires a medium, and early scien­ tists chose the name ether. What air is to sound, ether would be to light. Thus. light would always move at speed c

relative to the ether. independent of the

motion of source or observer. Furthermore. since light travels throughout the universe. ether would have to permeate space. We might even go so far a<> to

proclaim ether the universal rest frame-the one thing really at rest, relative to which the motions of all other things could be judged. Let us return to Anna running with the flashlight. We now accept thal because Anna is the source, her motion has no effect on the speed of the light.

To claim that the beam moves at c re lative to Bob is to claim that the ether is at rest with respecl to Bob and that Anna would see the light moving away from hersel f at O.Se. By analogy. were Anna to shout while running through still air

0.2vs' where \'s is the speed of sound in air, the sound i n front of Anna would be moving away at 0.8vs, while moving at 1 .0vs relative to Bob. On the other at

hand. to claim that the l ig ht beam moves at e relative to Anna is to claim that the ether moves with Anna and would therefore carry the light away from Bob at 1.2c. Again by anaJogy, if the air happened to be stationary relative to Anna and thus blowing at 0.2V5 relative to Bob. the sound of Anna's shout would move at 1 .0vs relative to Anna and 1 .2vs relative to Bob.

If ether exists, we might expect it to be detectable. Certainly. it is not as apparent as air. But by considering how we might prove the existence of air, we will see how we might do the same for ether. Even if observers cannot feel the wind at their faces. they can detemline the speed at which air moves by

timing echoes. Suppose the air is moving at Y ir relative to the stationary Bob, Bob shouts and listens for the echo from an obstacle down­

as in Figure

a

2.2.

wind, a distance L away. The sound moves at Vs +

travels downwind, arriving at the obstacle after a

Yair relative to Bob as it time U(vs + Yair)' After

- Yair and anives back at Bob's location after an additional time U(vs - v ir). If we add these and find a common a denominator, we obtain a total time of 2vsU( V; - v;iJ. Bob could calculate the speed Y ir relative to himself knowing only the speed V of sound in air. the a s reflecting, the sound returns at

distance

Vs

L. and the round-trip time.

The first experiments that attempted to determine the speed of ether rela­

Figure 2.2 The round-trip time for

tive to Earth involved liming round-trips of light beams. Appendix A dis­

sound depends on the motion of Its

cusses the most famous. the Michelson-Morley experiment, which used the

medium. air.

interference of light taking round-trips in perpendicular directions. All such experiments have failed to find a nonzero speed. Curiously. ether doe� not appear to be moving relative to Eanh. Maybe this is lrue. But Earth orbits the Sun. Why should ether orbit with Earth? To salvage the belief in ether. some scientists suggested that perhaps moving objects like Earth drag ether along with them. The results of experiments over many years showed no e\'idence of .... ith each new such an ability; indeed, the belief in ether was further .;;haken \

Bob

f

I>-

\. =

"

,,�

v.

+ -

.�

L

\'.."

"air

I

e 0: �ther's e)(ist

�nce, another failed anempt to find it. There being no evidenc . g possibility is that it does not exist. This is the prevallm belief: Light does not

require a medium. Were ether 10 exist, then

. . a frame stationary relative to Lhe ether onl), In .

would light move at c in all directions. In any other frame, light would move away from an observer at different speeds in different directions, like a shout on a windy day. MaxweU's equations predict that light will move at c, independent of direction. If ether existed, this could only be true in the one frame stationary relative to the ether, and Maxwell's equations would have to be of a different fonn in all other frames. Einstein's first posrulate would not be met. Perhaps it is possible to patch up Maxwell's equations to include the effects of a frame 's

motion through ether. to generalize them so that those we now accept are a spe­ cial case of a fonn correct for all frames. However, with no evidence of ether's

e)(iSlence, Einstein's second postulate is as plausible a solution 10 the "problem" a.o:;

as we IIOW have c in all directions

any. In essence. it is a proclamation that Maxwell'S equations

them are correct in all frames. Consequently, light moves at

relative to an observer, regardless of the observer's frame of reference. Still, the reader may find the postulate unpalatable. Even though Anna and Bob move at

O.2c relative to one another, each must determine a beam of light c relative to him- or herself. Is something wrong with our con­

to be moving at

cepts of distance and time or with the laws of mechanics? The answer is full of irony. Originally. no one doubted the laws of mechanics. It was the laws of electromagnetism that were Lhought to be in error, because they predicted the same speed for light moving in all directions without reference to a medium. But with Einstein's postulate that Maxwell's equations are correct in all frames, we find that it is the laws of mechanics that must be reevaluated. Classical notions of space and time must be discarded. In special relativity, distance and time become ine)(tricably related. Accordingly, the following definition is indbpensable: An event is anything with a location in space and a time. The birth of a child is a single event, occurring al a certain time and at a certain location. The regular eruptions of an ancient geyser constitute a of events. all with the same location but different times. In taking a snapshot of a friend, we record a collection of events, all with the same time but different

collection

locations.

Let us now discuss the startling consequences of Einstein's postulates. As we do, we will often refer to observers Anna and Bob. Understanding special relativity is hopeless if we can't keep straight who is doing the observing, and assigning names is the easiest way to do so.

2.2

Conseq uences of Einstein's Postulates

The consequences we now discuss are provocative. Distance and time inter­ vals are no longer what our limited intuition tells us they should be. We will quantify these claims later, but to gain the best grasp of their origin and mean­ ing, we begin with a qualitative discussion.

2.2 Con\CQl1eIk:oH. of Ein�in'1. Pmtulalin

Anna and Bob watcb two

Flgur. 2.3

Consequence I: Relative Simultaneity Anna is riding on a railroad flatcar thaI i� moving east at !;peed \', a!; depicted in Figure 2,3, In each hand, she holds a flashbulb, which she cau�s to fla!'oh at pre­ ci�ely the same in�tanl. Bob is standing on the ground, watching Anna pas!\, We have (wo events-the fIashes-\""ith different locations. According 10 poMulate 2, Anna will see both light beams move at c to herself. despite the fact that

9

Iighl!. na"ih Ann"

relative

.ub

she moves relative to Earth. The flashes occur at the same di!;lance from Anna. as Figure 2.4(a) shows. so the beams arrive simultaneously. This arrival is a single event. with II single location-her head-and a single time. According to postulate 2, Bob must also see both beams travel at c relative to himself. But according to Bob, Anna moves some distance to the east while the light beams travel, as Figure 2.4(b) shows, so the beam from the west fIa::.hbulb has farther to go. If Bob were to agree that the beams originalcd simultaneously, he would conclude that the beam from the east flashbulb will reach Anna firs!. This is a contradiction. A single event the arrival of two light . beams at the sume time at a single location-Anna's head--either occurs or it doesn't. Though Anna and Bob m.ight not agree on its location or it') time, they must agree on whether it occurs at all. So long as we accept Einstein's postulate,

Anna sees beams moving equal distances at c, Bob sees beam.. moving atc, but one has fanher 10 travel to reach the moving Anna. Figure 2.4

Equal di�lances tra\eled

there is only one way out of the contradiction. If Bob is to agree that the beam')

arr;I'e at Anna �imultaneously, that the single final event even occurs. he must detennine that the beams from the flashbulbs do not origin(lfe simultaneously­ that these hl'O events do not occur at the same time. [n fact, he must determine that the west flashbulb goes off first, because light from il has farther to go and the beams move at the same speed. Later we will use the Lorentz transfor­ mation equations to quantify the time interval according to Bob and to verify that the west fla"hbulb flashes first. Though the discrepancy in times is extremely small al ordi nary speeds, the effect is real. TWo events at different locations that are simultaneous in one frame of reference will not be simultaneous i n a frame of reference moving relative to the fir.')t. Before moving on, we emphasize a vital point: The preceding arguments had nothing af ,,/I to do with Bob's specific location in his frame or the time it takes light to reach bis eyes. The issue here is

(II)

What Anna see!;

Different distances lraveled

not which beam reaches Bob

first, which does depend on where he stands, but rather which begins fir...! . which has nothing to do with where he stands. Attempting to blame the effect on optical illusions is the comfortable yet incorrect way out of accepting a counterintuitive notion. The truth is more challenging. Bob could be standing right next to one flashbulb when it flashes and have an as::.islant. Bob Jr.,

\ I

standing next to the other flashbu,lb when it flashes. Each records the time of his flash and neither has to wait for the light to reach his eyes. They record diF

ferenr rimes. Anna, in her frame, causes the bulbs to flash !'.imultaneou.')ly, but Bob and Bob Jr., from their frame, detennine that she causes the bulb!'. to tla!\h

----='=-

•"

at different time!.. Later in the chapter, we will occasionally refer to Anna or Bob .....eeing.. something. However, our interest will invariably be where and when e\enls

v

'hi

Whal Bob �'lo

• CLeM"

"... ... ... .._ """" -...

MIl .....

..-It,

__ II) delay I. U

.""'"

r
n,- M nc'Q1 wdla1_ ) be W.rn lO mcaa "tw; .ouJd bco .a:a

l'betdOd'C',

b) .. � .. 4_ , '" BeI> , 1n- ,,1>0 ,- ,.. 10 be ncJu "" Ihe """ 10 It \\ DUI'N Itna,:JhC Rna aftd 8 c..:h Oft . wry loll, train.., lind �h on hi Of ha trata pot.1N 10 ft\:ord umn of n-rnb h.u an arm) of that ()( r n . ...C' VoIU nrhc II. lhI:u p;uur..:uLu 1oc.al1On1 for me lime beiOl Id)' rder 10 adt"hlJOIW �rwn 1n � b frame (Am) and Bob Jr,,1 \lotto sbnd fC'ady 10 (on.finn the cwm Consequence II: Time Oil.tion

•

fl&urc 2 , ihuw '\nna "am on l�r R:uL:at "'<WinS: til'il and Bob moJlng on the J1UUnd i\nrua nov. hHld • na\hll&hl llfhl lhm:b the �am to a l1um)r II her t�1 ." IlIu",rated In ''''gun.- 2 f,. lhc t-cam rcll('l.h taa..l Itl (he n...,hh�ht IlJ Iht ",m,

hI( Q/Wn in '\111101'" fOtme co.. enl\&

2J1k Hot.. J ' Accoruln, 10 hub, lht � f.trthn but II the UJmf' ham n.. JW'd HI IIme' tnt rul musl bt lon�r .... .. '

un

lhe (Ilher h;lOO.

11

lobi JI ..'.tn ..',,' '111 The 1,,11 .. 1 tllnc "

JoC'n • fTI('I"wi n;ak;ar. The �am retll"'..'h 011

"

(""Ilnl falli nt Vohcrt Il on,lna1w and returns hI the t'Ll!hll,hl lit a JX'inl ..till f.t!­ lher 10 Ihe e3.'>1 \1" " IIlS JI:lIOfUIII)'. the tIC m I,.,d

a tobl dl�l.ln",c 1(�'Urr

than �JJ It) poalul:u� 2. R� mu'il lte the hcl1l1l "\\"\" 111 c tn", ta..lcfl, "-, Bob

'-'III h;l\l." to "out

u

(neal lime If't'Ulf"r than �/lIc

\",. h....(' 1\lo\l c\cnh

the ","Te­

p.tlon 01 the hght bam .11'\4.1 11\ arrnul had at th,,' na...hhghl -.o..:�umng til the

' ame k,,;allon In \rma l/. traffiC'. anJ lhe' lime that (ld" C' in Bob" .. tr.1me 1\

II.ml:cr. ,We "til � later "h� "e cml'hMll\" Ihe ("ocn" Ixln!t .11 the 11,,"..lion III \lDC Iran'l(' Ilcrr ,",co

illlpl) n

h."

thJt

,�

II thert" I.. ,u.;h D tr.lmc,

p."C13.1 Ibe cn'nl'" "'lUIJ " ,,'ur III ,hllcrtnl h....' l tuJn' In allc)lhl.'f rramC'i. )

II . ..

1"0 t'Vcnb OIr,;curnn, al lhc: � 1000..Il\ln In one lramc "til he- ,cpa­

ralt'd hy " lon,t'r 110)(' inlcn;ll In I rrame 111\1"10, Tclall\e to the tir--I

"\S.ull. nhhuu�h the dft'd I" nlll nll{lccabk ,It (Irdlnary ..pt.�u,. II i, real 1.ll," hef 30 .. t'ly h",'r \\j).h:;h. Bob, \.\Jh.:hlng

'\cll'I""'" Aun'l hru ..ht'� her ',,"C'lh It 1n.lIll th" \lUt'ltk.

h

pr'·\':I'o('I� �'I,k' ,\nna " hen ,he ,t.ut, Dru,hlng hl.!f teeth,

"'lh t\.'I:nn.h t\lI' Illllt' ..,••11 11 11\10.11\. B"h Jr. ,tanding ,)0 Ihe ground C::t,1 {11 Bob. Iii �'�iJt' Anna

\�

h,,"n ,h,,· 1101 ..hc, B,'D 11 " III rcn)fJ

;t

hOle of l1Iorr than 30

1;

.Itler t"II,)1I AI:Olf\hng tl' I:hlo and S,lt'! Jr .lIltl all l1lher. III Sob' .. fmme or refer· �II"'''',

AnM I" i"ru..hlllg h..:r It't'th ,I,W, I). J\llllg all lhlll�' ..low I), ,lgmg ..10"1).

!'\\llt' th:!.t 4111 that I' d'C'''U ' I� 1\1T thl" lind \\thl.'r relall\ i ..tic eflecl .. to O\:lo:ur 1\ rddlll'� tIlIlll"ln, \nlla

I"

11111\ Ing f(1.11t\1.' t,,\ 81\0. hut Silb I" 010" 109 rehlt ... c

hl 'nnol Thu,. -\nnol mu't '''� Rllo hru..hlll� hi' leeth ..kl" I) and aglOg ,h.IV. I)'_

e the other aging more Ihe reolJ..:r 11\.1) '\hf'Ct:1 a JXlrn(ill'\ HII\\. ,,:all ealo:h ..e �hl'.\ I) than lht'm"",I",'," l!n\kr-.tandlllg thl' t"ne�l I' inhmalei) !ted to the prtn"',pk th.u 'llllUIt,IOc-it) i, nllt at....\)lutc-,

•

II'

\\ e.' \\111

,ee

In E\.ample 2A

Consequence III: Length Contraction \nna anJ B,lh (11.·..·Up) then u..u.11 frame,. but. a.. ....e ,et" In Figure :! 1 Bob nllV. hllk!.. a rlan� In an t';l'I""I',l lIriClitalllm. He could determine thl' length llf Ih,' plan� D) flndlll� th... • IImt' Inh.'nal t-.et\\een two e....ent.. Anna p.�l:lg

OM' C'Dd ot \hit pImL. and AM3 pan!tiI tbC' uthtf Ilk' kflIth oI lhe ItW'� � tit dw &1nw �«n � C\"C'ftb mulllplIC'd b) \QJ'IA' '('k�U) 1'('1;&'(1' .. 10 Doh n.ou.,b GM I«' • ""'UII rbnl- . .."OUkI dctcnnino 113 �"lh to • •he­ t� Ihc llIUr tnttnal �'vrr«n W wnr r-.o ('\ nt'i fiN \11\(' till \)

fIIture I 7 Iklh ....""* . plank. whM:tt. � mo.""" ",1.1".. "-, nna

J,. then the- other nd J'iL'l-,JJ" ht1'. lbe rl.tnl. ·\ k'l\I.th I' lhe plant. JII&UUlt lU1� �'ft'n tbc:-'IC' ('\"CI\b multlplic'd to, the J\1;1nl.. · h-h.-:lt) f\'I41I\e h' lin.. '''''. tht fYo'O f'\Cnb "oar In It\(' qJt'lC' 1�-1." In ,\111""" 11;1"": 01 ",kl\·I\\.·

�itk.U . \11ft:)' \U'\ k'atlOn , \,c-oNlll$ hi n,>t>. the pi.."l.. I' ,.!lIn.wl), .... I1 I�,II, 110 In cod;o. ....:' al ililrcrcm k auon.,- ) tNt ili-'4U....'h'fl I'"� Illne \l11.IIIIm th , Ih.u. n,�'" 1.ll.llIg k' , � d'IoII ltw lUt'lC' mtc'f\.aJ In \llnII ) hUnC'- "lll �' k!O. tUDe ILl PiIl-n. lhe rtu\� b .tkvWf Ih:UlIl.1l11.1 h') \nl'll.l ttuUl 11 I' 1Ii.\\lnhllll hi It,¢. "1l.lL.. -'"" tk'l\ 11,,1\ Itli Ih.tII \\ h..'11 ,1,\11\111011', The- ('UnI. an \1bjC'\1 �"'('UI'� In, I P ,,�ncd knKlh hlnll1&\lion l..Mh.·r \\ \' "Ill 'IUd"III, thl\ 1 . ..lh •.d \\ ,11",ul ....I"I\� if ,1ft tllllr !.hutlOf\. Hut ,,,l lItlUt..., " \' .1'1'f\�'h 1\, 'h' IIUhl II',' " nlt'II",,1 ,II \ \.\hlle II " IkII...II� ""','111&, Ill'I.U1n aUI'Rln, . kn,lh III an ,>hl'-' '1"'-"\\ )( ' I' as 1"'-""' all) I t " Ir,lIn 111..' log lit III m/!li l\"l!il'(" �\ 111 lUll " II'i kn!1lh "' �IUC3otJUltatoh woO III 1he- An'IC' .... alRI"' .prlle, hl lh... ell\" ..:I ." hI th..' 1\\\1 PIt'\I\II!' l'il ..'d' 11 I' • ,Ol'C' nm,w��' III dl '\UIi '\'i kn�lh l'Ulltt.ldl\lfl '" an \11'11..·.11 Ilhl'lllll l'.IU'l·d t>� !.kl .., '\ III h,hl Ira\('hn, Itl the \>h"" ",., IHIIll Ih\.' 1Ih1\ 1I1g \1"'ll'.;I The dl,-d " n.-al Wr{lOSt" ' lhe pl.lII\. " \11 kn�lh f ":\. tlntlng lu HI.t>, \\ hll hI li..!, II \I In ,\nna', Jr ..ulC, \fII'.1 ami hl.'! """1.1111 \Ill � l". h h.lpren II) ". ;lh�n\.·d \\ Llh .1Il (oJ \11 Itl.: ra-,'lOg pl..n\' _" /"1'\"1 Idl rll" \dm� t,,"I'. th"'n III Ih"'ll 1fi.lllw. "nlHI !Ukl .\",) "III t"t' a U"'lan..'\.· a" ..n It'\\ lhull l ..

1",,\

tune

'�Y'" 2.' Wh('ther Iln(' Ilhl�·t Ii" uhl\lc ,l1\llllll'r l.kJ"l:'lIIh I'll Ihe:' 1.t>'>('1 \\'1'" II"l1\e ul rl'lc.-cn..·('

1he I�ith Ilr an 'It'lje(1 In .. lrame Ihl,.ugh .... hlt,;h Ih.: \lhll.'\:1 1Il,1\,', . ' lI11llll('r Ih:!n Il'ri kn,lh IIlth.: Imllle 111 " h....h 1 1 I' III re I .-\�aLn, lIiJ I h.lt 1\ rc,-!uu!.'d I' rel.llt\'.: nHltllm If Anna \lR h!.'f 11",,;;\r .... '·1\· h'lld In, u pl.lnL. til kn)!th r� II \.\ llulJ ,x'\.·up;.- a di,tanl.'e h:" than f i.lI.:n)rdill� hI \1t'l\('f"\ I.'r-- In

"vh', Irame ,II n:ll'rcn..:e. tlc�au,� Allllil', plJnk mUH', rdativ!.'

\111 Pllrlcd

10 8"b

We nt'" ,,:ull'ldl'r " "Iue,uun thai In \lnt" hlrtn \)r anulhl'r " II ;':OIlHUIlII '1IUn.'l' \11 ;.:,m,lt'rn.tltl\n lor thl' l"ol'glOnll1!; "udenl (It "pcl:lal rda! l\- I I)' I n '0 J"ing. .... e "'ill t.e�1R tll '>('1.' till' inll'rJcpc.·l1lknl:c \11 th..' l'on,cqucm:c, \Vhal ap�ar III he paruJu,l" arc UI,O"II) JU�I nample, of applyi ng. (lIll' l'Un\equcn\:l!

01 'pe"'IJI �lall\'lI) and IglI\lnn� tht" olher,_ They arc a pa\,.'!..agl' ,",cal, and wc mu,1 buy till' ",hole p.ll"I.agl', Anna hi.!' dc\clopcd a re"olutlollJry new plane capahlc Ill' 1.I11aulL ng 'pt.·cll, near Ihal 01 I Ighl It i .. -lO Ol long \.\hen par\'cll lln thc runway. Bob ha, a 2() III

(b) A,"eonlm� 10 Bob

Illng iAlrplane hang.u- With \Ipcn dtxlrn ay .. at ea....h e nd. ;1\ LIl hg.un: 2.K(ill. ,-\nna. a yuung and lbnng Iype. t.lke.. oil. "I.'ce:!cmlc.. It) high ..peed. then '''(lOP' throu�h the hangar AI a high enough "�cd. Anna' .. plane fil'i entirely ""tthin lhc hangar all al on..:'(' a;':l,.'llnJlOg to Bob. (lCclIp)ing only 20 In .., ,huwn

tn Figure 2Jl(bl. In other ",onh. Btlh ..cc, the lail (II Anna's plane ill (JOe dour ....ay al pre"'J�ly nOlln on hi .. W.lkh. \\hiJe Bob Jr. ,.:C\ the p lanc ' .. no..,c al thc olherdoot'\l,'uy al�o pred,ely at nlxlO. Nnw ('()n�idt:r Anna... pt'n,pe,,·!lvc. Anna •

" at rt\t rtlati\'c to her plane and thu, sec .. It'> kngth 01.-" 4() m, She \I.'l''' the

lei A<;\,.on.1inf 10 Ann�

12 ChIptIr 2 Sped" RclatlvlC)'

hangur a.\ moving relative 10 herself. It mu\1 occupy less space in Anna's frame man il doe\ in 8ob\ frame, in which it is al rest. Thus, Anna will see the hangar a.. only 10 m long_ A!!o Figure 2.8(c) shows, her 40 m plane cannot possibly fit in all al oOl:e. How can we resolve this apparent paradox? Here is the connection: The t'vo evenl'>-the twO ends of the plane at the two doorways---occur according to Bob, so they cannol be simultaneous according to Anna. In facl, Anna ob..er\.:es that her plane's nose reaches the doorway through which it ex.its its tail enters the other doorway. According to Anna. the plane need not and does not fit in the hangar all at once. The whole idea of " all at once" is a question of simultaneity, which depends on the frame. A question often asked j..; , What would happen if the plane were to be .stopped suddenly-will it fit in or not? We note fir�t that no object can be \topped without force being applied to it somewhere. and a sudden stopping of the plane would be a violent process certain to deform the object in either frame. A mechamsm in Bob's hangar reaches alit and grabs Anna's plane, exerting suf­ ficient lorce at many poinl'l on the plane to stop it essentially all at once (zero Slopping dbtance would require infinite force). But simultaneous stopping of the different points of the plane in Bob's frame cannot be simultaneous in Anna's. Anna sees the nose stop first. Meanwhile, the other points are stiU mov­ Ing. getting closer to the front. reducing the overall length. The last point to stop is the tail. 'W'hen it i!. all over. the plane will have been squashed until it fits entlfely within the hangar. Now in the same reference frame. Bob and Anna would agree that il fits, and that the experiment was a harrowing set of events.

simultaneously

before

Evidence of Relativistic Effects The most famous early evidence supporting special relativity involved the behavior of muons. Muons are comparatively rare subatomic particles that are created when cosmic rays strike the upper atmosphere. It is well known that the muon is unstable. It will spontaneously decay-turn into something different­ after a period of time. The time elapsing before decay varies from muon to muon, but the average is known to be 2.2 j.1S in a frame in which muons are at rest. Given the altitude at which the muons are produced and the speed at which they are found to travel thereafter. a classical calculation indicates that essentially none should reach Earth's surface-but many do! The explanation involves either time dilation or length contraction, depending on the observer. To demonstrate this, suppose that Bob stands on Earth and watches a clock glued to a muon. Anna flies with the muon. I . Bob observes events-ticks of a clock-that occur at the same place 10 (he muon's (Anna's) frame. Due to time dilation, these events must take more time as measured by Bob's clock than by the muon's. On average, 2.2 j.1S will elapse before the muon decays in the frame in which it is at rest (Anna's (rame), so Bob's watch must register more than 2.2 j.1S. If, according to Bob, the muons survive longer before decaying-they age less rapidly-they will indeed reach Earth's surface in greater abundance.

:U The: Lorc:nu T,.,.dormacion EqIWiODS

13

2 . Anna, o n the other hand, says that i t i s Eanh and its. 1l1mo'phcre th:u mo\e. Due (0 length contraction, the distance between the upper utmospherc and the surface will be smaller in Anna's frame than in the frame in which the) are at rest (Bob's frame). If, according to Anna and her muon tr.wcling companion... the di'tance thnt pa�ses by " shorter. then more muons \nll suniH� to meet the �urfaee. The muon detection rate at Earth's surface predicted by special relativity i� in excellent agreement with the experimental findings. We will return to the muon problem after introducing Lhe relationships umt will enable us to treat it quantitatively.

2,3

The Lorentz Tra nsformation Equations

The Lorentz transformation equations are the fundamental equations relating position and time in one inertial frame to position and time in another. They are a modification of classical relationships. but accommodating Einstein's postulates. We will begin with the classical relationships. First, however, we

Figure 2.9 Two reference frames: Frame S' moves at velocity + v relati...e 10 S. and frame S moves at - I' relative lOS'.

introduce conventions used throughout !.he chapter. In relating observations in one frame to those in another, we refer to two representative frames: frame S and frame S'. As illustrated in Figure 2.9. frame S' moves at speed

l'

in the positive

x direction

S

relative to frame S.

s

Viewed from frame S', frame S moves in the negaril'e x' di.rection. (The x- and

,-

x'-axes are collinear and are shown mherwise only for clarity.) We reserve the symbol

II

exc/llsi\'ely for the relative

"

speed between the two frames. The sym­

bol l/ will be used for the velocity of an object moving relative to a frame. Quantities such as position, velocity, and time have different values in differ­ ent frames. In frame S', these would be denoted by frame S, they would be

x,

II,

' A"', 1/ ,

and ,'. while in

and r. Lastly, we define the instant at which the

Figure 2.10 The velocity of an object

two origins pass as time zero in both frames.

depends on the frame of reference.

Classical Galilean Relativity Once again, Anna is on her natcar moving at speed

II

Bob S

relative to Bob, who

stands on the ground. as depicted in Figure 2.10. We choose to the right as

"

being positive, so Anna's frame of reference is frame S', and Bob's is frame S. At time zero, Anna releases an object from the origin of her frame, and it nies eastward at constant speed

' II

assumption lhat "

= I,

=

we have equations relating position and time in one

frame to position and time in another:

x' = .t

-

W

t' = t

(2-1)

y'

Object

iJIl---x' o

•

relative to herself. According to Anna in frame

= II' ,. Classically, time is time, so there is no rea­ ' son to use separate , and r . Relative to Bob i n frame S, the objecl's position is denoted by x (no prime) and is the sum of the distances from Anna to the object and from Bob to Anna: x = lI't + �'l. or x x' + \'{. Togelher with our S', the object's position is x'

Anna S'

(

1. Ch.pUr Z

SpecIAl Rel.II\-lt)'

We say that equations

(2-1) are a transformalion between frames S and S'. If

the object beeped regularly, we could find the locations of these evems in frame S' from their locations in frame S. FurthemlOre, the velocity of an object is the time derivative of its position, and applying this definition to the transformation gives us a resull familiar from classical mechanics: II' =:;

dx'ld,' = d{x - "/)/dr = u

-

v.

(2-2)

II' = U - V The object's velocity according to Anna in frame

S' differs from

its velOCity

according to Bob in frame S. That position and velocity have different values i n different frames of ref­ erence is sensible, and it seems natural thai an object's position relarive to a moving frame could be expressed as a function of its position relative to the ground, the speed of the frame, and the elapsed time. But the assumption that time is absolute. also quite lIatural, dooms classical relativity. Equation

(2-2),

for instance, is at odds with Einstein's second postulate. for it says that if the object that Anna released were a light signal, such Ihal 1/ according to Bob would be c

+ v.

'

= c, its speed

If

Special Relativity OUf path to the Lorentz transformation equations begins with the application of Einstein's postulates to a very familiar law of physics: A free object moves at constant velocity. Figure

2.1 1

shows the essentials. We assume that at time

zero the object was at the point where the origins coincided. With no net exter­ nal force, it must travel at conSlant velocity; its position must be a linear func­ tion of time in both frames.

x' = /l' "

(2-3)

x = II(

Avoiding unnecessary assumptions. we allow that a given event in the object's life may occur at different locations and rimes in the two frames. As before, suppose that the object beeps regularly. We seek a relationship that would give us the x' and " of each such event in frame Figure 2.11 1Wo reference frames and

a moving object.

from its

x and f in

other. As it turns out, this implies a fairly simple relationship:

y

S'

"

ObJ(CI o

x' == Ax + Bt where A.

B,

(x', I')

must be

r) by a linear transformation.

" = Cx + Dr

(2-4)

C, and D are constants. It is leC! as an exercise to verify from

equations (2-4) that if ,

frame S. At

position being a linear function of time-should imply constant velocity i n the related to (x,

S

S'

this point, our only guidelines are that constant velocity i n one frame­

xII is constant, then x'/t' is constant. The main point is

that for a free object to move at constant velocity in all frames, we require only thai the u-ansfonnation be linear. It is not required that time be absolute­ that r

= r'.

2.3 The �nll Trandormatioa Equ.tiom.

To determine the constants

A. B. C. and D. we need to con.c.ider only three

special cases of motion. I. If the object were fixed at the origin of frame S'. it would mo\'e at

relative to frame S. so equations (2�3) would become :c' = 0 x =--= 1'1. Inserting these in the first of equalions (2-4). we obtain

speed and

\'

(2-5) 2. If the object were fixed

nI the frame S origin. it would have a \,elocity

of -)I according to someone in frame S'. so equations (2·3) would be

x' =

I'

-V

and

x = O. Inserting these in (2�4) gi\'es 8

D � -­

v

D=A

or

(2-6)

(The algebra thaI yields this equation and equation (2�8) below is left to Exercise 18.) Thus far. we ha\'e merely set up two frames wilh rela� li\'e motion. We have nOI invoked anything particularly ·'relativistic." With our third special case comes the crucial element.

3. If the object were a beam of light, Einstein's second postulate demands that ils speed be c in both frames. Equations (2�3) become

x' = ct'

x = ct

(2-7)

Inserting these in (2�4) and using (2�5) and (2·6) gives

C�

With

c'

(2-8)

B, C. and D now in terms of A, we may write equations (2�4) as

x' = A{x We can deduce for

v

--A

-

1'/)

(2-9)

A by a fairly simple argument if we first solve equations (2�9)

x and I. + \'1' )

The only difference between the frames is that S' mo\'es at + v relali\'e to frame S. while S moves at -I' relative to S'. It follows that (2·91 and (2�IO)

have to be identical except for the sign of I'. and this simply requires that their leading coeHicienL>; be equal.

15

.. 051 . ..... . ... .., A =

or

gi\en a 'Ipecial S}lTI bol: n.. (actor is ubiquitous In !'pa.-Iul n:lati\'it�, \0 II 1\ departure of reIUli\ ''''l Ihe of rc \u a e m 'Y A!Io we shall �. thi", j\ oflen Iht' tc

e�pectauons from da.\..ical l)ne\, It Int:rca-.e\ t:onllnuou\ly from esseOlial ly al small speed IOW� Intimt)' a \' approach!!\ c. ..

..

r---;:' 1

\

1

-

1

2 1

,­

(-2

Finally. in\erting (2� 1 1 ) into (2-91 and (2·10). we hu\'e the Lorentz lrans_ rurmation equation\.

( 2- l h)

(

t = )I,, + -:;- x + t c"

.

)

.

(2-1 3b)

N(lh� that equJlion ... (2-12) and (2·13) are aClUally equi\'alent but simply \oh�d for dilferent quanlilie\, We indude both merely for convenience_ It is

fairly ea.... I) argued thai \pallal coordinate.. perpendicular 10 Ihe direction of

rclilti\l� mOlion are the ...ame in both frame" (see Exercise 19). Thus,

(2-14) Di"tances perpendicuhlr to the direction of relative mOlion are not subject to lenglh contraction. When \' « c, which i... known as the classical limit, )I" is essentially I , and the terms in eql1alion ... (2-12) and (2·IJ) involving I'lil are usually negligi_ ble, (berei...e � discu...,e.. l1n exception.) In thi.. 1imit, the equations reduce 10 the l'ias...ical tmn...fomlution of (2·]). Cla....,ical phy\ics is valid in the realm of "mall �peel.b_

Consequences Revisited Now we are ready 10 quanlify the claim.!> of Section 2.2. Each consequence involved intervals belween a pair of e\ents--either spatial intervals or lime intervals or both_ Let event I have position and lime (.\' ' 'I) in frame S and 1 (Xi, ,; ) in frame S'. Event 2 ha� position\ and time� (x2' '2) and (xi, t;), If we

j, 'i) lnll) �ll'Mllon l�-l �,l) and ..epar-,III'I�

In ..!.'rt t '" ', ) .anJ ( \

; ;

m"l'rl l l.:" l) ttmt

\ \ , r l lnll) that e,,\u,llIl)11 uod thell ..ul:'llr,l(:( thl' twu ('\IU;\lll)n.., \\C \ll:'lhtio "

,

,

1

\\t' �ould c\aluJte U , l� \ll Ihe Llln:nlz Ir.m..fl"lnlhllilln e(ju Jlil,n .. il'l .1 p .u r \\1 e-H�nl " In Ihl' ..ame \\.I�. The J'llllll I" Ih.u l'quJlion.. t:!-11) ,\Ill! (.:! 1.\) c.m t'Ie' e J dlre..:ll� fIll a �ur of e\ cn(-. Ju..t ;,\.. fllr " ..In!!le C\elll u .. Ille li"l l'\lO"t'\tUl'nl'c \\1 FIn ..lclIl'.. p!.Nul.uc.., Jc;\l!n!! \lith relalJ\l' ..illlUI­ I.meit). \\\!' quantif)

III

,Ill c \ample,

EXAMPLE 2 , 1

Ann;! i.. un ,I fl,II..:.lr 11\\)\ lIIg CII'I at O,� rekltl\c tn Bllb, n.. ,hll\\ n in hgure 2 . 1 � . She hllll,h

1\

n ....htJulh in c,lch h.md J.lId cau,c' them 1\1 l1a,h \lllIuhilnl.'IIU,ly Anll;l\

h4lnd, lirt ::! III ilp;Jrt ,\lid her ,\On, arc (Iricnted III ;\n C;l't- \\ e,t dirct.:lilm. :\l'�'IIf\hn8 til

Figure 2.1 2 Boh \'il'wing two (" 'enb

(hut Ill'l'ur 'imultune�JU,ly ,Iet.:nrding 10 AIlIIII

Bllb. \\hieh l1u,hbulh 11.1,h(',> liN .1Ild O} hll\\ much')

s'

N

01'

Lei Anna\ frame

� S' J.IIJ 81)0', S h ent I is the eu.,! I1l.hh !Jnd eH'tll :1 thc ,,,('..t

;

,; - O. und

111.1,h, \ccordi ng 10 Anna. thc 1\\0 ('ent.. O\:cur \imultaneoll,ly. �o ,

the) lire II dl'lanee lipan Bob.

\\(

ll�e

-

a· Db).

�I

\:'

- tj)

1 _

ti

,

(0.&)·

[

+

1 m Ttl find the time mtenni llcl'ortling tll

-

(I;

0.6,. - ( , ,..

-

:2

]

ti)

11\)

,

]

+ 0

Bob

=

1 .25

0.6

X 2 m

3 X 10M

Ill "

-

b'cnt 2 ha� the ,nlilile-r (e,lrher) lime. Thai b. Ihc we�1 bulb na�hc� 5 n.. cilrlicr than the ea�t bulb. Note th"t if we calculated

ti

/� and 'I separately. from

\'2

I III ;\nd

+ I m. \\c would oblam ,2.5 nl. !Jnd +2.5 n�, re�pect.i\'ely. According to Bob, the

we�t bulb n!J,hc.. bdore the origin' cros)" while the east one flashes after.

Time Dilation Consider the �ituntion of Figure 2.13. Two events-ticks of a c1ock--occur at tilt'

x ; . Evalualing (2-l 3b) for this pair of x; would drop out. leaving '2 - I I = )',,( 'i , ; ) . or. in more compact notation, �I = )',.6/'. The special term proper lime is

Slim!'

ioc(l{ioll in frame S', so thut \""2 =

event'>, the term with xi

-

used for the time thaI elapse" in the frame in which events all occur at the

Figure 2.13

1\\'0 event�. lick!> of a

clock. at the !tame localion in frame S'

same location. In Ihe case we have considered here. that frame is S', but it need not be in general. so we choose the distinctive symbol .6.10.

.It

=

)',. �fo

(2·15)

The time inlerval tit that pas<;es between two events in another frame is longer by the factor 1'" than the interval .lIO in the frame in which the event'> occur at

the same location.

,.

s

1. a..,e. J ....., ."lhM' Foc any Krin of rt'laJed C'\enl". ..uch a.. IK:'1i.-. of a clock or beal\ of a heart, lhere IS always a fnunr \o\hC'l'C' evefll (,,-·\.·ur ill me ..arne place. or a frame thai foJklws !he e\'�nl uvund. for .n,lance. a dud. rna) di\e and [urn u.. it quacKs in Olght, bu, in the du.;;... ·' frame 01' rcft:'rcnl.'C, the quad . .. all occur al the same 1000'7Ifion. Sn:lion 2. 1 0 ,hm', Ihal ob,-,=nero in all olher frame!> \\ ill agree on

the lime lhal they � clap,in!! in Ihl\ 'pedal frame. a"hough each
ob
length Contraction ",..,. 2. t. Bob 1«1 11 m()",jnll
Com.ider Figure 2,14, To e,lah/i,h the length of an object Ihed in frame S' und lhu, m(l\'ing rclall\!C 10 him,clf. Bob ob,erve_� its end<" lIf the s(lme ,ime in hi .. from<. The�c twu (lD�cr\'Jrjon .. ure evenlS for which 12 = 'I' If we evaluate

( 2 - 1 2al for lhc�e e\tnh. 1�

L

0

I

J

drop' OUI, lea\lng x;

L'ly) Thc kngth nf "n ObJCl"

fraflle-" given the ..)'mboJ

-

x; = Y"('\l

-

x,), or

in Ihe frame rn which il is at rest-it" rel)(

Lyl ami i� �n()wn a,� proper length.

(2-16)

The length L of a mo\ing object !'> �horter D} the faclor Y" Ihan its length the frame rn \\-hlch II i, Jl re,1

Lo in

A, in the ca�e uf lime dilallon, no mall!;!r whal frame we occupy or who else

mighl be watching. we ne,er feel controcted. We never move relative 10 our·

'ch,('." II 1\ only ob..en'cr-; in other fmme.. who will determine that an object occupie� les, �p;1('e than nonnaL

Fitting It All Together Now let us take anolher look at the muon problem.

EXAMPLE 2 .2 A mUllO j, created I R Ihe Jlmo�phere J � m abO\e Eanh's surface. heading downward at speed O.9Nc II sun'i\-e\ 2.2 jJ.S in ils own frame of reference before decaying.

(a) Cr;:Micaliy. how far wOllld the muon Ira\'el before decaying, and how much

101<'OIU'lM:, "1UIoI In fWllt .\" ...,..,'I .j/lT't .hal Sob'I oIIcnatiom � lI",ullAnnJu. .. l'1l.I1 oJIco doc, ., Icau � lilal �, IF\' n-.k IU ,Iw nttb "ItIw

ubJ«', . di,Wk".pGf1equ.tl h) 'Il< knith ('If tIJc,

objtd fuN In fir, t�

longer than 2.2 jJ.s lIQuId it have to SUr\'ilc 10 reach the surface? (b) Relalivislically, a\"(,'onfing 10 an obser.-er on Earlh, how long will [he muon survive before decaying? Will it reach the surface? (c) Relalilisticall), according to the muon. whal is [he dis­ tance from the point in the almo1>phere \\here Ihe muon is created 10 Earth·s surf;'.e,
N

•

(.II) Thc- mlhM1 \h>ulJ Ir'lI\dJ - lf - t(lQK x .l ' IO'� I!\I\)(,� ! ' 10 · , ) - M7 111. f\1 tnHd l lnl y,\lUIJ �UI�

, .

It

.... \1U1J

•

O,qg

...

'\ X to·� III

,

•

HU �\

ha\� tIl h\� S �� I\lll$\'r It 'houIJn', n=iK'h the ,ur/,l\."(', h,lrlht' !.ii,·

tan..:(' " .sbout the time, 11'01) lar. Ilnco·tilth It\(: I'\"l.jUlrcJ hmt'

(lr

equl\;l/cntl), It\(- I1\UlIIl ,un i\t'\ llnl� dhlnu

(O QR}l,-j

5.03

,.,

No"

11 wnnlln� " in llr'Jef. H!I\ln� found the ubiquilou, 1,. \\c mJ.Y Hnt! COf­

rt'Ct Wlll' 11m! �r;\ce Inlena". but plugging i n blindly to u \lIne-lhlllhon or

length-colllnlcllon formula "III gl\'l� [he wrong nnswer SOrt- of the time. II ,hllUIJ br 1'C elnp�t' on thc muon', wn'lwlltch. tlcllllg I1l lhc 'lillie IOClltion in the muon's fmllle, \0 1110,.,. lime rnU\1 eli.lp,e on the ob�en('r\ own

d.. . We muSI multiply (not dl\ Ide) do

2,2 I-l' by 5.0.'1 01 course. equntlon (2·15) agree�. It comes with Ihe condition thaI �ro I� the mten'al in the frume II here the event;; occur !ll ihe sume locauon,

Thu" .lro mu,t be the 2.2 f.l' in Ihe frume oflhe muon, which wmchc\ Ihing\ that conleniently pa�� by ih \'t.'ry own IOC
the Earth ob,cn't.'r·, fmme, in which thc upper Iltmosphere U l ld Earth'� �urt:lce are

definitely ditTerelll loclltion�,

Dol

"Y,. DolO

E

5,03

x 2.2 f.l� '=' I Ll I-lS

From Ihe cla,sicul PCripect;ve, Ihe muon would havc to survive al lells[

10.2 '""� 10 rellch Ihe ground. Thm calculation was valid because the speed, diqance. and time involl'ed wcre nil according 10 the same (Earth) ob�erver. The Imp 10 uvoid is usmg a distance according (0 lin observer in one fmme and n time llccording 10 an observer in a different fmOle. Similarly. we can nnd a distllnce t1ccording to an EIlnh obser�er if we use Ihe lifetime oflhe mUOn us compuled in Ihe Emtll fr.ulle. AI 0.9&, the muon could travel (0.98 X 3 X r(jl mls) ( 1 1 . 1 X t o b S ) = ),25 km. According to Ihe Elirth observer. lhe muon survives 1 1 . 1 ,""S nnd could lr.lveI 3.2S km, allowing it (0 reach the surfal;e. (1;) The muon sees a moving "objel;t," specificlilly Ihe region of almosphere between Ihe poinl where il is born and Earth's surfllce. These points are 3 km apart aecordin� 10 an Elirth observer, bUI according 10 Ihe muon, Ihey are closer togelher. We \hould expect 10 divide 3 km by 5.03. The equmion L =- L,/Y,. agrees, It is Earth'� fmme in which this object is at reSl, so La is 3 km.

Lv Y,-

L= -=

3 km

--

5,03

'=' 597 m

Lenglhescontractionm,i� realn. Accordi y moving at occupi ue lyng p son.by thionls object = X 3 Xm m/s s muon. whil'h b) ItS own wri�lwatch jis, will the ground 10 Ihe mu

only 597

Co �eq

I=

ol

L ,.

.

il will

= 2.03 ....

597

lOR

0.98

0.98c

io

a�

survives :1.2

The

meet

afler 2.03 JJ.�.

is

worth noting thai according to the Earth observer. the muon could have been created al an alii tude as high as 3.25 km and still strike the surface, a/ihoughjusl barely. l f it had been creared at thjs h igh t, the muon would have seen an "objec(' 3.25 km/5.03 = m long, which would pass n exactly 2.2 ....s i .e.. 647 m/0.98c). The muon would agree that it barely survives [0 meel the surface. Both observers must agree on this! A collision between a must muon and Earth is a single event. an event occurs in one frame, occur in all frames. However, the explanations may differ. The observer On is Earth says that time dilation re"ponsible, while the muon argues that length contraction. Exercises 33 and 34 address the fact that the muon lifetime II

e

647 If

(

by i

it

it

is

actually varies. leading 10 the number of muons decaying exponentially with rime. ahhough still al a rate that is inexplicable without relativiry.

EXAMPLE 2 . 3

d pleaSIneaiwoul d Imagine Ihat such a an mspeeds manyparked) times those ofroday\ fasl r l i n en" l o ng (when and crui s es at 6000 m/s (13.400 mifh. or about At preci s el y noon, i[ passes o.,.'er LosirsAngel es. and seesobserver onocktheinground wireads th except iosnalely noon. eyesight peers wi n dow. t h ai t h e cl si d e a/ s o preci Howin e movikmnnorth, g plane accordingpltaone'thiss observer?i passes (b) Laloverhead. er, a groundByobserver /Sear oogt/ies. th1600 how much does Ihe reading SOLUTION e ground observer i n Los Angel e s must a l e ngt h shorter t h e pl a ne' s . proper length of m (6 X 103)' = 50m (I - 4 X 1O-IO)'" 4, J Here sbyesonlwheny is much ss an a y u iwey ihave n parenta sihtuatesesiondiffersoflen fromofarigreat amount. precihesiomargi n or some kind of approxi ­ mati o n. The bi n omi a l pro ti o n truncated power w h �. wi -4 X series) works well. To use it e. A propose

carry imerconlinental travelers

pl

e is

Mach

18).

at

50

an

(a)

He

through

checks the

clock as l

differ from [hal On her walch?

(a) Th

see

Ihan

50

L � - � 50 m YI,

1 -

,

3 X 10

thai

q an t

t

power requires

BinomIal approximation. When xl < 1 (1

..

x)" MI 1

ax

le

v

I

a tiny

either a calculator ap x ima n given in t her we replace x

th c. The Raising it to n

(a

th

10- 10 and a

it

Z,-' The Lorentz Transfomwioa EquaboM

The plane ....ould OCCUP) 1 0 nm le�� space than it does III �,t (bl The ground ob,el'\el'\ \\atch a single clock ticking off ewnt� m thr ,\{lim' hlC(j· tiOIl in the ptane'� frame, so mi� dock register.:; me proper time .l.lu' and the lime pJ.�,mg on thcir o....n \\atche� jl> the longer .l.t. We calculate �I in me u�ual da��ical way, because both it and me 1 600 kID are accon.hng

10 the same

(ground) ob\CI'\'er...

.l.x .l.1 = - = "

�to

=

tH -

)'"

1600 x 1O� m

6000 mls

= 267 s

J

1-

=

267 s

x

10'

(L.A. to Seallte in 4,4 min ! )

(6 X IoJ )' 3

Again using the approximation,

=

267 s - 5.3 X

10'8 S

The plane's clocl.. will read 53 ns earlier than the ScaliIe observer's watch. We see that even al speeds very high by ordinary standards, relativistic effects are quile �ffiall.

Now let us confront the perplexing question of two observers each seeing the other aging more slowly and contracted along the direction of relative motion. Before we do, we notc an important point: When two observers pass, although their wristwatches will. in general. read different values, they will agree on what those values are. For instance, if Anna sees Bob's wristwatch read 8 when he passes, while hers reads I I . then Bob sees Anna's wristwatch read I I when she passes, white his reads 8.

Figure 2.15 How Anna's passing ship appears to Bob. EXAMPLE

2.4

Anna and Bob arc on identical spaceships of length ( = 1 2 V3 rn ;;;o! 21 ffi. (The weird length value makes OIher values nice.) Anna's spaceship (fr-ame S') move� to the right relative 10 Bob's (frame S). Clocks are anached to each ship at its cenler, ils fronl, and ils tail. Figure 2 . 1 5 shows what Bob sees at t = O. The c1ock� at the �hip'.'>

wils are aligned, Anna's tail-end clock reads O. and the clock al me front of Anna's spaceship is precisely aligned with me clock at Ihe center of Bob's. (a) Ho", fa.'>1 IS Anna moving? (b) At I = 0, what do Anna'Sclocks read according to Bob') (c) Show mat the ship" centers will be aligned 20 ns later, according to Bob. (d) Find me reading� on Anna's clock!; when Bob's read 20 ns, .m ns, and 60 m, (e) Find pairs of

evenls showing lime dilation and length connaction from Anna 'f perspe<.:live.

Anna's �paceship, s·

jog !!�O� � �O� o 1>

21

• 02,. . .... .'•.,

..;.�I ..... a,���=::: ..

- :at .'

...,

_ _, '.... .. ..,

(I) ... _ ... ...... 01 _', tIIop c_1C1
- 2

-

,

-

2

<

(b) Whea AMI', frolIC lOb" IKkS. il lS an ntnt thaT anOOUrk:(,\ ( "I (hal dodo

I

i«IIMJII In hc1 frame We Wllh 10 knovt thl t' v.ht:-n , - 0, 14) ""� \Urr--''W: &h.t at I • 0, IOmif'()Ot In Bob', tramt'o uy Rob Jr I� nght on IIx �pt)( 1\\ Itc Krwl'Alng the CO,,'NU\4(C'S t t, I) (If thh hdJc\'cnl AnnI', frunl dod., • In ftamt S, VfC' can lind Ih lime In S' \I.. 12·12bl, •

it.

" - { (\��2)tl2\ -

("2)12 m

' X H)l m ..

-

1m

60 n�

) oJ •

AI.iIIIn. \11'(' OIJ'C nol ,luJ.,tng. vptl":lli lllush)ru.. Thn,

h 1M tl� II du .. Jocalwn tnl.l'Oe, .lnd Bvb Jr. i� Ihc� 10 c(lnfinn 11 (,'0It'. Tht tlm('·dJ!aIJ�lO and lC'nlUh...:ontTlCflOn f�Jf1llul.L'i art' flnt." ,'Of pall'\ 01 ('\cnh, bui lt a tmgJ.. (\('nl 110 the 4U('�'1,.1n. then the lurc:nu U'ilIl..lorm...llun �oalJon, are t� an�wcn For Annlll' center dnd" Bo� VouulJ �I) on an IbShlant at l • mil"m� lion !.)bscn.uon a' t • 0 f.qU.llltln l2-1 :!bl t�n } Itl� t' lO n, The 1\lP lhagt'o&m In !-Igure 2.ltI \h!.l',r., BlIb', ,*,«nalton., al l - 0, (WI! f'C'kr 10 tht numl'll!red I!"cnh lillcr I Ann.', dlk:b Jon', \lnke 0 'Imultaneou,l) *'-:>':l'NUla III Bob, anJ thc-),' \hllUldn'I, I"" thC'� d(J 'lnke 0 \lmultaneou,I}' 11\ A.n",,·,

!t

a.;>,:\",'\,bn): II' Ann.

(e) R\,b ,,«, IhI! (I!nler oJ Anna\ ,hip mme t4n.:� ovcr \peed, II> thus

Jt

The lime- according 10 Ollll, ctl\·

�, -

·W

It,n To find the rtadtng!. B\,b and ht� a"I�Ia.nh \� on Ann3'� doc..... al l - 10 n". � c\.lUld u� (1-1�b) Ilgaln, bul lhe fa...ter ....a)' il> Ilm� cldallon. Bob mU\1 "'et' Annl'� l clock &l.h.nclng hall J.� f�hI a� hi, ov.n (")'.. - 1) For c\cr) 20 n� (In hiS dt,lI\.·b. u.:h of Anna', mu�' til! '<en 10 atha.nl,:e 10 n�

rhul....

tel FI�un'

�.lb

figure 2.16 ..ho.... , thl!

ontrachon and IImc dilaoon ob,,·ious accordm,!! 10.) make, Icngth ...80b. but it also sh()\1.� Anna ob..enlng lhe� eifr:c!'). Al the p:1\Smg of docks

labelw � e\ent I and ('\enl 2, ob ..r:ncn In Anna'� frame at t' - -60 ns and ,. • - 20 ns-that is. �O n, apan In thclr frume--look al the samr clock In

1.l n. a.- 1\SIUIIf ..... . fTamt .... 1ft at ad\� 001)' � III Thi,,.,, tame II..... en" verify tu'Dt Wlaooa. 11M InIJ htf � 'M.1UW 1WWf .,..u. the ..... of ¥tew. UtI: tA,It,,",cllxh m 1kJ,tI. (ramt, fur '\Muan thal lklb' �1oI.-b MY um�Bt.:hroluftd I '.'flIh) and ". llbttn� al lht tr&mt " (} lutan' m .... fr.mt and onh h.t.1f hruhlp' Ii'n.Ilh.,." SICC' Ihr \'t'f) IC'QI.h of a"b'

".... .."

r11P ', *

--

....... .. .... _

- -I• ...,.

Wp. II " ('\wn.:k'd 10..1 half. k"onl

"',Ull" 1 1 7 stK,\Io Ii Anna' �'I)t1lrl('mtnlllf) ' Ie" llf lhe C',C'n'Ii. in tixample

� " (\mlJl'Vln� \Io llh .'I,Urc' ';:. I ft. \10(' � Ih�u. iL... I\l)(C\i (,lU'hcr. ill ('\lcr)' C\lC'nl whcrt" ,,,\1 �'I\""'L' p,u' \1\\)( ,,,hl the 1,lur 10it'lc1C1J1. Annu and Bllh 1ljR'C on th..� �·k...... � · tt:1ilhn�\ Tht' IInl'lll.lnam f'llim 1� th" It j!i. the fa....,·l lh•• !i.imul, 1010('11) \11 ('H·Ot... I' n\lt ;,Itoo..." IUll' that �1I\I\Io'" t',Kh lltoo�er"('r hI "'C(' the IlthC'r'!'I d\... �, runmn.:, ... 11'\10\'1 Ih...n hI' l'f h\'r 1\\\ 1'1 and Ihl' lIther'" di�tall\:(,� ,,:un· tru....tl'd h�ul't·, � 1(1 ,lIld ..1 1 7 r('H' al nthl"r IIlh�re'tlll� pt)inh. hlr in�llIn('C', "'l'n!, � .lIld , '''' .... or III II I.hll('r\·111 oil/a 10 Ihe 1\\11 fnun('�. SeC' Appcontl j .'( B . III Ih... '11Uallllll In E:\umpit!' 1 .J too)' a mul't" ltrurhku) f" t hu1hcI h .. ....·u""1n

dPph'lJ\:h

L" lng I"knlh:ill p.I"" IIl� ...hlp... "�alli. t'I�Un: :!. 1 8 dltrilie!i. Ein"lC'in'.� �C'l.'.

�;I-i

all dm:..·lItm, A"h\IU�h ll\lh �el!'''' the pul...e n';.lI,:h the end), \11' AIIIllI'" moving lie

the pul,(' 'prealilng 'ylmnetncally I'mi'll hl� (lfl!!in NtlW ('lm·

'l'C'"

...uJer :\no,,', � I(,\\' JudglRg b) the 110\'"

\\e ...t'e In hl'r frilllll'. ...he ,ee� thc­

pul\.(' n!.l....h Iht!' end, \11 her ,hip 'IIHuh.ll1ellu,ly .loJ Ihe

puh(' 'pre.)l,hn}1

1 0 at 40 n,. 1:;1I,:h \C'e"

.. ...

'Y mmetnl·ally at the 'amI! 'peed Inml Ihe ('enter of hi� nr

her o\\n ...hlp.

� I -10 II 'i1> Evrnl 2

EXAMPLE 2 . 5 Pe\lpk lIn l.lrth 1..011..... thai It \,ll..e, IIghl 40 )etln. to rc.h:h Plllnel

h,h J"'( been tW,,1m (';\11 ...he get

It'

PI",n"t \ b) Ihe I ll11e

""h,lt 'flCcJ .... rt"qUlrcJ ' CN(I/j' The tJI\lan\.'l: light trn..el... year. llnd the �ymhlll l\

I

I). [t happ..-n' 10 he 9.5 X

the \:onver<>lon here.)

X

from Ear\h. Anna

'h" 1\ 10 yeah ()Id') II \\1, 1Il

a year i, culled u light·

IO I� Ill, hU1 11 i, l'u\ier nOI 10 u,"

SOt n N It might

Event .1 Ih,II IO t>e 40 I)' lrom Earth by the time \he i,

10, Annn wlluld have

10 tra\'d 1'00'ler Ihan c But PI;!nel X I... .w Iy a....ay uccoruing 10 an Enr," (Ib'erver··· Bob, 01 (OUI"<e To Anna In her "'PJ(e�hlp moving at \. rel;lIiw 10 Eanh, the \«,Ill

Earth-Planet

X

" obJe<:t" Will t>e \horter

L � A,

I'

approiKhc,

rna) go 01\ fat

c.

a�

40 Iy

Ihe dl'IOln!;e appr03l'hcs O. The conclu�ion

\he hl..es II)

a.�

hule time

as

is pro\'(Icalive.

Anna

,he pJea�es by merely travelin.!l ;)1

Ii

-

e:�!J> {}

,,1Q \eolie' Ih.1I Ihl' ('\en" ,.ul'fy Ih(' Llln'nt.l Imn,f\lonallon t!'quuli()II�.) .\ t'lu1"1 lIt h�hl IlI:nll" ilt thl!' \lnglm Jo,t ,h they cl\)". The pul'ot!' Ihell �pread.� in

a1 4() n,

-lO=D

--

ImJ ""....lUl"t" In Ihl'" ,,:.I"'e. th�' ...hll"· n'n"..r-. Ilre tnt frame ori�ill),. ( E :(C'n.:ise

...hlp.u 1,!lllt-rent tllnn, II n'Jl'he' the ('nd, 01 hi' ,tJIiOnilf)' ,hip "'lIllUllllne(\Usly

11

Evrnt 4

•

24

Ch.pte, 2

Special Relalivity

sufficienlly large fraction of c.2 Here, the " object" must pass by in 30 yr of Anna'

FIgure 2.1' A Iighl pulse appears 10 spread the same dislance in all direc­

\

time. Using distance and time according 10 her.

tions in the �ame time according to both observers.

speed =

distance . lIme

Dividing by c, we have

�= \'

Bob

40ly 3 0 yr ' c

_

\, =

40Iy/y" 30 yr

� � = ,, - � ,, � 1

40

30

-

1

We have used the fact thai 30 yr ti mes c is 30 Iy. Squaring both sides and solving fOr v

yields

v =

0.8e

It is helpful to confirm that everything fils together in Example 2.5. As the reader may easily verify, for )I = 0.8e. r\. = �. According to Anna. the Earth-Planet X object is only 40 Iy -:- � = 24 Iy long. At 0.8e, it would pass in only 30 years. But how can Anna reach Planet X in 30 years when it takes light 40 years? We must be careful about who is doing the observing. Accord_ ing to Bob, light does take 40 years, but Anna takes even longer, for she moves at only 0.8e. Bob says that Anna takes I1ts = 40 ty/O.8e = 50 yr to reach Planet X. Still. he must agree that Anna is only 30 years old when she gets there, Whereas Anna's explanation is length contraction, Bob's is time dila­ tion. He watches a clock fixed in Anna's frame regiMer IltA' and the interval I1ts on his own is longer.

50 yr

=

3.lfA 5

A question to ponder: Becau�e Anna moves relative to the E:mh-Planet X system, though still at a speed less than c, she <;ees
A

L

2.4

The Twin Paradox

Simply slated, the twin paradox IS as foJi' w ; If I ;\ " ·;s stays on Earth while the other travels a....ay at htgh p '" hr.'), ,. ' �4 the other aging more slowly, then when the traveling tv ! "I ",1''S ·l i '.....in would t claim that the other is younger. Impussihlt fv.o l·corH�· c: �H Oi 'iit down together in a room and each see that the other is man) ,, '" j' �?t'r. rhl( in order for But there is no paradox, The crucial faclor (wt!'rllloxeJ the twins to get back together and discuss their belief" ' "H' :,it them must _llce frames. �:ce"""",. In so doing. the accelerating tWin must change rcft:r· ODe heading away from Earth to one heading toward it. An observer in the same inertial frame will determine that a moving clock is funning ' :

•

•

._

.

. ... _ - . <Jll....l:. h) the u...ual hme-J.IIJllllll t'onnula. e\en II tt\:lt "",,..-J.. ....uktat

IO...tant. the pn'pcr ,� h' u"-' in 'to' .... ,,old k the i.Jbblftlane\1U.'\ Ipccd In t\\,n p;.traJO\. the l*'....:r.ah,m... \ll the t.... m ....ho �in'j, on F.lIl1h

A' im

(h:­

t:arpru.\.t­

matel) mertlal) are the n:hahle I,)(k', But It IUl tlb..'f'rwr ...:'celetalo. . we cannot aprl) lime I.il lallon '-O "'lmpl) Clln...hkr a ..:a« .... htr(' O,� rnnain 'l.oo l-:Juth and Anna tOnb· ... fr.llemal l\\ln .I.lcr) male. � "-.OM trip a.........� MlI �.J.. v.tw:n Anna ;K.""Ct"ler.Jle... tmnlfll Ollh. ,he ffiO\e, to a ilinel"l"Ot tnertlOll frame In whtch

the trulh I' Ihal Bob 1\ C1bw/l4tt'h ollkr. n:l!an11e..... \)( am iliffcl"t"n.c .:' In 1&1"1 rotl'S. It .... the OlI,)mmctr) J.:Olu'-t.'l..1 h)" ,·\nna·, a..:..:ckrall"n lhut ' 11.. .1 .... , her to a£ree .... Ith e\er-rell.lble Bob ....hen ,he �el'" txt..: � -that at'" I' tllJer

Anna and Bob-and Carl Let U\ e)(amine the (win par-J\to," m lhe ":("Inte'-I l)f c\amrk � � A� .... c

�nuw,

Earth-bound Bob v.ail" 50 yea" hy hi.. 0.... " dod. for Anna hl fi"�t.:h Planet X but agree .. y..lth Ami" thm ..he i.. :\0 YCilr-. old v.hen ,he get.. Iherc-. hI[ I _ 0.8e, y•. . . and SO ) r :\0 yr But lIel·on.ling tll Anna. HolI l... mU\"Ing.

�,

\

'llYN 1." A. pend...,,,.

:1 ¥"\(" h� . ...h� . In the 30 )'eu� pa""mg on her wah;h a.. the Earth· Phme-t X \\hJ�·..

must claim that Bob age,> only :\O)"r

�-

18 yr. ThO'•. \\.hen Anna rcu�·ht,

Planet X. Bob ;,;a)' ..., '"I'm 50 yeaf'. old, ,md Anna i.. 10:' and .'\nna "''''', . 'Tm 30 years old, and Bob i\ 1 g." Thu... far, we have no paF-ll.lox None of thi... i.. cORlnlr) t\) \Iur "'f'l·rtC'nl<e, for no one hao., ever l..Ione thi... A, ..hov. n in Figure .:!. I I), \\ hat \\"fIU/J hl' a (lilr,,·

dox i, Ir Anna were to return to Earth and meet Bob, With BIIO "'I� m(:. "I'm J(X), you're 60:' anl..l Ann" ..aying, ··rm 60, you're 1�." To re..olvc the paradox. we introduce another ob"cf\cr. Cllrl. whll hap(lens

hI he In a ..hip heading toward Earth at OJk and pa..... ing Planet X Ju ...t a.... Anna

Jecs. Carl i... in an lOertlal lrnmc- alway... ha.. been .mJ ill\\.uy .. \\.,I1 �. When elll pa.....es Earth, Bob Will unarguably he 100 yeaf'. old. hccuu-.e Buh wail c.

yea ..... for Anna to reach Planet X and Will wait anoth�'r 50 fur ('..IfI

hI arrive.

�� � .ill . m � l. ni� � _� � ili. _ •• � � _ i ,b °uovmg at the ..,tHne relative ."'peed that Anna doe ....

Carl IllU!t>t alsu l"'!nmr: that Bob agc... 1 8 year,. If Bnb iii 1 00 when ('arl get, to Earth and , l< · ..:d 1 X year... dUring the night, thcn Bob mU!t>t be M2 year, old. uccording

l

.

l. when Carl pa"..c'> Planet X Thb i... truth

t t

m

"u

Carl's frame'

) return, Anna must accelerate: that i,>. she must c.:hange from one inertial another. (It make� no difference whether her ,�peed eould be kept con" I \1\ inertial frame ha� COn�tant I'e/ad,)·.) So far. we have not considered IS ollght affect Anna·... perceptions, but Carl doe!iiin 't accelerate. so we :'J

v nn hi.. observation... Figure 2.20 demonstrates the main point. Anna i'om one frame. where Bob is 18 years old, to another, where Carl

u: H l"

1

.:r that Bob I ... 82 years old. Although on the return leg of the journey. does determme that Bob is aging more slowly than herself, it is 100 late, )

•

'0; hu�e forward leap in age (according to Anna) has settled the mIder. Ju they reani.1e. l� � 2 1 Illu\lrales, Bob will be much older than Anna when

I v.o questions may still haunt the reader. ( 1 ) How is it pouible tbIl whee wbIIe CIII A'., lnd Carl are both at Planet X. Anna says Dob is 18 yeon old. lIIpJ4* says Bob " 82 years old? (2) Can Carl ",U Anna about Bob', fuIuIe?

A_. n" tOO ycar"lI gld IIut JutIf IooIc at you

yCHI·... GIlly 601

All tgg tnMl!, Bob What 0 differetlCe o little C'lCceleratign mokes!

Bob sends a light signal (rom Earth toward Planet X so that it arrives at the precise inSlanl Anna and Carl pass Planel X. If according to Bob il lakes light -1-0 years ro reach Planer X and it lakes Anna 50 years. then Bob had better send rhis signal when he i� 10 year� old. Let the light signal be an image of Bob blowing oul candle� al his 10th birthday party. When this signal reaches

Planer X. both Anna and Carl will �ee an image of Bob as a JO year old. Each

will ha ve infercepred all earlier signals. �o they will have knowledge of Bob's

prior activities. but none beyond thaI. The answer to question 2 is. No, because neilher has acruaJly seen any of Bob's life aJ!er his 10th birthday. However. we don't answer the question "'What is Bob's age?"' by saying,

"'The age he appears 10 be in the light signal that just arrived." Neither Anna nor Carl will say that Bob is 10 years old when they reach Plane! X, because each realizes thaI Bob wiJi have aged while the ligh! traveled. But Anna and Cart are

scrupulous record keepers. From their observations. they can calculate what Bob's age must be when they pass Plane! X, and the conclusions they reach dif­ fer markedly. Le! us investigate this and. in so doing, answer question I .

Figure 2.22 shows how each of the three observers sees the situation as Bob's light signal leaves Earth. As we showed earlier, Anna sees the distance befween Earlh and Planet X as 24 Iy. She sees me light signal and Planet X

both moving. and rhe separation bet ween them is decreasing at a rate of 1.8e. (This value is a relative veloci ty according to a third party, Anna. She doesn't

Figure 2.22 A lighl beam traveling from Earth to Plane! X according to three differ· ent observers.

4Dly

D.8e Anna

Bob

(a) According 10 Bob

fo Ihe righf at c C

E==> Anna

J.

24 1y

D.Sc

q Carl

and Planet X mo\·jng to the left at OJk _0 they approach each other at 1.8e·

Annil .se(';s the lighl �jgnal

mewing

o

0.80@ ,I

(b) According to Anna C.trl see� borh the light signal and Planet X moving with a fe/alive \'clm:iIY ofonl) 0.2('

W fhe right.

c

241y

6)

0.80

(c) According to Carl

<

Carl

2.4 The Twin Paradolr.

say that something moves faster than c relative to herself. The point i<; further

2.6.) Thus, only 24 ly/ 1 8c = 13� yr paso;.e'i for Anna as this light signal travels. If 131 year!> go by on her clock. then the time interval

addressed in Section

.

she determines to have passed on Bob's moving clock is smaller:

j,tB

=

1 3j yr -+- 1 = 8 yr. Anna concludes that when she reaches Planet X. Bob IS 1 0 yr + 8 yr = 1 8 yr. Carl. on the other hand, must also �ee a <;eparation

between Earth and Planet X of24 ly, but he sees the light signal catchmg up to

0.2c. According to Carl. the lime for the 24 ly/0.2c = 120 yr. He too knows that less lime

the moving Planet X at a rate of only signal to reach Planet X is

will have passed on Bob's moving clock, and so he judges that it marked off

O-tB

=

120 yr

-+-

�

= 72 yr. Added to the la-year age indicated by the arriv­

ing light, Carl deduces that when he is at Planet X. Bob is

82 years old.

It is interesting to note that if Carl jumps to Anna's ship. accelerating

away from Bob. Bob is

82 years old before the jump and only 1 8 years old

after. Nothing ominous happens, however. for Bob is far away. and 110 Olle around Planet X knows anything about his life beyond its firM

10 years.

The Light Signals Solution We conclude our discussion of the twin paradox with an alternative route to the same conclusion in Anna's round-trip to Planet X . Figure

2.23 shows Anna's journey from Bob's point of view. Anna travels c, then reverses course and returns at the same

away at a speed/slope less than

speed. To keep each other fully informed, Anna and Bob, by prearrangement. send out light signals at IO-year intervals. The signals Bob sends away from

Earth have slope +c. If he sends one every

10 years, the flfSt-Bob at his 10th

birthday party-will not reach the speedy Anna until she reaches Planet X. when she is

30 years old. The signals Anna sends back toward Earth have slope --c. So

long as no one changes reference frames, symmetry demands that Bob also

30 years old. and it must show Anna actually sees images of the other aging only one�third as

receive his first signal from Anna when he is as a to year old. Each

much as themselves. lmmediately upon lUrning around. Anna begins to receive signals from Bob much more frequently. Every

31 years of her return trip she

sees a new image of Bob another 10 years older. The images she now sees show

Figure 2.23 According to Bob. Anna move� at constant speed/slope away from Earth, then toward Earth. They regularly exchange light signals. all of which have slope of magnitude c.

Anna's age Anna's joume 40 ly - - - - - - - - - - y- - - - - 30 r

� �O.

o�� w �� �� � 70 � 80� 20� 30 .w � 50� 60 QO� 100 o 10 _= Bob', age

27

Bob aging three [ime� as rapidly as he�lf. As soon as Bob begins [0 receive sig­ naJ5. thar Anna sends on her rerum trip. he too receives one every 3t years and �!i Anna aging three times a.1, rapidly as himself. But here is the asymmetry: While Anna begins to �ee Bob aging more rapidly at the midpoint of her journey. the im,lanl she changes frames. Bob has [a wait 90 years (50 years for Anna to reach Planet X and another 40 for the light from there 10 reach Earth). Now there is only anolher JO yeaf'i left. Anna seeli Bob age one-third as fasl for 30 years and + (30 yr X 3) � 1 00 yr]. Bob three times as fas' for 30 yea" [ (30 yr X

j)

sees Anna age one-lhird as fast for 90 years and three times as fast for only

t)

+ ( l O yr X 3) � 60yrJ. When i, is all over, Bob has received 1 0 [(90yr X 6 signalli, confinning that Anna has aged 60 years. and Anna has received 1 0. proving mal poor Bob has aged lOO years. The danger in thili kind of solution to the twin paradox is that it emphasizes what is see" rather than what is actually happening. During neither leg of the journey would the observers claim [he other is aging only one-third as rapidly

j.

as themselves. As we know, 1',. is not 3. And certainly neither observer would make the mistake of saying that the other is aging more rapidly (as long as the ob.�erver is not accelerating). Both realize that what they acntally see is a com­

bination of time dilarion and the effect of their rapidly increasing or decreasing ,�epararion. The Doppler effect, discussed in Section 2.5, takes both of these factors into account It shows that the rare at which signals would actually be

received is decreased by the factor moving away and increased by

V( J v/c)/( I V( I + I'/c)/( I

+

j

vic) when the source is v/c) when the SOurce

approaches. For a speed O.Bc, these are, respectively, and 3. The light signals approach 10 the twin paradox is straightforward. One (win need only count the light signals he or she sees to determine the age of the other twin-and seeing is believing. Bur it does skirt the point that lhe asymmetry is in[(oduced i"",ranrly when the traveling twin changes frames.

2.5

The Doppler Effect

Just as for sound, if a light source moves relative to an observer, the frequency received by the observer need not be that emitted by the source. The effects are not idenlical, however. Sound requires a medium, while light does not. Moreover. in the case of light. we are u�ually interested in relative speeds comparable to c. in which case the time interval between the production (not .':iimply arrival) of consecurive wave fronts in the observer's frame is not lhe same as in the source's.

In Figure 2.24. an observer \rands al [he origin of frame S. A light source

mO\'e� at an angle 8 with respect to the hne connecting the observer and the

source and emits a frequencY!..ource in the frame in which it is at rest (frame S').

JI emit\ consecutive wave froOlS at point I and point 2. Let us call the rime between the production of these fronts in the observer's frame 8t . The fronts

i

would arrirf at the observer separated by this time interval were t not for the mOlion of rhe source. In Figure 2.24, the source is getting farther away. and

the time interval between the arrivals, 8/<1' will be larger than 81 by the time it

�

fakes [he second front to travel the extra distance. 1181p cos . (This resuh

2.S The Doppler Effect

Figure 2.24 A light �ource moves 1lI angle () with respect to the line connecting it and the observer. The wavefront emitted at

point 2 has ranher to travel to the ob"erver than the wavefront

emitted a! point 1 .

The dusheu line� a re puth, taken to the ob�ervcr in light from S Frume S the source (Frame S')

-\l .lI�

by

6Vi 4=::::

_

__ -

-

_

_

-

-

-

-

-

_ -+

-,-----� r \ r - - - - - � - --:.9Q ..·:: -

..

...---- "-.� )2'

- -

-

dr,

-

9O" ' 1

<

'1

••.

____

\

. . which move\ from pOint I to POint 2.

I

assumes that the lines connecting points 1 and 2 to the observer are nearly par­ allel. Since one period of light is about 10- 14 s. the source would not move far. ) Dividing by

c gives the time to travel the extra distance, so the time

between the arrivals of the two fronts is

The observed period is dfa. If classical physics were applicable, �t WOUld be the source period, and we could merely invert the equation to relate 0\',served frequency to source frequency. However. the source period

T

interval between the production of the consecutive wave fronlS

�urce

is the time

in frame 5'. The

fronts are produced at the same place in frame 5', so the lime interval between their production in frame of reference S is longer: d.t = p

Ill, =

y,.T,o"�,

(

1

,

+ -;: cose

)

T,o ",,,

=

1

�.T\Ource.

�

.':cos e C 1 _

+

)

The observed frequency is 1/1'a' and the source frequency is liT urce' Invert­ so ing thus gives

)

1

job�

=

'

-

.

C2

j"ourcc -'--' =­ •

(2-17)

+ -COS (I C

I

Again. this accounts ror two factors: ( I ) Consecutive wave fronts produced by the source have different distance:o; to travel to the ob!'.erver (a� 10 the classical Doppler effect in sound), and (2) time dilation.

Doppler effect

29

JO a..ter 2 Special Rdahvil), If the source moves away from the observer, ()

= o. v

1-­

fsourct

c _ _ _ v

1+­

(2-18)

c

The observed frequency is smaller than the source frequency, implying a longer period and wavelength than in the source frame. One of the important clues thar the universe is expanding is the redshift in the light from dista nt stars. All the wavelengths coming from them are shifted toward the red (longer wavelength) end of the spectrum. rn recent years, numerous redShifts greater than 5 and one as large as 10 have been reported. If the observed wavelength is 1 0 times the source wavelength, the source is moving very fast. It is left as an exercise to verify that if the source moves toward the observer (0 = 1 800), then fobS is given merely by switching the sign of v in equation (2-18). In this case, the observed frequency is higher and the period and wave­ length shorter than in the source frame-the light is blueshifted. Time dilation aJone would lead to a longer period, because an observer of a moving source sayll that its charges are osciJIating slower than they do in the source frame. The observed period is actuaJly smaller, so we conclude that the time-dilation effect is oUlweighed by the decreasing source-observer separation. If the source gets neither closer to nor farther from the observer (8 = 900), equation (2- 1 7) becomes fobs = 1.0"'" I - v2/c'. This lower frequency is cal/ed fhe transverse redshift. In the classical treatment of sound, there is no shift if the separation does not change. The sale reason here is time dilation. A simple example would be a source circling an observer. Waves all have the same distance to travel to the observer. but they are produced by charges that, according to the observer, are oscillating slowly.

V

EXAMPLE 2 . 6 The wllvelengths observed from a

I

star are four times their ordinary values. How fast

IS the star receding from Earth?

A longer wavelength corresponds 10 a smaller frequency. Thus, fObS

fob., = !...ource

, v ,

v c v

+ ­ c

I

"4/;;ource

=

or

� r:-;; 4VI + � = V I - � I

= "4I fsource .

/source

c v

+ ­ c

2.6 Velocity �formabon

Squaring and �ol\ilng,

.' c

2,6

I ( 1/4)' = I + { 1/4}2 -

""

O.882c

Velocity Transformation

With a firm grasp on position and time, we may now relate velocity in differ· ent frames. We know that the classical transrormation u' = II v is wrong. The correct one is a straightrorward application or the Lorentz. trnnsformation equation!'. We begin by considering an object moving along the x�axis. By definil.ion. the velocity of an object in frame S' is the differential displacement in that frame divided by the differential time interval in that frame. -

dx' u' · ­ dr' We may use equations (2· J 2) to relate the infinitesimal changes dx' and dt' in frame S' to those in frame S, fix and dr. u'

=

(;

)

Y,(dx - v dr) y,, -

,

dx + dr

Canceling y�, then dividing numerator and denominator by dr, we obtain dx - v dr

-:-­ u' � "'--v dx - -- + c2 dt

By definition, dxldt is

II' =

Lt.

the velocity of the object in frame S, so that

u - v uv

c'

(2-19,)

and

u �

II' + V u 'v +­ c'

(2-19b)

Equation (2·l9b) is simply (2· 19a) solved for u, but its form is no surprise. Just as for the Lorentz transformation. simply changing the sign of v adapts the rela· tionship to the other frame. If velocities are small compared with c, the denom· inators are essentially I . reproducing the classical velocity transformation, '

II

= Lt

-

V.

31

II CMpw 2

,..... 2.25

onte

pKw Ketall"lt)'

A 'J*e.hIP and a �IC'·

mavins relall\te to Earth

EXAMPLE 2.7 -

A \pace..hlp i\ traveling ,j\\,-ay from Earth at O.Se. as shown in Figure 2.25. Farther llUI. a meleorile is traveling toward Earth at 0.6<:. Both velocities are according 10 an

ob\tf\er on Earth. Alx'ording 10 an obserwr on Ihe 'paceship. what is Ihe velocilyof

the meleorite (relative to her<.elO? SOLUT'ON

Equation.. (2-19) involve a \elocity /I of an object according to frame S. a velocity

u' of that object according [0 frame S'. and a relmive velocity l' between frames.

The obJecl here is the meteorile, so the two reference frames are Earth and the

\pace\hip. The choice of which frame to de�ignate as S' is arbitrary. but once made, c\t!rything else follows. Let u� choose Earth as frame Sand lhe spaceship as frame

5'. By com:emion. 5' moves in the positive direction relative to S, so velocities away from Eanh must be taken as po!>itive. We know thal lhe object's velocity II

according to an observer on Eanh in frame S h -D.6c and that .' is D.Se. We seek the object's velocity /I' according \0 the spaceship frame 5'. Inserting in equation

(2-19a) gives II

'

=

-0.6(" D.Se -.... ,,,( , ., O.6c)(O.8c) ;-;--c,,,,,,, -

=

- l Ac

1/ ' = -- �

c'

1 .48

-O.946c

Not surprhingly. the meteorite lroveh in the negative direction, toward Earth. accord­ ing to an observer on the spaceship. Applying classical argument", we would

predict a relative velocit) between ship and meteorite of - 1 .4c (the numerator),

but \\e see lhat the value i); actually less than c.

A bit of valid classical physics should be noted. The meteorite's speed rei·

c according to an observer 011 Earth. If c 2.4 X 1 08 mJs and the meteorite moving toward Earth at 0.6c: = 1.8 X 108 mis, the two are surely ative to the spaceship is indeed l .4 .

Bob on Earth sees a spaceship moving away at 0. 8

=

getting 4.2 X 108 m closer every "econd. This logic is sound because all three of these velocities are according to the same (Earth) observer. To calculate such a quantity, the relativistic velocity transformation is not appropriate, for it relates velocities according to observers. But the l .4c is not an exam·

different

obsen'er at a speed greater than c. II is according to the Earth observer but relatiw to the space·

pIe of something moving relative (0 an the meteorite's speed

ship. The speed 0.946c is also the meteorite's speed relative to the spaceship but according to the spaceship. Because in special relativity we are not usuaJly Interested in a relative velocity between two objects according to a third party. when we say simply "the velocity of B relative to A," it is understood [0 mean according to A. ' gi....en by Provided thaI v and " (or II') are less than c, the value of u (or

u)

c.

equations (2·19) is always stricti) less than No observer will ever see );llme· thing mo.... ing relative to himself faster than c. On the other hand, if the "object" were a light beam moving at c in one frame. equations (2-19) wvu ld show its speed to be c tn the other frame, independent of v , as Einstein .�. pos· tulates demand. Tbe proofs of these claims are left as exercises. We complete the relativistic velocity transformation by considering veloc­ ity components perpendicular to tbe axis of relative motion. By definitioll ,he

:2.1 M....rnt'nlUm and Ene1"I)'

33

"-component llf \eloc:ity in fnunc S' invol ves lhe di'piaccmcnt along the

y'-a'i� and the lime in that frame.

Using v'

== -"

Dividing by

and equulion (2-1 2b) to rewrite the denonunator, we ha\c

dr and re
The corresponding equation relating z-components follows identically. Thus.

1/ '

,

=

T - \'

". "­

(2-20a)

Ii V x ,,'

Relativistic l,Ielocity transformatIon

II'Y

==

r,( 1 ;) lIy

(2-20b)

II:

co - "

II;:: =

r, ( 1

I Y) (;2 ,

(2-20c)

As always. solving explicitly or merely switchjng the sign of v would give

tI.\', It} in tenns of (u�, It;, . 1I�).

(ul•

2.7 Momentum and Energy

We now come to some of lhe most fascinating and momentous discoveries in the history of science. In a sense. we are at a familiar juncture. I n classical physics. once we learn to describe motion. via position and its time derivatives, we study what gOl'erns motion. The second law of motion is usually the initial

focus, but we soon find thai the more fundamental and often more useful law for a \)I\tem of panicles is Fnet C:\tcm.tJ = df'touldt, where Protal tS !.he system's total momentum. This tells U\ that if the �}stem is isolated from external forces. ..ed. the total momentum is con�r

PloW u

=-

L,P, = constant

(isolated system)

Cla�sicalty, the momentum of an IOdividual particle is mu. (Again, we use

for the velocity of a panicle because

\'

is reserved for the relative velocity

between two frames.) Why \hould the conserved property we call momentum

be given by the simple formula mu? It has several things going for it. Its con­ servation is experimentally verified-a must!-at least al low speeds. And it is reasonable that the vector property that is transferred but nol lost as one object

collides with another should depend on mass and velocity. But it has another mteresting property. Classically. If the sum of

mu

for all particles in an iso­

lated system is conserved in one frame of reference, then it is automatically conserved in another_ We ..how thiS via the classical velocity transformation. Restricting our attention to one-dimensional motion for now, we see that

or

(2-2 1 ) Given that the system's total mass does not change and that \I is constant, and

PI�taJ differ by a constant. If one does nOI change, the other cannol.

Ptotal

BUI the cJas\ical velocity transfonnation is wrong. Let us see what hap­ Figure 2.26 A collision seen in IWO diffe�nt frame� of �ference.

'" ° 0

EXAMPLE 2 . 8

Before:

OJI(

J

After:

0.6<

(a) 5«n In frame S

4

O.BB2e

3

Consider the collision shown in Figure 2.26(a) between pan.icles of mass and 4 (in arbitrary units). If momentum is given by p = mil, the total momentum in frame S before the collision is O. because the particles' individual momenta are equal and opposite. They also add to 0 after the collision. so the IOtal momentum is conserved. Using the relativistically corre{;t velocilY transfonnation, find the four velocities in frame S', moving to the right at O.OC, and show that "" m l/' is ltot conserved.

LJ , "

):UTION

We apply equation (2-19a) four times, using \I = O.oc and the four 1/ values given in Figure 2_26(a).

Bdore:

Arter:

pens when we apply the relativis.tlc tran�fonnalion.

0

0_94&

3 O. 38Se

(b) Seen in frome S'

1

Bero,",

After:

7:;-:,.",..;:; ',-;- � 0

::: 0.6< - 0.6< (0 .6<)(0.6<)

_ _

c'

---';""'-0-""'';--;- = e - O.oc -O.6 ( 0. 6 c)(0. ';' ·)

c'

-O.882e

-;;-c ;:;;C:'" = -O.946c --';("":0.8 : c)(0.6<) -0.8e -

D.8e -

c O.6

c'

0. 6c

(0.8c)(0.6c) c'

= O.385c

1.7 MomenNm and Eaerp

After:

� m Il' �, "

= 4( -0.882(')

+

3(0.38.5c)

II: ·;!.J!k

According to Ein ..tein·s lirst postulate, a law of phy'iics should hQld in all fmmes. E'l:.ample 2.8 thus gives u.. theoretical grounds for rejecting the cla.,sj* cal momentum formula. Perhaps it shouldn't be surprising that 11 abo fail.. the experimental test. At high ..peeds. m lI is not conserved III allY frame.

2: .

, ,.

(Neither "after picture" in Example 2.8 is correct.) Theoretically and expcri* mentally. mu i.. wrong. nle classical momentum II/U and the classical velocity transfomlation com* bioe to give equation (2*2 1). which says that the total momenta in the two frames differ by a conStant. Relativisticruly, things are not so easy, but the "complica* lion" leads to one of the most stunning claims of Einstein's theory. Using the rel* alivi�tic velocity transformation, il can be shown that there is no candidate for a relativistic momentum-no function of m and u-for which the total momenta in

two frames would differ by a constant. The difference would mvariably have some funclional dependence on the particle velocities

tli'

which may vary as the

particles interact. Therefore, the only way momentum conservation can be sal

*

vaged is if this other function of the particle velocities is ruso conserved. The requirement that momentum conservation must hold in all frames implies that momentllm cOrlsen'atioll is tied to conservation ofsome otherpropert)'.

Obtaining the correct momentum formula and " other property" by theo* retical reasoning is rather messy. We will take a shortcut to the OIher property by simply accepting the correct momentum as a given. Supported by experi* mental evidence of its conservation at all speeds, the relativistically correct expression for momentum is

where 'Y" = Note that II is

(2-22)

-�--,�=,=c'"

1 - ­

V112.1' + 112Y + It:<- in three dimensions and that (2*22) reduces to

the classical momentum at ordinary speeds, where 1'/1 == I .

The algebraic details leading to the equation that follows are left to Exercise 68. We are most interested in the result. Seeking something analogous to c las sical relationship (2-21), if we apply the relativistic velocity transfonnation (2-19) to our new relativistic momentum (2*22) summed over the system. we arrive at

V1'\.Li'Yu 1111 ,

or

(2-2 3)

Momentum

U

• 001 . ...... ....., If ..

Ii

IlUt IbM P _ p' .-e both C'On5t.Inlkomc1\rd. Itk"n

� '" "'

mUg

abo br C'OIlIIanI ""hal ('ould II be" " hal �,� momenfUm ntll" b.,.. ('on­ � III an uobkd )'*1Jl J There IS 001) l1lJe re-ooy i'kJ.J..iIC IUl.JJ cnctgy

'I c;MDOf be JUU klMlK mt'fJ). ba: usc llncw: energ) nl"t'd nul I'x� nn � ed. TlIe quaMll) "( "', IDUJI JOmt� KC'oun' lOr the (oul l"n\.�rg) 0' p..u1h:l� I,

..

,adw'lIJ boIh kUle'hC and tn/anal energl . r\CluaJl). of mau. bile rllC'lJ) 1,.

I. fib .., I

If ",," muJl,pl) "

m11 1

nil"

b) the um .. ,...1..,,

ITfw;o mo("';.JlIon lJl.1y

nl a

a..

It '(dlilh. it hJ'

c o m Imt rZ.

UOits

,,,-" o"'wm un'h of

"" 1 \\c...L 001 \10C '0(.10 'Ce huw

1111\ dd.m '1lI/v,", �frrl'("rJ)' ',"m h"'ll'ln\ (l(" luIJle', IhrolJ�h the l�H'cntL ,ran\(urmalJOn, lhe: rcqulfc(J)CI1I Ihal Ihe �lIIlL· fa" h",IJ In illl Ir.J.lllc', anJ "U un If mu'" nut be unJl'(('shnlalrJ Il\lo hcrc In d.J.'$H.�.J.I ph,.,,!"" " the ('J.urn

made rhal lhen:' h . I Inplc "',IV. • t llll) 111 1111. Il) lJ..:h.'nnint" the: IOldl energy 0' lin ,lbJC",'c fhc Ide.. 1 frh!luuofl.1r) fhc ffi05l Inllng cun\CqucnCC' 1$ thar ma�s nJ('!L�uf('� enl.'r�) . If an ohJc(,1 IS Wllt)nal'}, Illi lolal CIlC'i) I� "".1, nus 1f)I.:luJI." till mlt'mol t.'nerg) to " h�t·

(,'ocr

lonn It IIlIght toiLe

, c. 'r!1rtGaI

m,:

( 13.""'111), r rrlah\ llit:lcall). lJn(,II�- energ) " Ihl.' t.'n�rg� ,1" tXIJh:U

pI ) �lIh Ih(' nlt,f,HIn Ilf an nf\J,."d, (I

\\l·

" 111-

'lItllr,I�'1 from rhl" l'nt'rg) Ilf :1 OJo\lng

"''''J«t the lllllcrnal J (,IIl'rg' II \"luIJ ha"," til n" I, \\ c hJ\(' llnelK enl'r!:!).

y. tnc·

,

•

( y.

me

I lin'

'

(�-�6J

Ihl' pl.lu ....thlhl) Jnu l·'lI1'I'h�nl.:) of Em,tcin', clai m b) L'('"" l.knn� Ihe hll.11 (,11C'rg) (It d ,1\,\\ mmlng ohjt!cl. L!,mg the hIOomial \\t' begin hi

't'\."

arpfll\irrutlhlll.

, In,·· f

.,

•

. ' n u l'"

(II

<

d

Ill " til' pc.'r'<"l,'tJ), If u " ll'ro, thl'n tht' Illtal energy " ,imply the inlcmli t"nc.·r!!., IfU.J. Jntl if u j., nOfl/l'fP. the tOI.,1 �Ilerg) c\('ecd, the internal energy hy Ihe." d,t"'J('al kint'lir cncr,g) Nlm l'I'rl" J('r (he s I mple case of a complctd)

38 Ch.pter 2 Special Relillivil)'

Mass and Energy According to equation (2-25), the energy of an object at rest depends solely on its mass. For example. heating-adding energy to--an object al rest increases its mass: cooling decreases it. These changes are immeasurably small in ordi­ nary circumstances because ('2 is a very large number. A mass change of I kg

corresponds to an energy difference of 9 X

1016 J, which is more than enough

to put the Empire State Building in a low orbit about Earth. However, we can­ not simply make mass disappear arbitrarily. Generally speaking, exchange of energy requires a force. Chemical reactions involve the electrostatic force, which is attractive between the positive nucleus and orbiting electrons and repulsive between the electrons themselves. The electrons rearrange themselves. resulting in a change in the

;1Iternal

kinetic and electrostatic potential energies. In an exothermic

reaction, energy is lost to the surroundings, reducing the internal energy of the sample and therefore its mass. For example, as two hydrogen atoms unite to form a molecule. the lowering of electrostatic potential energy results in

10-19 J being given off. Using fiE = fimc2, we see that the mass decreases by about 8 X 10--36 kg. This i� only about one-billionth of approx.imately 7 X

the molecular mass. Understandably. such small changes went unnoticed for centuries. Nuclear reactions. on the other hand. are much more energetic for the sim­ ple reason that they involve a much stronger force. aptly named the strong force. We discuss this force more extensively In Chapters I I and

12. For now, we note

only that it is an attractive force shared by all constituents of the nuc1eus­ neutrons and protons alike. (II must be strong. for it is what prevents the nucleus from flying apnn. It overwhelms the electrostatic repulsion between protons.) When a proton and neutron unite to form a deuterium. or heavy hydrogen, nucleus. the lowering of the energy associated with the strong force gives off

10-1] J to the surroundings. The corresponding mass decrease is 4 X 10-30 kg, roughly 111 000 of the nuclear ma�s. Thus, the fractional change in

about 4 X

mass is about

6 orders of magnitude larger than in a Iypical chemical reaction.

Pound for pound. nuclear reactions are much more energetic. When discussing how the mass of an object may change, it is imponant to bear in mind that mass measures the internal energy of the whole system of particles. Consider a simple object: a helium balloon. Lf the balloon is heated. its internal thermal energy will Increase and so will its mass. But what of the individual helium atoms? Their mas..e� do not increase. because their internal 'Some m\tOduc,ory I"�h �JL of iIO-C.rallcd rduu"i,­

,Wid = y./", TlIc nOlion " unlle<;Q.l.1.fY and. w� )''''. lead. 10 ,onfla'iliS In!etpn:taliu',.i QuOllng fun,c..in rum",lr, " le i. no! g ood In ,ntro­ �f" duce the ('OnC"P' "fthe mAlI M .. ""I " of .. moving bod) for ",rueh no cl"ar definluon �"'n hi: glVCII, II I� ben,,'I/) ,nlmlll>« no otheT m.l-'. cooooepc than Lhe 'rr" mas' m_ In'lud 01 ,n1tOduc­ Ing M, it " bettel \0 menUon the nprr.�"on for th<: mo"",,,llIm and "nell)' of a body III mol,on.' In O!.hc:r word.. III _ need � p .. y�nw and £ = Yl"�' lie nwu:

energies have not changed. The mass of the balloon as a whole has increased

because its individual constituents have gained kinetiC energy, which is internal to the system. It may seem odd that a !.ystem can gain mass without its con­ stituents gaining mass. but

mass

is not con..erved.

energy is. and mass

mea­

sures its internal form". On the other hand. the ma..s of an object i .. not dependent on its ol'erall motion relative to an observer.] If we throw the bal­ loon at speed u. its overall kinetic energy would increase, but its

i1lternal

energy would not be affected, so neither would its mass. The mass of an object i ... the same when viewed from any reference frame. (Section 2.10 has a ml)re formal discussion.)

2.7 Momentum

In physics, we often speak of fundamental particles. A fundamental particle one is that is not composed of other particles. Most objects we study are not fun� damental. As we know, an atom is composed of electrons and a nucleus. Nuclei, in rum, comprise neutrons and protons, and even neutrons and protons are combi� oations of more-basic particles. Nevertheless, there is widespread belief thai certain particles are fundamental. The electron appears to be of this class. As far as we know, the electron has no internal slructure. It has mass (internal energy without internal structure !), but with nothing inside to jiggle around more rapidly and thus absorb heat or any other form of energy, this mass is not subject to change. Immutable properties that characterize a fundamental particle are known as intrinsic properties. Charge is one such property. Intrinsic mass is another. Momentum and Energy Applied Let us SOlidify our grasp of the principles with several examples. EXAMPLE 2.9

The kinetic energy of a proton is half its internal energy. (a) What is the proton's speed? (b) What is its total energy? (c) Determine the potential difference V through which the proton would have to be accelerated to altain this speed. (a) Kinetic energy is (y.. - l)mel and internal energy is mel.

(y.. - I )rnpc' � - rn c' = y � 1.5 .. 2 p I

VI

-

1.5 � u = O.745c

(ule)'

We see that for an object's kinetic energy to be a significant fraction of its internal energy, the object must be moving very fast. Internal energy dwarfs kinetic energy for slow objects. (b) The total energy of the proton is the sum of its kinetic and internal energies. £

= (Y.. - l )mel �

1.5(1.67

X

+ mel = YJ"/l

1O-27kg)(3

X

10' m/')'

= 2.25 X 10-10 J = 1409MeV

lassically or relativistically, the kinetic energy gained by a charged panicle is equal in magnitude to the potential energy change as it is accelerated through a potential difference.

(c) C

Iq ;'>VI (Y, - l)rne' � Iq �VI 10-27 kg)(3 X 10' mI,)' � (1.6 X 10-" Cl l�vl = � V � 470 MV KE �

( 1 .5 - 1)(1.67 X

and EoersY

39

40 Chepter 2

Special Relauvlly

1

h the same speed. 0.74 , al kinetic energy fonnula wit Had we used the classic energy increases vel)' ra · Ic y :::: 291 MY. Kinetic V J, d aine obt e we would hav er potential difference is required an ches e, and a much larg as speed approa ally. would be expected clas"ic

��

Let us now apply the

relativistic conservation laws.

EXA M P L E 2 . 1 0

with object 2. of mass 12m 0 and heading ea�t at O.Se, collj�es Object I ,. of mass 9mo ether. Find the mass and speed of the tog k stic twO The c. 0.6 al t and moving wes y mass umt.) . an arb"Ilfar (NOle: fIIo IS ' ' resulting combined abJect. SOLUTION

as east. The reader is encouraged to verify that if Let u� define the positive x direction re the collision would fonnula, the objects' momenta befo we were to use the classical t would be at h l t would be O--that is. t e fina objec al be equal and opposite. so the to ch would whi 10 me that mass is constant, RelativisticaJly. we must not assu lision e the col aus is Bec c t. stan m con is energy assume that the system's intemal we should expect the internal en rgy so . lost be will rgy ene tic kine pletely inelastic, entum-conservatiOn speed unknown. the single mom to increase. With both mass and erful new 1001' pow a e hav e the problem. but we equation is insufficient to solv rgy must be �o ene l tota and possessed by an object. expression for the total energy d object, and we bine a subscript f for the final com served. In what follows we use :::: � and 'Yo.lIe = 1make use of the fact that 'Yo.&

=1. :

:

a would imply that the As noted. the classical fonnul g. We see that it will be movin : Energy(,),1/me2) conserved

5 91110r,

-

)

'Y

final object should be at rest.

'",c2 + 'Y.!lII,-e2 = 'Y. lllrc?-

UI

+ 45 12m0c·' -

=:

' "" "'re("r!C )'

J _r=� , /J V

r

=

-

g the momenlUm equation by the

(wo unknown!>. Dividin We have twO equations in energy equation gives

110, The initial mass W3'> 2 1 mo' so ation yields t1If = 29.851 equ er eith in ing ert ins Re doer. indeed increa\c. the masslinlemal energy

2.7 Momentum and Energy

A

calculation of total kinetic energy before and after the collision In

Example

KEf

2. 10 veri fies that kinetic energy has decreased. Using equation (2·26).

-

- KEI = (-Yo. 1e-

1 ) 29. S5"'02 -

= -S.S5m c2 o

The kinetic energy change is exactly

[B - 1 )9moc2 + (� -

- 6m2 (i.e.. 2 1 - 29. 85

1 ) 1 2m02]

=

-S.S5).

Kinetic energy is convened into increased thermal energy. which regi!o.ters as increased mass. In fact. this relationship must hold for any number of particles colliding or breaking apart (see Exercise

96).

If the total energy is conserved

as particles interact, then the changes in internal energy and kinetic energy must be opposite.

(2-27) EXAMPLE 2 . 1 1 The nucleus of a beryllium atom has a mass of 8.003 1 1 1 u. where u is an atomic mass unit: 1.66 X 10-27 kg. This nucleus is known to spontaneously fission (break up) into two identical pieces. each of mass 4.001506 u. Assuming lhe nucleus to be initially at rest, at what speed will its fission fragments move, and how much kinetic energy is released? (Nore: The masses are given to fairly high precision because. as we see. the initial and final differ only slightly.) ;OlUTtOr

Relativistically or classically. if a stationary object breaks into two equal fragments. momentum conservation requires that the fragments move oppositely al equal speed. We could use equation (2·27) to find KEf and then the speed. but let us sim· ply apply energy conservation. Usi ng a Be subscript for the beryllium and and 2 for the fragments, we have

I

y � u

8.0031 1 1 u Se � ;;i; :;� ;;;: ;"--; � I .000012 2(4.00 1 506 u) 2m m

-:-;;= � ;= """ = 1 .0000 12 VI ( "Ie)' I

-

= II = 0.005c

For me energy released. equation (2·27) gives aKE � -amc'

�

=

-(2 X 4.001506 U 1 .5

X

- 8.003 1 1 1 u)(I.66 X

10- 14 J = 93

10-21 kg/ule'

keY

This is in good agreement with the ex.perimemally measured \-·alue.

It may puzzle the reader that none of the recent ex.amples has involved a second reference frame-a pattern has been broken. But views from different frames are nor so much the mterest here as is the fact thal momentum and energy in an\' frame depend on ma�s and speed in wa}" different from the

41

42

Chapt.r 2 Specill Re1Btivily classical The fascinating point is that it is the claim that conservation laws must be the same in all frames that led to this discovery. Before we move on, we note a general fonnula that relates the energy, momentum, and mass of a particle. It is left as an exercise to show that the speed u can be eliminated between

E = l'ume2 and p = l',,"U, yielding

(2-28)

Because this relationship is independent of the speed, it is often very conve· nienl, and we will use it at various points in the text.

The Particle Accelerator Einstein's famous mass-energy equivalence is central to the purpose of

particle accelerators. These immense scientific apparatus speed up particles to nearly the speed of light. They are the tools of high-energy physics, which seeks to identify the fundamental building blocks of the universe. The electron is the most familiar fundamental particle, but many others have been found. Still others are, at present, mere theoretical predictions. Often, these undiscov­ ered particles are expected to be fairly heavy. But how do we create mass? As Example

2. 1 0 demonstrated. Einstein'S mass-energy relationship is the key. If

particles collide inelastically, kinetic energy decreases and mass/internal energy increases. In high-energy physics, we smash familiar particles like pro· tons and electrons together at very high speeds to make available a large initial kinetic energy from which to create massive new particles. The greater the mass expected of a particle, the greater must be the initial kinetic energy. It is instructive to contrast two common types of particle accelerator. In a stationary-target accelerator, a moving particle is smashed into a stationary particle. In a collider (short for colliding beams apparatus), the two particles smashed togelher are bolh moving toward a head·on collision. To produce the maximum mass with the minimum input of energy, the collider is preferable. Why? Consider two identical particles. If both are moving at speed u in oppo­ site directions, the total momentum is zero. Therefore. if they stick together, the final particle must be at rest.

All the initial kinetic energy is converted to mass.

[f, on the other hand. one particle is moving and one is stationary, the initial total momentum is nonzero. so lhe final particle must be moving. Some kinetic energy is necessarily left over after the collision, not available to become mass. Although the simple example of a completely inelastic collision is not realistic. the underlying principle holds no matter what may result from the collision. In a collider. the final kinetic energy is a minimum and the final mass a maximum. Exercise 93 compares the two accelerator types quantitatively.

Massless Particles Experimental evidence indicates that there are particles for which

E = pc. When

these particles interact with other objects. the momentum and energy they carry are found to obey this relationship within experimental uncertainty. Comparing with

(2-28), there is only one possible conclusion: They have no mass.

E = pc

(m

=

0)

2.8 Ol:nl:ral RdlllVif)' and . Ftnl Look II COUIIC/Iop'

Bul shouldn't particles that move about and transfer momentum to other things have mass? There is no theoretical requirement. Thai applying F = rna

is problematic does not prove that m = 0 is impossible. It may "iimply mdicate that this classical law is inapplicable to massless particles. The topiC of Chapter 3 is the most well· known massless particle: the particle of light. or

photon. No experiment has ever demonstrated a mass for the pholon. If it IS not zero, it is immeasurably small. In faci. the assumption that it is zero is cen·

tral to our accepled theories of electromagnetic radiation.

Special relativity tells us some important things about the photon (and about

other massless particles). First, having no mass, photons have no internal energy. Their energy is all kinetic. Second, if m is zero. the only way p = 'YrI"U and E = 'YllmCl can be nonzero is for the speed u to be c, in which case both formulas are

undefined prodUClS of infinity and zero. It is true that all photons move at c. from

the instant of their creation to the instant of their disappearance. However. the expressions for and E are indetenninate. So what does determine the momen­

p

(Urn

and energy of a photon? The answer awaits in Chapter 3.

2.8

General Relativity and a First Look at Co!.mology

It might be said that special relativity begins with Einstein's deceptively sim­

ple pos(Ulate that the speed of light is the same in all frames. General relativ­

ity, a more general theory (sensibly), begins with another "simple" postulate:

Inertial mass and gravitational mass are the same. I n introductory physics, we

learn Newton's universal law of gravitation. The force between objects of

mass

m

and M separated by

tational constant. Taking

m

r is F = GMmIFl, where G is the universal gravi­

to be an object near Earth's surface, with M and

being Earth's mass and radius, the force on m is GM""",

rl""'"

_

r

9.8 ml'2 X m

m -

in introductory mechanics. we also learn the second law of motion. The accel·

eration of an object of mass m is proportional to the net force on the object and

inversely proportional to m.

a = F�

I

'm

On the face of it, these two properties of the mass

m

are entirely dIfferent.

There is no fundamental reason why the property governing how h:ud gravity pulls on an object should have anything to do with the property governmg the

Accordingly, it property. in the fir\t

object's reluctance to accelerate when a net force IS applied. might be safer to use an

equation and an

m' i

mg,

sigmfying a gravitational

for an inertial propeny,

10

the second.

43

44

Chaphr 2

Special Rtililivily

RgUN 2.29 The floor pu!ohe!o upward

a = Fnet

on Bob. who is at re'it on Earth. The

floor mUM al'iO push " upwanJ" on

I -

m,

Anna. whose frame at:("elerate� al K in Now consider what happens when an object is dropped from shoulder

gravity-free 'ipace.

height. In this case. the net force is simply the gravitational force. Thus,

a�

,nil,.FN '"' III," -

/II.�

{9.8 mis'

If fIIg and 1M, were truly different properties. there is no reason why we could 1 . 1 fIIg and another with fIli = 0.9m , in which g case they would accelerate at different rates. This is certainly not what we

not have one object with "',

=

expect. In fact, the equivalence of mg and "', has been experimentally verified to better than I part in

1012. Although we tend to take it for granted, their equal­

ity has an important consequence: It should be impossible to determine Frame 8

Frame A

whether we are in an inertial frame permeated by a uniform gravitational field or in a frame in which there is no field but which accelerates at a constant rate. Suppose Bob stands in a closet on Earth, frame

B, as depicted in Figure

2.29. He is in a frame of reference that is inel1ial and in which there is a uni­ form gravitational field of g = 9.8

mls2 downward. (This is a good approxi­

mation. although Earth rotates and the field at its surface is not perfectly uniform.) Anna is in an identical closet, frame

A, but out in space. far from

any gra\'itational fields. By means of a rocket engine. Anna's closet is acceler­ ating in a straight line at 9.8

mls2• For Bob to remain stationary. the Ooor must

push upward on his feet with a force whose magnitude equals the downward force,

FN =

1Mg

x

9.8

mls2.

For Anna to accelerate along with her rocket­

powered closet. the floor must push "upward" on her feet with a force sufficient to give her an acceleration of 9.8 Figure 2.30 In both the inertittl frame with gravity and the accelerating frame without. a downward force appears

to act.

"' , X 9.8

mls2.

If I1lg and

til,

mls2. By the second law of mmion, this is FN =

are equal. the forces are equal and would pro­

vide no clue to distinguish whether an observer is in frame A or frame

B. The

normal force is only the simplest indicator. The fact is that no mechanical experiment could distingui1>h the frames. In the linearly accelerating frame A. all things appear to be affected by a downward force just m; they are in frame

B.

The floor must push "up" on objects, and "dropped" objects appear to accelerate downward (becau�e, once let go, they do not accelerate along with the frame). All effects could be attributed to an "inertial force" of -mj3 opposite the acceleration. as depicted In Figure 2.30. Provided that mj equals 1IIg. this force would mimic a gravitational force mgg in all respects. By the same token. no mechanical experiment would

be

able to distin­

guish a frame that is accelerating in free fall 10 a uniform gravitational field from one that is inertial and without a gravttational field. As Figure 2.31 shows. an observer in the inertial frame would see all objects floating or mov­ ing at constant velocity, because no forces act. An observer in the free-falling frame would also see objects 'Oeemingly moving at constant velocity (though all would actually be accelerating with the frame). because the gravitational Frame B

Frame A

force mgg is exactly canceled by the inertial force -mia.

2.K Genttal Relativity

Uld. Fant Look at Cosmo1o&Y 45

Figure 2.31 Thing� seem Ihe same in a frame thai is free falling due 10 gravilY as in a fr.lme flrulling without gr.lvity.

m,g g 1-i'-".:t.I T/ m. _»I

�a

=

•

g

00;

0 = 0

Engine

off

Free-fulling frame

The Principle

�if----\'

tnenial frume

of Equivalence

Einstein's customary leap forward was to postulate that

all physical phenom­

ena, not just mechanical ones, occur identically in a frame accelerating in gravitational free fall as in an inertial frame without gravity. No experiment could distinguish the frames. Accordingly, he generalized the concept of an

inenial frame by defining a locally inenial frame: one that is falling freely in a

gravitational field. (It is "local" because it must be small enough that nonuni­ formities in the field are negligible, giving the same acceleration for objects within it. Also note (hat this definition includes ordinary gravity-free inertial frames as a special case.) We may now state Einstein's fundamental postulate of general relativity, known as the principle of equivalence. The form of each physical law is the same in all locally inenial frames. This postulate is only the basis of general relativity. Just as the Lorentz trans­ formation equations follow from the postulates of special relativity, a mathemati­ cal framework follows from lhis postulate. Unfortunately, general relativity theory is too sophisticated to discuss quantitatively here. It involves the mathe­ matics of tensors and differential geometry. Nevertheless. some of it>; astonishing predictions can be understood qualitatively just from the principle of equivalence. Three have anracted particular attention: ( I ) gravitational redshift. (2) the deflec­ tion of light by the sun. and (3) the precession of the perihelion of Mercury.

Gravitational Redshift and Time D ilation According to the principle of equivalence. light emitted at one point in a gravI­ tational field will have a different frequency i f observed at a different point. We see this by analyzing not a fixed light source and observer i n a gravitational field

g, but the equivalent case of a source and ob�erver in a frame w i thout

Principle of equiv:Jence

46 ctr.pter 2

Special ReJatlVlI)'

gravity but accelerating at g. In Figure 2.32(a), a source in an accelerati n frame emits a wave front when the frame has zero speed. Anna, a di stanc e "above" the source, observes the wave front afrer a time HIc. But by this ti gHIc. Thus, the light is observe . me she is moving "upward" at speed d 10. a frame that moves at velocity gH/C away from the frame in which it was red. According ro Anna, the Jjght will be redshifred. Provided that g and

�

II =

�rnu� �

e will be smaJl, and we may apply the binomiaJ apprOXimati o n 0 !he Doppler (onnul. (2-18). not roo large,

v

= /source

fobs

c

v

= fsource

+ ­

c

� fsource \.1

_

=

�

� )l � v) -I

!'o"�e

v - -

1 + -

c

-

I

c

v

) �\.1 - 2 �v)

2�

� - �)

Figure 2.32 In (a), a wave front travels from source to

observer in

an

I

accelerating frame without gravily. Because the

observer's velocity at reception of the front differs from the

source's velocity at emission, there is a Doppler shift in

frequency. !hus. there must be a frequency difference in

(b), the eqwvaJent stationary frame with gravity.

v v

! J H

=0

= at = gcH

h:

Observer

1=0

t

a=g

'i'

Source

(a) Accelerating frame

I

'i:!!

'i'

!!. c

l/

Observer

H

J

(b) EqUivalent frame

2JI Gl'neral Relativity and a Fi....1 Look al Cosmology 47 In (he last step, we have discarded the \.2ICl term. Thus (2-29)

Einstein's poMulate is that a gravity-free fm.me accelerating at g is equiva­ lent to a fixed frame in a gravitational field g. Thus, equation (2-29) must also apply in the frame as shown in Figure 2.32(b). As light moves upward. its fre­ quency must become smaller and its wavelength longer. as i l lustrated In Figure 2.33. We reach a lruly fascinating conclusion: This is a time-dilation effect, but it has nothing to do with relative motion between the source and observer frames. Consider 600 nm light, which has a frequency of S X 1014 Hz. As one second passes at the source, S X 1014 wave fronts are emitted. It might be said that S X 1014 distinct events have passed in the life of the source. But in a given second, Bob receives fewer wave fronts, because he observes a smaller frequency. To witness ail S X 1014 events in the source's life, Bob has to wait more than one second. Relalive to Bob, the source. deeper in the gravi­ tational field, is aging slowly. In fact, the time passing at the lower point must be less than the lime passing at the higher point by the same factor that appears in equation (2-29).

(2-30)

Does time really pass morc slowly on Earth's surface than at some altitude above? The weakness of most gravitational fields (gHlc2 « I ) makes gravi­ tational time dilation a small effect. But loday's high-precision clocks. whose time bases are atomic oscillations of ex:tremely short period, can reveal it.

R E A L - W O R L D E X A M P L E R E LATIVITY A N D T H E G L O B A L

POSIT I O N I N G S Y S T E M

Signals from GPS satellites can locate objects on Earth's surface easily within a few meters, but this requires great precision in timekeeping. Each satellite sends a time signal. and an error of just 10 ns would translate, at light's great speed, to a distance error of 3 m. This requirement has thrust into the mainstream of modem technology a topic previously of concern to only a few physicists and astronomers. Special relativity tells us that ;} Salellite moving relative to Earth's surface will run slowly. General relalivity tells us that a clock runs slower when it is deeper in a gravitational field. so gravitational time dilation will cause a satellite's clock to run faster than a clock at ground level. Equation (2-30) gives the simplest case of a unifonn field. In Earth's nonunifonn field. the corresponding relationship (see Exercise 98) i s

'" (:,., [1 satellite

_

-'-

(

GM

c1 rEanh

_

(2-3 1 )

Figure 2.33 A light beam's wavelength grows longer and its frequency decreases 35 it rhes in 3 gravitational field.

_ 1I 1o ..... _ Md , . ... di _ _ ... mw' o( ... Eonh. 1bio ia _ ..... .. .. i'W 12-30) .... II IDIP lint _. The ractor gH in (2-30)

.... . ,.• • _ � en
.

. ... ...'I ;W.,.,

A OI'S ,.,.,... _

•

III oIblude 0( 2 (}

)(

10'

m IIJd . .peed

oL 1.9

)(

1 0' mi.

' a' m. and a point II its eq\lIIl1l' has tangc'ntia.i �� of 460 mis, for bom IfIvilationaf lime dilJlion? (b) If unJ,-'\,'oumcd for. how � and toon would the tJlM � In ernlf by IO ns" "" s radial .. 6.4

x

e.., 8y eppu.,IIUICe'Y wlull frKtlOl1 mU..1 lhc lime be' adjusted to K\,'ounr

SOLUTION

(I) For repJar lime ddalion. we need the �lcIIiIC'5 speed rciathe to a pujnl lln Earth', &urface. Oi L"our'>C. dlft'erenl

pOInls lIn a rol:;llin,g gillbe move: ;.I1 diller­

ent speed\, and a r«c,..cr mlghl also be In motion. H()"�er. c\en Eanh',

equlluriaJ �pta1 1 mud saaeJlilc', peed

lmaller than the ulc:itile'" !ilpc'C'd, ,,0 u"m8 Ju"l lhc "ill ,IV(' Ui a eood appro:umahon. Thus, (l)r J d\)!.;k. Udmg on

boMrd I.hr satcUlle, we haw

\

(I

(3.9 10')2 "

I

.1 X

10"

.lf t-Mth

iilf

(I

1 . 7 ")( 1 0

+ X 5 X 10

10)

1 . °.l1

,aldlth:

") .l 'sellite

ft.,.- the gnmtJlionai del'l. wt'" CklJ f:;.:uth ... rJdUh I�) (he- limit "Irltm/t', and n(.)le thai GMt:;.,h = (6 b1 x JO I I N 1I1'··lg l 2)(51.)� x I� kg} .. X 1014 m'/"l

4.0

•

Thus,

I

I

I

I

.ltOCC-lIllrl l

�L�6 X )C)

The gr.mtallOrw.I e.t1"iX'( rruk.es the- soilcIlifc;'s time ,lightly latfer. 11:- dlX-'" run,

,,�'elI-d�pendenl effn·1 make... ml.' ,,;ud· hit" !!; ciul..-t run .fio)fe'T, hut b) Wl (''"tn timer ilI1l0unt. Adding the frJ(.:tional change\ fur each. yoc ha\C a tiny bit latta Ihan a surfxc dl"'�. The

:\.26 ")( 1 0

j,1�lIjle

I

i

10)

:0: -4.4 ')(

(b) Inserting 1:1 discrtpancy

i

of 10 ns.

I()

l(

H)

"�I

.l'Wltelhle

10

.lts.:Jreun(: = 23 ,

10

10

...

X.5 X

10 I J

"C i« tlwt In . "!) tbi.'(1 l11l,lt' cftOU¥h t m ..... VI\lt,Ikt k wnu\aW 1I.1 maLe me ')'SIt'm

.......

In prw:'",� ..k-=b � GPS ub;llllC) � presct 10 . rreqUC'QC)' thai ..",'OOftl!l fur th: lra\ IbIklfl4l t'ffo.'t n:l&tl\C 10 Evth's ,urf;ac(". *' II dc� ,ll\J:o 00 1M "n('l�u ,wtllt rMhus V�k""'I\C's aod ek:" "'IOO\ 01 re..""'CI\'nS \'&I), AAd ",,""\lUnhnt. 1\If thew trT.I! �u'C l'Cal·timc CD.:ubtio� In lhc ��h ins umt. f�1d !M\ffi('huv. wMp§ hnlC ml('("\.als. C\� when (hen' l!o nmti(m. \\h) lhoulJn'( II "art' '�l-e inh:na1s') Intk"l:d, lint of 1he lCntt'i of s.entruJ rctati\'I1� IS that a IlU�\l\l' h('u\('nly body warp)! sl""'-"Cotime nearby, Kcpn:!'OC'nting w:trprtl sp.tn'·tim\! 11\ Ihn'(.' Junensill1l i\ Jillkuh I! I.. t'ASIl'r in t"'ll JimclhIOO'i, 11\ \\'hkh :oop'KC l'i !In.'a, hgun.' 2. .l.l 5011l)ws a 11I.1.\M\(' If

J.

¥r.\\ Ilatiunal

no reblive

h(a\<'nl� �Ill� 1!L\lud"llllj; the:- rl'�ul"rit) III ,I !"II,dltnC'lhlllll.ll .l>PU..·l'. Inhiltal­

lants 01" Ihl\ !\\,I-JulIl'n,i,,";'11 uniH" '\,' "�J'n'l ult n'lb 1\, � (If ('11 11011 an':I, Wl'

IlLlloiltii!' " thl'n l'" l'an 't'C th.\1 thl..' "'l'lIs I1l"M till' hl"I\C,ll� t>txly ilrl." 1\";,lIy lar�l·1. but vul) Ifllm I'm l''\tfUlitlm'II'llllhll \'Il'\\ pi.lIllt, Silllll"rly. the \\"Il';!}!l' (If r�ill thrt'I..'-JI\1Il·u.... 'llnal �P;lI:l' h \\tIt app,trl'lIt hI hum.1Il bcin�' W... arl' ..:rt'lIlUn.·s 01 vue SIXI\,'C llf thn."t.' dlllll'Il'\Un, and l';Ulll�11 'land had, to \ il'\\ 11111' uni\t'r'l' lill

!our-dimcn'll)ll;ll ;nl.." �l·\l·nhl'k'>i. l'\t'n rl..'<.Iul'cd-Jil1lcll\l\lnal \It''''' ,u..:h ,,� Fi�urc �..\.l pnl\ Ilie a 4uahl"I1\� IIlllkr'tamhng nl 'lIme l'lh:.,:1 lit �l'nl'r;\1 n'la­ tl\ it),. Ol'i

we \\ III nun sct:'

Ihl' f.kl that grant) l"'1Il \'311"(' a rcJ.. hllt " ju...t lllll' !'>i�n III a I1Ulrl' 1'1;'-'11..' (ruth: (ira\ll) alk...·h h}!ht. A, "' ... �ml\l", h}!ht h.h ntl Illa". it� l'nl'l"g) I' ull l..inelil.: Su I! mu�t hi." thai gnl\ lIy pull .. on hlml� llf l'lIngy \\Ihl'r than lTliissI mll'mal l'l1t'rg� A, hght mmc, awa� from Earth, ii' �melil' energy mu,1 dl..'(.Tt'3.o;e ..., pOll'nl iJ.l l'ncrg) 1Il1.:n:;I'e' But thi, lead.. to the 4Ut',tL\IIl. Hl\W ":iln "'i!' u-.(' :-';1..'\\1\10\ Uni\l'r.ill l.l'" til' gr.l\ il.llitln. F ' vl'If",/r. to .l1.I.:l'\llInt f,'r Ihl' clllxt Ill' gr.l\lt�

"ilhllut an m for h�hr� The an.... wer i, th.1I thl....

law i, really II

Flgur.2.34 A h\IHJIUlt'n,innal rt'prl" t'nl.ltion 1)11l 1l1il",j\'C obJcl.:1 warping 'pa<:('.

.. Warped spac e-lim

II Chap,.. 2 Spetial Relabvily special case. correct only when the gravitational field is very weak. In this limit. lhe effect on light is negligible. and there is no need to consider an m. For

a shUng field. however, general relativity comes imo play. and Newton 's law is

replaced by a different view-warped space-time. The variation of light's fre­ quency near a ma.o;;si,,·e heavenly body is viewed as a natural result of its pas­

sage through a region where the regularity of space and time is disturbed.

Deflection of light by the Sun

We have seen that gravity "pulls" on light movmg directly away from a heav_ enly body. But what if the light is traveling laterally-does it curve? It must ! As shown in Figure 2.35. a laterally moving light beam would appear to Curve toward the floor in Anna's rocket-powered closet, so it must curve toward the floor in Bob's Earth-bound closet. Again. we don't try to reconcile the curvature with Newton's law of gravita­ tion. Rather. the light simply moves in the most natural way given the warped space-time through which it passes. A guiding classical principle stiU applies: Light always takes the minimum time to lIavel from one point to another. Figure

2.36 shows two possible paths or a light beam originating at one point i n space. passing through the warped space-Ijme near a large heavenJy body. then observed al another point. The darker path rollows what inhabitants of the flat. Iwo­ dimensional space might believe is a �traight line-one of the lines in a grid that would be regular if space were not warped. Smugly observing from OUf extradi­ men�ional perspective. however, we see that the other path is shorter. This is the path the light actually take�. 1>0 an observer with a "flat perspective" sees a beam that appeared to curve

as

it passed near the heavenly body and thus seems to

have originated at a different point in space.

Figure 2.35 A light beam curving in an accelerating frame without gravity and in the equivalent stlltionary frame with gravity.

t

a=g Curved

p',"

Light "".m

\II\\..,�

IT

(a) Acceler.lIing frame

(b) Equivalent frame

Figurtt 2.36 A light beam takes the shortest path. which to inhabitant' of warped space does not appear straight.

Actual p:lth

/ �.

Appnrent origiu of light -

q\

Ob5ervcr

Longer "straight-line" Palh

-

0

" AlIt pc�peclivc" (top view)

Our Sun doesn'l warp space much. Its gravitational redshift, for example, is only about 2 parts in a million. To have any hope of seeing deflection of light by the Sun, observations have to be made of light rays passing very close to it, where its field is strongest. This light, of course, comes from other stars. Their positions should appear to shift slightly as the Sun passes between them and Earth. The problem is that light passing so close is usually obscured by the brightness of the Sun. Therefore, the best time to make an observation is when the Sun is "darkened"--during a solar eclipse. An observation during an eclipse in 1919, shortly after Einstein formulated his theory, showed a deflec­ tion of about 2 s of arc for light barely grazing the Sun. Agreement with the prediction of general relativity was excellent, and Einstein was instantly famous. Progress and Applications discusses the bending of light applied to cosmology in the phenomenon of gravitational lensing. Precession of the Perihelion of Mercury

The orbits of the planets about the Sun are not exactly circular, but slightly elliptical. At one point in its orbit, called aphelion, a planet will be slightly far­ ther than average from the Sun, and at another. caJled perihelion. slightly closer. As long as a system is simply one object orbiting another, it is a direct prediction of classical Newtonian gravitation that the same path in space is retraced indefinitely. Influences of other heavenly bodies will cause the orbit to precess. The points of aphelion and perihelion progressively creep around

..... z.n An orbit pm:essing about

dJeSun.

�7"",,___of periheia l Prece5Sion

in a circular fashion. as depicted in Figure

2.37. Mercury's orbit precesses due

to the perturbing effects of the other planets, and Newtonian theory can pre­ dict what the precession rate should be. BUI much 10 the consternation of early astronomers, the predicted rate did nor agree precisely with observation. The problem was that Newton's law of gravitation is correct only for a weak gravi_

tational field. If any planer deviates from the class.icaJ expectarion. it should be

Mercury. for it orbits where Ihe Sun's field i� strongest Using general relativ_ ity. a correction to the classicaJly expected precession rate of Mercury may be calculated. The result of

43

s of arc per century is in good agreement with

obsenration.

Cosmology Cosmology is the study of the behaviors of heavenly bodies. 1t is invariably coupled to gravitation because gravity is the dominant interaction, being the one fundamental force in nature that is both long range and only attractive. As discussed in Chapter

12, the nuclear forces (the strong and the weak) are neg­

ligible al distances beyond about 10-15 m, and the remaining force, the elec­

tromagnetic. is largely canceled out for eleclrically neutral objects. Much of cosmology can be understood through classical Newtonian gravitation, but some behaviors require general relativity.

Stellar collapse describes the fate of individual stars. Early in a star's !.ife. the enormous gravitational attraction of its mass is balanced by its tendency to expand due to the constant generation of energy (via nuclear fusion. discussed in Chapter

I I). The radIUS remains constant. When the star's fuel begins to

burn out. gravity gets the upper hand and gravitational collapse begins. If the star is not much more massive than our Sun, the result is a white dwarf, and then ultimately a cold, dead chunk of matter typically no larger than Earth. For .�tars several times the Sun's mass, the extra gravitational pressure is able to force protons and elecrrons to combine, forming neutrons and tiny particles known as neutrinos. The neutrinos carry away a huge amount of energy in a cataclysmic explosion known a.C; a supernova. What remains is a cold neutron star of fantastic density, about J O J 3 times that of le.ad, with a typicaJ radius of only tens of kilometers. (The white dwarf and neutron star are treated quanti­ tatively ID Chapter 9 exercises.) For even larger stars, there is a third possible fate: becoming a black hole. One of the most startling features of general relativity is that space can have singularities; that is, discontinuities or divergences. A singularity occurs when a body of mass

M becomes so compact that its radius drops below the 's = 2GM!c2. The singularity would be separate from

Schwanschild radius.

the universe as we know it. It would simply be a silent hole in space. Moreover, its gravitational field would be so strong that not even light could leave it. thus the name black hole. (Exercise 10 1 discusses a crude derivation of the Schwarzschild radius from this assumption.) The presence of a black hole would be betrayed almost alone by its external gravitational effects and itli abil­ ity to gobble up things from [he outside. A black hole has an event horizon. Things outside can go in, but once inside the event horizon. there is no way out. An unlucky trespasser gets a one-way ticker ro the singularity. It should be

noled thaI not even the atomic nucleus-trillions of lime);. denser than ordinary solids.-comes close to qualifying a.-', a black hole. A typicaJ nuclear diameter is

10- 1:'\ m, but irs SchwafLschiid mdiu!' would be roughly

IO'5:! m. With so few

e.\lemal signs. obtaining conclusi\-'e evidence of black holes is not easy. Newr­ !.heress. maS( cosmologisb accepl not only that they exist. but also that they are a fairly common feature of the centers of gala.·des. As far as the universe as a \\-'hole is concerned. the most ba.sic obsef'\-'ation is that it is expanding. and it appears to have been doing '0 e\-'er since a Big Bang. an explosion of an unimaginably hot and dense collection of relativistic

particles. Chapter 1 2 discusses the evolution of the universe at very early times. In this chapter. we look at its expansion since the fOm\alion of galaxies. (We refer to galaxies because they move more or less independently. Stars do not, for they orbit each other in galaxies. Actually. many galaxies orbit in groups or clusters. but the word galax)' will serve our purpose.) If objects were to begin moving away from an origin at the same t.ime. each maintaining ils original speed. they would later be spread out according to their speed ... After I unit of time, those moving at a speed of I would be I unit away, those mov�

ing at a speed of 2 would be

2 unils away, and so forth. Perhaps surprisingly,

this would also appear 10 be the case no matter which object/galaxy served as the observer's frame of reference (see Exercise

102). Thus.

if the universe did

begin with a big bang, an observer in any given galaxy would see a propor­ tionality between the distance to another galaxy and its speed. This is just what we observe. The evidence comes from the Doppler shifts in the light spectra from distant galaxies. The farther a galaxy is from Earth. the greater the redshift in its spectrum, implying a greater recessional speed. We find that the speed increases linearly with distance.

This relationship is known as Hubble's law. and the constant Ho as the Hubble

constant. Because an object moving a distance r at constant speed

I'

would

travel for a time rlv. and because this is the same constant I/Ho for all galax­ ies. llHo is often referred 10 as the age of the universe. The evidence suggests a value between 10 and 20 billion years. The Hubble constant is not known with great precision. mostly because it is hard to determine distances to faraway galaxies. But even if distances could be known precisely, a guess at the age of the universe based on the Hubble con­ stant is approximate at best, for the expansion rate is not constant. Galaxie.. have not moved out independently at constant velocity. because they all feel the gravitational attraction of the universe as a whole. This alone would suggest that the expansion rate is decreasing. Even so. the rate of ils decrease would depend on the total amount of mass/energy in the universe. Analogies to escape velocity in classical physics are very helpful. A projectile launched from a planet at a certain speed would escape if the planet were of very low density but would return to a very dense planet. There would be a critical density at which it would barely escape; there is similarly a critical density for the universe. Present evidence suggests that the universe is very close to critical. but the cause is still in que�tion. LumlOous matter such as stan;, the easiest to document.

$4

a..pter 2 Special ReJ.IIIVily

accounts for only 0.4% of critical density. Theories regarding the production of mass in the early universe suggest that nonluminous ordinary matter (protons neU(TOns. and electrons) should total about 5% of critical density. This dark IIlat� ter wouJd be in the fonn of diffuse gases as well as dark stars�white dWarfs neU(TOn stars, and black holes. However, srudies of the orbital motions of gaJaxj in clusters indicate that there is much more matter that isn't ordinary�abouI 30% of critical density. On this. speculation abounds, with kinds of particles predicted but never actually observed being high on the Ust. But even this leaves more than 60% of critical density missing. A prime candidate to supply the deficit is a leon Einstein originally tacked 011 to his theory of general relativity in an effort to explain a universe, but then discarded when expansion was discovered. It has now regained favor. Known as the cosmological constant, it is a mysterious energy density that would permeate the universe but not dissipate as the universe expands. lr could explain not only the near critical density but also recent data indicating that the expansion rate of the universe is actually Cosmology obviously has many interesting open questions. We return to the topic in Chapter 1 2 after learning more about the underlying physics.

�

stable

increasing.

[!J

P

T

o

N

A

L

2.9

The Light Barrier

In rhe 1 940s, it was widely claimed that no airplane could be made to exceed the speed of sound-the attempt would end in disintegration. Many doubted this claim of a "sound barrier," for it had no theoretical support. Here we dis­ cuss a different barrier and a claim that is supported by Lorentz transfonnation equations:

No inJonrlotion may travelJaster than tile speed ojlight.

In Figure 2.38, Bob is at the origin of frame 5, and Anna is at the origin of frame 5', in a spaceship moving in the + x direction at speed v < c relative to Bob. At time zero, just as the origins cross, Bob has an original thought. In a

spasm of inspiration, he seizes upon the perfect finale to the symphony he had been composing. This event--event !-has position and timex] = 0, ' I = O.

Bob wishes to share his newly completed symphony with others as soon as

Figure 2.38 Bob has an original thought.

Amy

x,

s· ·

Bob

z

s

z

Bob Jr.

,.

�

2.9 Tbe lJ&ht ....

possible. He sends the completed symphony-the "information"-to Bob Jr.. standing at -"2 ' via the fastest method available. If the information moves at speed 110 by this method. its arrivaJ at Bob Jr.�vent 2-occurs at time

Anna's assistant Amy is riding near the front of Anna's spaceship and happens to be precisely aligned with Bob Jr. when he receives the information. Bob Jr. instantly communicates the infonnation to Amy. Occurring at the same location and time, the arrival of the infonnation at Bob Jr.'s location and its communication to Amy is a single event (again, event 2). From its position and time in Bob's frame. equations (2-12) give its position (Amy's location) and lime in Anna's.

( V X2) (

r'2' = ')'v - C2 x2 + "0

v

1

c'2

u0

)

(2-32)

= ')' - - + - x \'

2

As soon as Amy receives the information. she sends it to Anna, also via the fastest method. In Anna's frame, the time that passes as the information travels from x2 (Amy's location) to the origin of S' (Anna's location) is xiluo' It leaves Amy at t2. so it arrives at the origin-event 3-at time

"3 = r' + 2

x2 U

o

Now substituting from (2-32),

2

[ v("O -)]

(2-33)

c

= l' .x2 - I - - - + uo \ c 2c 2u0

If "0 is less than c. then ') is positive, and a1l is well. But for any Uo

there is a v < c for which ,) is negative. (See Exercise

> C,

103.) Recall that event 3

is the symphony reaching Anna at the origin of frame S' and that the origins. Anna and Bob. pass at time zero on both clocks. Anna would be whistling the

before

she passes Bob, who. by assumption, conceives completed symphony of it at the instant of passing. This violates the principle of cause and effect.

II

a" I • •

..- n: hils II _ ....... "'"' ..... _ poecede .!feet. then superluminal infomwioo .. .. . ........

..... ... c-. ..... "' ... ..' .... 110 ... .. ,..' " 1, _ .... ..... "' ... r-, _ .

.. ... .....-

'1110 .. "' ...... 1IId effect leads us back '0 a poin' raised in coonectioo willi e..... 2.4: Eveats may occur in a different order in different frames. If

thia II &rue.

lUI,.. cause pm::cde effecr in one frame but C'ffecl precede cause ia

IfiJDCbet? Sucb •

thin, would ccnainly be mind·boggling. but the LOrentz

IrUIfOllllllioa equabol15 do not allow it. Events can be reversed, but not ir they lOR causally ",'*
s,....,_ ..,..

could evenls spatially not 100 far from that uis. An tvent on (he �urface uf the pa.'( hghH:one. a dl-.tance �x = c j,/ away, space--lime origin at

•

t=

O. and

'0

could have been a <.:au� only If infomlallon from It were communicated to the

space.(imr origin al the !.�cd uf light. Thus. events 10 the past but outside the pasl light-cone are too far away to be c.:ausalJy related Thi", region is known as lhc absolute- eLwwheN_ AI tllnc!' lero, it c(lOslrh;t<:, to a <:'ingle point. becau\c no

event anywhere else

('all

ha\c!' an Instantanc!'ous effect on the event at the

space-time origin. The meamng of the future light-cone IS complementary. It arc not too far away 10 he affected by the event at the

contain" all ('\'ents thai

space-time ongin--<\'enh {If the absolute fulure. Naturally. it spreads ..pa­ lially a,<:, time prugre\ses.

III 0 V A N e E D

2, 1 0 The Fourth Dimension In thi" "IeClion. we will

more dearly why we speak of space-time

as though We also study limt were no different from the three sp .. llial dimension... other Sloe

1l."Jlt':IS of the four-dimensional nature of "pecial relUlivity. Rrcall fmm Section 2.3 thai for a free object to move at constant velocity in diffrrent frames. we require a hnear tran"formUlion. The logical generaliza­ tion of equations (2--1) Itl lhn:e sp' I Iial dimen'iions is

y' '

t

::::

==

A:nl + -"11.'"

AJl x

-+-

AI�Y

-+-to

+ AZ41 A�_l: + A:w1 A:!J:

We a�ain &.''isume Ihal Ihe relali\"e mOllon between frames is along the

x'

x

and

axes. Thi� would allow us to proceed as in Seclion 2.3 without any

explicilly four-dimensional appearances. but keeping all four equations pro­ vidrs bener insight. From equations (2-12) Ihrough (2-1 ..J), we see that most

of the A cons-lUIS � 0 and the �nu trandormalion cquatioas may thua he "Hillen L'i

..

:

-

)

- - cl .' .'

"

� 'y, " c

(, : ) _

.

or

[::;,:]

.'

Y,

o

0

-'1,.-

o

I

0

o

o

o

=

- y,.�

.'

c

(2·)41

o

o

0

Y,

'

'

The lir.t anu founh equatiun!> differ only in the exchange of ("1 for "� IUld for

I.

Mixing with !)(lsition as it does, time must

be treated,

CI

in a real sense, as

a fourth (lirnension. We refer to the three spatial dimensions together with time a'

II

ruur-\'Ci:lor. The miltri'\ fonn of (2-)4) emphasilcs the four-dimensional

ndlUre ()f 'pace-lime and i, often a convenient way of using the equations all at once, 41'

\,-l'

,ce in the following e'\ample.

EXAMPLE 2 . 1 2

hlr the

\ltualinn in h,alllple

1.4,

(II) dctemline the Ulrent/ lransfllrmation malrix

'erif), the frllfll Bllb'� fmmc tll Anna\ frame. then {b) u\lnS matri, multiplication, .. mUl.ll (I

51 LU

0) tllne (In Anna\ twnt dod.,

D

ON

(a) We hil\c

\·/c

\ J 1 and Y,

C:L'-I) 1\

(b) A I

tick

=

�

[L

2, so the Lorentz trnn,fomliitinn matri:t in o

0

I

0

o

I

o

0

0 in Boh\ fmme, Bob Jr., at.1

The time on Anna',

o

0

I

0

o

I

o

0

from cluck j, ,'

obtain II� location in Anna'� frame,

""

""

!f

""

6

vJ m, sees Anna's front clock

( - 1 8 m)lt"

-""

-

60 m., Note that we abo

Four-Vectors In special relativiry. certain quantities thai are classically vectors acquire a founh component; they become four-vectors. By definition, any four quanti_ ties

(As' Ay" A... A,) that transform from one frame to another via a Lorentz

transfonnation constitute a four-vectof. We already have one example: the

position-time four-vector (x, y,

z.

et).

An important characteristic of four-vectors is that each has a related quan. tity mal is the same for observers in aU frames. Following the fonn of

(2-34), the Lorentz transformation equations for an arbitrary four-vector are

(2-35) It is left as an exercise (0 show from these relationships that

At" - (A.f"

+

Ay"

+

A";: )

=

A' - (A':c + A'y I

+

A') l

(2-36)

On both sides. we see the sum of the squares of the spatial components sub­ tracted from the square of the "time component." Whatever may be the physical

nature of this calculation 's result, it is obviously the same in frame S as in frame

S'. Such a quantity is called an invariant. Jdentifying square of the three-component vector portion of

A;l - A ' · A ' = A;

-

A;

+

A;

A. we may write

+

A ' A = invariant associated with

leI us investigate the invariant related to the four-vector

A; with the A

A = (x, y,

Z,

ct).

We know that a difference between events also obeys the Lorentz transforma­

tion equatIOns. Thus

A; -

, A A

= (e!J./)' - (!J.i'

+

!J.y' + !J.z')

Because this is the same in all frames, we may find its value in the most con­ venient the frame in which the events occur at the origin. In this frame,

A;

- A · A is simply

(C8t)2,

where 81 is the time interval in that frame. Per­

haps the events are the heartbeats of a space traveler at the origin of frame S.

An observer in frame S' would see these events at varying spatial coordinates

and separated by longer time intervals, due to time dilation, but would agree on the number of heartbeats. This confinns what we argued earlier: The proper time for series of events in a given frame does not depend on who is watching from another frame.

Momentum-Energy Four-Vector We have encountered another four-vector, though a little work is needed to see

it. Equation (2-23) relates momenta in two frames summed over al1 particles,

hut it is also true for a single object.

2.101'be Fuurtb Dio

Using Px = 'Y"mllx and E == 'Y"mc2, we may write this as (2-31)

Invoking the identity 'Y,,' - ( I - I.txlllc2)'Y"'Y,,. which follows from the rel­ ativistic velocity transfonnation (see Exercise 67). we can relate energy in frame S' to quantities in frame S.

0'

Dividing this by c and placing it and (2-37) beside the position-time Lorentz transfonnation equations, we see a great similarity.

x' y,[x =

-

�(c<) 1

(2-38)

The relationship between Px and E is virtually identical to that between x and t. It is also true that

P; == Py and P� == Pl (see Exercise 109). Thus, p Py ' Pt.' and x'

Etc obey a Lorentz transformation and constitute a four,vector, just as do x, y. l,

and

ct.

As time (Cl for proper units) must be considered an essential part

of me position-time four-vector, or four-space. energy (Elc) is an essential part of the four-momentum. And what is the invariant associated with four-momentum? For the case

A = (px' Py' Pl, Elc), we have

A;

-

A·A

=

(E/c)2

-

p2

If we choose S to be the frame in which the object is at rest, the momentum is

o and the energy is simply me2. The invariant is

m2Cl, where m is the mass in

the rest frame of the object. Observers in all frames agree on the mass/internal energy of an object.

...

'

19

60

Chapter

2

Sp«I:I] Relillu\ll)

PROGRESS A N D A P P LICATIONS mechanic!> ....a!> confinned centurie!> ago--or ,>o ....e Ihought With modem technology. there ha!> been no pau'>e in the '>CrulLny of !>pecial relativity. The prn;-'>Ibihty of \iolation� of it!> fundamental po�tulatc", proposed by ne.... theorics of fundamental forces, have redoubled mterest. In 1002. a team of re...eatche� at the Unt\"er;ities of Kon�tanl and Dusseldorf in Gennan} confimled the con�lanC} of the speed of light 10 a high degree of preci�ion_ Their apparatu\ was a modem l....i,1 on the Michel,on-Morley mterferomeler, con\i�ling of perpendicular C3\ ltie\ made of a very �t..ble sapphire cryslal and kept at a temperature of " K to reduce thennal noi<;e, They monttored Ihe interference a!> Eanh revolved about the Sun. taking their laboratol)

2.40

Gra..itational lensing. The bright spot� In the: middle are pan of a gala:(y 1,:lu�ter_ The loop-shaped form.. lions a1 4. 8. 9, J.nd 10 o·clod, are Images of the same dt,. tant galaxy whose lighl has reached Earth by curnng in different direction!> around the clu,>ter Figure

Special Relativity Stands Up to tke Test Cla.,sical

from one frame 10 another mo\ing relative 10 the fir\t at about 60 I..m/s. AI such a low �pced. the effeci 01 the frame motion wa" expected to be ..,mall. but they ruled OUI eflech almo�t 5 orders of magnitude '>maller. Even the icon E =

lI/e2 receives regular attention, A team of ,cienti�" from

uni\'er,itie� and lab,> in ,e\'eral countrie' recently mea,url!d the ma'>s change .....hen a nur.:leu� captures a neutron to become a different nudeu� and Ihen emi" electromagnetic rJdiation (NlIIlIrt>. 22-19 I)e"emhcr 2005. 1096-1 097). The emitted electromagneti" energy was found to equal the "hange III IIIC2 of the neutron Jnd nuclei "IIhm 0.00004";( Eimtein\ cJaim� seem 10 be holding up quite well Gravitational lenses: Applying the Bending of light

n

If il gra\Ilational field "an bend Ji!!hl. then light ,>hould b.

T '

to. ween it and Eanh,

no

I-

, (

1•

.tblt" to tIa\'cl from source to ot-o,en'cr by III/litiph' palh cUound a lIla�si\'e nhJe,,1. Indt:ed thi, happen" and II i,

I

kUlI'" n a, A;ra\,llatLOnal lcn�jng_ Figure 2 .40 " a superb

l

�xamplc Tal.cn b) the Hubble Space Tcle,copt'_ it sho..... fIlultlrle Im.1ges (I' the ,arne gala,y. The massive

<1ries make ,kepticism. General ,I

no one doubts but

( ..It direct

It'JC

'" ('harged object when

'c ,he- b III lhl\ "a,e j, a clu,ter of galaxie'>, whl"h proou ..

ulalion that propagates

Spol\ al lhc cenler of Ihe Image_ Gra\itational len,ing IS an Indi\pen�able IXI :1 )

.Imilar phenomenon,

mology no....aday'>. One- application is in \artin� lIU h

krJling massive objects

.htft u':.uclatcd with lhe expan'>lon of the unh

r. :lie,e disturbances are

With lar�e fI:dshifb can be '>cen len\cd by obj<::-

t

w

smaller redshitb, conlinning Ih,lI the more-dlst<�.I
:n •

moving fa'>ter. Anolher application is to gauge tht" nl1�S i!

IS

lribull/m 01 a galaxy cluster; the mass den'ity mu�t bt

)st sy. t�ms, because gravity

LVl"') "lC "1'<. -ren ... tatie electricity stick h I 1".'1 tho.! only t,'!:'il
" ,,,( '-"'n th,

n

5 I • •

)Ive "mh, whICh t'> of tmmense mass.)

I!!rcatest where the lensing effect IS mmt pronounced.

lndir

Recau,e all mass, not jUlll luminous mas,>_ l'ont ributes to thl,;

of t"0 mutuall) ('rhiting pulsars was found to be losing

effect, gra\'italional leming is aha one of the mOl<>t impor­

ent"rgy

tant techniques in the search for dark matter. Not onl)' can

attributable to gra\'itational wa\'e� (Nobel Prize, 1993.

the presence of dark matter be inferred in galaxy clusters. but al�o light from a di�tant star can be perc.:eptibly

t t:vtdclle \\a� obtained III the 1 970�, when a system

II a rate predicted by general relativity and

Hulse & Taylor). However, direct detection is still elusive. In the vanguard of the search IS the Laser Interferometer

SummoAry

Gravitational-Wave Ob�nlatory (LiGO). ThIS ,aciluy has IWO location!> �eparated by nearly 2000 I m l c s to . . , d.M:nmmatc between local effects and those from a broad wave reaching Earth. Such a W',·e shou Id cause . . . . �hght vanatlOns HI length along different arm s of the mlerfcromele�. Among the hope< �e eonfi m nalion thaI

gra\lIiltlonal wa\e� move at (' and thaI the) are IranS\CNC In a way that agrec� With the dL\lmct!,c angular momentum they are predicted to carT). II is al-.o hoped to detect the lively r:tl.liatlon thai should a..:compan) colliding bInd. holes, which ....ould further \alidalc the theory of general relatlHty.

Chapter 2 S u m m a ry

p r- )'.,"'u

There is no evidence for the absoilltt' motion of any inertial frame of reference. All motion is relative. Arguing that an experiment should pr�eed identically in any frame. irrespec­ tive of Its motion relauve t o any other, Alben Einstein postu­ . lated that the laws of phYSICS must be the same in all inertial frames. I� panicular, Maxwell's equations, which yield the speed of light c. are the same. Therefore, light will move at c relative to an observer in any inertial frame. A direct resuh of Ein!>tein's po!>tulates is that an event's position and time in one frame will. in general. be different from its poSition and time in another. The equations relating these positions and times are the Lorentz transformation equations.

(2-12a)

61

E " )'u'''c;2

(2-22) (2-24)

The hlws governing momentum and energy conservation are inextricably related. Einstein generalized the concept of an inertial frame 10 include a frame thai is falling freely in a gravitational field. No ell:periment could distinguish the two frames. This view led to the idea of a gravitational field warping space-time and explains the effect of gravity on light. The speed of light is an absolute upper limit on the speed at which information may mo\'e relative to an obse.....er.

Indicates advanced questions

Conceptual Questions

Following from the\e equations are important consequences of Einstein's postulates: time dilation, in which an observer in an inertial frame delermine\ lime to pass more slowly on a moving

I. Explain to your friend, who is willing to accept that \ight moves at the same speed in any frame. why clocks on a passing train are not synchronized. If it helps, assume that Anna is at the middle of the train. 2. A friend says, "It makes no sense that Anna could tum on lights in her hands simultaneously in her frame but that they don't tum on simultaneously in another-she decides to do it, and irs done." By considering the tractable, if somewhat unrealistic, situation of Anna's

clock than on hi� or her own; length contraction, in which a moving object i� �h(lrter along the direction of relative motion than it is in the frame 10 which it is at rest and relative simul­

thought being communicated to her hands by light signals, answer this objection. 3. The Lorent7. transfonnation equations have x and ' and

(2-12b)

taneity, in which 1""0 even" !oimuitaneou,> in one frame are sep­ arated by a non..ero time interval in another. The demand 'hat momenlum conservation hold in all

frame.. lead� 10 Lhe relati\,l�tically correct expression for momen­ lon, ab$(nt 10 das...ical physic... for the lotal

tum and an ex

energy of a mo.. ng :mje...""t

' x

and ,' Why no v and v'? 4. You are gliding o,'er Earth's surface at a high speed, carrying your high-precision clock. At points X and Y on the ground are ..imilar clocks, s}'nchronized in the ground frame of reference. As you pass over clock X. it and your clock both read O. (a) According to you, do

61 Chaptw Z

Special Relativity

eJoc:ks X aDd Y advance slower or faster than yours? (b) Wbeo you pass over clock Y. does it read me same time. an cutier time. or . later time than yours? (Make

IS, You are strapped into a rear-facing seat at the middle of a long bus accelerating: from resl al about 10 mls2 (a rather violenl acceleration for a bus). As the back of the bus passes a warning sign alongside the street, a red

sun: your answer apees wim what ground observers sbouId sec.) (c) Reconcile any seeming contradictions bcIween your answers 10 parts (a) and (b). 5. A thin plale has a round hole whose diameter in its rest frame is D. The plare is parallel to the ground and mov­

ill&: upward. in the +y direction, relative 10 the ground. A thin round disk whose diameter in its resl frame is D

light of precisely 650 nm wavelength on the sign turns on. Do you see this precise 650 nm wavelength? DOes your friend silting al the front of the bus see the wave_ length you see? How could the same observalions be produced with the bus and sign stationary?

16, In a television picture tube. a beam of electrons is sen( from Ihe back 10 the front (screen) by an electron gun.

is also panJJel to the ground but moving in me +.r direction relative 10 il. In me frame of the ground, the plate and disk are on coune 50 that the centers of the bole and disk wiIJ al some point coincide. The disk is

When an electron strikes Ihe screen, il causes a phos­ phor to glow briefly. To produce an image across the entire screen. the beam is electrically deflected up and down and left and right. The beam may sweep from left to right at a speed greater than c. Why is this nO{ a vio_ lation of the claim that no infonnation may travel faster

contracted. but the hole in the plate is nOI, so the disk wilJ pass through the hole. Now consider the frame of the

disk. 1be disk is of diameter D, bul the hole is conrracted. Can the disk pass through lhc: hole, and if so, how? 6. In the {Win paradox situation. a fellow student objects to the argument thai Anna's acceleration is the root of

than the speed of light?

the asymmetry. "All motion is relative! Anna is acceler­

Exercises

ating relative to Bob. bur Bob is accelerating relative to Anna" Answer this objection.

Section 2.3

7. Does the asymmetric aging of an Eanh-bound observer and his {Win who travels away and back demand "rela­ tivistic speed'" (Overlook the fact that each has a lim­ iced life span.)

8. You are floating in space when you nOlice a flying saucer circling you. Each time it passes n i front of you, you note the reading on ils clock. Do you see ilS clock advancing f.ster or slower than your wristwatch? Does the space alien see your wristwalch advancing faster or slower than his clock? Explain. 9. 'The Doppler effect fonnula involves two speeds, 11 and

The Doppler fannula for sound involves three speeds (source. listener, and sound). Why a different number?

e.

10. A relativity enthusiast says. " If E = mc2 and energy is conserved, then mass is conserved." How do you

answer? U. Can a single object of mass 4 explode into two objects each of mass greater than 2? Explain your reasoning.

lZ. CUI lWO moving objects of mass 2 and 3 stick together and form a single objecl of rnbS less ilian 5? Explain

yow- reasoning. 11 TWo objects isolated from the rest of lhe universe col­ lide and stick togedler. Does the syslem's final kinetic energy depend on the frame of reference in which il is viewed? Does the sysrem's chang� in kinetic energy

depend on the frame in which il is viewed? Explain your answen. •4. .Particles of light have no mass. Does the Suo's mass dlange as a result of all the light il emilS? Explain.

17. Appearing in the lime·dilation and length-contraction fonnulas, ')',. is a reasonable measure of the size of rela­ livistic effeclS. Roughly speaking, at what speed would observations deviate from classical expectations by I %? 18. Verify that the special case x' = �vt'. x = 0 leads 10 equation (2-6) when inserted in linear transfonnations (2-4) and that special case x' = el'. x = ct in tum leads to (2-8).

19. If an object actually occupies less space physically

when moving. il cannot depend on the direction we

defin� as positive. As we know. an object aligned with Ihe direction of relative motion is contracted whether it is fixed in frame 5 and viewed from 5'. or the other way around. Use this idea to argue that distances along the and y'-axes cannol differ at aiL Consider a post of length Lo fixed in frame 5, jutting up from the origin along the +y-axis. with a saw at the top poised to slice off anything extending any higher in the passing frame 5'. Also consider an identical post fixed in frame 5'. y-

Whal happens when the origins cross?

20. Through a window in Carl's spaceship, passing al 0.5e, you watch Carl doing an important physics calculation. By your watch it takes him I min. How much time did Carl spend on his calculation?

21. According to an observer on Earth. a spacecraft whizzing by at 0.& is 35 m long. What is the length of the spacecraft according to passengers on board?

22. According to Bob on Earth, Planet Y (uninhabited) is 5 ly away. Anna is in a spaceship moving away from

7

'I

1ftClinI, AMa turns a. two ..... . :

Eanh at O.Se. She is bound ror P\aoet Y to study its

a,.

geolog)'. Unf�'r1unately. Planet Y explodes. According

oae

(Bob. of \:OUf'e. ha, to wait a while for the light from

(a) Which comes OIl final (b) How ,. .. .. ....

10 Bob. thi' IXcurTCd 2 yr after Annl passed Eanh.

the e\ploo.i�ln 10 amve. but he reache§ hi§ con<:lusion by . •·....ork in� back....ard.. ) Call the pa�sing of Anna and "

.. the fronl ... oa. . lbc .... Ace . . . ...

the lisMs come on • diIfeftat � 40 _ ..... movilll?

2'7. 8ob.nd Bob Jr. ,tand .. opel!. doaIways . rtl 'p ends of an Urplaoe haDpr 2S m kJaa. A-. GW8I •

Bob lline 1ere for both, (a) According 10 Anna. how far awa),

•

spaceship, 40 m looK as it sits on die ....,.. "-­

Planel Y when it e\plode�" (b) AI whal tilTK'

takes off in her s.-eahip, Ibm swoopa Ibrou&b Ibe

does it e\plodc"

hangar al COMlant 'Ielocity. At lftCiIeIy time .., 011

23. Anna i� on a railroad flat\:af rno\'ing at O.be �Ialive

both Bob', dock and Anna's, Bob sea Alma at .

to Bob. (Their dock' read 0 a' Anna'" center of rna!os pa��e� Bob',.) Arlllif\ arm I� out,tretched in the

front of her lipKCShip reach hi' doon.ny At time zero

on hi� clock, Bob Jr. sees the tail of Anna', spKClbip .

dlredion the flatcar rnO\'e�. and in her hand h a fla�h­

hi� domway. (a) How fut is Anna·, spacabip moviaa'P

bulb, Acwrding to the wri\twat\:h on Anna'� hand, the na�hbulb goe� oU at 1 00 n\. The time of thi'

(b) What will Anna's clock read w hc:n . &eel the tail

event accordmg to Bob difkr.. by 27 ns. (a) Is il

of her space!ohip at the doorway where 80b Jr. is stand­

earlier or later than 1 00 n�" (b) How long i�

ing') (c) How far will Anna u.y

Anna's arm (i.e., from her hand to her center

"hip i� from Bob at this time? l8. The diagram shows Bob's view of the passin& of two

ofmiw.)" 24. A pole.,'aulter hold.. a

16 ft pole, A bam has doors al

Identical space!i.hips. AMa's and his own. where 'Y¥ .. 2. The length of either !!opaceship in itl rest frame is L.o­

both ends, 1 0 ft apart. The pole.vaulter on the oulside of the bam begins running toward one of the open bam

What are the readings on Anna', two unt.belcd clocb1

door.., holding the pole level in the di�clion he's run­

AIII,," lpKclbip

ning. When pa..�ing through the bam, the pole filS

(barely) enurely within the bam all at once. (a) How

,

fast is the pole·,aulter running? (b) According to whom-the pole.vaulter or an ob..erver stationary in the bam--doe .. the pole fit in all at once? (e) According to

the other per;on, whKh occu� fi�t the front end of the

25.

(d) what IS the time mter\-'al between these two e\'enlS? long. The diagram shows Bob'!, view as Anna's ship

another, both dock!. there read O. At the inMant shown, Bob Jr., on bo:u-d Bob's ship, i� aligned with the very

• •

8ob',�p

100 m

pa<;se� at O.Se. Ju�t as the backs of the ships pas!> one

� § j

0 I I 0

pole leaving the bam or the back end entering, and Anna and Bob are In identical !'pace!ohips, each

the front of her spKC­

I

29.

Refer to Figure 2.18. (a) How long

1>

i!!o a spaceship'!

(b) At what l>peed do the �h ips move relative to ooe

another? (c) Show thai Anna·s times an: an accord with

front of Anna's ..hip. He peers through a window in

the Lorenll. transfonnalion equations. (d)

Anna'� �hip and look� at the clock. (a) In relation to his

of diagram� showing Anna's complementU)' view

own ship, where is Bob Jr,'! (b) What does the clock he

of the pa�sing of the liohips. Include times in boIh

..ee.. read"

SkddI a set.

frames,

30, You are in

0.8£' -

II

bu!> traveling on a stn.i�t road ar. 20 mIs.

A", you pa\s a ga.'i station, your clock aDd . c\oct in 1bc �Iation read precisely

O. You pass another gas statioa

900 m farther dov.'n the road. (in the frame of maalCe

of the: gas stations., all gas station clocks aK �

nized.) (al As yoo pan the: second SUIic& do you fiDel

ib clock to be Wad of, 01: behind your OWD dock. ""

(b) by how mucb':'

31. A spaceship travels at 0.&. As d'lis sptcahip CO'YCR 1be

u.

Bob I� watr.:hing Anna Oy by

to

her new high-..peed

plane. which Anna know� to be 60 m in length As a

�'··.n..01 ". , :.,� - ----. , -

.woo Un from coast 10 coast. by how much will die

ume interVOll R:giSlCTCd on an oaboud dock. differ rn. the time interval �asured on the pound'

..

Q ,'1 a ,.... ......,.

A ... _ _ . ......

traia. nwIina • • dcceot clip: 5 ... ."..,.

and the angle il makes with lhe x-lUis is

_ 00 ... .,- .. _ ..... _

.... ..... . .....CIpCII1iuj lip.. wbidI 11\\'111 5 Bive 7

h oar n:portL AI � noon on the train's dDcb. .... coMucllDf .. the froM of the train passes one __ ., .. . sip JeIIdina "All Clear," and another .....,.. _ dae bKt ,...ea Ibe OCher station and sees a ... ...... "'Severe EIec1ricaJ Storms Reponed! Slow

10 0. ."'- (a) Huw ..., is Ibe train? (b) Should i l slow

...., (c) 5uppoIe dial both reponioB signs display the

JIftdoelY. upto... ....,. _. 8y 1IIIIdI would the two obIctved time readinBs differ. II_ all" .. A '-'- apmimeat dete..:ted 527 muon.s per hour at _ ...,.

....

.... ..,01 MI. Wubiqtoa. New Hantpshire. elevation .910 IlL At _ level, die same equipment detected 39S __ per bour. A discriminator selected for IIIUOIII whole speed was between O.995Oc and 0.9954£' an. .... die mean lifetime T of 1 muon in a frame in wbidI if: i s II rIIIt is 2.2 J&.S and that in lhis frame the WI •" of IDUOftS decays exponentiaUy with time 8CCOIdiDl Io N "" N,-th, show that the mules ...... ill the experimenl are sensible. J4. fa drc hIDe in whieb they art at rest, the number of amoas .. time 1 i s given by

wIleR No is me DUmber at , = 0 and T is rhe mean life­ time 2.2 J.U. (a) If muons are produced at a �ight of 4.0 kID. bcwfina toward the ground at O.93c. what frae­ tioa will survive 10 reach the JroUnd ? (bJ What fraction would reach lhc pound if classical mechanics were

valid?

J5. A supersonic:- plane tnlvels al 420 mI�. As Ihis plane pu.ses two marten a distance of 4,2 km apart on the JIUUDd. how will the time intervaJ registered on a \'CT)' JIftcise clod: onboard me plane differ from 10 s? 36. How last muse , ptane SO m Jons tnvel lo be found by ot.enas on the pound to be 0.10 nm shorter than 50 m·) Aceordint 10 Bob. on Earth. il is 20 Iy to Planet Y. J7. AMI: has just p.ued Earth, moving at a constanr speed .. iII • ..,.:ahip. When Anna passes Plane-I y, she )'an oIda" tt.n when she pu.sed Eanh, Calculak v. • A .... fixed 10. lied .. rest in frame S. is of length 4, .. ....,. .. ..pc 01 lib with me x·axis. Later, me sled ___ .....,.. hme S .III • constant speed v paralJel lo ... ..-aiI. SIIow Ihai .c.cordin, 10 an observer who _ _ rat m frame S, me length of me plant i.s now

is 20

• 39. Bob. in frame S. is ob�r'\"ing the mo,·ing plank of E�t cise 38. He quickly fabricales a wall, fhed in hi... frarne Ihal has a hole of length L and thai i� !>Iamed al angle such Ihal lhe plank will completely fill the hole a ... it passes through. Thi.. ou;ur\ at the in..tan. . According 10 Anna. moving with the plank. the plank is of course n(ll of length L. but (If length Ln. Morem-cr, because Bob's wall mmc.. rclati\'c to her. Anna sees a hole thai i't Ie.... than I. in length; a plank longer than is heuded loward a hole ,honer than L. Can the plank. pa..s through the hoh! according to Anna'! If �O. at What

_

9:

t =0 o

L

rime/sf' Explain,

40. An experimenter dell!nmnes rh.d a panide created at one end of the laborollory app
a�

II

reacht:d the orher

md. la) Accordl"S to Ihe experimenter. hO\\. far did thc particle mo\e? (bJ In ih uwn fr.une of refcrenl.:e, how

long did Ihe piltllcle !>unI\C" Ie) According 10 the parti·

de, what was the length olthf! laboratory apparalu�� 41. A munn ha.'t a mean lifetime 012.2 fJ." in re..t frame.

II';

Suppu..e munn, arc ((a\'dmg at 0.92c relali\'e 10 Eanh.

Whal I� the mean cli ..ant:e r a muon will tr,Hlcl a� mea·

,"ured by an ob..erver on Earth·)

42. A pion IS �n ckmcntary panidc that. on a\'cragc,

If!grall!s 2,f)

J(

10

• s after l"feataon

di..cn.

in a trame al resl

relalive: to the pion. An cxperilTK'nter lind.. thai pi(ln�

l'rcaled cn the lahoratory tra\cl 1 3 m on a\'el'3ge before' dislOlcgraling H\1W I�.\I 4rC the PionS Iravelins throu�h

the lab"

.. 4_1, Anna and Bub ha\'c Identical sp.acestllps 60 m Ions. T1u: diagram shows Bob's obs.ety.llions of AM�'S ship.

\t 2,

,,\Ibll.:b p.a'>se-s " " !lpecd of c Clocks al the back (If al the center bo1h ship" read U Just a .. the}. p.us. Bob of his mil' and a' , 0 on hi'l wrisfWall.h peen al .� sec· and d<X'1 on Anna'ii ship.

IS

(., Whal docs thiS cI<X'k rcad'f (b) Larcr. !he b3Cll; oJ' Anrul's ship pas�\ OG':>. Al wb:J.t lime docl lhis occur ac,,'ording 10 Bobl

II a... . ., 2

Sper;iaI Relativity

die IMIIIIIIick in the positive direction al speed ". (-.) WJMD you reach lbe cenccr dock, it reads O. What do

_ odIer two re8d aI this insrant in your frame. (b) You

-.wy revene direction. 1be dock at the center is IdII IMdiq 0 mel so is youn. What do me others read? (c) How docs Ibis relate to the rwin paradox?

_ 2.5 51. $bow rbat for a source moving loward an observer, equadoa (2.l7) beco�

absolutely precise waveleng!h. they would be very diffi_ cult to discern. Fortunately, twO factors broaden them: the uncertainty principle (discussed in Chapter 4) and Doppler broadening. Atoms in a gas are in motion, so some light will arrive that was emitted by atoms mov_ ing toward the observer and some from atoms moving away. Thus. !he light reaching the observer will Cover a range ofwnvelengths. (a) Making the assumption that atoms move no foster than their rms speed-given by where is !he BollZJIlann constant_ I'nn, = obtain a formula for the range of wavelengths in terms of the wavelength A of !he spectral line, the atomic mass and the temperalUre T. (Note: vrms « c.) (b) Evaluate this range for the 656 nm hydrogen spec­ tral line, assuming a temperature of 5 X 1()4 K. 58. To catch speeders, a police radar gun detects the beat frequency between the signal it emits and that which reflects off a moving vehicle. What would be the beat frequency for an emitted signal of 900 MHz reflected from a car moving at 3D mls?

V2kaT/m,

ka

m,

.51. By what factor would a star's characteristic wave­

lengths of light be shifted if it were moving away from Earth aI 0.9c?

53. AI rest, a light source emits 532 nm light. (a) As it

moves along the line connecling il and Earth. observers on Earth see 412 nm What is the source's velocity (magnitude and direction)? (b) Were it to move in the opposite direction at the same speed. what wavelength would be seen? (c) Were il ia circle Earth at the same speed. what wavelength would be seen? S4. 1be light from galaxy NGC 22 1 consists of a recogniz­ able spectrum of wavelengths. However, all are shifted toward me shaner-wavelength end of the spectrum. In particular, the calcium Mline" ordinarily observed at 396.85 run is observed al 396.58 nm. Is this galaxy moving toward or away from Earth') AI what speed? 55. A space probe has a powerful light beacon that emits 500 run light in its own resl frame. Relative to Earth, the space probe is moving at 0.8e. An observer on Earth is viewing the light 3ITh·ing from the distant beacon and dccects a wavelength of 500 nm. Is this possible? .

Explain. 56.

With reckless disregard for safety and the law, you set your high·performance rocket cycle on course to streak through an intersection at top speed. Approaching the n i tersection. you obsero.·f' green (540 nm) light from the traffic signal. After passing through, you look back to obs�rve red (6.50 nmJ light. Actually. the traffic signal never changed color-it didn't have lime! What is the rap speed of your rocket cycle, and what wru, the color of the ttatJic signal (according to an appalled bystander)? 57. For reasons having to do with quantum mechanics. a given kind of atom can emit only cenain wavelengths of light. These spectral line ... serve as a " fingerprint." For i nstance, hydrogen's only visible spectral J ines are 6.56, 486, 434. and 4 1 0 nm. If spectra/ lines were of

Section 2.6 59. Bob is on Earth. Anna is on a spacecraft moving away from Earth at D.6e. At some point in Anna's outward travel. Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob. the projectile moves at D.8e. (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, " Greetings from Earth:' out to Anna's ship. How fast does the light signal move relative to Anna?

60. According [Q Anna, on Earth, Bob is on a spaceship

moving at O.Se toward Earth. and Carl. a little farther OU!. is on a spaceship moving at 0.ge toward Earth. (a) According to Bob. how fast and in what direction is Carl moving relative to himself (Bob)? (b) According to Bob, how fast is Carl moving relative to Earth? 61. Prove that if v and /I ' are less than it is impossible for

c,

a speed /I greater than c [0 result from equation (2-l9b).

{Hint: The product (c - u')(c - v) is positive.]

62. In a particle collider experiment, particle I is moving to

the right at O.9ge and particle 2 to the left at O.99c. both

relative 10 the laboratory. What is the relative velocity of the two panicles according to (an observer moving with) particle 2? 63. A light beam moves in the X)·-plane and has an

x-component of velocity of uf (a) In terms of u, and c, what is its y-component7 (b) Using equations (2-20a) and (2-20b). calculate its velocity compo[]ents in a frame moving in the x direction at speed v = ".t' and comment on your result. 64. A l igh t beam moves at an angle (J with the x-axis as seen from frame S. Usi ng the relativistic velocity

III

It,

transformation. find the components of its velocity when viewed from framc S'. From these. verify explic­ itly that it:> speed is c. 65. You fire a light ...ignal at 60" nonh of wC"..t. la) Find the velocity component.. of thi., .,ignal according to an observer moving eastward relative to you at half the speed of light. From them. determine the magnitude and direction of the light signal's velocity according 10 this other observer. (b) Find the componenh according to a different observer. moving westward relative to you at half the speed of light. 66. At' = O. a bright beacon at the origin flashes. sending light uniformly in all directions. Anna is moving at speed I' in the +x direction relative to the beacon and passes through the origin at I = O. (a) Show that according to Anna. the only light with a positive x'-component is that which in the beacon's reference frame is within an angle 9 = cos- I(v/e) of the + x-axis. (b) What are the limits of 8 as )I approaches 0 and as it approaches c:'! (c) The phenomenon is called the head­ light effect. Why'! 67. Using equations (2-20), show that

Section 2.7

By applying the relativistic velocity transfonnation to the left side of equation (2-23) and using the algebraic identity derived in Exercise 67, verify equation (2-23). 69. What is the ratio of the relativistically correct expres­ sion for momentum to the classical expression? Under what condition does the deviation become significant? 70. What are the momentum. energy, and kinetic energy of a proton moving at a.8e? 71. What would be the internal energy, kinetic energy, and total energy of a I kg block moving at 0.8e? 72. By how much (in picograms) does the mass of I mol of ice at ooe differ from lhat of 1 mol of water at O°C? 73, A spring has a force con<:tanl of 18 N/m. If it is com­ pressed 50 cm from its equilibrium length, how much mass will it have gained? A 74. typical household uses 500 kWh of energy in I month. How much mass is convened to produce this energy? 75. Detemline the momentum of an electron moving (a) at speed 2.4 X 104 mls (about three times escape velocity) and (b) at speed 2.4 X 1 08 mls. (c) In each case. by how much is the classical fonnula in error? 76. In the collision shown, energy is conserved. because both objects have the same speed and mass after as

68.

before the coUisioo.. SiDce me colldioa ....., ...._ the velocities. me final (loW) momoatwa " OW." the initial. Thus. momentum CD be � CDIy if i.t is zero. (.) U!!.ing the relati....i!!.ticaUy correct o.pressioD. for mllmentum. !!.how that the tolal momentum is len.�-that momentum is conserved. (Muses are in lU'bilnlry unit!!..) (b) Uloing the relati....istic velocity transformation. fiDel the four veloc.itie!!. in a frame moving to Ihc riJbl at 0.6('. (0) Verify that momentum i!!. conserved in the new frame. 0.80

After

0.60= '*'


�o."

77. Is it possible for the momentum of an object to be me'1 If not. why not? If so, under what condition? 78. Show that the relativistic e�pression for kinetic energy (yw - I )mc2 is equivalent to the classical �m,,2 when u « c.

79. At Earth's location, the intensity of sunlight is 1.5 kW/m2. Uno energy escaped Eanh, by how much would Earth's mass increase in I day? 80. The weight of the Empire State Buildi.ng is 365 kilotons. Show that the complete conversion of I kg of mass would provide sufficient energy to put lhi.s rather large object in a low Earth orbit (orbit radJus :a!: Earth's radius). 81. Radiant energy from the Sun. appro�irnately 1.5 X lOll m away, arrives at Earth with an intensity of 1.5 kW/m1. At what rale is mass being converted in the Sun to produce this radiant energy? 82. (a) A high-explosive material employing chemical reac­ tions has an explosive yield of 106 Jlkg. measured in joules of energy released per kilogram of material. By what fraction does its mass change when it explodes? (b) What is the explosive yield of a material that pr0duces energy via nuclear- reactions in which its mass decreases by I part in 1O.000? 83. How fast must an object be moving for its kinetic energy to equal its internal energy? 84. How much work must be done to accelerate an electron (a) from O.3c to O.6c and (b) from O.6c to O.9c? 85. An electron accelerated from rest through a potential difference V acquires a speed of 0.9998c. Find the value of v.

61

a_ptet 2

Special Relativity

each experiment. In which is more of the inihal kinetic energy converted to mass?

86. Wbal is Ihe momentum of a proton accelerated through I gisavoll (OV)? 17, A proton is 8('ccleraled from rest through a potential

�O,9 c _.-

ditfcren� of 500 MY. (II) What is ils speed? (b) CJas�i·

cal mechanics indicates that quadrupling the potential difference would double the speed. Were a classical

(a) Experiment A

analysis valid. what speed would result from a 2000 MV potential difference? (L') Whal speed actually resul l�') 88. A particle of rna..s mo moves through the lab at 0.6<:.

�II

Suddenly il explodes inlO tW() fragments. Fragment J .

mass O.66mu. moves al 0.8(' in the same direction the

original partide had been moving. Detemtine the veloc­

(b) Experiment B

ity (magnitude and direction) and mass offragmenl 2.

89. The boron-14 nucleus (mass: 14.02266 u) "beta decays," sponlaneously becoming an electron

0.00055 u) and a carbon- 14 nucleus (mas.�;

(mass:

13.99995 u).

What will be the speeds and kinetic energies of the carbon-I 4 nucleus and the electron?

(Nore: A neutrino

is also produced. We consider the case n i which its

94. A kaon (denoted KO) is an unstable particle of ma�s 8.87 X IO-2H kg. One �f the means. by which il decays is by spontaneous creatJOn of two pIOns, a 7T+ and a 7T- . The decay process may be represented as

momentum and energy are negligible. AJ�o. because the carbon- l oS nucleus is much more massive than the elec­ tron. il recoils ''slowly'';

y,, ;;;: I.)

90. A 3.000 u object moving 10 the right through a labora­ (01)' at OJk collides with a 4.000 u object moving to the left through the labor.nol)' at 0.6c Afterward, there are tv.·o objeds, one of which is a 6.000 u mass at rest. (a) What are (he mass Hnd speed of the other object?

(b) Determine the change n i kinetic energy in this coJlision. 91. A 1 0 kg object i� moving 10 the right at O.OC. It explodes into two pieces. one of mass m 1 moving left at 0.6< and one of ma�s m! moving right at O.8e. (a) Find

ml and ITI!. (b) Find the change in kinelic energy in this explosion.

92. Particle I. of mass 1111, moving at O.Be relative to the lab, collides head-on with particle

Both the 7T-<- and

71"'

have mass 2.49

pose rnat a bon moving in the 7T-

at 0.8e. (a)

2 10 · 8 kg. Sup_

+x direction decays by

7T+ moving off at speed O.9c and What was the speed of the kaon

{his proce�s. with the the

X

before decay? (b) In what directions do the pions move after {he decay?

95. In the frame of reference shown, a stationary particle of mass "'0 explodes into two identical particles of mass m

moving in opposite directions at O.6e. Momentum is

obviously conserved in this frame. Verify explicitly that it is conserved in a frame of reference moving to the right at 0.6e.

2. of mass tn]. mov�

ing at 0.6c relative 10 the lab. Afterward. there is a sin­

o�

gle sI.:11ion31)' object. Find. in terms of "', . (a) "'2: (b) the mass of the final .�tatjonary object; and (c) the

-�

change in kinetic energy in thi� col/ision.

93. Consider the col/i ..iom of IwO identical particles. each of mas� mo' In experiment A. a particle mo....ing at 51rikes a stationary particle.

O.ge

(a) Whal is the total kinetic energy before the collision? (bl In experiment B. both particles are moving at a speed u (relative [0 the lab), direcily [Oward one

another. If me total kjnetic energy before me colli­ sion in experiment B is the same as that in experi� men! A. what is u'!

(c) In both experiments, the particles stick together. Find the rna..'! of the resulling .�ingle particle in

�O.6c �

96. Write out the total energy of a collection of interact­ ing massive objects, break each term into internal and kinetic part.�, and show that equation

97. Show that & = p2C1

(2-27) follows.

4 m2e follows from expres­

+ ions (2-22) and (2-24) for momentum and energy

s

in terms of m and u.

Section 2.8 98. Equation

(2-30) is an approximation correct only if

the gravitational tIme-dilation effect exercise, it

i.. also assumed

is small

In this

to be small. but we still

k wIU

allow for . nonunifonn JRlvilaliODlll field. We start willi (2-29), hued on the Doppler clrcel in Ihc KC'elentina frame. Com.ider I'NO clevations, lhc lower at 'l and lbe upper a, rI + dr. Equalion (2·29) becomes

Similarly, if we consider elevalions r, +

:!.,Jr, we:: have

dr and r1

nepIi--. it ......

IiIo is. podoo of _

....... -.J. B

....... .... U.. ...... .. ... u - -(GMW,� _ M " _ _ of'" F \ body aad .. II _ -_ _ of _ _ ,..., Au_ ..... tIIIa _.. _ .. .... .., . - AI S_ ..... .... crilic:aJ .-. .... _ .... _ ""' _ .... __ pall of. ",,", " . 11 wilhin . ,.,." of 2 ofilia Sd:___ ..... ..

+

ltr, t}(r,

2Jr) + tlr)

Ihcchapler. (1bi. tindof ....... is 7• • general relativity i. lC)IDMjmea uefaI ... ..,. "1Il-

We continue the:: pnxe....... incrementing r by dr, until we

rea�h r2.

Now ima,!!inc multiplying the ldt "ide, of all the �ua­ tion.. and �ettlng the produ.:t equal 10 the product of all the ri�ht �idcs. (a) Argue that Ihe left ..ide of the prod­ u.:t I'" "'lmpl)"I(r�)/l(rT I. (hI As.�uming thai the term g dr".l m e:ll:h mdmdual equation is very �mall. so thai produ.:t, (11 �u.:h term� l'an be ignored, "rEuc that the right �ide of the:: product i\

I

-

I .J.I g(r)Jr

T. be reliable. predicIionI _ be .... ".. ...... nin, to end on the loaical. but uafuilw C

102.

Suppose particles brcPD �.. in ODe d' • .., from the origin at , - 0 with lite � WIIaddII 0. � I. � 2. �3 mil. and 10 011. (a) Aller I .. bow will .. velocities of the puticlcs depend . eM...... .... .. ori,in? (b) Now C'OII'icIcr ... obIa .. . _ tIl_ moving particles 1IOI &l 1he ori.... How will ... ,...,.

velocities of the 0Cher .,.uc1a: depend . ..... ... the observer?

Section 2_9

IOJ. Here we verily the conditioas uDder wbidl lj ill

equation (2·33) will be neptiw_ (a) ... ..... .; • .. ->< < I +

(c) Deduce gIrl from Newton'5 universal law of gravita­ tion, then ilr�ue that cquation (2-.1 1 1 follows from the

2 wi le'

(b) By construction. v cannot ex.ceed ... for il il did. "

re�ull.Ju�t a\ (2-.l0J duc... fwm (2-29).

.•

<0"

cquivaJenl lo the followina;

,

99. Accordmg In an ob...er"e::r al Earth', cquator. by how much would hi... clock and one on a \aleilitc in geosyn­ chronou... orhil differ in onc day? (Geosynchronous orbil me::an\ an tlrl"!il period of (me day-aJways in the ..arne pla.:c in the ..ky ) 100. If il i.. Iundamenlal 10 natufe thai a given mass has a criul·al radiu.. at "hil·h �omelhing extraordinary hap­ pens (i .e a hlad. hole forms), \\ie might guess thai this radius �hlluld dq�nd only on the mass and fundamencaJ conSWnI\ of nalore. A��uming Ihat r�Tltw;aI depends onl)' on ,\.1, G. anJ c. ...ho" that dlmen\ional analysis gives the equatlUn for Ihe Schwar7.!k:hiIJ radius to within a mullipJu:.iJlivc CQnslalit. 101. A projectile i� a di�l.ance r from Ihe l-enteT of a heav­ enly hody ilnd 1\ heBding dirct'lly away. Classically. if the ,urn 01 its kiM"lic and potential eM"rgics is positive,

.,. 0.....

fundamental equation. of poenI NIIdvity.)

information could noC cakb up willi Amy AI _ 2-

Use this 10 UJUC that if ... < C. dIeD Ii _ ... ,c.... for whatever value v i. alJond 10 bnc. (c) UIiII " fact tI:at ex - I)' " 0. "- ..... die ... _ of .. exptCSs.ioa in put (a) De¥a" esceech 1. 1lia "that when "" > C. v aecd DUI cu:eed e ID PIah' • DC,aUVC Ii.

11M. From Ibc Loraa ....... ..__ .. .. W

time idavals bec._ two � 41 _ U• • _

rr.ncs1Rr10FfJ1'_ .... ... ... _ _ _ ... _ in _ _ IiJr .... . _ _ _ . .. ocber

•

.a..... _ . - cbal cbea'

_ 2. 1 0

105.

(.)

DetamiDe

a- � " �

:• -

Ilte�::��==�i

position aDd Ii.- ill '" r

Et I1

-

(a) U�ing the rdativistically correct expre�sion for momentum. show that

F

F

...

lb) How much time goes by for obscnua on Barth u they " see" Anna age 20 YeaR? lc) Using the rt'sult of Exercise t 19. show thai. w_ Anna has aged a lime " . she is a dis1.lLnCC from Eanh (according to Earth observers) of

, du

y"m--­

dl

.., �

(b) Under what condition does the classical equation = mahold? (c) Assuming a constant force and that the speed is zero at t = O. separate I and Il. then integrate to show that 11 =

F 2 (Ft/mc)

gl' _t "" - cosh-

g

til

118. A rocket maintains a conMant thrust

F,

it an acceleration of g (i.e 9.8 m1s2). (a) If classical physics were valid, how long would it take for the rocket's speed to reach 0.99c? (b) Using the result of Exercise 1 17(c). how long will it really take to reach O.99c� 119. Exercise 1 1 7 gives the speed u of an object accelerated under a constant force. Show that the distance it travels is given by .•

giving

me'[) (FI)' J

x�-

1 +

F

-

me

I

120. In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than Ig. Here we see how far a person could go under a constant acceleration of Ig, producing a comfortable artificial gravity. (a) Though traveler Anna accelerates, Bob, being on ncar-inertial Earth, is a reliable observer and will see less time go by on Anna's clock (dr') than on his own (dt). Thus, dr' = ( Ify)dt, where u is Anna's instamaneous speed relative to Bob. Using the result of Exercise 1 17(c), with g replacing Flm. substitute for u. then integrate to show that

t

c ,

gt' e

= -smh-

g

,.

-

I

)

(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging anOlher 20. how far away from Eanh win she end up. and how much time will have passed on Earth?

r=+�",='" - I VI

(d) Plot II versus t. What happens to the velocity of an object when a constant force is applied for an indefinite length of time?

"

Computational Exercise 121.

In an inertial frame without gravity, the fastest-running clock is a stationary one. In a freely falUng fnune. it should be a clock accelerating with the frame. All observers agree on the proper time passing on a clock., so observers on Eanh should find that the path actually followed by an object/clock that is subject only to grav­ ity is one that maximizes the proper time from starting point to ending point. Here we verify that this i.s true. Assuming v « c. regular (speed-dependent) time dila­ tion tells us that the time interval dT we see on a moving clock is related to our interval dt on Earth's surface by

dT =

dt�r,

-

",\'12 c2)

::;

�r,

dr

-

1 "'\ 22)

Equation (2-30) tells us that the time dT passing on a clock ticking higher in the gravitational field is larger than the time dt on the surface according to

Taking both effects into account at once, we have

or

72

ChIIpNr 2

Special Relativity

Addinglintegracing these deviacions over a finite time

interval yields

4T - 41 C!

/gH /-1'2 � dl + r-

lions whose sum gives .loT -

, dl

.1.1 as a function of t.

varied. that will plot the two functions and their sum

2c-

versus time together on one set of axes, and H(I) and 1'(1)

This is how much the proper lime on the moving object

above the surface al zero speed. it!> clock would run faster, oUlpacing the ground clock. Were il to move at

H(r) -

ground level. its time would faJl behind a stationary

= k(t

1').

No matter what k and b might be, the object would be al ground level at , =

and ils derivative I'U) into the given integrals, then

(b) Write a program. in which k and b can be easily

eltceeds lhal on Earth's surface. Were it merely 10 float

clock. Consider motion described by

Hrr)

carry oul the indefinite integration to obtain two func­

0 and I = I . so we can vary these

-

on anomer, and [hat will also print oul the totaJ vaJue of ..11 when I = I . Use 9.8 mls2 for g. (c) Plot all

_iT

functions trom I = cases: (k, b) =

<0.2. 20).

(0.

I for lhe following special I). (7. 3), (-7. 3). (20, 4). ( I . 20). and

0 to I

=

(d) Discuss the "height effect" and "speed

effect" for each of the special cases of part (c). (e) From your understanding of classical mechanics. determine what k and b must be for a real particle leaving me

parameters and see what effect il has on the moving

ground aI ! =

clock's proper lime, while still gelling 10 the same

for Ihis case. then vary k and b independemly by sma!!

ending point at the same time. (a) Insert the assumed

0 and returning at 1 = [ . Run your program

amounts. Discu% your findings.

3 Waves a n d P a rti cles

E l e ctrom agnetic R a d i ati o n Behavi n g a s P a rticles

I:

Chapter Outline

3.1

3.2 3.3

3.4 3.5

3.6

Blackbody Rad iation : A New Fundamental Constant The Photoelectric Effect The Production of X-Rays T he Compton Effect Pair Production

Is It a Wave or a Panicle?

W

e now begin OUf investigation of quantum mechanics. 111 ...orne sense, quantum mechanics is the study of "mall things-so

small that it is essentially impossible to observe them without atTecting their very behavior. For example , the simplest way of observing an object is to look at il. But to do that, light must be bounced off it, and light carries energy. some of which will unavoidably he transferred to the object. Ordinarily, the effec t IS

inconsequential. but if the object is very small, such a.<. a single electron, it might be significant . Thus, we shouldn', be too surprised that the behavior might vary, depending on how the observation is made. A cornerstone of quantum mechanics is

wave�particle duality:

Things

may behave as waves or as discrete particles depending on the situ ation . ,

The

situation

"

"

might be imposed by a deliberate ex.periment or governed

simply by the dimensions of the region where the thing is confilleu. Two of the most important things we study are massive objects and electrom agnetic radiation. In classical situations, our observations reveal electromagnetic radiation behaving as waves and massive objects as discrete partic les We now .

look at the nonclassical side of the coin. In Chapter 4, we will lllve�tigate massive objects behaving as waves. In this chapter we study the complemen­ ,

tary lopic--electromagnetic radiation behaving as a collection of di'crete particles. We begin with a brief look at the discovery that sparked the quantum revolution.

73

74 a.pe.r 3 W� and Panicl� I: E1ectmmagnetic Radiation Behning a.\ Pankles ,....,. 1.1 Radiltion exits a cavity Ihraugb a bole, whicb behaves as a

3 . 1 Blackbody Radiation: A New Fundamental Constant

NKkbocly.

IntOftlml1 naJ\311t.n i� IINJtbed comph:tely ,-----, in multIple InlC'mal

. re'nC"\.·tiol\�. "

OlJr�uinG 11I,J1�lil>n

i. dl.iC' .>nly (" th("rmal motion ..1" .:hilfli�. In wall,

T

Figure 3.2 bpcriment �how� fhal (IS frequen.:y increases.

the blackbody

spectral energy dt'nsjly reache, a m31ti­ mum. then falls off C/as,ical wave theory predicts a divergence.

'" :s I �I !

CI:u..\K":tI wave Theory

bprnmenT...J data. and Planck's lheory

The quantum age dawned wilh the work of Max Planck in the year 1 900. Planck wa" trying 10 find a theory thill would explain blackbody radiation. All malerials emil electromagnetic radialion. because mey contain charged panicles Ihal jiggle around. and an accelerating charge radiates electromag_ neric energy. The amounr of energy radiared depends on [he average energy of rhe motion. which. in tum. depends on the lemperature. For example, coals radiale invisible infrared energy even when cold, but when heated. they emit more radiation. much of it in the red end of the .�pectrum. They visibly glow "red hot" Most materials, however, also reflect electromagnetic energy. A blackbody is defined to be any object from which eleclromagnetic radiation emanales solely due to the thermal motion of its charges. Any radiation that strikes it must be absorbed rather than reflected, hence the name. (The term must not be taken too literally. The Sun's surface, from which reflection is insignificant. is a blackbody.) While coal is a good approximation, fabricating a true blackbody might .�eem problematic. Imagine. however. an objeci with an interior cavity and a small hole connecling it to the exterior, depicled in Figure 3. 1 . Any radiation enlering the hole would refleci from the cavilY's inner surface many times, losing energy to the object al each refleclion. Essentially none would reflect back directly through the hole. On the olher hand. all areas of the inner surface contain charges in themlal motion. constantly absorbing electromagnetic energy and reradiating it as they jiggle around. They will furthennore be in equilibrium with the electromagnetic energy in the cavily-the charges and the radiation will have the same temperature, T The portion of the radiation leaking out of the small hole will be characteristic of this temperature, so the hole behaves as a blackbody of temperature T. Experiment demonstrate!. that the energy emitted by a blackbody, or equivalently a cavity. is small at low frequency, reaches a maximum, then falls again toward zero thereafter. This is illustrated by the experimental curve in Figure 3.2, which plots electromagnetic energy dU per frequency range df. known as spectral energy density. Classical theory, on the other hand. differed. If the electromagnetic radiation in a cavity behaves strictly as sinusoidally oscillating waves of arbitrary amplitude, the average energy ofa wave of any given frequency ...hould be kaT. (Appendix C discusses Ihis point and the other delails of classical wave theory and Planck's work thai we merely outline here.) MUltiplying this by a factor that accounts for the number of different waves per frequency range dfin volume V, the classical prediclion for spectral energy densilY i� dU - = kBT x -I­ 81TV ">

df

e'

Spectral energy density via cla-'i'�ical wave theory

Something i� certainly wrong here, for as Figure 3.2 shows, this parabolic function diverges as f increa...e ... without bound. If true, all malerials would radiate infinite power.

.\,21bc PhocoeJeI:IriC '"

Planck found that he could match the experimental dala with a I•.'Urious assumption: The energy at frequency f is somehow restricted to E = "hi, where " is an integer and h is a constant. The specitic error in cla.'.;sical wave thi..,(lry i!io in the average energy of a given wave. which b obtained b)' !Ote-grating O\'cr an a<;sumec.l continuum of pos�ible energy values. Under Plan(.'k\ a.o.;sumptilln. these ,;alues arc discrete, so the imegml be{'omes a Slim, with a notabl), different result. Repla�:ing the kBTof wave theory with Planck's result the spectraJ energy density r.u:omcs dU

81TV

lif

I

,

xf­ � ("

(3- 1 1

which filS the experimental curve perfectly. The value Planck quoted for h. the now famous Planck's constant, was 6.63 X 10-14 J . s. Naturally, it was the value that matched his fonnula to the experimental curve, While it is tempting to question why he couldn't derive it. the fact that it was even possible to match the curve was !ioignificant. Although varying II would change the energy density's magnitude at all frequencies. it would not alter its shape. Had there been no merit in Planck's assumption. it would have been the most binrre coincidence for that shape to match the experimental curve. With hindsight. we now realize that Planck's constant cannot be derived, for it is one of nature's fundamental constants (c.g., the uni4 versal gravitational constant G), all of which are a matter of experimental observation. For the discovery, Planck was awarded the 1 9 1 8 Nobel Prize. Planck's spectral energy density is the crucial link between temperature and electromagnetic radiation. and we will return to it in Chapter 9. Interest� ingly, although the assumption E = IIh!, on which Planck based his fom1Ula, might suggest electromagnetic radiation behaving as an integral number of particles of energy Iif. Planck hesitated at the new frontier--others carried the revolution forward. LeI us take a look at the next major step.

H·,·-" k·

. I',',I' ,,'

cnc''3) ,Jens,ty

h A63 to

PI,lf"'k" v"",t.)j,t

3 . 2 The Photoelectric Effect Classically. electromagnetic radiation is a wave. The energy it carries is dif· fuse, distributed continuously along a broad wave front, and its intensity­ energy per unit time per unit area-is proportional to E6. where Eo is the amplitude of the electric field oscillations. In the I 880s. Heinrich Hertz demonstrated that a light beam directed at the surface of a metal could liberate electrons. This is called the photoelectric effect-light producing a flow of electricity-and is depicted in Figure 3.3. It was also known that a certain amount of energy is required simply to free an electron. The electron is bound to the metal; pulling it loose takes energy: and any surplus becomes the freed electron's kinetic energy. The minimum energy required to free an electron, the work function 4>, is a characteristic of the particular metal. Table 3.1 lists some values (subject to variation. depending on impurities and other factors). If light were lOtrictly a wave, this effect should have several specific traits. First, if light of one wavelength is able to eject electrons, then light of any wavelength should be able to do it. Independent of the wavelength. the mte at which energy arrives (the intensity)--and therefore the rate at which electrons

Figurw 3.3

The photoelectric effect:

Light liberating an electron from a

metal surface.

•S(5

71

iIIiI.!'t�

.. . ...... t n " . ..

... .

-

2.3

7

...... P+niac. PIrtideI

Eo- Seooad.

if ... _.,. is low, then """" Ibough ....trons mighl sIiIl be .jecIed, a _

r

...... time lq should arise. Because • wave is diffuse. considerable

tUne

miPt be needed far enough energy to accumulate in me electron's vicinity.

1.9

2.2

•

_ .clOd couJd be ...... orbilnriJy buJe simply by _ing

• • oVl

..

.

(See &erciIe 16.) F'maUy. at any liven frequency. if the intensity is increueca.

die � electroas should be more energetic. A stronger electric field

sbouId produce a larJer acceleralion.

Imagine the experimenter's surprise when weak light of 500 om wave.. length ejects electrons from sodium. with no lime lag. while light of 600

3.7

wavelength cannot, even al

Drn

mon)' lim�s me intensity. Moreover. the energy of

the electrons liberated by the 500 om light is completely independent of the

4.3

intensity. Classically. this cannot be explained! In 1905. Alben Einstein proposed the following explanation: The light is

4.4

behaving as a collection of particles. called photons. each with energy given by

4.>

E = hI

Energy of it photon

(3-2)

h

where is Planck's conSIant. A given electron is ejected by a single photon. with the photon ttansferring all its energy to the electron and then disappearing_ multiple photons very rarely gang up on one electron. If the light's frequency is too low, such that the photon energy hJis le!'>s than the work function c/J. then there

is simply insufficient energy in any given photon to free an electron.

So none DrF

frrt>d. no moNer hoM' high 1M inlemily: no mailer how abundant the photons. (The phalon energy becomes internal energy or reflected light.) However, If the frequency is high enough. such that hJ > q,. then electron.., can be ejected. The

kinetic energy given to the electron would then be the difference between the

photon's energy and the energy q, required to free the electron from the metal.

(3-3) The subscript " max" arises because

cP is the energy needed 1O free the least

strongly bound electrons. Others may al ...o be freed. but less of the photon \ energy would then be left for kmetic energy.

Einstein's interpretation of the photoelectric effect explains not only the observation that a cenain minimum frequency i ... reqUired but also the other classically unexpected result\. I f a single photon-a particle

of concentrated

energy rather than a diffuse wave--doe\ have enough energy. ejection should be immediate. with no time lag. AI'iio. the electron's kinetic energy .,houJd depend only on the energy of the single photon--the frequency-not on how many !OIrike the metal per unit time (the IOtem.lty). In all respects. Einstein's explanation agrees with the experimental evidence. and the achievement earned him the

192/ Nobel Prize In phy\ics.

EXAMPLE 3 . 1 Light of 380 nm wavelength i\ directed at a metal electrode. To determine the energy of electron!> ejected. an oppo\ing elcctrO'ltalic potential difference I!. e"'lablJ"hed belween it and another electrode.

a,

"hown in Figure 3.4. The current of photoelec..

Iron, from one to the other i\ \topped completely when the potential difference i.,

3,2 n. �

1 . 1 0 V. D�tenninc (a) the work function of the metal and (bl the maximum-wave­ len�th light that ('an ejt.'Ct electron... from thi... metal. SOLVnON

(a) In the region between the electrode.... the electron... lose kinetic energ)' a' they � (1.6 X 10 1'Jl Cl gain potemial energy, II a potential energy difference 01" ( \ , 1 0 V) "'-' \ .76 X 10-111 J = 1.10 eV i, the most they ('an ...unnount. their

qV

.. ..... 'It

...... 1.. Cunonl_ .... 01..,..-. _ _ lbo _ilia potenlial encfl)' diffmmcc equIs 1be muimum kinetic eneru of the photoelectrons.

\..inetic energy k;\nng the fiN electrode mu,t be no larger than 1.10 eV. The potential diff�rence that barely ,tops !.he now is known a... the stopping po,rn. tiat Using equalion (3-3).

1 .76 X 10 1 9 J = (6.63 X 10-'� J . s)

�

q, = 3.47 X 10 IIl J

=

(

' X IO' m "

)

_1 -_

_ _ _

----=- __

380 X 10-Q m

2.l7eV

.- '"

(b) I f the wavelength o f the light were increased to A', the frequency-and thus

the photon energy-wou ld decrease. The limit for ejecting electron ... is when

an incoming photon has only enough energy to free an electron from the metal. with none lell for kinetic energy. Agai n u..ing equation (3-3), 0 = lif' - q, = (6.63 X lO-H J , s )

(

3 X 103 m1s

A

'

-

.

'

� A' = 573 nm

)

- 3.47 x lO-wJ

Wavelengths longer than 573 nm have insufficient energy per photon, so no

photoe lectrons are produced. The maximum wavelength for which electrons are freed is called the threshold wavelength, and the corresponding minimum

frequency is the threshold frequency.

The central point in Einstein'S explanation of the photoelectric effect is that electromagnetic radiation appears to be behaving as a collection of parricles. each with a discrete energy. Something that is discrete, as opposed to continu­ ous, is said to be

quantized. In the photoelectric effect. the energy in light is

quantized. EXAMPLE 3 . 2 How many photons per second emanate from a 1 0 mW 633 nm laser? SOLUTION

For each photon, E =

c

hf = h-A =

(6.63 X IO-" J · ,)

(

3 X

10' ml'

)

633 x I 0 9 m

=

3.14 X 10-I'J

To find number of particles per unit time, we divide energy per unit time by energy per particle:

number of particles time Clearly. photons are rather appear COnll PUOUS.

=

10

10-3 J/ ' -,-"" :-:-'= 3 . 1 8 X 1016 partide!J� 3.14 X 10 I' J/ partic\e X

-

-

"small " and it is easy to see how a light beam CQuid ,

"

t

+

71 CMptet 3 W.ves and Particles I: Elcctromagnetic Radiation Behaving as P'.utides FJgurw 3.5 In a nighl vision device, a light image becomes an image of free electrons. amplified in a multichannel plate and then reveaJed on a screen.

The photoelectric effect has long been used in simple Jight sensors, where light intensity registers as a photocurrenl, bUI it is also used in more sophisti�

cated ways. Let us take a look al one. REAL-WORLD EXAMPLE N I G H T V I S I O N

Incoming

light

By "replacing'" a photon with an electron. whose charge makes i l easier to "amplify." the photoelectric effect i!. a common fronl end on optical imaging systems. One example is the night vision device (NVD). A typical NVD is shown schematically in

Ob}ccti\'e len�

Figure 3.5. An objective lens focuses an optical image onto a thin piece of material. called a photocathode. where the photoelectric effect transforms it into an image of freed electrons. Naturally. the dimmer the light. the fewer the photoelectrons. Amplification is accomplished via a microchannel plate. This element has hundreds of thousands of channels per square centimeter. An electron entering a channel at

:;;::".,�!t". Ptu.!oeh....ul!n� ........ . /.k. •, pn'lloccd ...

.. :; • •

,

•

�."

..

., ••. .

.... �

;H'C

Pl<'I"('

PhOlocathode

one end. driven through by a potential difference. knocks off other electrons at each collision with the channel walls. emerging at the other end with about 10.000 fellow electrons. (Macroscopic objects that work thi� way are known as photomultiplier tubes and are widely used in astronomy and particle physics. as well as in medical

Mullichannel

plate

imaging.) This greatly amplified signal then strikes a phosphorescent screen to pro� duce the final visible image in the phosphor's characteristic color. Optimizing photocathode materials is front�1ine research. but we can easily

Elt'ctrun� mullipl)"

at ell1:h collhll.n in�ide the t:h;mrn:1.

\.....

"

grasp one of the basic constraints.

.... . ..... " " " .

��

,

,

Applying thE! Physics ,

(a) If we wish a light-sensing device relying on the photoelectric effect to be sensi­ tive to the entire visible spectrum (400 -700 nm), explain why zinc would be a pOOr choice of photocathode. See Table 3. 1 . (b) A common photocathode containing the alkali meta1s cesium, potassium, and sodium has a very low effective work function of about 1.4 eV. Whm wavelengths can it " see"?

(a) Zinc has a work function of 4.3 eY. Example 3.1 showed that a material with a lower work function has a threshold wavelength of 573 nm. Thus, wavelengths in the orange and red end of the spectrum would not liberate any photoelec� trons. Zinc would "see" even less of the visible spectrum.

(b) The work function equals the energy of the longest-wavelength pholon that can free an electron. Before finding this wavelength, we note that it is quite common to express wavelengths in nanometers and energies in electronvoits. so a value for the product he in these units is very convenient. Exercise 27 shows that it is

1240 eV nm. Thus, •

E=

h

�

he

A

-

1 .4 eV =

1240eV ' nm A

A = 886nrn

This material can see all wavelengths in the visible spectrum and well into the infrared.

3.3

The Produ ction of X-Rays

We use the name X.rays for electromagnetic radiation whose wavelengths are in the 10- 2 nm to 1 0 nm region of the spectrum. The name was coined by Wilhelm Roentgen, who first studied the radiation. Among other things, he

Figure 3.6 X-nlys are produced when electrons "boiled" off. hot filament an aceel· ertued into a metal target.

--:-:--c c-:--� HOIV iok nl -

� �

filament deceleration X-rt.� -

X-nay lube (evacuated)

_

'-- r---. ;

+

Aettlcnating potential

found that X-rays could expose photographic film after passing through a solid object. such as a human being. For his work, he received the first Nobel Prize in physics in 1901. Nowadays, X-rays are an important tool in many areas of research, but their production also gives key evidence of electromag­ netic radiation's particle nature. As shown in Figure 3.6, X-rays can be produced by smashing high-speed electrons into a metal target. When they hit, these violently decelerating charges produce much radiation, called bremsstrahlung, a German word meaning "braking radiation." The fact that electromagnetic radiation can be produced this way is not surprising from the classical perspective, but if it is strictly a wave, we might expect it to cover the entire spectrum. Although the tolal energy is limited by the number of electrons arriving per unit time. there is no reason waves shouldn't emerge with some amplitude at all wavelengths. But this isn't what happens. Figure 3.7 depicts the spectrum produced when electrons of kinetic energy 25 keY strike a molybdenum target. A broad range is apparent.I but none of wavelength less than '\

:=;

0.050 nm. This is called the cutoff wave­

Figure 3.7 The X-ray spectrum pr0-

duced when 2S keV electrons strike •

length, and there is no classical ex.planation for so sharp a termination of the spectrum.

molybdenum target.

-tl

?'

The nonclassical ex.planation is that electromagnetic radiation of a given frequency simply cannot be of arbitrarily small amplitude. If the radiation is quantized, the minimum energy allowed at frequency J is hJ. a single photon. We cannot produce half a photon, so if multiple electrons do not combine their energies into a single photon, no photon could ever be produced of energy greater than the kinetic energy of a single electron. Is this the case?

" •

! �'----::-:----:---:--A <

0.50 nm

Wavelength

Setting the kinetic energy of an incoming electron equal to the energy of one photon.

25

X

103 eV = �

he

=

1240 e Y · nm

A A = O.050 nm

A

This is fairly convincing evidence. Electrons apparently do not combine their energies.

I Not Wlown � d\aracterisIic X-ny spika " 10

i tendiOM specific 10 !be daaeIlIs ill tbt .... n malalal � Ihan Umpk deu:ktllioa a.ne­ tcr1sUc X·r.� � discussIId ill Secaioe. I..S.

II

0Iepter 3

Waves IDd Pwticles I: ElectrotnagntbC Radiation Behaving !U Particle..

3.4

The

Compton Effect

We now consider yet another phenomenon that couldn't be explained by the classical view of electromagnetic radiation as strictly waves, one Lhat uncoven anOlher important property of photons. The situation is the scattering of elec_ tromagnetic radiation from free stationary electrons. According to c1assica]

electromagnetic wave theory. the electrons would oscillate and therefore rera. diate. or "scatter." electromagnetic energy in all directions initially at the same

frequency as tire incoming radiation. Arthur Holly Compton, investigating this phenomenon with X-rays and using carbon atoms as the source of electrons. found that some radi­ ation scattered backward immediately with a wavelength significantly longer than that of the incoming X-rays. (The electrons involved are ini­ tially bound to carbon atoms. but so weakly as to be effectively free. See Exercise Compton's explanation treated the X-rays as a collection of

30.)

photons, each with a discrete energy but also with another property we usu­

ally associate with a particle-momentum. Signs point this way from two directions: I . According to special relativity. an object with zero mass should have momentum related to its energy by E = pc

2. According to classical electromagnetic wave theory, electromagnetic waves do carry momentum. However. for a diffuse wave, we speak of momentum deflsity. which is related to the energy density by energy volume

=

momentum volume

X c

With two such compatible clues. it seems reasonable that the momentum of a photon might be given by

E lif II p = - = - = c c A Is this true'! Compton provided the first experimental evidence. Let us set the stage.

A Two-Particle Collision In the particle view of electromagnetic radiation. the interaction of X-rays and electrons is simply a collection of separate two-particle collisions between pho­ ton and electron. for which we may now express momentum and energy can· servation. As shown in Figure

3.8.

we assume that an X-ray photon of

wavelength A strikes a �talionary electron, and afterward. the electron scatters

Fill"'" 3.' Momentum and energy when a photon strikes a fn:e electron. Al\l."r ,,:o\lision: 1bc I."i«troo lutters .t sJl«d 1I. 1lIlI11." .,.. A photon of w8"l."kngth A' scattl."n at an,ll." (t,

Bdore' .;olh�lon: A phOion of wavC'kngth A appro.:l..;h", an 1."1e<:lwn III re'�1

.'

(

•

I \I .,.

ltc/A

"

Bo'l."� ,how momentum (lop) and enl."liY (boltom) e)(pre�;'lom,

at speed It and angle 4>. while a scattered photon of wavelength A departs at 8. Using /ie/A for the photon energy (more convenient than the equivalent hI). we have Momentum conserved: '

x-component: y-component:

II

A

=

0 =

A7 cos8 II

h

--;; sin 8

+

-

'YIP'ell cos ¢ 'YIP'eu sin

(3-4) (3-5)

Energy conserved: =

c

11 - + .... m

A'

I [j

c-,

e

(3-6)

As it happens, the electron often moves very fast after the colli!'.ion. so we must use the relativistically correct expressions for its momentum and energy. The photon expressions are already correct; nonrelativistic ones do not exist for things that always move at c. Equations (3-4) to (3-6) have been found to agree completely with ex:peri­ mental observations when a photon collides with a free electron. An tests since Compton's original work have reaffirmed the conclusion: The momen­ tum of a photon is given by

h p = ­ A

(3-7)

Momentum of a photon

II Ch.pter 3

Wives and Particl« I: Electromagnetic Radiation Behaving a� Particles

The Compton effect's most striking departure from classical expectation is the large and immediate wavelength shift in the scattered radiation. We clar.

ify this by eliminating the electron

speed II and scattering angle

rP from among

equations (3·4) to (3·6). (The somewhat lengthy algebra is left to Exercise 38.

)

What remains is

C ·,,··,·

h m,e

A' - A = - ( I - cos8)

de

(3·8)

The difference in wavelength between the incident and scattered photons depends only on the angle of scatter, a discovery that won for Compton the 1927 Nobel Prize. We see also that the scattered photon is always of longer wavelength than the incident. Its energy is always less, which makes sense, for kinetic energy is given to the elecLTon. In particular, the maximum increase in wavelength is

for

backward scatter of the photon, 8 = 1 80°. because a "head·on" collision imparts the maximum possible energy to the electron.

(Note: In Compton's experiment,

was

scattered at all angles, because some X·rays effectively interact with a much heavier mass�the whole atom­

some radiation of the incident wavelength

giving a negligible wavelength shift. See Exercise

EXAMPLE

33.)

3.3

An X·ray photon of 0.0500 nm wavelength strikes a free. stationary electron, and the scattered photon departs at 90° from the initial photon direction. Determine the momenta of the incident photon, the scattered photon. and the electron.

For the incident photon. Pill(:id..,m

-

6.63 X IO-J� J ' s - 1 .33 X 10-23 kg' mls 0.05 X 10.9 m -

II

A-

We may solve (3-8) for the scattered photon'S wavelength.

A , - 0.0500 X 10-9 m = ( . 9 11

A' = 0.0524 nm

:=)

Thus.

' p -

6.63 X 10-34 J . S ( I - cos 900) 10 31 kg)(3 X 10' m/s)

x

h 6 63 X 10-34 J . S A' - 0.0524 X 10-9 m _

.

-

1 .26 X 10-23 kg ' mls

For the electron's direction, we use equations (3·4) and (3-5).

h

h

A

A'

- = -cos 90° +

o

'V m II cos '1I"'·e

¢ -

m 1/ sin ¢ = �sin 90° - '\I'1I"··e

A'

-

h 0.0524 nm

=

mil l "yU

� . 'I' SJO

Flgu,. ,.t Momentum COGIeI..... iD photon-elcctton co1liUoD.

0.0500 A. � Ian ... 0.0524

----

.==:.

A. = ...

'"..6"

p-

Rein<;erting In either gives the magnitude of the electron·� momentum.

0-

Thus. Pe

6.63 X 10-� J ' � ,-�in 90" 0 .0524 X 10- m

-

YJ"<:II

�In 43_6"

= \ .83 X 1 0 ' n kg'lll!s al 43.6° from direction of incident photon.

vector momentum addition is shown in Figure 3.9. Even though mus�less. X-rays arc very energetic particles. The electron is ··bumped" to 2 X 107 mls. nearly one-tenth the speed of light.

The

If treating X·rays as a collection of photons predicts !->o weB what we experimentally observe. why is the wave theory of electromagnetic radia­ tion still around? The answer is deceptively simple: The behavior of elec­ tromagnetic radiation is just as wave theory predicts. if the wavelength is long. Particle and wave behaviors converge at long wavelengths. Of course. this "answer" raises a thornier question: Can a line be drawn between wave and particle? This issue we take up in Section 3.6. For now. we simply note that if instead of a

0.0500

nm wavelength X-ray. we ';hit" the electron

in Example 3.3 with visible light of

500

nm wavelength. the wavelength

change predicted by equation (3-8) would be the same as before. The percent change, rather than the approximate 5% in the example. would be only

0.0005%.

Without great precision. the scattered radiation would

seem to be the same wavelength as the incident-the prediction of classical wave theory. Here we have one example of the

correspondence principle. This

principle, which should be viewed as a guideline rather than a quantitative "law" of physics. states that a nonclassical theory should agree with the previous classical one in the appropriate limit. For example. special relativ­ ity is a nonclassical theory that agrees with classical mechanics in the limit of small velocities. To this we add that the nonclassical particle theory of electromagnetic radiation agrees with classical wave theory in the limit of long wavelengths. As the wavelength of a beam of electromagnetic radiation is increased. the energy per photon decreases. A given intensity would then comprise a larger number of less-energetic photons and would begin to exhibit the behavior we expect of a continuous wave. (Occurring only for

short wavelengths, the photoelectric effect might appear to violate

the principle, but no "classical expectation" is really valid. because electron binding in solids. discussed in Chapters mechanical.)

9 and 10. is fundamentally quantum

p'

14

a..pter

3 Waves and Particl� I: Electromagnetic Radiation Behal'inl!l a.� Panide�

In connection with the correspondence principle, it is often said that clas_ sical behavior foHows in the limit that h goes to zero. It is true that photOn energies and momenta tend to be small because h is so small. If II were indeed zero, E and p (i.e., hfand hlA) would be zero, and light would never behave as

a granular collection of particles. However, Planck's constant is a fundamental

constant of nature-the constant associated with quantum phenomena-and as such, never "goes" anywhere. Thus, it is better to say that classicaJ behavior follows in the limit of long wavelengths.

An InelasticCollision The photon-electron collision in the Compton effect is necessarily elastic. Because the mass/internal energy of fundamental particles like the electron cannot change, neither can the system's kinetic energy. In fact, we may rearrange the energy-conservation equation (3-6) as E = E' + (Yu - I )

mec2. Photons have only kinetic energy (no mass/internal energy), so this is a statement of

kinetic

energy conservation.

(Note:

The electron kinetic energy is the relativistically correct form.) Let us reinforce our grasp of the conservation laws by conSidering a system in which mass

;s

subject to

change.

EXAMPLE

3.4

A neutron and proton bound together by the "strong force" i s called a deuteron. It is the nucleus of the hydrogen isotope deulerium, a150 known as "heavy hydrogen." A helium nucleus is two protons and two neutrons bound together by the same force. Suppose a very energetic photon strikes a helium nucleus and breaks it into two

deuterons, each departing at O.6c, as depicted in Figure 3.10. The photon vanishes in the process. (a) What was the photon'S energy? (b) In what directions do the deuterons depart? (The mass of a deuteron is kg.)

4.00151 u, where 1 u = 1 .66

X 10-27

2.01355 u, and of a helium nucleus,

.or

(a) Lei us use a subscript He for helium and D for a deuteron. Before the collision, we have a photon and a statIonary helium nucleus. Afterward, we have two deuterons moving at O.6c. Energy is conserved:

Figure 3.10 A photon disintegrates a nucleus.

Noting that

L+x ,

0.6<


h"A = 2(2 c

mD

•

Yo_6c = 1 .25, we have

I;)

= "'[2

'.

"

'�

mD

0.6<

X 1.25 Ino - InHe)

X

1 .25(2.01 355 u) - 4.00151 u J

= "'( 1 .0324 u)

X 108 m/s)2( 1 .0324 u = 1.54 X IO- IO J - 1 GeV

= (3

X

1.66

X

10-27 kg/u)

3.5 Patr PMIdoc:doa

.. ..

An energeti� photon. indeed! Thl.. ex:ecd th� entire ma's·�nergy of n pr\ltun,

( 1 .67 X 1�1 �g)(3 X

..

1 011 mJs)! =

I.S X

1 0 10 J . In the L'ollisilm we: are

IlnlllYling. allno..t all the photon'� energy goo:s into kineti� energy (If the deuterons; the �Y�lem\ final ma initial (4.00\5i).

(4,027 1 ) i.. only ,lightly higher than

the

(b) Direction.. arise only in the vector momentum-con,er\'alion equations. Cun· serving P" which i� initially O. would merely lell us that 8, mu,. equal 81, a� symmetry dcmand�. Thi,. in (urn. tell� Ul> that the deutemo, h.we equal I'..

Thu,

FlgUN 3 . 1 ' In

pair production.

I<

A

=

2

Multiplying by c and again u�ing Yo 6.'

result on the left side,

X ()'O.&-'''D O.6c) cos 0 =

whkb

II gamma-ray photon becomes an eleclnm. which curves one way In a magnetic field, and a positron. o.:uJ'\o'('s the other wa)'.

,,,,., ... .' . In

i"ible path of

in.:oming pho\('ln

"

,

p, conserved:

•

x',

x

P�i":'

x PII Ih

"1

1.25 enables us to use our earlier

B (into p8se) •

I<e •

1.54

X 10-10 J =

�

8

=

\ .5(2.0\355

X

70"

3 . 5 Pair Production

are

Figur. 3.,\2 Bubble chamber trails

are

..howing electron·positron pair produc­ two important ways in tion. Photons, uncharged and invisible. The photoelectric effect and the Compton effect which electromagnetic radiation interacts as a particle with matter. We now incident from the top. The lower. discuss a third. slowly diverging arcs one pair. electron and positron the upper pair in 1n 1932. a revolutionary new particle was discovered. Carl D. Anderson ha v e less energy and spiral faster (Nobel Prize, 1936) was studying the effects of cosmic rays, energetic parti­ cles bombarding Earth, when he noticed something behaving like an electron because much of the photon energy but of positive charge. It curved the right amount but the " wrong" way in a goes lo a freed alomic electron {the magnetic field. This positively charged electron was termed the positron. We straighter trail). now know that high-energy photons are constantly creating positrons all around us (fortunately not in dangerous numbers) through pair production. This process can be revealed. as depicted in Figure 3. 1 1 , by a bubble cham­ ber detector immersed in a magnetic field, in which charged particles leave visible trails of bubbles as they curve. From apparently nothing, there sud­ denly appear two charged particles deflecting in opposite directions. The energy to produce the massive electron-positron pair comes from a high­ energy photon, which, being uncharged, leaves no trail. Charge is conserved because the total charge of the pair is zero. Figure 3.12 shows actual bubble chamber trails of two electron-positron pairs. The revolutionary aspect of the positron is that it is antimatter. Antiparti­ cles in general discussed in Chapter 12, but for the positron. life is short and its end dramatic. It quickly finds an electron-any will do-and after a brief quantum dance, they annihilate together: erased, their entire energy suddenly transformed to two photons. The process, pair annihilation, is addressed in Exercises 4 1 and 42.

are

are

The

3.6

Is It a Wave or a Particle?

17

Its kinetIc energy would be appro:\imately 4 X 10-l� J, or ubout 6 orders of magnitude le.slO than the hJtal energy invo\wd in the process. Thus. we see Ihal

the nucleus l'an indeed ensure momentum I.:"onservation without signi ticantly atl'ecting the energy. The precedi ng di:.cus ...ion raises a poin( that crops up often in physics. Whenever "small" particles interact with "large·' one-so the small one s lend to have nearly all {hI.." kinetic energy. Why·.J The kinelic energy

\)f l:I. nonrel alivis­

tic particle may be written as ,,"":./"2111. sO if the interacting partic les have compa­ rable momenta (numerator), then a particle with a much

larger mass

(denominator) wIll have a much smaller kinetic energy. The rule also tends

10

hold when massless and mas!-oive panicles inleract. with the massless one fill­

ing the small role_ For instance. when an atom emits II photon, the recoiling atom and photon have e4ual momenta but ,lImost all the kinetic energy goes to the photon. .

Figure 3_'3 An "ellperiment" in which

3.6 Is It a Wave or a Particle?

a disturbance bc:ha\'es as a wave or a partide, depending on the relati\le size

This may be the most perplexing question for the student of quantum mechan­ ics. The simplest answer b that it has no predetennined nature. The observa� lion itself-whether the experimenter bounces light otT "it," places something in its path. or interacts with it in any way--detennines whether ·'it" will exhibit a wave or a particle nature, Much of what we discuss here applies not only to elec­ tromagnetic radiation but also 10 massive objects. and thus provides a valuable link between this chapter and the next. To cover either case . let u... refer simply to "

of the wavelength and the relevant dimension D of the apparatus.

"'

a .... avc· " \JcleC['xj

·'the phenomenon."

The Wavelength and the Experiment We advance the answer a step further with a rough criterion: The behavior a phenomenon exhibits depends on how its wavelength A compares with the " relevant dImensions" of the experimental apparatus, which we repre­ sent simply as D. All phenomena may, in principle. be described by a wave function. The familiar sinusoidal functions for the electric and mag­

Whc·o

to, rt,·tc,\Ca

netic fields in a plane wave of light are one example. As we find in Chapter 4. a massive object might also be described by a sinusoidal wave function. If Its wavelength is much smaller than D, the phenomenon Will exhibit a particle behavior: if comparable to or larger than D. it will exhibit a wave

3. 13

behavior. Figure depicts how the behavior a phenomenon exhibit-. might depend on a relevant dimension of the apparatus. The boaters are conducting an experiment-blindfolded. If the wavelength of the approaching disturbance (the phenomenon) is larger than D. Ihe width of the boat. these experimenters will certainly proclaim It a wave. because the boat is small enough 10 respond to its crests and troughs separately. But if the wavelength is much smaller than

D, the boat responds to the whole thing at once, and the experimenters might conclude that a particle, not a diffuse wave. has struck. By the same logic. pas­ sengers on an immense ocean liner would see even the top dislurbance as

A ,< /)

" p."[ICk

a

>.

• Ch.pter 3 Wa'VCI and Particles I: Electromagnetic Radiation Behaving a� P..utide, particle. because

A

«

D. while ants floating on a popcorn kernel would see > D.

the bottom one as a wave. because A

A simple application of the criterion to electromagnetic radiation is Ute ca
non of diffraction. the slit width must be narrower than or comparable to th e wavelength. If it is much greater. the wave passes straight through, as woul d particles.

Useful though the criterion may be. it is dangerous to view wave�particle duality too rigidly. as wave verSLIS particle. The natures are not incompatible but complementary. just two faces of the same phenomenon. We can't fUlly explain the behavior of a given phenomenon by either nature alone, and there is a close relationship between the two natures. The famous E = lif and p ==

iliA. which quantitatively link the particle properties E and p to the wave prop_

erties/ and A. are one part of the relationship. We now discuss another. A

Double-Slit Experiment

There is no more direct way to see the link between wave and particle natures than the double-slit experiment. Suppose light is directed at a double slit and is thereafter detected on photographic film beyond the slits. At high intensi­ ties. we observe a typical interference pattern, the intensity varying from a maximum (constructive) at the center of the film. to zero (destructive). then back to a maximum. and so on. The light is exhibiting the wave nature we attribute to it classically. Particles don', interfere or cancel; waves do. Now suppose the intensity is greatly reduced. so that a pattern is no longer visible on the screen. The film still registers the arrival of light, but sporadically at scattered locations. Apparently, the light is being detected one photon-one particle-at a time. Figure 3 . 1 4 shows the result if the film reg­

isters an arrival with a spot. Although the locations of the spots at first seem almost completely unpredictable. we eventually begin to discern a regular pattern. Understanding the link between wave and particle rests on two key observations: First, the exact location where the next photon will be found evidently can't be known, but logically. the probability of detecting it in a given region should be proportional to the density of spots there-high where density is high. low where density is low. Therefore. if the density of Spots assumes a pattern, the probability assumes a pattern. Second, careful study reveals that the density of spots in a region is directly proportional to what wave theory (physical optics) predicts should be the relative intensity in that region, which is. in tum, proportional to the square of the amplitude of the electromagnetic wave. In particular. no photons are ever detected al the loca­ tions where wave theory says there should be points of destructive interfer­ ence. Although the light is being detected one particle at a time. its wave nature is still apparem. Combining the two observations, we conclude that because both are pro­ portional to (he density of spots, there is a proportionality between the proba­ bility of detecting the particle and the square of the amplitude of the wave. This connection between particle and wave natures is a cornerstone of quan­ tum mechanics:

3.6 1. h . w.w or . �' .. Figure J.'4 Photuns produci.ng M

double-slit interfcrc-n,,-e pattem�rn:

partide- at a time.

10

Double-,,1it intensity

J\I\AJV\

When a phenomenon is detected as particles, we cannot predict with cer�

tainty where a given particle will be found. The most we can determine is a probability of finding it in a given region. which i!i proportional

(�

)

to the square of the amplitude of the at;sociated wave in that region.

�

�

?

prob bil ty of fi ding particle to a regIOn

0<:

Plitude of.wal'e '

to that regIOn

To clarify the nomenclature, if the wave in question is the electromagnetic field, its "associated particle" is the photon. Equivalently, if the particle in ques­ tion is the photon, its "associated wave," its alter ego, is the electromagnetic field. In electricity and magnetism, we learn that electromagnetic fields exert forces on charges, and we now claim that they also measure the probability of finding the associated particle-the photon. We point this out partly to prepare the reader for a possible shock in the next chapter: The wave a5$ociated with a massive object is indeed a measure of the probability of finding the particle. but it doesn't appear to have any other side to its personality. EX A M P L E 3 . 6 Light of wavelength 633 nm is directed at a double sliL and the interference pattern i� viewed on a screen. The intensity at the center of the pattern is 4.0 WJm"!. (a) At what

rate are photons detected at the panern's center? (b) At what rate are photons detected

at the first interference minimum? (c) At what rate are photons detected at a point on the screen where the waves from the two sources are out of phase by one�third of a

(�

cycle? (Note: From physical optics. the double-slit intensity varies according to J = cos2 q.,), where rp is the phase difference between the waves from the

'0

two slits and 10 is the intensity when rp = 0, that is, at the center of the pattern.)

90 ctt.pgr 3

W.IICS Mld Pltnicle� I: Electromagnetic Radialion 8eh:llling as Panicles SOLUTION

(a) Each photon has energy

h"A c

= (6.63 X 1O-)� J . s)

3 X 108 m/s 633 X 10 9 m

= 3.14 X IO-L9j

We must therefore have 4.0 J/s ' m2 3.14 X 10 19 J/photon

= 1 .27 X 1019 photons/s ' m2

(b) The first interference minimum is a point of destructive interference, because the difference in distances traveled to the screen creates a phase difference of 1T (one-half cycle) between the waves from the two sources. If the net wave (electric field) is zero, its square is zero and so is the intensity. No photons are detected here. (c)

Somewhere between the center of Ihe pllilem and the firsl minimum, the waves will be out of phase by one-third cycle: rjJ = We know Ihat 10 is 4.0

!27T.

W/m2, so we have

t

/ = (4.0 W/m2) cosl[ ! ( 21T)) = I.O W/ml Because this is one-fourth the value at the center, the probability of detecting photons here is one-fourth as large, so only one-fourth as many will be detected per unit time per unit area around this point.

3. 1 8 X 1O'8 photonsls ·m2 Although we detect particles, the probability of their detection is governed by

the behavior of the associated electromagnetic wave.

Fmally. we address a point central to quantum mechanics. In the double slit, it is not proper to ask, 'Through which slit did the 17th photon pass?" We haven't a1Jowed for this, and by altering the experiment so as to observe a particle passing through a slit-an experiment in ilSelf, requiring some interaction with light in one slit alone-we would alter the very behavior we wish to observe. Because interference requires two coherent waves, the pattern would be disturbed. We can't have it both ways. Wavelike interference can only be observed by allowing each "particle" to behave as a wave-passing rhrough both slits simultaneously. We will have more to say about this challenging idea in Chapter 4. 1s it a wave or a particle? It may behave as either, depending on the situa� tion, but the two nat ures are inextricably related.

PROGRESS AND APPLICATIONS Ejecting Electrons wit" X-Rays Section 3.2 discusses the photoelectric effect in which photons roughly in the visible

The light

wavelength r.lIlge eject the leru.{ tightly bound electrons in a

tuned from 0.1 nm to

University of California's Advanced Light Source (ALS).

"

"

in this case is X-rays whose wavelength can be 100 run. To produce the powerful beam

metal. Logically, �horter-wavelength. more-energetic phmons

needed, Ihe simple method of smashing electrons into a

Today thi� effect is being exploited to reveal much about how ela:trons are bound 10 atoms in marerials and how the atoms

synchrotron. Any charged particle emits electromagnetic

�houJd be able 10 free electrons that are more tightly bound.

themselves are arranged.

At the forefron t of the work

is the

target give.s way to a stream of electrons circulating in a

radiation when i{ accelerates, and at the ALS, specia] magnets, i ulating electrons so as to called undulatOT!., wiggle the crc

produce a very bright nanow beam of X-my!> When !he beam is directed at a sample, photoclttlnlns are produced. Studying their kinetic t>rlf!'rgi("f when free, k.nown a..� photoelectron spedroscop),. teU" u� about their energies when bound in the material. and �tud)'mg their motion when free tells us about their 1c)Cllliom when bound and, thus, the locations of the atoms. The lalter technique is lenned photoelectron diffraction. The' dltlractlon isn't that of the X-rays but of photoelectrons. A"" noted in Section 3.6. objects with mass also ell.hibit wave-particle duality, and these photoelectrons behave as wa\'es of such short wavelengths as to provide a very high-resolution picture of the atomic landscape from which they were ejected.

A lral:er mat�nal l.:ontaining a radioisutope that emits I"',itn.,m at " ,afdy 1\1W rllte il introdu\.-ed into the patient "ia the bh'M..�.htream. ",here It l:OIle..:l'!l. in ccnain tisti.ues. �Ra.dillal·ti\'it:,.' I� rJi'l:u�-.cd in Chapter 1 1 . \ 1ne patient i!io then plal.:C'd in�irJe a mal:hlOc liurT\lunded by a ring of . dcte..:ton. that ··Il'l.'... lor phuton" llf the .:haractC'ristic energy A {,("Itmn cminerJ b)' tt\e tr8l.:cr--aftCf no more than about Ii millimeter llf hllun\:ing anlund-flOd� an electron ", ilh whil.:h III annihilal.... F., Many inter,cl:ting line!> indicate a high density of annihilation� at a point-a \:on\:entration of the tracer-in the two-dimen�illnal "l'hce" of the patient viewed y a R X Produce to Way their of Because s New A penetrating abilities, X-rays have found a great host of uses by the detector ring. Al' the ring pas!>!:!> along the patient's body, look.ing at each ..lice. a three-dimensional image of in learning what is inside things without breaking them den),.;ty versu" p6,ilion emerge�. Figure :\. 1 5 "hOWl; (a) the open, probing everything from human bodies to construction basic layout of a PET machine and (M an actual image. materials to superconductors. While the century-old tried­ and-true method of Section 3.3 is still the leader, the conven­ Figure 3.15 Positron emis�ion tomogrolphy_ (a) A tracer emits a tional X-ray machine, with its hot filament sealed in a tube, is rather unwieldy. and the tube's lifetime is often short. The positron that annihilate, with a nearby electron, yielding two pho­ tons that place the annihilation along a line between the deleCtors. i huge and growing field of nanotechnology-applications n which some crucial element is measured in nanometers­ Many lines combine to produce a two-dimensional image of a slice. (b) Multiple slices produce a three-dimens.ional image-in may provide a new method. In work conducted at the this ca-.e, "hawing high tracer uptake in the hver and kidneys. University of North Carolina, carbon nanotubes are allied (a) with the quantum-mechanical effect of '-field emission" to produce a beam of X-rays strong enough to replace the Two phNl)n.. from an electron­ conventional X-ray machine in several uses. A nanotube is a regular meshwork of carbon atoms. forming a cylinder of only aboUi I om radius, and is closely related to many other all-carbon structures discovered in lhe 1980s (see Chapter 10). These continue to surprise us with new, remarkable properties and are a very hoi IOpic of physics research. In the X-ray source application, bundles of naootubes are deposited in a lhin layer on a metal disk. Electrons are coaxed from the layer toward a target not by heating a filament. which wastes power and produces electrical noise, but by field emission, a room-temperature way of producing a flow of electrons that relies on the quantum-mechanical effect of tunneling (discussed in Chapter 6). The resolution of the new tech­ nique is ell.cellent, and another potential advantage is faster response time for tracking moving objects. (See Vue et aI., Applied Physics Letters, 8 July 2002.)

r.�!!t;t·./

...., po,;lwn aIlmhilaliun are

.

(b)

Medical Imaging with Positrons As noted in Section

3.5, once a positron is produced, it soon engages in pair annihilation, simultaneously yielding two photons of a characteristic energy (see Exercise 42). This trait is ell.ploited in an increasingly common medical imaging procedure known as positron emission tomography (PET).

dch:l.:lcd �Imu\l.m�·ous.ly.

Chapter 3 Summary

fundamental daim� as possible. explain to your friend what evidence Ih.i.� pro\'ides for the particle nature of light.

3. You are conducting a photoelectric effect experiment by \hining hght of 500 nm wavelength at a piece of metal and detennining the �topping potential. If. unbeknownst

Electromagnetic radialion behaves i n some situations as a col­ Itttion of panicles-photons-having the paniclelike pmper­ tiC:
to you, your 500 nm ight l source acrually contained a �maJJ amount of ultraviolet light, would it throw offYour

wa\'c properties of frequency and ....a\'elenglh: .

E = Itf

h p=­ , where h is Planck'� constant. 6.63 X 10-,\.:1 1

re.\ult.� by a small nmount or by quile a bit? Explain.

(3-2) 0-7) s. Eleclromagnetic

radiation is more likely to c)(hibil a particle nature when the wa\e­ length is small compared with the relevant dimensions of the experimental apparatus, For a given intensity, a J.hort wavelength corresponds to u large energy per panicle and a correspond­ ingly _�maJl particle flu;II,-number per unit time per unit area. The

4. Suppose we produce X-rays not by smashing eleclrons

into targets but by smashing protons. which are far more massive. If the same accelerating potential differ_ ence were used for both, how would the cutoff wave_ lengths of the two X-ray spectra compare? Explain. S. In the Compton effect, we choose the electron to be at the origin and the initial photon's direction of motion to be in the +.t direction. (a) We may also choose the xy-phUle so Ih:1I it contains the velocities of the Outgoing electron and photon. Why? (b) The incoming photon's wavelength A is a....wmed to be known. The unknowns

radiation is thu" relathely partidelike. For long wavelengths. it is more likely to behave as II continuous wave. Electromagnetic radiation may e:o;hibil its particle nature

afler the collision are the outgoing photon's wavelength

and direction. A' and 0, and the speed and direction of the electron, Uc and tP. With only three equmions-Iwo

in several ways_ In the photoelectric effect, a phoron gives up

a discrete amount of energy to an electron in a metal. In the Compton effc!Ct. a ,han-wavelength photon scaner.; from an

components of momentum conservation and one of energy-we can't find all four. Equation (3-8) gives A' ill temlS 0/0. Our lack of knowledge of o after the colli_ sion (without an experiment) is directly related to a lack of knowledge of something before the collision. What is

es�ential1y free e]ei,:tron. and both electron and photon obey the usual con\ervation laws for panicles with discrete momentum and energy. In pair production. the�e con.�ervation laws are also obeyed � a high-energ} photon disappeat�, transferring its di�­ crete energy to a pair of mas�ive panicles, When electromagnetic radiiltion i,\o detected a, particles. it is uncertain where a gi�'en photon will be found, The most that can be detemUned i� 3 probability of finding it in a given region. which is proportional to the square of the amplitude of the associ­ ated wave---the oscillating electromagnetic field-in that region.

indicates advanced questions

Conceptual Questions I . Con�ider 11,1.'0 \cparate object' of unequal temperature.

internal energy drops. In fact, the process has a name; spontaneous emis�ion. Can an isolated electron emit a photon? Why or why not? 7. We analyze the photoelectric effect using photon energy

alone. Why isn't the photon momentum a consideration'l (It may help to reread the discussion of momentum nnd energy in connect.ion with pair production.)

8, A ball rebounds ela�tically from the nOOf. What does this situation share with the ideas of momentum conser_ vation discussed in connection with pair production?

What would you do with them and what would have 10 happen thereafter III enable them to reach the same

9. A low-intensity beam of light is sent toward a narrow

common temperature') Use Ihis idea to e'!.plain why the

single slit. On the far side, individual flashes are seen

ek.:tromagnetic radiation endo�ed in a cavilY has a lemperdture that i� the \3me as th;jt ofthc c3\ity walls.

sporadically at detectors over a broad area that is orders of magnitude wider than the slit width. What aspects of the experiment sugge"t a wave nature for light. and what

2. 'The ch� on a pieu': of ITk!'tal can be '"watched" fairly eas­

ily by connecting it to an elcc(I'"(N.:ope, a device with thin

leaves that repel \I.'hen a net chaq:e i� present. You place a large excess Ik!'gative char}!e on a piece of metal, then sepa­ rarely shine light MXlI't;e5 oftv.0 pure but different colof'j at il. The fiJ'\t <,Qun;e i� eJltremely bright. but the electroscope

Wlowli no change in the 1\\."1 charge. "The second source s i

feeble, bol the charge disappear'>. Appealing to � few 92

6.

it? (Imagine the two objects are hard spheres.) (c) Is it reasonuble to suppose that we could know this'l Explain. An isolated atom can emit a photon. and the atom's

aspects suggest a panicle nature? 10. A cohere", beam of light slrikes a single slit and pro­ duces a spread-out diffraction pattern beyond. The number of photons detected per unit time al a detector in the very center of the paltern is X. Now two more slits are opened nearby, the same width as the original, equally spaced on either side of it, and equally well

from I� blackbody, t: is not lhe c:onec1 speed. For ,..u.. ation moving uniformly in all directionl, the averap nm'ponl'''' of ,'elociIY in a Biven diftlctioo is

llluminah:d b)' the beam_ How many ph-Nons will he detected per unit time at the center d�tector now',' Why?

Exercises

Section 3.2

Section 3.1

11. For small ;:, ('- i� approximately I + ;:. (al U\e this to

show thai Planc\"\ spe<:tral energy den�ity (3-1) agree� .... lth the re,ult of cla�sical wave theory in the limit of

�mall frequencies. (h) Show that. where..... the c1:b)icul fomlUla divcrge� al high frequencies-lile so-called ultraviolet cata�t n)phe of this theory-Planck's fonnulu approaches O. 12. Al wh!!1 wuvekngth does the human body emil the max· imum electromagnctic rndintion? U�e Wien's law from

13.

!e.

Exercise 14 and al;sume a skin lempermure of 70°F, EqUation (3-1) e)lpres�e� Planck's spe<:trnl energy density as an energy per range dJof frequencies. Quite often, it is more convenient to express it as an energy per range (IA of wavelengths. By difli!rentiatingJ = ciA, we find that df = -clA2 dA. Ignoring the minus sign (we are intere.�ted only in relating the magnitudes ofthe r.lJlges dfand dA), show that. in lenm of wavelength. Planck's fonnula is

dU

-

dA

�

SJTVhc

16. In the photocleclric effect. photoeltcuons begin IeavinJ the surfal� al e�<;('ntially W inslIlnt lhaI light is intro­ du(cd_ If light bc'havcd a., a diffuse wave and an ekIcaua at the �urface of a material nluld be auumed localized to n1ughly t� !ma of lU1 atom, it would take far Ion&a'. E,timate the time lag, aM,uming a work function of 4 eV, an atomic radius of approximately 0, I nm, and a reason­ ahle light imensily llfO,Oi W/m2. 17. Lighl of �OO nm wavelength strikes a melal plare. and pholoclectron� are produ(ed moving as fast as O.OO2t:. (a) What is Ihe work function of the metal1 (b) What is Ihe th��hold wavelength for this metal? 18. Whllt is the stopping potential when 250 nm light

19. 20.

21.

ehr/H8T - I A5

14. According to Wien's la\\. the wavelength Anw. of

maximum thennal emission of electromagnetic energy from a body of tempernture Tobeys

22.

AnlnT = 2.S9S X 10-l m · K Show thaI thi, law follows from the spectral energy den­ sity {lUldA obtained in Exerci se 13, Obtain an expression that. when �olved. would yield the wavelength at which this funclion is maximum. The transcendemal equation cannot be solved exactly, so it i� enough to show that A 2.898 X ! 0 1 m . KIT solves it to a reasonable degree of precision, 15. The electromagnetic intensity of all wavelengths ther­ mally radiated by a body of temperature T is given by

23.

24.

'='

f = ar where a = 5.67

X

10- 8 W/m2 . K4

This is the Stefan·Boltzmann law. To derive it. show lilat the total energy of the rndimion in a volume Vat tempern­ lure Tis U = 8,"5.1.1 �/15h3c3. by imegrating Planck's spectral energy density over all frequencies. Note that

( "JO � dx -_ ,"4 15

Jo

e-'

-

I

Imen�ity. or power per unit area. is then the product of energy per unit volume and dislance per unit time. But becau�e inten�ity is a flow in a given direction away

25.

strike, a zinc plate? What wavelength of lighl is necesl;lU')' to produce photo­ electron� ofspeed 2 X lot' mls w ith a magnesium targeI? What is the wa�'elength of a 2.0 mW IIL'>er from w hich 6 X 101� photon� emanale every second? A 940 kHI rndio station hroadca,ts 40 kW of power, How many photon!> emanate from the transmitting aMenna every se(ond? To expose photographic film. ph()(ons of ight l dissociate \ih'er bromide (AgBr) mol«:ules. which requires an energy of 1.2 eV. What limit does this impose on the wa\'elengths that may be n:-corded by phoIographic film? Lighl of wavelength 590 nm is barely able 10 eject electrons from a metal plale, What would be the speed of the fastest electron� ejecled by light of one-third the wavelength? With light of wavelength 520 nm. photoelectrons are ejected from a metal surface with a maximum speed of 1 .78 X 10� rnIs_ (a) Whal wavelength would be needed to give a maximum speed of 4.81 x lOS mls? (b) Can you gue�s what metal it is" You are an early 20th-century experimental physicist and do not know the value of Planck's constant. By a suitable plot of the following data. and using Einstein's Irl - f/J, explanati on of the photoelecuic effect (ICE where fr i\ no/ known), determine Planck's constant. =

Wa�eteogth of U....t (om)

550 '00

0.000 .....

M

a...., I

WIvft and Particles I: EJcctr\'mlagfIetlc Radiation Behaving

l6. A socii... vapor lip emill 10 W of ligh' energy. I" � is S89 DIn. and it spt'elds in aU directions.

How IIlIDY photons pass through your pupil. diameter 4

nun, in I s if you stand 10 m from the light?

27. Using the high-precision values of h. c, and e given on the text's inside front cover. show that the product

can be expressed as 1240 e V · nm.

Soctlon

as

37. Verify that the Chapter 2 formula .lKE = -dnrc2 applies in Example 3.4. •

38. From equations (3-4) to (3-6) obtain equation (3-8).

+

tron speed /j may then be eJjminated between the

remaining equations.

• 39, Show that the angle:o. of scatter of the photon and elec_ tron in the Compton effect are related by the follOWing formula:

21. A television picture tube accelerates electrons through a potential difference of 30,000 V. Find the minimum

B

wavelength 10 be expected in X-rays produced in this

cot- =

2

tube. (Picture tube!> incorporate shielding 10 control X-ray emission.)

%9. When a beam of monoenergetic electrons is directed at a tungsten target, X-rays are produced with wavelengths no shorter than 0.062 nm. How fast are the electrons in the beam moving?

Section

Section

40.

(

I

'+meA

)

,.n 4>

3.5

A gamma-ray photon changes into a prolon-antiproton pair. Ignoring momentum conservation, what must have been the wavelength of the photon (a) i f the pair is sta­ tionary after creation, and (b) if each moves off at 0.6<­

3.4

30. A typical ionization energy-the energy needed to

remove an electroll---for the elements is 10 eV. Explain

why the energy binding the electron to its atom can be ignored in Compton !>Cattering involving an X-ray photon with wavelength about onc-tenlh of a nanometer.

31, A 0.057 nm X-ray photon "bounces off' an initially sta­ tionary electron and scatters with a wavelength of

0.061 nm. Find the directions of scatter of (a) the pho­ ton and (b) the electron,

32. A 0.065 nm X-ray source i!> directed at a sample of car­

bon, Determine the maximum speed of scattered electrons.

33. Compton used X-rays of 0.071 nm wavelength. Some of carbon's electrons are too tightly bound to be stripped away by these X-rays, which accordingly interact essentially with the atom a.� a whole. In effect.

me in equation (3-8) i� replaced by carbon's atomic

mass. Show that this explains why some X-rays of the incident wavelength were scallered at all angles.

34. An X-ray source of unknown wavelength is directed al a carbon sample. An electron is scattered with a speed

of 4.5

11

is ea.-.ie.:;! to start by ehmlRating 4> belween equation s sin2 4> = I. The elec_ (3-4) and (3-5), u.:;ing cos2 4>

he

3.3

Panid�s

X

107 m/s at an angle of 60° Determine the

wavelength of the X-ray source,

of an X-ray photon that . 35, Determine the ....avelength

can impart, at mo�t, 80 k�V of kinetic energy to a free

electron.

36. A photon scatters off of a free electron. (a) Wh.)! is the

perpendicular to the motion of the photon? (c) ASsume' that these interactions occur as the photon encounters a

lead plate and thai a lead nucleus participates in mOmen_ tum conservation. In each case, what fraction of the pho­

lon's energy must be absorbed by a lead nucleus?

4 1 . A stationary muon Ji- annihilates with a stationary antimuon Ji' (same mass. 1.88 X 1 0 28 kg. but OPPOsite charge). The two disappear, replaced by electromagnetic radiation. (a) Why is it not possible for a single photon

to result? (b) Suppose two photons result. Describe their possible direction.:; of motion and wavelengths.

42. In positron emission tomography (PET), discussed in Progress and Applications. an electron and positron annihilate. and two photons of a characteristic energy are detected (see also Exercise 41). What is this energy, and what is the corresponding wavelength? The pair can be

assumed to be essentially stationary before annihi lation.

43. As shown in Section 3.5, a lead nucleus can ensure

momentum conservation in electron-positron pair pro­

duction without affecting the energy baJance. But roughly what i� the limit on the mass of such a

"detached participant"? Assume again that it acquires all the momentum of the photon. whose wavelength is

10' 1 2 m, but the energy It "steals" is less insignificant. 0,01 % of the photon'S energy. What is the 1.21

X

mas� of this Ics<,-detached participant?

maximum pt)ssible change in wavelength? (b) Suppose a photon �cauers off of a free proton. What is the maximum possible change in wavelength now?

(e) Which more clearly demon.:;trates the particle nature

of electromagnetic radiation--colli�ion with an electron or collision with a proton?

Section 3.6

44. A beam of 500 nm lighl strikes a barrier in which there is a narrow single .. lit At the very center of a .:;creen

beyond the �ingle 'ilit, 1012 photons are detected per

r .

B

'>quare mitlimeter per o;«ond. (a) Whal is thr inrens.ity of the light at the cenler of the �reen'l (b) A second ml is now added \'ery do:;e to the fina, How many pholons

will be (!eleCled per "IU3re millimeter pet � at the center of lhc scrtt'n now'1 .5. Electromagnetic "waves" strike a single slil of 1 IU'I width. Determme the onRulorfull width (angle from first minimum on one side of the ceniCT to firsl mini­ mum on the other) in degree� of the central diffTaction maximum if the waves are (a) visible light of wave­ length 500 nm and (b) X-rays of wavelength 0.05 nm. (e) Which more clearly demonstrates a wave nature? '6. A bedrock topic in quantum mechanics is the uncer­ tainty principle. It is discussed mostly for massive objectS in Chapler 4, but the idea also applies to light; Increasing certainty in knowledge of photon poSition implies increasing uncertainty in knowledge of its momentum, and vice versa. A single-slit pattern that is developed (like the double-slit pattern of Section 3.6) onc photOn at a time provides a good example. Depicted in the accompanying figure. the patlern shows that pho­ tons emerging from a narrow slit are spread allover. a photon's x-component of momentum can be any value over a broad range and is thus uncertain. On the other hand. the x·coordinate ofposition of an emerging photon covers a fairly small range, for w is small. Using the single-slit diffrnction fonnula nA = w sin 8, show that the range of likely values of P:.:' which is roughly P sin 8. is inversely proportional to the range w of likely position values. Thus, an inherent wave nature implies that the precisions with which the particle properties of position and momentum can be known are inversely proportional.

Comprehensive Exercises n. A. phocoo has die lime ID('!IIIeID I lm . .. ..... ..,.

ina at Ir/J mls. (a) Determiae me pboIoa'a � (b) WhaI is the nbo oflhe ItiDetic � ol me rwo?

(NOk.' A. phocon il all kinotk 0ftCtIY.)

48. A. photon and an objed of maJI III haYe tho MmCI

momentum p.

(al A!>suminS that the massive object is mavio. slowly. so that nonrelativistic fonnulas are valid. find in terms of m. p. and c the ratio of the mauiw object's kinetic energy to the photon's kinetic energy. and arsue that it is small. (b) Find the same ratio found in part (a). but usins rel­ ativistically correct formulas for the massive object. (Notr: £2 "" p1c2 + m2c" may be helpfut .) (c) Show thaI the low·speed limit of the ratio of part (bl agrees with part (a) and thai the high-speed limit is I . (d) Show that at \'f'n' ruSh speed. the kinetic eneqy of a massive objecl approaches pc.

49. Radiant energy from the Sun arrives at Eanh with an intensity of 1.5 kW/m2. Making the rough approxima­

lion thaI all photons are absorbed. find (a) the radiatiOll pressure and (b) the total fon;:e experienced by Eanh due to this " solar wind." SO. A flashlight beam produce� 2.5 W of electromagnetic radiation in a narrow beam Although the light it pr0duces is white (all visible wavelengths). make the sim­ plifying assumption that the wavelength is 550 nm. the middle of the visible spectrum. (a, How many pboIons per second emanate from the flashlight? (b) What fon;:e would the beam exert on a ''perfect'' mirror (i.e one that reflects all light completely)'? 51. The average intensity of an electromagnetic wave is whcre Eo is the amplitude of the electric-field ponion of the wavc. Find a general expressioD for die .

.•

L"" posiuve P, p..

=0

�etftij.

L""

negative P,

Single·slit n i tensity

Ught wave

'"

photon flux) (measured in photonsls · mI) in ECmlS of Eo and wavelength A. 52, Show that the laws of momentum and energy conser-. varion forbid the complete absorption of a pbocon by • free electron. (Notr: This is not the photoeIcctric effect. In the photoelectric effect. the electron is not fn:e; the I!II!DI participates in momentum and encrgy CODgeI'YItion.) 53. An clectron moving to the left at O.Be collides 1riIb ... incoming photon movin. to the right. Aftr:r die coUi­ sion. the clcctron is mavinS to the riP at 0.& aDd aD outgoing photon moves to the left. What .. die wne-­ length of thc incoming pboloo? 54. An object moving to the rigbl at O.Be illIb'Ddt bad-cJ8 by a photon of wavelengd:l ,\ moving to tbe aft. The

96

a..peer 3

Wa\ICI and Particles I: Eleccromagnetic Radiation Behaving a icles .. Part

objcct absorbs lhc photon (Le., the photon disap�) and is af'lcrward moving to the right at o.&-. (a) Deter· mine the ratio of the object"s ma.o;s after the collision to

its mass before the coUision. (No": The object s i not a "fundamental particle," and il� ma.o;s is therefore subject 10 change.) (b) Does k.inetic energy inl,.Tease or decrease'!

55, Photons from space are bombarding your laboratory

and smashing massive objects to pieces! Your detectors indicate that two fragments. each of mass mo' depart

(;;uch a collision moving at 0.& at 60" to the photon's original direnion of motion. In terms of mo' what are the energy of the cosmic-ray photon and the mass M of

the particle being struck (assumed initially stationaryl'.'

calculus. For inslance, wa\'e." on a string obey the "wave equation," a PattiII differemia' equation in posillon and rime, and Ilght obeys MaxweWs «:qua. lions. similarly inml\'ing (:alculu.. wilh bolh position and lime variables. 1be dislinclion belween particle and \\'3VC nalure..; also govern� ":hal questions we ask. For particles. it is oflen. "Where is il going? When will II gel there?' For wave.". \\Ie ask. "Whal i..; il\ amplitude? Whm is ils wavelength? How '�PJead our is it? Where is il zero"''' We consider the.�e questions �oon, and afterv.'ard we introduce lhe SchrOdinger equarion. the equarion obeyed by marrer waves. Firs!' however, is the Big Quesrion: Just what j� a mailer wave. and how do we know il eveQ exists? Nothing illustrates the poinr better than the doubJe·slir experiment. Figure •. 1 'nlensily palfems when an electron beam �Irikes various �lils,

4. 1 A Double-Slit Experiment

a'

Imagine a beam of monoenergetic electronsl siriking a barrier with a slil beyond which is a screen rhal regislers each electron's arrival by prodUcing Ihf{'ltlgh 1I \\oIJ... llil small flash. When the slil is "wide." as shown ar the lOp of Figure 4./. the beam passes straight through and produces--clectron by electron-a stripe on the screen essentially the same width as rhe slit. Bul with a narrow slit, we find electrons registering sporadically OI'er fhe entire screen. Although this spread_ ing alone is hard to reconcile with the notion of electrons as strictly panicle\ if we add a second slit, the conclusion is inescapable. Suppose, then, that we add a second narrow slit. Again, with either Slit "". ",",I, open alone. electrons are detected sooner or later at all points on the screen. 11.[[1"\\ shl. But when both are open together, we see certain places, where electrons had tk'('llon, OlITI\C bee" detected with either slit open separately. where electrons are now never "'t ry one JII Uh, Ihl.' Krrtn, Thty detected. Opening a second " door" decreases to zero the number of electrons uon'j J \ " !U .lilly arriving per unit time at specific. regularly spaced locations on the Screen f'Illllls even at such low intenSity that rhey must pass through one at a time! This is impossible to explain if electrons are simply particles passing through one slit Single narrow Sltl 0 .... hUI whell ' or the other. A particle passing through one slit would not suddenly have rea­ ,hIs JI"l: npen. son to avoid specific locations on the screen just because another slit had been f\ thefl' lin' <:...11:1111 �j�linl' ut which opened elsewhere. On the contrary, this is destructive interference. Because ;i llot'i.:Clwn tl'er interference requires multiple coherent waves, each electron must be behaving MTlI'es. beh as a wave passing through both slits at once. Double sli t d':drvn WJ\'t h lbtTUl'lm:ly Figure 4.2 shows how the electron flashes accumulate with both slits in rftring ""itll open, and Figure 4.3 shows the final pattern in an actual electron double·slit itSt'lj. experiment. Figure 4.2 should look familiar-it is the double�slit photon_ delection pattern of Figure 3.14. The point is that both electromagnetic radia­ tion and massive objects exhibil the same kind of wave·particle duality. Associated with the particle of light (photon) is a wave of oscillating electro_ magnetic field. We must now accept Ihat there is ruso a wave associated with a massive particle. So what is analogous to light's electromagnetic fields? In a matter wave, what is oscillating? lThe electron is our prefcmd mM-'l� objr:cl of To identify at least one propeny of this wave. we return to the same two nlJdy b«au..e of 'IS relal,vely �mall mau. As we observations we made about light: ( 1 ) Although it is apparently impossible to will $OOf'I _, Lhu mU� ilS wa�elengLh not too small,.so its wave nature ,5 morc euil)' I'C\'caltd say where the next electron will be detected, the probability that it will be 1" .-,lfi'lli pil'�lllr

, ,

� j

41 It. Doubte-SUI ....L . . ..

I--

"

Fig...,. •.2 fJeclnms pn"JdIXina ' dou�Ie, �lil inlC'rf(rencC' pille-Tn-- (IhC pac1t\.'le al a

Itnle,

--'. �ftl't 1_ (l_ "' _ _ I� �

_ _ _ _

Ancr loon dct�·t...J

found in a given region should be proportional to the den-,ily of SpOl\ there.

(2) If we concede that coherent waves of equal

amplitude emerge from the two

slits, we would expect an mlerference patlern. In particular, we would C)l.pect the amplitude at the pattern's center, where the interference IS construclivc, to

be

twice what it would be if only one slil were open, and the "quare of the

amplitude. proportional to intensity. should be four times as large. And what

does the experiment show? It shows a denSity of spots that is four time ... larger at the center with both open than with either alone. In fact. for the entire pat· tern, there is a proportionality between the density of spots and the inlensity­ the square of the wave-arising from the standard analysis of the double slit. Combining these observations-lhat the probability of finding the particle and the square of the wave's amplitude are proportional to the same thlllg (the density of spots)-we reach the same conclusion as for light:

When a phenomenon is detected as panicles, we cannot predict wilh cer· tainty where a given particle will

be found. The most we can determine

is a probability of finding it in a given region, which is proportional

(

)

to the �uare of the amplitude of the a...sociated wa\'e in that region. probability of finding

particle in a region

DC

amPlitude of wave in that region

'

Although electromagnetic radiation and massive object-; share this funda· mental wave·particle relationship. matter waves differ in an important way: They

FlgUN 4.3 A\.·tuIII double-slit panem

produ!;t=d by t=1t=!;tron!"..

100

eM.,. " Waves and Patticlell ll: Malter Behaving b Waves

cannot be directly detected, Elt%;tric and magnetic fIelds can be isolated IIftd caused to exert forces on objecL<;-we can directly detect them. But we ha\IeQ ' found any ana1ogou� way to directly detect matter waves. No one has ever "seen! one. So how do we answer the question. What is oscillating? Apparently. \Vt have only a single candidate-prohabilirr is oscilluting.2

Matter Wave Interference: Evidence The double slit is conceptually the simplest experiment for verifying mane r wave interference. but unfortunately it isn't easy to do, The first evidence that matter has a wave nature was obtained by Clinton J. Davisson and Lester H Germer in 1927. Investigating properties of metal surfaces by observing how nickel crystal scatters a beam of electrons. Davisson and Genner were SUr_ prised to find that the electrons seemed to scatter preferentially at only cenaill discrete angles, Particles should not do this, Microscopically. a crystal is an arrangement of regularly spaced atoms. If a wave is incident, each atom retlects the wave in all directions. In essence each becomes a point source of waves, These sources produce an interferenc� pattern. just as do the multiple slits of a diffraction grating. with sharp interfer_ ence maxima separated by broad regions of low intensity. In the Davissoll_ Germer experiment, whose apparatus is shown in Figure 4.4, the experimental detection rale versus angle agrees perfectly with a theoretical prediction (see Exercise 23) based on the assumption that each electron behaves as a diffuse wave reflecting from many atoms and intetj"ering with itself The key to the experiment's success in revealing the electron's waVe nature is that the relevant dimension of the apparatus. the atomic spacing. IS very small. A crystal is thus an excellent testbed to verify a wave nature, But nowadays. it is the other way around-the electron's wave nature, takell for granted. is exploited to learn about the crystal. In essence, the diff_ raction pauern maps out the microscopic geometry. Figure 4.5 is a good

�

Figure 4,4 Dav isson and Germer's original eleclTon diffraction apparatus, showing the mechanism for varying a sample's angle,

�A common lnl'>Cooccption I' tlul lht dCC:lron's II ",mellow

/OI/W .. OiKlllal;ng, ptmaP'\ thal bIb or

JiUk �k IlDd rorth. n.c ",avc l'o',1hc parotic MUI doc,n', O§CIlLalc In a m31�r wa"c 1lD)' more

than photO!" o.cillate In an cle<:tromagnctlc wa'e. II rna)' be helpful to "ltW both phcnomcllil

cOD."'lenlly ti esK'nllally ",nc, of o..:illilhng probabilu)', add�"'lng other po5\1ble WU, onl)' u the need an"",

e,ample. Produced by electrons diffructing from a sing'� grain of aluminum· manganco;e alloy. it reveal!i. atom.. arranged in a fivC'-s.ilJelJ geometry. \ACIU­ all)'. ..uch a geometry came a.o; a surprise ....hen first seen in the 198(h. and study of "quasicry!i.tah·· wi.th such symmetries ha., been quite active ever .!.ince.)

The Bragg Law In a crystal. matter waves often penetrate to many atomic planes lJeeper than the surface. and the mO!i.t commonly u!i.ed quantitative relationshi.p for con­ strUctive interference, which we now obtai.n, thur> requires a hit mnre work than for a simple grating. Suppose, as depicted in Figure 4.6. a beam is directed 'Ctor i.. pO!i.itiuned .. an angle 8 with respect to a surface atomic plane. and the deh to receive waves reOecting at the same angle with the plane. An atom in the

at

top plane reflects a small ponion of the wave, ray I , which scatters in aU direc­ tions. But much of the wave penetrates deeper. so an atom in the second atomic plane also scatters a portion in all directions. As shown in the ligure. fay 2 has 2d sin 9 farther to travel than ray 1 10 reach the detector. where d is the spacing between atomic planes. Ray 3 travels the same distance farther

than ray 2, and so on with each deeper atomic plane. With the incident beam and detector at equal angles, the waves scattering from any atoms in the same plane­ for instance. rays I and 4--always have the same distance to travel ffom source to detector. So we view interference only between atomic planes at different

Figure 4.5 Inlerference pauem of electron waves diffracted by aluminum manganeloe

alloy.

•

•

•

•

• •

•

•

•

•

•

• •

•

•

• •

•

•

•

• •

• •

•

•

•

•

•

• • •

•

•

•

•

•

•

•

• •

•

•

•

•

•

•

•

•

•

•

•

•

•

•

102

Ch.ptv .

Wave!l and PanicIH II; Malter Behaving a.� Woves

Figure 4.6 Diffraction of a beam from multiple alomic planes.

2 4

3

Atomic plana:

depths in the crystaJ-very nice! Constructive interference occurs at where

2d sin 8 = rnA

angles

(4-1)

This is known as the Bragg law, for father-son team W. H. Bragg and W. L. Bragg. who won the 1 9 1 5 Nobel Prize for their diffraction work. Their source wasn't electrons, however-if was X-rays. BUI the same equa­ tion applies, because a wave is a wave. We noted in Section 3.4 that X-rays may behave as panicles where longer-wavelength visible light would behave as waves. The spacing d of atomic planes in a crystal is so small that even X-rays behave as waves. Anything will be wavelike in small enough dimensions, and diffraction patterns have been produced by beams of neutrons and even whole atoms. Of course, to exploit interference condition (4- 1 ) in the case of matter waves-to determine an actual plane spacing d-we would need the wavelength. So let us now lurn to the quantitative properties of matter waves.

4.2

Properties of Matter Waves

What properties characterize a wave? It should have a wavelength, a fre­ quency, and a speed. It should also have an amplitude that varies with position and time. The generic term for the function giving the amplitude is wave function. Because what actually oscillates depends on the kind of wave, we

use different symbols for the wave functions of different kinds of waves. For a

4.1 _ .. _ _

transverse wave on a string. we often usc the symbol ),(x. t). The string's trans­ verse displacement y varice;, as a function of the position x along the string and

time I. For an electromagnetic plane wave moving along the x-axis. we have twO wave functions. E(x, t) and 8(x. f), which describe how the oscillating

electric and magnetic fields vary with position and time. For a matter wave. the symbol we choose for the wave function is 'IT(.t. t). Strictly speaking, it should be referred to a.'\ the probability amplitude. for it is the amplitude of

the wave that (when squared) tells us probability. In practice. however, we usually just call it the wave function. In the next section, we will encounter the

"wave equation" that this wave function must obey. But first tet us study some

properties and behaviors that don't require us to have an explicit formula for

the wave function.

Wavelength In our discussions so far, wavelength seems to play the pivotal role. What is it'? tn

1924,

Louis de Broglie submitted the foHowing hypothesis: The wave­

length of the matter wave associated with a massive object depends on its momentum p and is given by

h

,\ = ­ p This relationship has been confirmed beyond any doubt, even for relativistic speeds, by experiments such as crystal diffraction, in which the momentum of the electrons in a beam is known, and analysis of the pattern establishes the wavelength. The contribution won for de Broglie the

1929 Nobel Prize.

and in recognition, we often refer to the wavelength of a matter wave as the de Broglie wavelength. Of primary importance is the facl that equation

(4-2)

This rela­ between wavelength and momentum is universal-true for an

is identical to equation (3-7) for electromagnetic radiation.

tionship

phenomena.

EXAM P L E 4 . 1 If moving at

900 mls. what would be the wavelength of (a)

an electron and (b) a

25,000 kg airplane? (c) Which is more likely to exhibit a wave nature? (a)

(b)

:":";':

10-34

6.63 J ."-s ::; =-X '- -"oo!O JI k "� /� s) g) ( 9 m

'\",,"," = -:(9 '. 11 X

=

8.1

X

10-1 m

6.63 X 10-34 J s Aairp1anc: = -;;; :C- --;- = 2.9 X 10-41 m ::-'-; 7.c00 ; �;-'; � kg ) ( 9 ( 25.000 m/s) •

, ..

(c) An apparatu$ with I reievanl dimension on the micrometer scale might

coax me electron to show ils wive side, AI;tuaily. Ihis is very slow as electron speeds go, so considerably smaller dimensions are usualJ}' ocedcd. lbe airplane's wavelength. howe'o'er, is a good 25 orders of magRi. lude ,maHer than lhe atomic nudeu!i (-10 u rn). Although "compo�ile" i objects much bgg er than dectrons-whole atvms, for inslance--hav(' ben. .. simple wa'o-es with a \\a\ e lcngth obeying shown to behave in ('xpcrimtnlS a equation (4·2). where p b the objC\.-l's center·of-ma!>\ momentum. in no conceivable e xperiment would something so big as an airplane e,'er beha\'c as a wave,

In ordinary �ituations, the wavelengths of matter wave!. are short enough to ensure particlelike behavior. because Planck's conslant is so small. How. ever. as an object's momentum apprO<.Il:hes zero. wouldn'l iLS wavelength be. come arbitrarily large? We might draw the unsellling conclusion thai any stationary object should behave a.. a wave. We shall face this predicamem in Example 4.5, but the upshot is that it I, hard to be sure thai something is indeed stationary, With our quantitative gra...p of wavelength. let us return to the Simplest eltample of interference.

EXAMPLE 4 . 2

-

Suppme a beam of elcclrom moving ilt 3 x I on rnl� �trikes a barrier that ha� two narrow sJils �eparated by 0.020 �un. beyond which are electron detectors.) Al the center detector, directly in the path the beam would follow if unobstructed. 100 elec. lron� per second are detected. As the delector angle varies, the number per unit time varies in a Iypical double·slit pattern between the maximum of 100 S-'1 and the min­ imum of 0 The firsl minimum occurs at delector X, an angle Ox from the center, (a) Find Ox (b) How many electrons would be detected per second al the center detector if one of the slits were blocked? (c) How many would be delecled per sec­ ond al the center detector and at detector X if one ,lit were narrowed so that il a/one would give a count rute 36% of its originnl value" (a)

'A. n<:Jkd "uha. i, I' turon.! hI do Ihc= ,,1n;U\ln double �lu, In foct. tile fiN SUCtn5, �hown in Figure oil, didn'! comc unlll the I %l:k---und c,,,n llYn. U rc<jUIt'N . few Imh. llIc: prohltm I. thai In.: u ll ..lk'l .dll(,�hlc .hl �parohon, are �tilJ mu�h l;ug"r thCUl tile ..Icaron ...a\'dcng!h. nen if the: clCUll'" m",'", roth ... .1""Iy. .. __«umed In Eumrle 4,!

At deteclOr X, lhe firsl point of deslmr.:live interference. the wave from one slit has 1 A farther to lravel than lhe wave from the other. From physical optics, we know that the difference in diSlances !raveled by the two waves in a double-slit e)(perimenl is d sin 8, where d is the slit ,epar,lIion. Thus. I . d sm8X = 2 A

Ar.:cording to de Broglie's hypothesis, equation (4-2), A �

" P

2.43 X

10-10 m

... ., - - u _ Rd�rtlOg.

I

2 \2.43

\

\b)

)(

10 1\) ml

W1m tkllh ..Ii" �Ir-:n. the o.kl('\.·lion t-olte 110 Ux.l Yo i.l\'('

0.3$

I, nns IS proportional 'll the

parth:k d.1.·I"tl�ln pn1bat'liIiIY and.. thus, 1\1 the lC/ItdFl' of

IPUltta WiII'c·. the tlltal

.. fI.x.

,h, cI'"IJiihMJco (11 1M

am\ ang ff\lm buth slih.

(Ntllt" Tll ;l\'\lid. di..tr"..:tilln...

Yolo'

lImit the unih on '4'. As d.iM:usst\l in E,ercis('

;!Q, indud.ing them ....uuht . merel) antnldu.;e a pnlPl1nillnulity ..:unslant that

..... ould �·al\..:d

111

the cnd.

l The waw:o. !"n'lll the slits ad.d. C\\ulllly at thi!!. Pl,int uf

COlhtrucU\e intl.'rfercnce, the total

,,\1

the u : nph\u.,k of either .....a\'e Illnne must be hulf

The dectrun detection rate would. ",ithout

econd 11 ..

tIC �S .. I at the center d.1.·t�tl)r. Nllie that 25 electrons �T l.ecuml w\)U1d uni\"e

..htJw;!\'e to interl·ere.

ill all detcl:tof"., With two "lih, \Wlce Ill. many

.I/lliuld t"Ie delL"Cted each

"econJ, but only fill (II't'ruge. At (Xlin... 01 constructh'e int('flercnce. HMol

are detected per ..econd. and at p
tel It only the

".ngle narrowed "lit \\cre open. all detcl:10f". would rcgi..tcT a dCle\:­

tion r:\te 0.36 lime!»

25 !»

'. or 9 s

':

The amplitude of the wave follows:

Thi" i!> 60% of the original amplilUde. Sensibly, an amplitude only 60% of the original implies a ,\'qu(JT"e of the amplilUde of only 36%.

With both slit!» open, we have two waves of different amplitude!», one pro­ portional to

5 (original) and one to 3 (narrowed). At the center detector, where

they o(ld eon!>lructively. the lotal amplitude is proportional to

At detector X, where the waves are

5 + 3:

180" out of phase, the interference would

no longer be totally destructive-their amplitudes aren't equal. Still. they

subtract, so the total amplitude would be proponional lo 5 - 3:

1 06 a..e- ..

..... -.I r.ticlel U: Mader Behavinc u WaVd

....... ....7 EJeccron detection rille WI'IUI aqIe

fot equal-width slits.,

uaequaI-widlh slill. aDd one slit BIec1roIsI per second (s I )

I

lbe averllge j.; �(64 s- 1 + 4 s- l ) = 34 s- I, which is the sum of the 9 s-t .....

2S ,- I «peeled from each slil alone. BUl lo find the probability, or detectioa rille, at a location, we do not add the probablliti�l from each slit. Probabililiet are always positive, so they can'l cancel. Rather, we add the waWl, which tnI)' add constructively or deliltUCllvely, 10 find the total wave, and then square tha 10 find the probability.

Figure

4.7

<;hows how the electron detection rate in Example

4.2 Varies

with angle for the equal-width, one-slit. and unequal-width cases. Note that. except for part (a), Example 4.2 could have involved light. We would simply replace the term

()<' Center deleClor

0.3.5" Angle Detector

X

nuJlter WQI'e

with "electric field" and

electron detection

rat�

or probability with " light intensity. We have analyzed electron behavior via standard wave theory. "

Having gained some quantitative feel for matter waves, let us take a look

at producing and using a matter wave beam.

EXAMPLE

4.3

To put the wave nature of electrons 10 use, an accelerating potentiaJ i s often the sian.

Obtai n a formula for the potenlial difference V required to give a panicle of mass In and charge q a wavelength of A. Assume thai Ihe potential difference is insufficient to accelerale the particle 10 relativistic speeds.

The accelerating potential gives the electron kinetic energy. and velocity is related to Ihe wavelength via the de Broglie formula.

qV =

Eliminating

\I

2" mv2 1

h

p v= = -

rn

rnA

between the two gives

h' V = -::-...,., -, 2mqA'

We see that a shorter wavelength requires a higher accelerating potentiaJ. As noted in Section

4.1,

the electron's small wavelength is now routinely ex.ploited. This

resull is an important element.

REAL-WORLD E X A M P L E U S E F U L WAV E L E N G T H S A N D A C C E L E RA T I N G P O T E N T I A L The transmission electron microscope

(TEM) i s a workhorse in biologicaJ sciences

and several other fields. The TEM replaces the opticaJ microscope's illuminating light beam with a beam of electrons accelerated through a potentia] difference, while mag­ neLic "lenses" take the place of glass ones, as depicted schematically in Figure

4.8.

After passing through the sample and a series of "lenses," the electron beam produces

..a _ .. __

_

'''IoaN ''1 In " Uocuoa � lmaat' on • \l."t'een. A" � \eam In upth:S. .. INCf06o.:opt'S resolutioo Is hnuled by .lee..... .... \IabI _ .... dtffncton; t feanares unalkr than � wavelensth art' blUl'Ted. lipl has wawleq1hs . _ measured In hundred) of l\Anometcn, 1M TEM'ljo ele\:tron" have mu..:h slkl(\Cr w.� ...._ lIckb ...._ l�lh. dlffrao.:t Ie". and thus reveal mu..:h finer detail in ktw.ftUIY electron dIIftKtkIo (l.EEDl. another applk..'alk'" n:lyma; 00 an accc:�rattng pcunua1, � far.:l that e�trons dtt'ft-., as wa� IS, ..... the I\AIIM � the whole POUlt Lik.e the' OavIS5On-Gcnner e�pm�n1. LF.EO �Oa:ll an ele\.-Uun beam from • crr-ta1. By US1Rl "'0V0" 1k."Ct'leranng poc.::ntials.. el(\;tr\"''' an t£ED d.,,,,', pene­ nte far. and the !"eSulbng diffract,on peak.", �aI tht geometru:al !>tnM..'1Uft' \�f the aklm5 ....,... 01 on the surfik."e. Let us taU • lool at tht a"\."C'kratina p:ltential, in lht� apph�·.rims.

an

....pptvi"9 the f'hy"lclo


should have a wavelength comparable to the separation betwt'C.'n the " "Ii,," ·the

aloms that scalier the �am. A typical atomic spacing in a �l)'..tlll l" 0.2 I\m. Appn".­

Imulely what potential difference is appropriate? (b) The accelerating f'ottntial an a

particular TEM is SO kY, If this were the only factor governing l'C'\olution, h\IW ..mall 8 detail could be seen?

(8) Using the relationship derived in Example 4.3. we have

V=

(6.63 X 10- 1 4 J · s)2 " ' ;-" � 2:: ( 9" .I:-:I-: "" X I: '" 0-:: :1 .6 :-"'-' '= C" )" (0" .2X " X I" )7: 0"" kg-;( -:

Accelerating voltages in LEED are typically good.

20 -200 Y. ..o our e�timate is 4uite

(b) By the same formula. the accelerating potential determine!> the wavelength

SO

X

10' V =

(6.63 X 10- " ) · ,)' = :-:-: -" ==-;-;,.--,.-:- -"-:=-::2(9.1 1 X 10·31kg)(1.6 X 10 " kg)A'

�A

0055 nm

= 0.

We conclude that such fast electrons. much less wavelike than in the LEED application, could, in principle. resolve details of about O.Ot nm. As it turns out. aberrations in the magnetic lenses limit even the best TEMs. which use nearly

to times this accelerating potential, to resolutions no finer than about 0.1 nm. Still. this is far bener than the approx.imate

Frequency

100 nm limit of optical microscopes.

Interference patterns give clear evidence for the wavelength of matter waves. The evidence for the frequency isn't so direct, but again, the relationship is the same as for electromagnetic radiation:

f=

E -

h

,..._ _ _ 1.....

'I is often

oflhe

waft

more

convenient (0 C':tprr�.. equation,

(4-2)

and (4-3)

numlln' and aapIar I'requro('l. defined a.. follows:

2"

in Inms

2"

k --

w B T

Wa\"C' number

Angular frequency

A

N(lte mal wave number .\ a "spallal frequem.:y." JU'i1 as angular frequency Q} i.. iR\!e�Jy proportional to the temporal period T. k is Inversely proponional lo the !ipaliaJ period A. Another very convenienf definition is Ii •

h

---- =-

2"

1.055 X 10-.14 J . �_

With the .. e definition.., we express the fundamental wave-particle relationships Q\ II

p

E

A

-

lif

III..

(4-4)

hw

(4-5)

=

Velocity We have left maller wave velocity for las I because we simply won', use It much. bUI if is worthwhile 10 sec why_ The famous I' = the

wow>

speed. Using equations (4-4) and (4-5), i l is I'WIlVC

=

fA

E ll

E

P

P

fA does correctly give

= -- = -

II

However. this may or may nol be the speed of the particle. In Section

2.7, we

suw that massless particles, slich as photons. move at c, and their particle prop­

erties E and p are related by E

=

pc. The above relationship then confirms that

electromagnetic wCII'es also move at c. Maller waves would move al

c if E = pc

aho held for massive partIcles-but it doesn'l. They would at least move at the particle speed if E were equaJ to

31 di.�cusses the point further.)

PI'particle' but this too is not the case. (Exercise

Wave and particle velocities. known respectively velocities. formula

r

are

as

phase and group

treated in detail in Chapter 6. The main point here is that the

= fA is of rather limited use for massive particles, because

v

is neither the speed of the particle nor the speed of light. The usual relation­ ships between strictly particle properties (p and equations (4-4) and not

hcl)'

nor

In'panicl/A .

(4-5)

are

=

ml" KE

= � m\.2, etc.) are fine,

universal. But for massive particles, E is

and p is neither

II/Ie

nor

II/Ivpartide'

(Exercise 32

focuses on the correct way to relate energy to wavelength and momentum

(0 frequency.)

4.3 The Free-Particle Schrodinger Equation

Ill!lcrmme the .... a..c fun(ti�ln '4',\. I) �lf a matter "'au" In �MtC ""n..e, all t)pe .. III \\;1H· .... arc the 'ame. h,[ eiU:h, thc� I.. an und«l)inS w.�

H(M 1.1(1

\.H'

equation, 01 \\.hich the VlII\e rundiDn hllln', !;"llnfu-.e the'>\: ten1\!>!) must he II at 1\\.0 familiar "::a'c.. ..oluUlln Let U" Il""-'�

Waves on a String

�

,-

iI�Y(\. t) \1,2

iI�_'{\. I) ih�

where \" i .. the wuve ..peed. The wave function is the ..olutilln \(t, I), which gi\\:" the ..tring', transverse amplitude as 1\ funClilln of pm.ilion .\ntl time. All

Fh4ll.lre 4.9 A "'-Iln" di"t"rh"n�'e on _ 'Inn�.

w;.l\e equation" ultimutely rcst on fundamcnlal laws. [loll 'hi-. one come, (aner mil. A OiIS;C !iinu..oidal solution of Ihi.. <.,Ol\1e devcr adaptation..) from Filet wave equation. illustrated in Figure 4.9, i, the wave function

vex. t) == A

..in(b

WI)

where

W "

Electromagnetic Waves The fundamental law� governing clectromagnetic phenomena are Ma'(wdl', equation�. In vacuum, where charge� and currents are ab'cnt, thcy arc

fE'dA 0 fE'dl _..'l... J B'dA

(4-6.)

=

=

(

a,

4-6c)

fB'dA 0 f B'dl -- J E·dA c

I a = -

c'- a,

-6b)

(4

(4·6<1)

Although we omit the details, these. like Fnet = ma, can be rearranged

into wave equations-one for E, one for B-involving partial derivatives in space and time. have two parts, case is a

E B. A

primary difference is that electromagnetic waves by nowre

and

plane wave.

The basic sinusoidal solution/wave function in this

By definition, a plane wave moves in one direction and

has a constant amplitude-it doesn't spread out. Figure

4.10 depicts a plane

wave moving in the x direction, and the solutions of Maxwell's equations that describe it are

E(x, I)

( I)

B x.

= =

A sin ( kx -

.!. A sin ( kx c

WI)Y -

wl)i

where

W = c

k

(4-1)

Figure 4.10 An e1e..:tromagnetic plane wave.

110

0 ,1

•

w.w. ..t Plnic.. n, ..... ..'VUlJ u w.vn

MMt•• Wav•• TIle wave equation obeyed by martcr waves is the Schriidinger equation. III

lhis chapter, we consider only the special case of free particles. leaving effects of c:II:temai forces to latcr chaplers. In the ab�nce of ex.lemal

forces.

the SchrOdinger equation is

-2m A'

, ',

the

a'''' (x, t) "xl

"' (xc-'-"t) , .a-,-,at

= ,fa -

(4'8)

This equation can trouble beginning students of quantum mechanics on two accounts: First. because its form is certainly not intuitively obvious. we might hope for a derivation from first principles. However. there simply is no more basic physical principle on which it is built. Its acceptance as, in essence, s "law" rests on its rendering correct predictions-probabilities of finding par. ticies. for instance. Although we can', derive it. we will soon argue that at least it has a plausible foundation.

The second concern is that the SchrOdinger equation is complex. involv. ing ;; that is.

Y-f. (See Appendix

K for a concise review of complex nUm.

bers.) It might be incorrectly concluded that a matter wave is not "real." Unfortunately, this would also seem to fit perfectly with the point we made earlier: The wave function is not directly observable. (Why should 11 be. since it isn't real anyway?) The reason for the

i

is not that matter waves

are unreal. but that they can't be represented by a single real function. Like electromagnetic waves, they. by nature, have two parts. and a complex function, carrying twice the mformation of a real one, enables us to handle them together. In analyzing an electromagnetic wave, we could treat and 8 as a single complex unit by including an

i

E

(see Exercise 35),

without making either field "unreal.'· We don't do this. because E and B have different personalities. and we like to keep them separate. But we haven't found similar reasons to keep the two parts of a matter wave separate. so we use a single complex function. It is simply a matter of convenience (see Exercise 34). It may be convenient, but can we

physically interpret

a complex wave

function? At the risk of sounding impertinent. there is no need. for the wave function itself cannot be physically detected. What ;s open to experimental scrutiny is the wave function's "square." which is a real, nonnegative quantity: lhe probabili!y densily,

Probability Density We have noted that for all phenomena. the probability of delecting the particle is proportional 10 the square of the wave's amplitUde. But what does this mean if the wave has two parts, an E and a B, or real and imaginary parts of '¥(x, t)? For electromagnetic waves, experiment verifies that the probability is propor­ tional to £2 + (cB)2, which, not coincidenlally, is proportional to the total electromagnetic intensity. Adding the squares seems to be the natural way.

f,penlllt:llt \cnlic,

the

, "line I;'"lr maller wa\"t!'''�u...w the 5I.Iuarelio \\1' th< rcal

,lJ\J I maginal") P;J.rt'

\Io hcre "'l.\,

tl

Re

Wl. \. t)

+ 1 1m "'l.\. t). Nllte that "'.

lioigmties lhe ,,'umplc:\

conJugate of 'I' and. conveniently, gi\e, the e'preliosi�"ln the uppe.\flln,,'C' \If u ,imple ... quatC. rl)rtunatl'ly. we \\ ,Il u"ually \\llrl with ,lust u lioin�k n:at wI",e

function. u piecc 01 the fun cllmpko'\ one, and "hen we SOl) " the squ.ul.' ",1' the W;l\C funclion:' it I ... i\ "imple "quare. But Vohen ....e ..Ill u,,� the lull I.:l)lUl'k,

funl.:tioll, thl" c).pres..ion IS undef';\(x"lI..I to mean the proJm::t llt the fundi\lIl Ilnll Illio comple'\ conjugate. Until no\\" we htwc been careful to "a) that the proha\:lihty ltml s\..juare \)1" the wasc's amplitude are proportiona!. SlIppo�e we h.lOl for thl.' r.\rtid� in ,I region of width

8

"urrounding a cert(lin point. For a

gi\'en

""we I\mpht\lllc.

there i" a given probability of ftoding the particle then:. But If il "

"0 "H.lU

thilt the wave'" amplitude IS essentially constant around llllll reginn, then the probability of ftoding the p.lrticle In a reg ion of width

26

mu't he twice a"

large, Thus, the square of the wave function'", amplitude must give u rmhahil­ \ty

per lI/1ir lell,r.:rh. In three dimen<;ion\ (a" we see in Chapter 7), it " pl'\)b",hil·

ity pl'r IIllit m/llme. The generiC term for probability per \Ilul length or volume is probability density. probability density

=

!'l'(.t. 012

(4-9)

The Plane Wave Let us now reveal the foundation of the free�partide Schr6dinger equation by considering ils IllOSt basic solution. A plane�wave solution is the complex

exponential

W(x, t) = Aei(b- a.rl

(4-10)

where A is a constant. To verify lhat this is a sollltion of equation (4�8), the question is

Taking the partial derivatives on both sides, we have

. fi2 - - (ik)2 Ae'{kx -WI) 2m

=

ifi.( - ;w)Aei(U - WI)

and canceling.

(4- 1 1 .)

Tbar !he tw.ctionaI dtpenderk:e (m �ition and lime cancels means thai funo. lion (4-10) obeys the Sc:hrOdinger equation for all value,> of x and I, provided only thai 4: and Cl1 are �laled a.s in condition (-'- 1 1a). It i� left as an exertist­ to YJow thal lhe more familiar A �in(kx - llJ) and A cos(k;r - Ctl)--applicablt 10 waves on a string. for e,ample-"imply don't work the same way. l'be) (A;.T - ta) is equiValent: to are not solutions. h is true by' the Euler formula that Aet A L"OSfU - llJ) ... iA ..inlh - llJJ. ..0 there is a similarity. Bur the compJe" c:\ponemiaJ ha ... (wo parts. an.J they are out of phase by one-quarter cycle. The requiremenl that (4- l la) must hold i" the key to seeing how the SchrOdinger WClI't' equation relates to the c1a..sical physics of particles. We hope rnal it isn·1 al odds wilh the fundamental wave-particle relationships, p >= hJ,; and E = hw. What happen.. if we insert them? Condition (4- l la) becomes

p

' =E 2m

(4- 1 1 b)

� ml·2, Ihis merely says that the particle\ Io.inelic energy must eq ual ih lolal energy, which is the classical truth, because � Given thal

p2/2m

t=

(mv)2j2m

free particle hJ:. no potenlial energy. Thus. rhe Schrodinger eqllalion is related

10 0 dostim/ (lccountinx ofent'rgy.

Although the wave function isn't physically detectable. Figure 4.11, which plots Ae,(b lLr) at t = 0, provides some insight i n to the mathematical

nalUre of a plane wave. The real part of '" is a cosine, the imaginary pan a

sine, and the Iwo parts are out of phase in such a way that the

magnitude is

constant-II varie.. neither in position nor in lime. The direct calcu l ati on of the probability density agrees:

or FI9U,.. ".11 A plane nlilller wave: The real anti irnuginary parh of "\t"O.' 1\1') plouctl ut I O.

/'I'(x. 1)/ = A That the probability per unit length is constant means that if we were to look for ii, a particle represe1lfed by a plane wave would be equally found anywhere.

likely to be

Obviously, a plane wave is not very realistic, but it is still quire useful. In physical optics. we speak of plane waves of lighl, because they are often a suf­ ticien tly good approximation of th e actual wave. The same i s true of matter waves. But the plane wave's importance goes even deeper, for a more general wave can be treated as an aJgebraic sum of plane waves-they are easily ana­ lyzed "building blocks." We will make use of this fact at several points later in

the text. Let us now turn to a topic where we need not know the actual ft i.� one of the mOSI profound ideas in all of physics .

'I'(x, t).

4 . 4 The Uncertainty Principle Th� men: fad that a rhCmlmCRlm ha... u \\ aw nalu� implies tnhc'�nl

".11 Sl.nale- lit dtfaactlob of an C'l«'uun plane wa'¥C

'.....

urk�rt"m·

. thn\Ugh Ii "'ingtC" ..tit I.:au� tic, in II... parth:k pn.�p'·rtie... Fllr (',ample. pa.....mg h. � tn'ma�nd"': P"\llI: W;\\C' t\'l .. ('r.:-.\d \1UI, "l ...t al'oll \:l\U-.c: ""'''('ftai",y i t mu an c _

In the momenta of the pastil"le, l phl't,'n...) delcxttd altcNard The ,lUll(' mm.' aprl) III an e1cdwo plane " .l\C. h�ure 4. 12 dcrll:ls a ..angle· ...ht puucm devcl· opt.-J. (ltle the l'arlier dlluMe ..lit) (me electron at a time. The H:" mp.1Ocnt " f ml)mentum of an elc\:tf\ln afh:r pa"" mg lhn1ugh l' ob'li iI..1u...I)' un,,;crtain. What \\ c mCi\O hy " unl'ertalOl)" in momentum " that if the' ('\penment is repeated many time .. it/I'II(inlih. the momentum d,'I(,I.:I('J ufter pot!'"ing Ihn_lugh the .. ht ..till \,.mc .. o\er a mnge of \'aloe.... BUI hi.I" J" v.c \.luanltf) II? Suppo...e that the P, . ",I\lc!. \\'� record fall wlthlll the r.mgc I \...g · ml.. hl I kg · mJ.... e:-lcept for one .\1 +50 \..g . . mls. What \.\Iuc llll we iN,.i�1\ 1\1 the uncert ainl y'! I kg. ",Js,? 25 \...g . mh;) 5 1 \...g . m/..? The definition of um:ertai nt)' b an ,ut"litrul') choil.:e, hut It lll:w iou..ly I,hould mea!.llre hl)\\ far dC\Hltion... are from the mC';,\n (uwrage) \·"Iu('. In ph)'!o,ics. wc define It .\... standard dc\iation For c)'llmplc. "UPp\N� re(lCllteu c:-lperiment.. are earned oul 10 dClenni ne n quanti ty Q, where Q lni�ht repre�

+

pO!'.ltlOn \. a component of momentum PI' or any other me., ..urahlc '-ttl,lO' Ql I!. obta� lcd " \ time.. the \alue Q� ,... ot"lt.\incd no! time ... illld so on. We find the mean Q b)' multiplying a particular "aluc Qj h)' the nUmN:f of tllne!> it II, obtained . "" I,ummmg. over all value.... then dividing by the hlti.\1 number of time.. for alt value!'..

senl

ut)'. The \alue

.

Q=

:L , Q, ",

L ,. ",

(4-\2\

The !'Ii.\fldard deviation tJ..Q i... defined as the square root of the mean of the squares of the values' deviations from the mean (explaining its alternative name. root-ll1ean-�quare devitlt ion) . For a more detailed. development. Appendix 1. Here we merel), present the fonnula that goes with the words:

ilQ =

L j (Qj - Q)2uj L ni

!>ee

(4·\3)

This definition i .. very welt suited to its role. It is the mosl lraclable one thaI is zero if and only if there is only one value ever obtained, which would auto­ matically be Q. and when values do vary, it gelS larger a... they become more spread out. Although it is important to know that uncertainty has a logical definition. as

we continue to investigate the uncertainty principle in this section, we won't

actua1ly use the definition. The point is that when we say, for instance, that there is " an uncertainty in the electrons' momentum," we aren't speaking of something nebulous but a specific value following from a concrete definition.

So let us return to the single slil. As Figure 4.12 shows, there is an uncer. tainly in the x-componenl of momentum of eleerrons detected beyond the slil for which the symbol is .lp\ . (The symbol 6. oflen means "change." but nO( here. Here it means uncertainly. or standard deviation.) On the other hand were we to conduct a different experiment, designed to establish the POsitio of electrons exiling the slit. [here would be an uncertainty in this quantity. too--.lx. The electron wave front i!i spread over the entire width HI, so there would be a probability of tinding the particle anywhere in this range, and the narrower the slil. the smaller would be thi" uncertainty. Above aU, there is a ljnk between the uncertainties in Pt and x. Because the width of a diffraction pauem, related (0 �.pt' is inversely proportional to the slit width, related to Ax.

�

the uncertainties are im'erseJy proportional. tlpx Figure 4.13 As a w(lve be("ome� more compacl, its ovcml! wavelengfh and momentum are less well defined.

- --

� . --

- ----_.

.b

-

pOSlflll!1 �(llIIrh:fC'ly UOknf)"II: .\ ,mJ p wc:1J ,ldiut'J

�

.

.1, fimte

P(hllil11l tJ�'[hl known: A IInJ p It:�s 1\l·1J ddint'd

� r\ L.-----',..1-

_ __ __ __

rO,ili{ln t'Vt'r, h�Un kn\l\\o: .\ ailJ p 1.'1'':0 Ie,_, \Iell dl'!jnL'J

1

ex -

dx

Although we have used the familiar single slit as a vehjcle. the particular experiment is not to "blame" for the conclusion. Regardless of the circum_ stances. it is an inescapable consequence of mauer's wave nature-whether obvious or not-that increased precision in the knowledge of position implies

decreased precision in the knowledge of momentum and vice versa. Figure 4.13 illustrates a simplified. qualitative argument. The top wave is infinite and regular. While there is no doubt of its wavelength, there would be a probability of

finding the particle at places along the entire infinite x-axis. Wavelength and, thus, momentum hlA are certain (6.p = 0), but essentially nothing is known

about the particle's posiLion (llx = 00). The center wave is regular over only a finite region. The COSI of obtaining a wave for which the particle's probable

whereabouts are narrowed down (LU "* 00) is that the wave is not regular every_ where. In any fair way of taking into account all of space, the wavelength cannot be said to be simply A, so neither can we claim [hat the momentum is precisely IliA (Ap * 0). The bottom wave gives an even better known position, but only by

further restricLing the region over which the wave is regular. Accordingly, it is even less fair to say that the wavelength of this wave as a whole is A, so t:J.p is larger still. The relationship between Ap and !lx is developed quantitatively in Section 4.7. Here we concern ourselves only with the conclusion, known as the uncertainty principle, and irs ramifications.

Because of a panicle's wave nature, it is theoretically impossible [0

know precisely both ItS position along an axis and i ts momentum component along thar axis:

ax and �.px cannot be zero simultaneously.

There is a strict rheorericai iower limit on (heir product:

h

tlp,6x > 2

Often referred to

as

(4-14)

the Heisenberg uncertainty principle, for i ts discov­

erer Werner Heisenberg (Nobel Prize 1 932), it is a shocking revelation. There

is a theoretical limir on the preci sion with Which some familiar quantities can

be known �imuhaneously. If we know a particle's position t'XDCtly. we can .1p. = 00). if momentum is know nothing aboul ils momentum (,1x =

0 :::::)

k.nown e1l3clly, position is completely unknown. The plane wave is II. good

example of the latter case. This fundamental matter wave ha.s a wa"'clcOJth perfectly regular throughoul space, gi\'ing il a perflXtly p�i� momentum, but It represents a particle equally likely to be found anywhere. A property in which there is no uncertainty is said to be momentum IS well defined (j,p( (6.x

::::

00).

-=

well defined.

For the plane W8\'e,

0), bUI position couldn't be mort' undefined

Don't be troubled by the inequality in {.t�I ·n-there is no unl:ertainty

about the uncertainty principle. The � rellecls the simple fael that then.: i)o. a particular wave shupe, called a

Gaussian,

also known as a bell I:urve, for

which the product of lIncertainties is a minimum. Figure 4.14 shows a Gauss­ ian wave form, a constnnt C times a " Gaussian factor" constant. It is mallimum at x

=

e-(,J1d

where

e

i)o. a

0, falls off toward 0 symmetrically as

.\'

becomes large, and the rate of fall-off depends on e. If e is large, the wave form is broad, falling off very slowly; whereas if e is smail, the W<.lve form is narrow. We leave the actual calculation of uncertainties from wave func­ tions until later, but it shouldn't be surprising that the position uncertainly

ux is proportional to

e.

It is also true that up is inversely proportional to I:,

and it is only for a panicle whose matter wave function is of this fonn that the product of the two uncertainties is the minimum theoretically po,�sible.

1 II.

For any other shape, simultaneous knowledge of the two is less precise:

up/J.x > ! tto

The whole idea behind the uncertainty principle is rather upsetting to a student of classical physics. Classically, we claim that we can calculate a particle's position and velocity for all time via

F

=

ma

and kinematics.

We need only know the forces acting and the initial position and velocity. But now we see that even starting such a calculation is problematic, for precise knowledge of position and velocity simultaneously is impossible. Fortunately, as we soon see, the uncertainty principle is of little consequence for "large" things. On the other hand, it is of greal consequence for the small things we study in quantum mechanics. As we saw in Example 4. 1 . while wavelengths of macroscopic objects are ridiculously small, an electron might well have a wavelength measured in nanometers or larger. Undoubtedly, it would behave as a particle in a situation where distances are measured in meters, as in a television's cathode-ray tube. But it should definitely show its wave side when confined in a system measured in fractions of a nanometer. Such a system is the atom, probably the most logical test of quantum mechanics.

and the simplest atom is hydrogen--essentially an electron orbiting a sta­ tionary proton. In such small confines, the electron must be treated not a!'t

an orbiting particle but as a bound three-dimensional wave surrounding the proton. Because the wave is spread diffusely. the probability of finding the

electron is spread diffusely. Our knowledge of the atom's approximate !'tile sets a rather small ma.ximum possible value for the position uncertainty, and there is, correspondingly, a rather large millimum theoretical uncertainty in momentum.

Figure ... ,. A Gaussian waft fonL .1, ,1p =

!�.

ift.... O} - Cl'-,*ne�

�K-"------,,

116

o..pter "

Waves and Particle� II: MaUer Behaving as Waves

EXAMPLE 4 . 4

The hydrogen atom is known to be about 0.1 nm in radius. That is. the electron's orbit whatever may be its shape. c.'{tends 10 about this far from the prolon. Accord_ ingly. the uncertainty in the electron 's position is no larger than about 0.1 nm. What is the minimum theorelical uncertainty in its velocity? An electron in an alom moves in three dimensions, bUI considering components of motion aiongjllst one of the axes should give us a pretty good approximation.
X 10-" m)

2:'

1.055 X 1 0-34 J . s 2

t:.p, � 5.3 x J O-25 kg ' m!s

Now using p = mv, 1\ I' � x

I:J.p\ 5.3 X 1 0-25 ko , m/s : -':'::_:: _ . '" ::::2 � o ::'.::. m 9. 1 1 X 10 3 1 kg

� 5. 8

X 105 m/s

From Example 4.4. we conclude thai an experiment designed to deter­ mine the hydrogen electron's speed, if repeated identically, must produce a range of values covering more than

I�% of c.

In fact, careful study of the

hydrogen atom reveals slight relativistic effects. Actually, the theoretical min­ imum

D.P:t of 5.3

X 10-25 kg . mls would apply only if the wave function were

a Gaussian, which it is not, so the [rue uncertainty is somewhat larger. But to be too concerned with this point is to overlook much of the power of the uncertainly principle. It governs all phenomena, for all have an underlying wave nature, and it may be used for order-of-magnitude calculations

in Com­

plete ignorance of the wave function. We shall see just how useful this can be in Example 4.6, which gives us an excellent estimate of the hydrogen elec­ tron's

energy. First. let us revisit the problem of reconciling the uncertainty

principle with classical mechanics.

The Classical Limit The uncertainty principle places no significant limitation on the use of classi­ cal mechanics in classical situations. In the following example, we justify this claim and also confront our earlier predicament A stationary object, with a corresponding infinite wavelength, should behave as a wave.

EXA M P L E 4 . 5 By simple visual inspection, we can establ ish the location of an object within an uncer­

tainty of about 550 om, the wavelength of visible light. Suppose the object is a

I mg

grain of sand. apparently stationary. (a) What is the minimum uncertainty in its veloc­ ity. and if moving at this speed. how long would it lake to travel the smallest djstance

perceivable. about 1 Imf.) (b) A wa\'dcngth of I nm would be �mall enough l\) en�ure

particle behavior in everyday drcun"tilnce�. How faM would the grain of ..and have t\\ mO\'e to havc ..uch a wavelength')

II

Ap�Ax � 2

(n)

�P,

=====-

AI'

=-

�IJl (5.5 X

-

Ap -

//I

�

�

.

10-7 01) �

9.59 X

9.59 X

1.055

10·19 k g · m/...

10 ·19 kg' mj .. 10 6 kg

=

X

10 4 J . ...

�--

9.59 X 1O--�' m/"

In Example 4.4. the electron's velocity uncertainty wa� nearly relatlvls.tic. It i..

a relief to lind it so much smaller here. Large mass is the re'lson. QU:.I1ltum· mechanically speaking, a grain of sand is huge.

t

distance ==-

speed

=

10-6 m

9.59 x

10 23 mjs

= 1 .04

X

10\6 s

3.3 million centuries

Clearly, we can know bOlh the position and the velocity of this object precisely enough to apply classical mechanics. (b) Using the de Broglie formula,

I' =-

P

III

hlA

= III

=-

6.63 X I O-" j · s

( 1 0 6 kg)(\O

9 m)

=-

6.63

X

\O-19 m/s

Part (a) calculates the theoretical minimum velocity uncertainty-more prech,e knowledge of

v is impossible-but measurements i n classical situations don't

even come close to such a ridiculously small speed. Here we see that

even a

(classically) liny object could appear absolutely stationary yet have a wave­ length so short as to behave particlelike for all classical purposes.

A Practical Application Chapter

7

introduces actual wave functions for electrons in simple atom!'.. but

the uncertainty principle alone explains a behavior for which classical argu­ ments fail. Classically, there is no lower limit on the energy a sman particle may have as it orbits a large body. For instance, a satellite may be positioned at any distance from a planet and, if given the proper velocity, maintain a circular orbit. The smaller the orbit radius, the lower would be the energy, though there is a practical lower limit in which the satellite simply rests on the planet's sur· face. By analogy, if an electron orbits a proton in a hydrogen atom, there should be no lower limit on the energy it may have. But there is a minimum energy. called the ground-state energy, and it is inconsistent with the electron simply resting on the proton_ The electron's wave nature-specifically, the inverse relationship between its momentum and position uncertainties-is the answer (0 the mystery.

1,.

CNpUr 4

Waves and Partid�s II: Maller Beha\'in,!! a.'> Wa�'es

)

FisJu,.. 4.15 If p$ has nonzero mean. a fypical value of p, diffel'i considerably

__p__ ' " O "-;"'-d---L � __ _ Number 01 limes

bUI if [he mean is lero. a value cannol be much larger

(rom .lP•.

Iypical

Ihan rlp,.

o

I

p.

Non''''N mean. ' p,

.... .. .

il PA

z,:rll me;!n.

Before we show rhis, we discuss a point that bears on various applications of Ihe uncertainty principle: If the mean of a quantity is zero, then the ul/certainty in the quantity and typical vollies of the quantity are comparable. If the mean is nonzero, this need not hold. For instance, a position mean of

100.0 m with an uncertainty of 0. 1 m implies typicaJ position values i n the range of 99.9 m (0 100. 1 m-obviously nol comparable to 0.1 m. However, the same uncertainty with a mean of zero would imply positions between -0.1 m and +0.1 m (or 0 and 0. 1 m. if negalive values aren't allowed), which

are comparable to the uncertainty.

Figure 4. 1 5 illustrates the point in the context of momentum. The top dia­

gram shows results of a hyporheticaJ experiment in which measured P values x have a mean that is large and positive. Essentially aU P values are much larger

x .lPx' The botrom diagram shows a case with the same ilpx but a mean of zero. Any P value within about t).p:r of the mean is likely. The chance of its x than

m

being I O.6.px is clearly quite small. and even the chance of its being within, say. dp of zero is rather small. Therefore, the mostly likely values of P are

x

those roughly the same order of magnitude as dp,.

x

EXAMPLE 4 . 6 An electron is held i n orbit about a proton by electrostatic attraction. Its total mechanical energy is the sum of its kinetic energy and the electrostatic potential energy belween charges +e and -e that are a distance r apart. E =

-nll.2 + I

2

(+e)(-e) r 41T£0

(a) If the electron behaves as a classical particle. it must obey F = ma. Assuming a circular orbit. apply F = rna to eliminate v in favor of r in the energy expres­ sion, and demonstrate that the energy has no minimum. (b) Suppose now that the electron behaves as an orbiting wave. The energy expres­ sion decreases as both position r and momentum mv approach zero. Assume that each is very small and thus comparable to its re8pective uncertainty:

r � D.r and p � D.p. The uncertainty principle then implies that pr � Ii. Use this to eliminate v in favor of r in the energy expression.

(c) Sketch on the same axes the energy expressions from pans (al

and (bl venu�

r.

(dl Find the minimum possible energy for the orbiting electron wave and the cor· responding value of r.

,oc
ma

_

I

e2

--- = 4"1T£o ,2

y2

mr

�

" v2 = --4"1T£o"'r

Thus,

We see that the negative potential energy is of greater magnitude than the posi­ tive kinetic, and the total strictly decreases (becomes more negative) as r decreases. There is no minimum energy.

(b) Assuming pr "'" fr" we have p = hlr or \I Em•ncr

....��c

Now. as

""

'

( � )2

2m mr

:=

hlmr. Thus,

l e2 li2 e2 - 411'£0-; = 2m,2 - 4"1T£0r

r decreases and the wave becomes more compact, likely values of the

speed increase inversely, and so must the kinetic energy. (c) The two plots are shown in Figure 4.16. While the energy of a classical particle decreases monotonically as r decreases, the energy of the matter wave reaches a minimum and then increases.

(d) To find the minimum, we set the derivative to O.

dEmancr = d,

Figure 4.16 As the radius of an orbit­ ing matter wave approaches

�

=

Reinserting,

(9. 1 1

32"1T2(8.85 X

ils

k.inetic energy approach infinity.

411'erft2 41r(8.85 X 1O- 12 N ' m2IC2)(L055 X 10-34] " s)2 , = --- = (9. 1 1 X IO-3Ikg)(1 .6 X 10 1 9C)2 me2 = 5.3 X 10-11 m

=

zero,

momentum uncertainty and. thus.

X

to

10-31 kg)(L6 X to-19 C)4 12 N " m2jC2)2( 1.055 X to

"" -2.2 X to- 18 J = - 1

3.6 eV

34 J

•

5)2

,

1.10 a.pr.r .

Waves and Panicles II: Maller Beha\ling iI.'> Wa\le�

I

I

This value happens to equal the experimentally veritied minimum energy. and the radiu� is also the correct most probable radius at which 10 find the electron (which doesn 't rest on the proton. whose rlldiu� is 10.000 times smaller). That they agree so closely is an accident-we have made many approximations-but it is no acci. i a dent that they are of the correct order of magnitude. The uncertainly principle s powerful tool.

The Uncertainty Principle in Three Dimensions The quaJitative idea behind the uncertainty principle is [he same in m ultiple dimensions as in one. The more compact the wave along a given axis, the less well we can specify the wavelength and therefore the momentum component along rhot axis. The result is a logical generalization of [he one-dimensional

result:

Note that the dimensions are independent. The single-slil pattern of Figure bears this out. Passing through the slil, narrow along x only. produces a large uncertainty in px, indicated by the subsequent detections being spread

4.12

over a large region of the screen. In the y-direction, the aperture is wide. so less is known about this component of position, and there is correspondingly little spreading of the pattern in that dimension. Thus, apy and Il.x can be small

simultaneously.

The Energy-Time Uncertainty Principle The momentum-position uncertainty relation is, a[ heart, a mathematical rela­ tionship. A width in space is inversely proportional [0 a "width" in the spatial frequency physics,

k

p =

= 21Ti"A (see Section 4.7). It is the fundamental wave-particle

lik, that takes it the final step. The same math relates a width in

lime to a width in the femporal frequency (j) = 21T1T. With E =

nw, the corre­

sponding physical consequence is

Energy.·Ime unc)rti inty prinCiple

(4·1 5)

How do we interpret this? If a stale, or even a particle, exists for only a limited span of time. its energy is uncertain. One example is the fleeting life of certain exotic subatomic particles. Their lifetimes can be quite Short-less

than 10-20 seconds-and this leads to considerable uncertainty in their mass! energy. Another example is the state temporarily occupied by an electron as it jumps down through energy levels in an atom. Because the state is occupied for a finite time interval related to

At, its energy is uncertam by an amount 6.£ inversely

at, which in turn gives rise to an uncertainty in the energy of the

photon produced when the electron drops down. This effect contributes to the broadening of atomic spectral lines (see Exercise

72).

4. 5 The Not-Unseen Observer Let us spend a little time ..ummanljng the limitations thai quantum mechanics

quantify

the effcch of places on our knowledge. Although we won't begin to external forces until Chapler 5. if the forces are known. the SchrOdinger equa­ tion may. in principle, be solved for the wave function of a massive ohject, which contains all information that can be known. But this isn't everything we might expect classically. The uncertainty prinCiple. for IOstance, ..ays that a wave funclion of simultaneously precise momentum and position is a theOl-eti­ cal impossibility. It follows that any experiment or measurement thai precisely

determines position must resull in a state 10 which nothing is known about the momentum and vice versa. Suppose we carry oul an experiment on a particle, experiment A. applying external forces in such a way as to determine both its po�ition and its momen· tum as precisely as possible, such that .6.x 6.p =

h/2. Assume. for the sake of A ' We have found

discussion. that ax is 1 00 lAm. and call the wave function '"

the wave function. but we aren't satisfied. for we haven't really "found" the particle-its ·'Iocation." All we have is this mysterious probability amplitude. We conduct another experiment, experiment B. in which the particle reg· iSlers its presence at a detector at a definite location. We rejoice-we have found the particle. However, there are no "point detectors." If the detector', width is smaller than the

100

lAm position uncertainty in \If

A' then we have

indeed narrowed down the possible locations, but we haven't established a

location with complete certainty. Yes. we have reduced the uncertainty in position, but if this is so. experiment B has changed the wave function. At the very least, it has increased !J.p. If we repeated this pair of experiments many times--experiment A to establish lhe initial wave function \If

A and experiment B to "find" the particle­

experiment B would find it at various locations within the 1 00 11m uncertainty

A' and the number detected at a given location would be proportional to j\f! 12. In essence, we would simply verify that I'" I:! is propor­ A A

of wave function '"

tional to the probability of finding the panicle after experiment A. But because

experiment B changes the wave function, we can't "watch"-repeatedly find­

A'

the same particle while preserving a single wave function 'I' The double-slit experiment, depicted in Figure 4.17, is a good example of these ideas. In effect, the slits are an experiment A, establishing an initial wave function

"'A beyond them, and experiment B is the detection of a particle at Figure 4.17 Eltperimem A eSlablishes

'ITA' which repealed eltperiments 8

Eltt""" beam

1111 W

verify.

112

Cftapter .

Wavea and Panicles II: Maner Behaving II!; Wavc�

the screen. By sending in a beam of particles one at a time, we are carrying out experiment A then experiment B repeatedly. Where "'A is large, experiment B registers particles in abundance; where 'IfA is zero, experiment B registers no

particles. We cannot conduct an intermediate experiment, determining which

slit a given particle passes through, and yet hope to observe the interference pattern exhibited by "'A' for this intermediate experiment would itself alter the wave fUDction. (A recent confirmation is discussed in Progress and AppHca� tions.) To observe interference at the screen. we must allow each particle's wave function to pass through both slits simultaneouslY--
until we actually look for it,

it is hard to justify the claim that it even

has a location before we look for it. Early in the quantum age, many errtinent physicists, most notably Alben Einstein himself, asserted that theories of wave­ particle duality must be incomplete, that some modification is needed to allow "real" quantities, such as position, to have definite values at all times. However,

the modem consensus, known as the Copenhagen interpretation, is that until an experiment actually localizes it, a particle simply does not have a location.

In summary, rather than the classical ideas of position and velocity, quan­ tum mechanics allows us to know only probabilities and corresponding uncer­ tainties based on the most recent observation of the "particle," and a determination of one property is liable to alter another.

(Note: There is a way

to precisely determine an uncertain property without in any way upsetting the particle. It involves "entangled" particles, a topic discussed in Chapter 8 Progress and Applications.)

[!l

P

T

o

N A

L

4.6 The Bohr Model of the Atom It i... instructive to take a look at an early attempt to solve the mysteries of the atom. This work, for which Niels Bohr won the

1922

Nobel Prize, is known as

the Bohr model of the atom, or simply the Bohr atom. When the fundamental workings of something are so obscure as to defy formulation of a comprehensive theory, we construct a model. We observe that a system behaves in a certain way, and the model is a simplified theory that tries to explain the behavior, If it agrees with further experimental observation, we cautiously take it as evidence that its basic assumptions are valid, and we =igure 4.18 A classical piclUre of an

.;lectron orbi!ing a proton.

move forward. If, on the other hand, the model's predictions are at odds with further experiment, the model must be changed. Still, we learn something worthwhile: At least one of the model's assumptions is invalid. The Bohr model predicted that the electron orbiting the proton in a hydro­ gen atom may take on only certain, discrete energies, and the predicted values agreed with the experimental evidence. Combining simple quantum principles with classical physics, it is based on ( I ) the classical second law of motion,

, ,

applied to an electron assumed to be held in circular orbit by its electrostatic attraction to a proton, as shown in Figure the energy of the orbiting electron; and

(3)

tion of the electron's angular momentum.

4.18; (2)

a classical expression for

a postulate involving the quantiza­

The classical second law of motion sets the Coulomb force belween electron and proton equal 10 the mass times the centripelll acceleration (or circular motion:

Figur. ... 19 Finilll wbole .aWl around • circumference. -

.-

,

,

,

,

,

(4-16)

or

2..,. • IA

, ,

, , ,

The work of combining this with the classical expression for energy wa... done in Example 4.6. The resu lt is

..... . - -- -

.,

', ,

Eclassu:lIi pll1icJe

where n = I, 2, 3,

...

Because L = mvr in a circular orbit, this condition may also be written

mvr

nil

, , , , , ,- - - -

It is negative because the negative potential energy exceeds the positive kinetic. In this classical expression, energy varies continuously. for r may take on any of a continuum of values. Now we add Bohr's main postulate: The electron's angular momentum L may take on only the values

L = nil

2," - 2.1

(4-17)

=

n = 1 , 2. 3, . . .

-

•

.

,

, , ,

,

, , -

<- '

-

21fr - nA bUI A

•

(4-18)

• Thus 21fr = nm•

Figure 4.19 illustrates a plausible basis for Bohr's postulate. Ifwe assume that the orbiting electron behaves as a wave wrapped around a circle and that it must meet itself smoothly. so that the circumference is an integral number of wavelengths, then A = hlp implies that the product mvr may take on only the values nk Between equations (4-16) and (4-18). we may eliminate v and obtain a condition restricting r only 10 certain values.

• mvr = n 2w

=

n = 1 , 2, 3, . . .

(4- 1 9)

or where ao -

( 47TE:O)/i.2 m e2

= 0.0529 nm

According to Bohr's theory, the elecb"on orbits at certain radii that are multiples of the Bohr radius ao' Energy, in tum, is also quantized. Inserting equation (4- 19) into (4-17),

E=

me4

2(47TE:

O

)

1

I

' ..2 2 = - 1 3.6eV 2" -n n n

n

=

1 , 2, 3, . . .

(4-20)

The allowed values of the eleClfon's energy depend on the integer n. known as the principal quantum number. As noted, these agree with the experimental evidence.

L = nli

•

114

CMpt.r "

W....u and Panicl� II: Maner Behaving � Wa�'e\

The Bohr model of the alOm is an excellent example of working with the knowledge al hand-Ihe Schrooinger equation hadn'l even been developed yel! However. it is flawed. Allhough orbiting electrons are most likely to be found al cenain distances from Ihe proton, mey must reaJly be treated as dif_ fuse wa'.es spread over a broad range of radii. This casts doubl on (4-19). which. in rum, calls inla question the modeJ's predicted energies. Funher_ more. in reality, orbiting electron waves have not only rotational kinetic energy. due [0 motion aboul the origin. but also radial kinetic energy due 10 motion toward and away from the origin. A fully quantum-mechanical treat­ ment of the hydrogen atom is found in Chapter 7.

r.J D

V

A

N

C

E

4 . 7 Mathematical Basis of the Uncertainty

D

Principle-The Fourier Transform

The uncertainty principle rests on a mathematical relationship completely independent of any physical application; The more spatially compact a wave is. the less well its wavelength may be specified. To begin, consider the left-hand plOIS in Figure 4.20, which show periodic Functions of position. All repeat within the same interval aJong the x-axis, indicated by .A I ' The top waveform is a pure sinusoidal wave. The other

FIgure 4.20 Three functions of the same fundamenlaJ wavelength bUI different spectraJ (harmonic) contents. AmpliTUde

TL

x

k,

fix)

/(.r)

x k,

__

•

x ,'

' 2k,

jJ t . �...L., ..,----...-----;- 3kl

Periodic function

.

�

....

....

Constituents

.r

Spectral

content

tWO are nOI pure. bUl lhey are algebr.tk· �ums of pure wa\'dorm� of dillerenl \\J,c1englh\. a... ..hown 10 the tigure\, center plots. These �um� ha\'e the sante fundamental (1onge�t) wavelength u... the top wilvefoml,

plus diff('rin�

amount... of wavelengths half a� long and oncHhird as long, Waveforms m:h in ...uch hannonic.. are common in musical instruments. A wave on D guitar ...mog. for e,ample. consi\\s of a lacge amplitude of the fundamemal WI1\·e­ length. coe,i..ting with ..horter-\\awlength harmonics llf VarlOU" amplitUde .., The important point here IS that a complicated periodic wavefuml can nnen be:' treated as an algebmic ..um of pure sine waves of dillerent wavelent:th... Tn lnow the ,lInplltlides of the dinerent wavelengths is to know the wavcl"uml's

·....pectral content," and these are the right-hand plots In Figure 4.20. Such plots may be familiar, for they are whnt the graphic equali/er on a ...tcreo dis· plays, with big spikes at one end when the ba\S is loud and at the other when the treble IS loud, And what does this have to do with the unccrt..unty prim:i­ pie? Making a wnve very compact make.. it a sum of pure wave.. covering a huge range of wavelengths. and lhis means n huge range of II/{)me"tll For what follows. it is more convenient to refer to wave number th�m to wavelength. Don't forget its definition. k = quency. For in ..tance, if

A

were

1.0

(radians is under!
m.

A

27T/A. and that

k would be

is

0.5

m, k is

it is u spati.lI fre­

21T m 1 . 47T m

n

whole cycle

I , two cycle.. per

meter. As a :-.impie but useful example of its convenience, the harmonics .!.hown III Figure

4.20 have wavelengths obeying All = A/II, The correspond­ 2,"/(A1'") = II 2,"/A1, integral multiple.. 01 the funda­

ing wave numbers are

mental wave number. It is usually more convenient to speak of something that

can be integral mUltiples of a basic value than a basic value Ol'er an IIlteger.

Now. adding pure sine waves whose wave numbers are multiples of a fun­

damental wave number always yields a periodic function. But what of a non­ periodic waveform? A wave pII/.\·e. like that shown in Figure

4.2 1 .

Figure 4.21 A wave pul..e.

is of great

interest in quantum mechanics. for it approximates a well-locahLed particle. It isn't pcriodic, but can it be considered as a sum of pure sine wuves? The answer is ycs, but not if the sum is restricted to multiples of a fundamental wave number. Figure 4.22 shows why. Waveform (a) is a pure sine wave whose wave number is the pulse's apparent wave number

ko' Obviously. it is a poor

approximation of the pulse. Waveform (b), a sum of just three !
infillite number of waves in the sum. We will soon

see how we knew what amplitudes and wave numbers to add together, Bul. as illustrated by waveform (e). the main point is this: A nonperiodic wave can be treated as a sum of sine waves of different amplitudes covering a comillllOlIs range of wave numbers. It isn't a sum. but an integral.4 In quantum mechanics, our basic function is the pure sinusoidal plane

Cd). wave describing a free particle. given in equation (4- 10): 'IF(x. r) = aren't interested here in how things behave in time, so we choose the con­ We venient time of zero. Thus, our "building block" is eikx. Now we claim that any

Ae'(.l(

'In ",me: -cn...,. an mt�r:1l ''J\er .l; i. hle a �um

o'·cr .l1l mull,pk\ (Of a fund.lJn.tntal waVe numm Ihal " ,nfinlte_imal. dk. II, infinite wlI,·clength al1o,,·, the wa�c·. Q�craJl pc1iod to � mfinitr A pul-c "t"I.�� ref'C'ali.

,..". 4.22 8ui1c1iq:. sinJIe ilOlaied pulse hom pure sine waves requites a continuum of wave numbers. Amplirude

I

I

(0)

Amplitude

I

,b)

•

"

ill

•

"

Amplitude

I

,

(0)

j(x)

�.�

·,1· . .

�

j(x)

I general, nonperiodic wave function "'(x) can be expressed as a sum/integral of these buiJding blocks over the continuum of wave numbers;5 +00

ti)n ,/,(xl 'lS � ,,-n FJn< ' '

J pf.me
",(x)

Tl ' ;,l,"ude A(kJ

�

/ A(k) eH<x dk

-00

(4-2 1 )

The amplitude A(k) of the plane wave is naturally a function of k, for it tells us how much of each different wave number goes into the sum. As we will soon see, it is the key to the whole idea, Although we can't, of course, pull it out of the integra1, the equation can be "solved" for A(k), See Appendix D for the details. The result is

_I / ",(x) +00

F'.IJr ,-

'n

41k) 'Jf 'U1t�ion .','<'j

'II is CIOII\'eotional ro U.M! 'V(x, ') when lime: l� I fac­

lOr,

i _ &nd y,(x) od!erw.M!

A (k)

�

2"

-00

k i e- x dx

(4-22)

The proper name for AV:) is. the Fourier tnDsfonn of the function �-,,). Fourier analys.i" i.-. the technique of treating general functions "'i sums of buic one... and. the woni lr(Jn,vorm ..ugg:es.ts. doing romething with a function. such as throwing I/I(x) in an Integral with �, and obtaining .-.omething else, A(l). related to that fun(:tion�whil;h is what we're dOing. Bdore applying it, two .), points are worth reiterating. FiN. although the building block rt' is nU1 a.'i ea.. to visunliz:e as a real sinu..oidal function, it is jU!;t two such fun..:tions linled in a "pecial way. co,,{kx) + j ..in(kx). Second. equation (4�21) d.etines A(Al "'i the amplitude of each building block. which means that. although ..:ontinuQus, it is really the ..ame quantity ..hown in the " amplitude" plots in Figures 4.20 and 4.22. Now lei us u..e equation (4·22) to see what it really tell.-. us.

Gaussian Wave Packet A plausible wave function for a reasonably well�localized particle is. the Gaussian wave p acket:

(4·23) Lei us inspect this one piece at a lime. The C is just a constant setting the func�

tion's "height" and related to the total probability, which doesn't concern m. r.)l is whal qualifies the whole function as "Gau.-...ian." As Flgur. 4.23 The real part of a Gauss-­ much here. The e-(·l/2 ian mailer wave packet. noted in Section 4.4. it is a maximum at x = 0 and falls off on either ,ide. The fall�off is fast when 6 is large and slow when it is small. In other word!.. the width of the bell curve is proportional to s. Multiplying the Gau.-.sian factor i.. e'A;tl, a plane wave of wave number A Gaussian is just a bump, but multiply· ing it by this sinusoidal function gives the product an oscillatory character. Figure 4.23 shows the real part of the product. From the cos(koX}-the real part of eikoX_it gets its wavelength A = 2'Trlko' and from the Gaussian factor, its fa\l�off. Figure 4.24 represents the entire compleJ( Gaussian wave packet in the same way that Figure 4.1 1 does a pure plane wave. The difference here is that our plane wave is modulated by the Gaussian factor. This certainly looks more

ko-

particlelike than a plane wave alone. but what else does this imply? To answer this, we calculate A(k). Using (4�22),

A (k ) � =

J

+00 h

Ce-(x/2e?eikoXe-ib: dx

-00 +00

.£ h

J

e-(I/4el)r+i(ko-k).t dx

_00

The integral is a standard form, known not coincidentally as a Gaussian integral, and is given on the text's inside cover:

Figure 4.24 A Gaussian wave packet; A plane wave Cei� of wavelength 2'1rf"-o­ modulated by a Gaussian bump. e-{xl�)l

...... -.u The Fourier ....form .

A{.t) of . O.uasian w.vr packet.

We have a = 1I(4s2) and b = i(kn - k). so

A( kJ =

I 2,

�,-"l' "t,,)' V4e2"

(4'14)

2"

A(k) tell us? It is p lotted . the fomlUla m;elf-that A(k) a1

Now we reach the pi\'otal question: What does

.Q,

I

in Figure 4.25. confinmng what we ...ee 10 happens to be a Gaussian function (which explains the s ape cho�n for "amplitudes" in Figure 4.22). Thi ... one. however, i.., a function of k and is ce n· (creel at k = ko' Equation ("-2 1 ) says th�t :-c can create o�r Gaussian ¢h:) by adding pure tlh plane waves, each multlplJed by the coefficlem A(k) given i (4-24). We conclude that in this �um, we would need large amplitudes af plan waves whose wave number is near ko and smaller amplitudes for other wave numbers. Furthennore, a!> [; gelS small, !/I(x) gels narrow, but A(k) gets wide, for c is in the numerator of the Gaussian's argument in A(k) and in the denOmbla tor in "'(x). The width oj t/I(x) is illl1ersely proportional to the width of A(k). (Wh y the width of A(k) is shown as 11(2&) rather than just lie we address a bit later.) Figure 4.26 plots A(k) and the real part of "'(x) for different e values When e is very large, the Gaussian factor in ljJ(x) falls off slowly, and ljJ(x) i

�

;:

:

�

Figure 4.26 The 'patial ",idth of I/J(..)

All r/! h;l\'1Ii the sam..­ b l..i..: \'0;1\1.' numhcr .lo ,

i, in..'eniely proportional to the wave number width of A(k).

A.

2'IT

-'0

i

_ ..'

, , �� , ,

..n all :1(k)

,

.,. ... __ R _ _ � j_ f :' __-� __

'"""""'

1,

" ""

_

But a� the width ,}f

�,

dc(;n�a\e\. the width of A(O inr.:rcnw\,

.. " 1.... ,I � l. , \'

--, " �.J,...t., 'l· """' -t ,,' l ji

�

"

,

__.

- �

•

, ' ,

I

,£

small

,

, I 2£

Ii

� " , 1I 1\' , ,

I

I .L... ......r . ... , "

" \

II "

£ small

, . I

I ,£

A(k) VCf'f>llS k

Ju..t a pure plane wave A.' The Alk) pll\t !>ay", eill&:t1y the: ..arne thing. It "a)" that the ""urn" in l4-21) I" 1u"t a ,ingle plane wa\'c of wa\'c num· ber All. At the other e,lrenle. when t: i, "cr)' ...mall. tJI(t) i... compa.":l. but AIll is corre�pondingly \'cl)' broad, We conduJe that making a wa..e fU1\(;;lilln 4tl \l more compact ma),.e, ih nmge of wa\eh:ngth.. les.. l'Omplll:l and ..ice \CnMI, As 10 the phy..ic,. we l..now that r :'> Irk. '0 a function iCon)!.i)!.ting 0\· a tun�e e number<; con)!.i,t ... 01 a range of momenta-a momentum mea�uremcnt W3,' of might obtain any of them. The in\cThe relatioQl:hip betwa!n the 'p;.Itial width ,,( u function and the "width" of it... wa\c number; {the width in l) thu... implie)!. an inverse relationship between .l.\ and �I. This is the uncenaint} prindple.

c ..'-I!ntlall)

The Inverse Relationship The Gam,sian wave packet is a very imponant special cU.,e. but Fourier aMiy­ sis confirms Ihe uncertainly principle in all cases. Before sUmmari/ing the general conclusion:::, let us take 1I luok at another spechll casco which �hnws how the Fourier transform underlie!> some rather familiar physics. EXAMPLE

4.7

We introduced the uncertainty principle in connection with the ,ingle �h(. A��utlll�, a\ in Figure 4.12, that the slit is along the x direction, that it is 01' width 1\', and thaI a� far as thaI direction alone goc,>. the probability amplitude tIJ(-�) is just a con�tanl C ending at the slit', edges, giving the simple function shown in Figure 4.21 Find and plol A(l). and interpret your result To inve�tigate thi� function via equation (4-22), we insert C and integrate onl)' between -11"/2 and +11"/2, for !jJ(x) is ° everywhere else.

A(l)

=

� 1 Ce-;b:

2

� J� +

dx =

-.../2

2

\COS(kx) + i sin(k:c)\(it Figure 4.27 The '>imp\e probabiht)'

-..../2

The imaginary plll1 dis;tppears. the integral of an odd function (sine) over an interval symmetric about the origin. Carrying out the remaining integral,

AI' ) =

C sin(+kw/2) - sin(-klV/2) -

21T

k

=

C�in(k\V/2) 'IT

k

(4-25)

amplitude of a single slit.

, ,0',,,,,_-\,,,

\

c

w

Figure 4.28 plots A(k), and its appearance is conspicuous. Evidently, there are certain wave numbers. when A(k) = 0, not "present'· at all in !jJ(.l). Wave number is related to momentum, and in the case we consider here, it is the x-component of momentum­ Figure 4.28 The Fourier tran.,form the "sideways" momentum. With certain values missing, there should be certain A{k) of the single-<;,Iit function. angles where particles are never detected-diffraction minimal In fact. squaring A(k) A{k) � ..",h.r.!\ gives exactly the same function that describes single-slit intensity. We already know . , that a diffraction pattern gets wide as II' gets narrow. but the easiest way to see it from result (4-25) is th;1t the first minimum is where kw/2 = 'IT or k 2"1(/11'. A<, w gets 211'/10' small. the k-value where this feature occurs gets larger-the pattern spreads out. That the width of A(k) is inversely proportional 10 the width I\' again confirms that the o momentum uncertainty is inversely proportional to the position uncertainty. =

=

k

110 Ch.pter ..

WIVCI and Particles

11: Matter B�baving "" Wav�,

At this point it is cenainly justified to ask whether this inverse retatiO D­ ship i!1o merely a mathematical peculiarity of the functions we have chosen to consider. Absolutely not! It always holds. As other examples in the end-of. chapter exercises verify. a more compact wave is necessarily one of Jess well. defined wave number. lo Seclion

4.4. we observed that the Gaussian .p(x) �

special. the one for which the product of the widths is minimum. Reexaminin g and for "'(x), we hav e we have [E{ k ko) l' and for W. appears in In Chapter 5, we will find see thai where E appears in ",(x),

the arguments of the Gam.siam� in

A(k),

[.
if:

(4-23) -

(4-24), = [( k - ko)/{2 · ,�)]'. A(k).

that the factor appearing here in a Gaussian isn't just a measure of the width but is ' in fact, the defined uncertainty, the standard deviation of the function.

ax = 8, Ilk =

axtlk �.

I

and the product is Although we won't go into the proof. it is true that for any other mathematical funclion. the product is larger, so that in general

Thus, for a Gaussian function,

Ilk Ilx 2:

2£'

I 2

(4-26)

-

Adding in the physics then gives the famous result:

math: physics: As a mathematical procedure, equation relurns a function of

k,

or

(4-22)

takes a function of

x and

277" over the spatial period. Replacing x by t, it 21T over the temporal

would take a function of ( und return a function of

period T, which is co. In other words, the Fourier transform of a general func­ we tionftt) would be A(co). Replacing the corresponding quantities in

(4-26),

may write

.6.w .6.(

Combining this with

2:

I 2

(4-27)

-

£ = flw then gives the energy-time uncertainty principle. math:

physics:

.6.w 81 2: £ = liw

4} -

---+

h 1l£J.r 2: 2

It is worthwhile to mention some nomenclature that in a very simple way distills the basic idea behind the Fourier transform. Figures

4.27

4.20, 4.22, 4.23,

show various functions of position x. We say that they represent the function in the spatial domain. Each one has a corresponding pial versus k that

and

describes the same phenomenon. the "amplitude" plots of Figures

4.20 and

4.22. and the A(k) plots in Figures 4.25 and 4.28. All these are said to be in the wave number domain. They convey the same information, just in a different way-as functions of the spatial frequency

k rather than x. For a wavelike func­

tion of time rather than position. we would say that a plot of fit) versus the time domain, and a plot of its Fourier transform, a function of

co,

I

is in

would be

in the frequency domain. See Progres<.; and Applications for an example of this

...... ... .... .M ,.

nOlllcndatun: dealing with �ent Nobel Pri.le-winning physics and accompa­ nied by a plot that look.... ...u...piciom.ly like !>Orne in this section. The Fourier tr.:tn...foml i... an indi"pen�ble tool in many areas of modem tcchnology--qu,mtum optic... (Ia,erst ,ignal tran...rni�ion. acou..tks. and meuol­ ogy (fumlamcntal mea,urement...). One e,ample is the problem of tnmsmining

.It i.. ..hon. Regardle..... of me intended ll\). the Fourier tmndonn Atru) feveah that the pulse really

a laser pube ft.t) whu..e duration "centrol" frequency

contain� a range of frequcnde... .lru around %. Though the centnll frequency has the large...t amplitude. other,. intended or nOt, arc: pre"ent Moreover, becau...e

.lw

,,-

1/ Ilt. the ... horter the pul'>e. the greater the range of frequenci�

that must be tran<.,mitted to accurately reproduce it. If any are left out. the pul"e is distorted. A similar "tuation arise... in lransmining any Mgnal that il' not a pure sine wave. Any modulation or higher-hannonic detail s.uperimpo!t.ed on u pure sine wave-as radio superimposes voice upon a "carrier frequency"­ results in a signal comprising a range of frequencies within a nonzero "bund­

width'· of the cenlral one. Both transmitter and receiver must be able (0 process (he entire range to reproduce the complex signal faithfully.

PROGRESS A N D A P P L I C AT I O N S

the bedrock ideas of quantum mechanics, still a quite active

ignored, in areas ranging from communication to basic standardS of mea!>urement, is becoming a central idea. In

pursuit, contributes in important ways to our understanding.

the past decade or twO, the topic of quantum informadoD

We claim that if in a double-slit experiment we were to

has exploded. In essence, the tenn acknowledges that we

Keeping a n Eye o n the Double Slit VerifIcation of even

watch whether a panicle passed through one slit or the other. we would destroy the interference pattem-fonnation of the pattern requires the particle to pass through both.

are now becoming able to manipulate much of the tenuous

Using a double-path electron interferometer, effectively equivalent to an electron double slit. scientists have

confirmed it (E. Buks, et aI., Nature, 26 February 1998, pp. 871-874). In the standard double slit, a path difference

infonnntion that quantum mechanics allows us to know-

not classical notions like panicle position and velocity. but

the quantum state. A promising application is in the field of quantum computing. The heart of today's digital comput­ ing is the data bit, which can be in either one of two Slates, usually a 0 or I . A dynamical analog would be a classical

to different locations causes a phase difference, which in

particle that must be in either one room or another, but not

turn gives constructive or destructive interference. In this

both. A quantum particle. however, can be in any of an

experiment, electrons may take two different paths to a

infinite number of superpositions/sums of two quantum

single collection location. but the phase relationship is altered

states. A simple example would be the superposition 'Jr =

by the asymmetric innuence of a magnetic field-varying the

a q.r \

field varies the interference. One path is essentially free, while

whose isolated bumps are far apart. and a and

the other passes through a tiny ··quantum doC (see Chapter 5

arbitrary constants governing the probabilities of being

Progress and Applications), serving as an electron weigh

found at one bump or the other. This more flexible bit of

station coupled to a detector. When the detector sensitivity is

infonnation is known as a qubil. Qubits can be combined.

set very low, thus concealing any knowledge of a passing electron, the interference pattern is clear. As the detector's sensitivity is increased, the pattern's visibility diminishes­ the important phase relationship is lost

+ b "'2' where q.r I and "'2 are Gaussian functions

b are

to carry vastly more information than the same number of standard data bib, giving hope of massive parallel

processing that may revolutionize information handling in the years to come. Many two-state systems have been studied to serve as a qubit-for instance. two different

Quantum Information

As technology marches on, the

energy states of a bound electron and two photon spin

realities and peculiarities of quantum mechanics assume an

states. These topics and some of their applications to

increasingly importam role. What could once be essentially

quantum computing are discussed in later chapters.

WorIdng with the Unc.rtalnty Prindpl. The unc:ma.int}' principle ties together uncerta.intin in Jk1-"iition and momenlUm. but if hu considerably wider applicatlon. The physical world ruibib many pai" of quantities, known as conjugate quanfitia. that al'C' inherenlly linked in the �me

I

I

way To kntlW one quantity e_,at.·tly i\ to be completely ignorant of it .. conJugak'_ ConveN"ly. if one quantil), i� known. an inadvenenl "ob..cI'V4I.tion" of its I;onJu!!ate will disturb mat knowledge. C{lOjugare quantities haying to do With lighl. �uch a.' pha..e and photon number. are of great ,,:urrenl imere,!. ught in which flul,·tuation, in one aspect rune ht.-en redu!,"w at the !,"ost of uncontmlled tluctuations in I fs conjugate i� known a" �ueeud light Wi!h cenain

.... i'"tcri�tKs. �quce..ed light promi�e� to unusually ,mooth cha enhal1l:e high·�pced communication. optical imaging. studies

ruot of jnlere�t in �o-ca"ed quanlum·nondemolilion (QND) mea,ul't'menh, Using QND technique,. a recenl experiment

wa' able 10 delect the �ame photon repe.lIedly ....ithout losing whit'h i, the usual fate of a phown aCluaJly "seen" fG. N(lgue�. el uJ. . Nature. 15 July 1999. pp. 239-242), Progre�.. in QND ..hould aid the advllnce of quantum computing. a., well as gravitation. Gr:hjlalional waves are predicted to e�j
Tight Control in Time and Frequency

the pulses is inversely proponional to the spacing of Ihe frequencies. Compnre these IWO plots with the plots in Figure �.:!�. which show exactly this relationship. differing only in thaI position replace" time. and wave number replace\ (temporal) frequency. Moreover. the broader the range of laser frequenciewmodes included. !he narrower ttlt pul<;e. which is the idea expre�sed in equation (4-27). The lower plOb in Figure 4.29 reveaJ that an offset in the �taning point of the regularly spaced lines. though not

changing the separation of the pul<;es, leads to the li ttle penks within each being somewhat out of phase, A major

achievement of Hall and Hanseh was to stabilize this phase

ofdC'licate atom proces_�e... and many other technologies. A...olding run·in� with Ihe uncenainty principle is also nt the

II,

unrelated plot. but il i� really the �ame information. �"pre..\Cd in a form more useful for understanding the application. In thi' ,,·jew. the sum lIVer frequencies give", a pulse. but regularly repeated. and the separation between

relationship. contributing to extremely precise control of the light. The hosl of applications is large and expanding. from global positioning systems (GPS) and astronomical ob�ervations to studies of matter/antimalter asymmetries and drift in fundamental constants. Figure 4.29

domain.

Shorl-pulse laser light in the time and frequency

Time Domain

FrequencyDom3in

Group delay "" phase de lay

As our desire for

e-,"'cr higher precision in ..crutini7ing the physical universe increase_,. new techniques must be developed. A great ad"anl'e. earning the 2005 Nobel Pri.le for John L. Hall and Theadme W. Hiinsch. was the development of the optical frequency comb technique. which provides an excellent example of Fourier analysis. A laser can be made to produce many closely spaced frequencies al the same time. (Exerci�e 9.80 discusses a simple case.) The upper right pIal in Figure �.29 depicts their varying amplitudes in the "frequency domain"-that i�, ploned versus frequency. Picronng them in the ''time domain" produces a seemingly

<>:::-<> /lrp

=0

Group delay ,," phase delay

�� !lip

*0

ExlTa-

polaled mod"

II

Active modes

,,11111,,

Q /"� 'if:'

l. In FIJUft .. 6. ray. are IIhowII � ...

Chapter 4 Summary

...

� I"ted up I. "� widll " __ . .. ....

JU�1 a s

eloctrom�,netu: radi.JIIOD -< buu: aUy a waw---�)' t1cha\� .� pmidc!o. • maui\l' Ilb�t -d�·t.lc.II)' . dI'M:n:tc J*11.:k-- nu� t't:h.:3\C ... . \toll'. Iu. ",ave num� and an,...I. Irtq�nl') af(' ,,,cn by the fuodamfntl' \Ioave p&rtu.:lco R'IJlilJn�hiPJ '

plane curtI) � IboIe ill ... ,... ...., A 7 -.

the ...., .. In ditremM pluth .. ....u,. "., ...... ...

W.Y

1'hcy nuJhl. for .....-e. ... be ....... . .. ...

..."omptln)I.., fi,lIft' [lues ilia .... dw \'IIIdIQ' f1I

the 8111" I•• t" II'J hpl.... )'OW' ..weI'

• ,

'\

_\ nMlter W.J\(' lunl'lil'" I'l:prc\l-nkoJ b� '1'( I. II. ,!no.! \Ih...�.. thl' S.:h,\xlln�cr CljUJlIlln, VIohl..h h bJ.�cJ lin enc'l:)- ,\OJ h.r II h�'C pan,,:k

1\

... .-\11 "tbn IhlftJ'l �inll f'\IuaI. 'It'hit:h wouJd be more WIll.,. It' e\hll'>ll ltJ .II� MIUro·.. Jll'llun ,. an elecbtw'l and

wh>" Hy mUJn. KlRIClhlna UIVfIWJI. hnw«dd you

Thl" ....uan: . ,'l lhe .....1\(' lundlvn', ab'l-.lulc \alue JI\t" the rn.babllll) �r UOit kn!1th 01 findmg the pattldc: pmNtlllll) Jen\II)'

•

I'I'( I. t)j!

to a par1ldc', untkrl)lnll VIoa\(� IWlurc. II� m ....mcrllum and J"INllfOn cannot be rro:l�l) lnow" \lmullaooou,ll_ 1llcrc '\ it thcon:u.:al lu"cr 11mll un lh< pnxiU(1 ,,1th< un.:enOlintle\ 0"'"8

(4 ' 4)

1

Thl\ ,\ LOlw, n J\ the unl.:cnamly prim:lple and \ II con'>6.juenr.:e - M. and the mathematu:ill pmpe:nlC\ of "3Ie, A ....a\e . lundlon for y,hu.:h nwmentum i� kno....n precl'cly loll' - 0) i� onc for I,\-hlCh the Pl-l\IIIOn " compltte ly unkno"n (.l...t .., -I; the pmbabllity dcn"ty I� 'rreild throughout ')pace Convcr.el)'. II I,\-;\VC lum:tion lur .... hu.:h the JXNtlOn i\ kno....n . preci\Cly " one lor .....hlch the momentum i� completely unknown.

01 I'

mdicates advanced que�tlon�

Conceptual Questions 1. E,pcrimcnh df'c<,:lIvel)' equivalenl to Ihe elcclron dou· ble \111 ha\'e been conducled III differenL no\el ....ay�,

produclIlg ob\'IOU� mal(lma and minima. Otten the pOlnl i'l \Ire\,cd Ihal thc mlen\ll), I� �,fln:m�h low. Why 1\ thl\ lacl empha"lcd M) mur;h" HoI,\- low i\ loy. enough 10 make Ihe point') Z. Generall) \pcakin�. ....h): I� lhe .....3ve nature 01 mailer \() countenntulli\c"

"",,(lflll"C'noute." M) a.. III mU.e (-.Je Ai ••vdikr &!' the ocher1 �. 1rw.'al\dMtn t t lighttLul� tal up . filament ....,.tUll' hot," pnLJunns Iletli t_t .al' wawlt'nJthJ OW has lidle to do with lhe li�lnml' \'OCtIp,lIIiltlllfl Clu vapM' bun.. wdI . 1l�lum and men:UI)' 5Q'eelhahb. produce !,.'Orkn thM do r.krcnJ I'n the Ial. an tht t'oulb. VII:WC'd with . diffTacbon ¥Qtln!: a simpk ('O'), u.'hm:al the incandcscem �ra.1nun I� ronliftLMiIJ.., lhat of. ps \.- (tW fIuorn.. ..-enll t'oulh hll!i .taanM:leri\tit: bnes. How d this tndimI."I l"\1Jcn,.'C tlf lhe w.,co NatUre of �-.bltlna electrons.. 6. An d¢l.trun beam In"e II barner With .. inale narrow �I I I. lind the Cll"\."tflln flUli ··number uf electn1lls per unit lime pcr uml ar �aJrI�'I('d .t the wry center of the r('�uItIllW tnt�n'll)' p"uem I" f'l Nell.1. two more tdenli· (,,1 ,,111\ "rt' "ren�, e"lUldl�lanl un either side of the lir,t lind �UIIJl) " ll1umlll�led" hy the hcam Whlit will he Ihe tlu, Jl the \ery (enter nuw" Does yuur answer IInrly that mllrc Ihan Ihree Ilmcs as many electrons pus thrnugh Ihn!t' 'hl� than through (Inc? Why or why not? 7. 11 II panlde's f'I"llwn uncenamly is uro. whal can be �ald ul I" m.lm�nlum un�·('rtalnIY? If . particle's posi. 11I1n \ln�'('rtainl) IS mlimle, •hat 1:'" be said of its m
1("01:11

111

I..

ScItIiJtt Wldllhe auumpbon dw • � 'Nne fuoc­ boa 1lIIY be uaced .. an aI,ebraic JUm 01 sinusouW functiollt of various ....ve numbrn. explain concisely why Ihere il an unccnainty princIple.

wilh the crystaJ f8l..""r and are del«ted at the .woe ... wjth the atomic plane. In diagram (b). the c,,"sW "­ been routed 4.'\" counterelock",·jse, and ""'aH�S 1It' __ Incidenl

upon planes compri�lOg diffnrnt �n.s cft:Uo.ta.

I f in the oriental/on of diagram (b). constru�ti\"e imer_ ference is noled only at an angle (J = 40", 81 "'har

Exercises

ang/c(s) ""'iJJ constructn;e IRlerference be found in the orientalion of diagram Ca)? (.vot�: The spacing belweca

_ 4. 1

I I. A tam of electronJ sb"itts • barrier with two rwT'O'N but equaI-wkfth lfib. A acreen islocatcd beyond Ihe barrier. and eIcctronI ft detected as they stnte Ihe liCtttfl. The "ceme(' of !he JC"Rlm i� Iht poim equjdj!rtant from Ihe sJill. When eidler slit alone is open. 10eJectron.s amve per ICICOIId in a very �mall region aldie ,"'Cnter of the screen. When both dil1 � open. how many eia.1rons wiIJ arrive

atolru is Ihe same in each diagram.) JJ. The �erup depicled In Figure 4.6 is used in a diffrac:fJOa experiment u\ing X-rays of 0.26 nm wavelength. Coq.. .�truCII\'e Interference i\ noticed 81 angles of 23.0" and 5 1 .4°, but none belween. What is the spacing d of atomic planes?

14. A beam of electrons litrikes a crystal at an angle () wilb the atomic planes. reflects off many aromic planes bel<m !he

per second in the wne �gjon at the eenler of tile screen?

surface, and then passes into a detector also making angle

11. Analyzing cryslal ditrnu:tion is intimately lied to the

8 with the atomic plane\;. (a) If the minimum () giving COn.

vlltious differenl geomelrie§ in which the aloms can be

srructive inlerference is 35°. whal is the mtio Ald. whe�

arranged in lhree dimensions and upon their differing

is the spacing between atomic planes? (b) AI what other

effectivenes.s in refl«ting wave�. To gra�p some of the

d

angles 8, if any, would constructive interference occur?

con§iderations wilhout l<Xl much trouble. consider the simple �uare arrllngement of idenlil.:al alom� shown in the tigure.

In dut�nun (at wave� are meldent at angle 8 Sou,,:e

Section 4.2 IS. How �Iow would an electron have to be traveling for ib wavelength

10 be at least

I flm?

16. A Bragg diffraction e:tperiment is conducted using a beam of electrons accelerated through a 1 . 0 kV poten. tia! difference. (a) If the spacing between atomic planes in the crystal is 0.1 nm. at what angles with respect

10

the planes will diffraction maxima be observed? (b) If a beam of X-rays produces diffraction maxima at the same angles as the electron beam. what is me X-ray photon energy?

17. Determine the Compton wavelength of the electron, defined to be the wavelength it would have if its

to'

momentum were mec.

Deteclor

18. A particle is "mermal" if it is in equilibrium with ils

leaT.

surroundings-its average kinetic energy would be

�

Show mat the wavelength of a thermal panicle

is given by

:-V-;:3�mk=B=T h

A

kBT.

�

19. The average kinetic energy of a panicle at temperature T is

�

(a) What is the wavelength of a room-tempera_

ture (22°C) electron? (b) Of a room-temperature proton? (c) In what circumstances should each behave as a wave?

20. What is the wavelength of a neutron of kinetic energy (b'

I MeV? Of kinetic energy 20 eV? (The difference is important for fission of uranium. as a neutron will be

_uS'

mt>tC' CDI" � "'Mn '" .a�lirnath It; ""' WVt'

"'""' of ....... al 35 .Y � _ II ..... • ••• ... ",," \U, o( 1 0 .... .,.... .. 111111 -=nen " m MyoNt _ ,Ut How .. tro. . ... �..I) In thi' II.. of the .... ... ftnt ..... 'W� nll.*·� are C"f.- � 21. l'lll(U,."'1 � ...."eIemed .. dw:wJh ' 20 V ,..... dln_nt,.'C'. pn�'UlJ I � ....... "... .. �1Cd aI I dI",Ne-• . lIl IppuQI' of 0.0l0 ... lilt tiit"pualil\n hIln� al clKtrm detomln � '0 1ft beyoa4 lhe ",,,,,\'lie "ht Wtth "'It I alone open. 100 ...... roer 'C'o..'OI\d "'ft' dctected II all dclel:ton. W\Ib 11\ 1 al" ne: "'ren. �Xl t'leI.·tf\>n, rer loCII:Und 11ft.'" M .n .k1c.:ll'r.&. N,I'W ho.llh �ht' an! open

...

,omparnI with lhC' di.mcn�, ,lithe I.n'aIli.lllTl au.:�

- l:'i fin In df«1.. a newrua of ""'I w;l.vden,th �\�, 1nOC" I,"'" a JIf'fUliC ...."C \hAn b�C' a lIxallJn1 J'&t\ide \ 21. (al Whal l$ tho: �" "f �Ibk .....\�kqttu. for a rcu In.., <.:0II'Tnf"I ",4m 1l' a tII\� "f q.eeda from ·'thermal" OIl }I..I ... ,� t \(""11< I Kl hI tl ll i("",toJ R�;&I pu1 (II. ... Ih" rllAll" I.r tout ....Ilh n't,'I<'n&:o;: \" 4n C'kctron. (d h �fC<'..h .... hal I'llnlllc 01 ..hnt('MI,'M I) �1..1 tt')clIl '..... . .... a\o;: nature "'[ 4 no.:Ulhll'l ' 01 an dr-..'In>n ' n Rl>ughl� ,,('C'al..mll. lur ....hal ranlt" "f ....1I\o.'\englh.. . ....flUIJ "" � I,l lr\'al �I\ d�ln1(\ I""(la\I\\..u,.lh an..1 "hill "" U\..1 t'Il" Iht' �·" Itt"f'I.'II'\...1mll nlJ\¥<, ,.\ 1I" d("ndmll Jl'\I<'n\l�I.."

1 \l'lam HlUt Il",um("lllvn,

23. A,olll'> In II 'l)'lal lunn "Ivnu,' ("II.If\(, al miln� Jln ...",nl anJj.lc, tl' the �urfllrt' Inc 1II.',,' ,: m('illl�IllII

....'Ih . �'I)('\,'I

fijlUrt' ,h,.,." .. the bl·h.l\ 1\1,.... " I rerrt''<'nta\iw IIKIJenl aml

"Cdt\crt'd "1I\'e.. 11\ the 0.,\ ",>\lll·(knl\cr ('\po.'nn1ent "­ hcam III c\e.:t!'l.ln.. ol�..:e\erule,l lhl\.\\Igh :'i-l V ,.. Ihn.-..:teU l1nnllllll), 111 11 !lId.el \uriIll:C. IInu ..\n>ng retl«:1111n "

q,III '\0 U"Ulg the Bragg 101". 'h,,", thal th,.. Iml'he.. a "pallllg f} III md .. cI
JelC\:led '\nly III lin angk

,urlilce in agrc.:menl with � \..nll"n value o[ 0,22 nm

tl"lo;,lknt bc�m

-,

•

l

-.

X 10� m, To

�mall. But SlIIall rtlullvc 10 what? Being II rough mea·

IS confined. Ihe orbit rndiu�

is certainly a relevant dimension against which to com·

pare the wavelength. Compare the two. Does Ihe Moon indeed orbit us a dassicul particle? ("'

£ar\h

ION kg and mM(lUn = 7.35 X 1022 kg)

�

5.98 X

25. In the hydrogen mom. the electron', orbit. not necessar­ ily circular. c�tcnds to a distunce of uboUi an angstrom a s

I

nm) from he proton.

If it is to move aboul

a compact classical particle in the region where it is

confined. the electron'S wavelength had better

be much

ulwu)'.�

smaller th::lIl an angstrom. Here we investigale

how large might be the electron's wavelength, If orbit­

ing as a particle. its ..peed at

1

}.. could be no faster than

that for cin::ular orbit at that radius. Why? Find the cor·

responding wavelength and compare it to atom be treated cl assicall y?

H" .... man), dl'cU\ln� per Ie&:Ilml wi\! bel detected at
2M. A I:>o:am ,\1 rarhdc•. l'lIch III nta�s m and \nonrel.Uvi..

II..:} \J'>«,j I'. ..tn�C'� 11. hlllTicl in which lhere are two nu· f\,.... 'hl� and 1:>0:)111\1.1 ",hl�'h i" .. b1mk. of deteclon. With

1

H\I"- lar apart III""( 'hI' �hb')

Ih) Ho"- man)' rarudC'!o. would be dele&:lcd 'al all

do ,0 11.\ a classical particle. it, wavelength �hould be

A = 0.1

\c)

(a)

24. The Moon orbits Enrth lit a rudiu, of 3.8-1

(I

dc.:If\ln" per \C\.'"tl1ld will bel detec\e4 at

IhC' �'fnlC'l dC'IC',,:III('

partide, f'C'1 'C''':llnu. Ill:<.:UI" 1l1 an anile of 30g from the Inlllal ..1lrcl'lllln 01 tnlllllln of lhe beam .

"'....... ..... . .... .

.

,ure of the region where it

mlC'rll'rc'n.:c rlloltl'rn'l \�l til..... I1\lI.n),

en�t' ("I,lIIl'm \\ Mtl'd In whi!;h the lint minimum. 16

•

•

•

tal fbI' hl"l nllnunum in Ihe 1'1«1ron \:O\In\ �""' .. ddcdlll X H,..... lar I� it "um the \:I'nlet of the

,lit I 1I1,lnC' 'lflCn. 100 rl\nlC'l'� art' detected per �nd al .III Jl'lel'lllr . Nllw '\11 2. Ill. Illul opened. An inlerfer·

• •

.

-

A. Can the

ucte..:\lU�) ('It'r ",,((Ind wilh lI.ht 2 alone open'

«() l"here are multlpk Ilfl�wen III pan (b). For each,

hllW molny partidell. would be detected II the eonter detectllf With b\,llh \I!b open?

29. In h,ample 4.2, neithcr 1"'1] nor 1'1'\ are g;ven uftita-­ only propllrtionillitie" W'e u�d. Here we verify that \be re�ull\ are unanec\ed, The actual value& liven i.n the example lire particle detection r8\1'S. in putic:lcalsccond. or s I for Ihi" quantity. leI us UK the symbol R. It ia lrue thltl the paniclc de�tion rate and the probability

den"it)' will be proponional. so we may write \"1'\1

-

hR. whe� h h lhe proportionalily constant. (al What mu�t be the unib of b? {b) What is

\'I'rI at the center

detectllr (interference mouimum) in terms oftbe cum.·

b' (cl wtw would be a n d the detection rate R aJ. the center detec­ ! I'" II. I'" II .

ple'� given delection rate and

tor with one of the !l.lit<; blocked'

30. Calculale the ratio of (a) energy lO momenblm tw .

photon. (b) k.inetic energy to momentum for . �

tic ma.,..in obje&:1 of speed II. and te) total eaerz:y 10

136

Chapter 4

Waves and Particles II: Maller Behaving as Wave�

momentum for a relativistic massive object. (d) There is a qualitative difference between the ratio in part (a) and the other two. What is it? (e) What are the ratios of kinetic and total energy to momentum for an extremely relativistic massive object. for which II � c? What about the qualitative difference now? 31. Classically and nonrelativistically. we say that the energy E of a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical pal1icle velocity I'particle is 2Elp. (b) Show that this velocity and th:lt of the matter wave differ by a factor of 2. (c) In reality. a massive object also has internal energy. no matter how slowly it moves, and its total energy E is rmc2, where "y "" 11v'1 - (vpa"icl.!cf. Show thllt vrmliclc is pc2/E and that "wave is c2/ "parlicle. Is there anything wrong with vW"ve? (The issue is discussed further in Chapter 6.) 32. Nonrelativj�tically, the energy E of a free massive particle is just kinetic energy, and its momentum p is, of course, IIII'. Combining these with fundamental relationships (4-4) and (4-5), derive a formula relating (a) particle momentum p to matter wave frequency f and (b) particle energy E to the wavelengm A of a mat­ ter v.ave. In Sectillfl -1-.). It is shown that 'vCt, t) = At"\A' flY) �a( sfic= the fr\."e particlc Schrooingel eQu..ttion for all and ', proYl(]·tt only lhdt ,he ,",onHan!. .If<.'' n ; tt b, flm..:jun 'l' fIw $t w m.·[ ,..1"! n IiJi.J}/1m lito t:L ·hnx ,' r ox I Viut! 1\ �I. t 'l' Iy to, • f ti"n t: res] � � Cf �..., (j

), ...

IU � � h'JW the ..: j, �. w.... ilan: f.n .In(! . 11 ·Jt

[Til:

a

'·I.·V ," t I.

"

It

a "

I, 'J'l

I , �,

n · .... l-e�-p PC

. ! U'll .; ...

ciU1

"

1t I, ,

1 ." re" t: IJ,

"

� ,<

:or, ....

r. "

r

�

n

.. II ... ,rr,

t:· ' Cl � l>..):: ' f ::.J

rrer'l !l ( Id

,·e .w. "

) 1,

,

I.'t'

.J,m,j·

"

... ....;.J:

I'

,

•

,

"

.

('I'

,

,

r

1z2 r1 '!' (..l.. I)

!Jr.

• 35.

",'

'l'lX, t) ]s bv ,Jdimtion "" I t) I 'Ii2(X, I How i� the- cumplt!x appro'".:h l:h"lsin In �eci n -L3 mor.; conwnienl than the a.lternative pmed hert! J In Set·tinn 4.3, WI! claim that in analyzing ek:(;tromdg netl!': wavc!-" we (;oulrJ hanrJle the field� E and B \Ahere

f

36,

37,

G · dl

'0J

= !... c a,

G ' dA

waves would have to obey lhIS' COil). Electromagnetic 0 . , oes thiS change of approach make plex equatIon. and/or 8 complex? (Remember how a complex n £ umber is defined.) An electron moves along the x-axis with a well.deli .fIed momentum of 5 X 10-25 kg·mls. Write an ev"preSSIOD " the matter wave associated with lh'IS eee. descnbmg I tron. Include numerical values where appropriate, A free particle is represented by the plane wave funcll, flJl W(x. r) =: A exp[i( 1.58 X 1 0 1 2 x - 7.91 x 1 016,)' where $1 Ul11ls are. understood. What are the po.' ....
Section 4.4

Section 4.3 33.

together with complex numbers. Show that if .... field" G .== E + jeD. the 1we an "electromagnelic llhe "". , of Maxwell s equatIons that link E and 8, (4.6c ..." la..t (4-6d), become just one:

(aJ L x: ;meut X i., carried out nine times identically, alue 5 IS obtained all nine times. Calculate \be
31t

[ n'lVing·) T 1(> un "l!rt;:ltnty in the position of a baseball of mass 0.14:'i kg IS I !lm What is the minimum tu

4

lIs s�ef 42. A mOMluilO of milS�

i ,'�, 0.15 mg is found 10 be fly" , �pccd of 50 cmJs within an uncertainty n"".' '''

(a) How preCIsely may its position be known? this wherent uncenaimy present appl icatiun of cJas.,ical mechanics?

43. The IXI..ition of a neutron in a nucleu!> is known within

aru:I that the proportioMlil)' r.cw .. . sue 1IUIII.IIet

we exped to find il moving" 44. Tll how �mall a region mu�l an clcclron be confined fOf borderline relathi'll.: speed,-f;a),. O.OSc-to become rca.\onably likely" On the ba.,is of thi�. would you expe<:l relali\'istic eff�h 10 be prominenl tor hydrogen's elec· tron, which ha.\ an orbit radiu� ncar 10 10 m" For a lead atom " inner-,hclr' eleclron of orbit radius 10-12 m'? 45. A 65 kg man walk.. al I mb., known to within an uncer· tainty (unrcah'lically �mall) 01' I 1Ullfh- (a) Compare the minimum uncertainly in his po"ilion to his aclual ph)',i· cal dimension in his direction of mmion, 25 cm from front to back. (b) Is It �ensible to apply the uncertaint), principle to the man? of the cornerstones of quantum mechanics is thaI One 46. bound particles cannot be stntionary--even at zero absolute lemperuture! A "bound" panicle is one that is confined in some tinite region of space. as is an atom in

"in fJ " tan 8 a 8.)

an uncenaint)' of -�

X 10

I � m. 1\1 whal speeds might

inl.kpcndcnt of 1ft,

.. .ad Ia.t ...... ....n ...-:

\I, "

51. In herci'il! 45, the case is made lhat lhe position uncer­

a solid. There is a nonl.ero lower limit on the kinetic energy of such a panicle. Suppose a particle is confined

L.

Obtain an in one dimemion 10 a region of width approxi mate fornmla for ill> minimum kinetic energy.

47. An electron in an atom can "jump down" from a higher energy level to a lower one, then to a lower one still. The energy the atom thus loses at each jump goes to a photon. Typically. an electron might occupy a level for

a nanosccond. Whot uncertainty in the electron·s energy does this imply? 48. The rfJ is a subatomic particle of neeting existence,

Data tables don·t usually quote its l ifeti me. Rather, they

quote a "width," meaning energy uncenainty, of about

ISO MeV. Roughly what is its lifetime?

49. A crack between two walls is 10 cm wide. What is the

angular width of the central diffraction maximum when (a) an electron moving at SO mls passes through? (b) A basebnll of mass 0.145 kg and speed 50 mls passes

taint)' I'or a typical malTOscopic object is generally ao much smaller than its actual physical dimensions that. applying Ihe uncertainty principle would be absurd. Here: we gain ",-nne idcu of how small an objccl would ha\'e to be �f(lre quantum mechanics might rear ilS head, The den�ity \If aluminum 2.1 X HP !tgIml. is

.

typical 01" M)lid� and liquilh around us. Suppose we could nam}"" dl'IWn the velocity of an aluminum sphere 10 ..... 'Ihin an uncertainty of I � per decade, How small ....ould . il ha\e U1 be f\u Ib po�ition un..enainty to be at. lea,t a� largo: a... of ih radiu�'?

-I,!"k

52. A particle i, conne..:ted to a spring and undergoes

one-dimen'ional m(ltioR.

(a) Write an e"pr.:s...ion for the total lkinetic plus

potential) energy of the panicle in terms of ilS pl)�ition .l, i ts rna,s m. iu. momentum force constant I(' of the sprin g .

p, and die

lb) Now treat the particle a., a wave. Assume Ihat the:

product of the uncenainties in position and momen­

tum i, governed by an uncertainI)' relation

� fl. Aho a.\!'ume that because x Is. 00

through? (c) In each case, an uncertainty in momentum

6.pSl"

is introduced by the "cxperiment" (i.e., passing through

a'\lcragc, 0, the uncertainI)' � is roughly equal to a

the slit), Specifically, what aspect of the momentum

becomcs uncertain, and how does this uncertninty com­ pare with the initial momentum of each?

,

typical "alue of �l"\. Similarly assume that ap Eliminate

p in fa

'\lor

;a

\PI.

of x in lhe energy e:xpftSSioo.

(c) Find the minimum possible energy for the wrIe.

50. If things really do have a dual wave-particle nature, then if the wave spreads, the probability of finding the particle should

:::::

spread proportionally, independent of

the degree of spreading, mass, speed. and even Planck's

constant. Imagine that a beam of particles of mass m and speed l', moving in the x direction. passes through a sin­

gle slit of width w. Show that the angle 6t at which the first diffraction minimum would be found (nA = w sin (In'

53. The energy

01' a particle of mass m bound by an UIIUIWIl

�pring io;, pl/2m + b:c.

(a) Clas...ical ly. it can have zero energy. Quantum

m.:chanicall)'. however, though both x aDd p .e "oo avcr.lge" lero. it.<;, energy cannot be zero. Why'?

lb) Roughly lopeaking. � is a typical value of Ibe ,..­ ticle'� pCI!1oltion. Making a reasoaabk: MSIJI!IIlIioo

from physical optics) is proportional to the angle at

ahout a t,.pical vahle of its momeotum, fiDd 1bc

which the particle would likely be deflected

panid.:·s minimum pouiblc: cocrgy.

(J =:

�p/p.

A(k). (a) How does the lJ!(x) plot demonstrate that the

Section 4.6 54. The allowed elel:lron energie.. p�dicted by the Bohr model of the hydrogen atom are correct. (a) Detemline the three lowest. (hi The el�tron ("an "jump" from a higher 10 a lower energy. with a phown carrying away the energy difference. From the three energie� found in part (a), determine three pos,ible wavelengths of light emined by a hydrogen alom. 55. Verify that equation (+ 19) follow.. from (...... '6) and (4-18). 56. What i... the densil), (If a ..olid'! Although the mass den ..ilie.. of solids var)' greatly, the number den�itie.. (in moUrn") vary \urpri... ingly lillie. The value, of coul'e. -but where hinges on the ..eparalion between the atom... does a theoretical prcdidion start? The electron in hydrogen muq be Ireated not a� a particle orbiting al a strict r.ldiu... a... in the Bohr atom, bul as a diffuse orhiting wave. Gi\'en a ditl"use probability, identically repeated expcrimenl\ dedi!.:3tl!d 10 " Iinding" the electron would obtain a range of ....alue\-with a mean and stan­ dard de\'iation (i,e., uncel1ainty). Ne\'cl1heless. lhe allowed radii predicted by the Bohr model are very close

partide'� momentum is roughly ( 109 m- 1 ) h? (b) How, plot demonstrate the same thing? (c) does the If you measured the particle's momentum, ( 1 09 m-I) 1I would be a likely vaJue. Would (0.9 X ) 0 ') m- I) II be likely "alue? Would ( 1 . 1 x 10') m- I ) h? Would (0.5 '1(' 109 m I ) h? (d) The bollom plots show two different Gaussian wave functions. Make sketches of their cone. sponding Fourier transfonns. The diagram shows the Fourier transform 59. of a Gau!>sian wave function lJ!(x) that reprcseniS a reason_ ably well-localized particle. (a) Detennine approltifllatt quantitative values for the wave functioo's wavelength and for the particle's position uncertainty. (b) Can You

A(k)

A(k)

determine the particle's approximate position? Why Or why not?

AI" I

to the true mean \'alue�

0

la) A\suming that atom, are packed into a ...olid typi­ cally j Bohr radii apart, what would be the number of mllle'i per cubic meter"? (h) Compare this .....ilh the typical mole den�ity in a solid of I (}� moUm1 What would be the value ofj?

57. In the Bohr model of the hydrogen alOm, the electron can have only certain \'eio!.:itie:-.. Obtain a fonnula for the allowed ....eklcitie\. then obtain a numerical value for the highest \peed po" iblc. Section 4,7 58. The top two plots In the accompanying diagram show a Gaus:-.ian wave function lJ!(x) and it:-. Fourier transform

�" 17

-6

l

·6

-l ··2

0

2;;

(nm)

+2 -4 ....6

3...- 4...-

.tfnm)

, 5

,

'0

k(nm- I )

0

+2 +4 +6

A 20

60. A 1 os pul..e of electromagnetic waves would be 30 em long. (a) Consider such a pulse of 633 nm wavelength la!.er light. Find its central wave number and the range of wave numbers it comprises. (b) Repeat pan (a), but for a I n.. pulse of 100 MHz radio waves. 61. I f a laser pulse is of short enough duration, it becomes rather superfluous to refer to its specific color. How shon a duration must a light pulse be for its range of frequen· cies to cover the entire visible spectrum? (The visible ..pectrum covers frequencies of -4.5 to 7.5 X 10[4 Hz.) 62. What is the range of frequencies in a I ns pulse of (a) 1060 nm infrared laser light and (b) 1 00 MHz radio waves? (c) For which is the "uncertainty" in frequency, relative to its approximate value, larger? 63, A I fs pulse of laser light would be 0.3 �m long. What the range of wavelengths in a 0.3 lAm long pulse of (approximately) 600 nm laser light? ( 1 fs = 10- IS s.) 64. Equation (4-2 1 ) ex.presses a function lJ!(x) as a sum of Equation plane wave�. e..ch with a coefficient (4-22) finds the coefficients from the given function

1:-'

A(k).

lJ!(x). The equations aren't independenl statements; in facl. one i� the in"er!>e of the other. Equation (4-22) give� when lJ!(x) is known, and (4-2 1 ) does the re"er,e. Eltample 4.7 calculates from a specific lJ!(:(). Show that when this is inserted into (4-21), the origina1 lJ!(x) IS retumed. Use the Euler fonnula and

A(k)

A(k)

-6 -4 -2

,

15

A(k)

the �ymmetry properties of odd and even functions to

_

simplify your wor\\. (Nonethele".., you will probabl)' have to look up the final integral in a table of integnll" l 65. Determine the Fourier tramfonn A�l) of the o-.cillalo!,)' function and intcll'rel the result

Ax) j{x)

{ o

=

Comprehensive Exercises

-�L s { :So + �L hI � �L

"t..·

66. Determine the Fourier lran"form A(k) orlhe o�inalor)

functionfi.:() and interpret the result. (The identity cos(koX) = may be useful.)

7t. {al Find the wavelength of B prolOD whole kinetic;

("nel']!.Y i, ("qual to It, mternal energy. (b) lbe proton is u·mall) �glll'\lN It'i belRg NUghly uf radius. 10-1' m. W\IUh) thl'i p"Itlin hehu't.e as It wave or as _ particle�

71. ACCMthng to th(" C'nt'Tgy·tllne uncenainty principle. the hfelim<- lot 111 11 'itJ.te anll me uncertAinty U in its cnt'll!) arc in't.("r.el) pnlpurtion_l. Hydrogen's, 656 nm

! (e+'Au' + e-''v') 1(x) s s + �L � lx\ > !L · �

{

-k

COSCkO"T) O

67. A particle is described by

�(X) � { �

n:-d �pci,:tral hne i, the re'iult of an elecb'On making . tran�iti\ln ··du\\n.....unr· fnlm a quantum "Itate whose

x

lifetime t'i atM.lut 10

lxl s + �w \x\ > � \:.

ted photon doe'i thi, impl)i'? INUIt":

{�

t:1\ce-··;,e., the energy of the photon-but he� it m..an'i the wu-t"fla;rll\' in that energy difference.) {b) Tn what •..tnge in wa\t'length, does this correspond'� {A... m,ted in Exerci-.e 25"7, the uncertainty principle is one C\)ntrihutor III the hroadening of 'ipectral lines.l (c) Obtain a general fonnula rel ating loA to �,.

lal s S

lal > ,

73.

where C is a constant. (a) Find and plot versus f3 the Fourier transform A(/1) of this function. (b) The functionft, ) might represent a pulse occupying = position) or finite lime (cr either rmite distance = time). Comment on the wave number speclrum f i cr is position and on the frequency speclrum if is time. SpedficaJly address the dependence of the width of the spectrum on 8.

a (a

a

69. A signal is descri bed by the

function

D(t} = Ce-l d/T (a) Calculate the Fourier transform A(w). Sketch and interpret your result . (b) How arc D(l) and A(w) affected by a change in 1'7 70. Consider the following function:

-00 <

Unfortunately.

we m;�ht u�e the symool lo£ for th� �nergy differ­

68. A function }\ ) is nonzero only in the region of width 2(5 centered al Q = O. �

.. 'i.

(3) Whm inherenl uneenainty in the energy of \he emit­

What momenta can never be measured?

a 1(a)

1.

x<0

O � :c < + oo (a) Sketch this function. (Without loss of generality, assume that C is greater than B.) Calculate the Fourier transform A(k). (b) Show that for large k, A(k) is proportional to Ilk.

(c) In general , ft:c is not continuous. Under what con­

)

dition will it be, and how does A(k) behave at large

values of k if this condition holds? (d) How does a discontinuity in a function affect the Fourier transform for large values of k?

The proton and ele..:tron had been illentifled by IQ20,

bUl lhe neutron wa...n't found until 1932. Meanwhile, the atom wa'i i.l mystery Helium. for example. ha... a mas� about four time'i the proton ma.... but a charge only twice that 01" the pratlln. Of coun.e. we n""" know that it� nudeus i'i two proton'i and IWO neutrons of about the same mas... But before the neutron''i di....:o"··

cry. it wa., sugge...ted thaI the nudell';. l:ontained four protons plu'i two electron'i, accounting for the rna!."

(electron... are "lighCl and the total charge. Quantum

mechanics makes this hypothesis untenable . A wnfint"d i'i a ,\tolld;IIR wa,'e. The fundamental 'itanding

electron

�

wave on a string <;ali..fie, L = A, and the "length of the 'ilring" 11\ the nucleus i, ilo.; di ameteT. 2Rnw:. "0 the elec­ tron's wavelength could be no longer than about 4R""". Assuming a typical nuclear radius of 4 X 10- 1') m. determine the kinetic energy of an electron standing

wave confmed 11\ the nucleus. lis. it moving "�ow" 01" "fast"?) The charge ot" a typical nucleus ;5. + 2(k,

so the

electrostalic potential energy of an electron at its edge would be -( 1/41T&()) 2(Uo21RIIIK (it would be slightly lower at the center). To e�ape, the electron needs enou gh energy to get far away. where the potential energy i'i

O. Show that It definitely would escape.

Computational Exercises 74, Here we study the plot!; of Figures 4.20 and 4.22. (a) Plot the point ( t ,II. Plot the function

f(.l)

=

I

e-� :c) from-t = O to x = 3.

sin

5.1 5.2 5.3 5.4 5-' 5.6 5.7 5.8 . 5.9

• 5.10

Tho

. 5. 1 1

rro ....... ...,.. .. .1. have • way

actioo of our objecl ol llUdy _ .. adapting the ScluilcIinpr .,.- '" applications to ....... _ A

is restricted by an extemll fan:e subject only to a Hooke'l law matter how much eDCIJ)' we to oscillate

reatcJriII

over an isolu:d bump is lubject 10 • bul if friction wen: oeali&ible, the boll

...

indc:finilely-il wouIda'l be bouad. Wo

Chapter 6. It may seem beckwmd to external forces before studyiDa

study of bound states tumI out ID be ... of the basic ideas

The slandard probIemo "" tic. and Ihus of little applk:obiUtJ, ... sons. Finl. as in aU of science, we of undeBtanding lhe quanlUm mechanics bY IIUdyiDa !be applications confronting ICiea&isaa complicated mathemItically or actions or (most often) bolla. A

the rnosI

frequ....y

�I ..1!II'i

1 . ChIpeI:r . Bound Swu: Simple Cases

5.1 The Schrodinger Equation

In Section 4.3. we discussed the SchrOdinger equation in the absence

nal forces:

h' .''1'(,.1) . • 'I'(X.I) .!. = ,h -:'-'or 2m o.r

-_.

let.

(5'1 ) =

'l'(x. t)

By substitution. we verified that the plane wave

of el

solUlion:

or

fj2{2

-'I'(X.I) 2m Using

p

=

flk and E =

I ...'. th·IS becomes mg �mt

p' -'I'(X.I) 2m

=

hw'l'(x.l)

licv and noting that p212m is just another way of writ_

=

£'1'(,.1) - KE 'I'(x.l)

=

£'I'(X.I)

We concluded that the SchrOdingerequalion is based on energy accounting_

without external interactions. the energy E is just kinetic. A plausible way 10 generalize the equation to include the effects of an external force is merely to tack on the potential energy associated with that force. We learn in classical mechanics that it isn -t even possible to define a potential energy for nonconservative forces, such as friction. so our generalization is good only for

conservative forces, such as the electrostatic force. Adding potential energy.

(KE + U(.,» 'I'(x. I) = £ 'I'(x. I) Thus, we might guess that the appropriate partial differential equation to replace (5-1) should be

h' o''I'(X.I) -+ U(X)'I'(X,I) , 2m ilr

Ji..' 'Th" " tb<' general fonn of tilt SdlJ'Odill�er equalioo. 10 the "ti.�-depernlenf·ljuaJificr could be amI/ted We mcJuJc il iO help d"nn/!lJ"h " {rom the: I'orthrom>llg 'IJeeial ca..e. the: lime--mdrpendc:nl fonn

This

time·dependent Schrodinger equation!

=

ih

o'l'(X,I) or

has passed the experimentaJ

test, so we accept it as the correCI law obeyed by a mailer wave. To determine the behavior of a particle in classical mechanics, we solve

F

=

m tPr/dP for

r(t). siven knowledge of the Rel e"'temal force on the particle. 1be analoJOUs task in quantum mechanic!'. is to solve the SchrOdinaef equation for 'ltV. I). �.i\en kno",ledge of the potential energy U(x).

5.2 Stationary States

The first step in solvlOg (5·2) is to use a standard mathematical techni.que: Stparation of variables. We as.s.ume that the wave function may be clI.pressed

a� a product of a spatial part and a temporal pan.

'1'(-', t) = �(x) <1>(,)

(5·3)

Why'! For one thing. it allows us to break a differential equation with two independent variables, x and t. into simpler equations for position and lime o;eparately. More important. although it does indeed reduce the generality of our solution. we shall see that these "special" solutions are the ones of greatest interest. Inserting (5·3) into (5·2) and then factoring out terms constant with respect to the partial derivatives, we have

h'

-2

m

<1>(')

.'",(x) �

,

+

U(x)�(x)(')

=

ffi,p(x)

.<1>(') at

-

The nex.t step in separation of variables is to divide both sides by the prod­ uct �(x)(').

h'

I

.'�(x)

-2m �(x) .x'

+

U(x)

=

I .<1>(') . 'n <1>(') ---at

(5·4)

The left side depends solely on position, the right solely on time-the vari· abies are separate. The final step in the process is a simple but subtle deduc· lion. Position and time are both independent variables. For example, the familiar traveling wave on a string is an oscillatory function of position at any given time. but it is also an oscillatory function of time at any given position, so a full description requires both x and t, which take on values independently. Suppose equation (5-4) holds when x = X l and t = t, . What about {x, r} = {X l ' 12 J? The left side would not change, so the right side cannot, no matter what t2 might be. In other words. the right side cannot vary with t. Similarly. the left side cannot vary with x. Our subtle deduction: Both sides are constant. Let us use a C for this separation constant.

h'

-2m H x)

d'�(x) dx'

+ U(x) = in<1>(') 1

d
- --

(5·5)

In general, the potential energy could be a function of time, U(x, t), and the technique we have used here would fail. The variables would not be separate.

,.. a.... I Bound Swa: Simpk' Cues

However. we will consider only c.::ucs in which the polenlial �nergy i!) time .... pendent Accontingly. we ha� IwO separllie. oRhnary �Ifterenlial cq� linked logether by the o;eparottion con�tanl C----one ,mvoh-mg Ihe spa�aI PIn ct the wa,� funcllon. and the other the temporal_ We take them one al a tllne. The Temporal Part

•

." (t)

The "temporal haIr' of (5-5) is I

ill --"'(I)

d"'(I) dl

.=

C

"'

d"'(I) dl

This qualifies as a fiN·order diOerenliaJ equalion because it involves only a firsl derivtuive of the function _ The phy-�ical a�pects of the i were discussed in Section 4.3, bUI its presence i�n'( a mathematical hurdle. We [reat it like aQy Olher constant. Although Appendix K discu�se� Ihe .<.imple method for Solving such differenllal equations. Wl' take whal will be our u... ual route of merely pre....enling the resull of the malhcmallcal solullon. then moving on to the phy�ics. The solution of (5-6) is ..

tj,(l)

"

==

,�I(Cltll

(5·7,

In one ...ense, Ihi� i'l a method of .,olving the differential equation. In fact. itl\ Ihe physicist's favorite method: Gue...... then ..ee if it works. That this funclion doe... ,alisfy equation (5-6) i� lefl a' a quick e:o.:ercise for the reader. As to the phy-<;ics. if we use the Euler formula 10 rewrite it a... e-I(CIt)t = cos[(C/h)rJ _ i 'infCe/h)ll. we <;ee that each part is a pure !.inusoidal function whose fre. Quency w (the coefficienl of 1) is C/h. A pure frequency implies. through the fundamenlal relationship E Ilw. a pure. well-defined energy Therefore. C - IUJ} 3 E III separaling I'oriables. Wi' !OCUJ on Ihou soilltions whose energy is well dejill'd. alld Ihe separation cOllslall1 is Ihal energy. This claim may seem a bit

weak. bUI as we now see. il fih perfectly inlO the resoJulion of one of the mOSt perplexing inconsi<;tencies in c1aS'�lcaJ physics_ According to assumption (5-3). the [otal wave funclion is now 'I'(X.I)

-

"'(x),''''Ar

(5·8)

No[e thai we haven'l even begun 10 consider the parlicle\ imeractions With I� �urround.ings through the polcnllal energy U(x). "0 I/J(x) i., still generaL Nonelhe_ Ie"!.. let us calculate the probabihl)-' dens!!),. '1"

(x. I) 'I'(t. I)

•

sa

..... a..

10 br plrylicGlly �pItJbk. Q WClvt' frmcliOll must IN- nomwIiUJb/�. If lie protIItbiIily per IIIliI disIance N'(x. I>P is multiplied by a small dislance d.t ...

daeD summed over all ..-ce. 1M result musl be I .

J i'I'(x. ,Wtt.

( 5-1 1)

=

�' '''''''

We might liken the probabilily to a feather pillow: II can be squa�hed Oat. fluffed inlo a rounded form. or compressed to a small ,"'olume. but even thOUgh the density and ex.tent may vary. the lotal amount of feathers is fixed. One consequence of thi!i condition is Ihat the wave function itself can't diverge. meaning that il can'l become infinite. A �cond consequence. addressed

in Ex.ercise 19. is that although the wave function might exlend over all of infi. nile space. il musl fall to 0 fa-.ler than inlegraJ

(5· 1 1 )

Jxl- If'! as x -+ :too: olherwise. probability

would diverge. (Actually. there is an " exotic" exception to each

consequence. See Comprehensi\.'e Exerci,o;e 73 for an interesting dir;cussion.)

Smoothness

To be physicalh llcceprabJe. a wove JUllctio1l must be smooth.

There are two aspects to smoothness: continuity of the wave funclion and continuity of its derivative. Ne\'er may the wa\'e function it,elf be dl\continuou�. Not only would this be contrary to what we expect of wavelike phenomena. but it would also represent a paI1icle of infinite kmetlt: energy. Figure

5.1

shows a wa\.'e

with an abrupt deviation Near the deviation. il rc'emblel, a wave of very shan wavelength. which implies ....ery large momentum_ If we measured the momentum, we mighl obtain a very large value. In the extreme limit. where the abrupt deviation become" a discontinuity. the '\hort wavelength" become.\

Flgurw 5.1 An abrupljump acts like . very short wlvelen,th.

zero wavelength. implying infillite momentum and kinetic energy. This is physically unacceptable. A dIfferent roule to Ihe I,arne conclu�ion is through

the mathematical form of the Schrooinger equation itself. Its kinetic energy term, the far-left term in (5-10). has a �econd derivatIve of 1/1(.\) with respect to

position. If the wave function had a discontinuity. 11\ first derivative would be

infinite at the point of discontinuity. and It\ second derivative would be a weird sequence of plus and minus infimties. We simply cannot make physical sen.!oe of such a pathological function. With one exception. the first derivative. or slope. of the wave function must also be continuous. If the first derivative is continuous. the second derivative-and thus the kinetic energY-II, finite. which must hold i f the potential energy U(x) and total energy

£ are

both finite. for the Schrooinger

equation i!'> ba!'>ed on the,e three energies adding up. The exception arise.s if we

do allow the potential energy

to Jump to infifllty at any point along the x-axis.

for then the restriction agamst the kinetic energy bemg mfinite at that point fails. Obviously. mfimte potential energy is not completely realtstic. but the notion of a solid wall or barrier is a useful onc. In confimng a partIcle 10 .some region of

are

stoUl �pace, we often imagine that it i!
5.4 A Review of Classical Bound States Before we consider our first simple case of a quantum·mechanical bound par­ ticle, let us briefly review the physics of classical bound particle... In clas..ical mechanics. we learn that when nonconservative forces are absent, total mechanical energy, E = KE + U, is conserved. Plots of energy versus posilion gives the simple example of a mas.. on a spring of show this clearly. Figure spring constant K. The spring's potential energy is a parabolic plot. The total mechanical energy never varies from whatever value is initially given to the system-it is a horizontal line. The kinetic energy is the "distance" above the potential energy parabola but below the total energy line. The..e "heights" are labeled for arbitrary values of x and E. Increasing E would increase the amplitude A of the motion, but at any finite E. the mass is bound within

S.2

!K.-c2.

the

region -A S x S +A. Now suppose that at the arbitrary x-value shown in Figure S.2. the ma.
Figure 5.2 Energy versus position for a mass connected to a spring. Spring constant

K

rM\1\MMO

Total energy (k ineHC plll� pot('nlialJ eomtant.

Turning

- ,),c� --\"':::::::;:: Ipoints

Al luming point�. kin('tic

ener!!)" i, O.

-

-:

//

'

1

I �.

OUI�id(' thl'm

it would have

10 be

neg"

" _ '_ ·_ _ :.,.-

_ _

"" .....

.

plot i� Ih('rdoT(, l line, a horiwnal

E = KE + U

d

. The ,1I('a ouL,i e

·

.

the potenlial

energy eun'(' ;� the da\'iically

-'>+�__"'_ U ___'_,_--c t,,"''-idd('n region.

_ _

-A

-

is

I"

0

Position

x

+A

141 Chapter 5 Bound Sa.us: SI1DP� Ca,-.es

mass reaches the poin! x = +A. the energy is all potentjaJ and the kinetic energy is O. This is a turning point. The mass is stationary, but there is a force

on it to the left. so it must return toward the origin. It cannot proceed to ValU es If it did. the potential energy would, of course. be larger,

ofx larger than

+A.

but since the total must be con�lant. the kinetic energy would have to be nega. tive. Because for any given E. the mass is confined 10 values of x " inside" the

pOiential energy plot. the shaded region outside is known as the classically

forbidden region. Figure Atom

I

5.3

shows a different system in which bound states are possible. 2 moves in response to their shared

is fixed at the origin. while alom

eleclrostatic interaction. We won't study the origin of this potential energy. We simply note that atoms, due to their dispersed orbiting electrons and compact positive nuclei, do not attract or repel as simple point charges. The potential energy is positive and large for small intera{Omic separation x, and has a mini. mum value that is negative. then asymptotically approaches tive side as x --...+

00.

At small values of x, atom

0

from the nega.

2 is strongly repelled by atom I

(F = -dUldx > 0), and aI large x, it is attracted (-dU/dx < 0). Accordingly, a bound, oscillalOry state-a diatomic molecule-is possible, but it depends on the total energy. If the tOlal energy is negalive,2 labeled Ebound' atom 2 will be

bound. osciJJating belween turning points X(l and xb. [f the total energy is posi. Eu bouml' atom 2 has only a single turning point. Were atom 2 n

tive. labeled

moving IOward atom I . it would momentarily stop at this point, then mOve away. never to return. Although Ihe potential energy does increase as x

becomes very large, so that alom

2 slows due 10 the attractive force as it moves

off, there is no outer turning point where the potential energy rises to meet the total energy. This situalion differs from the simple mass on a spring, for which there is no total energy large enough for the maSS 10 be unbound, because the

Figure 5.3 Energy versus position for the interatomic force

between a large atom fixed at the origin and a small one free to move. Atom J

o

'''''''

Turning / point

o

j 01-1-;""'-----1,--;''----+ I�-----Turning \:/poinlS-......, vio l-h � ;.,

�'iY ma} be n��ali'e due!u the u,ua.! fn:c<Jom 10 choooc lero poIenual en�rg}·. We ,'1Ioos
Atom 2

- - '

\�E

U(x)

.-­

, � " � _ _

_ _ _ _ _

Cla�"icaJly forbidden

regIOn

J

£""boUM

_ _ _ _ _ _ _ _ _ _

£_00

�

AgUN S.& In quaDbIm .. o c b .. " n , harmonic-o...cilialOr potential energy fl(r: increa.-.e!'. forever in both dirttlions. bound ,tatel aft' ItaDdina wavw. re hound ... tate ...ults when a panicle i!'. "caught" �lween IWO In �umm.u)', " IUming. point'" where the potential energ) ri..e<, 10 meet the 10tal energy. It i� worthwhile to take a brief look ahead. In l'ommon with d�sicat bound .,tate..., quantum-mechanical bound ..tate... have con...lanl tolal energy, shared between kinetic and potential. But in quantum mC'Chanil:.... we don', speak of 3. particle'... po...ition at a given time-a cla.....ical notion. Rather. we seek the wave function 'l'(x. r). We will "oon find that IR quantum mechanic.... bOund states are standI".': \\'lI\·e,f. Accordingly, JUM a... there are only certain discrete standing waves possible for wa\'e� on a string �tretched between fixed ends, there are only certain di"crete �tale... possible for a matter wave bound by a potential energy. Figure 5.4 show... the first four polisible ...tanding wave... obtained in a quantum-mechanical treatment of the harmonic 0.,cillal0r, plol· ted on four horizontal axes whose heights correspond to their fC!->pecti\'e ener­ gies. (We will lise this way of representing both matter wave... and their energies on the s(lme plot throughout the text.) Different energie... are po...sible. but not all energies. The "fundamental" (one antinode) standing wave is the lowest in energy; the "second harmonic" (two antinodes) is next; and so forth. Besides being restricted 10 only certain energies. the standing waves of quantum-mechanical bound states differ from the classical case Irl another intriguing way. Whereas 3. classical particle is restricted to location" "in ...idc" the potential energy plot, wave functions for quantum-mechanical bound states often extend beyond the classical turning points-into the classically forbidden region. Consequently. there j" a probability of finding the particle where, classically, it cannot he!

5 . 5 Case

1:

Particle in a Box-The Infinite Well

The situation in which the particle-confming potential energy U(x) allow s the simplest solution of the lime-independent SchrOdinger equation is the so­ called particle in a box. or infinite well. in which a particle is conflrled between insurmountable, infinitely rigid walls. Classically, it would simply bounce back and forth. bUI our quantum-mechanical solution of the SchrOdinger equation will be standing waves. The forces ex.erted by ordinary walls of wood or brick are complicated We choose to coniine an electron of charge q = -e between simple "walls" of electrostatic potential energy. Batteries are connected to two parallel plate An electron moves in capacitors, whose centers are separated by a distance .

L.

the region between them, as shown in the center diagram of Figure 5.5.3 Small holes allow the electron entry. The batteries' positive tenninals are connected to the inside plates; so once inside a capacitor. the electron experiences an

.

electrostatic force back the way it came (Remember that the force on a

negat;I'e charge is opposite the electric field.) Because ideally the electric field is confined to the space within each capacitor, the electron would move at con­ stant speed and kinetic energy everywhere else. Correspondingly, lhe electro·

static potential and the e lectron's potential energy are constants everywhere except for in the space within each capacitor. Defining the batteries' positive

'Wr ust' lbo: ran'lililu" � IO� """ (neIlY. bul,n pncUc:aI appliaIiaaa.. ad ....... ,ondu,"1<X �qu.nrum dob.� die � ... � on thr alUmIC�. mucb lhiaaa ... .y "rca\"' capa>;nor.

_

,III '

a

.... -= ..... c..

httide ID • boa

I

,

•• o

JIll; t1C'ct!t: new '03 each Cqat:llut t'u'TtJ a f.:rccc

.. , I-- Ir,lIIoud o" th.ttC�lrot\

•• o

",,"cor:. \.,

.tl 'I· r'C'�tntr"t J\ hitthc:r ct.!�!'.!: (.

,.,1

�ntJal tnc:rlY

,.

:!!..

�

:11 � th

d'lI

- ,.\"0

!lyE

,-r .he

s bo' :-:1 Its k.ruetk cper¥)

:u I> to ) i

U

lui

"'-'--_ U - 0 L ,

o

"'lUI" I e

t'

' 11\.1 .

t' It �:1II �,,�'n l

r t. J 'rr.)(lIe :hf'1:11111. UI(

n:lurn�

V = 0 and their negative tenninal� as V = - Vo' the eleclrOn's potential energy (U = qV) would be 0 when between the capacitors and Vo == eVo when outside the capacitors on either side, The potential energy is shown at the bottom of Figure 5.5. terminals

as

Now suppose the electron is between the capacitors, with kinetic energy greater thun eVo' Once it reaches the entry hole of a capacitor, it would slow down, but because ils energy is higher than the barrier. it would pass com­ pletely through, never to return. On the other hand. if its kinetic energy were less than

t'Vo. it could not escape. Its kinetic energy would fal l to 0 before it

reached the capacitor's negative plate. so it would return [he way it came. It is such a bound state that we consider. We will see in Section

5.6

what difficulties arise in trying to solve the

Schrbdinger equation without further simplification of our model. lndeed. it cannot be solved "in closed form" (wilhout computational approximation techniques. discussed in Section

5.10). We arrive at a problem [hat is solvable

by making two changes. First, to avoid the complication of addressing the electron's behavior in the small space within each capacitor. we assume the capacitors to be of negligible width. so that the potential changes abruptly. Second. we imagine that the capacitors

are

charged to infinite potential, that

the ..-ails IU'C' IDfinll�l)' hash. Euctly wby ahll adminedly UftI'MhItM: ...... bllfl simpliflC"s, 1M math "'c dc:fer 10 Section �.6. Here we SImply exploit tile

..... ... "" _ ...

,;1.-o:luslon: Tbe \\a\T (un..:lion cannot �.\tcnd beyond aI'I ;,vu.iwlY hiab pottn. IW C"nctIY banier· II mu)ol he' 0 outUde lhe �alll." With thne two chanp. die poccnt1al �ntraY funclion of FiBU� 5.5 brcomt the abruptl), ehanJ;lna and il1fLtlltcl y hllh U(.t) Soh(l"'n in fi,urc �,6. the famous ....... wtrII poIcntiaI ("ft(lJl'. The n.�{kr m" tlllO' ';lII\f), it 11\ the' I'Cgi\ln bt·twall the wail)o and Ihclt t'lisurilitt thal thc O\'C'r all ..... a\(' lun.. titln .' l.'ontlnul'U' In th\' (\'t:iul\ hclwe
d�l/J( \ )

d\�

Rearranging. we obtatn

,1'"" r)

_ .. -

til�

,

2mE h' '"

,

x)

'"

(S·121

..... he�. for convenlencC'. we have made the definition

Equation (5·12) demllnd� thai the second derivalive of I#(.t) be propor· tional to t/J(:d. and. importnntly. the proportionality con�tanl is negative, So what is I/'(x)? As we did to obtain solution (5·7) from (5·6). we adopt the method of gues\ing the answer, then checking that it solves the equatiun. (Again, see Appendi:ot K for further di"cussion.) What function hus a sC'cond derivative that i.. a neglllive constant time.. lIsclf? Any sinusoidal function:� lei us try sine. To yield the k2 in (5·1 2). we need a k in the argument.

t/J(X)

-= A sin Ia

(5·14)

The reader is encouraged to plug in and verify that (5-14) satisfies (5·12). no

maner what may be Ihe value of multiplicative con.<;[ant A. Other sinusoidal funclions, such as cosine, would satisfy (5-12) mathemalically. but one of our physical requirements rules them out. The wave function muSI be continuous. It is 0 outside. so the wave func· lion inside the well must fall to 0 at the walls. At the left wall. x = O. (5-14) already satisfies the condition. as cos Ia or any other function that is really just

·Rco:aIl . limil. duAcaJ ...... pftIbkIn: . -'llll I". n. .... F _ _ a-.. -Jr. - -. ... -I'.or - ... 4ll'ItIi'. Heft" too • fUlldioaa i. 1* .... 1ioMI . eo .. ...... deri'lMivc. wid! • tWpIiWl l*upoi......, ----' mid _ � m.t z(,)_ .. -'� _ Oft .Ipri", offon:e

' S C..e

I: r.nlc"Ia. a.-1"1111 ..... .. ..

probability density. Furthermore. although the spatial part here and in

every other ca� in this chapter is real. we will continue to use \I/t(x)p or "'·(:tWl(x)

rather than rp:!(:d. <'0 as not to get into bad habits that might cause trouble later.

The integral

fsin�((II1T/L)xJ tb. equals U2. independent of

F... 5.7 bdihi.1e well 'MaW functions.

L

o

R.

so that

Thu�, we have the normalited, continuous wave functions representing the allowed states of a particle in an infinite well and their corresponding. energies.

O < x < L

(5-16)

x < O, x > L Let us look more closely at our re�;ults. Figure 5.7 shows the four lowestenergy wave functions !J1(x) plotted on horizontal axes whose height.;, are proportional to the respective energies

En (ratios 1 :4:9:16). Because the temporal

wave at each level-the functions

are in many ways typical standing waves.5

parts of the wave functions oscillate in time-the reason for the fai nt inverted

Flgu,. 5.8 Infinite wclt probability densities.

classical

In fact. the quantum energy levels may be obtained solely from standing-wave conditions combined with p = fik: Wavelengths of standing waves are

given by

A"

= 2UIl and wave numbers by kn

= 2m>.. =

\.v4(x)P

_¥....:>..L�..L�+-­

7m/L, so that

Figure 5.8 shows the corresponding probability densities. \!J1(x)\2, and we clearly see one of the hallmarks of standing waves: nodes. where the probabil­ ity density is O. But how, we might ask, can the particle get from one place in the well to another if there is no possibility of it ever being found at a point in between? The answer is basic to quantum mechanics. It is a wave, not a parti­ cle. We know that a standing wave may be thought of as two waves passing one another in opposite directions, and while bouncing back and forth. they

,-0

x=L

fonn nodes where the displacement is always O. As a wave phenomenon, nodes shouldn't surprise us, and we cannot demand that the "particle" behave as a classical particle. In particular, we can't watch it, for the photons we would need to bounce off it would introduce potential energies that we have

�� standing wa..e.. an .�_ pa-...,

fta. quanrum.� SlaDdio& �--­ TaW pn:bbifuy mua _ v.msta. n.� of the situation prevents our having detailed knowledge of the particle's Yar)'tDI Nndion DlkipIyUc � iIa'l,. or 00II& &'I fof 'fr.I\a, (JfI • t&Jin&, but Alberis of(J) �-w. behavior. But this is no weird special case; lack of knowledge of preci!>e wbich I$. never O. WbeD Re � I) is 0. _ '9\x. ., .. whereabouts is a fact in the real situations to which quantum mechanic!<> i!> muilllUlTl. ItId \'lr;e wna. We miPl va.Iia • applied. We cannot, for example, say exactly what an electron orbiting a lluantllln-� aanding __ &'1 . .. _ lpinning ;d)out!be,,;.....u. ill die Re-1II1 ,.... nucleus is doing. Given these unavoidable limitations in the micro�opic

ignored in our solution of the SchrOdinger equation. The assumed simplicity

all

""

114

CNtpt.r 5

Bound Stales: Simple Casn

world. the predictions of quantum mechanics have succeeded. While those of classical physics have failed. Sight unseen. the "particle" is a wave. The standing wave of minimum energy is known as the ground state 8fId it.. energy as the ground-state energy. It is in the ground state that quantum. mechanical behavior deviates mosl from the classical expectation. Maq Important, the kinetic energy is nol O. A bound particle can�ot be stationary. As we learned in Chapler 4. this would violate the uncertamty principle, for having a position uncertainty comparable to L and a momentum identiCally 0 is impossible. Another deviation from the classical is that in its ground stale the particle is most likely to be found near the center. Classical1y. if we Were t� "rum on the lighls" suddenly and catch the particle somewhere in the course of its constant-speed back-and-forth motion. it is equally likely to be found anywhere within the well. On the other hand, the correspondence prinCiple (Section 3.4) says there should be a limit in which the particle behaves classi. cally. This limit is at the "other end"-large n. Larger n correspond to shoner wavelengths (An = Wn), and the shorter the wavelength, the more paniclelike and classical the behavior should be. As we see in Figure 5.8, the larger the n the more evenJy the probability of finding the particle is spread aver well-the classical expectation. Before we apply what we have learned, we reiterate that the wave func_ tion tells us probabilities of finding the particle at various locations. The inte_ gral of the probability density (probllength) over aU space is the total

th�

probability of I . So to detennine the probability of finding the particle in Some restricted region of interest, we simply integrate over that region.

EXAMPLE 5 . 1 An electron is confined in an infinite well. in the ground stale with an energy ofO.JOey. (a) What is the well's length? (b) What is the probability that the electron would be found in the left-hand third of the wel1? (c) Whal would be its next higher allowed energy? (d) If the electron's confines were roomier, L = 1.0 mm, while its energy remained 0.10 eV, what would be the probability of finding it in the well's left-hand third, and what would be the minimum possible fractional increase in its energy? , ,

(a) In the ground state, n = I . Inserting values in energy expression (5-16), we

obtain 0.10

X

1.6 X IO-L9 =

] 21T2(J .055

J

2(9.1 I

X

X JO-34 J ' sF

10 3 1 kg)L2

(b) The probability density is f¢(x)J2.

, (fL

I¢n (x)1

�

"'''

)

- sinL L

'

=" L = 1 .94 nm

2

"."x

L

L

� - sin'-

As noted above. we sum lbi!. over the region of interest, x = 0 10 x = U3.

probability =

J

pcobability length

dx =

J/� (x)/ n

'

dx =

J o

IJJ

2

-

L

sin-'

(",TX) -

L

dx

'., c.. I , _10 10 . ....-'""' - -

I'"

•

o

I 2n1f -- Sln�3 2mI' 3 I

- _.

�[� 6

L

-

L

';0( 2o"/3) 4,1'11"

J

Thus. for an electron in the ,. - I !>tate, , , I 1 21r probability - - - - !Oin:\

2.".

3

3&

I

3

- 0.137

�

0.196

The probability IS les!. than the classical exp«tation of one-third, and it a,n:<s with the probability density plOh of Figure 5.8. In the ground stale, 1I/I(.t)l! is largest In the center and smaller near the walls. In fact. the probat'lility of finding the electron in the Cf'ntu third must be I - 2 X 0.196 " 0.609.

(c) The energies are proportional 10 ,,2, '0 the n

ground-..tate energy.

...

2 stale is 0.40 eY. four time!'. the

ed) To evaluate the integral. we need fl. 0. 10 X 1 '6 X 10

19 J

n1112( 1 .055 X 10'34 J ' S)2 "'"

2(9. 1 1 X 1 0 lI kg)
�

,,

-

3.2 X I()�

We can use our previous resuil, �imply substituting the new value. probability -

I 3

I 2(5.2 X 1�)7T .- sin-- -- --2(5.2 X

3

IOS)1T

Whatever may be the precise value of n. the probability i� e��entially one-third. The smallest po!.sible mcrea!.e from the nth state would be to n + I Notmg that nothing ebe change�. so that other factors cancel top and bottom, we have

,

fractional change =

Erlnal - E,rul,al £ 011 11

1 1

=

Because II is so large. we can ignore the 1/112. 2

fractional change .. - = n

(n +

1)2

-

2 --::-=: -:-: :<� 81: 4 X Io-

5.2 X

2

II�

-'-� n.'

=

;;

+

1

nl

10----6

Even though a 0.10 eV electron is quite slow by classical ,>tandanl\. we ..ee here that when occupying macroscopic accommodations measured in millimeteN. it is a very high energy state quantum·mechanically speaking. Its probability denSity is a blur-a million nodes and antmodes evenly spread over the box---and il� "energy spectrum" would appear continuous_ The famous eVidence shown in Figure 5.9 may help dispel ...ome lingering doubts about quantum·mechanical bound stat�. Electrons on the surface of a piece of copper are bound within a circular "corra1" of roughly 1 4 nm diame·

ler. formed by 48 iron atoms ( the spikes at the perimeter). Though more com­ plicated than our simple one-dimensional cases, the result is qualitatively the

-

, .. 0.;:... . ... s..: ..... c.. Fig,," 5.9 Elecuon sWKfinJ wa�'e bound In a quanhlnt com.I.

�ame. We see a dear pallem of concentric antinodes-a two.di mensiooal .�tanding wa\le. The image was produced by a scanning tunneling electron microSl.:llpc! (disl.:u"ed in Section 6.3). Notably. It represents a probabiliry , den�II)'. not a wave function. and IS the combined result of many electron dt:h."cti()n�. It i..n't watching a given electron. Let u� now take a look at one of the mo�t important characteristics of the quantum well. il\ di ..crele energy Jump... and the most conspicuous evidence of t ht ....C' Jump,: the dhlinl.:tive wavelengths of photons they produce. EXAMPLE 5 . 2 -

Suppose .:m ele..:tron i, trapped in a quantum well 5 om y,Ide with walls that � high

enuu);h I'M the infimte y,ell to be a good appr(namation. (a) The electron make, a

transition doy,n to its ground ,tate from the ne...I-lowe�t �Iate. gl\ling energy 10 a pho.. tun \-Vhal are the photon', energy and wa\'elength? (b) How would photon W3\e.

len�th'i pmdu�:ed in other !ramiliom compare') Longer" Shorter? Both? N

(a) Energy j, con�en.ed. What the elel;tron lo,es mu,t go to the photon, Thus. the photon energy is the difference between the ele(:tron 's energies.

),,'( 1 .055 X I O � J . , )'

2(9. 1 1 x 1 0 .1 1 kg )(5

AI;I;ordingly.

A " (b)

- - . 7 . 23 X 10 2 1 J = O.045 eV

X 1O-9 m ) 2

( 3 )( IO� m,s)(6.63 X 1O-'4 J · s) 7.23 x 10"21 J

"m = 27.5 .-

A, we �ee in Figure 5.7, the energie� get farther apart. No ty"o are as close as the ground slate and n ::r 2, ,0 tramilion.. between other level'> would produce higher-energy. \horter-w3velength photom The characteri Mic of having a

ma\imum wavelength would help to di\tingui�h Ihis syMem from others-the

hydrogen alom. for imlance. in which higher quantum energy levels actually gel cloo;er together Spectral " fingerprinls" are mdi<;pensable W i n dows on the

mner �ork.ings of quantum \}'stem�.

REAL-WORLD EXAMPL. THE QUANTUM WEll. IA'EII .... . Iipl ....... 1M'" :

Many new optoe�ic cIn'.....

� OIl lhe pl«... ... -... .

7 ... ... ''"'J of .. ... ... ..... .....

kvels in . quanlWD well, i a � wiIIII " .... "" .. oI 'I e d•• __ rials. In a sc�'tor. 1IIU.y � cIDed "n. 1M. I. _ __ ... rather freel)" m\kh .. &hey do ia • ...-.IDe... .. , ,.,. _ . Mel .... b __ \'�nicnce, WIC define u O. OnSiMrily. Iheir -., � .. .... 0. 11Da...... buritd III deepn el'lCf'J� &rIC vaknf;:c e� .. ale __ dPdJ ... " ilia �ml.:ondlJCtor atonu.. U ooe of � is mi� . . ...... (it........ d«tnln �'an drop down to fill lhc hDk• • process *Pic" . Rpre ',lOa. 'nil . 1M �IlU(\;(' \If light in dwe diode Iucr and the noalueI u,tIl..,tai.. ... (l.ED). A I)pi.:a.\ �n�'gy Jump down inlOdwe hoie b allout 1.50 eV. wtUcb ai¥CS . � ..... kn�lh ,If atll.lUt �no nm, Bec.u� the ronduc:tion c*'roas ale ,.." bound ... ....u dlmen�i\ln'. Ihe=) hchav� mu.:h liu .:Iaukal partida that can like CIIII • ,...,...... 01" encl)!.lI':'. whu.:h hampers lbe establis.hment of "lui..... action: Ihat ia., tho produc-

li,m (II mall)' ph\llon, of �s.scntill.lly identical w."clen..... The quanlum wdi llber �mbed!O an extremely thi.n layer of one kind of 1ClDkoa· ductnr hel'M:'�n thid.er laye:n. of anolhiCr kind. 'The interf.:u betWCClft tho �a c\m,l1Iule l'Iectnl,latic walh. and the thin hlye:r the quantum w�n, shown in Piawe �,IOn, Not \Inl)' doe, thc well mne: the �nlClJ)' of the upper level in the elcctroa,'a d\l....nwani tran�itllln more p�d't<, but it Bls.o increun the energy IOina to the phD­ Il'n b)' the dillerence bet.... een the quantum well's. ground "talC and 0, which can be tuned a,:cl>rdmg III the ....ell Width-a \-�I')' nice: fcaturc=.

is.

Applying the Phyma

Gi,cn a :\ nm ....".k quantum ....ell in Figu� 5. IOb, what is. the w.velmJth of the em1tted phUlon-/ •

TI

N

The ene..,.)' 01 the 11

-

I ,talc i,

w1( I ,055 X \0 :\4 J _ __ :, ')l � � 2 ... 6.1 X 10- 1 J 2(Q, I I X 1 0 .ll I.g)(3 X 10 OJ m)!

-

o'042 cV

Adding thl� to Ihe energy diUerenclC, 1.50 eV, from thIC zero level down to the

valen!.;e level. the neow phown energy would be 1 .54 eV, or 2.46 X 10- 19 J . A

-

(J x IO" m/s)(6.6:l X 1 O- :W J ' s)

._..

2.46 X 10 19 J

-

801 nm

The model we: have: di..cus..a\ herc= shows how characteristics of the quaatum we:1I are e),ploite:d, but II doc\ ovc=rsimplify. In reality, the da:trons in the semkoft. du(tor, due 10 their inleraction� with the !Oc=miconducting material, behave u 1houah their ma.... is ..maller, "'hich would increa� Et and thus the photon erter'JY, Mole­ OH'r. the holes in the: \'aknce ,tate:' are in the Foan1C physical region as the conduetioD de(tron,. '0 they too gt.:1 trapped in the: quantum well and thus also Usume quu­ tiled energie:\, (Semi(ondu(tol"\, induding holes and effective mass, are discuuc:d iD Chapter 10.) In fact. more: ellon i.. now going into lasers based not on the quantum well bui lln the quantum dot (di\(u,<;ed in Progress and Applications) in which elec:­ tron, are bound in all three dimen�ion�, Another factor O\'Crlo\'lked in our trc=atment of the quantum wel1 luer is that me we:1I really l,n't deep enough for the infinite well energies to be an especiaUy aood approximation. What do we do ,f it isn't infinite?

- _ . _ _ _ c..

_ .. " ..... - � ""'IY. m

•

--. --�-..: - 1tE

lJ - Vo

5.6

Case 2: The Finite Well

Our .second simple case, the lIolle weD, in\'olves a more realistic � encrJ)' function rhan the infinite well. so it is a more accurate model in many circumstances. As the name suggests. we simply drop the assumption lhiI lbe "height" of the polential energy walls is infinite. The price we pay for this. however, is thai we can no longer wive the SchrOdinger equation completely for me wave function... and energies. AI .�ome point. we must resort to cotnpt,.. tational approximation techniques, known as a numerical solution. BUI lhere is much 10 be learned about basic quanlum mechanics from the finite well befOtt facing this difficulty. First. the polenfial energy function thm defines the finite well. depicted in Figure 5.1 1 . is given by

V(,)

O<x L

(5-17)

Ir still jumps abruptly al the walls but to a value Vo that is finite. Because it ;s finile, a particle could be given enough 10lal energy 10 escape (E > Vo)' bul we restrict our anenlion 10 bound Siaies where £ < Vo' On Ihe other hand, though the potential energy is finite, Ihe lingering. somewhat unrealistic assumption of an abrupt jump means that the force, F = -dV/dx, remains infinite at the walls, so a classical particle bouncing off the wall would still do so instantly. For easier reference, we rewrite the time-independent SchrOdinger equa_ tion (5-10) as 2",( Vex) - £) "'(x) /,2

(5-18)

As before, we have a proportionality between if;(x) and its second derivative, but the proportionality constant varies as we move from one region 10 another region in which Vex) is a different constant value. The sign of the constant is crucial. In the region between x = a and x = L, where we have defined Vex) to be 0, the constant is negative-if;(x) and its second derivative must be of opposite sign. Inescapably. this describes an oscillatory, wavelike function. As shown in Figure 5 . 1 2 , if if;(x) is always concave down when positive and concave up when negative, it must continually return to the axis. Thus, when E is greater than the local value of V, if;(x) is oscillatory, as it is inside the infinite well. But now we must also be concerned with the regions ou(side the well, where Vex} is defined as Vo and is greater then E. The proportionality constant in these regions is positive, so I/J(x) and its sec­ ond derivative must be of the same sign. Oscillatory functions like sine do not meet this criterion, but exponential functions do. It is left as an exercise

H«I" E.

t M) ( r '!\o rl('jOf,IJ\"C• •,,,J • • no.!. II'.[.it tI;I,' " "rl''''''Mtl!" '"'1M. BC'::lU� " III al.... ; l\� CO"'':'''II:
I

d'I" rf'iIC"

1

•

Where U ' E , L.' r > n,
,\'Uft \hl" lU\;1\ \Ill" and I't \ 1 1'1,'b all­

. . .....

.. •••••

.. .

.11-

�h,,,,,, n h'�cth'·I. ttl", life I,httelem eLllnK:\. �" neither Ih..It rel.lul � hnFhb ,,,,r thl'if m"'r'c�'I\."" ;He

1I�1(,:ld il j, 3.

I1lc:lllmglu!

dc,;l)'mg e\p',ncnllal,

to verify that when U(x) is simply Uo' equation (5-18) is satisfied by a func­ tion of the form "'(x) =

Ae'�

where

a

IE

{2m (Uo

'J

h'

E)

(5-19)

But t/I(x) was 0 outside the infinite well-why do we worry about these classically forbidden regions now? The answer is that besides the troubling nolion of negative kinetic energy (which we return to later), we have no reason to exclude the wave function there. Exponential functions are smooth, and they don't have ro diverge. So long as we assume e-CU to the right of the well, where x > D, and e +tU on the left, where x < 0, both sides would be decaying expo­ nentials outside. Thus, they fIt our physical conditions outside the well, and in fact, we would never be able to satisfy the smoothness requirement at the well boundaries/walls if we did exclude them. Then why was t/I(x) zero outside the infinite well? It is a limiting case. The limit Uo -+ 00 implies that a -+ 00. The decaying ex.ponentials faU off infinitely rapidly outside an infinite wel1. The wave function depicted in Figure 5.12 clearly shows the difference in character between the region inside the well (U < E), where !/I(x) is oscilla­ tory, and those outside (U > E), where it exponentially decays. Because it sat­ isfies the Schrodinger equation, does not diverge, and is smooth everywhere, it is physically acceptable. So where does energy quamization come in? Figure 5.13 shows an acceptable finite well standing wave with its "exponential tails" in the classically forbidden region, while above and below it are marhematical solutions of the Scmodinger equation for slightly higher and lower energies. As we know, solutions inside the well are sinusoidal, such as sin kx. But from equation (5-13), the wave number k is V2mE/ft, so

FIg_ 5.13 UnieM lI'Ie enel!t)' .' ju�t righl. a �oJutjon or (he Schrodinger l'qu.uion "..iJJ djH�'l!e at infinity.

1

.......

:1!

........ Fu", h;.l'h. \ 1010 'h.lrt.

... .. (",'ITC,·t E. ,I"PI-' 1<1<1 rI¥lu at wall

4{' <0 U '

d '.

>0 --,-;dr-

a higher £ implies

II

shorter wavelength. At thc left wall. each of the three

functions is an exponential smoothly meeting a sinusoidal function, but their

slightly different wavelengths cause them to be progressively more out of step [hey approach the right waH. The middle one hits the wal l al just the right angle 10 join smoothly with a simple decayillg exponential. Mathematically, an exponential that increases with x is also II solution. as the :!: in equation (5-19) as

Jells us. and the higher and lower energies, hitting al the wrong angle, can be made smooth only by adding such a function. Thi s

increasing

exponential

CIlUseS r/I(x) fO diverge as x -+"", so these solutions are physically unacceptable

Although these "bad" solutions do obey the SchrOdinger equation in the classi­ cally forbidden region-their second derivatives are always the same sign as r ....atures are just wrong. We could nail down ¢(x) on the righl ¢(x)---e-lh ir cu

side. but then the left side would misbehave. In fact, there is no way around the problem. Wave functions of only cenain energies can satisfy the smoothness conditions withoui diverging at one end or bolh. Energy is quantized. At this point, those interesled in a detailed solution of the finite well to the point of obtaining the energy quantiza tion condition should refer to A Closer Look: Solving the Finite Well.

A. CtOM!" Look The potential

Solving the Finite Well

energy gi\'en in equation (5-17) and Figure

5, I I di\'lde.� our one-dimemional space into three distinct region\: region I. 0 <. l < L. where Vex) is. by definition, 0; region II, ' D. where V{.t) is the constant Vo: and region III. x � L where U(.t) i'i also Vo' When the potential energy

change1 abruptly from one region to another. we solve the SchrOdinger equation for !/I(,l) piecewise. meaning indepen­ dently in e;l<.:h region. Then, by adjusting the arbitrary con­ stant!> accompanying each ¢(x), we ensure that the wa\'e

function, �s a whole. obeys (he physic�l requirements of smoothness and nonnalization. We begin our methodical approach with region I, Region I, 0 < x < L In equation (5-12). we already have the Schrooinger equa­ tion for this region of zero potential energy.

where

",. _ _

• • ew

�ously. "'t &fJ1tN that �Lnb

sabsf.es Ibis difrtttDbal

cqIIltlm and that il ths lht 1"C'qUII'tlnCnt of btina 0 al W Itl't .a11. matl;hing the zrro ",a\'e fUIk:tion OIIl1lde me '*CIt But In !be finite ",til case. t� wave fu.ncttoo is lkJl 0 outs.ide thr ",ell. 10 Inb alonC' cannot be COITt'Ct. A more p:nenl soIu, tion or such a sccund.onier (InvolvUlg the second dtrivallw)

Jiffcrtntiai C'qUIIlwn will haw ,"'(1 arbitrary 1.:0I'lStanU. HIm' tialh l'Onstanl'l of mk,nllion It IS l'Ommoa to npeS5 the5e �"\Q�t:u'lt!o as multiplintlVe "oeffaciC'nts 0( two i�ndenl IURl"lions Iholt IlJt' summed. Stt Appendix K for more detail. but the )1e:ne:rul olullon t""t ,IXs alung "'ith these words i�

...... .. . .., &. B«MlIC' U'{c) II the same. ... ...... ...... at ... S<...... _ " .... _ _ u .. ..... u. �

But nnw Ihe pbym-ally � term IS ..... ....

dl\-etJC'!. as .l _ +x fot any ,,&lUi of Vo- , .....

G,

RJthe:r tlMn lou", lR)! the: matht'matu,:al reas,}ft, hlr why this ,hl)ulll tic the: �cne:ral ,(Ilutiun (II' (5·1 �I, II ., much murt 101p',n,lOt hlr our PUI'JII.I'l'\ \Imply III t'C cunliJ�nt thiU it 1,1 III

1I(ln IS cenam l) 'moolh. anJ II JI\�rges nuw!ltre so it (lbelS But to be- cenalO that the ",01\1: o

(Iur phY'I"al rc4Ulrcmenh

fun.:t!(ln hit all 'pa':e " ,imllarll

we11

bcha\'CJ. we must

(on,.lIer the other regl(\O'

Region

whert

a

_

L

Ud\'in, lhruwn 001 di\'erJenl terms, our ..xl i5 normaIil.abIe.

and ib pieces � J:�fNJfd'f'h' IomOOlh.. All thai

remaia.. it. to

tnsure that they obey the ImoothncS5 requi�mcnb AI the buundllries where the re.ioos mC"et. Reaions II and I meet at

- n. and re,iuns

I

and III meet at .l '" L Smoodloeu

nitt well. we hall only Iwo-
arle:o>-I:'C'cOluie the conditIon on the dcrivAlive doesn°l apply when U( \\ is mfinite, as noted in Section �.3.)

�( ,)

I/I(x) conlinuous .1 x - 0: �lJ(O)

£ _} !2m(U _1 � "

It'"'"OI1

-

_

A. �in(l<))

+

= ....,(0)

B cOS(kO)

dl/lu I

�

Ic "" Bl

By con"truf.:tlon Vo '> E, \0 tht proportionality con\tant between !/let) and Ih \C(.;{)nd derivlltive j\ now jX)\iti\'e, which

�I/I(X) conUnuous .1 x

<;cluhon i� the ,um, eill.:h multiplied by an arbitnuy multiplica­

I/I(x) continuous al z - L: �I(L) "" I/IUI(L)

implie� an eltponf.:ntlal function There are actually two such

function\, dltltring by a \agn in the �ument. and the gtneral live con\tant

!/I(.t)

Ct'-co.' + D�-
Sine anJ co\ine art: phy�ieally acceptable

10

their

region. region L Are the!oC tltponcntial\ phYSically aCf.:ept­ able in regiun 1]1 Both are .. mooth. but no matter whal the

\alut of Uo' t" ,n dwtrge, at; x _ -.JI:. Thtrefore. although it is correcl mathematically, the reqUirement that the wa,'e

function bt nurmali/able makes thi\ ttrm phy,ieally unac­

etptable. D inuit � O. Note that Wt have no similar rea�on to throw OUI

(5.21 )

. > L

�

t"'(l t-,Iunllaries, which ,ivts us four conditions. (In the lofi·

·

Or, rearranging

.<0 O < .l

requIres that 1/1(-,) and ill derivative be continuous 81. Ihese

Un' '0 the tlme indepcnJent SchroolOger Uere. V(II tquation (5-18) be.:omt\ E)

Bcoa(u)

""d

\

II. 1C < 0

be 0.

AIt.lher, we ha\'C

Inlb I +

,(lluu.lo. nu mailer whilt the \alue, 01 arl'll tTa1) "unManl' :\ anJ B ought be. VenlicauCln i\ left .h an CJlcr"i'iC'. nus lullC�

...

Ct'+n.'o I{lr though II doe.. divtrgt 01\

.t

_ +�,

dx

alt'·.... l

dt/l(z) dx

_

0:

dx

� A.A co�lkO) - leB sin(kO)

.1-0 �

=

contlnU0U5 al z

1M cos (JeLl

_

L:

dt/l l

dx ..-L

lB sin (U) -

dx ..-0

laC " UJ

fA sin(leLI + B ct (JcL) - G�-aLl K

dl/ll l

=

dtfIm dx

a(d-aLl

I

.l-L

Although it ctrtainly isn't obvious yet. 1hcie (our conditic-. lead 10 the same qualitative conclusion u lhe contimaiIy

region II co\"ef"\ only ntga\lve valut.. of x, where this func,

condilions did ror the infinite well: quantization of1hc J*1i.

tion il> never inliOltc,

de's tntrgy. Showing it invol\les some algebraic: gymn.....

_ .... .

__ ..... c-.

... .. .. _ _ _ _ _ A ..c.e .. • 7 '._ .. .. .... _ _ .... ·C'-AI. •

••

•

-
...,'. .. ... u P'

..... ".

Cmo(AI. ..,

Go "

.. .. .... .... .....

CWl(U) +-

•

Q

.. ... oI ... .... lO tre lClr .... .. 011 ...... ., ,... ct ..... . . ,.... .. ... ... . .......

• •

c .... .lL)

0

ANI! • .., .. .,.,. IIII'IIC1J\Ic dUs .... 1ft flCl. ... cntI'JY � ...'ICC\II 'j·ll. and l\..lO). wt k.ntM' m.. daa n..eltlt· Pmm ... t IM . � WI £ Md UfI' IUId for • JMI'I Yl'CIJ drpCh 1'0' ..11 1• (",12) hokiI for DIlly {'CIftaaft � vafun, 0( £ U.lbl.".. ctu. as ....heft our "'UlICIft .-1thoul oIpp01.I __ .... lot II .. . IP-alIed uws..:crdt.1t&I �kln and

� be toIwd by pm..;11...... pIfICf _tWqun.. One meaI­

.... au... c:.. be -.n fatrly co..I I)'. how� The- I1mll l'o • Ie ....... lhII ,.. .. . and fiq\Mllon (�·2::' lhen tl«omn CGt lL Tlul. In lum, ""fU11'Cto INt AI ... It .. ..h"h .. pNdtcIy «JaWhoft ('\ 15) from fhto Infmite well TM UlrmllC wen .. Indeed . hmluftl (a. of .... HnllC' , ..�II.

..... - "' .. - � - - "

.. ....... £ . . .. z.) A fC'P1WlIIIab'We U. lme I *-' ... four I:W'Wn. " fCpii!!iill:Db • PlkDliai well ... ... ano.cd \"ahIn of A, MId 10 four encrn tn-cb. AI'r. ilk t\O Ibt ....kJw nl t. lane wooId lIdcrIa,:I Iht ...... a.rw- • flJUle ..C'II ahrray, hu aI au OM' bound ... .. It&fIrJIoJ .a"C' A tJw. P:'4IMU
�

.. In 01(' flnllt well: ".gure S 1 • •\l}0\1or..t "'Ive number

Inlena.lrotIs 01 <.;ur. 1,"10 ""11h horllL'ntal ""dl·.kJ>lh '*

0rIphUaf: .1 0fW w.y W 1ft' how lIIqUIuI oo � 'l-!l) Impl.a

.... y quantl, ..,lOn It I, �tl all. an PCTl"IlC' 10 �)W thai aflC'r

lAIfttlhJ thco delllnlllo", of .t and ell from eqll:lIIOn, (� I " and

('\

U,

r"o«'U Al,1

� ,-..;

,

:U l

,

C·

I ,W

lL .

I.

Il.

lL .

•• ••

•

I. l :\,

)

••

20) and "")'1"1 out \Orne "pot'll&. '0\(' lI:.:an .. ka.s1 I3OLtIC'

Vo an oq....Uon (5.�l). ""

..

... ... 0( _ __ .. .. .... ..... ...... .. . ,S-ZJI IkJkb for 0lIl) � .. � , u ....,..

....., .. II " .. .....,. a.a&MI C CUttk. and . t• • baIIt ... ..,. ... AI. ...t rnmns.. a bat. w� hAve Z ,.UL

:.:=����

...... 1 . ..... .. .... _ "' C,.2l»· ·· t "' ''' .. ... _ _ _ 100 . _ ... .

'T

. ". or

,

(:1) ' ,

.. _ !, -l, t>.

-

1.t.'1 u.. no\\ I.."ompare the tiMe and infinite \\-ell .... Figure 5.15 shows allowed \\il\C fun..:ti\'n .. . md pmNblhl)' lkn"ltie.. Ill( Nlth well ... " here. as usual. rach thlli.;wmal a\'I" j... at a hl'lght proponlonal to that ..tatc·... energy. Becaut;e aU 3t� ..ituatlon... imllh� ...tanding W3\e.... certain characteristics are the oound- ..t -.am.:: Ene�) i ... qu�tlleJ, .tnt! there i.. 3 ground ...tate of minimum tOIa! energy tn \\hl�h the �ine(K energy I'" nm 0--3 "fundamental" ..tanding wave wilt! some m;t'\imum \\avelcngth linllled h� the \\ell'... \\Itlm. But what of the differences'!

t64

aa.pew 5

Bound Srares: Simple Cases

according [0

(5-24), this in tum is Uo

-

E, or Ihe magnilude of the ("pOi", .

kinetic energy Thus. no experimen! could be certain Ihal the particle il finda:

. negative kinetic energy.

..

Funhermore. we must be careful in interpreting the penetration of classically forbidden region. We cannol conclude that a panicle is passing oulSide the well and then going back inside. Once oUlSide. where !be potential energy doesn', change and the force is thus 0, what reason ....0Uld . . have to tum around? More 10 Ihe point. we can't watch a quantum-mec

consra:: �

particle undislurbed anyway. We must be content with Ihe meager kn OWIM _ Ihal there is a probability of finding the particle outside the well ifan atte � were made to find il there. If no attempt is made, we simply have an

turbed standing wave that happens to extend beyond the waJls.

un:

Matter wave penetration into the classically forbidden region has an el twmagnetic analog. When light waves undergo IOtal internal reflection atec. interface between two media, no sinusoidal wave propagates into the medium. All energy is reflected. But there is an oscillating electromagne ti field thaI decays exponentially with distance from the interface. FUnhennorec

see�:

if th� seco.nd medium is thin e�oug� that the �eld doesn't decay too

IllUch

. . . WI thin, a 51gmficant amount of hght Intensity WIll get through. If light wave:l can do this. �I shouldn" s�rprise us that mauer waves might be able to escape

. through a thlfl wall . TIm IS the phenomenon of quantum-mechanical tun nel. ing. which is discussed in Chapter 6. Related to penetration of the clas�ically forbidden region s i another

important distinction between the finite and infinite wells. As Figure

5.15

shows. the energy levels in a finite well are lower than those of an infinite weI! of equal width. This is true simply because the wave functions in the fiDilt well do not stop abruptly at the walh. We noted previously that the penetrati on depth is greatest for the highest energy. and to meet �moothly such a gradUally sloping exponenlJal outside. the wave functions inside must be nearly horiZOn_

tal al the wa/l. As we �e, this implie� a longer wavelength in�ide the well than

for the corresponding infinite well function, and so a smaJler

k and E. At tht other extreme, in the ground state. the penetration of the finite well's clasSi_ cally forbidden region is small enough that the wave function inside mu�t already be nearly 0 at the wall. �o i1'. wavelength is only slightly longer than in

the infinite well ground state. and j", energy only �lightly lower.

Figure 5.16 C1a,�ical oscillator: ma.��

on a �pring_

5.7 Case

3:

The Simple Harmonic Oscillator

Of our simple case�. the harmonic o�ilJator i5 the mmt realistic. Once again, the potential energy define, the problem. The harmonic o..cillator potential

i energy s

U(x) = � d. where 1(i, the �pring con�tant. as dq>icted in Figure 5.16. U(x) i� a continuou, function. Moreover. II

This case is more realistic becau\C

is a good approJl:imation of the actual potential energy whenever particl�

undergo small oscillations about an equthbnum pm.ilion

• -

In the immediate vicinity of a local mimmum. all contJOuou� potential energy functions "look'"

parabolic; mal ilj, there IS a parabola that clO1C!'ly iipprmimates the function about

its local minimum. The bond between twO atom, in a diatomic molecule j, a

5.1 c... 3:Tbe

good example. Figure

5.17 shows the potential energy sbared by two atomJ m

a diatomic molecule. plotted versus their separation x (discussed t'w1heI- in Chapter 10). The force is strongly repulsive (large slope) for small separations

and weakly attractive (small slope) at large separations. but in the immediate vicinity of the equilibrium separation xu' U(x) is nearly parabollc. 1berefore. the lowest energy vibrations about the stable equili.brium point will be essen· tially those of a simple harmonic oscillator. Inserting the harmonic oscillator potential energy. the time·independen.t SchrOdinger equation is I - h' d'�(x) + KX'.p(x) 2m di' 2 -

�

E.p(x)

(5-25)

This is one of the few cases that can be solved exactly�without numerical methods-for the wave functions and corresponding quantized energies. Unfor­ tunately, it is still rather complicated. We leave the ex.act, methodical solution to a higher·level course, hut the arguments that energy should be quantized

are

similar to those for the finite well. In that case, requiring smoothness a( the

E. we !JI(x) exponentials that diverge as \xl - For only certain. discrete values of E does !JI(x) have a simple decay to 0 outside. The

points where

U(x)

jump1. would demand that. for almost all values of

include in the wave function

00.

Figure 5.17 For small oscillations about a point of stable equilibrium, all potential energies resemble a parabola-a !.imple harmonic oscillator. The trw interatomic pOll!l1tial ..... I� �tf(lngly repUI'l\e btl!cp) at 'ma l1 .�

U(xl

\\

-.... \ '0-+ o r+-T --------� .--------\ \ \ mH\lmUn, �«n \ , . high.:r end�. ,I "

•

so a

�:HSl.\

the 1....0 .

•

""II,., I the mHlImum a!

'(I"

the true

potllti 1 agn.'I:' ..iln the modd P :thcT

lat�r_

polentiJ.1 ,)f a h.uTI'Ionl.

...... . . '... _ .k l "C O "" . .. ... ,.

.. -01 . ... .... .... C...tiaJl .. ... Jio, .; _ . ... bere. excepI dw tUrtina .-ilh. nice. smooth po IInI pIoce. .... ___ ..wuon. "'" _Iy ...-10. Stili, _ ... ' c:erIIID £ wIues IS the C1IF\'aR1ft' outside JUSI ",chi so lhal tMy settle do.u • •

for.':

bodI ea1Iedlei.lnd these IR" the QUIIIbl.cd mer;ics. (See Section S_IO lhal demonstrates tMse behaViors,) AhhouJh we woo'l solve the SchrOdinger equation (or lhe OSCiUator � we pin bil lIl(Jfe e.periente w1lh it by lun"ng (he, q U�llon around_ .... Ibu soIvua, (SolS) (or \It(z), suppose we rlOd a fundlun tit a certain (1lI'II " aI8IIOII of . simple compuICf I«hnique

�

wort

. HO'N f... c.an we go with II"

EXAMPLE 5.3

'lllution or;:

mau m . one "lan l"O:nlercd at the onC­ Dcu-nlllrlC 1M en�'l!Y and the properly nllrlT\alJl�J "3\<: lun..:llon m thl\ "Ute For . �lImplc harmtlnll' oKiJI;.tlnr (II "pnn, l"11n'lanl �. ond

Schrtldmgtr c=quatilln IS of w hlfln 1/1( t)

u

A,' .....z. il Ga

":I

The liven ..tttI mu�1 ht a �ClIUliun 01 (�.l."i).

h;' .iM-·,r '2m d,l

The (\rxlntnllal now ul,,) l-anah, and aller I"('arrangtog 3 blt. ....e ha\'e

) (��lr �K}l

( h:

t

ThiS ,an be made 10 hold for all "aloe, 01"

.

1

�0

by adJu�llng: a \0 that the coeffi,ient

0(

This in lum. tells U\ E. be..:au,o: the (erm to parenthe<.e\ on the far lefl l1\Uil U]'l) be O. NOlo: Ihat il I" no a'l"iJcnt that the l'nlldi{lOIl\ to be met are fairly \impl� WCn"' the lum:tion nnt a'tual ly a \olulion. ....e . would have quite a me..� at this POint.

.2 11 n

E·

\O lh;ll

h'a m

h

�

- -

2

m

To pf\lperly nl'rmahLe the ....a\"c funr.:lion, .....e calculate th� {(lIar probability.

J l"'l.t I! d, J )C

+ )0

x

(.o\t'·.....z )l dt

-

Al

J

..

' ... (b � t'-�

A2Jf;;

The mtcgral lablc on th� 1(:,1\ lO'lde cmcr hai �cn u..cd 10 evaluate the Gaussian intt'llral SClling thi$ equal to I . ....c . oblain the normalintion J;on!)lant. A

£�

_

(;Y (2v��K;�r/� (��1)'18

h I.

2\:I m

�

_

and

•

�(x)

( ) �

' l ,,'f,

, '"("\r,;�I�)xl '"

.

The n,� lobation of (S-U) ..... 1bII .. ...,..",.

tnOIlK OSClllalor aft .iven by

E - (n +

IlAwo

(n -

0. \ . 2. 1.

)

7

_ ... .

.... 01 .. ....

�

(5-16)

... . oi llIM)' SOle that lhc: mctJ� 1ft tqually IpKed. .. kmpaatalll chis.. _-;Jlalln, �)'�Ien\\"fI (lhen:l!c 53 otuins a \lay cae resub by a .� yel iB5tn,,:b't'e., aprn"lOll.:h.) h"" C1ICh mcIJ)'. then: is. wUque waw functian. 1IINCDl of \Iohl.:h are 11"�n m Thhk: �,I P.ach is Ihe product of a norrnali.uIion 0llftSIMt. a OaUMQJI f;lIo:tl'l{. and one uf a Rt of relalcd functions known .. the Hermite poly­ [ll.l'!nials. &}\�n lhI: ,ymhul l1�(ln). From the table Ind equaIion (S-16). we see IhIt 1""'1 I.' Harmoatc oaemuor 'lan c\ln"-lderN in \he: eumpk is the puund. state. tht simple tiau,, .a'uUo... Sc\cral """(' fun..:tll1n� ..nd probability densities are ploucd 00 the • . ..�) en('r�)'·le..el diagram (11" hgurc S.Ul. Becausc the potential enerJ)' "waU . illl:n'l IntiRlt�·, all "a\'c functions penetrate into \he clusically forbldden �.ion. o Ahll. ih U\ thl." infinite nil" limle wells. an OlCmator in it, ,round state Is mosI h\.el� 10 t\e hmnd ,ll the center_ C"lass.ically, an oscillator is �t hkely '0 be \.... • Illund "h\"re Ito. '�ed 1\ ,malh."l<>I-ut thr extretnCs of its travel. But once agam. "t:: 'CI." thai 01\ the pat1u.:le energy l:'I«omcs 1�. the pttdiClions of 'Vi ....ical R"lei.:hanics con...el'JC. For 'qe ". quantum me.:han;c� and th(l� of da

( )111(.)1

the probablhty den,'ty i, largc:,t near the dusical lumin, points, F'slur. 5.1'

(l'>(111a.ll1f

A," ltll:n;:nc" to..,wllf\l llw cla 1I..11Imll. the

At '...It C'nc.." \en ')� A.. .' + l. I J

I

�hl

-'-"

l

)' 1),;.:.... �h1 hlshn' 11!C'''' t1\1" lUmina """l'Iit IC'1.tr.;1l1C' \ \.

..

\"'tl�l

( ) ..•Vi

Itl (ab'''' - '.)1 ,..,..

•

lo

I\ UIlJ!.JlIt funcu.;m txi\I,.

" Wh"'�

� ....

lAWn lAWu !AaIo

l .... � 11"'0

·1

"ntttI ll b ...,... • • O . .. _ ... ..... 1' ... ...' I . 10 _ ...... .... . '

o

..., ....... .. .... � .., .. - l· - 2�-·.. -··

dae.n',WMtIlM The proNtnbty \ mpll at lhe tCntC'1 In tbc Jround lblc.

(.j

(ooU'&ry 11) lhe c:la.\.�I�·a.I U""" (b)

..ypoUiM ..... ...... ..... ..... ... .-.:r o f dte ..... ......

'"" hacDoe .

f, . . .

n

...

•• _

. cs

,

... _ ..... c-

HAL·WO.LD EXAMItLE THERMAL BEHAVIOR OF A

I

DIATOMIC GAS .. ........, � ..

..... ... �

M

-

learn '" ditl"cmll kincb 0( ps.es luw�

� .- hne OM nlw, eumtiaJJy illck Pi.... II

me ..... .we � JUft MImI W du�.cr around a IN' different: vaIaea. n..

....... .,. IbaI IheIe vaJun � 10 tram;lauonal moboo 01 .. • whole. IOI8bOII 01 _ mokc_, Md VJtotk.. due to the iDIu __ .....N ... RJr Ihe COIIIIOOD duIomic pscs Nt' 0:- and H!, the- klual \alucs . roo. apa .... .,ree quilr Mil with tM arpmrntJ--bul onl) If the atom" don'l vibnIe. .. n.r.-swa 10 this cl&ukally puuli", lilUMKIfI j, thai lW.:IUalllr C'ncrglC'\ � qaaa.. OI'CCK:.I

....-air •

tiad. In r:oIli.... � and "Itntitlfllli C'lX'rgy �'"llI'I be e\.:hanged, and f i theat weft III('ft enc:rv in ttans�1IJ Chan in \,Ibtati..... wC' ml}:hl e�p<'I:l 1M C'�l'hangc 10 lead 10 cquaIit.c: them. Howcwr, II quantum O'oI.·IlIalllf r.:annol dmp belt"" It, ground �

8Ad iftbe nmlaliooaJ energy is m�uffu:iC'nl lll l'oump II up 1Il1� ne�1 allowed C'ntrgy. 1

b l"hang... in C'lkrgy prr IIIIiI: c..... in 1rmpcr8lUfe, If tI\'er "orne nUl)!e III h'mper,Uul'l" the o<.r.:llla[(l!""'; ne\er budet from their puund staln. '·Il"ntiun.aJ ITItltiun "III r.:"ntobt.He n()lhing [0 the heat c.aparny. Ikfore tailn, II quaniltilll,e IOllA. "" adJn:.." a rather mlll(�r pom[ hert. but Oftt Vltnbon8I enaJY will nevrr dwntlC'_ HC'al r.:a�·Ily

1\ II

th8r ....e ....111 101<;(' III II /TIl1f(' �ener,d lorm In I.lh.'r r.:hilplep., Wh...n nl'(l panlde� �� II mutua! fun."C: bul neuher 1 \ IheiJ. the f('lall'''' Il\(llll," can b..- analY/ed a, though on.: partiCle II fh.ed lellc..'li,d) mlinlle mb') ....hlle the (liher fIlO\e, aboot il under the lfInucnt.:e uf the mutu.al fol\.'C, We nn,'d unl) U'iC' I� proper ellectl\(' rna\\ f{lf the movlfl' panllC1..- W�n the partll"ks are j'ql4<11 ma�\. th" "n:du.;eJ ma\\" i, half the robs of e.t�r pollud.... (10... mnre }:cner.al ca.� i\ dl\t.'u\\ed III C(lmpn:hen�i\e Eter. CI\C 7JtO_) N"",· Itt U\ e\pldln the lad.. t.,f \ .bratlun In a dlillomll' molel"ule,

Applying th. Physks The df�-U\(' �pnng .;on,tant llf lhe H� molel"ule

51.l N/m Should \lbralt(l�

�lt) al room temperature')

t'tlf''lJ l"tJntnbu[e ttl h)t.ift.l�en\ heat ca

1\

hWl).

A(�-onJlng to equalll'n (5·;!61. allll....ed (N.:IlJat{lr energle, are ..epar.noo b) USing a redu.;ct.I rna , of hall the h�dnlgen nla" . the da"ll'al anguli\/" frequency {)f Vibration I� ... w

(I

_

iK Vm

_

I

\' !

57] N.Lm X 1.67 X 10 27 kg

_

8.28 X 101 4 radj�

Thu,. It,) Jump from one \lbrati(lOal \tate 10 another require., an energy inpul of

The a\er.l.ge ((amlational L.inetlc ene�y of a ga\ particle i, BO/llmann con'lant At a room temper-nure (II ]00

HUB X

10 v). K)(]OOK)

K, this i� -

�kBT. where kg is the

6.2 X 10-� I J

A, .....e \!!'e, the ...nern needed lor the \ibratlonal Jump i, more Ihan 10 limes what tralhlalional energy can pro\Ide. Crude though thi, e,tlmate may �eem, it is no coin­ �'Iden!;e that hydrogen's heat capacit)', which i, fairly con,lant around room temper_ ature, be£lIl\ an unml\talable In!;rea'tC in Ihe range of

Section 9.9 for a more detailed treatmenl.)

IllOurtlnd.f of kel"'in.

(See

' . ...... 'MIa U j ·• • • , . ""' :. .. ... . , , _ _

.

'01 onl) do mdh idual mokculn hive �na but III kww IItI .r ..... �twl) Dli tn;ltm al" £orm solid!.. and solids an: full of ,1..." An ,rnr.�l ��(' of the: ",JUnd lak' rM.lCUCfU k,.InettC merI)' I tt.I f\� AI ahsolUIC lnu Kmpcr;alure. molion donn' , stop. Ttl \.'\Xll • rol\cocuon of �11l:1tlftJ partlClrlio lu IIbsolute un', W(' would �mU\e all e� ptlulltlr. aUI the result woul..t not t\C .. !i:liIle \.1' ICtll mOllon, h ....,JUkJ only be "lfte In .hteh .n l'�":lllatot\ In'( in their ,nlund stalCS. StiU \hc:n! h somt''''"R thaI ",ould Ix' I('lW en1n1r}. In su..:h II lair. t�f('t \.'OUw. tic nu random. dllKlnIcttd diunbution ,,11 l-nc:lg) II I � \"dcreti L' II ..:1U\ he. .

5 . 8 Expectation Values, Uncertainties,

and Operators

\\Ith " �'lI.xI �r"'r ,11 the ha'l� h.,lund ')'tc"" , \\C nl'\\" C.h'<':UM. ,.ther illl,"",\{'

t;,lnt elemcnt' ul qUiUltum me,,:h;ml<':' Quantum mt.'\:ham..:all}. \\(' (un·t "nu"," "II th.Jt \\t' n"�hl hl'\lC to "1ll1\\, hut in iHldlh" 1\ tu \ lm"le pn1hat'lIhul'!rio, ilnulher "�J) u,dul thmtt wc \:.10 "ktermir,,: " an expfdatlon "alUf. ll1\c n::adcr nut alr('aJ�

ocqualntcd

\\ Ith

the: h,""'1\:, ot pwh.&hlhhe' and. 1I\'cragr!ii

IS

strontt1),

('n":\luflll1,cd t", lo,l" U'cr (\rre nd l ' J �fllrc rn.lI.:«ding_) M.lthc mal l<.:al ly. an e '{lCdatton ,aluc I'" an a\era�e_ The rmhahiht)' IJf:r 1I'III If"nX,h I'

II./t (,)I�. ...0 ....c

may ",me

pmbahlht)' 01 be mg found In the mte�al d\ around .l

•

tilt! \)I� ,h

A.. \\( "no\l., II \I.e ..umJintegrolte thi .. prohability mer aU ...pa..:c, we mu,t �el ,

tnl1mlaI1l.llion). The 'tandard pre..criptlun for an a,'crage i, tu multiply Ihl" prob;lblht) 01 fmdmg the particle, at \ by the actual value .\, Ih�n intrgratc

"'I!r all 'pace. The ..ymbol lor the a\'emge. known a... an cllpt..:tatiun value. I" an 0\ e"core bar:7

J x\",(x)I' d,

\

aU 'I'lh=t

The term {'If'I'C'wI/elll

1'£I111e I... carefully cho...en To refer to (5-27) a... the "avcr­

age value'· 01 the pt1...ition would be to ..ugge...t that the. particle alway... hu.\' a po..ltion. contrary to the modern "iew of quantum mechanic.., Murcu\,cr, a

..mall particle cannot be watched without a1tering it!; "ate, ...0 it wuuld he impo,>..ible to veflfy an a':crage po'>itlon m any ca...e. The expectation value is

the ,."Iue ",e \\-ould obtain If we were to begin with a particle in ...tale A, do an experimenl lo " find'· the partJ(;ie.

begin lI�ujtl with a particle in ..tale A, fmd it,

and ..0 on. repeating the identical experiment many lime... and averaging the location'>. Note th..t the expectation value of -"C 1\ a numerical \'alue, not an explicit function of -"C, hecau..e the integral ha\ been taken over that variahle. By the \ame logic. the ex.pectation value of the \Quare of the lXNtion IS

, 1 .

/ "\"'(.\)\'

all 'pact

nu. la n.mll:ally equUlD tJ-ld) &em �J.

IL,

WId! I �!be pnIIrIt ...., Q __ ...., .

IN aacrk probtbI.Ilty deaIay D\Qt. �. fonh<.'Oft'IftI apibOa(S-lI)II: t ,...., �

Our interest in the iqWll'e is lhat il is part of how we choose to define UDCer. tainl)'. In quantum mechanics. we define uncertainty as standard deviatioa. which. in the case of position .-:, is given by � •

"i f

(x

-

;;)'II/I(x)I'
(5-21)

., ""'"

This IS the ..quare root of the mean of the squares of the deviations of x from ils expectation value-the rms deviation. It is the SlmpieSI quantity thai is 0 if and only if x never deviates from its expectation va1ue and that increases as deviations increa..e. We ..ee this more clearly by rewriting it as


J f l(x

-

;;)I/I(x)]'<1<

(5-29)

.U spact

0

The Integral can be only if the integrand is everywhere O. for the integrand is nonnegative, But what does this require? Imagine a wave function w(.-:) that is centered on the value = Xo and that falls off symmetrica1ly on both sides. Clearly. its expectation vaJue i would be .10, But for the integrand In itself would have to be everywhere except at the single (5·29) to be = i = xo. where multiplication by - j would cause me integrand point .r to be O. The wave function would have to be an infinitesimally narrow spike, (See Section 5. 1 1 for further discussion of this peculiar function.) Thus. equation (5·29) is if and only if there is just a single point where the parti· cle might be found, it is nonzero for any spread out (nonspiky) function, and it increases as becomes broader-a good definition of uncertainty. lr is usually easier to calculate uncertainty by putting it in a slightly differ­ ent form. Noting that j is a value, not a function of x, we may write (5-28) as

x

0, ijJ(x)

0 x

0

ijJ(x)


Jf

(x'

-

2'-' +

P)I",(x)I'
.n �pace

The first integral is the expectation vaJue of x2, the second the expectation value of and the third, due 10 normalization, simply I . Thus,

x.

Ax =

Vxl - ax + P

=

Vxl - P

(5-30)

Although it is no longer obvious that the quantity in the radical is positive. this form is usuaJly more convenient. It also makes it clear that 10 caJculate uncer­ lainty/standard deviation, we need only IwO pieces of information: the average of the square and the square of the average.

. .. ae ,. ... .. 'c .. . ", . ....... u ,.. ... ..1

EXAMPLE 5 . • -An ekdtOd i l l ft . state liV'tft by die ••"C tur.c:tion

�ne (.l the

... . h .e

of

Ie) � ufM,.'Ctt.mly In position ,

(.l

We dclenmne the

+(.1)

_

A. (b)

Alldl.-.)"1t�

the

npedatioo

lue of

....

the

Md

.rtmrary constant A Vla nonnaliulio.a

f

1';-{x)P dl

-

1111 .1*'1

/ (M-Il' W'#l\ly .�

/ �.(.-.)If�d.t

+,,-.

_

1/(2a1). Its v.lue is

Al

_

ch

1

.:/ntegral. liven on lhe tnt ... in",,� I;over, With

11\15 I S . standard GausSi a a _

poubOl"l.

-,""

'

2Trc1.

-

, (b)

Insenin, in

(S-27). x

-

f

.r\�(x)lldx 1p.te a11

-

Al

f x �-t.-Wi2r dx.

-01>

This is also a Gaussian integral, whose value is bY2�;i. Thus,

The result makes sense, for the function is symmetric about the point x (c) According to (S-30). we need to calculate �,

Xl ,.,

f

.r\!JI(x)IIdx all 'lIKe

""

Al

J

.�

_00

xl t'-(.-b)'{lct dx

Again we have a Gaussian integral. evaluating to

Now insening in

(5·30),

(el

+

blrV2TI'f::1,

-

b

,. '"

TABllE. 5.2 Basic operators --

• • £ - ,, -

"

with ib u...e. we ...imply pre�ent the result. For momentum. the operatOf­ which .... hen put in the proper loc3tion in the probability intqnU gives \he e,pect3tion vulue-b

. p

� . -in'Ix

-

M

Strange though thi... operator may appear. we �oon � that it gives very �nsi­ ble resulh. Function... of an operator are al�o allowed. For example, we obtain the operator for kinetic energy a\ follow!.:

p' 2m

-

.

KE

-

pl

I . 2m

(

)( a )

" I . - jll�

-

2m

.'1.1'

-ill··

,1\

_

- �, a'

-_ .

�'"

� .'1-,\

II not ba.-.ic the1ll�lve.... ma..., quantum-mechanical operator.; are combinations (If tho..e for the bru.ic dynamical properties: momentum. po�ition. and energy. In Table 5.2. we complete the Ii!)t of bu....i' operator.; by adding that for total {a., oppo� to kinetic) energy. Note that. with operatoo. in hand. we may write the Sdu-ooinger equation (5-2) in a fonn clearly iUu\trating its fouOOatilln-cnergy:

1<£'1'(" ,)

+

U(x)'I'(x, ,) - f:'V(" ,)

Our general method for calculating ex.pectation values also enables us to generalize the deflOlllon of uncertainty to arbitrary observables.

(5-32)

Ur,' tl " ." ,

Using the infinite well as a model system. let us calculate some uncertainties and see if they accord with the uncertainty principle.

F

EXAMPLE 5 , 5 For a particle in the ground ..tate or an infinite well. find (a) .l.1'. (b) product .ix �p. s

�. and (c) the

•

(3) Equations (5-30) and (5-32) agree that to calculate uncertainty. we need the expectation value of the c;quare and the square of the expectation value, Equa­ tion (5-31) IS the general fonn for expectation value. !>o ....e can certainly u<,c it to find x. But a considerable simplification is that. a!> in equation (5-91. the + temporal part of 'tI drops out (because the product e ''''' e-"'" i!> I). Thus. the

expectation ..alue of position relies on the spatial part I/I(x) alone.

x

-

f

"" (x)x"'(x)/L,

all �p;u;c

,;",. -• .

--------

'lis S a

.� , ..

.... . . d

_ 2 ("

_ _ II ... ..

..

�l"- (�) .)

�\!!d! . _ :(� "

_ " ...

_ ... ... .... _ (s.n�

.. - Vi! - il -

....,.., .. _

) _ ,,(1 - I )

II•

,

zr

G 2�) (H

,..-;-; .,..-c-;-:""" " -

o.IIIL

.... --,_ .... ... ponido - _ ... _ - .. l1li. .. .. -. . t .. ...... iI . ........ ....... A'" Gl . "111 . ... .... ... .... .. ,... .. .,..... il*-·' I I ,U II , R' .... lAlp..,. .. .... ... 1I r . ....... . .. ... 11 ... ....

.. .... ... C$:3. � .. _ .... ...-,. .,...,.. _ .......- io ... '7$ C . ...

176 CNptw 5

Bound SIaIe$: Simple Cases

(C) The product ofthe uncertainties is

ir .1p

=

O.18 1 l.T ""

=

0.568h

Here we ha\'e a nice example of the harmony of the quantum mechanic� we've slUdied thus far. According to the uncertainty principle. ;lx Ap �

This fits perfectly! While the momentum uncertainty decreases with

�",

increasing l.. because longer wavelength implies less kinetic energy, the position uncertainty increases, for the wave is more spread out. and the Prod Uct is simply that is independent of l.. The reason the product isn't equal

ro !1i

t/J(x) isn't a Gaussian, the function with the minimum possible product.

It is len as an exercise to show that in the infinile well, both ax and tlp increase wilh increasing II. Thus, the ground state is the least uncertain Combi_ nation of position and momentum. II is also an exercise to show that 4.t approaches the classical value L/ \112 us n - <'XI, I n agreement with the COlTe_ spondence principle, quantum-mechanical predictions approach the classicaj as 1/ becomes large. [!]

P

T

o

N

A

L

5.9

Nonstationary States

In separating variables to solve the SchrOdinger equation. we restrict Our aUention 10 stationary Slales_ in which the energy is well defined and the prob­ ability density 'P*'i' is independent of lime. A quick look at a more genera) case gives us important insight into the process of jumping from one station_ ary state to another. A bound particle, as we know. will have quantized energy. Let us refer to its allowed energies as £t ' £2' £3 , ' and the corresponding solutions of the " time-independent Schrodinger equation as rjJ I' rjJ2' "'3' . ' , . The total wave function in stationary state n is given by equation (5-8), 'VII (x, I)

==

¢,,(x)e-i(E'/")1

Now consider a sum of fWO such solutions, but of different energies. (5-33)

Because the Schrodinger equation is a linear differential equation, any sum of solutions is also a solution. Each individual function is a solution of the time­ dependelll SchrOdinger equation, so their sum is also a solution. But neither is a solution of the time-independent SchrOdinger equation of the olher. and because n and m involve different value� of £, the sum does nOI have a well­ defined energy. (For a quick fonnal proof, ba...ed on the ideas discussed in Sec­ tion 5. 1 1 . see Exercise 67.) Solution (5·33) also lacks the other important

171 _ . ..... _ _ Caoa

swting region of the x-axis, then we find its values point by point

as

9ft

move away by imposing the (act thai I/I(x) must obey the Schrtidingef equa­

(ion. 8ul how? We first observe that by 1M definition of a derivative, evaluating dfltb aI IWO points .1.f apan gives dfl

dx

�

=

lim

iT-o

fix + .1.1) - fix)

dll

-

and

4x

dr

_� •

�

.

lim

f(x) - f(x - .1x) •

_

�x

.1.>--o

By the same definition, if we lake a difference between these two, divide by

.ix. then lei .1x approach 0, we have the derivative of the derivative-the

second derivative.

tP/

lim

.1..-0

- =

dr

(/(X_:-�·1 I(X)) .lx

lim ----

.1.<....0 .

.

fix

m -= li l.<--tl

-

+ h .. :) -

--

_

--

21(x) + fix .l.r

hm

it--V

(

1 f(x - ax) f(x -

Ax

)

6.x)

Before puuing this to use. we nOle thai the numerator would be identically 0 .ffix) were a linear funclion-a �lraighl line-becau,>e its value at x would

be the a.. :erage of its values on either side,

(fl.r. + .lx) + fix

-

.lx»I2. Thus.

Ihi .. "finite difference" form of the ...econd derivatIve is indeed nonzero only

ifjix) has a curvature at the point

x.

Now, after replacing/by

I/J

and drop.

ping the limit ..ymbol, we in ...ert It in the time-independent SchrOdinger

equatIon.

h' .p(x + 2m

.1x)

- 2.p(x) + .p(x . �x)

Rearranging a bit, we have

.p(x

•

.lx) �

N(x)

.. + V(x).p(x)

.1xl

.p(x

.h) +

2m..lx2(U(X)

h'

_ .

-

�

E.p(x)

£) ··.p(x)

(5·35)

This is the form of the SchrOdinger equatIon that a computer can use. And what mu�1 we give the compuler'1 Of cour...e.

U(x)

mu ...t be speci­

fied. b well a ... the particle's mas, m. Then we muM choose a certain energy £ [0 te!>!. After Ihal. equation (5- 3 5) simply !'Ioay'" that to find I/J al the point x + �. the computer need only k.now its value ... at two earlier points, I/J(x) and "'Ix ..u). But i t can then find I/J at the point x + 2..u. becau<;e it now knows its values at the IwO earlier point... I/J(x + .it) and I/J(x). The process .

contInues untIl the computer finds q, at all poinh .1x apart along the entire

.J-axIS.

L ",. ,.. c"'_ . te • *

But where do those fino! two ....alues. of." from. and. wby do we DCICId fK'O anyway� We need IwO because the Schriidinger equation i, a ft'cond-order come

differential equation. involving a foeCOnd derivative. For example. the fint.. order equation dj(.r:)/cb; - bftx) has solution ft.x) ;;: u-lu, where A is. an con .. tant. c",scntially a constant of integralion. Solutions to second­ arbitrary order equation... be'ide<' being harder to obtain. have two arbitrar)' conslanb. The con'tan'" may be "peciflcd in different ways-ways that doD', appear equivalent-but in any ca-.e. they detenninc the solution for all 5pKC. In an "analytical" ..alution to a differential equation (see A Closer Look: Solving the Finite Well on page 160), they u�ually appear as multiplicative constants in front of two independent function�. In OUf pnsenl "numerica'" solution ....e take them to be the value... of the \olution at a slarting point and a point nex1 10 it. In effect. by ..pel.:ifying the�, we specify the "height" of the solu� lion at a starting point and it� ...lope-how it change!> or doesn't change--at Ihal point. From there, the numerical !iolution simply charts "'(xl according to the curvature imposed by the SchrOdinger equation. By an eS!>entially equiva­ lent route. we may aho plot I/I(x) in the negative direction from the starting points. The I/I(x) that resu!" ;!i automatically as smooth as the point-by-point solution allows, so the only remaiRlng phy!>ical condition of concern i!i nor­ malizabihty. It must not diverge, and thi<; i<; the key to applying the procedure ""'

•

and spotting the answers. Given U(x) and m. we choo..e a .1x �mall enough that q..(x) !>hould change only a little from one point to another. a <;mall fraction of what we might guess to be a typical wavelength. Then we choose an energy E to test. In almost all cases. what we find is that for tln�' initial I/I(x) at the Maning point. no matter how wc choose the slope there-whether I/I(x + .lx) i.. greater than. les� than. or equal to l{.I(:f}--the point�by-point solution diverges as x got... to +00. -_. or both. We can choo'>C the mitial slope just right M) that "'(xl die!>

nicely to 0 at one end. but thi" use� up the arbitrariness of the ..lope. and ",(x) still diverges at the other end. We conclude that this E is "wrong" and move on to another, At only certain values of E. we find that for an} initial ",(x) at the "tarting point (unles... the \tarting point is by chance actually a node). we can choose a slope there <;uch that I/I(x) die<; to 0 at one end. and it automati­ caU,' does so ar lire ollrer, 100. These are the allowed. quantized encrgie....

�

Not that because thi!) work� for any lOilial value I/I(x) aI the �tarting point. we <;till have freedom to choose Ihis value. Thi') doe')n'l change the character or �hape of l{.I(x). but merely Its "vertica\"' �ize. In fact. it �imply allow... u'" to normalize it. In conci<;e term<;. once a correct energy is found. the two arbi� trary constant... allow us to fit the boundary conditions that q..(x) go to 0 at both end'). which u...e... up one constant. and to normalize the function. using up the other. It might �em that the computer approach would take forever. a.<; there are infiOltely man)' po')�ible energies and infinitely many slope<; to try at each. It tum� out that the di\ergence 01" ljJ(x) changes drastically a!i the trial E passes through a " rea\"' value. "'0 we can zero in relatively fast. Moreover. if the potentIal energy function ;'" ...ymmetric, It greatly simplifie... choo...ing slopes. Se\'eral end-of-chapter e\erci'-C... apply the approach to the finite well. the har­ moOlC oltClllal(lr. and other ca'-C\,

'"

t. ett.ptw 5 Bound StaleS: Simple C8§C!$

a D

V A

N

C

E

D

5. 1 1

Well-Defined Observables: Eigenvalues

A well-defined observable is one for which the uncertainly is zero; thaI is one for which an experiment designed 10 measure thai observable can Obt ai only a single value. There is a mathematical tool thai allows us to deten nine easily whether a wave function describes a slate for which a given obse rv.

�

able is well defined. Before introducing iI, however, lei us Conside r expect 10 have well-defined values of the bas ic observables. what (unclions we might

Momentum: In Section 4.3, we mel the basic quantum-mechanical

plane wave, the complex exponential

e'u. Being a simple sinusoidal

function of pure wavelength and spread equally over aJi space, giving an infinite position uncertainty, it is the prototype function of wel!­ defined momentum.

Energy: In Section 5.2, we asserted that the temporal part of the wave function,

e-,(jJ/, has a pure frequency and thus a well-defined

energy.

Position: A wave (unction of welt-defined position would allow no

probability of finding the particle anywhere but at a single point.

Calling this point xo, l/!(x) would have to be 0 everywhere but at xo' We give this peculiar wave function the special symbol l/! "<x). It is x

crudely pictured in Figure 5.19. Now the mathematical tool:

Definition: Given an operator Q. the functionJfx) is an eigenfunction of the operator if and only if the operator acting on the function yields a constant times the same funclion.

Q

j(x)

j(x)

=

Eigenvalue Eigenfunction

Operato!"" Eigenfunction The constant A is said to be an

eigenvalue of the operator. rn genera],

an operator will have multiple eigenvalues and cOlTesponding eigen­ functions meeting Ihe condition. Figure 5.19 A function for

which po�jtion i.\ well defined.

This tool's utility in quantum mechanics is as follows: A wave function qr is one for which observable Q i s well defined­ for which there is no uncertainty in [he outcome of a measztrement

I

o

of Q-if and only if it is an eigenfunction of the operator Q, and the

eigenvaJue is the well-defined value of the observable. Q is well defined

--"

Q

¢::;>

.lQ ::..: 0

¢::;>

Q'-Jf

=

Q'-Jf

Note that is in the place of the eigenvalue, which is correct, for Q is a constanl, and jf there is indeed no uncertainty in the observable's value.

112 Ch8pt:ef' 5 Bound Swes: Simple Cues

EXAMPLE 5.6 Which of [he following wave fllnC[;Ons have Ii wen-defined momenrum? For Ihose that do. what is its value? (a) At'-{"�)'. (b)Acos kw' (cl A[cos (y) - i sin (k,r) l' (d) Afcos (kO't) - sin (ko-rlJ.

SOc'JTlO,,"

(a) p ",(x) = -iA�Ae-(',�f = (·· iA) Ae-(./24")�

ilx

(_-_,,x)2 2

This is not a constanl times Ae-{·,�)l. so [he momentum of this wave functiOft is not well defined, Ofcollrse. we wouldn', expec, a Gaussian "bump" like thil '0 have zero �p. for 6.x is not infinite. (b) Again. because [his is not a constant times Acos kw' the momentum is not welJ defined. It may seem odd thaI an infinitely long cosine function doesn't have a well-defined momcnllrm. os it would seem to have a well_ defined wavelength. The onswer is basic to quantum mechanics and reflected in the fonn of the momentum operator. The fundamental plane waves arc not sines or cosines. but complex exponentials. Only they have weJl-defined momenta. (c)

-m

� A[cos(klfl - i sin(klfll = -iM()"I.[ -sin(klf) - i cos(koX)l =

-M()"I.[ -i sin(klf) +

cos(klf)

]

This is -Aka times the Original function. The multiplicative con�tant, the eigenvalue, is the well-defined value of the observable, so the momentum

is a well-defined -Ako' in fact. this function is A,.-'L¢. Because the momentum is negative, we Set that thi, comple� e�ponential represents a panicle moving in The negative direction: whereas for Ae+'*c-<. it moves in the positive.9 This further Jemon�trateS why Acos klf does not have a well-defined momentum. Writing it as cos klf = + e-iko-t). we � that it is really equal part� of two waves of opposiTe momentum. Its momentum expectation value would be 0, but the particle could be found moving in eiTher direction, The same arguments apply to the �ine functions of the infinite well.

1(,.+i*O-<

Momentum is not well defined, The "mi��ing" i is crucial. •,1. /1d.aromagnc:"� ....c·, d,fI:<;uOtl of mouon

,,,u ,, of l11li1 be =n (.om a ··. .... ("hot·· II " !/Ie d,,,,, E ll: B U..., ..toif, E ...I."ve to 8 b) 180°. we haw a "'.ve """,nil ,n rhe "I'JX"I'II: d"�,i",,, SilDllarl}. by :
Categorizing wave functions as eigenfunction.� of operalOrs has many applica_ tions. II is widely used in more advanced analysis, such a.s matrix quantum mechanics. Probably its mOM useful application, though. is the one we have highlighted: It is the si mplest way of detennining whether a function has a well-defined value of a given observable.

......- ... ..1" t',

,•

• RoaRasl AND APPLICAnONI Quantum Wells as light Sources With advances In 10 choose . frequency anywhere In the visib&o ... _ abo a unall and cffiClenl PKu,c wcll wiled lO '" iKIua.ia& fabricating mlcrOM:Op" �tructures in recent decades, the UloC of hghl 1hat 1'10 anticipated In � IIDII c0m­ ph),iC� of quantum well� ha.. anumed a major role in modem technology, The ","ell con,i'h 01 e\eClroMatic puter dc!>iln, bamer. in �emiconductor cl')'�tab (or. In \ome easel>, Fi� S.21 1Iohowt. two umpks produc:cd by a lC*m . carbon nanotube.., diS\:;u��ed In Section 10.10), and the Sancha Nauonal � tt.l lc::atu.re In � MId usual caplive� are the electron and Its �emlconductor alter .-.ec:mingly OPPOSite applil;aOOo or quamum doU: cfti,;ienl ego, the hole. If the particle'.. quarter. confme it in only gener-dtion of normal ",",liar h&M. Althoup. III more cfficiad. one dlmenl>10n, leaving It ef>senllally free in two other,., we than the old �tandard Intandacent. tudaY'l 1luocaccnt bulb. be"dcs being rather bulILy, 1\ mll typiuJ.1y \eu than 50% effi· call it a quantum well If the partide IS confined in two dimen�lon\ and free in one, II is in a quantum wire, cient. Sohd-'ta� lighl 1>OW\.·u--1acILina naawcd cbambeR. When confined in all dimensions. it i5 In a quantum dot. tran�formc:r.., and hut fllamenl\-are already ut.ed in many An three are being studied for a host of applications, but low·power appli.;allon!l. but they may one day n:plKe the the dot is particularly attractive because, meeling standing­ �tandard Iightbulb a, the tran\1',tor did 11.\ vacuum tube pn:de. ce��r. Coupled With the efllClenl LED, the quantum dol is. wave condition<; along all axes, ih levels are the mOSI contender. In sc.veral different appmaches being llUdicd. lightly controlled ultraviolet light from an LED "triung quantum dot5 has been In many ways, a quantum dot i� like a designer atom. 80th have quantiled energies. bul quantum dOb are more convened 10 a broad '�lNm of ....avelengtiu. by the inter· fleXible in Ihal they can be tailored to the spttific applica. play between the geometri," propnties of the dots and the lion by varymg their size. Among the most promising appli· chemical properties of the material uf whKh they 11ft: made. Efficiencies att already better than !Undard n� b.ltt.. callons of quantum dots is as a medium for producing laser light. MO�l lasen rely on transitions between eleclron Oetecting ONA. with a Quantum Dot Tbe tunability energy levels '>pecific 10 the material-gas 1asen on lev'els of allowed energie\ in quantum dot!. has eained many In Ihe gas atoms, and traditional solid stale lasen on fairly broad energy "band�" in the semiconductor malerial. 8uI application.. in biology and medicine In one, Iobov.'n in Figure 5.22. • quantum dot i� thc hean of a \'ery sensitive the energy leveh In a quantum dOl are more easily con­ detector of DNA. Fir-t. IWO I>petlal molecules aU' trolled, for they also depend on the simple georneD)' of the introduced-a ","apture probe'" and a " reporter probe"­ ....ells. As we find for a particle in a box, a ..maller L implies higher energies. Many early difficulties in producing unifonn which, when encountenng a DNA molttule. fonn a three� molecule �and""'lch. The quanlum dot i'!i the crucial dots have been ov'ercome. A major advance was the develop­ indicator that the landwich has fonned and thus that ment of a "self-assembly" method in the 1990s. in which DNA is pre-.ent 11 if> clothed in speCial molc<;uJes wi.th streSses in a growing cl')'�tal spontaneously fonn pyramid. an affinity for the captur!' probe. If the sandwich fonru.. shaped dOl!; of regular Sile and distribution, Numerous layen this attraction bring� into clo'!ie proximity the quantum can be used, and when placed between mirron to produce a dot and the �portu probe, which only then are resonant cavlly, as depicted in Figure 5.20, the many dol!; fill role as atoms in other lasen. When they mature, same the quantum dOl la\e� should offer not only a convenient way Figure 5,21 Light from giow'lIlg quantum dots.

Figureo 5.20 Quantum dOlS fonning a laser. QuiUltum dOIStmJcondUCtor mIrror

Scmi�l)�uctor muror La�,

IIghl

.

"' 10 . ...t.. by • proceu knowD as fluornc:entt

example is the heavy-hydrogen oucleus-a neutroa ...

a

pboIoo of one enellY euiles a YdoDOr"-the quantum

which the p8I1ide�

classically forftiddeo region than in the classically all�

dot. or QD. in the figure-which transfers . ponion of

region. Cases where �nt'tratlon IS unusuaJly high

the ene1JY by • non-liaht-producing rewnant interaction

10 . nearby "acttplor" -tbe CyS--whkh then deexcites.

continut' to attract quilt' a bit of alieniion. and tht' outJriDt faih are referTed to as halos. Althougb electron� form halo'> in alom�. and aloms form halos in WaH! function

producin, ils own distincfive photon of lon8t'( wavelength.

Mol«uJar-based malerials thai talk in this unmistakablt'

molecules. the most commonly !'iludied halos are in certaia nuclei. where an excess of neulrons forms the halo. For instance. the common helium-4 nucleus binds two pl'Olonr,

waY-Ibsorbing one wavelength. exchanging energy.

emittinB another-without somt' kind of garbling � not

easy 10 find. The versalilt' quanlum dot can be tuned to the

and two neutrons In a very light umt. but helium-6.

specific application and provide!'i a nice solution. Halo. In

the CI...iCllly Forbidden Region

depicted in Figure 5.23. add� two more neutrons. which

haunt regions quite far from the nuclear core. Forces in tht atomic nucleus are very complicated. and halos Provide an important view in helping unravel the mysteries.

Wave

functions of bound panide� usually extend inlo the

classically forbidden regilln. sometimes very far. One

Figure 5.22 A quantum dOl DNA nanosensor. Sandwiched

DNA "i�'k� to a quantum dot (QD). " E�ciUltion" energy excite�

the dOl and t'merge\. affer FRET. as the emission (Cy5) photon

.... .. Sandwl!;hed hybrid

•

Sueplav.Idlh· conJugalcd QD

E�(;ilation (488 nm)

�

Nanosen�or assembly FRET

·':;'�ii,:;:r.;;:;'m

iSSion (Cy5)

Emission (QD) (805 nm)

their shatt'd "strong force"-. are more likely t o be found in the

proton bound together by

raoaIJICe CDClJY lranlfer (FRET). In this clever process,

(670 nm)

Figure 5.23 A two-neutron halo

around helium.

"""".. " - .. ..--I by ...._ ...

Chapter 5 Summary h' unJo:Nand

I b.:- "!uantum-n)C\,hmlt.,.'a.I tw:oha\'ior (.f _ masslvc

l¢,.e-.'t. Il lS rn..,:�"UaI) I,. lou.... I� ....,I'C' fUl\\.-tioo

hn!1t'f ("quAi,,", Il\lO ,.r tho: S.:h"... J"."..nI1al cn('rg� (I( 1), 1'1

AII -,"

mc-nl }tC'ldina • "alut b an ot.er...... MII:h • 1'__ GI'

tu�·h. In

...

'f'l.{,

n. I

tulu­

ltwo rre�n..�

"I'

$"hlllg th.. S..:h""hnger O:llualn'o hlr "'\I,tl 1:\ r\lU8hl� anill,... n/ ,/:rlJr: for 1"('), gOU\ 10 the ct ••" •.;••1 pn.:,,:o:lIure 01 \lll\ln� I' SCpaf'JUlln (If \'ari.lhk, )ieIJ.. tho: tlllll:-lIhJl'llCnJent S..:hrodingcr c1l until11l

IIkVnthlUm 'M.1UJd In � �k1 • dlfItftaI vUue 11 III 7J ldc'nti�·.ally The CJ.ptrilnml ("&MOC be .amply • &Mer ...... th'oR of Ihc wne a;ylWm. .. My obaecwdoa dlICurba it nw �ult \It 1lWl)" Idcntk:"aI fCpCbtioM would produce .. ....... knl'''' n liS lUI c'ra:tautln 'alue, IlIkI a .WIdan:I *"Iadon, wbidt lju41Uum rM\:h&ni,,·s dk.'lCa; .. it» deftRllion or w ... .. ... 'linl)'.. Tltc>y mOl)' tit nkulatC\l frum thr- Yra� function "Ia <.l und

.It

.I

or-'"

If'.( I. ')(lIf'( l. ,)dt (5-32)

(�·I O) The WIlle IUIII:llon '1'(\. I) bc..:ome... IJI('I)r ,(LA), Known ;\\ II ,tallon,lI,) �lilIC. It .. energy f. i\ ....dl ddined. I1ml il� probabilJl� den,it)' d{lC� not viii)' .... Ith IIiIlC A qu;mtum-me;,:hanu.:ill treatment i, required in \mall ,),�. [em.. ....henever the ullnCIl\Wn\ are C\lmpilrdble 10 or le,� than Ihe wiI\elcngth or WI.t, 11. Such II \)'\\cm cannllt be COII.\lilIltly \\atched to keep trud.. 01 l}(),ition, momentum, and \0 un, bc((lU�e an ob,erver would have to imcrJ.r..:t wuh II \OIuchow,

whc� U I� IhC' llJ"Cr.ltllr !ls�\"II.·i!lteJ with thr {'II:o!lerval:ole. A I:o\.und panicle may 1na�C' u Irllnsililln bc-tw�n one &111· \;\lnar} �tUll" and ullI1ther, ut whil·h tim(' its p",!:labllity densit)' dtn"I \ur)' in lime; It tl�l·lJlIlIt'\. Wh('n II chllJ}l!cd particle drop. 10 a 1(IWeT enC'r�} ,laIc, the t'\l·illnlit'n rretju('ncy of it!! chuJo denMIY " Il1,t ri!!ht to �enl"rate " rhnl"n thDt canies away the energ) d,ltl'T'C:Il\;e.

Indicates advanced questIons

d)qurbing the very behavior being ob�erved. If no IlUempt i;,

mnde to find it, a m:h,i\e obJeet in a \mall 'y�lem behave\ not a, a panicle but a, a wave. When a mB�sive object i� bound, {lnly cenam wave func­ tions sati�fy the phy�iciLl condition� of �moothness and normal­ iZBbilily.. and Ihey form \lUnding waves, This is the rOOI of energy quantizntion, in which energy is rCMricted 10 a cemlin di�crete �ct 01' allowed value�, each with corresponding wave funclion, In all bound Cll;,es, the mmimu1n-energy stBte ha� nonzero kinetic energy-if a panicle h bound, it cannot be absolutely stationary. Three simple cases of bound panicles, often used as model� of morc complicated system>;, are Ihe finite well, for which (I numerical solution is needed to obtai n energies and wnve functions: the hannonic oscillator, whose energies are

E = (II

+

�)'u"'o

where

o

"'

-

l;

-

m

(5-26)

and Ihe infinite well, or panicle in a box, whose energies and wave functions are

I/In(x)

f2 n1TX V L san L _

"'"

(5-16)

In the limit of large energies, thc predictions of quantum mechanics and classical mechanic� converge.

Conceptual Questions I. Quanti/atinn I� an IInpctrtunt chanll:terislil: (If sys�m, in which a panide is htlund in II small region, Why ",m;III," and why "l:IounJ" ·' 2, A ,tuden! 1)1" dll�sical phy�ics says, " A charged particle, likc lin electron orhiting in II simple atom.. shouldn't huvc nnly r..:cnain stal:lle energies: in fact, il should lose ener�y by electromagnetic radiation until lhe utom col­

Inp�e,." Answer Ihc\e two ("omplaints qualitatively. appeilling \(1 1l� few fundumentul claims of quantum mechanic� as po��ihle. 3, The term j"lerll('ti(ln b sometimes used interchangeably with/orn', and lither time� interchangeably with {'flle/llia/ l'IIC"RY, Although fmce and potential energy cenainly aren't thl" same thing, whatjustilication is Ihere for u�ing Ihe same lem to co"er both? 4, Explain to your friend, who is skeptical about energy quantilation. the �imple e\'i
115

116 a..- 5 8auod SIIIeS: Simple c-

5. JUJI what is ltaliOllary in a sWionary state? lbe paticle? Something else? 6. When is the temporal pan oftbe Wive functiOll 07 Wby il this important? 7. We aay Ihar tbe ground Stale for the particle n i a box has nonzero energy, wtw. goes wrong with '" n i equation (5·16) ifn = 01 .. Equation (5-16) gives infinile well energies, Because equation (5-22) cannol be solved in closed fann. there is no similar compacl fonnula for finile well energies. Still. many conclusions can be drawn withoUi one. Argue on largely qualilltive grounds thai if the waJls of a finite well are moved closer logether but nOI changed in heighl. lIlen the well musl progressively hold fewer bound Slates. (Make a clear connection between !he width of the well and lIle heighl of the wa/ls.) 9. A half-infinite well has an infinitely high wall al the origin and one of finile height Vo at x L. Like the finite well. Ihe number of allowed states is limited. Assume that it has two stales, of energy £, and £2' where £2 is not much below Uo' Make a sketch of the polentiaJ energy. then add plausible sketches of the two allowed wave functions on separate horizontal axes whose: heights are £1 and £ ' 2 10. Consider a particle in the ground stale of a finite well. Describe the changes in ils wave function and energy as the walls are made progressively higher (VO is increased) until essentially infinite. J J. A panicle is subject to a potential energy that has an essentially infinitely high waJl al the origin. like the n i fi­ nite well, but for positive values ofx is of the form V(x) = -b/x. 1'.'heR: b is a COnstanl (a) Sketch this potential energy. (b) How much energy could a classical panicle have and still be bound by such a potential energy? (c) Add to your sketch a pIal of£ for a bound particle and indicate the oUler classical turning point (the inner being the origin). (d) Assuming that a quantum-mechanical description is in order. sketch a plausible ground-state wave function. making sure that your function's second deriva­ tive is of the proper sign when U(x) is less than E and when it is greater. 12. Simple models are very useful. Consider the twin

states decrease. Based on changing wavelength at-. argue that this is reasonable. (b) The energies oftbe first and third stales increase. Why? (Hint: Study bow the behavior required in the classically forbidden rta:iot affects these two relative 10 the others.) (c) The distaat. wells case might represent two distant atoms. If eacb atom had one electron. what advantage is there in bringing the atoms closer to form a molecule? (NOlf: Two electrons can have the same wave function.)

l

-

l

""

finite wells shown in the figure, at first with a tiny separation. then with increasingly distant separations. In all cases. the four lowest allowed wave functions are planed on lUes proponionai to their energies. We see that they pass through the classically forbidden region between the wells. and we also see a trend. When the

•

UD

U

13. In the harmonic oscillator wave functions of Figure 5.18

there is variation in wavelength from the middle to the eJttremes of the classically allowed region, most notiCeable in the higher-" functions. Why does it vary as it does? 14. Summarize the similarities and differences between the three simple bound cases considered in this chapter. 15. Consider a particle bound in an infinlle well. where the potential inside is not constant but a linearly varying function. Suppose the panicle is in a fairly high energy slate, so that its wave function stretches across the entire well; that is. it isn'[ caught in the "low spot." Decide how. if at all. its wavelength should vary. Then sketch a plausible wave function. '

16.

The quantized energy levels in the infinite well get far.

wells are very close. the four functions and energies are

ther apart as n increases, but in !he harmonic oscillator

what we might e�pect of a single finite weJl, but as they move apart. pain of functions converge to intermediate

they are equally spaced. (a) Explain the difference by considering the distance "'between the walts" in each

energies. (a) The energies of the second and fourth

case and how it depends on the panicle's energy.

.

(b) A '1cry imponant bound ,ySlem. the hydroacn atom. hu eneray le'leb, that actually pt clOlel' together as. It increase•. How do you thlGk the KpIhlion between the potential eneray "wall"" '" this "Y" tern unu rclau'o'e to the other two? fupwo. 11. tn se'o'eral bound sYltemt..lhe quantum·mo:bbically allowed enerales depend on a "nile q�tum number We found In Sccuon S.5 thal tht enern Ie'Ieb In an utfini.r wtll are II'Ien by E,. - a,n1• where ,. - 1,2,3, and 0, IS a constant. (Actually, we know wluit 0I is. bui lt would only dlstnci us here.) Section 5.1 "bowed

11.

i).

thai for a harmonic oscillator. they are E,. - a1(" where ,. - I. 2. 3. . . (Using an ,. - with ,. !>u1ctly

\

�

,.,

A ct.wcaJ .,.nide ""*"-d to .. ..... ...... ....

rieIw;cs a force ...... .,......, ..., is

U(.l -

� I

-

�+ \

2

(SI aniIs)

(a) By Mdina Its milumunt vahc 1N14uenailua ilia beha'lion at . - +_,.u.cb Ibia palla­ till enetJY (b) SUppo5e the particle has an eMIlY ofO.S J. Ao4 any tumina points, Would the putitk be bouDd'? (c) Suppose the puuc\e hal an meru 1 _ F_ any tuminl points, Would lhe pUtic\e be bouD4'?

0 and .

of 1.0

Section 5.5 posili'o'e is equiulent to It + with n nonnegali'o'e.) Finally, for a hydrogen atom, a bound system 11. A study of da. ..,ical waves tell, us that a standiq ....ve that we study In Chapter 7. E,. - -aJn2, where It I. can be uprt:ssed as a "um of two tra'Yelifta Wa'lH. . Consider particles making downwvd transi· 2,3, Quantum·mechanic,,1 tra'lehna waves, diKuSled in 111 tions between the quantized energy levels, each transi· Chapter 4, are of the form 'V(•. n - A r"b-- 1). Show tion producing a photon For each of these three that the infinite well', "'tandinl·wave function can be systems. is there a minimum photon wavelength? A u,pres!>Cd a!o a ,"um of two tra...elina waves. maximum? It might be helpfUl to make sketches of the 23. Wnte out the total wa'le function 'i'l.t. J) for an electron relative heights of the energy le'lels In each case in the It - J .....te .. of a 10 nm wide Infinite well Odler 18. Quantum.mechanical statIOnary states are of the than the "ymbol� x and I, the fUl\(,;tion Ulould include ,1111 general fonn "'(x. I) - .,(x) e For the basic plane only nultKncal valuell. wave (Chapter 4), this is 'V(x. I) = A eih e-'·' "" 24. An electron in the " - 4 nate of a 5 nm wide infmiU: Ae'{b-1II11, and for a particle in a box. it is "'(x, r) '" well make ... a tnu\\ition to the VOUnd state. Biving oft' A sin(h) e-,AI'. Although both are sinusoidal, we claim energy in the fonn of a photon_ What s i the photon's that the plane wave alone is the prototype function wavelength1 whose momentum is pure-a well-defined value in one 15. An electron I, trapped an a q1laGtum well (pnctically direClJon Reinforcing the claim is the fact that the infinite). U the lowelot-energy tnn.!i.ttion i, to prod.uc:t: • plane wave alone lacks features that we expect to see photon of 450 nm wavelength, what should be \be only when. effectively. waves are moving in both weU's width' directions. What features are these. and. considering 26. Becauc;e proton� and neutron.. are "imiiar in mus, stu, the probability denSities, are they indeed present for a and certain other characteristics. a coUecti.ve term. particle in a box and absent for a plane wave? ltuctt'OfU'. has been coined that encompasses both of

.

""

Exercises Section 5.3 19. Under what circumstance does the antegral

fx; xbdx

diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than �- 1f2 does as x gets large.

Section 5.4 20. A comet in an extremely elliptical orbit about a sw has. of course, a maximum orbit ndius. By comparison. its miOlmum orbit ndlUS may be nearly O. Make plots of the potential energy and a plausible total energy E versus radius on the same set of axes. Identify the classical turning points on your plot.

theloe constituents of the atomic nucleus. In many nuclei. nucleonlo ate confined (by the suang fora:, discu'llOCd in Chapter 11) to dimeMion' of rouably 15 femtometer;. Photons emitted by nuclei u tbc: nucleonlo drop to lower ene1'8Y levels are knowII as gamma panicles. 'The" ene1'8ie1 1ft t}'J»cally in the MeV nnge, Why does this make Knse'? 17. Whert: would a particle in the: fin.t tJ.cited stale (ftnt above ground) of an infinite well mostly llkcly be tou.r. 18. What i!o the probabilify that a particle in tbc: tint C1Cited (n = 2) <;tate of an i.nfinite well would be fOUDd m the IlUddle third of the well'? How docs Ibis � with the dasloical npectation'! Why'! 29. A tin� I 111 partide is in a I em wide eadcl5ure aod ilia a �ear to boutu: from one end to the c:Ah:r aod t.dL (a) Haw many nodes � then: in ib eaclosure'! (bl How woukl

• Jt.

Uo R"ache<. ftz(n I )2/8mL�, la) Argue thai this __ belhe<.:� ba,.cd only onk V2mE/1I2. the shape .­ !he WI\'e in\ide. and the degR"e of penetration oflbe d...�icaJly forbidden R"gion e'ptcred for II slate IlIbaae energy £ i� only negligibly below Uo' (b) How many Mlle� wlluld be found up 10 [his same "heighC ill" Infinif.. w'ell.

yow __ daDp!rdle,.rtK:1e _ me. mas.\I� IDIM"I tutor' A pMil:le it bound by • jXJIe!dw etlCfJ'y or !he !'Imn

-

'"

cr

o

u {0

0"

If! < �II. Ixl > 1"

Ellercises 38-41 refer to a particle of mass m trapped in a half-infinite well, with potential energy

given by

U(.,) �

Thi� Jill,,!"); from the infinite welf or Section 5.5 in being �ymmC" m: IIMUI r 0, whi(;h impJie� thai the probabil­ ity de"\ilie, mu_, also he symmelri(;. Noting Ihat either �jl\(' or co,inc w(luld fit rhi' requirement und could be mlldt' l'lmUnUOU\ with the- len> wave function oubide the weU, delennine fh(' uJloweu energi('\ and COTTesponding nonna/l/ed Wi,lH! funt:lioll" fur [hi- potential 11'('11.

{�

x s O O<x
x�L

U.

"'"

,

S.ctlon 5.6

0

x "' L

JI. V('rify IhJI �"Iulton (5·)9) �Jli�fie, the SchrOdinger

cqUJ[jnn in form (5-111).

32. A limit potential enc:/}!y fundion U(x) aUow\ l/t(x). the

solutioll t\f the time-inde�ndent SchrOdinger equation,

10 penelrale the cI;J"IcaJly forbidden region. Without il"uming any partil'uJar function for UC.I'). sholl thaI 4-(1) mu,1 have an jnOe<.:l;on point al any value ofx

whert' j[ enter. II da\\iclIlly forbidden region.

33. Vcn(\' [h;ll A �jn Ik.\) + n co,; (.\:.r) is a solu[ion of equa­

tion (5-12). 34. A 50 eV elc<.:lfl.ln i.. [rapped between electro�[a[ic walls 200 eV hiBh How far doe\ it, wave function extend beyond the W'alls� 35. An elcctron i� tmppcd in u finite well. How "far"· (in eVI is il from being free if the penetration length of ih wave functiun into the c!as�ically forbidden region j, J nm'', 36. Wherell� an infinite well has an infinite number of b'lund .qOl[e\, a finite well does not. By relating the well height Uo to the kinetic energy and the kinetic energy (throu,e:h A) [0 II and L. �how that the number of bound \Ialcs i.. ,e:ivcn roughly by \/SmL2Uolh2. (Assume that the number i.. large.) 37. The deeper [he finite well. the more stales it holds. In fact. a new �tafe, the IIlh. h added when the weII"5 depth

Advance an argument based on p = h/A that there is no bound Siale in a half-infinile well unless Uo is alleas[ ,,2/32mL!. (Hint: What is the maximum wavelength pos�ible Wilhill the well?) 39. A finite well always has ll! least one bound stale. Why does the argumellt of Exercise 38 fait in the case of a finite well? 40. Write �olutions to the Schrodinger equation appropriate in the various regions. impose required cootinuity con_ ditions, and obtain the energy quantization condition: 38.

VEcot

( VimE ) ,- L

-

""

� - .vVo E -

41. Using the result of Exercise 40 and a computer or caku_ lator able to solve (1f3oscendental) equations. find the two bound-�tate energies for a well in which Uo = 4(rr2ft2j2mL").

42. By largely qualitative arguments. Exercise 37 shows that a finite well can hold an 11th stale only if its depth Uo is at least h2(11 - I )2/8mL2. Show that this result also (0[101'.'5 from equation (5-23) and the accompanying Figure 5.14.

...' "". ONam (" r�u.i'ln l�-:!'\ I(\'m �\latiQl'l \�';:2).

h.lN:t�� ira l'e11Oft l.

l � 8) - ,m:t;8) Arnhm8 "" l",m, "",8 - h: � ��8). liN .:,10,'('11 lM argument uf the :! ..md tI) ':''l ' (otml;Cnl ll\lm iL I,. lI,m

: � o. wh",h �Ulrn that dw prob.wlh)' daNll} be ")'m�tric and thUi lha\ �:) be cllbct a&'& odd � O\'ODI

In !.IuaJulll\; I"m), IUlll then \
fu"",""oo of! 8y (\)lhparinl the fClion n aNi npw. m

pl''''\L\(' b� .. kfmlll,ln

fUl'II.:tions, arvue lhat lhlli in 1\UT1 demands lbal (......) ..."

4-&. U..mg ,-quatlon c�·n\, lind tne C'nC'flI.�' t1f a particle' "-"'11-

Ill) t ';0';' tll.) muu be either + I (even) 01 - I (odd). Ne,,'. �IW Ihill thelOC �OhIlilkln� can be c��. 'am.un) IU\d. all - -coa(tUl). �"re�ti\et)'. as nil HnaUy, plull theK lOCPl.ratC'ly tw.ck inta Ihto reaion 1

fined 10 It linlle v.e1l ....h\l� waU.. � half 1M hcltJ:ht ,If llf

In c :. < + Ul. 1lw cbup of

... . naMes lhitb the � '110 .... it " .,...u\c ..

�ll.. Nut. put the fC'II.u1till8 a{ua­

lhe gn.1und·'t,uc intim!e well eOC''lt). f:"

".

,A cakulahn

computer able 10 ,uh(" e''4uatwn, num('rh:al\� rna)

bo: u�ed. but thi.. happen... ", be II ca<.e whe«: an e\a.:1

�tlIU'l\lnS II.nli IohO\lo' that

un"wer cun be l.kJucC',j ....'lhouI IO(\ mu�h 1«lubl('.)

45. The� are mathemalKul

" llutwn.. Iu Ihe Sd'lr(-..iins.cr

-\L - � I. :: t < + \ L t --" +�L

equoltion for the fimte .....::11 hlf Uti," cnerg�', an'" In

t o(

facl. thc) clln be m.uk ..mouth ('\'('f)'....here. GUide..! by A Clo�cr Look: Soh-lOg the FInite Well, �how Ihi� as follow�: (a) Don't lhrow oul nny nmlhcnmt1clll �olution!..

Thill i�, in region II t � < 0), l\�sumc thai 41(.,) l\��\lme Cr+(U + De-U." and in region 1\1 (.� '. thnt �I\.') fj>+tu; + Gt! n, Write the �moothne�s conditions.

Lt

A.

��

(b) In Section 5.6, the smoothne�.. condition.. were combined to eliminate

B, and G in favor of C

In

r·"{�ll \ . ( ,)

e x

the remaining equation, C canceled. leavi ng an equation involving only .l and (}, solvable for only

Ie)

cenain value� of E. Why can't thi.., be done here1 Our solution

physical ly ? (d) Show Ihal

is smooth. Whot is ..till wrong with il

D: F�

� (B - ; A )

and that


Nme thai C i� now a �tandllrd nmltir1ica\ive nonnaiiza­

tion eon!.tant. Setting. Iht: integral of \III(;::)\! over aU spare t�1 I would p\l(: il in Icnn\ of l and R, but bttausc we can't �llhe (5·22) tAaetl), for k (or El. neither can we obtain an e,act "alue I'or C.

41. Com.ider the delta well lKltential energy: and

�' .{(A - <},;n(kL) H 8}00*L)] +

1

:-in( U.) �,

-\L - � L < , < +�L :: > � � l.

,<

- :-in k:

+

selling these offending coefficients 10 0 repro­ condition (5-22),

duces quanti7.:ltion

U( ,)

�

{

o

.\ � 0

-00

Althoug.h not cumplelciy

X "" 0

reali!'.tic. thi$ potential energy

is often a convenient approx.imalion 10 a very strong. vel')' narrow attractive potential energy well, 11 has only one allowed bound-�Iale wave function. and because the top 01' the well is defined a... U "" O. the cOlTeSponding bound-Slale energy i� neg.ative. CaU its value -Eo.

46. 11 is possible to take the finite well wave functions fur­ ther than

(5-21) without approximation. eliminating all

bUI one nonnalizalion constant C. First. use the continu­

ity/smoothness conditions to eliminate A. B, and G in favor of C in (5-21). Then make the change of variables : '= x - U2 and use the trigonometric relations sin(a + b) = sin(a) cos(b) + costa) sin(b) and costa

+ b) = cos(a) cos(b) - sin(a) sin(b) on the

u-o

u-o

U

I

- -QII

,., Applym, the u�uaJ argumenl� and required tonll­ nwry cundilion� (need /I be �mooch"). show thaI the "ve funcdOfl t i given by

•

IbJ Sketch 41(x) and Vlx) on the �amc diagram. Does thi$ wave funcllOn eUllbll tbe expetted behavior in the cla.uically furbidden region? .g. FIgure: 5.15 show� the allowed wave funttions for a finite well whu
-

Sealon 5.7 49.

For the hannonic oscillator potential energy. V !�. ground-swte wa\'e function i� 4J(x) "" A�--{\.·�}r, and jt� energy is �JII V'.Klm �

the

-

Find the das�ical tuming points for a particle with this energy. (b) The SchrOdinger equation says that t/J(x) and ils sec­ ond derivative �hould be of the opposite sign when E > U and of the same sign when E < U. These two region� are divided by the classical tuming points. Verify the reilltiomhip between I/J(x) and its second derivative for the ground-state oscillator wave func­ tion. (Hlllt: Look for the inflection points.) (II)

SO. A 2 kg block oscillates with an amplitude of 10 cm on II

spring of forte constant 120 Nlm. (a) In which quamum state is Ihe block? (b) The block has a slight electric charge and drops 10 a lower energy level by generating a photon. What is Ihe minimum energy decrease possi­ ble. and what would be the corresponding fractional change in energy? 51. Air i� mostly N,. diatomic nitrogen, with an effective spring constant �f 2.3 x loJ N/m. and an effective oscil­ lating mas.� of hruf the atomic mass. fur roughly what tem­ pern.tures !ihould vibration contribute to its heat capacity?

52. To a good ilppr"OJtlmauon. the hydrogen chloride mole. cule. Hel. beha\'e� "lbrationaHy as a quantum hannooJc

of )pong f;onStiUlt 480 N/m and with effe.:ti..,. O';(:iltaltng ma�) ju,t that of the hghter alom. hydrogen. If it were: in ih ground Vibrational state. what wa\'e_ length photon would be JUS! oghl to bump this molO!:\tlc up to Its next-higher vibrational energy state" 53. /n Section 5.5. it was shown that the infinite welt cllCIJies follow SImply from A = hlp; the formula for kinetic energy. p2f2m; and a famou.' standing-wave condition A = 2lJ11. The lIlXumenls are perfectly valid when the potential energy IS 0 (inside the welt) and L is strictly mnstanl. but they can also be u�fuJ in other cases. The length L allowed the wave should be roughly the distance between the classical tuming poinls. where there is no kinetic energy left. Apply thes� arguments to the oscillator poten. tial energy. Vex) = � ",x2. Find the location xofthe clas_ sical tuming poim in terms of £: use rwice this distance for L; then insert this into the infinite well energy fonnula. so that E appeun; on both sides. Thus far. the procedure really only deals with kinetic energy. Assume, as is {rue for a classical 05Cillator. that there is as much potential energy. on avemge. as kinetic energy. What do you obtain for the quantized energies? 54. The potential energy �hared by two atoms in a diatOmic molecule, depicted in Figure 5. I 7. is often approlli­ mllted by the fairly simple funclion Vex) (a/xI2) (bI.x6), where constant� a and b depend on the atoms involved. In Section 5.7. it is said that near its minimum value. it can be approximated by an even simpler funr­ tion-il should "look like" a pambola. (3) In terms of a and b. find the minimum potential energy U(xo) and the separation Xo at wh.ich it occur.;. (b) The pamoolic approximlllion V,,(x) = V(xo) + ! ",(x - xol2 h3S the same minimum value alxo and the same first derivative there (i.e., 0). Its second derivative is /c. the spring constant of tllis Hooke's law potential energy. In tenns of a and b. what is the spring constant of Vex)? 05tlllalOr

.

=

-

Section 5.8 55.

56.

ClasSically. if a particle is not observed, the probability per unit length of finding it in a box is a constant ilL along Ihe entire length of the oox. With this, show that Ihe classical expectation value of the position is ! L. that the eKpectation value of the square of the position is! L2. and that the uncertainty in poSition is L/m . Show that the uncertainty in a panicle's position in an infinite well in the general case of arbitrary n is given by

0112 -

�

2n 1T2

'"

.

[h1l.."U:U I.ho:

.�c. 11'1

�,

L

hat I:I�"C dotoI It

..

.,� .... lm the I:lanical �ty

of

\!

12 k\B5ed

P.xetd!ot y'i ' SboW Ihal tnc Uncata:lnty In a panidc', momentum '"

11\

an

s.ct5on S.,O Sec the Cl�tuoU Eaea;... .....

,nfimte .'\"lI ln the aeMraI (ate of I.tbllrary .. l$

s.ct\oft S."

"ri/l. � \\ n.I 11 the J'f'",,"ul:l of the uru:enailltlo dctcnnioed in I"Cft by

E\('n.:i->c' �6 and �7'1 O,....:u." the resull. �9. Octcn",,,," th� e'rc..:b.lion ,alue of the po.l$itllWl of .

h.unum".: IKl.:llIah,r In 11'0 ,round state. 60. SIlt'''' that the um:o:rtamty In the po�ilitln of v. If\�tmd· ,laiC hann,'m..: o....:llIalllr " l , , \.- 211"l' ml( ) 1 � 6'. Sl'M,.... that Iho: un..:ertll.ml) m Ih.:: m'�IlI.::ntum "I' 1& �round 'iale hamll\nn: \,....,llall'r " l \'II '2j(ml<)' � 62. Wh,II " Ihc jlrndu..:t III the un(crt;lint,e\ lktemlin.::d m £,el\:I'e\ 60 and 6 \ " E\plain I) 63. RepeAt E\en:l\e, Nl--62 tl�r tho: 111">1 e\..:lIc,J {If ,1(1\0: 1\1 a hanmlnl": ("cillalor,

''7. Ptow that the tnDlltk1ftal-stale wa'e'C � ($-)))

don 1Il1I ha..-e

"It-defined nnJY

I:U\(ftIl) hIIw . well-defined mftJy1 bpWn.. the ".W fURl,"tKWl �ll - A(,"" + tt ....... hIrte . ..eU-der"ved mtlmeR'um'� F.J.plall' 1be \�.tf" fur anau11lf momentu m dKMat the ::-u.is in pherical polar �l\onllnalC$ " Walil/�). find • fIInt-

... O!..� 79.

�

... It a partido: In a \lallOnar), \Iale 1\ "'IU,..J. the upecta· lion ,;llue 01 1\\ momentum mU'1 be 0 ta) In \\ll(,J, .

11\\1'1 ft.,4I\ lhlt w\luld have . well-defined ::-compoaeal llf II.nlular mumentum, 71. Sn,-,w \hat �1'

boulld� a f�e parude certamly may ha'e a nonlero momentum (Note; Wlthoul lo,,\ 01 generahl)'. .pIt} may be cho'en to be �al,)

Section 5.9 65. In equatton (5-3:\). the twO �olution� are added in equal amounh Show Ihat if we m�lead added diffen-nt per­

.

centage, of the 1....0 \olution� il ..ould 1'101 change the Importanl conc\u,ion related to the mcillalion frequency of Ihe charge den\ity. 66, Con�,dcr .\ wave function that i� a combination of

twO dillerenl mrmite well \tationary "ale�. the ,.th and Ihe mth.

- p+�J) - P+lJ)· Thali.-.Yerify

m..mWnlum 11pCrator. wre will be • nomero unoertainty In momentum Stan by !l.howin. that the ,",uarrtity

Q. mlegr.lte h)' parb. �n argu.::

thai the �,ult " l,Jentically O. Be ca�ful lhal )our �ument I' ,omehow ba\ed on the partIcle bell'll.

()

thai unle!O!\ \he ..ave f"UlKlilm ill an elaenfuDt.:uon of Ihe

....h'·, Ib) Pro,e It Slartmg from the I.eneral e'(p�'\lOn 15-�II ""'th i' III plil<.:e 01'

•

... Tu dncri be . mann ••'ft. doH lhe f\InctioQ AUntb)

J

.1 i� I�l! Then. u,ina the di(fe�hal operator {ann of p an,J Inle�ralltm by pan�. !JK-rN tnat it iii also

J

"'-

(N"tf'.

ii;;

/')�(.'II ·I{;;

-

pl+(A)ldtt

Recau\C momenlum is. real. p is real.) Together

Ihe\e �hllw that if .1p is. O. then the preceding quantity mu\t be O. However. tnc Integral of 1he complex square 01 a tunctllln nhc quantity In brlll:k.eU) can only be 0 i.f

tne lunctwn IS. i,Jentically O. ,n the as-setti.on is proved.

Comprehensive Exercises 72. In a study of neat tmnder. we find lhal for a solid rod. there is. a �Ia\ilm\hip between tne second derivative of (a) Show that "'(x. I)

i., properly nonnalized,

(b) Show thal lhe expectation value of the energy is the

. 1 (E� + E",). (Be a,'crage of the IWO energie�: E = 2 careful. The temporal part of the W3"e function defimtely doe�

'lOr drop out.)

(e) Show \hat the expectation value of the !>quare of the energy ;� gi\'en by

£2

=

1 (£2

�

n

+

£2 ) '"

(dl Detem\;ne the uncertainty tn the energy.

the temperature with re-.pecl to position along the rod and the lif\t w,lh �s.pect to time. (A lint'or temperature: change With pmihon ..ould imply

M

mucn beat flowin&

into a region as. nut, M'l ihe temperatu� there would not cnange witn time.)

-b

OT(A.

/)

0/

{a} Separate "anables. That is, assume a solulion that is. a product of a fun!;tion of ;{ and a function of t.

... it .n. thea divide by it. Obtain lWO ordinary

diIremIti&I equalionI..

(bl CCJIIIider . rairty limple. ihorncwtw unrea.li�.:.

ease. SIIJ'PO"' the �re is 0 AI .¥ ""' 0 and x "" L. and posIlM in between. Wrirc- down lhe aimplelc l'wIcUoa of.¥ lbaI ( l i lill these condi lions

and (2) obey l lhe dirrm:n lial eqlWlOO in....oh·in' x.

Doe. your choice determine the value. n i cluding

u,a. of IOIIIC comlanl? ec)

Obtain the hill R,. II for thil cue.

7.1. In Section S.l we Ic:amcd thai 10 be fJOfTlllIhmble. a wave

0

hlnctioo ( I I mUll not iuelf di....erge and (2 ) musl faJl to fuIer tIwI W- In as '" plS large. Nevenhc:less. we find lWO (w)ctiOllS IIW §lighlly violarc- the-e requ jrement� very

Atilt. -.11

useful. COMider Ihe quantum mCl:hankal plane WiI\'e and !he weird furn:lion tP (x) pictured in " fiaure :'i.llJ. which we here call by ils pwper name. the Dirac delta ful":tion, (a) Whkh ol-lhe Iwo (klrma!ilabiluy �uiremenfs i� viol.led hy the plane wave. and which

by lhe Drac i delta f'uIII:tion? (hi Normaii/.ation of the plane

wave could be accompll\hcd if it were \imply truncated. rntricted 10 the n:Jilln -h < _l < +h. being idcnti"a1ly

o uul\ide. Whal wQlJld then ]Ie the rdatiomhip between

" andA. and what IIIo'ould hoippcn lO A ii' h approad��

a i tinil)"" kI R.ilber than an I01i000cJ) tall and naJT()\llo-' ,pike "kc W Dirac delta funCtion. consulcI' a funclJon thal t� OC"\�� e'�'ep the' narrow region -.£ < ,( < +6. where ill YaJue t$ it "onstanl B. Th" too could be nor· mali7nt Whal �("IuIJ he: the- relaliol"hip between 6 and B. and what v.ooJd h.appetl IO B .IS e approadle, O? (What we gel tS not elO ....11), the Dlra.: della funclion. but the

distim:lI<m in\\Jhe\ comp.mng mlimtles. a dangerou\ bu,if)ess thal we will a\\'ld,) td) As we�. the two

"el.:eptiooal"· fUndtOn� ma} be \Iewed a_� limit� of nor·

mah/.abJe one�. In ttK>!.C hmi N. they are also comple­

menial)' 10 etach ulhcr in tenm 01 their po,ition and

momtnlum un�·enillnlic.., Wtthout gelting into cakula·

lion,. dc."scnbe how Ihey are wmplemenlary.

74. 1lIc: figure �h(Jws a pulentiaJ energy functton, {al How mu"h energ} wuld a da�si�'al particle have and \till be

£'(t)

bound' ' . (b) WhcR would an unbound paltKk have its muimum Ir.inetic enerJY? Ic) F« �'hal l'Wlgc: of tneraia mlghl . cI�;';al panide be buund in either oftwo differ. ent regtOIl$" td) Do you lhmk thar a quanIUm-lTIeI:hazhc:.! i pan Ie) p&I1i1.:1t with energy In the range rderTrd 10 n IIIo'ould be btlund In one re�",n or the uther? Explil.ln. 7S. Con\jder a panIcle 01 m.ns m iUld energy £ in a region where Ihe pUlenlial energy i, a (.'On'tant l./Qo grealer!han E. and the fC!!IOn utend.. 10 X " +-. lal Gue\s a ph)'5I"all) Kcepable \()lu uOll of the Schmdmger equation m thl' re�I\," and demnn,u'ale thai It i.. a \olution, (hi 1lle ref:.on nok'd IR pan (a) elOtend." from x "' -+- I nm 10 .... To the left of.1' .. I nm. Ihe panicle's wave lunUlOn 1\ f) co�1 IO� m It). Is V(x) also greater than -JIl kg_ By how E hcfC" (f;1 The panlde\ ma\' III IS much (in eV) doc\ VI)' Ihe potential energy prevailing from x - I nm to -+-_. e�cced the panicte's energy? 76. F.xerCl,e 5] outline\ a procedure for predicl1ng how the quantum.mechameally allowed energies for a harmolllc o\clllator \hould depend on a quantum number. In _.

10

e,o;enc�. allo....ed ltn�ti" energie" are the panicle-m·a_

00)( encfJle�. except the length L h replaced by the di�.

tance belween da�,ical lumi ng potnh. expressed in term\ 01 E. Apply Iht� procedure 10 a potenlial energy

of the fonn c"hl - ·Nx. v.hert' b i� a con"ant, AS�ume thai al the ongm there I� an mfinl!ely high ....all. . making II one tummi! point. and determme the olher turning

pomt I n term) 01 E For Ihe a\erage potenlial energy. u-e It\ valu� at half",ay between the turning points.

again m term, of E. Fmd an e)(pres\ion for lhe allowed

energie.. m lerm� of III. b. and n, (Allhough three­

dIRlen,ional. Ih� h)'drogen atom potential energ), is of

Ihi\ form, and the allow'ed energy le\-els depend on a quantum number e)(actly a\ thi, \imple model predicts,)

77. The harmom" o\("illator polential energy is proponional to x2, and the energy I�\'et\ are equally \pac�d:

E�

(rt +

!). The �n�rgy leveh in the infinite well

be"omc fanher apan h energy increases: En ex

8ccau\e the function limh...., ..

infin itely large for Ltl

0, I

0.2

0.)

> L. Ihe infinite well potential

\I ( L

'

I

"lI;

energy may be thought of a<, proponional 10 !xl.

JO

,\

n2

LIILjh i\ 0 for Lt! < L and

How would you expecl cne'!!y Ie'ch to be "paced in a pcMenllai v.cll ihal ,, (al pnlponlona! to)41 and (b) propor. tional IO ·!-ti- t.) For the harmoniC ()�illator and infirute 0-1

0_5

,

0_6

well. the number 01 bound-'wt� energies i� infinile. and arbttranly

large

bound·swtc ellC"ryin are pcbsible.

194 Ch_pter 5

Bound States: Simple Case..

After choosing initial values w(O) and !/I(axt and assuming

U

is known and a vaJue for i1x selected. we need onJy pick an E, and w(24.t) can be found. Thereafter, taking x to be a point

2i1x to the right of the origin, we have ",(3.1x) = 2",( 2.1x) - ",(.lox) + 2( .1x)'( U(2.1x) - £)",(2.1x) and the process can be repeated indefinitely. Choosing t./I(O) and y,(i1x) is simplified if U(x) is symmetric about x = 0, for this implies a symmetric probability density, which in tum requires that !/I(x} be either an even or odd function of x. An odd function is 0 at the origin. so 1/1(0) must be O. Since the slope is, in general. nonzero, !/I(�) must be nonzero, and we define it to be I (affecting only the vertical scale of t./I ovemll). An even function, on the other hand, has 0 slope at the origin. so 1/1(0) and I/I(tu) must be of equal value, which we can define to be I . 99.

Finite Well: The goal here is to find the energies for a fmite well whose width L is 4 and whose depth Vo is 5 in the simple system of units discussed above. To exploit symmetry, assume that the finite well extends not from x O to x = 4, but from x = -2 10X = +2. For Vex). the function 2.5·sign(x"2 - 4) + 2.5 can be adapted to almost any computer. (a) Plot this Vex). (b) For 6.x, use 0.00 1 . Now, following the above guide­ lines on choosing 1./1(0) and 1/I(1lx), test both odd and even functions at different trial values of E by finding 1/1 at all positive multiples of ilx out to x = 4 and plolting the resultS. NOIe that because of the functions' symmetries, there is no need to plot negative values of x. Find four allowed energies. (c) What lells you that an energy is correct? (d) Using the definitions of k and 0:, the finite well energy quantization condi­ tion, given in equation (5-22), can be written =

2 cot

( Vb;i;) L

fI

=

IE � V Uo - E V�

compare tho.se re!o,ults with a numerical approach. Along with the values of m and h discussed above, we choose our unit.s so that the spring constant 11: is also I. The potential energy function U(x) is then simply For ax. use 0.001. (a) Following the above guidelines on choosing 1./1(0) and t./I(tu), test both odd and even functions al different trial values of E by finding I/J at all positive mUltiples of ax out to x = 4 and plotting the

�,t1.

results. Note that because of the functions' symmetric" there is no need to plot negative values ofx. Find four allowed energies. (b) What tells you that an energy is correct? (c) Compare your results with the exact valu� given in equation (5-26). 101. Half-Infinite Well: A half-infinite well has an infi_

nitely high wall at the origin and one of finite height U 0 at x = L. Like the finite well. it cannot be solved exactly. Here we find energies numerically for one whose width L is 2 and whose depth Vo is 5 in the sim. pie system of units discussed in the above guidelines. The half-infinite well potential energy isn't symmetric about x = D. but part of the symmetric-case discussion still applies. Because the left wall is infinite, I./I(x) must be 0 for x � 0, so 1/1(0) must be O. Assuming a nonzero slope there, we can define I/I(tu) to be I . (In e%ence, we are keeping only the conditions that apply to odd functions, which must be 0 al the origin.) For Vex), the function 2.5*sign(x - 2) + 2.5 can be adapted to almost any computer and is correct for positive x, which is all that matters. (a) Using 0.001 for Ax, test different trial values of E by finding 1/1 at all mUltiples of IlxOUI to x = 4 and plaiting the results. Find two allowed energiel:>. (b) What tells you that an energy is correct? (c) The "unsolvable" energy quantization condition for a half-infinite well is given in Exercise 40. Do your values satisfy it reasonably well?

_

It can't be solved exactly. but do your values satisfy it reasonably well? 100. Hannonic Oscillator: The hannonic oscillator can be solved exactly for the quantized energies, and here we

Va

x

0

x

L

6 U n bound States:

Steps, Tu n ne l i n g , a n d P a rticl e-Wave P ropagation

Chapler Oulline The PotenlLai Step

6. I 6.2 6.3 6A

The Potential Barrier and Tunneling Alpha Decay and Other Application.. Panicle-Wave Propagation

�

C

ommon to aU topic... In thi.. chapter i .. the ab<.eo!.:e of a force c pa� ble of confimng a particle to a region of "pace. 8ecau-.e the partICle::

i�n't bound, �Iandmg wa\-'c.. do not (ann. and energy j.. no{ 4uanli/ed. Ncverthe­ ]e\), the c,urprising bchavioroo we di'>Cu�... are further evidence of an mherent W3\'C nalUre. Among other thing... we find that a panicle may be tum� bac.:k by ....ical mechanic.. \aye, it ..hould proceed. and it may prevail a fon:e when cla ....icai phy�ics says it ..hould be rurned back.. We begin again.., a force where cla by ..tudying the effects of very simple force.. on plane wave", fol· in Section

6.1

lowed in Section

6.2

by an introduction to the fa<;,cinaling idea of quantum·

6.3.

we sludy M)mc applications of tunneling, mechanical tunneling. In Section ..�ically baffling phenomenon of radioactive alpha decay. The wch a.. the cla.. chapter ends with a ..ection devoted to undcf'l.tanding the richer behavior of nlll.­ so-simple, but more reali..tic, free-particle wave functions, clarifying the n=la­ tionship between the wave function and the motion of the particle.

6.1

The Potential Step

The fi....t ob..tade \I.e place in the path of our fTee particle!'; h simply an abrupt .. a light wave ,triling increa....e in the potential energy--a polential step JU�I a - 'orne tran...miued, <.OnlC reflected·--so does a maner ..... plate divide.. a gla divide \l.hcn It em,:ounter. an abrupt change in condition.... This dlle..n', wa\'e a ma .. ,ive partu.:!e divide!>. any more than a photon divides in the ..:i1SC that mean of light. But becau\c: in either ca� the wave is related 10 the probability of fmd­ ing the partic.:le. the dlvl'lon (lf� wave governs the proltability of a ,Ivcn inct­ dent partid�phtllUn or ma.'iSi\e obJcct--being trarbmltted or reflected. To 'ay anything about thi,. howe\'er, we mu..t be able 10 dlstlflguish belween partidc'S rom'lng one way and tho� movlflg the other, and to do this

195

196 Chapter 6

Unbound States: Steps, Tunneling, and Panicle-Wave PropagilllOn

we must be careful about the wave functions we use. In Section 4.3. we dis_ cussed the plane-wave solution of the free-panicle Schrooinger equation: right-moVin g This represents a panicle whose momentum is in the positive x-direction. It is an unrealiMic function for a single particle in the sense that, although its momentum is well defined, the position uncertainty is infinite-the panic le is equally likely to be found anywhere. However. in the cases we consider here we will use it to represent a steady beam of particles all moving in the positiv direction. with A related to the flumber (rather than probability) per unit dis_ tance. By replacing k = p/ft ) by -k. we represent a beam with momentum in the negative x-direction.

�

(

'¥(x.t) R. I"

L ft

l.

r.;;;•.·· '

Figure 6.1 A free dectron encounters a pOiential step

(5-12)

mountable.

-, -

Enero

E > Uo

t

KE

V a: fVo

U

_ _ _ _ _

2mE

� -fi2"' (x) � - k'", (x)

where

k

=0

2mE J 1>'

(6-1 )

All four functions-sin kx. cosh, e+11:.t, and e-1.lx-satisfy the equation. Each is a sinusoidal wavelike function whose second derivative is opposite the func­ tion itself. [n the bOllnd systems, we could have started off with e+i.lx and e-1U, but sin k.x and cos lex satisfy the physical requirements and are, in fact, sums and differences of the right- and left-moving e+11a and e-ikx (see Exercise 13), just as we expect for a sfanding wave. Complex exponentials would have been an unnecessary detour. In an unbound system, however, where we need to keep directions of motion separate. they are the correct choice.

E�pcrimental o,ctup

I '=' O

left-moving

Because we will concenrrate on stationary states, in which the same e-l� appears in all '¥(x. /), the spatial parts are our main concern: I/I(x) = e+l.Q moves right and I/I(x) = e-ikx moves lefl. Facing complex spatial wave functions after an entire chapler on bound states in which they were real. a brief digression is helpful. Here we con­ sider particles that move freely before encountering a force. Between the walls in the infinite and finite wells, the potential energy was 0, so a "pani­ cle" would also move freely. But there we used sinh and cosh, while here we use e+il:.t and e-1b. Why? The time-independent Schrooinger equation for U = 0 IS

that is clas�i(;ally sur­

£

= Ae- 11:.t- iwt

I

.i _

_ _ _ _

6.1,

a rep­ Now consider free particles encountering a potential step. In Figure resentative electron moves to the right in a region of zero potential energy, until it encounters a very narrow region. where it experiences a large force opposite its direclion of motion-a backward " kick:' Passing through. its potential energy jumps abruptly to Uo and stays there. Because a force is a change in potential energy (F = -dUld;r;), the narrow region near x = ° is the

only place where the electron experiences a force. The potential energy i!io given by xO

Let us start with the cas.e in which an incident particle has kinetic energy it . £ greater than the potential energy jump, as Figure 6.1 !'>hows. Cla.....ically. should merely slow down abruptly at x = O. its kinetic energy dropping a,> potential energy jumps. then proceed at constant reduced !;peed. There is no classical turning point where its kinetic energy falls to 0 and at which the force would cause a reversal of its motion. The particle would not rebound. But classical physics won't do. To understand the behavior of ,>omething as small as an electron, we must apply quantum mechanics. In the region to the left of the step, x < O. we win have a right-moving incident wave, and we alloW for a left-moving reflected wave. Each solution of ( 6- 1 ) is stH! a solution if it includes an arbitrary multiplicative constant, which will eventually tell us how they compare; so for the incident wave, we use Ae+ilr, and for the reflected wave, Be-ib:. In the region to the right of the step, x > 0, where U = UO' the SchrMinger equation is

,p",(x)

2m(£


-

�,

Vo)

(6-2)

",(x)

Note that because E exceeds Uo' "'(x) and its second derivative are still of opposite sign, as in (6-1), so the same functions work here as when U is O. The only difference is the coefficient of "'(x). making it convenient to rewrite (6-2) as where

' k

�

�2m(£

Vo)

h'

(6-3)

The mathematical solutions here differ only in having k' in place of k. To again distinguish directions. we choose the pair of functions e+ik'x and e-ik'x. How­ ever, we have a physical reason to throw out one of these. Beyond x = O. there is no change in potential energy-no force to reflect anything moving to the right. Because there can thus be nothing moving to the left in this region, e-,k',l is physically inapplicable. Altogether, then, for x < 0, we have incident (inc) and reflected (refl) waves, and for x > 0, we have only a right-moving trans� mined (trans) wave.

"'x
+

Be-ik.t Ren

,�,> ', o( , ) .

=

Figure 6.2 represents these various parts schematically.

Ce+ik·x

(6-4)

Trans

The square of Ae+ikx is proportional to the number of particles per unit dis­ tance on the left of the step and moving to the right-incident particles.

Figure 6.2 lneident particles may be

reflected or transmitted. Ino;

-

-

R,n

-------L\

_

.-0

_ _ _ _ _ _ _ _ _

+ i = I). so tho de 1be square of any complex exponential is I (� t: X � ; I�. nSlt» is particularly simple: A A. (Remember that. in genera1. when we -

0

wave function. we mean the complex square-the function times itssquart·, • conjugale. Also. with complex spatial wave functions. we mUSt allow t cornple� c --o � art.iCle s coefficients.) The com:sponding quantity for reflected p n left of the step and moving to the left-is a-B. and for transmitted particl es , It IS C o C

� 0

l"'l�n = B'B

I"'I!� = A 'A

I"'I� = C' C

�

(6-5)

11 is worth reiterating that because E > Vo' it would be absurd to II Ow for reflection of a classical particle. But a light wave divides whenever i ters a different medium. giving us some precedent. In fact, without Wlng for reflection in our maner wave, we simply could not satisfy the sm o hness requirement. b The wave function and its derivative must e continuous at th e where the two regions meet. These conditions are

���co , �n ��

POint 0

d"',
i

= d"'»o d.x

X'-O

i

A+B=C

or

;

(6'6)

rsO

k(A - B ) = k' C

or

In the case of a bound particle, imposing the physical require ments I eads . hal e v ue 0f k (hence " to quantization. However. no reSblctiOn on t ' £) art. s se. from conditions (6-6). Because no particle is bound, we don't have the c ' ends that would inevitably lead to a standing wave. An inc tions at I particle may have any energy. .

�;:�

nvo

us are reflection and transmission rob bo p a ties. We ca1culate these from ratios of numbers per unit time in the corr,. sponding beams:

(6-6) give

What conditions

number time Using v =

p/m

=

Iik/m

ex:

=

0

Ih.

number distance distance

time

time

DC

1",I'k

The transmission probability, given the symbol number transmitted

= � - =

-

1",I'v

k to express this in terms of k, we have number

T=

DC

time

number incident time

T, is thus

1",1�"k.>O = --C*Ck' A*A k 1 ",lf.i.
ility R " and the renedion probab number reflected

R

•

e tim _ ,-:" " _ -, -: -:number incident time

Now. eliminating eliminating C yields

T

=

pl�nk ,�_o = B"B = !.\I/I\; n.:k,,_ O A"'A

B from condition!\ (6-6) yteld.. C =

B = Hk - k')f(k + k'))A. Thu...

I�k1V.;

+ k' HA, llnd

[2k/(k + k'}]A·pk/(k + k' }]A k ' 4kk' A.A - k ==- (k + k'}2

-

and

R =

[(k - k')/(k + k')]A'[(k - k '}/(k + k'}]A (k k')' A'A ' = (k + k')l

We could have deduced one probability from the other. for the total number of particles had better be conserved. so the transmission and reflection probabili­

Uo E.

ties mUSr add to I . Verifying that they do is left as an exercise. Finally. we express the probabilities in lenns of

nitions of k and

T

=

k'

•

we have

VE(E - Uo) - U"0)'' ( vE 'E::+'-:":- V r.E:=� 41.

R =

and

Using the deft­

(vE - Vi U�)' (vE + V£- Uo)' -

(6-71

Notably, expressions (6-7) are essentially identical to those giving reflected and transmitted intensities of a light wave normally incident on the interface between two media. Those for light differ only in that

VE

and

VE - Vo' proportional 10 the speeds in the two regions, arc replaced by the speeds

cJn, where

'1 is the medium's refractive index, The behaviors are com­

pletely analogous, The most important point here is that, contrary to the classi­ cal expectation, the reflection probability is nonzero,

EXAMPLE 6 . 1

An electron of kinetic energy 5 eV encounters a probability that it will be reflected? ,01

2 eV

potential step, What is the

TI

Its total energy is its initial kinetic energy, because it starts where given to be 2 eV,

R

(Vs - vs.::2)' vS + vSr:--:2)' � 0.016

= (_ r:

_

-

U = O. and Uo is

Something about Eumple

6.1

might ....:em lTOubling. Planck's constanl is

nowhere 10 be found in expre!i�ion� (6-7). Apparently, the probability obtai ned would apply ju�t as well to a bowling ball with a kinetic energy of 50 J J. But never would we ex.pect to � the

encountering a potential Jump of

20

bowling ball reflect. It is a cla.ssical C3!>e, and it should simpl)· slow down to a kinetic energy of

30 J. What's wrong'?

We have assumed that the pOiential energy jumps abruptly in an infinites. imal region at the origin. Thh IS a valid approximation a� long as the region where it actually changes i.. much narrowcr than the \\-3velength of the inci.

�

�

Subatomic particles have rather small mo ent and thus long . wavelengths, so cin.:um!»tam.:es 10 which the assumption IS vahd are common.

dent particle.

As we have come to expect, wave behavior prevails because the wavelength IS large compared with the relevant dimenSIOn of the apparatus-the step\

abrupt jump. But a calculution of the wavelength of a moving

rll(lcroscopk

object. as shown in Example 4. 1 , gives n ridiculollsly small value, fur less than nuclear dimension,>. Our simple assumptions and analysis fail in such cases. A common misconception is that a particle may reflect at a potential jump

illcrease.\·, but it Illay do so even if the potential energy dmpl. A force ill a particle's direction o/motion may CUUM! it ..imply because the potential energy

to rellec!. Reflection and tran...mission probabilitie... for a potential drop follow from the same analysi ..., merely replacing Vo by -Vo' and in particular, the rellectlon probability for a drop is

flot zero (see

Exercise 15). The conclusion

really shouldn't be too �urpri ...ing, given that, with

E greater than the potential

energy at all pomts, renection is at odd� with classical mechanics

case.

ill either

The analogous behavior of light IS again helpful. Just as any increase Or

decrease in refractive index cause� partial reflection of a light wave, any change in potential energy results in partial reflection of a matter wave.

E < Uo

Now let us consider what happens when particles encounter a potential step

whose height Vo is greatcr than the particle energy. As shown in Figure

6.3,

the situation is the same as before, but the potential jump is higher. An elec· tron behaving classically would ,<;low to a stop �omewhere in the tiny region around the origin, then reverse its direction. It certainly should reflect. The analysis actually differs very little. For x

< 0,

we will still have two x > 0, however,

solutions representing incident and reflected waves. For things have changed. The Schrodinger equation is

"'", (x)

FI"ure 6.3 A cJa ... ... u:ally insurmoUnlab1c potential ...tep.

. ,' u.\

This is equation where

E<

(6-2)

=

2m( Uo - £) " , r

",(x )

(6-8)

Iightly rearranged <;imply to bener suit it to a case

...

Vo. Now rJt(x) and Its second derivative are of the same sign-a

crucial distinction. Oscillatory functions don't satisfy this condition. but expo­ E

--
____

_ _ _ _ _

nentials do (see Section 5.6). In more compact fonn,

U

l

_ _ _ _ _

where a l!!

2m(Uo

h'

-

£)

(6-9)

6. . ",, '01 ... ... .. T\\'o

funl.:tions work: 4/I(x)

C

t/I(.t) = ,-tu

and

t'+tu

«(>-10)

Ho\\e\'cr, the region to the ri�ht of th� step e:\t�nds fort''t'C'r. so we must throw out (,1-(11. for it dl ....er�c.. as x - +.:.:. Thus. in pla,,� 0( 6-4). we now have

1/11< 0( \ )

:D

At'+ill

+

Bt'-iLI

Inc

Ren

(The con!'lpicuou ... absence of a "Trans" label we address soon.) Th� smooth· n"�s I.:ondilu:ms analogou," 10 (6-6) become

,',

"'. - 0

d
0

1

I

(0) ' A ....,+,HI +

,',

(0)

"'. -0

0

d
'0 1

d.\

'

O.-JA:O

or

.1 -0

=

C.. •--utl

A + B .: C

: ikAl'+,·I;O - ikBe-"'o -oCt'-(10 or k(A - 8) = -aC �

It might 1\cem thnt a full calculation of reflection and transmission proba­ bilities R and T .. hould follow, but a clo!o.e look at the magnitude of B answers many quc..tion\. Eliminating C between the smoothness conditiuns gh'cs S = -ila + ,k)/(a - ik)JA .

101

=

=

'(_ V J( ik )( VB 'S =

-

a

Cl' +

ik

A

,

)( )

ik ' A ik a -:-; a + ik A a ik

a +

-

- a_+

I��)

a - ik

roo = IA I = _vA'A

The magnitudes of 8 and A are equal. (8 merely poinl" in a difl"erent "direc­ tion" in the complex plane.) It follows that

B' B

R=-= I A'A

Because R and T must add to I , it would seem that the transmission prob­ ability is O. indeed, it is. If a panicle were transmitled, it should continue mov. lng 10 the right at constant speed indefinitely, since it experiences no force beyond x = O. But the wave function fall� exponentially to 0 beyond the �tep; nothing survives to be found infinitely far away. Although it doesn't do so abruptly at x = O. the incoming wave does reflect completely. Inferring T from R is the best way to detennine it, because it isn't proper to clllwlllie a trans­ mission probability from Cc. Our ratios involve the number per Imil lime, 1¢121" but in the region x > 0, the speed v is not "real" (KE = � Im,2 is nega­ tive). Because no number per unit time can be specified. neither can a trans. mission probability. On the other hand. a calculation of number per UtI;! distance is still justi. tied. The quantity !I/I1 2 i� �till real, and because the wave penetrates the step. there is a probability of finding electrons on the "wrong" side of the �tep. But it can't be concluded that par!icleJ are constantly getting through to x > 0,

Agure 6.'

Reflection and lraruomi!lihOfl

probabilities for a potential �tcp.

LO r---r---'--'_--�

I

O.

o. 0.4 0.2 ()

1

I�

then going. back through to x < O. for there i� no force beyond the step lhIt could reflect anything. We ..imply accept that as long as no attempt is made 10 lind a particle. we merely have an undisturbed wave that. though it reflects completely, happen.. to peneuate the classically forbidden step. The "penetra­ tion depth" Ie; the "arne a.. in Section 5.6, where the wave function outside the finite well wa.' abo ..imply a dying e;(ponenlial. I S = - = a

,

EILu

-;= �== h

V2m(Uo - E)

The clo.. er E i... to Va' the greater is 8, and the �Iower the decay of the wave function. In any case, the wave function is very small for x » 6. As always, eleclromagnetic waves show analogous behavior. Just as a Vo > E step completely reflects massive particles, a smooth metal surface completely reflects light-making it a good mirror. The light wave doesn't propagate through the meta!, and no photons are transmitted, but there is an electromagnetic field within the metal, oscillating with time and decaying ex.ponentially with depth. For an electromagnetic wave, the penetration depth ie; often called skin depth. Figure 6.4 shows reflection and transmission probabilities plotted versus EIUo for a potential step. If E is less than Vo' the wave is totally reflected. When E exceeds Ua' the reflection probability falls rapidly with increasing E.

6.2 The Potential Barrier and Tunneling Tunneling is one of the most important and startling ideas in quantum mechanics. The simplest situation is a potential barrier, a potential energy jump that is only temporary. If a particle's energy is less than the barrier's "height," it should not get through---classically.

E > Uo

Let us first consider the case where E is greater than the barrier height Uo. As shown in Figure 6.5, at.x ::::; 0, the potential energy jumps up, giving the pani. cle a "kick" in the backward direction, and at .x ::::; L, it drops back to 0, giving it a forward kick.

U(x) Figure 6.5 A clas�ically surmountable potentia) barrier.

=

{�O

x < O, x > L O < x < L

Classically. the particle shouldn't reflect; it should merely slow down between .x ::::; 0 and .x = L, then return to its previous speed. Let us see what quantum mechanics has to say. In the region x < 0, we should again have our incident and reflected waves. Between x = 0 and x = L, the situation is ex:actly as it was to the right of the potential step when E was greater than Vo' Thus, SchrOdinger equation (6-3) applies just as before, giving the same mathematical solutions: e+ik'� and

t--"" ,l. In the Cllse of me simple step, we threw out the h:tl.-mo....ing t

a '. (or beyoml \ ... 0 able to renect a wa.. e. Now th�� is-the nothing was there pOCential drop at x ... L Therefnre we cannot jUstifulbl)' thn\\\' l\ut lhi" solu­ tion. Finally, In the region ,\ '> I.. where the potential �nerg) is a�ain O. the SchrOdinger equation i'io the �ame as fl1r \ < O. a... Ilrt tts mathemattcal solu­ tions, t'*/lr and t-Ill . But here W I! d11 1hrow one out, r'- loll , for O\\thing mo..�s to the left in this reg\On. Altogether. we h:\\'e

tII1
= Ae+,b + Inc

Be--'u ReO

Trans

I/IO< l< L(x) = Ct·+ik'�

+

De-,·f�

The smoothness requirements at the two points where region... meet iue

�\ dx

x"O

< X< L

= dl/lO

dx

\

\A + B

or .{""O

C+

: ikAe+'/.:{) _ ikBe-iW = ik ' Ce+ifO or

\k(A - B) = k' (C

L ik'L = Fe,kL\ !/JO<xo(L) : \Ce+I"' + De

�o/IO<X< L dx

\

.t= L

= do/lX>L dx

\

x"'- L

: \ik ' (Ce+'k'L _ Oe-.n)

=

-

01

ik'Dt' I�'O

0)1

ikFe,'L\

As before, these give us the reflection and transmission probabilities, but now "transmission" is finding a right-moving particle in the region _\ > L. Thus,

B'B R=­ A'A

I /II�", k F' F T= o =-

10/11'

me

k

A'A

Between the four conditions boxed above, we can eliminate C and D, then solve for B and F in terms of A. The algebraic details are left as an ex.ercise,

k'2k2

sin'(k'L) R = ----��� � k" k' sin'(k'L) + 4

(k'

_

k" )'

4 _ k" '� ? " "' ("" k';� � --.:: ..-!".-;:T= (6-12) k'2k2 sin'(k'L) + 4 , " (k - k -)' -

­

Finally, using the definitions of k and k', we obtain

R=

sin'[ \ .hm(E - Vo) L/ft] -:"-'-, -;= -' J V 2m = = = � = -:"i ft ' -:" [ ( -: E/-: V-:" 0)-' (E V0) L/ ] + 4( E/ sn [ 1]

-'v-:-'i o)

4(E/Vo) [ (E/Vo) - I ] �� � T = --:J [V ' = 2m = = = � � � � L � ft -'� /(E V: /-:" o)sin2 Vo)l -' V0) / ] + 4(E (E I]

(6-13)

2M

CIt.,.. 6

Unbound Swes: Scept. Tunneling, and Panjd�· Wave Prop;ipllon

As they should, R and T add 10 I . and jusl as in Ihe E > Vo polential seep. reflection is possible. contr.uy 10 the cla.!.sical eJtpecl8rion, Obviou�ly, with rhe width L now a faclor, the probabililies for a banicr_ more complicated than those for a slep. bUI Ihis richer detail admits an inlier. esting phenomenon: resonant transmission. Allhough both probabilities 1ft in general nonzero. the numeralor of R involves 3 sine. When this is 0, the� is no refleclion. The condilion i.s �. � U­ \ /2",(1:: o) L = -

-.-.--- --

�

1/""

or

16-)4)

This i.1> not energy quantization. 'ncident particles may be sent in with any energy, bur resonant (complete) tran.')mL�sion occurs 31 only certain energie�. A familiar behavior in light (yel again!) is completely analogous. Nonreflec. tiYe coalings exploit the same wave properties. If !.he wid!.h and refractiYe index are cho.�en properly, a thin film will pass lighr withoul renection. In both applications, rl!llections within the film ("over" the barrier) preci!;tly cancel lhe wave reflected from the first interface. As always, it is important 10 remember that wa\'es. nol particles. interfere. A particle-photon or massive--encoullIering an obstacle may later be detected as a parficfe. having been either reflected or transmiued. bUI the process inyolves a probability that come!> only from the It'(f\'e function. E<

Ua-Tunneling

A panicle may escape confinemenl that classical mechanics says il cannot escape. This is lunneling. a fascinating phenomenon whose application we discuss in the nexi section. Here we analyze it in the simplesl case, for which we have already laid the groundwork. Shown in Figure 6.6. the situation is a polential barrier with £ < Uo' A classical particle would rebound at first

Figure 6.6 Tunneling.

encounter, x = O. The solutions of the Schrodinger equation in the regions outside the bar­ rier, x < 0 and x > L, are identical to the E > Va barrier. Things are different between 0 and L, which before was classically allowed and is now classically forbidden, but the solutions in this circumstance are the ones we discussed in connection with the E < Va pOlential offep. The SchrOdinger equation is again (6-8), and the mathematical solutions are e +aA" and e-u..f. In the case of the slep, we threw oul e+(u. because it diverges as x --+ +00. The bamer does '101 extend to infinity. so we have no reason to throw it out. The solutions alto­ gether are thus:

,". ', > L (x) = Fe+ik.x 'Irans

(6-15)

The solutions inside the barrier may still seem troublmg. because a is real. Comparing with the other regions, we might say that the wave number and

[lll.IRlt'nlUlll are ,maginary. In

"

WilY. Ihi:- Ii,:-. h'IJ the kinetil; energy in...idc- the

NJrio!r i", alter aiL negat,\c, In any 1;;l.'>C', there ,:- no 'real m�lll\C'ntum in....ide ' Ium::llon.. 110 IlIlt rerott'>C'nt nght- llml le{t.mo\,ing ['W1'­ the b;lrrier. ilnd the '''''0 .:il''' Still, bllth are necl1et.l tll cn..ure ..moIJthne..... It i.. lert [I" an c'erriloc to ohtain thc 'mllo,lthne" I;onditilln... fnlm whkh the rdle..:tlon and lnln"ni�"'lm rorohahilitie... lE).C'rri-.e 1� :-.how thllt find ", eo they l;;Jn ;\l;tu311) he dcdul;cd by a ..impic f('pi;Jl;l'menU R

T

'

t

..inh� \/�m{ Uo

,inh)1 \/2m(Vo

/:')L/IIj

+

''

E) VIII

�(EIVo)l1 - E.'!Vu)

4(£/ Vo)(\

�Vi,,�un

"i nh

E, Vo) E)i�hT+ 4(£!Uo)(l

FIgure 6.1 Wa\'e 'un.;ti�\n� (or pani. in gcneral. nOtl/,ero--a particle can escape through a barrier thJ.t it cle .. �'I dillerent energie.. inddent fnun can't �urmount cla....ically. H i<;, important to grasp that il can do this 1101 due to the leh on I!. potential hllfTier. some my..,crioulo t1ucluation in its energy. allowing it to go "over the tl'P'" On the contrary. from the start. we assumed a stationary ,tate, in which energy i!. c-'� _ , +"", -; ,_ � _ _ , well defined. Tile particle lIew'r Ii(l.\ .wllkiem energy to .WmlOlml lll{' hurrier. t., _", In':>lc'ld, il "tunnels" through. and the principles of quantum mechanics demand that there be such a possibility. The �olulion to the left of the barrier is ..orne t.� -----�c--"'"'--c��_.,,-� combination of incidenl and reflected wa\'es. of po!'oili\'e and negali\'e momentum. At :r: = O. it smoothly joins a function inside the barrier that tends to die lc.J ------'c--�--���­ off (the C in the eKponentially increasing C,e-+a..'being u!>ually quite ..mall). And at.� "" L. thi.; "moothly joins a transmined wave of positive momentum. There ;!. f' l -'""'-­ no physically acceptable solulion thai is identically zero beyond the barrier. Figure 6.7 show!. wave functions (real part) for particles of different enert:, _ .... gies incident on a barrier from the left. Note that the wavelengths decrease a!> kinetic energy increa<;es. At energy E 1, the exponential decay wilhin the barrier is rapid. and lillie transmission occurs. At E�. barely below the barrier height. the wavc decay.; less rapidly. leaving a larger transmitted " Iail." Energy E3 is above the balTicr. but we see evidence of significant reflection-the wave Figure 6.8 ReOeclion and transmission is of smaller amplitude to the right of the barrier than to the left. AI.;o, the probabilities for a polenlial barner. wavelenglh is longer when it is "over" the barrier, because the speed is smaller 2nd n::wnance there. Energy £4 happens to be the first lTansmission resonance-the wave on 1 " .0 the right is of precisely the same amplitude as on the left. Note that the bar­ c ' , , , rier's width is 1,\. Not coincidentally. this is just the condition we would , , , ex.pect for maximum transmission of a light wave through a thin film sur­ 0.'$ " " ' rounded by air. At £5' though not a transmission resonance, little reflection 0.6 occurs-the amplitude on the right is only slightly less than on the left. Figure 6.8 shows renection and transmission pTobabilities for all energies incident on the same barrier as in Figure 6.7, with the above energies E\ to E5 OA indicated by dashed lines. Although reflection dominates for E < Ua and transmission for £ > Uo. both occur at all energies eKcept transmission reso- 0,2 R nances. In the limit L ----+ 00. the barrier becomes a step and Figure 6.8 becomes o 3 Figure 6.4. 4 5

80lh are.

--\-"'--+

\

206

Chapter 6

UQbound SIaleA: Steps, Thnncling. and Partide-Wa\'e Propagallon

fig..... 6.9 Frustrated total internal reflection-an electromagnetic analog of quantum-mechanical tunneling.

The inevitable electromagnetic analog to tunneling is light striking a tbia metal film. If the film is thin enough. the exponentially decaying electrol1llc. netic field within the metal will \till be significant at the far Side of the filrD. and some fraction of the incident Intensity will emerge. A half-sil....ered mirror

operates on this principle. Another analog 10 tunneling is the phenomellOQ 0( fru'ilrnled lotal internal renection. repre�ented schematically in Figure 6.9. lq

the top diagram. an electromagnetic wave traveling through a prism strikes the interface with the outside air. which is of lower refractive index. The incideru

angle exceeds the crilical angle, so the wave is totally internally renected. BLII though there is no light per se transmitted into the air, there is a time-varying.

exponentially decaying electromagnetic field in the space beyond. To this point. the behavior is analogous 10 the £ < Vo potentiaJ step. except thal lhe

.-

incident and renected waves move in perpendicular directions. (f a secon d prism is broughl near. so that the decaying field is intercepted before it falt� off too much. the step is effectively replaced by a narrow barrier (the air_

Reflected --

\

"

luudem .-

�I

rlan'lllln�d •

space), and the internal reflection is no longer total-it is "frustrated." The incident wave is partially lfansmiued. and the narrower the barrier, the greater

the transmission.

Tunneling Through Wide Barriers "

-

In many applications. the transmission probability for tunneling is very smaJl.

Inside the barrier, the exponentially increasing component Ce +ax is negligible. so !/I(x) is essentially just De-ax, or De-u8, where {j ==: lIa. If L » 5, very lit. tie of the wave survives

is thus

L

- �

5

10 x = L.

<xL

�

The condition for a "wide" barrier

V2m(Uo - £) --'-:-'--'-L » r.

We see that the barrier is " wide" if L is large, or both.

E

(6-17)

is much less than UO' or

It is left as an exercise to show that if (6-17) holds, then the transmission probability of (6-16) assumes the simpler form

(6-18)

EXAMPLE 6_2 An electron encounters a barrier of heigh t 5.0 eV and width 1.6 nm. What is It\( transmission probability if its energy is (a) 2.0 eV? (b) 3.0 eV?

First, we see if condition (6- 17) holds. For 2.0 eV,

L

lj

�

V2(9.1 1 X 10"" kg)(5.0 - 2.0)( 1.6 X JO " 1) -'= = ---'-'--"-;-: � � = -;:;c :-=-'= '---'-'-' '� L6 X 10--9 m = J .055 X 10 3 4J·s

14.2

6.3 Alpha Decay .��== lmqine: Bob is stancti0S OD shoWn in Figwe

ball would have a pavilllio1lll bill. Anna rolls the ball, wll>cb tioD is negligible. WheD the lmpoIsible' The ball', toIII

been ., leGIt 900 J-its added ro its kinetic etIOIJY I... tban 900 J of kinetic

II

with a kinetic ellOlJlY of 200 the hin would have had ro be Something quite like Ibia

Certain :·: ta:'· :.:torm e �::= away by panicle

alpha decay

.

excess energy by

•

carried the _ stable state. The IIId cles that may be omitted a-n as IIII IIIpIIa

�

doph:ted in Figure 6. 1 1 . It

146 neuIroDs, for a toIaI of

201

Chapter ,

Unbound Slates: S�p!II. Tunneling. 0II1Id Partide-·Wave- Prnpa,91loo

what remains is a nucleus of 90 prolUns (92 - 2) and

1+-1 neulTOns ll46 - 2). �

234 nucleons: thorium-2]-l. Both being positively charged. the thoriuna nucleu'i and the alpha particle repel one another_ A3 the alpha particle moves away from the thorium. eltx:tros[Jtic potentiaJ energy is converted to k inetic and the alpha particle speeds up. Howevcr. experiment reveals a »deficiency ; In the final kinelie energy. Lei U\ cak:ulate the minimum energy we mighl

clu.Hically expect the alpha particle to have.

To understand aJpha decay. we need to introduce a few ideas belonging to

nuclear physic.... the topic of Chapler I I The con.-.titucnts of the nucleus are held

logether nol by electro\latic force. for this alone would violenLly scatter the J><Xi­

lively charged protons. but by the altogether different both proton... and neutrons. are found

strong force. All nucleon�.

10 attract one another via this parricularl)"

strong force. but only when very close together. It may be said that they attracl each other only when in contac!. Thu.!.. the nucleus may be viewed crudely & a collection of nucleons stuck as close together as possible in an approximatel) spherical arrangement. Now. a3 depicted in the potential energy curve of Figure 6.12, an alpha parricle far outside the radius of a large nucleus would experience only the electrostatic repulsion from the prO[on� in�ide. so its electrostatic poten.

lial energy would increase if it were brought clo.�er. Upon reachjng the surface however. the attractive strong force would be overwhelming and lead to a rapidl),

decrt'lJSing potential energy. Accordingly. the alpha particle would be at a poim of maximum potential energy when at the nuclear surface. I

Scattering experiments-a refined tenn for throwing things at each other

and watching how they rebound-have determined nuclear radii fairly accu_ rately, and the mdius of the thorium-234 nucleus is approximately 7.4 X 10-15 m.

At the surface of a thorium nucleus (90 protons), an alpha particle (2 protonq would thus ha',:e an electrostatic potential energy of

VeiN: :=: :=:

�

(2 X 1 .6 X 1 0 " C)(90 X 1 .6 X 10-19 C)

q,q,

41Te r

:=:

41t{8.85 X 1 0 · 1 2 C2/N ' m2)(7.4 X 1 0

15 m)

5.6 X 10-12 J = 35 MeV

When a uranium nucleus alpha-decays. the alpha particle at the surface of the

daughter thorium nucleus would have 35 MeV of electrostatic pOlenlial energy and would !opeed up

it i� repelled awny. Classically, the kinetic energy can be no less than 0 at the nuclear surface. so it can be no less than 35 MeV when very far away. where the potential is negligible. An alpha pani­
cle with less than 35 MeV of kinetic energy should never be found. The exper_ imental truth is tbat an alpha panicle emitted from uranium�238 and detected

far from the thorium-234 daughter nucleus has a kinetic energy of only 4.3 MeV. Quantum mechanics provides the answer: The alpha particle never

a1trxtS 1111

In.c hd,a,·ior of. m"Lo:.;uk In a ",al�' drop " "m­ w O!M"", 100 c'"'I'3Y I� nccded 10 i . It bring 1110 W ,un,,"_ Th�. the: drop·. iowc-.I cne'K) " ",-hen II, .II.1."C an:a .. . m,01mum. ",·)u"h " !he hMi. of .u,f:o,,� ",n"/)n. n.. apl.ll named md ,uc:cn.tlll ··"quld drop nlOde"· of the .... . (�er I L) d.IUmelo thai nucleon, behav� nuck hk( mol«ule" In • liqUid d,,'P

has to possess 35 MeV of energy.

The horizontal gray line for E in Figure 6. 1 2 represents the true situation.

Initially. an alpha particle moves relatively freely within the uranium nucleus, pulled equally in all directions by the 234 other nucleons. As i t approaches the

barrier at the surface_ however. the other nucleons exert a net force on it back toward the center. Although the alpha particle docsn·t have enough energy to sunnount this barrier. it may quantum-mechanically tunnel.

F1g� 6.12

fo �).l'u('<' the nudeu... an alpha piU'\iI:\( mu..t tunnd

lu,,";.an! -rom th�

....... .111

the stl�'nl !iut1 nlC\c',r Un..:: ·Ill� I: ·J'
" O<1htJc 1M ,,;..,.:I,-u�, ,he _l·tI"",,,k ""I,",:rn:n. ,:. I.'i ,,, d�.In.>''t.\11o: I\'PIl\""'\"

,

A calculation of the transmission probability for our multifarious potcn­ is tOO involved to present here. But c:akulated decay rates (dt!c;.,ys energy tial per second) agree well with experimentally determined values. \n simplitit!d form, the calculation is as fonows: number of decays

number of times a strike). barrier

time

time X transmission probability

one strike

= -,-,:- X -

time to cross diameter

T

I

=

diameter of nucleus/speed

=

v -- T

X T

2rnuc

Of course, once the alpha particle escapes, alpha decay of the original nucleus is no longer possible, because the element has changed (e.g., uranium becomes thorium). So refen·jng to a number per unit time for a �ingle nucleus is nonsense. But with the huge number of nuclei in even a macroscopically small sample of radioactive material, simply multiplying by the total number yields a proper average decay rate. As noted, agreement with ex.periment is very good, but we must not overlook the simple fact that the probabilistic

nature of alpha decay is evidence of its quantum-mechanical basis. Watching individual nuclei. we would find great variation in the time required for decay, and this is exactly what we expect if the decay is governed by quantum­ mechanical probabilities. Example 6.2 provided the key to understanding another classical mystery of alpha decay. Most alpha particles have energies E in the 4-9 MeV range, yet their decay rates differ by more than 20 orders of magnitude. as shown in Table 6. 1 . The probability of the alpha particle tunneling in a single encounter with the nuclear surface is very small, and as we found in Example 6.2, when

this is true, transmission probabilities are extremely sensitive to the value of E.

210 ehllpter 6 Unbound States: SlepS. Thnneling. and Particle-Wave Propagation

TABLE 6.1

Energies and decay times in 0: decay

.emitting nuclew

a-particle energy (MeV)

ct;;;

o Mean time t

;&

Po-2 1 2

8.8

4.4 x 10-7 ,«

Rn-220

6.3

79 secon�

Ra-224

5.7

5.3 days

Ra-226

4.8

2300 yurs

U-238

4.3

6.5 x 109 years

-

-

In conclusion, according to classical physics. alpha decay-in the unex.. pected low energy of alpha particles. in its probabilistic nature, and in the vast range of decay rales-would be completely baffling. Its explanation is a tri­ umph of quantum mechanics.

FigUnl 6.13

Tunnel diode.

AV = 0. / := 0

+

ill'

.-

',

_

I(� - f-,

I

Net electron flow

The Tunnel Diode A tunnel diode is an electronic circuit element whose response to applied volt. ages is unusual and very fasl. In a narrow region between the device's two ends (leads), there is a change in [he maleriai's physical properties thai pre­ venlS simple conduction of electrons from one end 10 the other. In essence, the electrons a[ the ends are separaled by an electrostatic potential barrier they can't classically surmount, crudely depicted in Figure 6.13. By design. how. ever, the barrier is fairly thin, and significant IUnneling occurs. With no applied voltage, it occurs equally in both directions-there is no net flow. When a potential difference is applied. the situation becomes asymmetric. Right and left tunneling rates differ, and a net currenl flows. The tunnel diode may seem to have liule utility, as an applied VOltage will induce current flow in many materials. But one of its distinctive features is how the current varies with voltage. It doesn't steadily increase as applied voltage is increased: at some points, it decreases. Moreover, changing the applied voltage changes the transmission rates almost instantly, and quiCk response is very desirable at high frequencies. The more common devices thaI control current via changing voltage rely on relatively slow thermal diffusion of the charge carriers. (See Section 10.8 for further discussion of diodes.) Although its early promise as a high-frequency switch in integrated circuits ha,s dimmed. the tunnel diode has found use in a varielY of modem electronic circuits. A more complex device. the resonant tunneling diode, is discussed in Progre"is and Applications. SQUIDS A tunnel junction. two conductors separated by an insulating barrier. is a k.ey element in many electronic device�, and when Ihe insulator separates two superconductors (see Chapter 10), it is known as a Josephson junction.

fl.) A.lpha Decay atId Otbcr A.ppIiatioea

through which electrons tunnel in pair!'. The Josephson junction is at the hean. of devices known as SQUID� (superconducting quantum interference devices). The insulator serves as i.\ weak link between the charges' motions on the two sides, coupling them together as would a weak spring connecting two pendula.

111

FIIiIU" 6.14 "The effect of Oat: osdlla­ tor on another depends on the pbMe

relationship.

As illustrated in Figure 6.14, with this kind of coupling, energy/current flow is intimately dependent on the phase relationship between the things that oscil­ late on the two sides-pendula or charge-camer wave functions-and there may be constructive or destructive interference. In the case of charges �pa­ rated by the weak link in a SQUID, the phase relationship is also a very sensi­ tive function of any nearby magnetic field. Depicted in Figure 6.15, a SQUID is. in essence. a loop with two Josephson junctions. Interference caused by

1. ­

No '_ No mcrzy tRMfer

..,

Weak link

'I;'I;VVVI/VVIi,VI/V1.fU

small changes in the magnetic flux passing through it produces easily detected changes in the current I. SQUIDs can detect extremely small magnetic ftelds. such as those produced by the human heart and brain.

Energy tran�rer

-

FOlXe --_

Field Emission As we know. to remove an electron from a given kind of metal requires a

certain minimum amount of energy: the work function cP. In effect. the metal's electrons reside in a potential well, due to their attraction to the

positive ions. and they are " reflected" at its waHs by a potential step they cannot surmount. as shown in Figure 6.1 6(a). Owing to the random thermal

Figure 6.15 Elements of a SQUID­ sllpercondllcting quantum interference device.

distribution of speeds, at any given temperature, a small fraction of the electrons have sufficient kinetic energy to escape the metaL Those that do escape participate in an equilibrium exchange of electrons with the sur­ rounding space. Heating a metal filament to enhance this effect. known as

Figure 6.16 Electrons in a metal (a) behave as though in a finite well. An electric field (b) alters the "wall," so that tunneling may occur.

o:or \

Metal

....

Electn;n� bound by pt) enlial �tep

a � j

+ Vo \ 1 -:0]

\f

° 0°

Cathode

\ Anode

Tunneling through a potentia. barrier !

la)

(b)

Insulating junctions

(tunneling)

212

Chapter 6

Unbound States: Steps, Tunneling. and Particle.Wa�e Propagation

Figur. 6.17 One pixel of a field emis·

thermionic emission. has long been used as a source for electron such as those in the conventional cathode�ray tubes used in televisi

sion display. Applying a positive bia ..

X.ray machines.

turns on any of the three different col· on of subpixel.

Indi�idual pi:\el

---

--�- -

Rod

r

pho'>phllf

phosphor

Blue pho�phor

�

ons

tid

But there is another way to coax electrons from their potential WeU .u positive electrode is brought nearby. the potential step may effectively � changed to a �tential barrier" In �igur� 6.16(b), the positive ele�trode 1IlOdi. . fies the potenllal energy function seen by the electrons-Iowenng it OU ts

ide (Remember: Negatively charged electrons are at low PO tential energy where the potential is high.) Now electrons moving too slowly to

the metal.

s mount the work function barrier may tunnel. The technique can be u�1Jr· t generate an electron beam and is known as field emission, after the ele(; o trj field in the region of changing potential energy between metal and electrodec In many applications. it is preferable to thermionic emission, for he aling ·

metal filament not only wastes power but often produces considerable ele ctri� cal noise. Field emission is the heart of an alternative kind of flat-screen display i

2.5

active development, the aptly named field emission display (FED), depicted

mm

t �

Figure 6.17. Its potential advantages �ver the tr�ditio�al �iquid crystal di�Pla}

commonly used in laptop computers mclude Wider vlewmg angle and quicktr response.

Microtip\ ,

Column line

The Scanning Tunneling Microscope Figure- 6.18 Sc.:hematic diagram of an STM in operation. ,

IIL-,

Scan

. _'

'

Barrier width (variable)

Tip

•

by an STM.

to better use than the

scanning tunneling microscope (STM),

'

whose initial Rohrer). In thl\ design won the 1986 Nobel Prize in physics (Binnig and device, a slender metal tip is positioned near the sample under study.

Becau�

tip and sample are not actually in contact, the free electrons in each cannol pass between them in the usual classical way-a potential barrier intervene� . However, in the STM. the separation is made small enough that significam quantum-mechanical tunneling occurs. Variation in the tip-sample separation

smaller than typical alOmic dimensions translates to easily measurable

changes in tunneling current. Thus. as the tip of a tunneling microscope i$ scanned laterally over a sample's surface. as shown in Figure 6.18. it is able

Sample

FigUf1!! 6.19 A mi" ing atom.

Tunneling probability is very sensitive to barrier width, and nothing puts thl

as ..een

to "see'· individual atoms. Furthermore, calling the tunneling direction thr and the scan direction the x. by repeating the scan with the tip displaCed

;::

�lightly in the y-direction. an entire "topological map" of the sample's SUr.

face can be generated. Phenomenal results have been obtained. Figure 6.19

�hows a lone atom missing from a pattern of iodine atoms adsorbed on a pial. inurn surface. The hexagonal geometry that characterizes the iodine bondtng IS qUite clear.

ogy.

The STM ranks as one of the most indispensable tools in modem technol.

Its uses are already legion.

includmg studies of the geometry and compo.

sition of an endless list of �urfaces; locating Important biologicaJ molecular groups such as the fundamental building blocks of DNA: mapping micro.

scopic vortices in superconductors; nudging atoms from one point on a Sllr. face to another. And no end is in sight.

REAL-WORLD EXAMPLE S E E I N G SMALL DETAILS WITH AN STM The metal tip of an STM has a work. function, and the sam� is true of samples we might study with an STM. The two work functions are not, in general. equal. and in !.he STM. a potential difference is applied between tip and sample. giving a .-.Iopc: to the potential energy function felt by electrons moving between them. Ne\icrthele$". by trenting the space between tip and sample a!', a simple rectangular barrier who� "heigh'" above the electron energy is typical of a work function, we can understand the tunneling curren! should vary in a detectable way. even for the: e'l�mel}'

why small variations in tip-sample separations "seen" by an STM.

Applying the Physics Electrons in an STM tunnel between sample and tip. Using a Iypical metallic work

function of 4 eV for Uo to change to produce a

-

E, about how much would the tip-sample separation have

\0% decrease

in tunneling current? Use the wide-barrier

approximation.

In equation

(6-18), only L changes. Thus,

The tunneling current will vary as the tunneling probability varies. Setting the ratio

to 0.9 and taking logs on both sides gives

'" ( 0.9)

�

::::::) AL = 5 Atomic radii

10 31 kg)(4 X 1.6 X 1 0 1.055 X 10 34 J . S 10-12 m = 0.005 nm

-2t>L X

\12(9. 1 1

X

1 9 J)

are on the order of O. J nm. In practice, the resolution of even the

best STM doesn't exceed about 0.01 nm. but our estimate certainly suggests that this remarkuble device should be able to "see" atoms. As an alternative to translating a varying current

to an image of varying height, STMs often sense the tip location

needed to keep the tunneling current CO/lSlallf, and the motion of the tip thus tracks

the height of Ihe sample.

6.4

Particle-Wave Propagation

Thus far, our study of unbound states has concentrated on the effect of simple

forces on plane waves. A plane wave is not the most realistic malter wave for a single particle. To represent a reasonably compact moving particle. a tra\:eling else­ wave pulse is much better, It is broad in one region and essentially where. But the behavior of a pulse is considerably more complicated. even

0

with 110 steps or barriers, so we will have plenty to keep us busy as we restrict our attention to pulses moving in vacuum or in homogeneous media. A wave pulse may be tre:lted as a sum of plane waves, and to understand the behavior of the whole, we consider the behavior of its pans. Each constituent

214 Ot.pter 6

Unbound Stales: Steps, Thnneling. and Particle-Wave Propagation

plane wave moves at its own speed, or phase velocity, giving rise to two important consequences: ( 1 ) The speed of the region where the probability t: density is largest-the speed of the particle, or group velocity-may dd plane waves; and (2) an mitia distinctly from the speeds of the constituent l well-localized wave pulse will spread out with time, known as dispersion. its constituents get progressively out of step.

� �

Phase and Group Velocities Figur. 6.20 Wave group: The crests

and envelope move at different speeds.

Figure 6.20 shows a traveling wave pulse, which we associate with a traveling particle, at several successive times. Regarded as a sum of plane waves, We refer to it as a wave group. The light red arrows indicate the motion of a panic­ ular crest. The thin red outline traces the "envelope" of the wave group, a shape that moves with the group and defines the maximum possible displacement al each point. The dark red arrows indicate the motion of the envelope, which clearly does not move at the same speed as the crests. The speed of the enVe_ lope is known as the group velocity. If we are interested in knowing When the particle is likely to arrive at some destination, it is the speed of the envelope, where the probability is large-group velocity-that is of interest. The speed at which an individual crest moves-phase velocity-is rather unimponant. Nev_ ertheless, an understanding of the group's motion comes only through study of its parts. Let us begin with the simplest possible wave group.

A Sim ple Wave Group

Consider a wave group consisting ofjust two plane waves of equal amplitude A. Using the form we first met in equation (4- 10),

(Note: Despite the i and the use of '1', the main ideas in this section apply to aU wave phenomena, including electromagnetic radiation.) We wish our group to have reasonably well-defined wave number and frequency, so we choose k and w values that deviate by only very small amounts dk and dw above and below central values ko and wOo

k, WI

�

=

ko + dk Wo + dw

Thus,

,/

Factoring out common terms leaves

"" ko - dk W2 = w0 - dw �

Now, using the Euler formula, we obtain a fonn es..�ntially identkal to thai of a classical beat frequency. The sum of two sinusoidals b«umes " pmdm.:t of two who�e wave numbers and frequencies are va!'>1ly different.

F&gu.... 6.2' . ..., ,

We have a complex exponential that moves at speed wdko' the phase \elocity central value!\ ko and wOo modulated by a cosine function appropriate to the dwldk. A!\ !\hown ,In Figure 6.21, it is the latter tern, th,\t speed at moves that defines the envelope-the group velocity is dwldk. We see this more clearly bility density. by computing the proba 'I'·(x. I)'I'(X,

t) '= 4A2 cos2{(dkh

-

(dw)ll

(6.20)

The phase velocity disappears. Because a particle must move as the probabil.

ity of finding it moves, we expect the speed of the particle to be best repre·

\'l!F,

dwldk. Our surprising conclusion is that the sented by the speed of depend on the actual central values Wo and ko' doesn't group/particle velocity bul rather on how w varies wilh k from one constituent wave to another.

Although this simple wave group exhibits an important feature-that the

e�d �wldk- it isn't very particlelike. it is peri. probability densit� moves at sp : odic, as is any jillile sum of penodlc funcl1ons. So let us now tum to a more

particlelike case in which the probability density isn't spread out all over space.

A Particlelike Wave

The most general way of expressing a wave group is +00

'V (x,

t) �

100

(6·21 )

A(k)ei{h-w,) dk

This is a sum of plane waves of all different wave numbers, each multiplied by

its particular amplitude A(k), and each including its time dependence. (It is equa·

rion (4·2 1 ) with time dependence added.) Position x and lime

I are the usual

independent parameters, and we will address A(k) soon, but where do we get w'?

This is a central question, and the answer is that it can be considered a function

of k, for each phenomenon has built into it a relationship between w and k, or

equivalently between E and p. Consider two familiar cases: electromagnetic (EM) waves and matter waves. From equations (4-7), we

see that for plane·wave

solutions to Maxwell's equations in vacuum, w = ck. equivalent to E = cpo For plane-wave solutions to the free·particle SchrOdinger equation, equation (4. 1 1a)

showS that w = hk2/2m, equivalent to E = p2/2m. A relationship expressing frequency as a function of wave number is known as a dispersion relation. EM wave dispersion relation: w(k)

Matter wave dispersion relation: w(k)

=

ck

� -

2m

(6·22)

(6·23)

Slmple wave IJOUp.

216 Chaptw 6

Unbound Slain: Sleps. Tlllmding. and Partide-Wa\'e Prttpalafion

Before looking closer at the group behavior, consider the phase velociry in these [WO cases. By detinilion. phase velocity is just the standard fonnula for the velocity of a plane wave.

I'

pha\C'

W = A( = . k

(6-24)

Thus.

EM waves: I'phase

Mauer waves: \I

ph!N:�

= W

k

w k

= C

(6-25)

lik

= -= -

(6-26)

2m

Whereas electromagnetic plane waves. unsurprisingly, all move through vacuum at the same speed

c. matter plane waves of di fferent wave number

move at different speeds. Here is where the distinction between pbase and group velocities arises. Our main object of study is a wave group Whose In the case of malter wave numbers cover a range of values centered on

ko-

waves, the phase velocity corresponding to this central wave number would

Vo

= po/2m = vol2, which doesn't equal the velocity of the par­ ticle. However, the phase velocity is not the important one. (This is quite a be flko/2m

relief, for as we soon see. it can exceed

cO The wave function (6-19) for the

two-wave group bears this out. Its complex exponential moves at the central, or average, phase velocity

wr/ko•

but (hal part of the wave function disap. pears in the probability density, which is what really corresponds to the par· ticle's mol ion and moves at a different velocity. Does this also hold in a more general group? As written, wave group (6-2 J ) is general, but almost always we are inter­ ested in a wave function of a particular shape, and one of the most conunonly considered shapes is a Gaussian wave packel-a single bump. At

t=

0, we

desire our wave function to be of the form

(6-27)

Note that this is a right-moving plane wave multiplied by a Gaussian bump that falls off away from the origin. Because of the Gaussian factor, it is nOI infinitely broad (� *-

)

XI ,

�o its momentum is not perfectly well defined.

However, the oscillatory complex exponential gives it an approximate wave number of and thus momentum of Mo. In fact, its real part, involving cos

ko kox. would resemble the top/starting wave

in Figure 6.20. The way we

ensure that integral (6-2 1 ) agrees with (6-27) at 1 of

= 0 is by the proper choice

A(k). Appendix F, applying aspects of Fourier analysis covered in Section

4.7. discusses what choice is needed, then tackle� the integral. Here we simply study the result. For the wave function given at

1 =

0 by (6-27),

equation t6-2 1 ) gives the wave function at arbitrary lime r. and ils IU\lbabH­ it)' deosity IS

1"'(\, 1)1' where �' -

(b-2Rl

dw(k) \ dk '0

--

and

Given the many factors in (6-28). it is worthwhile noting tiNt that it is cor­ rect at t = O. It reduces to

which is indeed the complex. square of (6-27). So what doe!". it do gresses? Taking it one piece at a time, consider a ca�e where D

==

as ,

pro­

O.

(6-30) This is just a pulse "sliding" along the x-axis at speed s. and given the defmi­

tion of s in (6-29). we see that this is the same result as for OUf earlier two­ wave group.

Note that nonzero D wouldn't change this conclusion. As we soon find out, D governs not the speed of the probability density, but how much it spread!;

in time.

Let us calculate the group velocity for the familiar cases. From equations

(6-22) and (6-23),

EM waves: v group

Matter waves:

\

� dw(k) � !!.. � dk dk Ck k C

� dw(k) � !!..hk' group dk dk 2m

v

\

(6-32)

o

flko

'0

� -

The group velocity of an electromagnetic pulse in vacuum is

III

c.

(6-33)

because all of

its constituent waves share that phase velocity-they must move as one. Equa­ tion (6-26) told us that a matter wave's constituent plane waves move at differ­ ent speeds, but group velocity (6-33) is just what we expect it to be: Maim = Po/Ill = vO' the velocity of the particle. No matter what may be the

Z1. CIwptar 6

Unbound Scala: Steps, 1\mncling. and PmtJde-Wave PropagatIon

Figure 6,zZ Dispenion relations for

EM and matter waves. 'The phase

velocity is the slope of a line from the origin. while the group velocilY is the slope of the: tangenl line. EM wav6

w=d

Mauc:rwavb

w \'pb&M

-

wilt.

/ /�

w �

"iI?l2m

'J ,

•

phase velocity of constituent waves. the region of high probability moves • the speed we expect of the associated particle. How do OUf results relate to Figure 6.20? As noted in connection Wirb equation (6-27), the wave function starts out looking just like the lOp plOi in the figure. As time progressc!\. its little crests do indeed move at a phase veloc. ity different from the speed of the envelope. (The actuaJ 'I'(x. f). which is a bit messier than I"'<x. t)F. is given in Appendix F.) But these features disappear in the probability density (6·28). leaving just the envelope. a smooth pulse mov. ing at the group/panicle velocity. Figure 6.22 illustrates the relationships between the phase and group velocities for our familiar cases of EM waves in vacuum and free-particle mat. ter waves. The two plots are disperSion relations (6-22) and (6-23). The dashed oval on the matter-wave plot represents the range of wave numbers and frequencies in a group--a continuum, but still a restricted range. The slope of the line from the origin to the central wave number is the phase velocity of the plane wave whose wave number is ko. Lines to other points in the range, of course. have different slopes, or different phase velocities, but none of these is the velocity of the probability density-the group/particle velocity. As We have shown, this velocity is the slope of the curve at ko. We found that the phase velocity of the central wave was vr/2 and the group velocity was vo' and the diagram clearly shows that the group velocity exceeds the phase velocity. The electromagnetic plot is comparatively easy to interpret. No specific range is indicated. but it is clear that the phase velocities are all the same and must equal the group velocity. The interested reader is encouraged to take a look at Computational Exercise 59. Using a simple sum of real cosine functions in place of equation (6-2 1 ), it obtains plots. suitable for animation. that resemble those in Figure 6.20 and that clarify phase and group velocities and their dependence on the dispersion relation.

An Electromagnetic Pulse, and a Surprise Let us consider the behavior of an electromagnetic wave pulse traveling through a medium where plane waves of different frequencies move at differ­ ent speeds. EXAMPLE 6.3 For waves in the region of the electromagnetic spectrum used for GPS (globaJ posi­ tioning system) Signals. the interaction between the electromagnetic fields and Earth's iono!>phere leads to a refractive index that varies according to

n(w) where

w

=

JI

-

b

w'

is angular frequency and b is a constant. (a) Find the dispersion relation.

I.

(b) For a pulse of central frequency wo' detennine the phase and group velocities. (The

reader may quail al seeing a refractive index less than for it implies a phase veloc· ity greater than c. We confront the seeming violation of special relativity afterward.)

SOl\,;,', )N

(8) By definition. the refractive inde� of 8 materiai is the ratio of the speed of liaht In vacuum to the sp«d of 8 pure electromagnetic plane wave in the malmal­ that is. to the pha� velocity.

e

=

n � --

\'phase But the speed of a plane wave is also

w k Solving for

�

wlk, so that e

VI - b/w'

w. we obtain the dispersion relation. w(k)

�

Vb

+

(ke)'

(b) Using (6-3 1),

We now reexpress this in tenns of the given

=

wOo Using (6-34). we obtain ko

= � Vw'0 - b e

Thus,

- b/w"ij Finally, evaluating the phase velocity also at the central frequency, we have e \lgroup

= cYI - b/wij

The example seems to make a very unpalatable claim: The pulse's group velocity is okay. but its phase velocity is greater than

c.

We don't have to look

far for this "problem" to recur. When mass/internal energy is taken into account. the same holds true for matter waves, even in vacuum (see Exercise 44).

Is special relativity violated? 1t is true that any individual plane wave may

travel faster than c. But a pure plane wave-of infinite extent in space and infi­ nite duration in time-cannot transmit

infonnation.

It doesn't vary in any

significant way, in a way capable of conveying information from one place to another. On the contrary, to transmit any intelligence, the wave must be modu­ lated in some way, perhaps varying amplitude or frequency or simply turning it on and off. When modulated, it is no longer a single plane wave, but becomes a

220

CIu!pter 6

Unbound SlalCs: S�PS. Thnneling. and Panide-Wa\'t Proj)a!!lIlion

combination of plane waves-a wave group. The information travels at the group velocity. which is less than c. Under somewhat exceptional conditions which have understandably garnered considerable attention (see Progress

Applications). even group velocity may exceed

and

c. but this requires further

rethinking of what it means to transmit information. and in no case has a vio­ lation of special relativity been suggested.

Dispersion Not only can differences in the phase velocities cause a wave group to move al a speed quile different from its constituents. but if can also lead to the phenomenon of dispersion. the spreading of a wave pulse. Dispersion arises whenever the dis­

D in equation (6-29}--the second

persion relation is nonlinear; that is. when

derivative of w with respect to k-is nonzero. Probability density (6-30) assumes

D = D, and it describes a Gaussian pulse simply sliding along the x-axis, unde_ fonned. at the group velocity. However, when D is nonzero, we have to go back

to probability density (6-28). In the denominator of the exponentiaJ's argument D causes the moving Gaussian to become broader, ultimately a constant of I . In

the factor multiplying the exponential. it causes the probability density as a whole to decrease. so the pulse flattens out. Thus, the probability of finding the particle

spreads over an ever larger region. while the probability per unit length climin­ D is caJled the dispersion coefficient.

ishes. Governing this behavior,

Dispersion would occur for the GPS pulse in Example 6.3, because the (6-34) for the medium is nonlinear; for the same reason, it

dispersion relation

occurs for a matter wave

even in

I'GCUWlI. As shown in Exercise 48, probabil_

ily densily (6-28) becomes

1 'I'(x, /)1-,

=

VI

C'

+

" '/'/4m'.'

Note thar because the factor

1I.212/4m2£,4,

portional to

e-4•

[ -(X

exp

2.' ( I +

-

Sf)'

n'/'/4m'.4)

]

(6-35)

which causes the spreading, is pro.

the narrower the pulse's initial width, the more rapidly it

spreads in time. Exercise 49 investigates the phenomenon. It is worth reiterating that our wave group is a solution of the free-particle Schrodinger equation, just as is a single plane wave, but unlike the plane wave, it is a good description of a well-localized particle. Figure

6.23

shows

the time evolution of the probability density of a moving free particle. Disper­ sion invariably leads to increasing uncertainty in a particle's position.

Figure 6.23 Dispersion causes a matter

wave to spread.

1'1'1'

..

o

'I

.... ... AfPII .... ..,

P ROGR ESS A N D A P P L ICATIONS Resonant Tunneling Diode Photon Detector

,\pplil:ati(ln� of tunneling in modem ek�·tronics are muillplying rapidly 1ll1waday\, One holding great promi�c. tlc\eloped by ;'I:ienti�t!> at To�hiha Re�an:h Europe anti ('ambrid!!e llnivcr<.lty. e'lploll_\ tunno:!ing characleri,til:;' along with tho;.e 01' the ubiquitous quantum dot (1 C. Blale,[ey. et al.. Phy,ir.:al Review Leuer.-. Fehruary 18. 2005). A!> ...hown in Figure 6,14. the device is a diode with 111"0 barrier<. that electron, must tunnel through. �cparaled by a das;.ically allowed region. Perhaps not �urpmingly. the beha\ior exhibil� features 01' both the E <. Uo lmd 1:; > Un barriers. Tunneling through the barriers <;.IIOII'S rt'iOnllIlCC' that correl>pond 10 dIsCrete energy levels in the intervening well. (Exerches 39, 40. and 56 investigate rt!sonant tunneling.) A, we know. even very small changes cun alter tunnel ing probabilities-and thu� currcnh-drastically, and in thi, device, a single photon cau,es the change. It create, a hole (a state mis,ing an electron) that becomc� trapped and alters the tunneli ng rate via the electrostatic influence of its positive charge. Device� able to detect �inglc photons would be very useful in application!> ranging rrom quantum communication to medical imaging. and thi� one appears to recognize single pholOn� with considerably greater reliability than other

method�,

Figure 6.24 In a quantum dot re�onant tunneling diode, electron!> tunnel from the emitter through twO barriers. A photon promOies an electron and the hole left behind. trapped in a quantum dot, alters the tunneling probability.

� � ) �

�==�

Q""","m doc,

�

mer; Double ba

'" ",m" E"" --= ,--States where free electrons nlove

"

, -

Emitter

� Collector

Filled �Ia!e) where hole� can be made

h"

STM March.. On The STM continues 10 make news. In ill splendid developmenl of the slI.ndard technique. �se.n:hers III the Colo...do Scbool of Mines ha\'e added II new dimension. In lhe standard STM ima�e. electrons \If the highesl enelJies in I conductin, sample dominate the lIicene. often obscurina the bebavior of li1wer·energy ones. To filter among them. the new technique!' uses a tip made of It specially cleaved semi�'onducting �·rystal. Whereas a metal lip cannot "look" for specific energies. the semiconduclor tip. with its band� of allowed energies (Section 10.5). can sift tbe tunneling t'urn:nt according to electron energy by varying the lip-to-sample bias voltage. Figure 6.25 shows on the left a view llf a silit'110 surface produced by standard tel·hni({ues. with 1 2 atoms prominent within the diamond. Six olher atom!>, differing in both location and energy duc 10 Ihe different nature of Iheir bonding. are e,�cntii\lly in\'i�ible. When the new technique is applied to IllOk for the\c energie!>. a\ shown on the right. it is the other atoms that di\appear, (P. Sutter, et aI., Physical Re\iew Letter\, Apri l 25. 2003.) It i� not only the eyesight 01" the STM that continues 10 improve. We often wish to gra!>p and manipulate microscopic Objects. and the atomic-scale dexlerity of the STM i� increa!>ingly an�wcring the call. Researchers at the University of Berlin ha\e employed an STM to carry out a chcmical reaction with single molecules. As shown in Figure 6.16. an STM is used a!> a source of electrons (a) to loosen an iodine from iodobenzene. leaving a phenyl (b). The iodine i\ coaxed to a convenient. out-of­ the-way location (c) by van der Waals interactions and chemical force\ between it and the tip. By the same

Figure

6.25 An STM with X-ray vision.

,..... 6.26 Sin,.moIccuk SUIJUY via STM

very Impor1anl 10 the pnx:essing and !>Iorage of quUItUIb information. At the olher e�lreme. \e\-eraJ experiments In Ihe pa.\1 dec;Jde or liO ha\'e demomtrated a group velOcity for hght greater than (. The efleet (l(curs in medIa at frequency ranges chanu:teri/ed by anomalous dispenlon ' in which the refral;ti\o'e mdex. contrary to ilS usual bt'ha\-'lOr. del,;realiC' With frequency. Whenever somcthmg-sul;h a� pha�e velocity-appears to mo\-'c at wperluminal \pc:edli, the first queMIon is always whether it can be u�cd to transmit information In aI/ l'aSe\. the answer hih been no, Some feature of a pulse may \eern to travel through a region faJ;ter than light could through a vacuum; il may even seem 10 emerge before the apparently corresponding feature in the inci. dent pul�e cnlen. BUI it� information i", encoded in corn. plex ways. and even the "front edge" of an incident pul�e

force!.. [he phenyl is slid O\i�r Cd) 10 anolher phenyl. Ihen. with e:U'ilalion pro\o'ided by another ..hot of lunneling elel.'lrons Ce), the IWO ar� "wcJtled" into a biphenyl mole­ cule

10. (Saw-Wai Hla. el .11

Seplembt'r 25. 2000. )

M.ny Spe.ds of light

.•

Phy�il;al Re\'iew Leners.

carrie\ information about the whole, so Ihe arrival of any feature " not really Ihe question. When anomalous dis­ per"'lon prevails, II i� Io:ommon to speak of a "signal veloc, ity" dislincl from the group velocity_ and never bas this been found to c,'(l;eed c

The group \'e!ociry of Iigh! has

bet:0lTl( qUite a hOI I{lPU: in m:enl yem. a\ new experimental tcchniques are probing all e\lreme\. In the lale

1 m... light'�

group velocity wa., ,Io\\-ed to les� than I mI� by �ending il

Figu,. 6.27 A /ighl pulse "stopped" and restored.

IOto 8 ruther exolic medIUm. a Bose-Einstein conden\atc

(or BEC. � ChOJp!:cr 9 Pro�rts� and Applicatiom), in which lhe rrfnKIJ\'t' inde'( inl;re.I-\e$ abruptly. Other methods of slowing Iighl pul'o('s 10 petle,trian speeds an:: also being !otudied, Ughl of sUl;h slow group velocily may lead 10 new /ighl swil,he, orother opt�It!(:tronic devices. 1I may also prove applicable 10 nonlinear optic.s. in which a medium doesn'l respond linearly 10 lhe electromagnetic field. This effeel l" exploited. among other lhmgs. to double laser frequencies. Presently, nonlinear optics requires high-power

(. )

mm

'-:::----'

lasel'i. but slow light may provide a low-power alternative. The abilily to slow lighl ha.. recenlly led to a further Iwis!. In \eH!ral experiments carried out at Harvard, MIT,

(.) '----=--'

and the Air For..:e Research Laboratory. using both gaseous and solid media. the group velocilY of a light pulse has been s/o","ed ,really as il inferacts wilh the media, at which POlO! a separate la�er �ignal beginli a lransfonnation of its information from light to atomic lipin �Iates. a\ depicted 10 Figure 6.27. and ending in a final "image" frozen in the "pin slale.\. known a� a polarilon. A later laser signal retrieves the light pulse on demand wilh little di\lortion. Thi� ability [0 stop a lighl pulse may prove "

"

/

U"f f

« ) '----'

10 '--_----'

Chapter 6 Sum mary Just a.'. it has an effect on II massive particle. any change in potential energy ha� an effect on ,II maner w�ve �nction. A ..mall particle subject to II force Will often manIfest Its wave nature. behaving II.. II classical panicle would not. It may be reflected by a force that would not cla!>sically reflect it, even by a force in the particle's direction of motion. Conversely, its wave function can pass mto II region despite a force thai classically <;.hould cause il ia rebound. and a panicle may thus escape where hmically it ..hould be confined, a process known as tunneling. The probabilities of a particte being reflected or transmitted are governed by the solution "'(:c. t) of the SchrCldinger equation for the given potential energy and the imposition of the physi. cal requirement of smoothness. To understand the relationship between the motion of a par­ ticle and its matter wave, it is useful to represent the particle as a wave group-a sum of plane waves. The speed of an individual plane wave is known as phase velocity and may exceed the speed of light. However, the probability density moves at a different "peed. the group velocity. which is the same as the speed of the particle. Speed differences among waves in a group also cause the group to spread with timo--the phenomenon of dispersion. i vacuum, and leads to Dispersion occurs in mailer waves, even n a in icle's part position. uncertainty increasing

�

indicates advanced questions

Conceptual Questions 1.

Could the situation depicted in the following diagram represent a particle in a bound state? Explain. U(x)

2. For particles incident from the left on the potential energy

shown below, what incident energies E would imply a possibility of later being found infinitely far to the right? Does your answer depend on whether the particles behave classically or quantum mechanically?

o -�

L

u,

3. Why s i m. "'Pi< of _00 pnctieaIly -.. from Sections 6.1 and 6.2'1 ... The equations ror R and T in the E > Uo bIniIr ..­ essentially the same as ror li,b. pusillJ throuab • tnnI. parenl film. h is possible to rabriCAte • thin film lhII reflects no ligh!. Is il po5Sible 10 rabricalC one thai

lJ'ansmil\ no light? Why or why not? Your rriend has just finished classical physics and taD" wait to know what lies ahead. Keeping �traneoul ideas and postulates 10 1& minimum. explain the process or quantum-mechanical tunnelin&:. 6. How should you an!liwer someone who asks. "In tunnclm, through a simple barrier, which way are particles movm, in the three �gions--befo�, inside. and after me barrierT' 7. Given the same particle energy and barrier heighl and width, which would tunnel more readily: a proton or an electron? Is this consistent with the usual rule of thumb governing whether classical or nonclassical behavior should prevail? 8. A ball i!o thrown straight up at 25 mls. Someone asks, "Ignoring air resistance. what is the probability of the ball tunneling to. height of 1000 m?" Explain why this is not an example of tunneling as discus!liCd in this chapter, even if the ball were replaced with a !limall fun· damental particle. (The fact that the potential energy varies with position is not the whole answer-passing through nonrectangular barriers is still tunnelirl8.) 9. In the wide-barrier transmission probability of equation (6-18). the coefficient multiplying the exponential is often omitted. When is this justified, and why? 10. An electron bound in an atom can be modeled as resid­ ing in a finite well. Despite the walls. when many regu_ larly spaced atoms are relatively close togetber-a.s they are in a solid-all electrons occupy all the atoms. Make a sketch of a plausible multiatom potential energy and electron wave function. 11. The diagram below plots w(k) versus wave numbec for a particular phenomenon. How do lbe phase md group velocities compare. and does the answer depend on the central value of Ie under considentioa? Explain. 5.

Me 0011 ' .,..... -. ...... T

..... ......... .....

U. ,.. ... _ _ ... _ 01 . _ . "" aI _ .. . ....... CI)WIal "'• •f ..,.,....... _ ... _ Mlcd'..- ... .... ..loaIy� (A"t: . ....... _ _ .... ...... ..,. heIp ) 11o< .... '7# 1 , .., . ...... CWW• • �, but d tl I '1..... a., • c:wto. decoM......,. bowII . . ..... _ (_ a..- IO� _ """ .. .. "_ ",_lllt .... .. ..... 0UI .... 1hc 1«l_� 01. _ _ 10 t It _I) prupottioftaI lO tho etlectiw .... of 1M .�, A.fJW lbat 1ft this cry�l. dill etfet..'1i¥e .... .. dw ..... rOl' __I vaJun of .t. but CUl M It diffcnal for IOIIW \'Ill.... .... in one rq:ton In •

(b. Vaify by ��II cakvJaioa ... Ibe nDo of ......... proOobild) _.y ., .... _ ...... btbf)' drmIf) .... I

... .,. • e I&. A beam of pu1M:1n 01 cKt¥>' E and ID _",''' ..... of t. " d . .. : ribcd by lbo _ ru. , : _

Ie.

The- amplitude of the �iI'e (n:I.ued 10 !.he DUmber ID('nkn� rcr wut dl�tance) I arbur.lI'1l) cho!.cft ... " I la) [)demunc cumpletel) the renected &ad ',",n...mined \U\C-s hy enfl1rciIlg Ihc fC\Iuirt\J conllnuil) ('omJilion� hI ��1a.1O thClr 1P"s�ibl) com­ ple:t) ampliludes, ltl! \'�'flr) thal lhe ruliu of rt'f1«led pn�t)thl)' oen\ll) 10 the ml.:idcllt pC\It\.lhllir:; dcnllt) u�rcc:s \l.llh ClJU:llIllP' tfl-7). h l) thl�lI ln Sa'ilon fl l lh"t fllr Ihl' E « Cn PQlCm..J /In ... �.t)l(n .A:)�-\. IT-.e this 10 cak.--uWc ,tcp, H the pnl�"lllt) d�'n'll) Ilnt'luJlng bulh In\.'IJenl and re!1c\"(t"d \1.11\\'31) III Iht Idl M th� >;ICP

wet)' .enap w.)',

•'.

,

� :

= ta) Shu\Io th"l thc rc,ult i... -l :\ 'm:!(b 81, ....herr 1 tI liln (.(:IIr ), Be\.'au,e the reflel.'led \1.01\( I 01

I

Ihl" ,a/11e amplitude :1\ the:' 1II\.'.denl. thl' I' ' I)�

Exercises

5_n 6.1 IJ.

!olandlng "ave palh!'m. \at)mg pcriOdlClU),

ix-I"cl'n 0 and -l,\.,.t.

(b) l'>etcnmnl' II and [J (the amplitude of the

l . I' eljui\'.tlcnl [{) 'ft' , l ,han(ol nf;cl,I.b)' PUl\'IIJCJ that IA ) ,."" 8" • + d)

Show thul lP'( t I

IjIt_> A' ·

!til

+

A'rA'

t

J(II

14. Verify Ih...t Ihe nan, 11\t.'I\l!! and rell�,�tlon pmoabdJllc, aiwn in cqualltlll II'!,7) udd hI I

IS. C"lt;ulale Ihe rt"IlCClulII pnl",,""JI) hll a � (,\' electnm encountering II ''It'P In whl�'h tht' p"lenI131

lIrvps by 2 c \' 16. A p4J1i�,:le mtWUli In a rt·8 1\lu {If lero fon:e t"IKounle,.... a precipu:e---a sudden drop 10 the jXllenllJI cncrg)

10 an

arbltranly large neSJ1I\1," \;lIUt', What is the pn."abiIHY

dUll i t will "�{l {l\c:'r tht' CtI�C"'1

; f; I' dt" �'fl� h) the waw

17. A beam of patlld� of C'ncrg)' E and ,"{" den! up,m .1 pc.lll"flli�1 Mt'P llf U(I fUnt'li on

(/11M: ( .-)

wa\c 1O'".1e the ,Icpl III the hmll� A

I ,.al'

(.) Detemlinc l'{'ll1pklely Iht' ret1e�'fcd wave and the wa\.'e ,",,,.Ie Iht' ,tep hy enf'()rl'ing Ihe required

if

...

O. and mterpret your re,ult,

...

\

)0 0

0 ;lnd

20. P.Lnldc' �ll energ) 1- an: incidenT from the Icft. ",t\t.·rc

Up)

t:.

O. and ill the orig i n encounter an abrupi dr'lp In 3£. (a) CI"'�lcally,

jltllentlill energy. \\how <.Jeplh llo \10

hill woulJ Ih,,' panicle, uo, and

\10

hat would h.lppc,'n to

Ihe'lr I..lnelil: energy" (b) Apply quantum mechanll.:,

A" umin" n_ Ct) • l('b C' .tn IOt,.denl wiI,e of the form '"" " .

\10

here the normah/'ltjon con'tant has been given the-

'!nlple value of I , detemltne completely the wave furl{>

tion eveJ}\\hert:. ,"el uding numenc \alue::. for the mul.

liphl:athe con,IJnh_ (c) What I' the probability that mcident panicle, will be reflected,)

Section 6.2 2 1 . Whal fraction of a beam of SO eV electrons would gel through a 200 V. I nm wide electrostatic barrier?

22. In the

E "> Uo polenlial barrier. there <;hould be no

1..'OnllnUll) nmdillon , 11.1 oht31n their (po�\ibly

reflected wave when Ihe incident wave is al one of the

t'llmpll'\) amplitudc�

tmmmission resonances, Prove Ihl� by assuming that a

"

I!dI'I � p.tio;k\ . _.... .. w t\nI "Pf. . . ..

. ,. ... .. .· h \rA� �.). .. 1M (GamwII) (U"dn� kl d'IoN; "- • 0. �..... 1 .. panllto:u1Mty t.imf"r 10 "'" tpC'Cw ,.... _Nd\ ....w tUftOlltne y\'Ut ""'"" \ � .. � "*" ... ph ucaI 0JIb0. - t\brI _ ft . ... � 1; .� ICIII1e -a'ftkttc- oIbchl. II) toe lI1I\lnI1) td\raN ....tuk � an: POI R'tD."tctJ .. all Sep:w.. ... ....... an ",hi. boD k) 10 1lNl ....,. \'W thr: � . "" � til" rcrk\1IO.l ..-r """'I')' � (.) Ftlf a lUm o(

1.\.

..... ,1$17" ... . .. ... ,' ... ... ..... so . ... . ....... .. . ... .. .. ..... ......., ..' .1

t, ·

�ot. l

\N>: \nr:M ! aum-..ndcl.l t>�

au.

:no

....hId: ".,� )" ("'ht.�

Ibr1 aft' wtthin the' mml '*111 � ,,",,,I)' ,, tnruGniltN' i�) "bat ....;I' �.... l....trl\e the') .... .'\l'tIfII" lhr NmCf\ llf m:lttt1 � Uot",} ('llnd.m,on \t-- ,,,)1 \'"' COO\Nft\C un Ihe relt\IUN\>ot"" tct\\t'Cn the 1"'Yftn 10 \&\ u..t {hI ..., . ,...'tc:n1tll\ �.UH"'I \It ho.'t�hl �O C'\' ,.\ hnd • I '''\lUI. , :H. ( ....IJIIl an'tIl"l \ HOU mu ltlf ",I",.:h Ihtre will NI t\I,' f('n,",ll\.n ,.1 " C" C'k\II"!l� IlldOCII\ ul"" 1I thc NonlC'1 \�l \\ hilI .... 'IUIt.l t'-..: Ih(' ,clkdl\'11 rh\�."lhl)' h.r .\('1 ('\' ...'�lf\'lh \1,"'10 Ihe woc' �rTi...f' INo,l"· n", \l.nC' ,...",.h hI " 011l(1(,\\';C' ,n \t.p«d III \(b lhan ) l�, \ ,.( \1""' ,,, ,1110 ('Iwr¥-) f. nlll\ 'n; in a relilll\ .... Mrt' Ih.:n: 1\ Inlt aU, I'"� PO'IC'I1I1,,1 C'tIC'l):) C'n�,\'lIlIlen It f":otcnuiI\ .hl' " I .... IJtn I .md \"'rlh f (."

11I(l
pal1l.;k

,�..

l

('(l)

\"

\

"0

(I < \ < I

,

lL

I*'tiliRl.'d "'..." 1f\I\I&\t\",.. lb- \ 2)

Ullt the Iltlln""1l\1I ttl'. and Q tollbuln� (6-\6).

19. Ob"llI the l"" w.."hnc..ll co",b1\ol\& at the �

�I"'�n ((!Ii,w" tur 1M I:

Uo burlOf (t.e • \\IftftOlina)

,., . •

• .lO. Here wt take . dim:I IlfIp'UItCh to calculaUDJ ref\ectkJft prohI.blhl) rOf lunMUnl. mcuwhUCI obI.Uftlnll ..11liuR"hir applied In I.lIhct IJ.en:I....

tal Write out ,he I�l\Cn cund.ihOR' II the bouftd. burlcr. Prom IlIit� betW('CR re.i�'n. fo, lhe E; � lhem, ..how In.! lhe coeft\t;\enl 8 uf the rcnact04

Uo

- ,

w.vt \l li�n by

H - A

All inilt ''0 needed IS an appropriate sub,tituti(ln

In a known pwbability.)

(a) a.s 26. Given the situallon of E:\ercl!oe 25. Uo -to oc. lhe renel:tion probability approaches and

show that

(b) ib

". "nl Iitep b. 1IlbdiNl

lCl ft... l· l.I Why'r lb)M_lho� lMn

tnl

[ V2111(F. + U\JI,/h\ 'in�V2"'lF +Voli/hl + 4(f;/Uo>0£/uo) + 1 \

(Nlllr:

A'A

und thull i.krive �u"tI"n' \b - 1 2 1. keno..tkll\ II.nd ttrmllm.I""',,, prohlb\\\Uea {t.. 16) 0Ift be

I\nh {oL)

V .'nh'(oL) + 4a't

{'"�k

'in!

R -

f"

"'hcore A. II. anrS I .n: the """'",,ktodw � or ,tw "\l:i'-'o"I, ",00.:,", and ltar&M\\\Ued w....... """O IhC'\)'. h.'ln\ t� h"" .mtw.1lhnou CUftCMUDM.lOIw fur . ..nd ,. In tc-m\t ut A. \n� them In the " UKl T NiOl,

,.J r ,

T

A'A

-

, - \I

V. ..IP,,"' ..... .. ...... .. � ...... .. "" ....

"- .. ,

,'.

.,. tI

, .:: I

()

... 1 ' 1I,.1.)'j

'""

\, L -to O. the rener.:tion probabihty approocnes O.

Ie) Con!oider the limIt in which the ....ell becomel> infmitely deep and infinite�Imally narrow-that is. Uo _ x and t -11 �bul the product Uri- is eonstanl. (fhi.. "delta well" model appnn,imate'o lhe effect of a narrow but strong

/('"

+ .. )

� "�

)

'oth (aL) a.

thl Verify Ihlll the retloc\iol\ probabkUty R a1veo In equation (601 b) follow. from \hi. ruuil.

�uo (a) About how much "'''

31. Suppo� the tunneling probabUi" hamer when

F. ill

would it be if E wen: inalCad

\. \0 12 for . wide

�Uo7

(b) U til cut doc, nul ,"upport the aeneral N\e \hIa trIIIIIftbIlcIG probabilily i. a lI�ry .emitive functioo of E, what ma­ il ClIceptiuna.1'1

The pavitatioMl potemill ....,. aI. I q object i& plotted '1Cf'Sua position from BIdb',

32. Jump If) Jupjl�r

IUtfate &0 the surface of JUPlift. MottJy il is dale to the Sua. bul &here � downturns at each end. dbe 10 the aunctions 10 tile 1"'0 p1uJc:ts.

�--�l

Sun'.

....

'!!

pWJ

x

-

lo�t'r\\�Irt"led by ..uong ama.:tmn)

o\�·lual Ujrl

MOOtI U{r)

,...•. •

. -_ . . . .. . - . . . - _ ... - 6 1C J(}l I m

Make the crude Hppmximalion lhat this is a rectangular

101 1 m and appro"imulc height 4 x I()II Jlkg. Your muss is M 19, and you launch your­ self (mm Earth at an impre.�"'\ie", mls. Whal i.s the probability thai )'ou (.'an Jump to Jupiter?

JJ. SUPP()� a barner qualilit'\as wide. and its heighl and width are sUl:h that :U V2mUrl" • 5. (a) Calculate the Iransml.I;slOn probabililJe� when ElUo is 0.4

and when il l" 0.6. (h) Re�:H part (a), but for Ihe CIl\e

\hmllri" I S 50 ifl'ilead of 5. (c) Repeal part

(a), but for U

,-

ClluJumb po(("nLiaJ

....

barrier of width 6 )(

where 2L

•

V2mUol" - 500. (d) How do your

result� \Upport the claim thai the lunnehng probability is a far mort scnsilive tunction of E when the tunneling probabililY j� �milll"?

34. Oblain equillion (6-1 M) from (6·16) and (6-17).

Section 6,3

1 (bl The Sun's core i.. ani)· about 10 K. Ifnud�l l.:an't make il "over the lOp," they must Wnnel. Con\ldcr the followlOg model. illuslr.lled in the figure: Ont

nuc/eu.. i.. li:I;ed 01 the origin, while the other approuchc.. from far away with energy E A, r decreu..e�, the Coulomb potential energy innea'Cl , until the ..eparation r is roughly the nuclear nidllH

(he potentjal energy is U and 'nu,.' whereupon �l then quickly drop� down 1010 a very deep "hOle"' 11!) the strong-force attraction takes over. Gi\en tb.tt

E <<.. Urrw, ' the point b, where tunneling mUM begin. will be very large compared with rn�' 1;0 "'co

approximate the barrier'\ width L as simply b. Ill;

Uo, we approximate by the Coulomb poten. bl2. Finally, for the energy E, which fixe, b, let us use 4 x � kB T, which *' a rea. height,

rial evaluated al

�onable limit, given the natural range ofspeed� in ll Ihermodynamic system, Combining these appro"i. mations, �how that the exponential factor in the wide-barrier lunneling probability is

35, Fusion In Ihe Sun: Withoul funneling, our Sun would fail us. The source of its energy is nuclear fusion, and a crucial step is the fu..lon of £I lighr-hydrogen nucleus,

which isjusl {I proton, and a heavy-hydrogen nucleus,

which is of the same charge but twice the mass. When these nuclei get close enough. their shon-range artracrion via the strong for(.;e overcomes their Coulomb repulsion. This aJlows them til slick rogether, rCoSulling in a reduced tolal mas�inlem81 energy and a consequent release of kinetic energy However, the Sun'.. temperature is sim­ pJy too low (0 emute that nuclei move fast enough to overcome their repulsion.

(c) Using the profOn mass for Ill, evaluate this factor for 7 u temperature of 1 0 K. Then evaluate it al 3000 K. about (hat of an incandescent filament or hot flame. and rather high by Earth standards. Discuss the consequences. 36. To obtain a rough estimate of rhe mean time required for uranlUm-238 to alpha-decay, let us approximate the combined electrostatic and strong nuclear potential

energies by a rectangular potential barrier half as

(a) By equating the average [hennal kinetic energy that the nudei would have when distant,

�kBT, and the

high as the actual 35 MeV maximum potential energy. Alpha particles (mass ..j. u) of 4.3 MeV kinetic energy

Coulomb potential energy Ihey would have when 2

are incident. Let us also assume rhat the barrier extenrb,

fm apart, roughly the separation al which they stick. show that a temperature of about 109 K

from the radius of the nucleus, 7.4 fm, 10 the point where the electrostatic potential drops to 4.3 MeV (i.e..

would be needed.

the classically forbidden region). Because

U C( lIr, this

-

.

I"-.ot lli. �"14 l blnl:'!. tht nkltU\ ,l( \M DU\.:\tu the pomI I at ",hi\:b t,,) s

" Me\

\

� �!> ......\ ('Q. �

,

1 :1) l',.....

tamp.. ,.) st.w. IhIl ltlie ftlklo Tiff, . . )9•

<

" .!del l \.,

,

,

, .'

Ih.....,..... etUlk ilp\,[,I"m;t\I\'n,. th..... nll."ttw�.1 ,ug·

8-e,II."\1 in Se\.·II,'1\ to..\, ,\lid the \\ lUI.'·NllTier appII'\' IIn,IIIml In \lhl,lIn 1\ "nlll\" h,t the 111111.' it la\...1."'

.... ....

,..... .. lhe ..... .. � ... _

,... \t�\

,."

�b) Oo lht """"" ddIer "' __ T ... ..... tIlht)' � �"" \'O'Y '"'" Of ...." .. .., ... .. ..... ... ,at T'm ..... Hen ... ....r 1"1 t...... ,hal. ....h'''' a ,,,,,, � \wrier ......,. "- . R't\l!\:kd .....\11. '�ma • .....-oH buntr CIIl pw It �t\e\.'\1M m\b&N\hy ,If l.\'ftI. knawn as � .... Ina and Imp..'IfUJ\\ '0 RW\y mo6em e\cctroNc ..,.. t'.qU-l&tlon (�\�) It'"'- tho w..... f\a.'tklra � . ,,\o,lc hIunot ,We! ."ml.ny waft', MOd \be tohadoaa ,""Woe ,be NrtlC'r 1 1\1 mw f\'Oftl (or .. teeaad. 0110., we hill thl "n"hl han\C'I" • d'.'�""e J \0 ,he n&hl. to .. .f " L. whl\:h roquUet m.t � lu .\ It C" l\'n.h Iwm , \\\. ,,1\\1 ..1\111 1M \o\1l\'C' !Unch\",ll " d's.'allCf: .I.

\\1

UI)

Rll!.htb,uI ,\·r

J.l·e!l),.

(hl "1.1 g.,lin "'Illl.' IIPl'n:ciutwn lit the utl!iellltlc, in a Ihl."llrl.'tKal rreUH,;\Il11l, wO\\"' lhe ewrfN." ··\"I'le\... \\'Mu" Rather than II"Utnlllg a \!llm:- for UII, 'N' Ihe I-no\\ n ""ltII.' !If Ihl.' tnl.',m Ill1\e III ueell) II.." ura· "alu\.' "I tin' mum- :!.\S ;\nu mler Ihe eI1ITC'I".)nhng .. Rl.'lllm ;\11 IIIher ,,�,umr\llln' Id ClllUmcnl on Ihc 'cn\.ih\"lt� 01 the J.c..:a) lime til the hcig.ht 01 the potenllal l:larril.'r, 31. A method for finJ.ing tunneling pmhal:lihty lilt a barrier that I' " w ldc" but who,c hei ght varie, \1\ an arbitrary way I' the \ll-called WKB approllimalion.

"'

"

,

�. II \\�

''''\\

_

At,jAl'

_)

+ .

_(I

.\

,./

lC'rlll\'c , b), - ,. we ohtain .u\u\t\)RS ur \he

Ittnc,uhlc\lC'nl.1o:m Schrt\dillycr filuation vahli fOf . "loft

bantcr" c\\C'Il"hnv; ''''in ,

,. \0 1: .

1.

"

�',

'\

� ,.

Ih'\o\<:\e[. 1l\l\C thai the c"lmr1e.. conJu8,llte of the t,me'

indc['Cmlcnt St:hrt'l\l!n\.lcr cqualion i!l. the same equation but \\Ith iJlt In pla..:c ur iJI. �ilt.I\C f.', U(,t), and.U other la..:hU\ 11� n:al In "lthel word.-... \IIt i!o Ill> valid 11 t.alution �I' �I lol". "'r the

IC"t\ bllmer we can instead write

l.elt barrier tit � . Here

. ,.

U(x) is thc height of the arbitrary potential energy

barrier, w hich II particle Ii""t penelrale� at

, 0 ami

tinally ellils at \.' ""' L.. Although nOI entirely rigorom. show lhat thii. cun be obtained by treating the barri�r a" a series of rectangular slice�, each of width (L-o; (though each i� still a "wide" barrier). and by a�!.uming that the probability of tunne!tng through the total is the product t

of the probabilitie� for each ...Iice. 38, The potential energy barrier in field emission Ii. not rec­

tangular, but fCi.emblei. a ramp, as ..hown in Figure 6.16. Here we comparc tunneling probability calculated by the crudest approximation to that calculated by a better one. In method I , calculate Tby treating the barrier a\ an actual ramp in which

U - £ is initially
off with a slope of M. Use the fonnula given in Exer­

We chumc the\c heI,.'IIU!'tC thc f"" -term represenb puti· clc� on Ihl! Iclt Ill" thl: leh bilfrier IU\d movins righl­ im:idcnt I,artide!o, The F·term in the righl·barrier �oluli()n.. repfc""n'" pllrtide!O lrllnsmincd lhrou&h the right barril!r. Tu�elher, they imply perfect trmunias\on.

hlf WI

W"\· We t:l1n IUlti!ofy the SchrMinger equation

cverywhere I'm both barriers if we can make \he solu­

tion, compatihle in the region between them. x "" -f to

.t to' + .\. By equating the two solutions in this re&ion and ui.ing the rclll1ion.-..hip between B and A given in

Ellercii.c 10-a hit me�!oy to derive. but noncthele&l uue

I"m the f." <: Un harrier-show \hat the width 2s of the ..pace between the barricf3 mUII1 obey

cise 31. In method 2, the cruder one, assume a barrier

whose height ellceeds £ by a COllstalll tPJ2 (the same as

the O\'erage excess for the ramp) and whose width i<; the

See Ellercl'>C:"io 40 and 56 for applications of Ihi. result.

.. Eun::i. 39 Jives the condition for resonanl tunneling "'""'Bb IWO burien ..".mod by • •pace ofwufrh 2>.

43. Example 6.3 gives me refra(.'uw: index for high.

frequem:y electromagneti(.' radiation passing throop

� in lenDs of. flCtor P ,i...en in Exercise 30. (a, SuppoIC dw in SO11K' system of units, J. Dnd a 3.R'

Earth's iono)ophere. Thc constant b, related (0 the SO­ called pla.�ma frequem:y, \'anes with aunosphcric c0n­

ditions. but a typical value is 8 X IO'� rad1/s1. Giveo .

boIh 2". find two ...alues of 2f that give �sonanf tun·

nel;n., What are lhcsc: dis� in ICnns of wAvelen,gths

of Ill? Is the tenn rt'.JOfUUll 'unn�/jng appropriate']

(b) Show mal lhe ,ondition has no solution if J

= D. and CApJain why this must be so. tc) If a drus;nil parti­

d� wanlli to �urmounl a barrier without gaining energy. is adding a s�ond barrier Ii good solution?

Section 6.4 41. The maller·wave di)opersion relation given in equation 16-23) is COn'Cct only ut low spce<J and when

GPS pube (If fl"etjuency 1.5 GHz traveling

lhrough

H !un of ion()oph�re. b)' how much. in melers. would the

wave group and a parlit:ular wave crest be ahead of or behtnd (a' the case may be) a pulse of light passing through me 'arne distance of vacuum?

E p k) values Eo and Po' Although it comprise.. many plane

44. For a general wave puhe. neither nor (I.e.. neither are weI! defined. but they have approx'irnatf w nor waves, the general puhe has an o\"eraU phase velocity corrc�ponding to these values.

mas!Jintemal energy is ignored.

Wt)

Eo/Ii poIh

Eo Po

� - � -- � -

ko

(al Usini! the rel:lIi\lj),tically I.:(lrrect relation)ohip among energy, lIlomentum. and ma�s. ..how that (he COffel.:1 di)'pcrsion relation is

If the puJl.e describes a -'large" massive particle. the uncenainties are reasonably small, and the particle may and momentum be 'laid to have energy Using the

Eo

Po'

relativi,lically correct expressions for energy and (bl Show thai in the limIt 01" I(lw 'peed. (small

p and k)

and ignoring l1la.�oJinternal energy, this expression

momenlum. show that the overall phase velocity is

greater than c and given by

agrees with that of equalion (6-:!31.

42. From e4ulliion (6<\]), we I:onduded that the group "'dOl:il) or II mailer wa\le group equals the velocity

of the mas�i\"e p:lrticlc with whil:h it i� associated.

Vo

Howe\er. ooth the di)'persion relation u�ed 10 show that �' �f\>U

�

-

I!!'rjm nnd the formula l'fWlh:k = plm used 10

relal this to the panidc velocity are relalivislicaJly

incorrect. II might be argued thut we proved what we wished 10 prove by making an even number of mistakes.

(II) Using the relativistically correct dispersion relation given in Exercise 4 1 . show Ihat the group velocity of a wave pulse is uClUulJy given by

NOle thai Ihe phase velocity is greatest for particles whose speed I� lea.')!. 45. For wavelengths less than about I cm, the dispersion relation for waves on the surface of water is

w = V('Y/p)I(3, where 'Y and p are the surface tension

and densily of water. Given 'Y

p=

= 0.072 N/m and

l oJ kglm\ calculate the phase and group velOCities

for a wave of 5 mm wavelenglh.

46. For wa\les on the surface of water, the behavior of long wavelengths is dominated by gravitational effects-a

liquid "seeking its own level." Short wavelengths are

dominated by surface tension effects. Taking both into account. the dbpersion relation is

w = Vgk + ('Y/p)k3,

where yis the surface tension. p is the density of waler,

and g is. of course. the gravitational acceleration.

p = I!k is universally I:orrect, so IIko i� indeed the particle momenlum p.

(0) The fundamen"," relatiomhip

lit is not well defined. but this is its approximate, or cenlral, value.) Making thi� substilUlion in the

expre�\ion for VS >U from part (a). then using the R , relationship between relativistically co momentum

p

rric

and panicle velocity v, show that the

group velocity again is equal to the panicle velocity.

(a) Make a qualitative sketch of group velocity versus wave number. How does it behave for very large

k? For very small k?

(b) Find the minimum possible group velocity and the

r = 0.072 = 9.8 m/s2.)

wavelength at which it occurs. (Use

N/m, p

= 1 0� kg/m), and g

47, For wavelengths greater than about 20 em, the disper­ sion relation for waves on the surface of water is

... =

...... ..., .. ...... mwmt., tb) Wb.I. "' .... _ wuId ha.... and yeI: bI bouIrd' (e\ Wbu .. .. ......

\. �l. la) <':llkuh"� the ('h.t� and IrouP veIOl."ties.

1\>1 Il Wa\C \,1' � m "",,,,d('nllih It» \\�III � "".'"e SPft*'

.l� \1

et'ICfI)' tbt PM'k" could .... _,. "" ... hr Ii

tr.l\d.. ' Ju"UI) )\'I\Ir an"""'er

..s. Fn.ln' �U.\ll\ln'1 ,to':l:") and Itl-1Q) \,Nb,in the dil{lershlft

4•.

.:\X'llIncnl \\\r matter ""'ll\n Un , ..:uum). thrn �how , \! ; d : ,,\u) .xn',I) (t>- :"�) h'lIo\\ , from (bo21"', th;ll p.... .... · leJ.n1ed in E'anlrok 4.1, In Il GaUM1an fun.:u\ln A, , .,l lhe 1,lnn I$<{,\) '.'\. �.( ' 1.0:)).

S "

lmath of tllM? (4) la. poaUbhi .. .. panK:w to M'IIC aft enltl"JY .,.,... lhaD. .... lrt. ,.. (e) . and 'tUl be ''bound . (ur IOmO I*iod. ot UQW'I' Ib.plaI.n Coo"'de, • .,.rude of maos. III InMdo the well dIowa. U

1.""OaSidenb\c

S4.

IhI! ..lIlI\Jard denl­

li,'n,or un.:ertainly. in pt.NIIl)n, The prubattilil) den,it) I"r II Gau">lan WI'''! tun.:h1ln "'",'\lld � prop',rti,'nal l.... � \) ..quareu t'-I "; ! .... ) C\lnlpJ.ring "'" 'Ih the IIme­ IkpcnOelll G'1U�..lan pmhabihty 01 �uJ.hon ttl-'�). ""'t \et. Ih,ll Ihe un.:el1ainly in pt" llion of the hnle-(','ohing Gall"i;ln wave functllln of II free panicle i, gi'cn b)'

"'-1W¥l. il" \\"I'Io'ft1 eneraY ltale would.. of couno. be \be p:t\lUnd ,tate, bul ,.,ould It be bound'? AuuI'M 1hM• •,

letit tur 110 Yohtle. It OI,..,.-upws. Ita "round stale. wbleh l& much I'Met Ihlm 110, and lbal the burien q\lelity as. \1;ide, Sh\'IW

thllt 0. t\l\I,h avenae time it 'NOUld renWR "'-,und i, klm:" by T (mW'/lOOMLl)a2... . where ,r

-

1.\'�mllLvfa

L

11ml ,.., 11 ,u ut. .. ill f: and in.:reru;e, wiln lime. Suppose the wave I"un.:lion of nn electron is inilially delennineu lO he (I Gau....ian oj 500 nm uncertainty. How long will il lake for Ihe uncertainty in Ihe electron\ po�;tion to rellch 5 m. the lenglh of a typical automobile? 50. Show that the qUite genernl \\ave group gi"en in equation (6.2 1 ) i:. a �olulion of the free-particle SchrOdinger equation. pro\ided that each plane \\ave's wdoes sati,f)' the matter ....ave uispeJ'!;ion relation g;,'en in (6-23).

Com prehensive Exercises Solving the potential barrier �moothness conditions for relationships among the codTitienls A. B, and F, giving the reneetion and tran�mis5ion probabilities. usually involves ralher mes�y algebra. However, there is a spe­ cial case that can be done fairly easily, though requiring a slight depal1ure from the standard solutions used in the chapler. Suppose the incident particles' energy E is precisely Uo' (a) Write down solulions to the Schro­ dinger equation in the three region\>. Be especially careful in the region 0 < .f < L. It should have IWO arbitrary constants, and it isn't diffLcult-just different. (b) Obtain the smoothness conditions, and from these, find R and T. (c) Do the results make sense in the limit L _ �? 52. Show that if you attempt to detect a particle while tun­ neling, your experiment must render its kinetic energy so uncertain that it might well be "over the top." (Hint: Apply the uncertainty principle, and note thaI the parti­ cle must be localized within the penetration depth.) 53. A particle experiences a potential energy given by U(x) = (.r2 - 3)e-� (in SI units). ea) Make a sketch of Uex), including numerical values at the minima and

51.

F:\cn:i�e 54 �\\('� il n,u�h Iilelime for a trapped panicle t�l e�(a� an cndllsu� by tunneling. {Il) Cunsider .n e\el.;lmn. Gi\'en lhat W � UK) nm, L - 1 Itm, and ", nd «- Uo&.'\liiumption Uo - � cV. fiN \erHy thllo! the: f.. hold�, then ("'aluat!! Ihe hfetimc. lh) Repeat part lo.), but flu a 0.\ p.g panide, \\'ith W '" 1 mm. L =0 I. �, and a harrier heighl lio tnal equal� the energy the panicle would have if ,t� �pc('d were jUlii! I mm per year. 56. Exen:i\oC )\) gi\'h the: ..:undition for resonant tunnehR8 through two bLlll"ief"\. l"oeparuted by 110 I>pace of width 2s, expres,e!.l in term, of 110 fa..:tor 11 given in Eurcise ?>O. Show thaI in the limit in which thebarrier width L _ :xl, thi� .:ondition become\ exactly tnergy·quanlilaliQR con­ dition (5-22) ,'Of the flRile well. ThUlii. resonant tunneling occurs at the quantized energies of the interve"'R8 well.

�5.

Computational Exercises 57. A computer can solve several equatioRs in several unknown� ea..,ily, and here we study a particular E > Uo barrier problem, where all the values it needs are real, Once the tompUler finds the multiplicative CORSlants of all the function� involved, we can verify equations (6- \3) a� well as 'OCe what is happening when the particles are "over the barrier." Sti\l, it helps to simplify things as much as po��ible. With length. time, and nuw. 1ll our &a­ po�al, we can choose OUf units so that the panicle mIlS m and the "a1ue of Ii are both I. and the barrier width L is exactly '71". Suppose that in this system of units, the energy E of the incident panicles is I.. t 25. and the barrier height Uo is I., Furthennore, because only ratios 1ft ever

7 Qua ntu m

Th ree D i mensions a n d t h e Hydrogen Atom

Chapter Outline

7.1 The SchrMinger Equation in Three Dimensions 7.2 The 3D Infinite Well 7.3 Energy Quantization and Spectral Lines in Hydrogen 7.4 The Schrodinger Equation for a Central Force 7.S Angular Behavior in a Central Force 7.6 The Hydrogen Atom 7.7 Radial Probability 7.8 Hydrogenlike Atoms IJ 7.9 A Solution Examined IJ 7.10 Photon Emission: Rules and Rates

P

erhaps the most profound failure of classical physics is its inabil­ ity to explain the simplest possible atom: hydrogen-an electron

orbiting a prolon. According to classical physics. the atom should be unstable. Any lime a charged particle accelerates. it emits electromagnetic radiation. so the continuous centripetal acceleration of the hydrogen atom's orbiting elec­ tron should cause it to lose energy and spiral into the nucleus. What we observe, however, is that atoms are USUally quite stable, emitting no radiation. They can be induced to radiate electromagnetic energy, but they do so only at certain frequencies. And while easily demonstrated with a simple diffraction grating, this too defies classical explanation. One of the first great triumphs of quantum mechanics was its explanation of these observations. But before we can understand the hydrogen atom quantum mechanically, we must extend the SchrOdinger equation to govern matter waves in real, three-dimensional space.

7 . 1 The Schrodinger Equation in Three Dimensions

to one dimension. the SchrOdinger equation is

a U(x)'l'(x, t) = ih-'l'(x, t) at

231

As we know. this equation

IS

ba."iCd on an

ene� accounting of the t general. a

rur.:.:: I!.I! . ��.... Sec�'" GIIIt

U = E. !qJBliaI coordinates.. U(x. _v, .:}. as Iii me wave fuocllon. 'if! t, Y. to I), In Ihrte dim
the kinetic energy term on the left to twe generah/ed. A� noted in leon i!',r,f2m in OPC"rdtOf furm. 'The logical generali/Jtilln i... to

S., Ilia re I P;+P: +p? - that is. add terms with )' am.! ? In pla,,:e of \:, Thus, P � p! "" _ ___

(

, �

2m x �, (I ;;2

+

�

ih' ;;.�

)

+ . ., 'I'(_l. \' . : . t) + d� ;I'

(II ,

.

,

.

�)"'( '. ). �. I)

= i� ,iI "' ( . ., -, . .. •. I) vI

To "'pcclf)' pm,ilion in three dimen'jon.... we are nOI required to Use rect ��. ..." gular Canesian ctxmlinalc..., and they would be particularly aWkw d the "'phcricall) synunetric hydrogen alOin., To ,lVoid biu... the . nger c4uatIOO i... u'iually expre'ised In cooroLlHHe- mdependent rorm u , SinD ". , . . - -e generic symbol r (boldface) to represent po".tlon ar1e�illn (.\, v .;:) . ... 0- 'III: htl'l. cal polar (r. O. tb). and .. on-U( \ . v. �) becomes U(r) and "'(.t \P

s��t

hewme ..

?

.

,

.

"

.

to t)

'I1( r. I). For the kmetlc cnergy term. we adopt the generic sYlTlboI Vl

from vector calculu... Ex.pre ....ed m Carte..lan coord mate...

Ex.pre�....ed I n other coordinate ..y .. tcm ... It may have a quite different a p�iU". ance. but il't mathematical functIOn i'io the same. Altogether. we have

-

,,2

bn

V'�'(r. l)

+

U(r)"'(r. l)

a

=

"'-"'(r. l) at

Now let U'io briefly discus.. other important changes as w e mOve t o

dimensions.

(� l )

three

Probability Density and Normalization In one dimension. the complex square of the wave function is the prohabT I ity per unit length. It is naturally the probability per unit volume in three di sion!>. Both are called probability density.

men_

,

probability density

o.

probability =

volume

=

1'I1 ( r, 1)1 '

We see that the dimensions of the wave function are now (volume)-112

(length)"-312.

Or

7.1 ,.. » ...... -

Ii i wu\r funl.'lhlll dcsc n brs l in,lt putklt. the local probability uf ftnd. _'" it 5lMllC" hi.'rr m thn'("-(.!imco'lillnal 'race must he 1 . In three dimetWoM. Ilu .;,lR4.IIUOll �-'\':onl( ..

J

all ....�..

I'I'( I)j'dV •.

(7·2)

I

TheT ime·lndependent Sch.oding •• Equ.tion

\.. In \11l!." dUlIcn'llm. \\(' 11)\1..1 01"11.'0 \:nn!l.ukr ..Ialt')o, In Whh:h the' c:nclJ,Y I�

�dl lklinl.'''' IInLi u"';hall�lI\g, Th" ,:t1mes fmOl aSSllll\lIl�

t'It'
'VIr. t) •

the \0\1"\'(' fUlk: liun 10

lb(r)tP (t)

8, 'IC'P' I(knlkal hl ltl\hC in SCI.:lillll ) , I . In!.cn i nlt this prudm:t inhl elluu-

Illlll (7. 1 ) h'lI .. U" thai 11l1' temporal polrt ., II i.lglllll

111110\' \ Ihal

c/l(t)-
I .tnd

iP{t)

1'I'(r.

oj!

�

I(fifl)" \"\adl)'

II!'>

l\c.·ftlf'(',

[';'Ir)j!, The!' " !Iopalial !tOitlc"

III thl' ,ep.lraI Hln of \'ari.\hh:, then gi\'c" u, Ihe tlOu:'-imJcp.'ndcnl Sdriki i ngC'r cquJ.lIl1n

,,2

2",

" ", ( . ) + U(r)"',r) ; EI/I(r)

1111" is ll' far as we c.m go until we Io.now the potential cnerg),

(7·3 )

U( r).

Ouantum Numbers A' in Ch.lpter 5. we concentrate here on bound slutes-potenliitl energies (;Ipaph::: of cnnlining a particle to a region of space. In one dimension. the physical conditlOn� we imposed on the wave function led to standing wuves

and quantiLcd cnergies. Now with conditions in ,href' dimensions. we will obtain thrcc-dimen�ionul slandlOg waves. and three quantum numbers will arise. Thc�e quantum numbers invariably yield quantilcd energies. but they

govern quanti/utiQIl of other physical properties as well. depending on the par­ ticular

U(r).

7 . 2 The 3D Infinite Well The simplest three-dimensional bound system is the infinite well. It is a good model of the ubiquitous quantum dot, and it highlights several important con­ clusions in a rather familiar environment, conclusions that can sometimes be obscured in the hydrogen atom's different geometry.

As always, the �tarting point is the potential energy. In the 3D infinite well.

pictured III Figure

7.1.

a particle is bound in a box-shaped region by infinitely

1 ''T'I

,n,!o

l-,t,,,,1 fit

c, i"." f,i1';J r e'.jl!

lion

-

...... 7.'

"high" walls II x

A _....ionaI .

_ ...0.

ollOflY ,

• •

• • ,

•

•

•

: v-o

... . .. .. .. ..

i. �ron:

U(r) L.

-

-

0 and L,. Y

{!

o <

0 and L.,. and l

-

x

<

-

0 and L" The peg ..

L,. O < Y < 1.,. 0

< : <

otherwise

A§ tn one dimension. the wave funclion cannot penetrate o. walls. so we concentrate on the region tnslde where ,

t.,

U=

L,

outside �

Whenever we face a differential equation Involving multiple i

�

,,·ariables. our firM \tep i� \eparation of variables. We assume that the s Pllill

W8\1C function i!> a product of three functions. each of a different indepe aclaa

variable. and here the ohvious choice is Cartesian coordinates.

o/I(x. y, ,)

=

F(x)G(y)H(z)

Following the u�ual recipe for separation of variables. we next insen this pr1Jd. uet into

(7-3), then divide by the product Expressing V2 In Cartesian cOClf'Ib. U IS 0 Imide the well. we thus obtai n

(

nates, and noting that _

�' il'F� X)G(Y)H(:) 2m

+

" F(x)G(y)H( ,) ill

ax'

+

a'F(X )G(Y)H('» az'

F(x)G(y)H(z) =

)

E F(x)G(Y) H(z) F(x)G(Y)H( z)

Canceling top and bottom where possible and mUltiplying both sides by -2mih2• we arrive at

-=o'-:F.;:. (x",) F(x) ax' I

+

g

o'G(y)

G(y)

01

+

-� = H(z) az' I

a'H(z)

2mE

(7-4)

("

Each of the spatial variables is now in a separate term, so the usual argument applies: If the equation holds for X ' YI' '\, it can hold for x ' Y I ' z \ only iflhc I 2 term Involving x does not vary with x. Each term must be a constant. Thus,

d'F(.<) d..(l = C,F(x)

d'G(y) d' )"

=

C,G(y)

d'H(Z) dz'

= C, H(z) (7-S)

Things now look rather familiar. Just as in one dimension, each function

must have a second derivative proportional to itself, and because the wave

lOne dl'I1I1CtiQll �re " that U\(o CQ'hlllnt_ C are l'Iot .. ,q>1,ntly oqall�_ W('re \,h(') I»" h�e. the sollJ·

t,om wou.ld be UponeOllal nther Ihao "nu"",,,jal Ellcmw 17 dl!'mon\lnlC1.. hov,·c'cr, thai C\por1I!'n·

lial� cannot be 0 al both wall\

function IS 0 outside infimte walls, continuity demands that it go to 0 at both

walls. These are exactly the conditions that gave us standing waves in the infinite well-the solutions are essentially identical.I

F{x) =

nx7TX Axsin L ,

G(y) =

ny7TY A sin -Y

Ly

H(z)

=

nz1TZ Az sin T

,

ID

(7-6)

'Tbt only rn.1 difftf'ml.� is thai �ach dirnens.eon is lIadependeni. 10 the COD­ ,WII' • ond L I1
0'

n,'

+ ',-

(7 - 7 )

l.�

The overall wave function IS th� product of the mdi\'idulli vnes, und u!'olng sim· ply A for the product AI Ay A.: we have

"'n" /I../I- (x, y, � )

(H )

Each set of the three quantum numbers (nl, n\, nl) yield, a unique Wjj\'e func� tion, a unique three-dimensional standing wave. Perhaps the most important lesson here is that whereas houndary condi· lions 10 one dimension lead to energy quantization according to the vulue of a single quantum number, imposing conditions in three dimension!. nluurally leads to three quantum numbers. In the infinite well, they are nl, ny, ":' but it is true in general that for �ach dimension, the" i$ an independt'nl quantum

l1umbu. As in one dimension, there s i a ground state. Consider the case l.�

::=

I . Ly - 2.

and L. = 3 (in arbitrary units). None of the three quantum numbers may �r this would make the wave function O. The lowest energy is (n, , n.'", II'>. ·

be 0, f

(1. 1. 1).

The corresponding wave function is . 1 1TX . 1 1TY . I1TZ

sm-- sm-1/I1 1 1(X, y, Z) = A SIO -2 3 1

With three quantum numbers, it isn 't quite so obvious what state would have the next·higher energy. It is left as a quick exercise for the reader to verify that itis (n..., r1y, IIz ) = (I, I , 2) .

Chepter 1

236

Quantum Mecha

TABLE 1.1 States in

the

3D

fr�� / £�,J'.#I, 2"';/) 6

2. I. I 1 . 2. I I. 1 . 2

6

.

11

3. 1 . I 1 . 3. I 1. 1 . 3

11 12

2. 2, 2

14 14 14 14 14 14

\ . 2. 3 2. 1 . 3 1,3,2

2. :l l 3. 1. :!: 3, 2. I

2 , 2:: ., . .:.

.J. I, I

1 , 4. 1 4 1 ..:

1 , 4, 2

2.4. I 4. 1 . 2

4. 2.

1

2. 3. 3 3. 2. 3

3. 3, 2

4. 2. 2 2.4. 2 2. 2.4 1.3.4 3. I . '"

1 . 4. 3 3.4. 1 4, 1 , 3 4.3. I

3. 3 . 3 5. I . I 1 . 5. I I . 1, 5

nil · m

18 18

19

-

19 19 ...:.:,

_ _ _ _ _ _

21 2' 21 21

21 21 22

22

22 2' 2' ,.

25 25 25 25

25

25 27 27

27

27

bers corresponds

10 a

. 51Tx

17r)' h rz sin -s' 10 sin L sin ---=L L = A sin L L x 1 7r)' 57TZ . 1 1T l-rr z 57T)" SIO -- sin--s' A l-rr x 5 sin L L L L sin= A sin L L coordinates.(x, y, Z). mum values at different maxi their have these else. enero-y-IS call d same g the If nothin ce-different wave functions. having The coincidenand energy levels for wh.ch .t .s true are said to -rrbe-h2dege�ial' 11 ) .IS The energy 27( ' /2 degeneracy, l). contex this in enl judgm l 2 ) are mora L 2f. no '/2m 1 2 ( -rr ' plying and sa.d (im -rr2f,'( 2mL ) degenerale. Levels 3(onds funclion. wave said to be 4-folderate single 10 a to be nondegen -each corresp }1TX

37TY

3-rrz

= A SIO-"'5. 1 . 1 L '" 1 . 1 .

:.-.. " ::

_ _ _ _ _ _ _

1 . 2. 4 2. I . 4

•

17 17 7 1:: ..:

_ _ _ _ _ _

3.3. :: '

energies. Of those ers for many allowed numb um quant 2 of sets WS j2rnL2) correspond 10 l 2 ( 7T2f't Table 7.1 sho the energies 3(-rr2fl2j2mL2) and . , 2 2 I ) and (2 I , ( I , only y shown of quantum numbers, respecuvel (- The different sel,' four sets 0 from quan. que lts uni ce, resu /2mL' ) , for instan set of -rr '-h ' e(lch qUOnt 27( y, 11m energy ng Ihe same. energ BUI despile havi ers. numb . tum cflon. The wave funclions are different wave fim

II

-

1.3.3 3. 1. 3

real three-dimensionaI World '" our in ons cati appli any to m e the same energy and the number of equaI-energy An idea crucial s can hav metrY 0f Ihe syste state ' m. that multiple . sym the w.th s ease possi ,Iales incr the box is as symmetric as ble: a cube. Suppose Deg eneracy

Energy (7-7) becomes

9 9 9

, " ,. I

I.

Hydrogen Atom

6

1 2. 2 2. I. 2

3, 2. 2 2. 3. 2

well

3

I. I. I

nsions and the

Dime nic� in Three

10_

=

0

•

EXA M P L E 7 . 1

'

e

cubic 3D infinite well �f I nm ( I . 2, 1 ) state of a the (n.r' n�_, n�)on's energy n. (b) Where is I e P
�

\

FI,... 7.a De� in a cubic btfi·

Note: To be complete. we would need the normalilation conloolant A. It tums out to be (1JL)'fl and is the \ame h)f all �ets n . n,. n.. The proof l!l.left as an ,

nilt well: Equal energy but different �lat�.

(b) Wbere doe� the probability density ha\'e ih maximum value'l

I n.\""

L

,211"."

L

. . , .. A loom- - Sin --- �m2

. �

� . .

In!

L

�L.

The mtl}limum of sine �quared is I . so this product il> maximum ....hen � or and .\' "" and z "" Thus. there are two points (.\. �'. !) where the

�L.

�L

�L.

.

�

electron is most hkely to be found.

,

,

,

,

(c) By symmetry. £2,1.1 and £1.1,2 equal £1.2. !' and the locations of the corre�pond­ ing functions' maxima follow simply by permuting the coordinates.

----

..'

..

��.

Figure

7.2 represents the

probability densities for the three standing waves as den­ sity of shading. with two regions of maximum probability for each. They have the same energy. and the shapes of their probability densities are similar. bur they are

"' . : -

,

�

different states. Splitting of Energy Levels Degeneracy results from symmetry. For example, if instead of a cubic wen. we had Lx and L equal to L, but Lz equal to O.9L, much of the symmetry y would be lost, and the result would be splitting of formerly degenerate £ , 2 1 ' and energy levels. For the cubic wen. equation (7-9) shows that ,. equa\. are

£Z, I ,P

£1 .1.2

,

For unequal lengths, equation (7-7) says mat all three levels would rise. but , would rise higher.

£1 .l 2

(O.9L)'

E\.2,1

<

EU.2

,

\.vu.N

240 CMpter 7

sions Three Dimen Quanlum Mechanits in

and Ihe Hydrogen Atom

heavier than elec atoms but far tr than gold er light es backward. Only a c0ll�151011 ced boun charged particl sionally e alphas occa · this, and the n � . . core couId ex.plam found !hat som ellOtht ated positive entr conc ly a planet. orbit high . moons ier as nu core a heav !he mably orbited lose energy by elec� electronS presu electron should ing orbit estimates that !h.IS cO\l was !hat an deficiency us. (Exercise 26 ap,. into !he nucle al spir Bohr, addressin pog the radiation and In 1 9 1 3 , Niels d.) PtIlb. secon !han a nano um ideas, boldly tulated would take less and early quant tho of classical ture mix untered in Chapter enco first lem wi!h a we e elo:. which , states stationary state, but it could do so in existence of Slt.on, in a stationary not radiate is discUSSed would the hydrogen atom q of Iron model . Bohr e states. The ""'. aWS, but .ts stationary States fl s .ts i · between thes revea .. . . H·tocis·.ght ion Sect , and its quantized energies are titatively in nics mecha tho� tum n feature of quan ment that arrived i th still a vital mechanical treat tum911h quan the fuUy predicted by SchrOdinger equation predicts at the the 7.6, tion . in Sec. As we will see followt og energies; allowed only the IS ron elect atom' s hydrogen

�

mOC\ol',

�

.

En

46

= -

�

!

\�

me4

I

where

2{ 47T£O) 2fi2 1l2

(7- 1 11

fl = 1 , 2, 3, . . .

this is not peculiar to quantu negative, but m energ ies are the ted, e gravitational negativ " As expec the y e nh al to Earth b satellite bound mechanics. A total mechanical energy. Its e II ive negat m hc E","mS",'h"lr has toward 0 as the orb t rad energy -Gm iul which increases ial, potent · negative ses toward 0 b t as increa less than its arly We ) simil ' � y. Energy (7-12 obtains quantized en g approaches infinit 27 e ercis "S .of qua"lIIm levels n. (Ex move to higher approach, sidestepping the co ple"t) ctive instru simplified yet this form by a 5dinger route.) slate, neither the probabir den· of the full Schr In a stationary puzzle solved? tr Itm wave would vary Wi: Is Balmer's e. electron standing density of the charge downward a as state, d an�h groun sity nor !he on state above the on is in any does oscillate. In fact But if the electr density bility proba g which the can occur, durin frequency to generate a ph ton Ole at just !he right ates oscill it 5.9, Suppose the I on states. Secti in een the difference betw ," · on the energy IS the energy equals The energy given the phot "r· state to n, ion from state (7-12), we obtain makes a transit l . Using equation y minus its fina energ initial electron 's

r

�

�

For a photon,

or

I

A

Ephoton

=

)f ( 2(411'.0)' h' nl m-t(.!, "f)

hJ he/}.., so =

he "T

=

� ��wn ::

me'

I

I

-

7 1 .097 X 1 0

n

nf

- .!..

)

(7-13

We infer that the wavelength.. gh:en by (1-1 1) result from transition" down to the /If ::= 2 level from the "L :., 3,4, 5 and higher levels-and Bahner's pUI..IJe is �ohed! Logically, there should be wavelength... corresponding to tran ...ition!io ending at levels other than " f = 2, and the..e too have been observed. Thll!'le ending at the fIr = I level, for instance. in\oh'e a greater downward energy jump. so they are of relativel.y. ..hort w.a\,'elength. All are In the ultraviolet region of the spectrum. TranSition.. endmg at IIf = 3. 4 . . . mvoh'e ,mailer energy differences and are consequently in the infrared and beyond. .

Energies and Wavelengths

Figure 7.5 illustrates the quantized energies in the hydrogen atom and the tran­ sitions possible between them. The combination of constants out front In equation (7-12) works out to be 13.6 eV, so the energies can be written as (7- 14)

n � I . 2. 3. . . .

Energy increases, becoming less negative. as II increa.ses. asymptotically approaching O. In its ground state, the electron is 13.6 eY " deep." a rather famous value. We would need to add 1 3.6 eV to free it-to reach the lowest unbound energy-which agrees with the well-known ionization energy of the hydrogen atom. In Figure 7.5, arrows signify downward transitions, in which the energy lost by the atom's orbiting electron is emitted in the form of a pho­ ton. Transitions are grouped according to the energy level at which they termi­ nate. and each group is named for a pioneer in spectroscopy. The Balmer 2. As noted series. for example, comprises all transitions ending at nr above, hydrogen's four visible wavelengths are due to transitions to this level, so they are members of the Balmer series, the fIrst (II, = 3, Ilf = 2), the second (/lj 4, rlf = 2), and so on. lt is natural to wonder what an electron in an n· = 3 state would do. Would it drop to the IIf = 2 state, producing a photon corresponding to the first spectral line in the Balmer series (rli = 3 IIf = 2)1 Or would it drop fully to nf = I)'? IIf ::= 1 . contributing to the second line in the Lyman series (IIi = 3 The answer is that it may do either. governed by probabilities. This issue we take up in Section 7.10 and in Chapter 8, but there is an even more basic ques­ tion: If the electron is indeed in a "stationary state," why would it make any transition at all? Our treatment of the hydrogen atom assumes an isolated atom. However, even in what we might consider a vacuum, there are always spontaneously fluctuating electromagnetic fields, known as vacuum flUctUBa tions. In effect, these change the all-important potential energy. While we won't go into the underlying theory, known as quantum electrodynamics (QED), vacuum fluctuations perturb the atom so as to cause the electron to seek a lower energy, if there is one. Consequently, an electron not in the ground state will jump down through lower energy levels, producing corre­ sponding photons, and eventually end up there. An atom may also be induced to jump to a higher energy level. Sensibly. the excess energy it sheds in its otherwise inexorable decay to the ground state ==

==

,

�

�

142 awp.., 7 Quantum Mecbanics in Thtee Dirneasioos aDd the Hydrogen Atom ....... 7.1 Hydrasen·, eneraift and .pectraJ Jines: A photon ill emiued when the electron "jumps downwanJ "

£� = 0

£,

- 1),beV

11

5: 13.6eV t.-_, '"

£

_

-

, "

'-v--'

3' - 1 .5 eV

PaM-hen

- D.beV 6' - 13.6 eV .'

U"n�\

(n, :II :n

'---v---1 Balmer sencs 2) t"l •

Ll6eV

'---v--i Lyman series (tl f "" ! )

must somehow have been given to it in the first place. One way to do this_ es!<;entially the reverse of photon emission-is to strike the atom with a pho­ lon. To cause the electron cloud to resonate just right, the photon must be precisely the energy difference from one atomic energy level to another. and il is fully absorbed in the process. Another way of increasing an orbiting elec· tron·s energy is to strike it with a massive particle. I f the colliding

panide�

have sufficient kinetic energy, an atom may absorb a portion of the collision energy and jump to a higher level. The colliding particles naturally depart with reduced kinetic energy after such an inelastic collision. EXAMPLE 7.2 The atoms in a sample of hydrogen are aJl in their ground states, far too cold and slow 10 bump one another to higher levels. Now a beam of electrons accelerated through a poten· tial difference of

12.3 V is directed al lhe sample. Whal wavelengths will the hydrogen

aloms emit? (Assume lhat downward transilions occur so quickly that there is no chance for multiple accelerated electrons to combine their energies on an excited atom.)

'3 ..., 01 . .. . .. ,. ... ...... ..... .. ,1$*&17

.. \,c ('ncrlJ)' 1 3,b tv. Each ck-ctron in \he beam bu kmotk .'\ll the atoms inuillll)' h of 1 2.. 3 tV Sl\ in IS collision. it ,-"OUid impan AI muM 12 ..\ c\/, rai"", lM fflC1&)' alOn\'S t'IlCfS) to - ' 3 6 c\/ " 12.3c\l - 1 .3 C'V. Bul \ . 3 c\l ' lkll one 01 the allowed higher cllC'",i�. l:l\)n\ C't\uutu.'In (1-14), these �

l:t.t'! !tV

13.b cV .- - \ .51 eV, 3'

" U R� cV.

1110.... a colh..illn hi!... �'nuugh energy ill rai\(' tht' Iltnm no higher thM the II :\ lew" lnl'rcafta, It "111 emil \:h;n·.ld�l llo-tk ph"'tl1lh alo- ItS cI�tmn drop.. bm:\. 10 the llmund ..'olll.'. Thrce tn,Ul',ItI\m.. ,If .. 'Plh"lbtl.': It Illil)' l1nl(l l1irc..:tlv I'l\un the thinl to the first cnerg.y k\et . \If It llIay orop hum the thirl1 hI the ).,(":\)1\11. �nd thcfC'at)er it w\luld \'l\IP ' fiN III all l·"..\·... thc photon cner�}' 1.. lhe .. hlleren..:c betwttn thc I\hlll1k .. n hl th.. l1w.

('llerS>

k\cb

f. ,

I�',

E:l

. .

,

E,

1-: 1

,

f-'l

r·z

,,

cl

-- 1 .�1 eV " \ - n,6 e\l) "

1-.' 1 -

-- 1.51 1.'\1 - .\.4 cV

t-- .\.� cV)

\ ,\)eV

� - I Hq::\I)

" lU.J. eV

�-

\2,1 eV

E"huhll'l

::J hdA.. l1r A -' hdf.' The photon wilvclength\ folk)w Irom pru'\Qn' Rel.:lllling tholt the prOllu..:t lit' In the pre.. cnt unil.. i!'. 1 240 eV • nm. we ha\'c

1240 eV

.

nm

1 2 . 1 cV

A:\-_l

:=

;

1 2�O eV ' om -- -- = 1 .9 V

656 nm

1240 cV ' nm 10.2 eV

(Notr:

- 10-' nm

-=

122 nm

Some values may not appear to work out exactly. hut this is due only to

round-off error. ) The first and third wavelength" are. respectively. the ..ccond and

first me mber\ of the ultraviolet Lyman series. The second wavelength i.. the first in

the Balmer series.

Spectral observations are one of the primary "handles" by which we test the claims of quantum mechanics. but they are also an indispensable tool in chemistry. biology, astronomy, and many other discipline!'.. We discuss e\e� ments other than hydrogen in Chapter 8. For now, we !'.\mp\y note that each element's electrons have different quantized energie!O. and thus emit different wavelengths, a unique spectrum that serves

as

the element's spectroscopic

"fingerprint." Sending an electric discharge through a tube fined with a dilute gas is a common way to reveal such spectra, Another way is simply to heat a gas, although this requires very high temperatures (see Ex.ercise 3 1 ). Cold atoms emit no radiation-all electrons are in their ground states and stay there. If the sample is heated, interparticle collisions may become energetic enough to cause the electrons in some atoms to j ump to the first state higher

1M Ga;""

0

I. ....

n . n.. [lo " [ _ _ "' Ik H� Atom

..... ". "" __ ..
than .he Jround

A' 1/". pom� ....

raised. (Jther hlghC'r Intis become 8l;,,-e

be

....

••!e.

p<eUUm

.. uuld

be

fairty ...

ISlinl or nwnually a mele line. cOI'KSpon(ilng to tranSitIOns ,... .. flnl elClaed 'late bad, tkMn to the:- ground tale Ai the tempe l....

1.:00

Iblc. allo\lo mg mort

proolk.'CCJ In ubscqucnl dowll"anJ tr.In�UUlR" and gioving.

Ittn p:c.:trum.

"t�

..... i.I

richer �

M.ltlC'T may 4bo reveal Its 4:0mp
1,:hMgcd

m vIOlent mouon, ('mit cssenliull) a �unlIOUlIU'i !>re�lnml uf ...."J...clenau.. l>1<M.;U ed 10 Sc'·llon 1 1 . Ihl' Iii u bl.u.:ll"ktJ�· s�,":lrum. and II depend, 01 the !\lur'S lemrcraturc rather than It ... \7l\mjlll!\IIHlR Hll\\('\Cr, II thi hpn CIt It.. WA)' III I:arth passC!\ through .1 l'\1()!t:r IOh:rWntng gu.... a ... \le-Plcted .. Hgure 7 .f" wu\·ckngth l·lIrre""pt1ndmg t\l .,hnercn�e ... 10 the ('nag)" I('\rt

of he ab'mrtl(:J (\, Ihl' al\Hn'" drop bal.·l do\\ n to Iht l roved Male. Ihey recnHI tht""'l' wu\ clt.:ngth ... In all lhr�clilln ... I"h .... ret' mtlll'J Iighl

the gU\ .I!t\ln,

'A I II

Ml !IocJlh:red I.:tllnparcd

�I1no hpi mu1U�

hyllrvll;tn pccuum

lS

"-'llh th.: r(' ...1 \11 the Iighl-1,;llnttnulng unafl(Clcd Ol

I" "-'il� tWill Ihe 'Iar III Eanh- Ih.1I it , ... c"enIH1II) ffil ...... tng In Ihl! Iighl t.ha "'pcl ·lrum ru... l Il\ cl y ILJent l fit' , Ihe element Ih;u is

ilrtl\t'S. rhu..., a ou ...... lOg

dlll Og Ihe utt"'llJhll1g. Fi gure 7.7 dlu ...lr.lte ... Ihe phenome non The tliP plor. "hu\\ , an cml ...... nm ...pcdrum for h)drogl'n The buttom phil ...ho..-., wbl ..-. il\ c kngth...

\11

a

l'l"lnllnutHl'" "'pel·lrum would rea.:h Eanh after pa!$iDt

Ihmugh �I doud \11 h)Jf\lgen. The B
..-.

ill be e ......ellll
'1 z: 2 ...talC, SO

pholon... Whtl...e ellefglc", I:urrl!'pllnd 10 tran"'llllln\ between the '1 - 2 lnd h'gh(·, k\"eJ... I.·oull.! nol he
Ihelr �hamcteri'lil: \�'i.l\dcngth...

Lei u... now "'c(' where hyumgcn·... quanli/etl cncrgie, come from and find 0Ul

\\hat olhcr quant1l:uioll, tim thrcc-dimcn ...ioni.ll problem ha� in ...tore.

Figure 7.1 U�Jm�cn · , effi."ion 'per.::trum--bright linc'.It I.."ertain '\-Il\elength ...-and its ab'orph(lo 'pc�·trum- "'iI,,:length, mi" ing from a conllnuum 1I).Jro�('n ('mink," �rc-:lrum \\ ;\\(·knglh Innll HyJnlge-n at-.\llrpI111n '�"'nml

III I

100

III I

L)ffian �rie..

'00

100

II I I 400

I

500

Balmer �erie ...

600

I

100

�

7.4 The

Schr6dInger Equ....

...;

}OJ

iI.IOI

.. + .... .. . ' .11.7 + ....- 'hi d on r. whicltil 'I1It I ... . ian cuurdi .... .. would lie .... . poiItM. . .... , tII t n ..pIIIo" in 11 Pi .. ..., ....." of the ",,"" 0m naturally ...... IpIIowicoI ...... ....,

C

....'('. . A M(,( re\'irw of Ihis coordinate )'IMm lI .....wIIIIe. In sphtrical "dar 1."tl(JCdi....� all 0( thne .tiE'."'" .... II ..... .rnlcd tty \'IlIUC's (If the- Ihl'tC' c.-oordi1llleS r• .. aDd .. .,.. Ia ..... 7."

�

sun�-hins. I"nll1\ I� utili'" 10 an arbitrary locatioa iD IpfICIf i • pall'" vtOtar r It.. Icn�lh I' the \"'rll,lrdimlle r (the .I('Ular discu.;e from the ortain lid DCM to

be �'Im'-\l�d wilh the \'C\:'lur r) and is ,,'alled the ........ Tbe coord:luIe .lI tIw .1Il1!1l" t'lcl"ccn the p'..itinn \'rI;lur and the :·u.I. and i called tho ...... ...... \\llh th<' "f'llk" t'It'inlt the l'-Il:\is. The c(lOI'dinak" is the 1liiie. about tbt l-Mla. �w«n the: H" '" and the pmja.'tiun ofthe po6ition wctor In the�. II " \ alletl thl!' azim ut hal anale,.j Do they cover all !!lpKe? For • •iven I'IdiUI , and ptllar an�11!' fl. \':U)'in� the lUimulhai anale '" through 2" would iumbe I lad­ tude ..t .ul�l(' till" a 'phcrkal surfac.:( of radius r. Varyina the polar ana'" ' from o III Jr (nllt � rrl ,,\mld Iht'n fill i n all latitudes from the 1Iphere', north pole to III '

Figu� 1.8 Srh�ru.:al polar cuordinates.

t

r

()

. . . . .'."ip (J

.... x

tl lJ -f 1T 1/ . . . t.- 'COl '

'/ . ...

.,.. - .. ... . . .. ,

.... ..

246 ClMpter 7

en§ion� and the Hydrogen Atom Quantum Mechanics in Three Dim

r conver,ion .. TABLE 7.2 Ca rte \la n-p ola

\' ;1

r,m tj , m c/J '" - Ian '

: ::: rcOS 8

,

�''---

----- -----

from 0 to infinity would include all conce ing ntr;c south pole. Finally. vary · nsh·IpS between (x, y. z) and (r, re1aUo The e. pac ... all ring ove 6, til) spheres--c d Figure 7.8 and summarize in Table 7.2. are color highlighted in , such a.... in normaliLation integral (7-2) To calculate probabilities in spherical ite... imal volume element
r

dV

rd9

9 10

r

dr.

.10 8

r

p��e need

dV = ?- sin Odrd9d
r on fen. �?ng

electron's potential energy depends onl r, In the hydrogen atom, the origin. not on (} or
r.

V2

\72 =

�[i!... (,.1i!...) ,.1

ar

ar

+

esc 8i!... a8

(

sin

8.i!... a8

)

V2 .

+ e sc'

8 -aa4>'

J

I;ger a�a.

price we pay for simplicity in the poten,.,,1 This hefty expression is the factors between two derivatives in the r · . . y 0f haVlOg energy term. Its pecuhant but it merely gives a more compact fo and terms may be disconcerting, act .on everything t� its right-i ncluding f Each derivative IS understood to . . . mind that this exp . IS applied Bear course, 1/1. to whIch the whole thmg in Cartesian als parti nd seco simpler sion does the same lhing as the t arrive at the time-independent SchrOdt· nger nates. With everything in place, we . equation for a central force.

9

10

�, 1

[

_ __

2m ,.1

a ar

_

( )

a + ar

,.1-

esc

a 8a8

(

. Sin

a 8a8

)

+

]

� COO����

a esc' 8- ojJ(r 8 4» . a4>"

+ U(r)ojJ(r. 8. 4»

=

E ojJ(r. 8. 4»

(7-'51

To understand what this teU\ us. we break. it into pi�ce!> thal we ..:an ,.:rutinize �eparately. A." in the 3D inrlOi.te well. we sian by writing the wa\'c

function as n product of three functions, each of a different independent \:triable:

Here we IOtroduce some 'Very helpful bookkeeping. The \'ariable is in lower­ case. while thc correspondingfonction of that 'Variable is in upptrcase When

we insert this product function into {7-IS}. we obtain a �parale differential equation for each 'Variable. The mathematical steps by which the equation'!. come about are outlined in A Closer Look: Separating Variable'!. for a Central Force. In the following sections, we concentrate on what the differential equa, tions tell us,

It. Closer look

Separating Variables for a Central Force

variables in SchrOdinger equation (7.15). we first rearrange it slightly,

To aid in separating

'

(

0

)

0'

1{1(r, 8,q,) cse 8 a sin 8 a V1(r. 8,q,) + csc2 8 8 8 aq,2

AU dependence on 8ngle� 8 and q, \!i on me left and aU dependence on r is \In me right. The u...ual deduction appbea: Because the variables can take on values. indepc:ndcnUy. both sides must equal a consl.D.nt. for which we choos.t the: symbol C. Setting the right !Oide equal to C. festoring to R its argu. ment, and rean'anging a bit yields the ndlal equation. d

��

dr

(7-16) Now we replace V1(r, 8. q,) by the product of three functions, R(r}0(8)<%)(q,), and note that the partial derivative in any term acts on only one of these three. Suppressing function arguments for simplicity, we obtain

(

)

'0 o R¢ esc (J- sin 0-

o = - 0 <1> - "ar ar

a(J

0'<1>

+ R0 csc2 (J--

( 'R) - 2m" fl, (E d9

-

2

1

o

o ( 00) ( ) - -- ,,- -

=

09

sin (J-

09

1 0

OR

Rar

ar

0$'

2m" -(E h2

- U(,))

dr

(

(7-17)

c;

CRlr) (7-ISl

)

Setting the lefl side of (7-11) to C gives

1

acsc 8 a8 '

sin (J

I �
39 + c sc·8 til 3q,2 ,e

�

"" C

(7-19)

To separate e from ¢I. we multiply both sides of (1-19) by sin28. then mo\'e the term dependent on q, to the: righl side and the rest to the left.

itO

- UV))R0<1>

I a2q, + csc2 (J�--

)

d ""� 2m� ,2- Rtr) - --ER(r) ... -- Ulr)R(r) ,.2 ,.1

0 1 o sin 8

aq,2

The next step, as always, is to divide by the product, RE>¢l. This cancels twO functions in each tenn having a derivative. leaving only the one on which the derivati'Ve operates. Thus,

- csc (J-

(

(

ae

sin (J·ao

)-

.

\ a1cll

C sm1. 8 = - til (7-20) at¥-

Both sides must be a constant here, \00, and we choose the symbol O. Equating the left side to D and rearranging gives the polar equation.

a

sin (J a

8

(

)-

06 (9) sin8---a9

C si.n2 98(9)

=::

D8(9) (7·21 )

and setting the right side of (7-20) to D gives the ....... equation.

=

- D ($)

0-221

e [)imensions and the

ill '(bre JtI a.,w 7 QuantUm Mechaokt

Hydrogen Atom

al Force 7 . 5 Angu lar Behavior in a Centr

nding conclusions about the an front !>Orne a."IO� , . . In this !'.Celion. we con chamcal partIcle. It IS I m�rtant to me tum an qu g tn motion of an orbitin atom a� a specIal case they . apply to the hydr�ge� y the gh ou art alth t mind tha ce IS strictly radial. We take the angub, �� where the for applicable in all ca. variables one at a lime.

�

on The Azimuthal Equati

is ng azimuthal motion The equation governi

(7·2 21 ounter. this one is familiar (Al l equations we wil l enc ' a,. Among the differentia ' ' Proportional IS ' e 0f the funellon ' atlv to cond denv e :;; e Th !) not are the others usoidal (0 > 0) or exponen k . this could imply sin tlal . function. As we now out by the particular nature of th. ed ru are r latte the i\ 0) solutions. but (D angle goes from a given val ue lhrou h '7l' J tha mu azi the Ao:. . spatial coordinate and it must do so smo th function 4>(4)) about the .,:-axis. the utIOn can do this. But the reqUi.re y.. that no exponenual sol Exercise 35 shows . n see soo we as ' ns, , usot'daI functto sm menl restricts even the solution should be sine, cosine, Or rne me is whether so A remaining question mistry. sine and cosine are on the application. In che nds depe It used . tion bina com mical bonding, where one atom reaches OUlto che to ed suit l wel are often. all they '" ' ' ' ' s. The functton cos'Y has Its largest magn'"ude tIon d uec rent we d'� ong others aI ' ''' ' Iargest along the y-axis ( wh'Ile sm 'Y IS , 0 or 1T. aIong the x-axiS. IS when , ifically a camp usually choose a combination spec Figure 7,8), In physics. we IS smusotdal !) We discuss the rea = cos Z + f sm Z . exponential. (Never forget n. Noting that . a ctio h fun suc r side con first us but let if son for the choice shortly, ution of (7-22) is sol a e, itiv pos be st D mu n cl>(4)) sinusoidal, the

,

<

�e

I

�lIeels it�elf.

4>

'

��

:

ell

is

(7·23)

� mteger.

shows that if this functi n is to meet itself smoothly whe n cjJ must be an The smoothness condition changes by 21T, then _' this angular situation. a periodicity condition-has therefore given us a qu tum number associated with the 4> coordinate/dimension. We choose the s Exercise

bol

36

Vi5

m[ for this integer. Attaching a subscript to distinguish one

function from another, we thus have

For each

� uantum

� ber

nu

me

�

tha

=

0, ::t l ,

::t2, ::t3, . . .

I�

an allo�:�

(7·24)

arises-and two dimensions are yet to

come-a physical property IS quanuzed. The property quantized according to

Figure 7.9 St.mding wll\'es on me ..�·llXis...

I

,, I, , , ( , ,

,,

,,

,

,'\, -

'----"

"

"'t -

, -�....

,

,,

mj

-

%. 0

,

,,

�

r

\

,

,,

,. ,,

,

,, I

,

) � ",f' ,I I, , I

,, ,

�"

-

- � ..

,

no,

,

,�

,

, -�

IIIl == 3

,-

,, ,

,,,

0

2

,,\ ,

�, ,

,\

; -Y

'

the value of tnt is the z-component of the electron's angular momentum. Lz: This may not strike the reader as even remotely intuitive at first, but consider Figure 7.9. which crudely pictures the real part of einl/4>. The dashed circle repre­ sents a "cJ>-axis." For mf = 0, eiQ4l is simply 1 and is thus the same distance "above" the cJ>-axis around the entire circle. For mf = 1 , eil4> is cos 4> + i sin 4-. Its real part passes through one whole cycle, varying from positive to negative and back. as goes from 0 to 27T. Similarly, the me = 2 wave. ei 24> = cos24- + j sin 2cJ>. passes through two cycles in an interval of 27T. In general, the circle's circumference would be an integral number of "wavelengths": 27T1" = meA. If we make the independent argument that a "wavelength around the 4--axis" should be inversely related to a tangential momentum mVt in the usual way (A =

hlp), we would have A = hlmvt' where m here is the electron mass. Combining these two observations gives us 27Tr = mth/mvl' which if rearranged becomes mit = mv{. and mv{ is simply the classical ex.pression for angular momentum (r X p) in circular orbit. The orbital plane for the azimuthal angle is the :
mf

=

0, = 1 , =2, =3, . . .

(7-25)

The above argument has several limitations. In particular. it would seem to apply as well to sines or cosines as to complex exponentials. but only the

2JO 0..- 1

Quamwn Medwucs In Three DimrMlon\ and the H)'drogcn Arom

latter represent well-defined quantized Lt, Although the fact that Ii�Gt momentum p is well defined for �Jb bUI not for sin lex or cos lex is some �. forcement for thjs claim, the more rigorous justification is Ie£( to Exercise 40 (II is based on the Section 5. 1 1 concept that the complex exponential alone is � eigenfunction of the L: operator.) In any case, that Lz is flat well defined for sine and cosine is the main reason physics lends to focus on the complex exPDoen­ tial. Another is that, whereas sine and cosine "stick OUI" along .thex- and �axes. m m the complex exponential gives a probability density (e-J �e+J �) independent of t/J. It is the same in 811 directions, as we expect for an isolated alom. We will discuss lhe consequences of lhe quantization of Lz after addresn si g the polar angle 8, but for now bear in mind the similarities to our one-dimensionaJ case5 of Chapter 5. There we had a quantum number n, which determined the value of the physical quantity energy, and for each n there was a corresponding function. Here the quantum number is m(, determining the value of the physical quantity L::., and each has a corresponding function t.1>m/t/J). The Polar Equation

Equation (7-21), governing polar motion, depends on D, and we have jusl argued that \ID has to be an integer, me' Making this substitution, we have Po'

I,

•

(

00(0» a sinO- sin O 00 00

)

- Csin' 00(0) =

m1 0 (0 )

(7-26)

This is a far more challenging differential equation than others we have seen and we won '[ go into the analytical solution (found in any dedicated quant� mechanics lext). However, the vital conclusion. which should have a familiar ring from our one·dimensional studies, is this: While there are mathematical solutions 8(8) for all vaJues of C and m(, they a11 violate the physical require_ ments, diverging at the limits 8 = 0 or 8 = 'Tr or both, unless C and m( are cer tain values.5 11 turns out that C = 0 and me = 0 give physically acceptable solutions. The values C = -2 and nI( = - I , 0, or + 1 also work, as do C = -6 and nit = -2, - 1 , 0, + J , or +2. Happily, there is a general way of expressing all the allowed values.

.

C = -((( + I ) mt = D. :t I , ± 2, . . . , ::! e

Don't miss the point: The

i slruC­ �ComptJtalJonaJ ElIerrJ'iC: 91 lays out a very n live nurrn:rical route to verifying lhi.\ claim, based on the computer qpproach of Section 5. 10.

( = 0. 1. 2. . . .

(7·27)

requirements have given us standing_ wave conctitions in our second dimension 8, and our first quantum number me has been joined by a second, f. The functions that satisfy equation (7-26) depend On both quantum numbers and are distinguished by two subscripts, 0e,mtUn. Known collectively as associated Legendre functions, several are given in Table 7.3 on page 257. Because we so often need the product of 0" m (0) and m (
quantum number t determines its magnitude, ILI. Unfonunatel)', 'he� nm', a cute and easy way analogous to Figure 7.9 to Ju�tif)' this clatm. E,.cn:,�c "" \ discusses a simplified approach. but the rigorou� one is based on the formidable-looking derivatives in equation (7-19) being pro{M..'rtional tl� the quantum-mechanical operator for the squarr of the angular momentum (demonstrated in Appendix G). Exercise 42 use� this result to !o.how that equation (7-19) can be rewritten as Equation (7-27) tells us that C =

-e(e +

L,2 0<1>

1). so that

= e( e + 1 )h20<\>

The upshot is that L2 can take on only the values €( e + I )fi.2. (In the language of Section 5. \ 1 , the product 0(1) is an eigenfunction of the operator, implying that L2 has the well-defined values €(e + I )fi.2.) Because the square of a vector is the square of its magnitude, this is equivalent to saying that the mag­ nitude of L obeys the following: ILl

= Vt(e + l ) h

e = O. I. 2

• . . .

(7-28)

f

to its relationship to orbital angular momentum, quantum number is called the orbital quantum number, while, for reasons discussed in Chapler 8,

Due

mt is known as the magnetic quantum number.

The case

e =

0 is troubling. How can a particle orbit without angular

momentum? In general, the motion of the orbiting "particle" is a combination of rotational and radial motion. The e = 0 case is one in which there is no

average rotational motion, but there is radial motion. The electron might crudely be pictured as oscillating back and forth through the origin, as would a ball dropped down a tunnel through Earth's center. Still, we must not throw out fundamental principles of quantum mechanics. The electron isn't cle following a definite three dimensions.

path,

a parti­

but rather a diffuse standing wave occupying

Having considered the two angular dimensions of the Schrodinger equa­

e

tion, we have obtained two quantum numbers, and me, which detennine two quantized physical properties, ILl and L ' As noted earlier, until we actually t use the specific hydrogen atom potential energy, everything we say applies to

any central force. Let us examine these claims of quantized angular momen­

tum more closely.

Quantization of

l

The first thing to note is that we should expect some relationship between ILl and L -and we have one. Equation (7-25) gives the allowed values of L,. any z integer times Ii, and it arises from the azimuthal part of the Schrodinger equation alone. But the polar part gave us equations

(7-27). which restrict the

....

.. a..- 1

,

.....

, .. . r

,

..

.

,,

,.... 7 10 P1."., .. � p""" Irk' .,...",.nc

II .c lon . 1I!

I

" "

P

'rII,_

J

Ih,j/l

",

�

t

8uI ...h, n·::t C'.. 11 r, " .up ,Jltr fl."" . Th ..

II K "' � "' , �.

C"1ICC \lthal � �m .' liII .... .rdl �.,mp.;'1)Cnts .. u " � ntr;J'utl .. quanliUlJOft " iJ. f't)fl1 the 'I: hf(Idll 1 Ci ', ' ''Y atbttr.u)' tlM:UItIDII .. umpuMnt ,.1 I .hl h .e ( ,li the • rnp l 1P.1nC'tIb *uuJ.1 hJl' kl1' nl .Jd tttm.lll C\lC1 lhl I .lIl rh" .hull/ii , ull .h."'i _oukJ '"'_ um mu01&'nr I;--:l: ... ,1 p: t .� anti tllte IInull�uu "r'!(lVo 11weD u.1rr., til p. It! .. 'tt\ c;rtalnl} iIloAl ' the U",; ,1114 01"J I"'" ,�i I I ,,1 "lXPdk.-ubr lit " ;t1 ,he pbnt � £ , " ft rumlll}n ...nuld h.:" " WII' '"l ":IC O. c ha\' .: MJ!;. • plomt bulh I&nd 11.\1 ( .•1111' Iht lhe I un-.11OG ... the ...� W· [ri "nC'!';1. thl - Ivl Imrlllt� wu; r1. . tlIlh ,,"n l 1 ) 11 ho\I. ; ,n an d.r.:'r" !,' -...--Iprt-:ui ."Jl hl 1 m kr;r; " i be n an) of an 1\ "tlmp',;r, 'It 'tIC rl � I 111.1 mluJc ..nd ,"'� � In F JUre ' Ie T\M .. .. . . I all 'h '�I ..n Jll'n'.un hct "I dirt untl • rtMUl tude r l Jut: lhe nu: n' , Jtr�'! I.. J I!-:-n lhi 1tt:1\lmllm \; • .rnp..1tI uftt} Pi" :'I JrC J .. .:h It. Utk:1:'ru U,! ..Ivn, lh:!t a' Ih.JI I • .an I� , pU l ArrtYW � il:J ( .. I� Ibdlt" 0' t h ure � I I h.1W Ih. { ... � " 'n :1IIr 111 l !he �Ilf " . Ion, 1" f11 "tu me nltJ ,I,1I -' ,lngl ::91. � " U'1'OW !U'C 1 t' .. . ' ,J.'iC (. " In 1.. . I j )ffi; -!!:d inl nlun I 'J, ,,1 '" ( Ihe .lIo"c.J vaJw:� l'iUffibe" lftrmnc 1II r. :l .. �un :I II YoIlh .)nl), d� " (Ie' ,.)m '

•

.. 1 111

l

'

t (1.

II

,,1 I

ll.-nlUm '.ure 7.1 1 An):ul:u 1Il,11 .. f J.nd I 4Uanll/.llllHn ftlf ( I

I�

.

"

1

/I

/I

1

"J

>I

" f Jlrn;oon

,;,' Nol "nl" I II tft, 'r�; . h:.,; I I, '" ...
It

,•

;n spt:ak or onl:/lI,lilun EXAM PLE 7 :3

," What I th.' InIlUllll:m

m lhC' �� IJ , l ,al When t

_.,.1

I, 11K an

II "'III be mosl ncJl\

I lJ I_� • Ji - I I

..

')nenb of an;ubl

,.0 t_t p#runr',

•

discuued .. ThaI a.\).' oolv .Ilun, 11

.&.,

ax!' .J

,ten (0 !,;osll the 'i

little sense ILl

l •

t

•

I •

v

•

•

.

,

,.. 1<'

•

"

at e �

Nm �-'--

ntum

"

'I) It • .-,"-�.

• I

• fond

. .. .... . "' lor", \

ntum of • p:nt

If> mJ

111

, .)1,

L

•

�,

' 00 I t

I

I .

'"

I

l' m

, ' c., I( '0 t

l

.

{

yOdIIIt .

lumm, JuruhtnJ our r-er

•

h••� anguw �

�

1\

diS "

"""" "

..,...., tfCIII

'

.

I

."unu'

� eJ,...

Directional PrC' hav Bc<.JlI,;c we tv:re 10 pace

!.he

aI"o-ay

•

an tU.

plobah'

,.

I

..""k!

f
.

J \ tnfd

'

I of t

b I

....... _n.ru,.

.. .. .. ..: I,' Ie Ilr

.., , "' oo.c .alll<' '' uLl Ihu> ..... . .... b I� h cd anv ,.1Iue from (� U> ' ''' " nnt \i f t . \ Il1l l

".we

. ties

1 ul lhlDl . we clll\llOl ", r.siuered Ihe radial � pe< 00. bul we can II \cui dI.­ rlccu.m would be 100 c As ell
noted earlier. the complex square of tb(q,) is I . giving a probabilit . y dent of me azimuthal angle, and 8(8) is real, so the probability de S t i B ut why should it depend on either angle? We have SPh � y s jllsl ",cal sYZl}.. mctry. Shouldn', il be equally likely to find the particle in any dire etlon') l'\. . . ' es Lz• it is � aJ I Iy delermm Out answer IS thaI unless an expenmcnl actua u I tktJ)' add the we if and probabili ties to be any of ils allowed vaJues, . fOt au m, from - f 10 +f. the result is indeed independent of 8 (see

:

ei"",(8).

.In�

. (�j �.ml '

82

e . OJ. In short, if we are ignorant of mf. the probability of finding the i eSI_ f art.icle $ spherically symmetric. However, in most situations of inter r cal reactions to applied electromagnetic fields-there is some kind o extern al influence that distinguishes the different slates, so it is impOrtant t 0 an idea of how they differ. xercls S

. .

:� chenlJ_

have Figure 7.13 shows 8i, r' from Table 7.3, plotted versus polar a n e m all ca�es through e = 3. Several features are noteworthy: First, in th gl fOr

case, the probability is the same in any direction. in contrast to a p

e

. e ::::: 0 artIcle w·

!th.

ough �e

out angular momentum, which would oscillate strictly along a line thr origin. Second, in all e =I=- 0 cases, as me increases, the probabiJjt

Y

densl

!), changes progressively from mostly along the z·axis to most1y in the lane. XY·P momentum would also be more nearly An orbiting particle 's angular paraIlel to the z·axis as its orbit plane approaches the .xy-plane. The COrres pondence is

Figure 7.13 Angular probabililY densilies for a central force.

[·axis 6

..ry-pJane

e = 3, lIle "" 0

92(6)

e = 3, mt "" +,

e = l . mt "" Q

e = 3, m t = +2

()()

e = 3. mt = +3

e = I , m, = ± J

, . - ..,-

Flgur. 7.14 A "rude "��SP'lndeTb.'e to orbical

nll'tllln.

f

= 3 cases., but it is Important to nOI� that the depicted in Figure 7.14 for the be o l wave antinodes and are irreconcilable with standing. typical are sepafJled � a

particle's strictly planar motion. Third. in all

ml

*- 0 cases, the probability

density is 0 at the z·axis. If a particle is 10 have nonzero L.. il must never be = :t( cases have equato� ial lobes, ...ugge ...t· found on the .:;.axis. Lastly, all nil

iog orbit near the xy-piane. which become flatter as ( IIlcreases. In Example 7.3, we found that for large e, the angUlar momentum vector may point more

nearly parallel to the z·axis, suggesting motion more closely restricted to the .n'-plane. In the classical limit where e approaches infinity, L can be precisely aiong z. so the probability density should become more planar.

7.6 The Hydrogen Atom

Having dealt with the angular parts of the wave function, we complete our study

of the SchrOdinger equation for hydrogen by considering its remaining radial

part. Radial equation (7-18) depends on C, which we now know to be Inserting this and then rearranging gives u s an enlightening ronn.

�' )

d d - 'r -- R(r) + d 2m r r dr -h' I

KErad

h2f( C + I )

2mr'

KErot

R(r) + U(r)R(r)

-

( f + I).

= E R( r)

(7-30)

PE

From clas�ical mechanics, we know that rotational kinetic energy may be written L2l21 and that the moment of inertia I of a point particle is

tells u"hat f,2f( f +

I)

m,1. Equalion (7·28)

= L2 , so the second term on the left of (7-30) accounts

for rotational kinetic energy. The third term is. of course, potential, so it

_ ...

ZI6 Ch.pter 7

QuUllum Mechanics in T'ItIu DimeMions and the Hydrogen Atom

seems mat me first term must account for radial kinetic energy IOWani . . ' . away from the nucleus. Indeed It does. While we omit the lengthy de . t l'lid � 2 may be shown mat the tirst term corresponds to �m(dr/dt) . Overall e al .' it ua� (7.30) is much like the SchrOdinger equation in one dimension. e e ,il<1II1& of and involved. more course bit a is term it energy kinetic diffe n In the rotational energy term. but the latter is merely a scalar function of and ib effect on the wa..e function i!) therefore much like that of an additio 1 tial energy term. (KEIl)\ + PE i .. sometimes regarde� as a single . t potential.") In particular. the rotatIOnal energy term diverges at r = 0 tt l\'t ever l is nonzero. thus excluding all l "* 0 wave functions from the gill (see Figure 7.15 later In thiS section).

Th

�

:�

"ert�n. :�n.

Finally. we insert into (7-30) the hydrogen atom potential energy 0-1 0).

As in the case of polar equation (7-26), we leave the analytical solution . of thl' - her·levei course B � to a hIg - 1 equation I erentta mtncate d-ff . ut Just as in tha t c :e. a the crucial point is that the solutions are all physically unacceptable ' . ' d,vergln . g at r = 0 or mfulIty, except when the constants take on certain valu es. 1'he energy must be one of the values given previously in equation (7-12): _

_

E, �

II = I , 2, 3, . . .

and ( must obey

(

=

0, 1 . 2, . . . , /I - I

(7-32)

(Note: Computational Exercise 92 lays out a numerical approach to theSe

conclusions.) Our third dimension has yielded a third quantum numb � L called th� prmclpal quantum number, and the correspondin g quantize d property ]s energy. _

_

Recall from Section 7.5 that the solution of the Schrodinger equal- . �s . . polar pan placed a restnct]on on the quantum number from the preced' ng azimuthal part-mf could be no larger than C. We argued that this made se Something similar ha� happened here. Whereas eq f?r L� cannot �xceed : . lion (7-27) said that ( could be any nonnegallve mteger, equation (7-32) sa that it can be no larger than n - I . Again there is a physical reason. The qua energy, and f is related to the rotati tum number n specifies the on I energy, so it is reasonable that f might be limited by the value of II (see Ex a.

�

: ��

L.

cise

65).

lotal

;�)

Taking the restriction into account, we must replace equation (7.

by the following:

CJi 1tiZ
1dlti�n fe·

I LI � Vf(l

+

I)h

l = 0, \ , 2, . . . , n - I

(7-331

, . Tho K""- _

for each II amI t . equation (1-31) has a solution Rill

;].�"ociJted Laguerre [unction. Table 1 A lists several e'l're�,ed \0 term, I..)f the Bohr radius.

"

()

•

(4"'olA'

----

me2

of

(r).

known as an

these functions.

= 0.0529 nm

This combimltion of com.lunt, IS pervasive in atomic physks. !<.o the uetinition I' \'ery convenient. Moreover, the Bohr radius is a gOlxi gauge of utomic

rlimem.iolls, u, we win see. Our look at the Schrodinger equation for hydrogen is complete. We ha"e found lhree quantum numbers and three quantized properties. and in Tables 7 J and 7.4 we have the corresponding functiom. Let us now turn to two important related topics----degeneracy and normalizmion.

TABLE 7.3 Angular solutions: Spherical harmonics

TABLE 1.4 Radial

1.0

� .. �16" 5

2, I

- loin 91'- ' 8"

1, :!

I)

3,0

cos 9 loin 91':'::'

3, 1

(3 cor.2 9

2,0

1

{15 'is;; � 15

3277"

_

sin2 ge :':: 2;41

3,0 3, :!: I 3, :!1

3, :t3

�os -

,

- cor. 9 Sin 3277"

' n "'li, !,1 e -

1

-_ ")

r ' IUn),, � .

2,0

'i4,;

1.0

olut i om. ot' n· 3 t )

,

0,0

f3 cos 9

'i

3, 2

.1.•"

...,

Degeneracy

there is consequently \ymmetric. and mUch is ..phericaJly atom gen hydro The that the energy of an electron rh' n us lelh ) on (7-12 ''' g , f. degeneracy. Equati us Ihal for each equalion (7-33) lells BUI or II. rI. on prolon depend, only mome ntum L/-d ifferent ar angul different ah<>n , L" there are states of there ru:e states of differen t -di Ue ch f. ea for t tha l'tnl R 8 (7-29). JO rum. tells Ull > has a um qu e wave function mc , l (II. set h eac (JII �"" "'t ' Thus. because arc degenerate. I n fact , ShOWn the ground ..rale pt I'f! ce ex els ' lev y T Ie 7.5 de mo all energ nstr.'e' Increases as n-, lab acy ner th ge de Exerci"c 46. the Increases. . . b er 0f states as 11 ' qUlcidy grow mg num Gi ve n its spherical sy m mct We do is rather special . Actually. hydrogen entation-a componen t ori al ati Sp an , gular m( pend on not expect E to de ears i n eq uation app C? of at wh t Bu (7,31 ), 't be a factor. momentum-shouldn The fact that they do n ot e it? on nd depe es d energi pend Shouldn't the allowe In general, the allowed energles y. rac ene deg l nta ide acc fOr on f is known as nger equation before the h dro 6di Scnr the of t par ial gen. equation (7-30). the rad " is only in Ihe special case of e sim e. on nd depe do . rted ple specific U(r) is inse entally" does not A II)' de ial energy that E "accid ent pot via m ato . en rog IIr hyd simple case would lead to qu m this en m fro rgy ergy ene ial tion of the potent ctrons orbiting e nU In particular. additional ele Cleus levels that depend on e. s destroying the accidental de�eneracy thu , rgy ene ial ent pot . would alter the ental to chemlslry of energy on C are fundam e enc end an dep and d y rac Degene . apler 8. are dIscussed further 'In Ch

V��I �

I

;;,sf

e

07'

It

�

,:

�

:

Normalization

complex square of the wave funcl'Ian. The tern· The probabil ity density is the ' . . S drops oul In the usual way. As we noted m ee· poral part of the wave functIOn . . r the pola part IS and out, s real drop also . Now . 7 .5, the aZimuthal part tlon arrive at we , real also is h whic al. radi incl udin g the

1'I'(r. 8.
()I'

=

2 R'(r) 0 ( 8)

(7-3 5)

TABLE 7.5 Degeneracy increasing with n n

I

[

0 0 0 0

Degeneracy: Number of .st3tes

I

3

2 0

I I

4

0

+1

0 -I 0

I

2 +1

-2 9

-I

0

+1

+,

-

}u...t a... in one dimension, the total probability of linding the electron �ome· .... here til "pace mO",t be I Using, the "olume element in spherical poIlU' l.:04.lrdi·

1" " 1·>"

nate... from Section 7.4. nOfmahl_'l1ion I.:ondltion 0·2) bc"ome�

1" .� r"'O

or

1 r

�- ......

,0

tjaoO

4;1.-0

' R (r) "dr

' ' R ( r) 0 (8) " ,in Bdrclad

1'" 14;1"'2l1' ,

9-0

_

4;1-0

�

0' (8) sin 8d8d

= \

We could assign a single normalization constant to the entire wave function, but it turns out to be more convenient to consider the overall com.tan! to be n product of two conSlants--one that is kept with the angular parts and one with the radial part. We choose the constants so that each integral is indepen· dently I .

].'

(7-36)

0' (8)21Tsin 8dO = \

(7-37)

Note that the trivial integration over

EXAMPLE

(7-36) and (1-37).

7_5

e =- \, /lie =- + I

Verify that the Table 7.3 normalization constant for the spherical harmonic with is correct. SOLUTION

From the table. we have o 1. +I (8)<1> +1

(
S; sin e e · . '\j(3 +

whose complex. square is 0'l, +l(e)

=

2. sin' 8

81T

inserting in (7·37),

r'(2..81T

Jo

2

sin 8

\} 217 sin 8d8 � {Tr sin1 8d8 4 10 =

The value of the remaining integral is 4/3. so the normalization constant is correct.

eabo uts The Electron" Wher

.

electron ought be found . Figure locn', h'd � v n s where 7 IS I . .. Now we can d'scu� " . '" i ties a s den,J(Y 0f �h ad'109 lt'Or aU \1",.__ . uen"i lhl\' probab the . . -.q . repre.. . . enbo . notatio IC n. 1 0 which .. are "labeled u,ing spectrosCOP through n - J. Slate. Ule ' I and the leller the value of f. Table 7.6 g \'3Iue 0f , the 1 ... ate . ue"lgn 1 r numbe V es alphabetically beyondf}· Thus. a 3d ' . (leiter, ad"ance Siale I. e 'it,:heme for l w •• . () "tale. mt takes on IOtegral _. For each (n. , val one WI"m n - 1 and ( = lie" he scales are all ihe same-the rIghl ring i n the 2.p ( (T + d ( an between is- at the 3p Dt (wo') . 1110 and Ihe rIg hI ring in . 7. 1 6 pro ' i. com,'an' probability density shown 10 Figure 0 ces urfa .. The \" t"t' �_ . ' 1 5 .. F 7 f Igure plots 0 the . J' they are spun . a W3'" , ,_ In " VIC\\. an allemall\.:e abou'" . ' -_"bori ng" spheres-are not sh own. ) Note . that prob x-IS (The j: stalC" the "'·a aThe surfaces bruptly. enclose only . , d0 not drop to [ero .. . " ities - , " dens blilty the .... . ' Iy to be foun . it\1 " d Expone IS like ntial t";1 majority 0f Space where the electron ... s extend to infinity iO all cases. . . . . F'Igures 7. 15 and 7. 1 6 paint nice pIctures of the probabilitIes. but Sc verI u , feature\ are e.speciaJly noteworthy.

.

.

_

I . As always. the probabiJily density IS a diffuse cloud spread av er space. . . 2. The probability density e:ttends farther from the ong mJproton as 1/

increases.

( = 0 and the kinetic energy is solely radi al. density probability is spherically symmetric. With no orbi the tal mOlion. there is no hint of an orbital pla�e. In p states (both the

3. In the s state,s. where

2p

3p). three possible approximate orbits are suggested--one with motion roughly along the z-axis (m( = 0) and two roughly in the xy-plane (mt = ± I ). The d states, e = 2, suggest five and

possible orbits. adding two between the z-axis and the xy-plane (mf = ± J ) to the extremes of m f = 0 and me = ±2. (See also Figures 7. 1 3 and 7.14. ) On the other hand, as noted in Section 7.5 the probability should be spherically symmetric if we are igno ran of fill ' in which case each value of m would be equa lly likel f y. Fo chosen n and (, if the reader imagines y adding the probab ili ty de � �fl . nSlt for all values of 111(. a sphenc Jes . ally symmetnc average sho uld seem quite plausible. In all ( -::f:- 0 states, the pro bab ility density van ishe s at the origi n A panicle with angular mo mentum can never be fou . nd there.

;

4.

TABLE 7.6 Letter

Value of (

Spectroscopic notation ,

o

p

d

2

f

3

g 4

h

5

-

,.6 Tbo ttydlcP= A.- ..,

�2. 0. m

t2 . \ ' O)

tl. 1. ' I )

'-----�----) '"

,<

·O '� . ,

'L_ .

(3.0.m

(3. 1 . 0)

(�. I, .. I )

L \ ----� -----� ) __ V

'p

(3. 2. 0)

(3, '2, :t I )

(3,2. ±2)

\L------�V�--�) 3d

S. If we look at states of a fixed e but increasing n (from

1s to 2s to 3s.

or from 217 to 3p), we see that the number of radial anti nodes increases. Increasing the total energy but not the rotational energy implies greater radial energy, and thus more radial antinode!«. By the same token, for fixed n but increasing e (2s to 2p, or 3s to 3p

to 3d), the number of radial antinodes decreases, because increasing the rotational energy at a fixed total energy must decrease the radial

QuAntum ,MechanKi

n Alom e HydrOge

and lh IOIlS

.n lbret J)inJtII5

urfaces 0 . ure 1. 16 s fIg

( consUUlI

probability density in the

· hydrogen alOm

(2. 1 , 0)

(2. I. ± J )

V�----"

9

)'

(3,

(3, 1 , 0)

I, ± I )

--v�----�I '-_Jp

(3, 2, '" I )

(3, 2, 0)

(3, 2, :t2)

� -----v � -__ __ �

3d

anti nodes , may be thou ll e, with more radial ghl energy, Orbits of sma ard and ay tow aw tIOn mo fro ch mu m the with of as rather ellipticaJ, e ch and me whi are for h es bot slal n - I hand, nucleus. On the other ial ino rad anl de one and only e hav are wed, the largesl values allo its al a fixed radi us. (The ground Slale is more like circular orb with the "Iargesl allowed" angular momemu m obviously an exception, '

for n

7,7

=

I being 0.)

Radial Probability

only in the electron's radial whereab oUls, Quile oflen we are inleresled ple is this question : Typically h exam le A simp tion, direc of dless regar (7w far 10 d stale? Proba bili ty inteara is the electron from the proton in the groun .

'

tlng IS the key to any such calculation. Inlegra .

-36) � over all space gives a UOl. I tOtal

"

probability. so integrating o\'er a re'>1ricted range give.. the corre'\ponding

fracllon.

probability of finding particle between

T l

and T

2

:t:::

\Ioreover. assuming that T I and r2 differ by only dr. the probability dP of find­ ing the electron becomes

dP

� R'(r)

" dr

Thus. the radial probability P(r). a probability per unit radiCJ/ diSlOnct'. i..

P(r)

(7-38)

Note that integrating thiS from zero to mfinity must give I . as 11 would merely reproduce (7-36). The fact that the radial probability requi.res an "elttra" ,l with the square of the radial function R(r) IS sometimes confusing. but there IS a simple argu­ ment that (7-38) should include this factor. Imagine a situation 10 which the probabihty per IInit voillme IS constant-that is. R. 0. and are al\ constant. al least over a considerable portion of space surrounding the ongm. Findmg the particle somewhere 10 a thin spherical shel1 of radius 3 nm would be nine times more likely than finding it in a thin shell of radius I nm. simply because

the area of the larger shell is nine times as large. (Remember: The area of a sphere is propon.ional to ,-2.) Equation (7-38) agrees. If R were constant. then ? would give the appropriate "area factor" of mne. Figure 7.17 shows radial probabilities plotted on axes whose heights are proportional to the hydrogen atom energies . All but the ground state are degen­ erate, so there are multiple states at the higher energies. Note that all the'ie standing waves extend into the classically forbidden region. Also, as noted in Section 7.6. at any given n. smaller f implies less rotational energy and thus greater radial energy. resu lti ng in more anti nodes and a radial probability that extends farther from the origin. By contrast. the highest angular momentum siaies. ( = n - I , have only one antinode and do not extend as far from the ori­ gin. They are more like circular orbits at a single well-defined radius. Finally, note that Per) is 0 at the origin for s states. in which t = 0, despite the fact that

R(r) in such states is nonzero there (refer to Figure 7 . 1 5). While we could sim­

ply blame this on the r

2

In P(r). the underlying physics IS worth rel1eralmg.

Consider the ground state. From Table 7.4. we see that R(r) for the

I s state I� a

0 and reaches a maximum before falling off. The probabihty per unit volume does fall off monotonically. but the "amount of space" in all directions grows as r increases (the "area fac­ tor"), which makes; the probability of being found at a given radius actually increase for a while. Again we see that if the radial direction is the only con­

simple decaying exponential. but Per) starts at

cern, P(r) is the better measure.

.......

_

Z6I

264

Dimensions and Chapter 7 Quantum Mechanics in Three

the Hydrogen Atom

es in the hydrogen atom, Radial probabilities and energ i spond'mg Tad'lal a corre and value , adds an e (Note: EftCh h'19he' I 7 . 17 , Figure

function.)

2200

"

0

E,

3d 3,

3p

E,

···:::r·:::·:::::···· 2s

r········

,

.:::'... . .. .:" .....

-, .

2p

E,

. ........

....,.

For a given II, IOlVer I is smaller ro[;Jlional em::r!!
. . ... .. Higher I j, like a circular orbil at ,\ single radius. The 3d ha, only one ,mtinoJe.

1

I,

UCr) (X- r

EXAMPLE 7 , 6

An electron i � a hydrogen atom is i n a 3d stale. What is the most probable radius at . to find It? which

"N

)

The most probable radius is where the radial probability is maxim um. Looking up

RJ,2(r) in Table 7.4, we have P{r)

==

(

2V2.-' ?R12{r) = ,.1 3ao . /: e-r/ (300) Jj ' 27 V Sao

I

-

,

'

the fo find

we maxil11um.

$el lhe derivalive to O. The nonnahzation (:on$lant may be

. :: -dP(,) - .. (6,-5 + r6 1-2Qo) (O-2r:3oo dr

,go'''''

_

">(6 +

0'

.n ... fh e . ,,>,lf

-2 )(O-2rI3oo = 0

30 0

0 and r = -. but these are obviously minima. wht� P
. at oallon holds

IS O. The

r=

r

0

6+,

ees WI'th the 'fhis agr

2

-

3"0

= 0

�

r111<)<,1 proba,

blc

-

900

3d plot of Figure 7.17 .

= n - 1. the most nearly circular orbits. there IS an for which t . most probable distance from the proton. According to , Item to the pa

S 10 stale

Ortant imp e 7.4.

Tabl

In ge

neral .

ility obeys • the radial probab r Therelore

In. S IeIt as an

exercise to show that this function has a maximum given by lor

e = n - \

(7-39)

3d plots in Figure 7.17 all agree. Though I < n - 1 the Is. 2p. and Note thai handled. they follow the rule not so easily staleS are

mat

.

fairly well. We conclude

ately as the square of 11.. rbit radius increases approxIm

�

Per)

useful for finding simple probabilities. but it al...o allows ot only is late an expectation value of any function of r. As usual. we put the uS to calcu ',y to be averaged in the integral with the probability density and inte­

quanti

over al l gflne

possible values.

(HO)

7.7

EXAMPLE

r. , - 1. 'rPI"dr - 1. ' {p"oi)(�7)'5a('-"''''}r

Here,j{r' i�JU' 1

found I n Example With the nllhal pmbabdlty

Inside cover as nle integral I� giH�n on the Ie"!''' ThO'., \Ve ha\'c b =- ll.lao and ", •

7

r

In ['(ample c'(pedillion

..

8

_

--

(.3ao).1 (27 )25"6

7'

(JllO)' --

2

1.6, we ha\'t

f;: r"f'

b'dT ::::;:

mtlb'-+,

= lO.5a

0

Per)

7 6. \\c found the mo,' probable radiu, of a 3d electron 10 be 9"0 An d" value. on the other hand, " an a\cnlge 0r ra II we �ould fmd if Vt�

3d

, In slales. The plot of Figure 7.17 L repeJtedly looL.eJ for eledron\ In lias Its th e I at y rupt b o a fenmnaiCS ngm II while , it extend but ' 9" \1n£le m:uimum al 0 s infj. certainly appears 10 be the lar ger value OIIeI)' far the other \\li) The a..erage r.ldlUS

7.8

.

.

Hydrogenlike Atoms

Our ...o/ution of the hydrogen atom was based on one a.lI-important pie ce of rhe polenuaI energy in t'ormuhon: ' '

U(r)

=

---

Any applicalion in which the potential energy is of the same form wou ld � ol� low the same way. Hydrogenlike alOms are those with a single elect! On. Clearly. hydrogen IS hydrogenlike, but another example is singly ion. e helI U m. Normally. helium has twO electrons orbiting its two�proton nUCleu,z d s, If ' removed. the remaining one h as potentl'aJ energy ' ' one eIectTon IS

U(r) =

(2e)( e )

singly ionized helium

In general. if a single electron orbits a nuclear charge of +Ze the pOI ' enllal energy IS •

,

U(r) =

Ze'

one-electron a(Qm

' a H___.. ....... 1'tIu", aU our fQmlul.." for hYllrogen JPpl) to h.ydrot;enhke a.tllm, if we 'Impl)' f('rl� t"2, �here\'er it I' fount!, by Zr In punlcular, the eneqw le"eI, fllllow h\"lIYl equatilln (7· 12),

hydrogen lZ

-

I ): E" ..

hydrogenhke:

For t

-

I.HI eV (7.41 1 II',

E" ..

II- I . th� uppro\;lmate radii obey

(7·42) Bound-state energies are dee per by

22, and the orbit radii 'mailer by lIZ.

EXAMPLE 7 . 8

Wht\! are the ionization energy and the mOM probable mdius of the electron 11\ the ground state of He -+ '1

The energy in the ground state is given by ('7-41). with Z =- 2.

£1 =

22

- 1 3.6eV

\ .�

=

-54.4eV

An energy of 54.4 eV would be required to remove helium's second electron. In the n = I state. e can only be 0 (i.e., n - I), so ('7-42) is applicable. r\

=

2\ 1 (O.0529 nm)

= 0.026 nm

Singly ionized helium is considerably smaller than hydrogen.

Strange as it might seem, the hydrogen like. one-electron atom provides an invaluable simple model for understanding behaviors in multieleclron atoms, the topic of Chapter 8.

REAL·WORLD E X A M P L E Q U A N T U M C O N F I N E M E NT A N D TA I L O R I N G P H O T O N E N E RGY

F in semiconduclOfS, the fabric of much quantum dot research. an electron can effectively be freed to move about the solid material, but in the process. a free positive charge car­

rier, called a hole, is a1so created. These £wo attract each other just as the proton attracts the electron in hydrogen. forming something like a loosely bound "atom"--<:a11ed an exciton-whose energy and radius are important factors in quantum dot applications. In particular, electrons bound in a quantum dot won't ex.hibit the tightly controllable

"7

energi� \\e lk<.rre---quantwn eonIlIwml'nt-lf the d(l"� dimen�ions an:: harger -. the ortut radiu, of the ('''1100, While (;ondltJOn� � of course dlffrrent In a whd hlIIia'. lal than in free 'p3I.:e, propertlr.. or the C:'I.(:iton can be calculated with only a r", _. oltJOn) of hydrogen alom Ihrory. For one thing, dlc:lec.:tnc polarization lend!. to dilule IIId. \\eaken the Coulomb allr.k:tJon bet\\�n elC\:lrOn and hole, and simply di\,lwna: the dwrgc Z by the matcnal'.. dlcloxtnc.: con�tant accounts for this reasonably well. Anocber ra..:IOr, !holt both the clC\:trun and its "nudeu,," the hole. movc. is handled rather �lI) by replacing the elet:tron ma.�\ \',l!h a reduced efTeclJ\'c mass, also a pro�rty of the \Cmiconduc.:lJng material. The Orbit radlU� 01 the exc.:llon In common semiconductors. fortunately. is mUch larger than t)pical atomic dlmen�lom. but il is still submicroscopic. which explain' why quantum dot technology took off only after significant advances in Our abihtlt\ to manlpul:ne materiah on a very small scale.

Applying th. Physics The dielectric constant K and reduced effective mass for a particular semiconductor roughly 10 and 0, Imtle<.lmn' re�pectively. (a) Estimate the dimensions necessar y for a "dot" of this material to exhlbll quantum confinemenl. (b) The very presence of the hole, actually an empl)' slate buried far "below" the electron energy. means that the electron can Jump down and produce a pholon (see Real-World Example: The Quantum Well Laser in Chapter .5). In this particular semiconductor. !he Jump I� 1,7 gi\ing a far.red photon wa\,elength of 730 nm. Confining It to a smail dot forces the electron to an energy higher by the dot's ground-state energy. as de:picted in Figure 7.18. Roughly how small a dot would give a 450 nm blue photon? are

cV.

-10' (a) Dimensions at least as ..mall as the exciton's orbit radius are needed This and m by the: comes from equation (7-42). replacing the umt value of Z by reduced effecU\·e mass.

11K

100 Bohr radiI. 5.3 nm. is indeed an order of magnitude larger than real atoms. (b) A good model of confinement 10 three dimensions is the cubic Infinite well.

whose ground-state energy. from equation (7-9). is 3( 'Tr"!.f.2/2mL2). We need to

"squeeze" the electron higher by the difference between the photon energies.

Figur. 7,18 A small dot rorces an electron to higher energy. tA '

_

Unconfined e:lectron

6£

= =-

=

,,,(_1- _ __1 ) ( Ablut

1240ev . nm

1

450 nm

_

1.06 eV

1 730 nm

)

Equating this 10 the ground-stale energy gl'r'es

1 .06 X 1.6

x

10 1 9 J =

::::)

Squee:zed 10 hlghe:r energy 10 a quantum dol

A�

3

'Tr2( 1 .055

2 (0. 1 x L = 3.3 n m

X 10.34 J . s)2

9. 1 1 X 10 I I kg)L2

The calculation assume!!. or course, a cube. wherea!! quantum dots can be of many dlfferenl shape!>. It also overlooks the independent behavior of hole energies wilen they too are confined. but II is still a good eSllmate. A<; we see. control or very small dimensions i� essenllal lO tailoring pholon energies.

7 9 A Solution E·em1ned

7.9 A Solution Examined

o

}-la\'lng skipped the detailed solutions giving us the function.. 8(8) and R(r) in Tobles 7.3 and 7.4. II would be easy to lose sight of the fact that they do ..atisfy the ScnrOdinger equation. Here we demonstrate this fact for a reprc\cntative wave function. then we study the relationship between the function and the . . e\eClfOO'S whereabouts conSider the 2p state for which (n. ( mc) = (2. I . + I ). case. test our For have we 7.4. and From Tables 7.3

,

�

In Section 7.5. we verified that the compic)( exponential eimt

(

as;n e a sin 8 � a8

)-

C sin-, 8 sin 0 J:

i sin 8

m

In the far-left term. we have one derivative already acting on sin 6, with the other poised

to act on everything to its right. Thus,

-

a sin 6-(sin 6 cos 6) ae

sin 6 (cos2 8 - sin2 6 )

C sin3

-

6

1,

mi sin 6

0

C sin3 6 1. mz sin 6

- 2sin28) - sin3 6 In,? sin 6 sin 6 - 2sin3 6 - sin3 6 m� sin 8

sin 8( I

C

C

1.

1,

state has me = + 1, so the sin 6 terms cancel on the right and the far left. leaving only the sin36 terms on both sides. The equation holds only if C = -2. so our e = I function satisfies (7-26) for C = -C(f + l), jusl as it should. The radial part of our function, re-rl2ao, should satisfy (7-3 \ ). Does it?

Our chosen 2p

- ( ) -2m ", I d , d -- r ,.2 dr dr

,_ r/...., o + "'( (

_

re

..

+ I)

2m,2

r. !Do

_

re

,

(7-43)

270

nsions and the:

DIme Mco;:hanics In Th:ree ChIIpt., 1 Quanlum

Hydrogen Atom

re-,1200 ») d( r' ( � =£. -'-dr ? dr

far-left tenn. Consider the

2m

.

iate &' W intermed O_ � a ,e becomes '

reader can easily work OU, . steps that the

� .

or

. n unctlo nole Ihal [he f

re ) _ _I_ e' � £ ) + I r, ' l(l + --'--+ 2. � ( r47Teo 2mr2 4a5 o,. a 2m e := I , ) r,' e' , " (mao � -;: 8ma5 1, £ (47T£o)h2/me2.

7-43). this into ( As we reinsert have we Thus, ls. term and cance _

;"

,:z

_

has Because OUf stale leaving

with the tWO terms

,:z in

the denominator cancel ,

I

-

-

By definition. aD

/ 2a -, • appears in each

parentheses is O. and What so the term in

=

remai ns is

an n = SIale says it should be for what eq uation (7- 12) lly The energy is just does CorrespO.nd rea on cti resentative wave fun rep OUf if see w let us .No 7. J 5 and 7.16. Figures to

2

EXAMPLE 7.9

;

I. +

[ ) state, Calculate the probabilit that is in the (2. A hydrogen atom electron spective of radius : ( b) be ween hin 30° of the A)'-plane, irre wit (a) nd fou be uld wo it hin 30° of the xy-pl
�t

bability for the region w' h I' n , or !'IT, We seek the pro (a) The xy-plane is 8 = 90° find a probability over a restricted region' e 300, or J,-rr, of i1l', As always, 10 this case (7-37), e nonnaliz3tion integral, in merely restricl the appropriat 7.3 gives le nalizalion comlanl from Tab Inserting 8(8) with its non probability :==

:==

J." !'(Jl;8," ) J. - sin 8

' 21T

sin 8de

"./1

3

_

2".13

4 "./3

. 8de SIO-'

:==

- = 0'688

3 11

-

4 12

1.10 Phocon EmiIIion:

RuIes .xl ....

271

we have considered, coverinl 1rf3 < 8 < 2m3 ' the " solid angle" A� II happen , space, angularly speaking. (II is 2'fr steradians out of half is , of J. . _ lue ¥'. all Va ) Our probability is considCl1l.bly greater than 112. d 4TT. of SO space ' . l for 1111 or a lota more likely to be IS found nearer the _ty-plane electron the see thai I , m, = ::t I diaanroJru of agreemen! with the t in SO we is 0"the -. ILXI', which than 6. 1 . Is and 7 Figures . ,

7

""

•

Table 7.4 into the restricted radial probability integral , g R(r) from InsertIn 36). we have

17-

final result involves somewhal ledious integration by parts.) As

. g the . . IOblal. nln . . probable radiUS for thiS state IS 4ao' Figure 7.15 section 7.7. the most noted I.n . hal Indeed '
�%:d in that vicinity.

and the angular ranges are restricted is just ity if both the radial . tWO fractions.

The probabil I') educt of the the pr

probability

= 0.688 x 0.662 = 0.455

consider a mther small region of space. the probability is Although we . nearly 50%

Emission: Rules and Rates 7 . 1 0 photon

rJ D

photon-producing transitions from one quantum . 7 3 we noted that In Section probabilities . Tbe question of why a particle ot r are governed by stal�.to a. n y" state would choose to leave it and enter a transitional phase to . ll a r . '·n a sia d, In
�um

� h�

� �I

fl U

tional

.

photon, for te generates a

.

.

it explains why some transitions are highly

and rapid, while other seemingly plausible transitions essentially do Cavore at all. nOI occur

Transitions Allowed silion from one quantum energy state to another, a panicle occupies In a tran . . lenow, a wave functlon ' ,s lerna combmatton af
n"

�

po" p�'

V

A

N

C

E

D

272

Chapter 7

Quanlum Mechanlc� in Three Dimensions and the Hydrogen Atom

Because the charge density'� lime dependence .is our main focus here' w ...e n ly. we crudely repres et(i part. According to include each stale's temporal enl th " , e ' transitional state by the followmg combmatlon.

'" IronSI1JOn (r' I) � 'I'I (r,/) + 'I',(r,/) (7-44)

'

proponions where hi" and "f' are the initial ,and final states, In general, the . of I "Ictty, we pick a simple eq the two stales would vary with time," but for Simp . ' u al sum of initial and final. The probabthty densHy IS

"':( r,l)'"f (r, I) =

(tPi ( r)e-i£,I!" + tPr( r)e-i£rl!" f (tPj ( r )e-i£,tI"

+ !/If(r)e-iErl!'' )

(7'45)

frequency w = (Ej - �f)/fi, w !he complex exponentials have angular hich is . . energy fiw IS Ej - Er ThIs answ whose a photon generate Just nght to ers the simple question of how it is even possible to generate a photon--charge d n sity does 1I0t oscillate in a stationary state-but to quantify the rate at Wh� c energy is lost, we need to consider the actual process of generating elec 0magnetic radiation. Some of the physics we now introduce comes not from quantum rnech an· . ' ics but from c1assl�al ele�tromagnetlsm-namelY, the pow�r radiated by an . . . charge dlstflbut]on. It turns out that the most efficient radiator osclilatmg is an in which positive and oscillating electric dipole, depicted in Figure nega· ucmg w �veI'k ocattons. �ro I e oscill ations II I ' d ' ap y s ' d' tive charges peno Ica � . of electromagnetic field. The tmportant quantltallve result IS

�h

7 .19,

Figure 7.1 9 An oscillating dipole is an antenna emitting radiation.

power radiated by an oscillating electric dipole =

(7-46)

For point charges + q and -q separated by distance d, the electric di oje moment vector is by definition p = qd, the direction being from the neg iv charge toward the positive. In equation (7-46). p is the dipole moment's am P ltude. and (;) is its oscillation frequency. TD be an efficient radiator, a hydrogen atom should have an oscill atin dipole moment. but it isn't simply two point charges. Whereas its POSitiv charge, the proton. is compact. its negative charge is an electron cloud . Still the dipole moment has an expectation value, and this is what we use for With charges of +e al the origin and -e at the point r. the dipole mome t

!

+q sin wr

- q sin wr

t

Elecltic P dipole antenna

/ ! :

�

'.'O""*-r.�:I.Wa __ m

would be t'f Multiplying by the probability of finding the electron at r and Integrating over aU r then gives -

.

p�

J

(7-41\

(-er)\'i' (r. r)\'dV

all �pac�

Now. if a function changes sign when r is replaced by - f . we o;.ay that it ioo, of odd parity. A function of even parity is unchanged by the replacement. U '" t!ot even or odd parity. its square is even parity. and becau� f it..elf to;. of integrand is of odd parity. Thus, the integral over an !otpace would be odd. the 0, as is the integral of any odd function over a one-dimensional interval sym­ n\etric about the origin. There would be no net dipole moment. A crucial ques­ tion. then. is whether 'V has allY parity. (A function need not be of either parity. e.g . \ + x.) Let us consider the effect on hydrogen atom wave functions of replacing the position vector r by -I"" . In spherical polar coordinates, this involves replac­ ing cP with (
eir/ler

\\

.

-

8

R(r)

r

r.

R(r)0(8)(4»

p

�

(

-eRe e,,,,lh

J

)

r
all �p3ce

(7-48)

The product I/Ir*(r)l/Ij(r) will be odd and the dipole moment accordingly nonzero only if the initial and final states are of different parity. Thus. becau<;e parity depends on C. we conclude that e must change by an odd number. Actually, even meeting this restriction does not guarantee that an electric dipole transition can occur. Further restrictions can be deduced from the char­ acter of the wave functions. but another crucial factor is the very nature of the

(discussed 10 Chapler 8). which PreVen phOlon 1�lf. in particular ILS spin ts II a cenam amount of angular momentum AIto. from carrymg away more than j,( to z I and �m{ to 0 or :.:!: I . Tht' l, gelher, the various re!.triction� hmit >e known as selection

art

rules.

EXAMPLE 7 . ' 0

(3, 1, + I). Which downWard

(n. (, nit) A h)'drogen 3p electron IS in the \tnle rules" n ctio sele the siuons are forbidden by

lraQ.

ed, as is a transition to the 2$ stale. for bolh haV Is stale i.. allow e , to n 2p �tate. regardless of mc' is forbidden 110n = O. But n tr'.lnsl

A transition to the

(' ::: 0 and me

because Af would be O.

energy has great symmetry, selection rules are Any time the pote ntial ecule. the ioterpar , t1cle instance. in a diatomic mol liable to crop up. For age separation . F r I aver m a l cer a at lS a(Qn g ' ar e I atm foree keeps the aSC·11 g ulsion than fo llra asymmetric. stronger for rep is e forc the . ions c oscillat ns . it behaves esse Olial as . But for small oscill atio a lion (cL Figure or. and by arguments very simi ar to illat osc nic mo har ple symmetric sim occur only between sta dipole transitions can then tes those above. electric . I. for which 6.11 == 3: .

�

5.17 ).

Transition Rate Equation

it (7-46) is the key to estimating how long

�

�

\

should take an electron to

another. drop from one state to energy photon photon

time transition l i me

=

her the ole moment or the hig The larger the dip n to occur. expect the transitio

=

_"ti"m::e,-photon

=

(7-49) cc-pci' 2w 3 l 2eoc3h

frequency, the more rapidly We

EXAMPLE 7 . 1 1

th ( ,{', r) the ground state from e n m a photon as it drops \0 its em m ato en rog A hyd of the electric dipole moment the expectation value (2. 1 ,0) stale. (a) Calculate n time (b) Esti mate the transitio ""

,

IU 10

(11.e,me) = ( 1 ,0,0). Inserting dV in spheri

.

the final state is (a) In equation s gives cal polar coordinate

(7-48),

p

=

_,

Re

& f .•,./'

r�;.o.o(r)·p,.l .o(r)"'

aIL space

�

in .drd.d

1.\0 PhoIon Emis.wn: Rules ud' " \ Table.. 1-(,(111

1. 3 1\nd 1.4. the wa"!! function'\, Il�

.p;.o.o(r).v1.1.0(r) ,-.:

. 0ul the wnllng r

=

(

I

-

'e· r. 1Io

a�< ! "

) � � ( V -�. 4"7f

I (laO)J;1

r ,ector in II� Cartesian componcm lonn

+ " )- +

i

+

\...t3�()� r

,

r' �\ �"" ) co.. 8

rev!!al.. it �.at "Im ' pIIficatl�ln.

+ r co.. tI i -.urv i\"e nt ... pon for the e othe lT r ·.CO l two Contain sin .J. t. .h , c '. u .f.. an '+' or ..,.." . dU ..." ,+" .• 0 I\1) _ l'Cre re in the ;ntegra1-'mtegrall.llg from 0 II) 1\ no e 217" give:. 0. Moreover bec h w e cls .k au' . 'the In k.' th real illS e real part of the who , ion!. are nct le fu expr essio c n , \ merely replace.. the . I.).Erl" with a cos;ne. M a..." 'I' \\a lOg thoIS replacemen e'(pOnentlaI e t, in..erting the wave roOlPIc'{ nnd what remains of r. and then factOring out constants \.\oe have A ' .-::!-- co, ( , co, 0 i) e". , ,-" "'. o, 0 " p �

[unc"oo�

2j/21rCl04 0

t

= r sin 9 cos ..p i

nin B \inq,i

( :\ J ,

)

�

on 'pace

"0

ral over cP gives just 211". and the separate r and 8 integral'\, are

1M loteg _

"

,Ae-lr/2IJodr = )(� 0

ThUS.

)0r c

4' ( /�orJ 3

os2 8 sin 8 d8

o

"" �

( '; E'\ _

= -0.14 lIOfCOS --';- )Z

is sensible on two accounts. First. the 0.74ao suggests that the charge our re�uIt comparable to a Bohr radius, which is ble. Second. the dipole sep�a l is the z-axis. Referring to Figures 7 . I Sreasona and 7 . 1 6, it is reasonable thai osc�l\a . along een the spherically symmetric ground betw state and the (II.c'mt) oSC1J\al1on the z-directi in on. ated should produce elong a dipole moment along (2, .0) state.

�::

=

1

� d�

To use (bl

.

(7-49)' we need w. The two energies are - 1 3.6 eV and - 13.6 eV/2·•

.

ency IS n -3.4 eV. so the photo frequ

W =

( 1 3.6

- 3.4) X

\.6 X

10-19 J

\ .OSS X 10 14 j . s

=

�

moment is I ude of the dipole The ampI't

P

�

9 0.74(0.052

transition time SO that the

:.-.

X 10-9 m)( \ .6 X 10-19 C) � 6.3

X l O-w C ' m

is approximately

/ 10-12 C' N ' m')(3 X 10' m/s)'( \ .055 X 10-34 J . 5) 2(8 85 X ' (6.3 X 10 " C · m) ' ( \ .55 X lO" s-' )

l

"

2 os

. . known to occur in about I .S ns, so our estimate is very good. Typi· . ' n IS ThI. s transItio transitions occur in the range of I to 10 ns. cal electri.C d'IpOIe

171

In concJu"lOn. \I.e nOIt: th.u tr.J.n..ltlllnl> forbidde� 3\ elet.:trlc dipole tran.�. lion' rna} occur b} other proct:....t:... Charge may. lor rn'tance. O\cllIate

ma�nt'ti(" dipole. where a current

bI

loop penodlcally ..\l.ap.. t.hreetlon... But \lII;h an "anlenna" i\ much le\\ efficient than an electric dipole. \0 If'oln\ilrons ou;umng by lhl\ pnxe�.. � much .. lower, typically b} a ractor of 10'. O�llIiJlion.. e"'en more complrcated are po..\ible (e.g electric quadrapolel. but Ihesf" tend to be yet more InelTicient and ..10\\ Tran\ltion.. that can <J..:l,:ut only by !.IO\\< procc::...e::. cau\e cenarn atomiC \tate" to be unu\ually long h'ed .•

These are known as metastable \talel> and are Imponam to the operation of I la�r, di«:ussed r n Chapter 9.

PROGResS A N D A P P L I CATIONS

Figure 7.20 AI hIgh II, '0 many Matt:)

High,n States: Rydberg Atoms AlOm\ In which an t'1ci,:lron 1\ t'�l,:lted 10 \ery hIgh quantum Slale�. typicall),

lIon could 'Ipell

40 nnd hIgher. art: common lopic, or di)cussion

n

1 I I

art: av:uloble thol lhe proper combina_

oul a word !

nowada}.. Known a\ R)dbc'l: aloms. the), arise in many Ltetl\e d�n, 01 phY'K\ re\earch. While Ihe t'lectrons In gTrWII(J-IfIltt' mulilelectron Jlom� inter3CI ::.trongly wilh

eat:h other. ir e"Clled to a very high le\"el abo\e the ground stall'. an ekctron

In tim' alom

orblh a� Ihou!!h in a hydro­

genlrle ;ltom. wrlh the nudeus and relatl..ely compact

�mdlnder ol lhe electron, comlllutlng an ilppro..imately poinlille po\llne co�. Wllh we3k.ly

bound electron..

orbitIng \0 far rrom the nudeu\. Rydberg mom!. are:

t:leCln.I'tallCdlly malleable dnd thu, well suited to �Iudie!> In many Mea'l or sludy. Rydberg Slate::.

or diamagnell\ffi

or panlCular Inte�!>1 dre Iho\e in w.hi(h t and nit

the mnxirnum allowed (both n

art:

I). for in these !>Iates.

the elet:tron\ orbit rnO\1 re,emble\ a clJ�\ieal circle Figu� 7.16).

(cf

Owing 10 the \Iow me3n� by which they can

..hed ener8y. the�e circular \tales of hIgh " tend 10 have

Once the subject or theoretical IInlihydrogen atOnl� moving slow enough 10

\cry long

Antimatter in the Lab

energy ph)'''lc\. Funhel"nlOre. le\'eh al

debate alone.

electron In occupy

con�iderable quantitie..

Irfetl me... with imponant 3pplications in high­ high " are closely �paced. and ellch level ha.. a large degenertlcy. allowing the :r

wide rilnge or very Intere::.!ing

combillllt/Oll.f (wpcrpo\ition\) or \t
the particular It hu\ been ,hown thaI an cleclron rn \orne

combrn:1I10n\ or \Iate\ would behave very much like a stable orbllrng Cllh\lcaJ panicle ralher than a dlffu::.c wave. The comiderJble Oexlbihty allowed a R}'dberg wa'e packel " well lllustr.lIed

III

Figure 7.20. which depiclS how

Ihe probabllrty den"'ly mIght.

rn prrnciple. be tailored 10 �pell OUI ,} word Similar. Ir le\\ fanciful. wale packet sculpling may aid the de\elopment or quantum compullng.

allow prcci\e rnea..urcment� lire now being produced in III

the

Illb.

All particle.. h3\'e

antIparticles. who�c mass is Ihe same but whose charge is opposite. The po..rtJvel)' ch,lrged po�itron. Introduced in Section

3.5.

1\

the anti panicle or the eleclron. and the

proton 's antJplItlicle " the negatively charged antiproton. Th31 a

po..itron �hould be able to orbit an antiproton in

es\entially the \ame way an electron orbits a proton has long been 3\sumed. but a mlljor hurdle in bringrng It aboul is that antiparticle.. are very hard to hold. Should an antI panicle meet ils partIcle. the two quidly 3nnihil3te. RedUCing Ih� odd, thill antimatter will run Into m3tter. with 11\ dangcrou'l eleclmn\ 3nd prolon\. demands e.."eme \-acuum condition::. 3nd containment ,ria maller­

in the ATRAP

where the abIlity 10 control combination::. or muluple states

rree eleclromagnetic field.. Scienti\1\

is croci31

(Antlhydrogen Trap CollaboratIon) (eam at CERN hold

p'

_A

arbi,w qwifttum nun�r t. whll.:h &(1\� UlC magOiluili' \\1 t� • nplM mommtum. arkl the pn� lpal �u:mtum numlxr ... \lth�h I"vern!> meq:). Ihn� val� Afe gl.. en h) ..r

�l4lTr'IJ}:'hl ll:Z

I,

I I. t L

"( I + 1\ " .., A

"',

"

I

•

I. :!. '. .

0, I . �,

II, ·� I . ��.

.."

.I

l1· 12)

n-.n) n':��l

h., ul,:h �t (",f,mi). thl'n: I' it uRique I,\,a\'c fum:lion ..l�), Hllwe\'a. the energy of R... ';( , H. � }

r

the- hyuf\lg<'n

tltm�h up, approuchtng a 'Ingle l imitmg \alue, lt. . the �hl.m-wa\ elength llr the \(mg-wa\ciength

tad ct• ,eo(''' th.lt the -.erie, Ilmlt lll.: �'u". anJ v.hat i!l. it ..., .. hydrogen', altol,\,l.'d energle, thLIt IC;ld, 10 thi�

�

/tit

c

.. a\� [un",tlOn 1:0. not llllowell for It p l"llO�tant \ anltl

box. What phy...ic, ;\c.:l"ounts for thi� difference'

8. Section 7.5 argue, thut k nowing ull

' c 10,

three comp

of L would violate the uncerta inty pri nciple,

the J.... t.Lnl'... fr\ltll the (lrig inlpr\ltlln. IIml 8. th... polar angle \\ ith

� �

uhen KnuWi

its magnilUde and one component does. not. WhlLt magmtude and '1\'0 components') W knowing

( ....ed Clln· wh:lt...\ cr 'pn:lal a' .... " \·,tal:!h...heJ I:!) e'temally Lmp ... dedn,ln v.-oulJ nll)'t Iikel)' be ULILnn, rh... nu.lLu... .It \.\hll·h th l\lUnU i n....r\·a'c' w ith n appm"m:ltt'l) a........onli ng tll

�

second dem;Lti\'e are proportional. But �.l 0 i� a t'on,t,lnt lind IS allowed. v.herea, �U b . "­

.\OlI liS

d"'pend, ,'nl) l'n II, Cc," ...cquc- ntl )'. the

dt'l!... ocr. ....y �ml,\, a... n! L The pnlbahdit) t.leo...it)' in the hydrogen (tlom \ane... v.lth r.

Is

·1

7. \t:lthemoltic.:ally. equ.l1iull (1-22) I� the �ame diff --.. e\{u;\tlon ,1" w� hnd for a partlr.:ie in a btlX-tht

,;lr)HI,III,l8)
__him

�

n(ln'! �s the IOtimte well h.we scocs IlInus") 6. -\ ga, C;U\ be t\IO colu to athNI:! B__lmcr "trlClI li thi... llho t"le' h1r lhe p"...",hcn "cries? (SI!'C' Figllrt Ftlf the Lyman ,e ne,'.' E 'plain

liS

lJUld l be left any freedom at nil. and if so, do you lhink. 1t would be enough to satisfy the uncertainly prln�'ip\c"

9. Knowing preci ...eiy all componentl> of II nOnll!'ru L

would violate the uncertainty principle, but kll()\\'ln I that L. is preci,ely :,em doc' nol. Why not?

r\

ljlLilnhlm-mcch__nil"al treatment llt the h)'Jrogen ilium

e'pl:un, Lll:!"'l!r\.ation, that tannot be UnlkNO(xl da,,�ically. ml"iuJi ng the alom':-. ..I.lbility and the h l ...crete wayelengths of hghl thlll It emit'

Indicates advanced questions

I I.

Conceptual Questions I. What j, a quantum numt'lcr. and how docl; I t ari�e'J 2.

Some degencr.tcie.. are ea�y tll unde"tand on the basi" of

'ymmetry in the physical "Itu:ltion. Othe" are �urprising. or "accidental," In the ,tute... given in

10.

Table 7,1. which

d�gener.ll"ies. II any. would you call accidental, and why?

J, Con\llier a 2D infinite well whose side.. are of unequal length . (a) Sketh the probabi lity density-as density of �halilOg-for the ground 'tate. (b) There are two likely

choice... for the next lowc:-.t energy. Sketch the probabil. ity den...it)" nnd explain how you Imow that this must be the next lowest energy. (Foeu .. on the qualitative idea,

avoiding unnecessary reference to calculations,)

-'. Cla""ically, what happens when a moving object has a head-on elastic collision with a stationary object of

exactly equal mas..? What if it strikes an object of

,mailer mruos'J Of larger mass? How do these ideas relate to Rutherford's conclusion about the nature of the atom?

S. In hydrogen\, characteristic spectra. each series-the

(Hint: For C = 0 ,tate... the momentum vector p IS radial.)

What are Ihe dimension.. of the spherical harmonll:� o I. un ct> (tP) given 111 Table 7.3? What are ,''. K;" U '"1

'"f

dimensions of the RII,( ( ,.) given in Table 7.4. and \I.'h}'

What are the dimensions of Per). and why? E:\plain to your friend, who hal> juS! learned ahout \1111..

pie one·dimensional standing waves on a stri ng n�ed It:. ends, why hydrogen's electron has onl), certatl) �n II ... gi es, and why, for some of those energies, the ele.;1rQn can still be in different S\(Ites,

12. A particle is trapped in a spherical infinite well, Tht potential energy is 0 for

r < a and inflllite for r :;,. Q, Which. if any, quantization conditions would yOu expect it to share with hydrogen. and why?

= 3 states a!'. representative. explain the relationship between the complexity-numbel\ of nodes and antinodes--of hydrogen's standing wave\ ln

13. Taking the /I

the radial direction and their complexity in the angular direction al a g iven value of II. Is it a direct or inverse relationship, and why?

14, At heart, momentum conservation is related to the

universe being "transiationa11), invariant," meaning thai it is the same if you shift your coordinales to the

right or leff. Angular momentum relates to rotational invariance. Use these ideas to explain at least som� of the differences between the physical properties quan.

Lyman, the Balmer, and so on-has a "series limit,"

tized in the cubic three-dimensional box versus the

where the wa\'elengths at one end of the series tend to

hydrogen atom.

_

I�

.. tI particle to a bool'. _hat ill ,iiI' kX .. u ne-JU1 �I\'II", !

the me.lrun, ,., ft,' '\p:'\:I!I,'alh, ....hal ,j,,�s �no.....ma . " wll us" 1M \\h:tl l I� IIlC!U1'''' ,.,

" t,., .. h,.h\',C'n

awm" Id hll a h)dn1t:\-n 1.1,'Ill, '"holt I' tho: n�.,,\U\1lo ,>f f1 01 '"t"

_"'I-lQftIO \)I� an)

t.. ' ,

$. \ S

v

Eltercises

�hR'liIl\Jft ffoIuauun "

2M\f, . .'

{Ito)

�U,IIh.ln \7."i) "' ...1'(' [II."II,'C, tne �n('ral m'lthcnm\I�'ul '1llulLon ","uld he til

nll"llInh ( 1M 1 \1, dn

Sh,.w \hal lhl.. Iu\\!';\llm ca\llmt � 0 al two point' Thi<.

m.u.c, ,\ un unacc('plablc ;;,,!uI'Im tof the infini te well. \It'lce it ,',Inm,t be I.:OnI1llUOu\ w llh tne wnw fum;llon,

t2l1.)·I�.

19. \'e.nly Ihal the <;oluilon.. gwen in equation" (1-61 �ati!>fy ,lIflerenlilil equation, (7-S) a� well as the �uired

th� energie)' correlopond"

An e\(:ctron i, trapped in .. quantum dot, in which it i& (onfined to a very .:,mull regIOn in all three di mensions,

If the lowe�l-energy lmmition is to produce a photon of 450 nm wavelcngth, what should be the width of the well (assumed cubic)? 22. Con�ider a cubic 3D i nfi nite wei\. (3) How many differ­ ent wave functions hove the <;.\I\1e energy [IS Ihe one for

",. II�)

�

(5. \ , I)? (b) Inlo how many differ­ ent energy level� would Ihis level split if the length of one side were increa!>ed by 5%� (c) Make a !>cale dia­ gram, loimilar to Figure '1.3, illustratmg the energy splitting of the previou'lly degenerate wave funclions,

(d) Is Ihere any degeneracy left? If so, how might it be "destfoyed"1 23, An electron is lrapped In a cubic 3D infinite we\!, In the states (II... "I' II�) = (a) {2. I . I), (b) ( 1 , 2, 1), and (c) (1. 1, 2), what IS the probability of finding the electron in

)0 l"

t) <. , < I,(I < �· < L

I�thC'rwiM'

.. I.)') be \Iuhl,je Ihil> well" ,

'llitJlmu .. hlf ft..,) aml .llt") In'ldC' th(' wC'II'� Ar(' Ci llnu C, pi,)"ti\i(', negll\tve, l,r lero') hllJ'f\"lIIl!. ul"rmpmllC' condillon,. lioli Ihe \l\low«i \ialue .. �If C, and C,. (0) H\lW Illan), lIIuC�ndenl quantum numbers JUt: lherd d , l hnd the all�lwC'd eMrJ,el> f I') Arc Ihl'1'(' cnerlpe� r�n which Ihere " not a unique Cl)ft'1:'pondmg wavc lun..:llon')

,h(lW thilt the I.:omcl nomlah/iltlon con..lnnt i,

gte� po\,>lble? (b) To how many differe", statcs do these

•.

What lun!';li,'n, w�luld be I:tc!,;('pll1� le ,tundmg,wBve

.".� \, \', .:)

boundary condillons, An eleclron " confined 10 a cubic 3D mfinite well 1 nlll on a 'ide, (a) What al'(' the Ihree lowest difft'rr>nr ener­

, Iln\l ('

II\linlle- \\('\1.

\\'\\<11 'lmuld If. \) lind

ou",dc Ine walls, whlth [\1'(' 0, UI. hlf the cubic ]D infinite "'ell w,we function

which (II..

.,.,(\. \'\ - !t\lR\\'l.

the-n 111\ llim¥ �\ J't, l�\ ).) Call 1hco liepanbon

11, II lncn'n,lant (',

21.

(;I\('n Ihlll l' I a ,'un"llIInl. �f'ar.tco \'anablcoll by tt)1II1/. II. l\llu\t\ln �lt lht funn

s.ct1on 7,2

20.

s

u.. ...... lD ....... w.8: In two dlltlllmaklN.. the

_ I \ In t� h,J.ro�('n &!UIl\ \."-:c\lt to) ('it-.ltIC dll"',k radlall\'n" (b( l,tcltll\( ,'I the �\ ,,, ...1 " II unu,ulIlh ,.hol'1 \If I,mg" I..Dl...n \11 \'0(' unu,u

..

41.. 0 S I S L)'P

dlftenllCft In It. three NMIha.

,.. t...... tbc Ilan" I" .n .!,

-\

179

2S,

Con,iul'r " l;utUt: ]0 mfmite well llf �,de lenglh L.

There
Section 7,3 26. Cla!>�icall)', an orblllllg charged pdrtie/e

radialb elec­ tromagnelic energy, and for an electron in atomic dimemlom. il would lead to collap-.e in considerabt)' Ie,,), than the wink of an eye. (a) By equating the cen­ lripelal and Coulomb force!.. show that fo' a da'lsical

eharge -� of mil!>!> m held in CIrcular orbit by it� attrac­ tion 10 II fixed char ge +� ...the following relationship hold�: w

'"

u'

<; '!2jV41'r£rP' (b) Electromagncli,m

lelh us that a charge who..e aCl.:eleration i, Gl radiate..

280

Chapt.r 7

po....er P

::

QlI�nlum Me.:hanlC� in Thn:e Dimcn\IOM and the Hydrogen Atom

r1l2/&:fF'. Shuw thnt thi' can ah.o

30. Calculate the " ,erie, limit'" of the Lyman \Cri� of spec. lr.ll line!<.. This i� defined a') the shofte�t wa\elength �

be e'pre"cJ in tenm of the orbit raJlu;.. as

rbJ(9fl'lT1s-t1mJf*).

sible of II photon emitted in a transition from a higher

P ... Then calculate Ihe energy 10,1 per orbit in terms of r b, multiplying this power by the period

pan (al lo eliminate ClJ. (c) In ..ueh a classical orbit.

the tOlal me(hanical energy is half the potential energy

l� Section 4.4). or E"rhlt

:::

-e2181T£or. Calculate the

change in energy per change in From thi� and the energy

determine the change in

r:

dEOfb,/dr.

10\1 per orbit from part (b).

r per orbit and evaluate it for a

typical orbit radiu� of 10· 10 m. Would the electron's radiu� change much in a �ingle orbit? (d) Argue that

dividing dEOfb.ldr by P and multiplying by tlr gives the lime required for

r to change by dr. Then. sum these

times for all rudii from r,"n,�l to a final radius of O.

Evaluate your result for 'uullnl =

10- to m. (One

eventually be moving relativi�tically.) a bound panicle are those of a typical standing wave,

2UII. where L i;. the length of its home. Given that

,\ = hlp. we would have p = ,,/rI2L, and the kinetic

energy, p'll2m, would thus be n'2h2/8mLl. These are

actually the correct infinite well energies, for the argu­

0

(inside the well) and L is strictly constant. But it is a pretty good guide to how the energies should go in

(a) To how high an energy level must the eleClIon� be excited"?

(b) Energy is absorbed in collisions with other pani­ cles. As!<.ume that after absorbing energy in one collision. an electron jumps down through lower levels so rapidly that it is in the ground stale befoft

L allowed the wave should be

energy musl be available in a single collision')

(c) If such energetic collisions are to be effected sim­ ply by heating the gas until the average kinetic energy equals the desired upward energy jump . what temperature would be required? (This explains why heating is an impractical way to

observe the hydrogen spectrum. Instead. the atoms are ionized by strong electric fields. as is the air \\hen a static electric spark passes through.)

roughly the region classically allowed to the particle,

2 state absorbs a photon. (a) What should be the photon wavelength to cause the

which depends on the "height" of the total energy

electron to jump to an

other cases. The length

E relative to the potential energy U (cf. Figure 7.4). The " wall" is the classical turning point. where there is no

kinetic energy left:

E = U. Treating it as essentially a

one-dimensional (radial) problem. apply these argu­ ments to the hydrogen atom potential energy

(7-10).

Find the location r of the classical turning point in terms of E. use twice this distance for L (the electron

32. A hydrogen atom in an /I = "

= 4 state? (b) What wave­

length photons might be emitted. by the atom follOWing this absorption? 33. To conserve momentum. an atom emitting a photon

must recoil. meaning that not all of the energy made available in the downward jump goes to the photon. (a) Find a hydrogen atom's recoil energy when it emit� a photon in a II =

2 to /I

=

I transition. (NOle: The

can be on both on sides of the origin), and from Ihis

calculation is easiest to carry out if it is assumed that

obtain an expression for the expected average kinetic

the photon carnes essentially all the transition energy,

energies in terms of E. For the aventge potential. use its

which thus determines its momentum. The result

value at half the distance from the origin to the turning

justifies the assumption.) (b) What fraction of the

point. again in terms of E. Then write out the expected

transition energy is the recoil energy?

average total energy and solve for £. What do you

obtain for the quantized energies?

28. Show that of hydrogen's spectral series�Lyman,

Balmer, Paschen, and so on-only the four Balmer

29.

Seclion 1.3. We wish to cause hydrogen gas to gloy..

with its characteristic visible colors.

If an electron is to be raised to the level found 10 part la), how much

27, In general, we might say that the wavelengths allowed

ment is perfectly valid when the potential energy is

four Balmer series lines noted al the beginning of

another collision occurs.

limitation of thi" e�timate is thaI the electron would

,\ =

= I fmal lcvel. (Nolr: In . Figure 1.5, the speclrdl lines of the seriC\ . cro....d together" at the ..haft-wavelength end of the serie\.) 31. The only yisible spectral lines of hydrogen are the initial energy le\el to the "I

T � "17rfw and using the formula from

lines of Section 7.3 are in visible range

(400-700

nm).

34. The space between two parallel metal plates is fIlled with an element in a gaseous stale. Electrons leave one plate at negligible speed and are accelerated toward the other by a potential difference plates. As !l

� V applied between the

V is increased from O. the electron current

(a) What are the initial and final energy levels for the

increases more or less linearly. but when

lhird (i.e., third-iongesl wavelength) line in the Paschen

4.9 V. the current drops precipitously. From nearly 0, it

of this line.

4.9 V. (a) How can the presence of the gas explain thc!.e

series? (See Figure 7.5.) (b) Determine the wavelength

� V reaches

builds again roughly linearly as .l V is increased beyond

t\b!;cI'''\IOIh � tM Tn ... Ilh ('11\11'- II rathcr ' lu, ... " h�hl"

..hrn .l\' ("

��\", 4 Q \'.

s.ctiof\ 1 5

.I�. ' 11\;,ut\o:rnJtl....\I

,k"

td \\ h,11 III..... .111.... '1\ hI "'" lhailoul II.llmulhai "" Iu· \1111\ f • hOI" It .... \'II .kllllt\! . \\Imp.lnenl "'t An»ulllJ 11\\111",'11\\1111 11I.1 Iholt 11\ , .. I\I�

l) " n...!MI,'''' \n) Sh,'.... th,II Ihl' 'Imph

�',Uln,>I. 1\1\'("1 I\'(,U 'I\ll,,'lhh ",h\'n 1\

\"ml'he, i\ 1\)11\1,1

lin I

\.I!ti.:'

"I

m;h', .11\\1 Ih\,' ""'I.I�': \,11 1

" I\,,\\o C" 1\\\1)

1 ,hl'uIJ tIC" Ihe

Ih('

,um

�

I}

�� II"

Nt·\ � I)t.!.\'.� I )/('>, ..h,,,,..

..hl'ul,1 h\1'1... til\' 1\\cnt�(' hut the

" 1\\

I Wh� Ih" I' 1\\1\

\1t'lI·cll'/im·a \,I\lll' ni l � IcaIU\I\" Ilw l'i�llnlu, "PPf\,\\\'h ,d.:rrcl1

41,

hI III S"dill,, 7,' )

In 'NII.· nJ". to. Ihl' "\....r,llIlr hn the "lUM'\! \11 Ihc ;'1\Ij!ulur 11I11lllCIIIIUll ,� ,hllWfl l\l tlI.:-

What

'L " (

:",''lInPlnlCnL� 01 .U\gulllr momentum ,... 1\ J'O',ible tl>
er,i1 one that need not

,..

II, ,11\\1\\ 1'.1 \ ,llu.'" 1!t\\,\('J ll� II\(' I\Umt1C"l \,It \'1\1"1:''''

I (t ... \ )/1:

.'8- A partldc Ilrb,llllg due til all a\truetl\ C cenlrul lol\:c

,

l It\l,'J\ ,t-. ,.,1-1,11'('

Ih,,\ h� Ihe,,: M�uml'nh. U\l' ,\\.:ntJ:l· 11\ I '

,;,:..:wr 1I1;ll..e \\" Ih lh... ;::-IIX"')

It \..g . 111/..

• •

ll\\�\\ UIl" \l1I\

""I 'ttltl' of Iht' hydl\lgen .UOIll

1.00 X 10

I�

�I .. I I\c\ ;'I\I,\· thcr� l\'u\1\ ,.. 1\\\ i'1\.'1<",,'11 IlU,¢o,;linn U\ '1',\\'l', lhl: ,I"·HI)!.'·' \11 / ,1 \11\101 ..h\\\lhl be 11\,' "I\I" . un,\ till' ..11m Ill ul\ U\ICI' ,11\I\lhl �,\'� Ihl' tl\'l'fll�C 01 1.-

Wl\J\ p(\�'lble lIngh;:, [\lIght the angular momen\lItn

h.ob angular 1I\0111<::ntUI\\

n, + I ,

ftI(

m\('gcr " U\

,..

,,-1.11,:.1 hI .m�ul,u \1l1'1\l(,l\llltn Jue' h ' \" \\. �'il\\nni ,UI\\ Kldl,u,1 1 '\1\"':111 .;"n t1C" '\111\",\ It, hlll1'.... ... U I ,'.111 ',I�\· " n II\\I� th"lIt \;,hll'\ fill II, .... \Wlt'

tflr ,IN-Iut ttw r"' " rho:- 'Iml'k�\ Ilrl'l"\l,tdl i, 1" ,'\In ,Iokr ell - 0 .lnd cP - .! 11 lb) \I 0 \H'I"(' O. c\\llall,-.,n l"·�l \hlulJ ,it) �HIlI'I) th"l lh(' \':0.:,)1,,\ dl'mil"' ... ,'I 1Sl\�1 I' 0 ..\rgllc Ihlll \hl � I,X' lead, 1" 3 1,hY'Io.:nll) Ima\'...o:-pl,lt'o1c ",)\Ull\I\1, (",,,,0:-1'\ HI the ,t'M.'o"I"'N� 0\ (%\(tP) bt-mg 1,:l11l,llIlU. Whll'h " e(lI CreU by Ih... /tIl 0 col'l' " \ " I\ulmn, (I-l.\). ,16. Vm,e lh,l\ Il lhe iUlWtill1\ (,j, 1I0Il 1, 10 mecl l"l'lI '11 11,,'lhh .\hen 1/1 dlil1\�e� by :).11", \ 0 IllU,t tIC lin .\"7. \n deo.:lron

MI.A r

.... \ 'I1UI'"I1\-,\ "1'1,,,,,,.h h' 1M "u�'!I\'n \11 hI,\\o f

I... . "'hId'

(� + 8,"

...,unp.-..nenl nfthit.

p th'n\ the'

hUlh"l)I 1..1111Ulll aoo lhe Iknml,\,lft ,,' Itw ''fIOhlt'>n tor I'. \In.\ p�, .lIllUC th,,' I� 1'IlC'fll"" h" I ,.. III ,\ �.

'\ltUt!llI"l 1'1 Iho: IlI ll\luth,\\ C'qU,'u\,'n

I'��.!) I' "'I�) 'rl'hc' \, ,,,-,,,

thI Rc-...·;t.lI lhol\ I . • r

What " II' \\.I\('kn�lh·'

11-

\'iO\;lIe Ihe ...nlUt)thnc�' "ond,tlOn,

CIC

/I

,m

...11\ II "

)

ilfl

1I,e Ih... III n,'"ule C\luat Lnn \1, 11)) ih j � O �)

and thal m \!1I.:t co\'cr� mit unly complex exponentin\-.. bul al�o, \'dtll ,ultable rerJelinitlon� of multlp!tc:lti\c

C"�(l(11

':\,In,tant..., '1I1e [lnll co'!ne:

Section 7 6 ,

l a) Show thut the complex square of thi!; function h nOI, 10 general, independent of . (b) Whnt conditions 1I\U�t

be mel by A undJor 11 for the probability densily to be rOlaliorllIlly 'ymmelric-UHlt i" independent 01 4>"

(Thi� hLghllght... another rea..on, be...ides the;r being 01 .."ell-delincd L.:, why we like our preferred Solulions.)

40. Here we pur-.ue the more rigorous approach to the

cla,m that the property qunntiled according 10 1111 is chain rule,

ax a f1y (I + -- + = 04> 01/1 a.1 <14> oy

iI -

u�e the lran...formation... given in Table 7.2 to show mat

""

Pxplia:"Iy verify tlmt Ihe " mple I'llllo,;t\ll!\ �11Ll

R(r)

A t' II,

he mudc 10 '1I11\l'y rudiul clj\1t\\ioll n')\), IInl! ,n ..o

doing, dcnm1\'tr:lle whal ,Is an�wl.�r momentum Ilnll energy mu..t he, 44. t low many d,\l�rent 1d ,laiC' are there" What phy!-..ieul

propeny (II� uppme\! to quantum number) di�tingui�he... Ihem, and whal Jiflercnl values mfly this property

L:,

(3) Starting with a "Iraighlforward application of the

o a -v + x­ ax OJ'

43.

45_

a...,ume? An c\ea:tnm i, in ..Ill /I

'" �late lIt the hydrogen alom.

(a) Wh.u '" 't� c1\crgy"l (h) What propert\C� Oe!<>idc\

energy lIrc quantI/cd. and what "lIlue... might be found

it the..e properties were t�, � mea...ured'�

46, In Table 7.5, the pilt\cm that develop!-.. with I'

tn...re a!-..\flg

.. uggC..,I, that Ihe number of dillerent �et� of «(, '",)

values lor a g,i,'en em:rgy

IeI'd I' i.. n2

Prove this

mathematically by ..,ummiflg the aHowed value.. of III, lor ;l given f ovt'r the gLven II

allowed value... of f

for a

·7. Show lhal the aIlf:Ulat nunnllliLalion con�tanl in � J!ecl,_ Table 7,J for the ca!>e (t, "'I' = ( I . (» i� � .... Show thal lhe nonnaliLaliun con\tam \115i J2'11" !!i\en "!:o! In Tabk 7.J forlhe anllUlatparh ofthe ( = 2,Inl WoII\'e funelion i� CO�I. 4'. Verify the co�lne ot the nonnahlalion con�lanl of the 2p radial wan: funelion, giH'n in Table 7.-4,[1, '0

54, U,in!! the fum:llon, �i ...en in Table 7.04, ...·crify tbat for Ihe more-':Jn:ulolr electron orb;" in hydrogen (i.e . ( - /I - 1 l. lhe radial probability i� of the fonn

Show that Ihe mo�t probable radius i� given by , rmm' pn1Nble = ".°0

55. For ,Iale, when: l " - I . the radial probability a\\ume, the general form given in ExercIse 54 The propoolonalllY constilnt that normnlize� this radial probabJllty J\ �1"'l:n In Exerci�e 64, (a) Show that Ihe I.')(.pcctiltion value of the hydrogen atom potential energy j, eXlKtly twice the tOlal energy. (It turns OUI th3t lhi� holds no mailer what ( may be.) (b) Argue that the expcctalion value of the ki netic energy must be the ""'

•

SO. A hydm!!en i1!t1m l'il.'etrun i, in a 2/, 'Wle. If no experiment ha\ ba:n dQIlt' to e\Iabli'h /I :·eomJX>nenl of angular momentum, ttk' alUm I' equally likely to be found WIth any llll(lwed \·alu(,' of L,. Show that il the probabilIty dcn�i� lies for lhe'ol' dillerent p!)\\ibJc ,tale\ are added (with equJ.1 weightin!:), Ifle re,ult i, rndependent of botl! ¢ MId (J. 5J, A wave tundlOn wilfl J. nonrnfinnc w;l.\elength-howe\cr appl'IJ)(imale it might be-ha\ nOllLero momentum and Ihu\ non/em kineti.: enc'l!). E,·en :l 'ing/c "bump" h,t\ "ml'ti\." enc:f!!l- In eltiler Col.\e. we nn \3y !lUI thc function hOI, kincli.:: Clk.''l1)" hec:lu\c it ha\ cUl"\alUre-\l ,econd demalne. Ind(,'Cd. 1he !.anetu; energy operator in any WOrnJnolle \}�tem imohc_\ a ,econd deri\ati\e. The onl)' functi�ln Without kinetic energy would be a ,tr.ught line. A\ a \p:t;ial ca\e, ,hl\ Include, a con�U!nL whi�'h OJ:ly be thllu�ht ot a' a fUnctIOn with an infinite wa\elength By I�l('king at the cun·atun: In 'he uppmpnau' dim,mioll(.{). an\wer the following: For a given n, I� the kinetic energy '>Old) (a) '.JdJa1 JO the \tatc of I()\\.e,t t-tha( I�, ( "'" 0; and (b) rotational in the \Iate of high('.\( (-milt j�, ( '" II - I ? 52. We h,l\·e noted thai for a given energy. a,> I increase� Ihe mallOn " more like a circle oIt a comtant r.Jdius. with ,

the rotational cnelllY im:rea\ing 3S the radinl energy cor· n:\pondingly dcaea,\{'\ Bul l\ the mdial k.inelic energy o Illr the large\1 � valuc, l Calculate the ralio of c)(.pccta· '

tinn value\. rJdial energy to rotational energy. for the

(n.

f.

111/ )

(2. I , "" I ) \tale. U�e Ihe operator, KE"" KE""

'"";Jilr<) ( '") 2m

10'1(1 + I ) 2m,2

ra�

whll:h ....e deduce fwm cquallOn (7.

.10).

VCr)

(b) Show that. d:lssically. an eleClron with this energy

\houl d not be oble 10 get farther Ihan :Wo from the pro­ ton. (c) Whal 15 the probabilily of the electron being found in the cla\\ic:llly forbidden region?

58. (a) What I� Ihe expectalion value of the distance from Ihe protOn 01 an eleclron in a 3p \tatc? (b) How does [hj� compilre with the e.'Cpeclation value in the 3d Slate, calculated in Example 7. n Di�cus" any difference� 59. Imagine IWO cli.l\\ical charges of - q each bound 10 a .

central charge of +'1. One -q charge

b in a circular

orbil of ri.ldiu\ R i.lbout it� +q ch:lrge. The other oscil­

Jate� in an extreme elhp\e, e�",enlially a straIght line from ih +q charge oul to a minimum distance rma.,.

The two orbit, h..,\"e the ",arne energy. (a) Show

Ihal rm., = 2R. (bl Considenng the time spem at each orbit radlu\. in .... hi(:h orbit i� the -q charge farther from ih -+-q charge on :lverage'J

60_ Con,,,.!er an elet:lron in Ihe �round \late of a hydrogen

atom_ (a) Cakulale the e�pectalion \ alue of its potential

energy. (b) What i\ the e1li.pectation value of Its kinetiC "

5J. An elcctron i, in the .1d 'tate of a hydrogen 310m_ The mO\1 probable d,\Wnce of Ihe electron from the prown

9(/0. What ,\ the probabilily Ihat the electron would

be found Pclwecn

57_ Con\ider J.n electron in the ground state of a hydrogen alom. (a) Skellh plols of E and on the same axe�

energy ' I Hili'· What 1\ the c"(pecl
Section 7.7

j,

negati...c of the IOlat energy. 56. For a hydrogen atom In the ground slate. determine (a) the mo�t prob:lble location al w hi ch 10 find the electron and (bl the mo�t probable radius at which 10 find the elcuron (cl Comment on the relationship between your :lmwe� in pan� (a) and (b).

!lau and

/O(/n'.'

IOlill energy·}) 61. b Ihe potential energy ot an eleclTon in a hydrogen J.lom well defined·' b the lonctic energy well defined' Ju\(ify your an\wt::r\_ (You need not actually calculate uncertJlntie\.)

E� 213

62. Calculate the uncertninlie� in r for the 2_1- and 2p �tate.. u�ing the fomlUla

What in,ight doe.. the difference belween these two uncertainlie� convey aboul the nature of the corre­ sponding orbits? 63· The kinetic energy of hydrogen atom wave functions for which ( is ih minimum value of 0 is alt radial. Thi� is the case for the Is and 2� �wte!>. The 2p state has some rOlalional kinetic energy and �ome radial. Show that for very large II. the ..tate� of largest allowed C have essentially no radial kinetic energy. Exercise 55 notes Ihat Ihe expectation value of the kinetic energy (includ­ ing both rotational and radial) equal!. the magnitude of the total energy. Compare this magnitude with the rota­ tional energy alonc. L2/2111�. as�uming thalli is large, that C is as large as it can be. and that r =: lI!flO. 6-1. For the more circular orbits. ( "" 11 - I and

,

be gi\en by KE ,2/thn:or. (b) According (0 equation (7·]0). the rotatIOnal kinelie energy in hydrogen is •

," -/( 1

1

+

1 )/2mr. Ofcou....e. r i' not well defined for . " doud,� but by u'ing r iii ",]uu' argue that the condition that not e\cced ,.. i, �a,onable_ Section 7.8

Whi<.:h elcf:tmn tram.lion' in 'ingly ionized helium yield pholon.. in the 450-500 nm (-blue) ponion of the vi.. ible range. and whojl are their wa,·elengths') 67. Doubly ionited lithium. U!·. ab\orb!> a photon and Jurnp� from the ground "ale to ii' n = 2 level. What wa!> the wavelength 01 Ihe photon? 68. Roughly. how doe\ the <;i/c of II triply ioniLed beryllium ion compare \.\ilh hydrogen? 66.

Section 7.9

69. For an c:le<.:lron in the (fl. l. III,) "" (2. O. 0) Siale n i a hydrogen tHOrn, (a) write the ..olulion of the time-independent Sr.:hrOdinger equation. and (b) verify' expliCitly that it i, a �ofU(ion wHh the expected angular momentum and energy'. 70. An electron in a hydrogen atom i!> in the (II, (. mil (2. I. 0) �tate. (a) Calculate the probability thiil it would be found within 60' of the :-a\is. irreo;�tive of radius. (NOlI': Thh means within 6(P of either the +:- or -.::­ direction. wlid angle:. totaling 2rr ster.Jdians. or �halfof �pace:·) (b) Cojku/ate the probability that il would be found betv.een r = 2110 and r "" oon' irrespecth-e of angle_ (c) What is the probability that it would be found .... ith.in 60� of the .:-axi� WId betv.t.-cn r = 2110 and r = 600? 71. CaJculOlle the probability that the electron in a hydrogen atom would be found within 30' of the .ry-pl:lOe, irre'pectillc of T3dLU�. for (3) l := O. mj = 0; (b) ( = r. 1/1/ .., =- r ; and (e) l = 2. 111( = =2. Cd) A� angular momentum increa.o;e,. what happens 10 the orbit' whose :-eomponenl.' of angular momentum are the maximum allo....ed"l =

(3) Show that the coefficient that nonnalizes this probability is

( ) 2

/1(10

"'''

I (2/1)!

(b) Show that the expectatLon value of the radius is given by +

�)a 2

'

and the uncertainty by

.1 r =

110" v

R'4 -

2

+ -

(c) What happen" 10 the ralio .1rfr in the limit of large n? I� thi .. I:Lrge-" limit whal would be expecled cla�sically')

65. Here we in\,c\tigate the link between " and f. reflected in equation 0-33). (a) Show that if 01 clas�ica.l point charge .... ere held in a ..:m:ular orbit about II fixed point charge by the Coulomb fun:c, ih I..mctLc energy would

Section 7. ' 0 . .lm, = = I eorre"poocb to a 72. Show thai a trnmition ....here dipole momenl in the .l)·-plane. v.hire .lm, =0 COITC­ �pond� to a momenla/ong the .::-a"(i�. (You need consider only the c,b-parts of!/I, and !/Ie' which are of the form �4) 73. Verify for the angular �olu[jon� 9( /JIIl>(,p) of Table 7..1 that replacing d> wLth c,b + rr and 8 ....IIh rr - 9 gi�e� the ,ame Junelion \\-hen ( i, even and the negal,,'e of the function when l 1\ odd 7�. A partLr.:ular vibrating diatumic mulecule may be treated 3.\ a �emple harmoni..: o<;o;ill3tOl'"_ Show that a

_ 0.1&

,

� � " _ o. " ! ,,, " " _!D c I " "' _ _ "''' ''''''' '''-

2 Is! a- .. " l .... ...." ., .. .. 0 -- - - - .,. - .... . C r• • h "' . ...... ...... ... ..... . .

...... ... ..., ..;p. 01 _ 10 ,.,

"

"

..... .

lot w.. .. dIIrIr ., (.) � dw '._.., ,,.. .. ... p,c.ktd u.ae 10 me p-.t ..... . dII.. .... iI ...,., by cl«UK .... ,... .... (b) WbII .. dIr wlYI'kn,dl of I� 1

m

..... -,

,.. eu:u_ ... .-cnc .. _hi p aod 011J'Mk' die ___ ... tor . h)'dmpn 810m ck\:lron mOl", .. aIsrctri.: .... .,.".,bllfI I'rum tht (:II, I "', I (2 I, ... H "' 10 dw ".......", IbIk Commt'nI 0l'I 1M

NIaUonIbtp PI cfw ,..,11 to thaI .D Elample 7.1 1 t7, C� lIw .kIccrk tti".w IhIlInt'ftC p 800 C'AhllWl'e dw rrul.Uoa IJnw (or . hydruJCo ilium tl«lr,,'1 lIlitkin¥ .. .tedric ltipoW lnIIIlIilKIII (rllm IN If!. , . 11l, J • _'. �.I)I 10 &hi f2. I, 0) !I&iUt'

'71.

WhIa appl)'m, qWlJllum m«hanu."l.. \U' uf�n I.'tlfk.'cn­

Iller on .e.ael tMI "Iu&hf) III> "tIfthonormoJl .. Tht m.un P'l'nl ll WI I( we ,,"alullr " pmhthihl) mltgral " H::r all .,..." 01., '., ur u( "'J.�J' lAC gel I .urt,urpns, In,I),). bul" we- n81ua1t Ilkh an mlC,rill Ill! til. -"'1 tIt �J-.1 � p. 0 Thill fg�n" III hr lrue ttlr.:.lll Ihe Y'1fl'IlI wJwI'C' WI' h.1I\C IlINI.IIIC1J "r "Iwlll akmnl ... o( w8Vf I'utl('lon!l. ll' B , Ihe partlde In a �'\' lhe hlrmonK: I)KIII.,m InJ ltat h�Jr\lg-en ..111m) fh . mlC,,.IIft, U\o�.al1I 'pace, l'k,,,,, Ihal t'\prL'��um (?,oW) 111m" nl.lnnalll(d unless a 'J('lilf 1.1 I' induJed "" IIh cht p,uba�I"I�

j

Comprehensive Exercises 1'9, lllt 1/1. /.II �Iak'-t��, 'I..I�' III "' hll'h m, - ll--h'l\ 1ll1" ! 01',111 pnlbu�IIiI�' ,/cn,l!) ...hll1.!1 lhc ,NI\I', .)11.1 '0 11 I' ollen f'('t('rrnl lll a.\ " 21'. 'Illlt' Ttl .dlllil ih prllO-lhlhl)

di:nltily 10 "lid; tlUI In tl.h...r '" J�'. .Intl tllU, till'l!Il"'e I'-In'

1M,j.� ki'kh \11 nkll«'ulat I'Illlkhng \1 Ith (Ilha ,lWnh, Ifl ,

""Imil' c/('I.:lfUII 111.1.\' /1"Uflk.' i\ "illt' IUl1dltl11 lhal I' UJl

.!Illlt'hnu� 1....lmhm,lllIm LII nUlltlpk' ".lll· fundllm, 11pt.'n 10

+1 + "':',1

it On!.' su\"h hytmJ ,1.1It''' I, lilt' 'UlIl t/I�! I· IM'/r'. 81.'\.'lIu!< Ilk' .sdmiJln�t'r l"lu.llion I' a 'inc,if Jif'....n::-ntmJ l"\juIIllLln, .. ,um llf �tllulI{ln' II Ilh Ihl' ",Ime ('nctR\ j, ... §lllulilln wllh Ihat ('nt'�� AI'II. nllmu/i/alilln l'tlfhWJb ma� � IgnllmJ in lhe /t11/1l11 II\� 4Ut" lllllb I "

Iii)

Wnle Ihls """H' llIn,'!llln anJ I" pR1hahihl) Jen-

sit) in tl'nn, til r.

Ihl

0. imJ I/J. Il\.! Ihi' Fult'r fi1mlUia

'" slmphJ., )\lur I\',ull,)

In whll."h tlf Iht' fllllt'" IIlg \Ia), dflC' Ihi, 'Iall"

tiiller fnml Ih pOln, 11 1."" "'!, 1.+ I .\OJ tP2_ 1 I' and fRlm Ih� 21'" �Iull': EnL'!}!) , RaJra.l lkpendl'nce of

t" hi}

... � deaslIY" A.,w.dtfMA .... ot .. �1iI)' decw,y" nu. "* I unto rderred 10 IU the- 1p.. \\'by" How ml&hl IIIo e prodlk� .IIi Jp. "'.lik'''

_ c.1.lf\Nder l'a11 puuc:b thai c'penent.'C .IIi mUluat lOI\1t

loren Tbr ..-I.uk.1 C\lUoIhon 01 rnot-. "201. ' "', . ancJ 1(11' p.rude 2 .. f, . JI'"1' wkno the Jt-c mean� a lime deln'� '

hid DO nltmal

(or putid(' "

I

•• ,

Show � Ihni: are "Iun "knl 10

"

.�

- I:llnlolJlnl

onO

'...1 - ."--a;1Io

....� ".. . lm,' 1

,', .. ,. ..oJ

m lm�

mr

+ fIIJ

In lllht'r \\I'flh. l� mulllln I:..n tlc: an.. lyu:J In '''''0

Pll'�-(:�: Ih.: �L'nl.:r lit rna" mollon .1I I.'on,I
\c1u..:Lh

HflJ I� n:l.dlh' IIwmm. 001 IIl ll'ml'. of .... lml'-ranld; l"\juall\ln ""h':I'l- lhal P;lflKk 1''(fX'nen�'I'''lh(' mulual IMl"l' antJ hob rhl' "reJu>':I'J m".., p 81. l \Crl'I"C />in tJl''':U''':'' IhL' IJea "r l'C\Iu':L'tI 111.:1\, \\ hl'n I"ll oh)t.'>.:b mll\': unJer Ih(' mnuence of lhelr mulu;IJ /111\.'(,' alonL'. \\e I:an Ire.1I Ihe "/11111'" motmn ..., "

it Ilnl"-I",Inll"ll" ..),Iem of m""J-l "' 1m/lint " m,l. , >:l'ounl ror ihe .-\mnn� (1Iht'r Ihmg'. Ihl .. ;1110"'" u, III I I....:! Ih ..t lhl' nudt'u" 111 ;1 hyJrog('nhke alom l)n'l pc::-r­ ledl)- "
11t'l,IL'd III 01,1"

m, and �h
(.

orhH�
,Int.! t:h;l;gL' + 7t· By apprnprime ,ub'IIiUllOfl\ mill /ilrmula, l!l\cn m lhL' chaplcr, �h,'\\ Ihtll (Il) the Ill.t" III r

i.IIV\\t'J l'nt'Q:ic� are (7�J-l/III)EI/1I1. where Er j� the hyt.!rogt::: n !!rtluncl 'I,ttt:::, tint.! (11) the "Bohr radiu�" for

Ihl' �) ,tem 1\ (mIZp'Wt). wher\' ((0

I' tht: hydrogen

Bohr r:IJHI'

82, [: \Crl'I'1" S I Ohlillfl' ftmllUl;I' for hydrog.:nhl..e atom� in which tht.' nudclI' I' not a!,>,umeu inllnlte. '" In the rhapIl.'r, Ilul " of m..,.. ttl I , I:.-hile 111-; I' the mass orthe (.rhlung negallie charge, (a) Wh;.)1 pen:entage t'rror is

lntnxlun:d 10 Ihe hyt.!rogL'n grounJ-..t::"e energy by

a"ul\llOg th
Ihe pttltrln in the nucleu ... makrng the lIudeu... Iwice ru. ma�"lc. Tal..ing nude:!r ma" inlo aCl'ount, by what

/'lCrcen l do Ihe ground-..ttllc energic, of hydrogen and

lIeutl"oum dltle,-I

83. E\i'n:I\C 8 1 obtaim formula" for hyd rogenlike ::ltom� in " hich the nucleu, i" not as\umed inlinite, as in the

Ib:. � ....

\�. k1' I� III ma� M, ..-.tuk "'J lIthe- m.a..... I.f tIw OI't!tllnll nl")l::ih,.. ' \:h.lr'Jt In 1,,,,InIO\um. an I:"k..:ln>n •

crl'I'" i Ii.ul,k' J'l"'\tl\'("

cbIu'1:" , 1l>4 III h",tfllit!:". but 1'Ift('

\Jll bo.� � \� IhI: '-1.1\\\ ' , ... th.11 1,1 the \'I("<:\n\(\- .. ItnlQ l'*" p,....

fOC'f)i'

I" 1l111lk;IKIlI ,aluc� I"t lho:' In,un..l'''hll(

oJ Hnhr r.l�hu,' III J'lhlt,,,01um

J.L 1.'-llM-I..::..II). 11 "'oU e'ro:�·I('\,\ 11\;11 lin \,ft\'I\"� dc,;ltl."

'A'\l\,1J emil Ilhh.II'�ln III Ih". '.1111C h�"cn�') ii' II" Olt-,I 1I\'\Ilk·n..·, \\c h.1\t' " \II:n nl>lch,," II" ,,\>h..cncJ til Ih�' IImll "l 1. 1 1)1(' qu.mlu1II numl:l\'r.. l),.....lio ,I \Oo" ll.. III til" ..:.1'1."" \itl Slu\\" Ih.1t Iht' I.hllll.n (n('j�) h'f II\( '1Il,IUc" I"" ",t-k ... n"'l):� .lt11l1� .11 til... "knI t! C'n�I" 1.1 Ih... h�Jh\!;...n elll'tJu:.. " 'l"-IJi-'. \\ h,ll' thai lI'r Ih.' ,malle..t IlImp ,It Ihc ··Iugh-II cm1"' I" lIfJlnl. \\hc!\.'" III I' h) \h\l�...n·.. }1.WUlIl.hl,II(, C'IlCT�) I/hl 1\1 ,hll" IIMI Ih... J.n}1.ulaf wl{\(lIy 1,1 ,I \t-I l''>t' t' d.I"I"Jl l...\ml ..:h"r�... hdJ In mhll llPI'ut u r"cti

",m\ dl>ll�'" by Ihc ('mlk'mh ttlr..:e I' �'\l·n b) l \'f'�i·'h1't·llmr {d (1" ...l1 lhlll r _ /11 uti. i, \h,� \tI • In!!u\;u rl'\.�\ll'n..:)' e'II!(II IH lhe lllill1t1l�Ull'il,lnlp pholtl\l Irn{u...nr) al l.'Hhl:r �LlI.I 1,1 hy\ll'\lgco', allowc\! encfg'�'·)

s.c. Ik e'pcl'\Ullun ,,,lUI! I.llh�· c\ed"'"" ).,I11C\1..: �ncrg) in Iht' hydftl��1\ };I"unti 'lilll: 1'1111111, Ihe ""'�IIIIIIIIt' ,'I the 1I1t,11 en�fgy (,ee I',el\: ...e flO). What mu', be Ihe \\o,l1th 1.1 II ,,:l,Ib,� 1I1tinll'" well. ill IeI'm, Ill' "0' for II, �round ,IJlc h) hale Ih., ,amc cnt:'rgy'l

86. �f'C''':lrill llllc" urt' IUlly due In lw,1 ellcd�. Doppler bnlJlknillg >Inti Ihe unr.:�n'\lnly prull;illle, The rdJII\e e e ",,,e!cngth l1ue to the flr'l ellel:t ( .. ) " gl\en by E,cl\:"(' Z,57

\Mt;IIIOn

HI

.lA

A

\i )'/"flTfm

c

",here T I' the lempcrmurc ot Ihe ,>ample lmd III Ihe light. The Vllriation due 10 Ihe wcond cHel'l (�cc F.xen.:bc 4.72) t" given by

m.l" of Ihe parllc1c� e1ll11lmg lhe

\

_

1(111) �a. w1lh I" nunu,,\lm urtN.t 1Wbu. � .

rmhtol"___ �II" lOl l m aNS ,,� IMlUmum. or IPheboG.

1\10 unlt'i .. fal. Wtwn ... tbole minimum '" mul· mum radII. II� I'MIIU' ", lit ,,"1W"Ioe, noli chaftll,q. MI '� rilJla' �U�lu; t'MlV) "O, .no.!. \I ""�,tc eneIJ)' I ('Ilurch "".II,,"�I hunl "" ."'11:,,1 mn:hanl\:' rob.lioaa' ,,'Il('Il) " ¥I\CIl h\ " �/. ""here I I", lhc mumcnl uf U'ICI" llll, ",hl\,t! hI( . "'''Int .. """,," Iii. !IoImpl) ",,.2 ,.) 1bc �" '''\l'n, '1'Ot'C'\J ,1.1 f'('lIh1:"hu" I� f'I !�� 104 mil. ('111";\1 \3\\' Ih II.I\v;uhu fl\l\n\('l\Ium 1M \lenh that the ,um of Ih," filil\ 1\;lllIllIa\ ,""I('l\lla\ ('nt'lJ) lind n.t.tlunal t'l'IC'raY 111'1.'

(,,\11;1\ .II \"C'flhC'ht'lI 1111,1 "phdl"n

,Hl'ftIf'mltf'r'

\1I�tlh" tl\t>mC',,\um I' n'n"f'''N \ (d ('1I1I;ul.lt Iht '1,11\1 ,,( Ihe l/-,a\U,l\l\llllll ,,,'tentl,l.\ C1\('I)I' 01"" nltll.lI\lml.l \'II':: \�� ",h,'n Ihe ,llt'
'HI\\lI\ ' �\ll lI lhe ..:I11lI!,'t hu,1 thl!' \lIllie Illllil ent",)' hut \k'..:nt>r\\ iI ":11'\:111\1\ I'rlllt, III '" IUI\ 1,"It".. wllul\,l it ,Ifhit, \11111 h�I\\ \\'\'1,1.1 .h ""lIul,1t I\Itll\l(I\\II11I t,;,lmpun:' wilh Ih(' \;,I\uc ,.\ pall Ia)" \('1 N.l'l;,Ill' �,I�II tlh"",,,,,llun, lo the .11"'11)\1 (1\ \.. 111.::11,' ('IICI!:), III 11)11(\11,1.('11 de",I(\1II u�ih III' thc ';\lI1t 1\ t'
�

\.\here �I i' lhe typil:ul lransilion time. (a) Suppo,c Ihe hydrogen 111 II .,tar has. a lemperature of 5 X IOJ K. Compare Ihe broadening of lhese two ellet:h for the fir.,l \me In Ihe Balmer "eries :\ _ II, 2), A.,�ume a lransition lime (i.e.. ", of 10 � �. Which effect ... more important? (b) Unl1cr whul condltlon( ..) mighllhe Olher effect predominatc? 87,

A comet of 1014 kg mass describe� a very elliplKal orbil aboul a star of mass A Gra\'italional Analog:

[IV) Wilh the further dcfl1liliull g(.l) ;-. fir), !.how lhal thi., become...

286

. Chapter 7 QUlUlturn . ,ICC h'a . , nu::\ In Three Dlmen$ions and the

Hydrogen Atom

(b) Assume. as is quite often the case, thaI the

deviation of the atoms from their eqUilibrium separ ation i� very small compared to thai separation-thal is, � « a. Show that your result from part (a) can be rearranged into a rather familiar form. from which it follows thaI

(d) Impo:,ing the periodicity condition fi nd the t allowed values of C. (e) What property is quanti:ted according t o the of C'! 'ttlIJe (0 Obtain the radial equation. (g) Given thnl Ulr) = - blr, show that ak.. functi the form R{r) t- r/(I is a solution , vu\ on Of only I"r C is a certain one of it., allowed values. value of a, and thus find the (h) Determine the ground-state energy and wave function of lhe Oatrogen atom. '

n = 0, 1 . 2, . . . (=

""

O. 1 . 2,.

(c) Identify what each of the two terms represents phYSically. 89. A spherical infinite well has potential energy

Computational Exercises Since this is a central force. we may use tne Schr6dinger equation in tne form (7-30)--tnal is, just before the specific hydrogen atom potential energy is inserted. Show that the following is a solution R ( r) =

A sinbr ,

Now apply the appropriate boundary conditions. and in so doing, find the allowed angular momenta and ener­ gies for solutions of this form. 90. Residents of Flatworld-a two-dimensional world far, far away-have it easy. Although quantum mechanics of course applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron natrogen atom is

91. The Polar Equation: Here we verify the claim a \

tatting (1-21) while obtaining plots that with the functions 81 _ ' " ....(6). given in Table "1 .3. 'Fl I .l rs review Section 5.10 and the Guidelines at the beginning of the Chapter 5 Computational Exerc' If we expand the derivatives in polar equation I and also divide both sides by sin26, we Obtain

10 equations

O���·

The symmetry of the hydrogen atom argues that the probability density should be the same as we move away from the xy·plane in either the positive or negative z-direction. This, in tum, demands that 0C.m,(6} be either an odd or even function aboU/ the xy-plolle. where 6 = 7rl2. Therefore, let us change variables, with (3 ;: 6 - 'TTf2, so that 0fJrl,(l3) must either be 0 (odd) or have 0 slope (even) at (3 == 0 (i.e., the xy-plane). With this replacement, cos 9 = - sin f3 and sin 6 cos (3, and the polar equation becomes ==

(a) Separate variables by trying a solution of tne form I/!(r; 6) = R(r}8(6), then dividing by R{r)0(6). Show that Ihe 6 equation can be written

�0( O) dO'

�

C0( O)

where C is a separation constant. . le, 0(11) must be contlnu­ (b) To be physically acceptab . about an axiS, ous, which. since it involves rotation t must be the means that it must be periodic. Wha sign of C? ntial is an acceptable (c) Show that a complex expone solution for 0(6).

(a) By writing the second derivative in the "fmile difference" form of Section 5.10 and the first derivative similarly as [9«(3) - 8«(3 - 6.J3H16..B. show that the differential equation can be rewritten as 0( � + ;�) �

2

0(�) - 0( � - .�)

+ � � tan � 10 (� ) - 0( � - t.�)J +

6,(31 (C +

m� sec1(3)0«(3)

e...m...

the one-dimensional linear cases, leuing (b) Ju,t a' in {:J '" J.{:J and choo"ng values for 0(6.{:J) and 0(0) would allow calculation of 0(2.l{3), then 0(3.1p), then 0(4j,p), and so on. For .1{:J, use 0.00 1 . Now. choo�e the simplest values of C and m( allowed by equations (7-27). decide whether the corre­ sponding function 0(,rn,(8) from Table 7.3 is an odd or even function of {3. choose 0(0) and 0(.l{3) accordingly (again. see the Chapter 5 guideline!.). then plot 0 at all positive multiples of .1{3 oultO {3 = 1.57 (i.e. , 1"1'/2). Afterward, vary C and III! by �mall amounts and describe the effect on 0. Repeal the process for two other allowed values of C, and all allowed values of m( for each C. tlnd verify that aJl lhese sets give physically

Wh�n �llpre..\Cd in lenm Or lilt Bohr radius. the hydrogen ground·�lale energy or (7-12) is

-fl2J2moJ. Thu.., we now work in a sy�em or

UIIII" where the ground-,tale energy i\ £ = - I , and Ihe definition r .. all"" means thaI an ,.or I Bohr radius correspond, to .f

for larger C may seem to have a slight divergence near the end. This isn't "real." Rather, it is due to the fact that our limited precision fails to accu­ rately "cancel" the Ian {3 and sec (3. which grow

very rapidly near 1"1'12. ) 92. The Radial

Equation: Here we verify the claim lead­

ing to equations (7-12) and (7-32) while obtaining plots with the appropriate radial behavior. First. review Sec­ tion 5.10 and the guidelines at the beginning of the Chapter 5 Computational Exercises. Now, let us make �ome convenient changes of variables. (a) With the definition ,. =: ao x. show Ihat radial equa­ tion (7-3 I ) can be rewriuen as I

d( , d )

- -- ), -- R(uoX) + xl dx

f(f + l ) xl

dr

R(arr)

2111aijE ,�,- R(uoX)

-

-

unltsl Definition\ r a llf1r and R(O-w) =ftxllaw for it meons thlltflx) iii/HI be 0 at the origin.

(b) By writing the �econd demative in the "tinite dif­ ference" foml of Section 5.10. show that our �im­ plified radial equnlion can be rewritten as

[e(1

/(x + .lx)

i)

-

'J

=

+ 6.x2

2Ax)

-

+ I)

)...2

Ax

-

-

_]

!lx)

- - E Ax) ,

x

As in the Chapter 5 guidelines. letting x

=

.ir and

choosing values forJt,lx) nndftO) would allow calculation ofJt2ar), thenj{3:lx). thenfl4!\x). and so on. For .!lx. use 0.01. As noted.flO) has to be O. Because normalization is not our concern.flax) isn', crilical. Use 0.01. Now, choose the simplest values allowed by equation� (7-12) and (7-32)--

'£ =

-

I and e

of 6.x out to .r

'"

'"

O--then plotfixl at all mulliple�

10. (Nale: This is a numerical

technique with limited precision. so even "correct"

functions may appear 10 diverge. which is the rea­

son for limiting x to 10 here.) Afterward. vary £:

and f by small amounts. and describe the effect on fix). Repeat the process for two higher allowed values of E (remember the system

� unils!). and

for nil allowed values of e for each E, and verify

that all these sels give physically acceptable solu­ 20 for the second energy nnd 30 for the third.)

,EI"� w. and (7-34). show that this 2111tlo

becomes

I. Con..�nienl

tions. (Here you should plot to higher x. perhaps

Then. with the definitions R(tloX) =:Jtx)lurr'

£

="

olso say thlltflx) is Just ,.R(r). This is imponan•.

acceptable solutions. Note that the symmetries

make il unnecessary to plot negative values of {3. For ench set of C and 111(. discuss how your 0 relates to the functions in Table 7.3. (Note: Plots

-

:; /(x) = £/(x)

217

Bearing in mind that Figure 7.17 plots the

square

of ,.R(r). discuss how your plots and those in Figure 7. I 7 compare.

8 Spin and Ato m i c P h ysics Chapter Outline 8.1

!.1 fJ !.1 !.1

8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Evidence of Angular Momentum Quantization: A New Property Identical Panicles The Exclusion Principle Multielectron Atoms and the Periodic Table Characteristic X-Rays The Spin-Orbit Interaction Adding Angular Momenta External Magnetic Fields and the Z-Axis Excitation Spectra

that in quantum mechanics there I bound-Slate problems meaning Iapproximation solvable "in closed (onn," e harmonic is orten said

are only three

without techniques: the oscillator, infinite th well. numerical atom. Each is basically a one-particle problem-even and the hydrogen the proton contributes to the potential energy bul is other­ hydrogen. where must now confront the "unsolvable." Most real appiicalions wise ignored. We systems. and a logical place to start is an atom with multiple partic1e multi are once multiple electrons orbit the nucleus. the that true is it Still. ns. electrO problem is mathematically unsolvable. It is impossible to solve exactly a . SchrOdinger equation that takes mto account the potential energy shared by each electron with the nucleus alld that shared between pairs of electrons. The prospect is not as bleak as it might seem. Principles we have already encoun­ tered will explain quite a bit. However, the first section of the chapter focuses on a "new" intrinsic property of individual particles: spin. Spin must be under­ stood first because. while of little consequence in a one-particle problem. it is vital (0 understanding muhiparticle systems. We introduce this property in the context of a �earch for evidence of angular momentum quantization in the hydrogen atom.

8 . 1 Evidellce of Angular Momentum Quantization: A New Property

that a co�ponenl The claim in Chartl'r 7 that angular momentum is quantized. may lake on only certain values along an axis estabhshed by e�penment. IS eVidence IS ba..ed on a startling. How do we �ub!.t:1ntiate It? The most famous 289

'Igur.

e., A charge With angular

momentum ha\ a magnetu: dipole momen!.

('("!vt'I'UOI1.:d

ClllT'C'n• •s

'\ m·)hOIl_ \ CUfTeOi I � , .---- " PJ"Hlh!

tlc,-Iron

, .

,

h..o rj�ht-hil",J ru!c,; �

L I

1\

r x p

simple hypolhesj�:

A charged particle with an angular momentum forms a Cir.

culating electric currenl and (hu!> a ma� nelic dipole m�m�nl. and if an orbiting . quantized. lIS magnetiC dJpole momen! electron's angular momentum IS �hOUld be 100. A magnetic field can exert a force on a magnetic dipole. so shouldn', the magnetic force on the alom in some sense be qU3.nlized?

Angular Momentum and Magnetic Dipole Moment

�

Imagine a classical electron orbitmg counterclockwise in a circle of radius , as depicted in Figure 8 . 1 . Being negatively charged, it represents a clock�i

- � - -

,

-

,

••

(conventional) current of magnitude

I =

elT, where e is the

fundamental charge and T the period of revolution. Magnetic dipole moment is given by

I-L = lA, where A is the area encircled by a current loop, so J.L

=

fA

e ,

e

, - 7r/ = -7r, = -

T

21T'r/I'

=

e

-vr 2

=

e (

--

2me

)

m vr e

The last step uses the classical formula relating the angular momentum 10 the mass. speed, and radius of an orbiling point mass, and me is the electron mass. By the usual right-hand rules. the electron's angular momentum is up. and its magnetic dipole moment is down. Thus, we may write

e

IL L = - - L

(8- 1 )

2me

(We will address the reason for the subscript o n J-t a bit later.) This derivation is entirely classical, treating the electron as a particle. It is meant primarily to illustrate mat a charged orbiting particle has a magnetic dipole moment. But it happens that the result is quantum-mechanically correct. It correctly describes the relationship between the angular momentum of an electron matter wave and its magnetic dipole moment. We will soon see how to exploit this rela­

I

tionship 10 reveal angular momentum quantization via a nonuniform magnetic field. but first we need to understand a peculiar behavior that OCCurs in allY magnetic field.

A

magnetic dipole in a magnetic field experiences a torque

'T

= I-L X B,

and this causes the angular momentum to precess about [he B-field line,

as

does a gyroscope about a gravitational field line. Suppose B points in the .:-direction, as shown in Figure 8.2. By definition of the cross product, torque ILL X B

is perpendicular to B and ILL' which, because ILL and L are opposite, means that it is also perpendicular to L. The change in angular momentum dL by the rolational second law

'T

=

dLldt

is in the direction of the torque, so it too is

perpendicular to B and L. Thus. the angular momenrum vector maintains the same angle with B, while its point describes a circle-it precesses. We obtain the rate of precession, known as the

Larmor frequency, from Figure 8.2 as

follows: lin the djscm.'lon� that follow �c"e/31 references to the angular momentum ilnd magnclic moment really should be \0 their elpecliltion value,. but the eonelu�ions are sliIl lohd.

I-_e L X BI 2me

8.1 E"" de""e or AnJular Momentum Quantiuuon: A New PI..,.".

Figure 8.2 A torque. � cau..e..

X

8, perpendicular to L

L 10 precess. By righl.MII.! ruh:.#J x " I lhl� \\J). ''' ''' (' ,tt',ll) .mJ

1\

,II, mu,!!'Ie, h'" •

dL ,

,

,

,

,

,

I �Y

,

+,

+

+x

------ -

-� dL dt

dq,

, ,

e 2me

-LBsin 8

eB

(8-2)

- � --

Overall. we see that while the l-component of I'L is constant, its

x-

and y­

components change continuously at a rate proportional to the B-field strength. And where does this come into play? Whatever direction 8 points becomes a special axis, which we call the z-axis by convention. Only the com­

ponent of

ILL along that axis is fixed. so we cannot simultaneously observe

quantization of ILL along another axis. Because J1.L and L are directly related.

it follows that we cannot observe quantization of L along another axis. We

have uncovered an intriguing truth: The lheoretical conclusion arising from solving the hydrogen atom Schrodinger equation-that only one component of angular momentum is quantized-is in accord wilh our experimental

inability to observe quantization of more lhan one componenl. All means of probing possible angular momentum quantization would employ some kind of inherently directional influence. so there is none lhat does not itself establish a preferred direction-lhat is. a z-axis.

191

2 92

0.".., 8

Spin and Acomic Phy,iCS

FlgUN 8.3 Stem-Gerlach apparatus: An alom with a magnelic dipole moment passing through a nonunifonn magnetic field.

The Stern-Gerlach Experiment

In the Stem-Gerlach experiment. illu�traled in Figurc 8.3, th� poles of a mag. . the field IS nonuni net are fashioned to produce a channel along which fonn I hopes of ob�erving a "quantization of force:' a beam of atoms is sent thro g the channel and deposited at a screen beyond. If force may take on only err_ (ain discrele value!l, then the atom� passing through the channel should be deflected 10 only cenain discrete points on the screen. The origin of the force on a magnetic dipole is illustrated in Figure with the analogous electric case alongside for comparison. As we learn electricity and magnetism. the net force on a dipole in a tlllIfonn field is O. 50 we see why the Stem-Gerlach setup employs a nonuniform field. The ex.pres. sian for the force comes from the potential energy of a magnetic dipole ,... in an external magnetic field B.

�:

8.4 �I

•

U = -w B

Path of al0m beam

In general. force is the negative gradient of the potential energy, so

(8-3) (8-4)

Figure 8.4 (he

Dipoles experience a net force only if

field b nonuniform.

Ft

E

-4� F F

8

\

•

•

I

_ I'

•

•

Dipole� in uniform fields F nc�

=0 F

___ -

p

...

� F F

__ I'-

I

.....

F Dipoles in nonuniform fields F

""

*- 0

.8

\lat:.�IIr.: t.hIX11e m()ment I" an (l\cr.J.1I prupcrty \If a current distribution-not II lundll)O 01 jXl"III()O·-l'ooO aBam we ..ec that if B were Ct.lRslamJuniform. me

I�,"e \ululd he /ero. In the ..c=tup of "'igure 8.3. the fon::e neduces 101

...... .. . ac� dipo&e milht ha\'e IlnY orient..... iII! . ftekI. tht r-romponent of force would vary

f1pN 105

conlinuou.ly.

We relate

th", to angular momentum Via equation (8- t). Note especii\lIy. how­ e\er. that becau..e equation (R-51 m\·{lhe.. onl)' the ;:-comlXlnent of 1£. we n�d tlllJ) the :-CllmjXlnent 01 (8-1).

(

F

Even da....icall)'. an orbiting electron would po....e'" a magnetic dipole moment and "0 ..houJd c){pcricnce a force. but there i.. no cla!-.\Ical reason for quantilation along an a)ti\ c�tabli,hed b)' the ob..erwr. Accordingly. for a �i\en ,,,Iue 01 L. the :-component L, could be any of the continuum of \",\\ue.. lrom -L to +L. cau"lng atom.. to be deOected vertically o\er a contlOuOU\ band. a.. depicted 10 Figure 8.5. It L. " qUilIItized. howe\cr. the f(lrce \hould be. too. USlOg the quantiLation condition L� = m, h.

F

e

-

" "m

m

I

h

8B. a..'

i

m(

= -i.

. +(

A\ \I.e �ee. m, i.. the important factor gO\'eming the effect of the magnetic

field. which i... Why it is known a, the magnetic quantum numbu I ..tate.. Now suppo...e hydrogen atom� who...e elcl.:trons happen to be In ( �, through the channet. precessing a� they travel. Because m, may take on three value.... we ..hould expect to see three lines at the <;creen. corre...ponding to an upward force (m, = - I ). a downward force (m( -=- + I). and no force (m( = 0). as shown III Figure 8.6. In general. we should expect to ...ee 2 ( + I hnes-an odd number. When a beam of hydrogen aloms wa<; used with the Stem-Gerlach appara­ two line� were ...een.] Were thi� not baming enough. ground-Male however. tus. hydrogen atom... have ( = O. <'0 we would e)tpect the expenment to <;how ju�t a Figur. 8.6 The force on a magnetic dipole depend!. on it� component along the magnetic

�ur.;.w 1f1 I�v. aul \hE MCpoIIbM ..... dw. h>n;lf t",'m uncnlloh,m mc-rn- (8-3). M'� ckmml " Uu.! �on avcra&n '" � l
field, which i� quantized.

F

...

;.

o

) In!be oriI111011 192� sla'D--GcIr\Idl �

1_lnte", �n:: -. but � -.ut& sitWT Mala. lb."na _ �, dtcm:In beyond. l cIoIal .. . "' .tIc:ll IbtJ t.bouSd bdIaYc bk.c: � f:JI , .. 0 1'tIiPr' Uld T.tyt<3f � die c� in Iq�5 liMn, hydro&cl to Nk out .., t:lClmphc:a­ twn Uu.! Uluhipb: ekcIrom rrupr. �

IN 0,0 ' .. .. _ _ ..... .. ' No _ _ .....

__ ' ..

' • • lie -pm : .... ... ... .. . .. nk _ IWO .lbtld C!OIIIIJUMtIlL

1 n. ... .. .. on FlJIlft ,.,.w _Oectod luoe. ....o__IIIuIa ., ...., 1m: '** ,.. � _ OJOUS lar .... ccnm Spl....... . .. f cue does oc:cur. buI 'pbmlll �n an the ( O,...."C meal the eft'etts 01 . ...

nctJ&: rNXnCnt

....e ha\'ftl'l pn:'\ 10l1.'-I)' I,.-on.'idcTtd

Spin Elccuuni and Olh.:r patti,,;b po�� an Inlrinsk rnaatM'tk dipolt . • and an mc�tnl,;abl) rdated intrinsic anlfUlar motnf!ntum. and 11 1\ tilt ... tallon uf thi", ralh
(Ro1lt'm"a

givcn ttl.: nalll(' spin anJ the ,)mtx)1 S. __ S h an angular roocra. lum. I An InlnR.de property ., onl! thai 1'- rum.l.lm(·ntal hl " panlde'" nat1lt" thOit l'annOI tx- talt'n aWJ� nil mallcr ho" v.e nught try lndeed. mtnn..,,, prop. I!rtlCS at\" whal lll ..llOgUI,h (In\! I'undamcnhll partK'le Ir\)111 .Inolher--an de.: IrtJn '-flIm a photon, for tn ..("nn,' We are alread) t.\Ouhar Y+llh t"ll mlrlMic

t •

U llh�nllu'"

prup.:'rtll!�: rna.... .tnd t:har);c To Ihe...c

\H'

nml, add 'pm,

Although ilIl e\.penOk·I\I�1 f;l..:t. the e'\I..I�n�l� of ..pm

I"

not pn..'llll'tl'd fQ b lb

naluf(' c.\.plalOl"d \ 1;1 a OllOrdltl\ l'lll' theol')_ I t llrlC'\ iln � n'llur.III). blll lNlI, !�

a quantum-mcchanlcal thelll)' thai adhere, 10 the pnnclpk.. of rclutl\ uy, LII piIrtr

ul.U'. whcrca.. lh'O! SchnxllO�cr clIu.ltlun fur l i lree partlcie n.-.." on £ - rf!n&. ill �-, + � .... . • on E� - P-( rd.ul\-I ...IIt:i.dl)' l'flrT'tl'l geller:liI/Jtlon mu" he ha..cu !We Reb11\, ..lH: quantum ml"l:ha mc'" i... oc)unll the: "'C\)pe of the 1I�\.t. )et "I.' n«O � ,9

(lu.uUllall\C gm...p {If 'pill. lor It I'" ('ruc.::I(l1 10 undeNandlng mUl.:h 01 the �,l\J

U , '()unJ lI�. inciudll1g. nOlllbl),. llw..t of cheml ..t!) Thu,. \1,-1.' mu .., be ront,,"! \\I(b \lmpl) accepllOg the folk)\\ mg re..ul" 01 relaU\ ....lIl" qualllum Illechanl{"

\ l ( I ..

I . Analoge..)u, to the rdatlon ...hlp L

I ) h belween or/ntl1l �'U--

lar momentum and the dllllcn\lonlc\\ value I . the magOltude of a JW\l. de', imr"',\I(' anguhlf lllolll('nlulll dt'�ll(h on a dilllen..lonle...... \alue \.

Unltke l. ho\\ c\cr. \ I'" nut a quantum number ,h.,t m.,y take on diller·

enl \-lllue,_ In\lead. II t..Iepend, on the klild of particle. It I!; a given Ii'\cd valut! ror u gl\'en kllld of parllcle.

For the electron. \\ hich 1\ our mJIII objeci of IIltereSI. l =

proton 31 ..0 ha, s =

�.

�_ The

particle . lIl",cu,\\t!d In Chapler I�. ha,

The W

.\ = I . Other e\ample... are gl\cn In Tablc R I Importantly. II I ';; s thu ..

we regard

il�

'he ml'lI.mre of thi .. nt!\\- IIItrin... ic property. We sa). for

�

in..tance. that the electron 1\ a "'Pln4 particle. Th1\ usage is common but rather ambiguo"", for we ha\ I.' alread) cal led S. the intrinSIC angu· lar momenlum. .. ...pin." and from (846). Ih magllltude h

V1(1 + 1) h

"Jh. not �. We Will fllld that the ambigUIty rarely cau,es confu�ion A g iven panicle \, IIItnn"'IC magnetic dipole moment i!; related 10 i�

or

,

tntrin ...ic angular momentum by

,...

�

- !( .

q S 2m _.

18-7)

"" '"'

fAaLI . ,

'""n of � 'K'kdN {'Art" \n

...,nllf

HIlt

--

(httr"p'al .plftl

aI \PIa)

h.t,,-k

-"

�

\

. 11

h\'fi' 'I

, ...

...... -,

II

\lrtw rart"k, " O\Cbum .-:1cu,'

"

! I I

'.. "UtnlR, ft

.-

•

I

[I.",,... •

'f\IlImoH,

....rtk·...

)'hoI\'ft. J' l\';uknlh. J \'-"lI,I!lJ t\ 1'1 2

lilill " on

the- (',\111\': k' ... t.; h.\f��.

III ," 11\"'-... ...,

.\I\U g

tl' romaKIU."lh:

,"

rutlo lh\.' ,uh'crlJ't S " \�r) 11l1J"'rtaI11, Il)r IA ... , ... 1\.'l.'ll'{t h\ .,"1111\11

ung:u1.11 nhlm�ntUIl\ \,

\\ hill"

nhlllh.'l\tUI1\ /

hlf tlw c-h ..'dru!l. ,l:(' , ... � U..t',), .utd 1/ " 01 \'our-\.'

1',

" h.'\llh.'l.t h,l (lrhlt("

'\Il�uh\1

lout e\�\Clty. !:'Ilit d()-.\· \'!ll.'ugh h't 0\\,,1 t'. '\l etl"atlOll (8 7) gl\l!'" ,

S

N\)I� llMt It 1\ only In the tactor ul :2 that thi ... diller, IHlm C-(jUiUH'1l

l� 1 ) I\)r orolt.\1 motltlll

h)r the rnlton. 1: turn ... out to be 5.6. It ... chnrgl! (I i... t- t', ...n 1(\ li s • (\llll S .tre (lilrallcl rather than 0ppO\lle. MoreoVl'r, hccau...c 111 1 . . /fI C'. \ the rrot nn \ magnetic moment is quite ... mall, which I'" why the­ predominant eHl.!ct

•

I n the Stern Gerlach ex.ptl'ilnent

i� due

lU

the

h}drtlg�n atom'.. e lectron .

1 Analogou ... to L,

"'r

II. the �-component 01 mlnn"'lC nngulilr momcn

tUI11 " mherently 4uanti/cu. tilking on only tho...c value,," given hy

nt,

The quantity

m\ " J

.f + I .

-.f •

.

�

1 . ,\

new quantum number. the spin Quan,um

number. It", alloweu value... are from - s to + l III Integral step... and are thu, dependent on the kind of punic Ie.

The W

panicle ha",

electron. with .\

=

�.

s

"'s

1 . !i.O tni may be - 1 , 0, or + t ror an

may take on only two value",;

� or ;. �.

In another example of ... cemingly reckless term confusIOn, ttl, i ... al ...o we say that the electron· ... olten referred to as .....pln:' When m IS ,

-�.

spin i... minu ...-onc-halt or that the electron i� .....pm down," and when

"'J

i\

+ �, its spin IS plu,-one-half, or it is "spin up," But neither doe,

th" further leveling of the terminology cau,. trouble. The,e c'pre,· sion.. mu,t refer to allowed m.f values. for s and S are fixed '1 It. re'peClively.

-Y

i and

-

experiment. For ( == O. th Now let us rerum (0 the Stem-Gerlach e ' 0 'It Shou Idn ' t be "u b'�ect to a o�"'" magnetic dipole moment �L IS , "0 for e . tic magne ic dipole inmfls the momen!. there 'iihouJd slil l be a force on • , than (8-1) rather and (8-8) Insert (8-9) i ns . we (8-5) on equati fore In - ead Of L; m, Ii.

.;;.:::

•

=

F

(_.!..-S.)ilB'i

=

me '

iJ:

=

e

- - 111 f III

,

fiB.

'i !i a-

ms = -s, , · . . +s

-

for an electron, tile force should be res. Because mS takes on two values ' . nCled hnes. correspondmg to III � _ I to two values. There should Indeed be IWO ; where bor" or (Section 8.8 discusses the case of and til,f = itu1 and . _ intrinsic magnetic moments come Into play and more hnes would be •

+4.

e * O.

•

•

•

b

seen.)

EXAMPLE 8 . 1 Delenmne (he acceleration of a hydrogen atom in

II.

change is 1 0 Tim.

magnetic field who r se ate of

�

'N

Equation (8-10) applies, w ere the eleclfon's spin quantum number nls may be either of the two val ues ::t j.

F= = -="'

iJ B'. e - -m h m� s iJ;;.

1.6 x 1O- 1 9 C

( I ) ( 1 .055

;, 2 9. 1 1 x 10 " kg

X IQ-" J ' S) ( I O T/m)

::!:: 9.3 X 1 0-23 N

Although the force is on the eleclfon's magnetic moment. the entire atom accelerates. a

=

F

- =

m

9.3 X 10-23 N 1 .67 X 10

_7 kg

.,

= 5.5

X 10"' m/s2

Exercise 28 investigates the actual distance the atomic beam would be deflected under typical conditions. suggests thai an electron has angular momentu m due to spinning on its axis, in addition to that due to its orbit about the nUcleus a . s does Earth in its orbit of the Sun. Helpful though the spinning sphere nOlion may be. the intrinsic angular momentum and magnetic dipole moment of an The term

spill

electron cannot be understood on the basis of a spinning charged Object. Spin is Ilonclassical lo its very roots. No assumptions of mass and charge densities

circulating about an axis can explajn the observations (see Exercise 25). Spin has nothing whatever to do with spatial motion, something that could presum. ably be altered-it is truly intrinsic. Indeed, this would seem necessary in

ilil)'. -.ugge\ted by the "bsem:e of upcome-nw evi,""0 '" ro��ib .

,,'� e- _ 0I ,lh�, -i m HIS. th,lt the electron occupu�s no 'Plice! l' ,I till e d . , ""f1ct l ' POIfII angular momentum-angular momentum wlthuut .plin. .. t: ' (lU ;C � .

, Ie"

n t,ICn ,-the

."

re
1

_

_

pcd"l:�d "'l ,h' I

quantum number fill specifies II., spin state. 11'1 'late is . ) wh i le the function , thefoll r q uantum numbers n, L "'1' and III ami tely <,pccificd by n comp e ' .,cntcd " .. pre ,l-.. tJel' ll.

Ill;))' be re

(8-1 I ) tal..e on. only two values, related to two oppo.. ite componenh, it convenient to use an arrow ror III•. ,'ie ollen tiInd

Bec·(luse

. cao lt

1/1". t. "'1' +!

I (her ", NCI

= I/I/J, f. ( r, 8, if» "'I

i

Spin up

Spin down

(8-12)

nor the arrow should be regarded as multiplying ",n. (' '" (r. 8,

cJ»

1, sense. because :,palia! and spin Slates are of entirely different chara ,

. tM usu . . In p n of the overall d�scnptlon of the stale. so II 1 \ spin is a necessary aC,ler. But the wave funcllon, alongside symbol the lace P f!llr 8 iJlu\trales the electron's quamized intrinsic angular momentum Figure 8. . . dipole moment. That II has a d'IpOIe moment mean:. th at It .I S, bY ag etic and m uch like a bar magnet. indicated by the north and soulh poles, The nalllr e, 1 1 " . nnguIat momentum 5. w lose opposi' te 1he I· nlrinStC magmdI·pole In D ,nent is . , 0 11. And JUSI as the orbJlal angular momentum L may be found to lude IS 2y , certa i n angles with the z-axis in the hydrogen atom (cf. Figure

I�

�

�

��

.

.

experiment to determme the z-component of S has only two pOSSible . are I expIott ' th'IS up-or-down trrul ' 0f eleetron Spill tions [lat outcomes. Applica countless. .n 7. ")· "

Figure 8.8 The two possible spin states of an electron. Spin up

� .

5: -- +-� " .. s = l!" ,

J'

-

� .

, . 5, = ' II t. --

Spin down

5 = 13-, "

""the ....ave funcunD. and �ptn �A maID1 can n:pl.. el"mmt>..

Ju�l doubk� the num""r

of

Z98 Chaptw .

Spin and AIomJc PhY'''''

REAL-WO RLD E X A M P L E

E L E CT R O N S P I N R E S O NA N

CE

many \(;iences i� �I«tron spia A n eff�t used as an integrative Ilx)1 i n with Ii' built-in bar magrlet . ekclron an field. (ESR). In a magnetic or A photon of t� c energle\. fxmlhle two ha\ allowed orientation... OfTect 0I&Iy " hit the lower-energy orie " nant," frequenq can ('ICClie U lTan�lt!On from n tit 10 a rna pa or I cular, In \peclfic atom. nl on. higher one. In ESR. a malenal_ clic held, eleClromagn radiation Ifl the subjected to an e:demal magnetic range is applied. and ab\orptlon 01 the radiatIOn .\ monitored as either its e In ab\orptlon Ind, or the extemal field I" \·aned. A marl..ed Inr.:rea.. cafet thai Ibt photnm are of the n:,"-onant frequency. Th... frequency is proponional to he � \ the eleclron \\· ... matcnal', by e�pericnccd field hich magnetic U of the Ua�I)' I 1Ic1� the e:l.temal field and fIeld.. � ntcrnJI to the mnterl3l ESR can be u�d to 1Ud) t.bt mternal field... or. for a malena! In which the effects ol lhe'\C fields are well k, l n may be u!>ed merely 10 reveal the prc..ence of that material To beSt grasp thclWt ,J1 b I ..\ Cl of ESR. let us analyze a '>Imple ca�e.

�

�

�

� �

lrtnctb

�

Applying the Physin

A magnetic field i.. applied to a material In which an electron is able to nIp liS 'PIn orientation and in which internal fields are negligible. (a) If an electron IS ' Inll1 all) m the low-energy orientation. which way I'> it!> �PJn? (b) ElectromagnetiC d ra lalion . , ' Of frequency f IS now applied. Whal magnctlc field sirength would lead t O this field for (c) Evaluate f 9.5 GHz. a typical absorption? ESR frequ
reson:Ql1

"'"

) U

(a) The external field establishe!. the ;:-axis. so we write 8 = energy (8-3) thus becomes

- ..Ii

U

B i. Onentatio

n

:

A� we leam in electricity and magncli'!Om. a dipole i\ in a low-energy M te when II IS aligned with a field. which. in this ca�e, is when IL� is POSitl'.. Equalion (8-8) tel" u.. that the �pln of an electron i s oppos te Its magneli c . dipole moment. so the low-energy state has the Spin 0ppoSJle the field. W e �)' that the electron i.. spin down. rclative to the externally imposed louis.

�

(b) As we did to obtain equation (8-10). we replace J.L_o in the orientation en part (a) by ( - elml')S and use quantilotion condition (8-9).

I

U�

:::!: �.

,

-

me

111

ergYof

fiB

J

thiS give.. [wo orientation energies whose Because 11/, is restricted to dLf­ ference is e1iB/m The photon mu�t provide thi .. energy difference.

l"

lif =

I

,

m,

fiB

::::::)

In'ierting \"alue�.

(e)

B�

2,,-(9. 1 1 x 10·" kg)(9.5 X 10' Hz) ( 1 .6 x 10

" C)

.. � 0.34 T

�oth;� 10 i�

ESR is useful only when the material under \ludy hob an unpaired electron Th mean.. that the spatial �tate occupied by the electron 1\ not aho occupied by electron wilh opposlte l:.pin. If it were. then neLther electron would be able

flip

:\. 1 bu.Jl"nu' til Ant(ul41 Mlln�nlllm QuanuUluoo: A N"", ,pm.

j"(lr Ih"

\\(,uh.! pul l\\o e lectI'llll\ III the \,Ime Im.hvnhlJ.l-piLMKle ..litle. ,\, \H' .

.

Ihl\ would \'wlllle a !undJ.mcot.ll pnnt'lple ()j" qUlntum mecham�'\ Although Ihere nre mlltcriah in which the internal fiehh are negligitllc {u\Clul for apparatu\ cahbnllion), Ihe real apphl,.ltion\ of ESR invohe LIllemal lil!'l.h, \u\'h

,000 ..n,

6..\

tho,e

pnxlut"cd by Ihe elcl'tron\ o\\n orbit (di\cu"ed in Seculln R,b) and tho'!! III

more complicated. btll the added det... 1 the aWOllc nudel. Anal)'si.. ;\ natul',lIl)' fIlil}..e.. the technique more di�criminlllLllg, Mllny uf the \{H.:alled tran"tllln elemenh

HAl hlWt' unpaired dectron� lind ure common large" of ESR in phY'LC\ !lI\d chenu\\I)'. Biological applicallOn\ olten target molecule.. l..nown .1' free r.ldlcal,

( ..ce Section

A \'cl)' clo�ely related technique. nuclear magnetic re�onance. the Oa,i, of magnelic re,onance imaging (MRI), i� dl\cu\sed an Section 1 1 .4

Spin and Degeneracy By �imple argument<" �pin should increu�e the degeneracy In tht::. hydrogen atom. Were the energy of the atom independent of Ills' then for each set of II, ( , and fII(, there would be two possible values of "'s and thus two different states with the same energy, Doubling the number of states with the same Ii would

112

to 2112 . However, with spin comc� the possi­ bility of new interactions in the atom, which destroy some of the synlllll!lry and so red/lce the degeneracy, Several sections later in this chapter are devoted

increase the degeneracy from

10 studying these more complex inlcractions.

Electron Spin: A Two-State System

We have said that an electron can be either spin up or spin down. It i s true that an experiment must measure one or the other. But as our acquaintance with the fascinating nature of quanlum mechanics lells us. this doesn't mean that the spin

is either up or down

before such a measurement. For example. if an

electron's spin is measured as up along the z-axis, then it certainly i\ up along the z-axis. But a subsequent measurement along, say, the y-axis might find it either up or down along that axis, in fact with equally probability. as the sym­ metry about

z

in Figure 8.8 suggests. In effect, the measurement along the

z-a.xis leaves the electron in a state that is an equal combination/superposition of spin up and spin down along the y-axis. The y-component S is 1/01 O--that

.

value can nel'er be obtmned-but the values that are allowed, are equally

y+21 II and -2I I .l t,

likely,

In genera], a spin state may be an arbitrarily weighted superposition of spin up and spin down along an axis, and one promising application is to quantum computing (see Chapter 4 Progress and Applications). In conventional digital

computing, data bits-zeros and ones-must be stored. Any "two-state device:' like a simple on-off switch, might suffice. BUI quantum computing uses qubits (quantum bits), in which the state is an arbitrary superposition of IWO states. One of the first ways siudied to produce a quantum logic gate-a basic element of quantum computing-used the spin of a trapped alOm's lone valence electron as a qubit. Yes, a measurement must yield either up or down, but the number of combinations is infinite. While this increase i n information makes quantum

ProprII'If

Z99

300

CMpt.r 8

SPin and Alomic PhY\I(\ computing intriguing, It also bring" with it a problem. Noi� i \ mUch to obscure a continuou..ly variable combination than a ..imple On

Itkt� � .off. (�

Progres.. and Applications at the end of this chapter for further discu

� .'\Ion.l

The Spin of the Photon Allhough the spin-� electron iii the central player in the chapter. the ' ln oflh . Havmg ' s = I , 1h e phaton is Iipin_ l�P � phOlon deserves some altenhon. .A con\t. quence is that if an atom undergoes a transition in which a photon III r �" . d. there .IS a I"1I1lIt on how much the atom s anguI ar momentu m may chp Odu"'" a l\ quantum number may change by tlgt:-I momentum no more tha angular n I . Thi, r resrriction manifests itself in the selection rules that often a ise in di of spectral emissions.

5cU\SIOI\.\

8.2 Identical Particles

As we know, quantum mechanics limits the knowledge we may have of a one· particle system. It places further limitatjons on our knowledge of mult i lcI� systems. In Section 8.3, we will see that these limitations are ine x linked 10 spin, but their foundation can be understood without it. To things, we put spin aside for the time being.

;

� /,Cably Illlphf)

Imagine that two identical particles share the same space. Whethe \\' r e ' 1CS, two waves, or a smg ' 1e combine ' d wave regard them as two partlc , , l' t 'lS shlJ . ' " reasonable to speak 01' two " 'partlc Ies, In th at we may Isolate a system Ih . 31 has ' 0f a known fun damentaI Particle, . . ' . properties lh e IOtnnslC IWlce For inslanc e system of lwo electrons has a charge of2 X ( - 1 .6 X 10-19 C) no malt h · a Cr Ow we regard it. By "share the same space," we mean that the particles ar e physically isolated. There is some region of space in which they "overla P. region where either particle may be found. By this definition. two ole . . clrons rau 109 around m a box share the same space.

�Ol a

l

To proceed further we need a two-panicle SchrOdinger equat ion 1 ; One dimension, the Schrodinger equation for a single particle depends, o c urse, . To specify kinetic and potential energies in the two-particle ca on !; .e, It IS . Stili, we seek a true two-pan reasonable that we would need an X l and an I'cle . . - d completely indepe equailOn 10 w h'ICh Ihe partlcI es are not treate ndent] b ), much as a function de are described by a single wave function ljJ(x I '

x.

.

r

x') .

x2

� .�

ing an electromagnetic wave might represent many photons. The folio n WI� . has passed the expenmental tcst:

(

,,2 a2

112 a2

)

--- - --- .p(x I ' x2 ) + 2m a.ti 2m ax�

(8'13)

where E is the energy of the entire two-particle �yslem.

As a simple context in which to confront tbe central point in multiparticl

systems, let u.;; consider an infinite well in which the particles exert no forces

o�

each other. (Students interested in the details of separating variables and obtain_ ing solutions in this case should read A Closer Look: Two Particles in a Box.)

_

u_ _

lWo Particles In a Boll

T.... wl\c �ualiun I IH:\l. we apply the VII'I.blt:,. lS,ummS that

_Al 2IIt

ly IepU'Ilion 0(

lN�

t8-14)

••l.lt)

�.l .,)

a.tf

,,

) !!. '' 'i' : .( � 1=-"' - ',-' ::-7 + .., :t. ••( . , ) + U(" ) +

a.j

U(.,) - 1£

ow

...lK'rc ';.. lUlJ 1/1" 1lf'C In gcncr"l hffercnl fun..:ti(lns. As usual. ..

""c IH\(rt IhI:: rroJu�1 mt{� tK·' 'l. lht:n d,\',de by it.

+

U("X

\'�I/J,,(.ll)I/J,,(_,y I/J,l'I )I/J (. 2)

_

. ,

Now canceling. _I'll

-

2n. IP,,(l,)

(l11/J..(l\) -- +

(1-16)

E.p..('-.!1�b(.ll) tV..(� L )tbbtll)

fj11/tt>(x1) 2m .p"(" 2) il9: h1

iI·ti

+

U(\" t21 - E

(S· I S)

when: U is the infinite well p.ltentiai enefJ)'. By the usual argumenb, each tenn in �nlhelt' must be Ii constant, which we have designated el and Cl' Setting each \enD to its constant yields a differential equation identicai to the one­ particle Infinite well SchrOdinger equation. except that E i. repllK:ed b)' C Thcrerorc. we can limply write dooNn the same ..ulution,. given In equations (S·I6). willi the C�­ ,ponding replacemenb.

As In the one-particle caq:, !.he pou:nllal energ)' dictate\ (8-15) a\ It stand� does nOt ha\c ,t, and '1 ..epatate. becau..e U{X1 (2), In general. need not be a

the �olullon Equation

• .

\Um 01 'oCpar.ue fum:tlOns of x I and Xl' The potential energy

u\OCiated With an inlemal force between two particle!. In a

\)�Iem i� u�ual1y a functIOn of their separatIOn. �ll - :cll. and

does not

ond

�parale mlO a sum For example. the electrostauc

potential energy ..hared by two elel.:lron!o i$

The quantum number. need nol be the "lme, \0 we u� an n and an n', and hecau<;t: they '>C"rve equally well 10 distin­

guish the mdl\ldual·particle [unction,. we dispense with the earlier " and b. Wa\e [unction (8· 14) becomes

If, on the other h,ind, the particle.. experience only external fon:es, the sYl>lem'Jr, potentiaJ energy i!o a "urn.

(In the Cli'le of

gravity, for in\tance. it would be. m.IO'l + mg.\"2') We consider ooly external force� here. It I� true that this simplification.

..uictly "peakang, preclude.. applications to charged panicles, .. our mve<,ligation of the new fearure of mul­ but it streamhne "

"

uparticle system\ rno<;t important to us in this 'ittlion. For two particles exertmg no force on one anomer, experiencing only

the external force at inllmte walls, equation (8-15) become..

and in\cnmg C1 and C inlo (S·IIl) yields

1

E.

Perhaplo not \ufl)ri..ingly. the total energy of the two-panicle

)y�tem i.. the ,urn of th� mdividual infmite well energie5. in

'

state� II and n .

For two particles m an mfinlte well. equation (8- 1 3) is !:>atisfied by a \I rn. pie product of pamcle-in-a-box W3ve functions for each particle. each "'ith its

own quantum number.

where

"'"( x) =

fi . 1I7TX '<Jr. ""/:

( 8·17)

Before proceeding. we note some I mportant nomenclature. Becau<;e the tenn stalt> can be ambiguou�. the term individual-particle state is often used for a slate occupied by a single particle. such a� I/In(x l ), while multipartidt state u<;ually means the slale of the whole sy<;tem. as in !/I(x I ' x2)·

A Not-50-Small Problem Let u-" examine (8-17) for the case " cle wave function would be

=

4 and

2

=

-

L

fI ' =

3. The system's multipanl_

. 47TX I . 37TX2 Sin -- stn --

L

L

We would expect the square to give us the probability density P(x l' x2) for thi�

two-particle system:

P(x,. ', ) = ",'(x,. x2 )

figur. 8.9 Probability den ...llie� of the fhlrd and founh mfinile well �Iale,_

(8·18)

As a 10gicaJ generJ.lization of its meaning in the one-panicle case, this would be

the probabiliry of simultaneously finding particle I wilhin a unit distance at xI and particl� 2 within a unit distance 3t x2-that is. proba�iJity/(distance l • distanc ). � . If we mtegrate (8-18) over both XI and x2 between limits 0 and L, we obtai n a sen..ible result: The probabiliry of finding both particles somewhere withi n the box is I .

Figure 8.9 shows the individual probability densities for the fourth and third particle-in-a-box energy levels. and Figure 8. 1 0 shows the total probability Figure 8.10 An asymmetric probability density.

L

o 0

}W , o

L

"( "' •.

11 )

4

.,

L

L

•

nnot b<1 In \,: Idlfn Ih.J.l p.anic� 2 rnay be f...-.. t 1 fhl' .-. ... . .-.x. .--., at a 't " .... L U ; u 1 h:lc I 1ll.1)' n�\'cr � tnuoo lhtte I to clalm ""haI c " " . _.. tollI 11'131 1 ano ' .1fC' 1,1,.""cuI. "1 . '_ t.. _ _ _ no I �" l � di\t I in..: I I'\i f' ._ L... . .. I , _rllll,," ... � -... ' . r ...· heJ III �(.'ep Irm:1r. ul them K�tc:I). � tbelt .. \e fuucUUm. ,, the - '" sptll.-e· IJc nt H.-dl r;' l rtl� ln hanng the Wlme Pk'e m a&id to be: IllClhtift •• ,arnejlblt- f an e 'r�nm(' nt ,\ pcrt ornlaJ and a PiUlIc\e found, _C' I I mpl y h � . \0\ li."h partldt" It IS. Bct.:..u� It Ii. Impoulble to wnf) . h � tr tr , � ) mt ,..,. IYoO pann.: I e ul t h cs. we demand that the Ix"ha\'Il'r... dm t} I e Id h tndlOn' h• f llh}\.\ Ing r('� "" . ) , , , lt)- mu ..., � unl.:hangcJ It" the label "f mdl PHlt'lahdlt) l..h:n' tlOlUWt Ih.:hed. '''' arc .. ldc art able p

p«

llity

The reader nlny be une",,> . with thl' reMnl."tlon It 1 ndcf\ predu

on(' thin;: lu aher a .:1I0n\ that can be apc:f1nl(nLlJ ly \'cnhcd h'(lr) I t,;, . ui It I '" q U ite another to alter It merely }:Ie�au� It rC'n�n praJ.ll.:l101'l1 \'l 1/ h(' ,'rr/ ed expenmcntally (be..:au\e \\ c L-:.m't lnn.. V.hl\;h �tde h t f/lflll( · rc ... the tnctlOn e r , I ... nec.:e.....a[). wev . The tar-rC3(hmg nnd over I . Ho ,' .'\ e t"OU ld) "e diSCUS we ... oon ... cannot ... be ence explained uthervd\e. equ tiab1c cOn .... I 0 h t t lt e . h" r we a t P(x x ) "' a (; 'WI panu.:lc lahth \ ana.] 2. lea.\oe!o W might \' 2 o

fi'/"I.,

' " "Ju ...e � l �'"

li

H

II

I I re

_

_

tog

ngcd') We might try adding the �ame funcllOn .. Ilh the I nod 2

uncha

4

sm2

1:= P(x" ', ) L' .

.

4 1TX 1 L

�

, 31TX2

sm " - + .

L

4 L'

,m_

.,

\\ Ib.:ho.J

41T\} )1fX 1 01 I. L '

. symmetric-the same fu nc tion-under IOh::rchaogc of the rani. 111l s IS I deed B ut it is unacceptable because. quite \I mply, It I\n't the klUa.re of a ele labe ' S f the Schrodinger equation. To produce a \)'mmelnc prnbahlht)', den­ solution r t with the wave function. There are two ....a)" of modltying 0 _, 'Ia . . we mu'" � slry, rod · olutio i n equation (8-17) to meet the requiremenl.

�

(he p

ue ,

.1.

n

( ' '2) o//...(x I )"'..'(x,) "" .(xI )"',(,,) .1. (x I ) .1. "" (x,) (x , ) '" '" (XI ).,. (X2) "',' "'s

'fA

s

+

'.

1'-2

n

"' "

-

Symm
/il/eor

diffe ren liaJ mger equation i� a e at Be�;au\C the SchrOd qu ioll It uljon� ) b l a �o ulion aha re � .. ol of .. ence differ h l o Or and rn'" "hieh "u . . d of Ih<'0 ,_ s (8- 1 9) B UI In <.,0I UllOn ). (8·17 3,� y l J l ca a a m e a n th -, (8."'0 &\ '"3I1d m - I � <;rale fl and the othe IC� OC'Up e", J I.: paItI " r the s, - n· I� · · ., one of -� ,h. . . . . a� if/(lIlOflJ of particle I In '!Iate II, Particle 2 in Sial comb are 80th . cd , don e Il' . ' 0 II. ur Imme \Wle m 2 diate e l ic artd part CO n , ne Slale Of in J particle I� wh t� 9 and (8-20> lead 10 a :",ymmetric prOba W3\t!' funcljon� (8-/ ) ity de n\ll) .. thaI they do. If we ,,"wllch the panicle 'abe . A �imple w:gument. �how Is Ih · 1e t · e h h l d anrlsy W . ge e \}n. m mcrnc s w ' ,.. unchan n funcllo WJ\e 'q. metric itChes Sign...... wouId the change. W · , hence their name'" In neilller ea,e rJ tlJlg o . UI � n i t O p Ihc fies II)' demitie.. c1an

i

'

��

probab"

Ps(x, . ', )

'

_

= ",](x,. +,

p.r.,! , x,) =

', ) =

squlIre

'

•

",,�(x, )",;.(x,)

",;.r.,)ifrH',; + 2"'. (x, )",,,.(x,),,,,,.(x, ),,,.(x )

"',(x,. x,) = ",;(" ),,,;.(x,) + ",';.(x, )",,;(x,) - 2",,, (x ) '" ,

•.

,

I I

(x, )",,,. (x ) , '".(x,)

Interchanging particle labels ' and 2 leave� both ex press ions un h convenience. we refer to these probability densities as "sy rnme � ?,nged. For Or " ant!. "yrnmelric"-thal, is. according fO, (he correspondmg bOI probabdllY densltles are symmernc. NOie thai only the probab'l' both I !ly . , . de OSII.)' IS reqUired 10 be symmelnc. lor, a� aIways, the wave functions (he

WOl'e fimcfl:C_

m selv es c an. not be observed. and anw;ymmelric probabilit symmetric Both the y den..iti reserv t . eC r lam fealUres of the III/symmetrized probability densily (8- 1 8) . en p roperly normalized. they yield J when inlegraled over all value, ofx and a x�. nd they may be Jnrerpreled as a probability of finding particle J Wi th n Unu dislance at xJ and particle 2 within a unit dislance al x2 , Now. however ' I e probabili y ;'1 the same jf J and 2 are switched. t

i

�� �

To illustrate simiJariries and differences. Figure 8. 1 1 Show p � sex,. x2) an PA(x" -l1) for our special case " = 4. fI' = 3. Like the unsymmetnze d d pr Obability

Figu,.. S.11 Symmetric and antisymmelric probJblhlY den..itie�

�lllfes re occupied.

a

When n 3 and ,. � _

I I

4

L

I J

I

I I

8.2 ,..."""

Figure 8. 1 0. they have 12 bumps. but in both Ps and PA, they are ymmetricully between t"I and x2, Nevenheless. there is a crucial dl\ti",:· tion between Ps and PA' The large,1 va1ue� of Ps occur where lhe particle.. are clost: together. near the line x, = x2, On the other hand. PA has it!. large.., bum� \\hen Xl is near L and x2 near 0 and when XI is near 0 and x� near L. It i.. idl'nlic(llfy :.ero when xI = x2" h may be said that the particles are, on average. farther apart in an antisymmetric spatial state than in a symmetric one.

de:1l�lty of

dinded ..

EXAM P L E 8 . 2

TWo particles I n a box occupy the

/1 :'

4 and n ' == 3 Slates. Equatiom (8-19) and

(8-20) say that the �ymrnetric and anlisymmetric wave functions are of the !"orm

v'2

(

,

41TX,

. 31TX2

.

= -- SO" -- ""--

L

v'2

= --

L

The nonnlllization constant

(

31tx1

411',"2

•

)

+ sm -- ""-L L L

L

)

. 41TX, . 31TX2 . 3"..1'\ . 41TX2 "" -- ""-- - sm -- s m--

L

L

L

L

Yj! en\ures that the total probability of finding both par­

ticles somewhere in the box is I . (Verification is left as an exercise.) For both 415 and

1jI/.: calculate the probability that both particles would be found in the left half of the box. between 0 and !L. Then, compare with the case where the symmetrization

requirement is ignored-thai is, where the panicles are treated completely inde�n­ denlly. The following integrals will be helpful (j and k are integers): '2

J L( =o

Li�

J o

. hTX

.

)7TX

,,"--�m-dx L

L

27T

jrrx

L

sin2 -dt == 4

L

Sin[7T(k - ))/2] k-j

_

+ ))/2J +)

sin[7T(k k

)

SOLUTION We integrate the �quare of the wave function only over values of XI and x2 between

Oand}L. Using a ::!: to treat both cases al once, we have

-

..

_IOI<�ra1'_ 1!'\lll

4. and the e 10 U

ual

, ("" 111< ""

:! L L

"'" probabilil) ==

i}:i' 4

.!. 4

:!:

+

evaluate 10 4Ln"', Thus la..t tWO

2 LL

L}44

0. 1 32

:t:

4 4L 4L L2 7-rr 7-rr

{O.382

==

Symmetric Antisymmetric

0. 1 1 8

icle individually be'109 foU eimer part nd ob,bili'Y ' pr ' both being there would L , ,h' of n, h'Y bab, , m" uo "" ,' _, sO ,he pro . '' ' ' 'l"' . , d be ;, s, ,he symmetric ' Which oul I�nonng sla,e ' w 'he , ,I', haJJ ,wo I y.eldS ui«"" n' c, 10 which it is less' NOle lhat 'he req to !he be" etn m ,-"n ym m n",,"0. 13 Iv 1 and the an " ' The ,) FIgure 8, 1 1 , than ' , er " ....,. ely wllh auv � .. alit - " qu t>Jb,IIII l<e ag rn, babili'i" . , pro le<.>

of

.

"

,

'

In

'

cillo lIa .

have equal energy, the same as and '"A " Ihat "', ' (see ExerCISe 34), Accor lng lY, It funcl lon anilSym metric state as . the , " In Ihe system x ' 10 fin d pa i c les e erted fofees ely the t h,t O lik case ' . eq ual ly be the _ nt symmetnc s ' WO tate elec trons, the a . ,,. , l l les pa '" WO ' pa t , verifies 8,2 tliC I cl le t e ,y e S In ' as Ex amp bec,use, the repulsive e h e oget ,, 0 nd close ar be 10 be fou he n te p t Ie Ie pot S leS even Ihough ds hol nt argu me , unsoI vab le except y numencal ' d, uce equ allOn red be the

d"

on, we notewave ng ovi m e Befor Un;;yrn me"ized f Ihe , UId ..ee_ mmc Bul this wouldn I c r h m e lher' Were on one ,noof lower ener.y, .ould likely are " ,ta , ,ueh Th;, d ul wo schrtidIOger s r nde re rgy tial ene method

'f

rt

i

r t i r

t

'

-

r

nergy en·

b

"

' is not sufficient to c onSider J' ust sity, den , Y ilil ric probab We must d a given locations. a sy mmet e ' g f ' Ies b m d ' ie an part to In given SPin ted . . at g, ven loca io ns state. rela , nd d109 th Ihe spatial of ty 9) ' (8· 1 a tions (8·20) probabilt f iabels Equa thai the change a ex der un " symmetriC stales S foJlows: as eral ,zed gen be may Sy mm etr ic (8·21 ) Includ in

g Spin

To ensure

ensure

em

fin

it OU� t

t

,

I

"'5 " ",,,( 1 ),,,,,,(2)

+

",.. ( 1 )"',,( 2)

A n isy rnrnetric

t

(8-22)

�; ;

specify the ind' v , dual Parti bers needed to num m ntu u s, standin particles ' I a esents all qua Here, II repr the f nctions are the of ts men argu opriat e. o� example, the te system is app s ale. and ver coordina hate w in ' lS actuaJly leS ground state, n a hydrogen's in for coordina in u up spin with an elec tron ' and, in accord with eq atl On (8·12), 10 represent +i}'

r

cle t

'

_

g

0, } { I, 0, the set { II, e 4>2) 1 ' t "',,(2) would mean '" I . 0, o( r2, 82, metry to refer to he ch aracter af a state ,

lilt, Ill,

=

s

sym

ange ' when Iab els We use, the term exchexch n't change SIgn anaed, If it does . are s label when partic le etric', if it changes sign ' its exchange y mm etry is symm s hange exc sWItched, Its . ' , . anllsymmetrlc symmetry IS 0

"

are

8.3

The Exclusion Principle

ready for one of the central ideas of the chapter: Why i ... ..pin "0 behavior of multipart;c1e systems? 11 is fundamental to nature the crucial multiple indbtinguishable particles will he in a muhipartide of m that a �)\te symmetry, and II'I/ether it is symmetric or allf;.\vm­ exchange ,[�IC of definite of the system's parricles. All fundamental particles !ipill the 011 nlt'lric' depends is either integral or half-integral, and this divide, them into hilH� �pln s that predicts and experiment verifies that: Theory ories. categ \WO

We are now to

Sosons Particles for which s == 0, 1 . 2, . . . manifest a symmetric nlultipartic1e Mate " I 3 5 " 1es for Wh"Ieh !i' = 2' 2:' 2' . . . mum"rest un Feruuons Partlc anlisylllmcl'ric multiparticle Siale

Table 8.1 in Section 8. I gives several examples of bosons and fermion:;. It also reveals lhat all the familiar building blocks of nature---electrons. proton�, and neutrons-are spin.! . Being thus fermions, they always assume antisymmetric l\1ultip,rrticJe states. This fact may seem innocuous, but the consequences are re,ounding. Let us investigate the most far reaching.

Fermions: The Exclusion Principle Consider a system of two fermions occupying individual·particle Mates n and n'. If n and 11' are equal. the anlisymmetric two·particle state, equation (8·22). i� identically O. "',,( 1 )"',,(2) - ",,.( 1)"',,(2) � 0

We conclude that two fermions cannot have the same quantum numbers­ their spatial stale or their spin state or both must differ. This is known as the exclusion principle: indistinguishable fermions may occupy the same individual· particle state.

No two

To specify a particle's full sel of quantum numbers is to specify its state, so an equivalent way of expressing the exclusion principle is that no two indistin­ fermions may have the same set of quantum numbers. Although we ouishable • have considered only the simple case of two fermions, the exclusion principle holds for any number. 5 It was discovered by WOlfgang Pauli in 1924. and the achievement won him the 1945 Nobel Prize. The exclusion principle applies only to indistinguishable fermions. As noted earlier. this means that they must ( 1 ) be of the same kind and (2) share !he same space. Of course. all electrons are identical and all protons are identi­ cal. but we can tell electrons from protons. Their charge and mass serve as labels. We dOll't demand exchange symmetry for particles that can be distin­ guished, so the exclusion principle does not apply to a system of different

Exc\u!'oion prinCiple

�Se--.'c.'ral c.'nd-of-chaplcr e.'erci� dl""':Uo;$ � geIKr. aluauun of c.'quation {S·121 app1i�abk to

any nt>m­

tier of p;l111Ch:� and leading to the \>arne C(ln..;ilNOQ. .... n as the Slatc.'t detcrmi.rwnl i L It � no

IV ' .. 04

�

"' ,., AJl1Iff"

. ' and. e le tron The .<eCOnd COnd ' ' l real SIIuat,o . lunctlOn... n, du tal l on c,ponentoall). "e

c .n n oto pr n'" Wge I )JIn. ..much a a\:e . . .. h . h n'. ", , , I<" I , " ueI
(l

01<

.

,",

' ,,..0 <

"

...

ea

\ ....

10

t e

\\

· Ole be a ,hould mult, Panl · e. there on de ,he cas . under exch C melTi ange Th " o m f y .. U' an, \er.e· an arlut g' the unl a1l do,n Sla'e. However. for II pra\.:tal·� ... In e ,arn < .. .. doe .. . "OIl the I,'. do symme ange c .. ," f a m< t", \l can req re the e .clu.s'on pnn . ,0 hu,. " ", . ) , ..e e T on me C'pl, ft)l l p doe .... .. .u ' discu", . ' 1 '_ ,he sam e cupy mdl\, . y oc I' dUal. . ' oos ma ",,1 te,,"' E S\'sIe,// I e<... v . " ,.0 ,den -" ISO (lIed . ' • u a�. . , a'ed A\ lJ.e re , l lc cannol be ove� e l be! p l 0 nc . . pn 10.'01 , , l l< //I " . ' n l ple, nalUre wou d be profoUndl n ) on p ,. " .. e,e lust . < < ,m her e , " y· e, er chenl1" r >I<

. 11.>11 "

II

"

anl t. sy mmet . the nature of frol11 ric tly direc . .. ( 8 . 22) 'a cause; ,ha' ple 0 If " S SIgn pnnci ,he Oli OU . "a'e (8·21 !U,'0n etnc IficallY ), symm , eVc w pec h s ICh I'f elC ,Ia'C perfec'lY acccp . I, ,., tt le par . ua , d,\'I eq al. 0' are Ill< IllS. bOSl 10 s 110' applY .0 ,, 0", 0, /P/' doe ll(' p,i Ixc/llSiOIl able. The

an e anl, • Th ,he Slale is os' an m ons os b , er IP , " ' " c mu lle l HO d u m me sa and occupa'ion of s Boson

be

ing an d Pair les rtic composite ' pan'ocl" osite Pa or boson" bu, Com p fermions r o,he · lisled i n T are e es able I q' I are 8 maS! af t he part f ' compo d of uarks neutron. WhIch are e " olo and lso be r � lC Hlr '1' ' addlllO g ular - mon n,un f an the fru11l . a are I ru "

Fundamen,alonparorttIchelesodoer. In ac!n. c,n a ..ites. even IIvbeu' c0I11poha (see in de,a,l in Sec"on .7, are p adeyd nee'a dnlno'ge be 'd I r' untl-,h . h or , er e mlCg can addand odd number a bo,on, S u th the ng atnl
dt 'eusse alwa

n".. The qua . pler 12). 8

d

i

e

r"

cl

se

;

bosons sO when .

e,'en

n muS!

. , re su lt is ,his ' I nlegers . an IOnportan t "glued" ,oge her i n 0 a campaSlIe

. ' a boso",. and half·iOlege " -, ha' un" IS ' I enllca nm of fe ons r numbe f arm even sa

rS, so an al .,nlege ' . ng 'he pro,on, campnSi . For x amp e . 10 r . , erm lon . . subatornle p . an . a short- lived There " ' n , lo . erm , . 3 " a , . . ptnS t id, ·, fer mio n, ', quark', tha' IS a sp,n ,rue< 'pm f rk a q ree same allowed 'alai (Ihe d e al s o which ller sma e l t t t 10 no ca;t can he hree Sptn· can· n I ch). sear a ,i e re are n boson. . i lCg r.s i an form vely speaki ng) in ph Y'"cs re;eareh ra,i (figu sll,uen" ,opics of ile;. In partic u ar. �� cl �nnl. ons u d compos . , af superfl U1' ds and oxo'ic. and certain kindo focUS l n Ucltl' I'y rco supe 10 rther tn Chap er 9 s are discus>ed fu are eenl r IOpic e Thes conden»les, . ' . (elect rons .1 0 the case of' form boson , Bose.Einslein '0 rtng pat . " s . Accord",g y, Ihey 1 . u' 11 ' in the other case - ) mo;; ato ' n IO . ferm . lIY. whole In lac' b050ns, ever the o,uperconducllV lUSIon principle. exc an by d e lraln eon, are no I anger

l

· artIcle s but the ns l

_

l

l

I

•

,�lunterpolnt to fermion". actually "prefer" to be in the ..arne "Iale. and Ih,.. i\ I,\hat S"e.. n<;e to the clas�icall) weill.! propcrtie.. e,hlblled In the..e ..y..ttm..:

lero visco...ity. monolithiC beha..ior The mecham..m b) whll;h Ie'" re\i..tance, iring occur<; IS rarely obviou.., and 11 1\ often de!)troycd by thermal a�plati(m pa (\en a.t \Cr)' low temperature... but the more we Icarn about ...ud me�:ham ..m...

... there will be to put characteri<;tlcall) quantum Ixha\· the more opponumtie y In everyda technology. Ill"" to use

8.4 Multielectron Atoms and the Periodic Table To grasp the basics of quantum mechanics in three dimcn<;ion\. the hydrogen atOm is a splendid object of study. But i<;olated hydrogen atom.. are rare on Earth. so if we are to underst�md our surroundings. we must under.. tand mulll· electron atoms. The exclusion principle is of prime importance: No two dec· trOns may be in the same individual-particle state. Thus, the lowe..t energy �tate of an atom with Z proton" in its nucleus would have liS Z electron.. fill up

the Z lowest-energy quantum states. one electron per state. It may fairly be �id that the whole of chemistry is a consequence. Before we uplore thl" claim. however, we need to discuss a "new" factor In energy quantilatlon,

The Dependence of Energy o n

t

The presence of multiple electrons in an atom bnngs With it many changes.

For one, the spatial states are not the same functions as in the hydrogen atom. Nonetheless. they have much the same character and are again distinguished by quantum numbers 1/. e. and m(. Another change has to do with the quan­ tized energies and merits a bit more altention. In Chapter 7. we found that the energy in hydrogen depends only on the principal quantum number

1/. We noted that this "accidental degeneracy" is

due to the potential energy in the SchrMinger equation having a simple IIr dependence and that any deviation would cause the energy to depend also on £. Repulsive forces between electrons are a deviation. How does the energy vary with as a result'! The key is in Section 7.6, where we observed that a \mallcr

e

C may be thought of as a more elliptical orbit. For instance, a

35 ({ 0) clec· 3d (( = 2)

tron might be found either near or far from the origin. while in a

::z

Slll.te, its radius is more nearly umfonn, more circular, as depicted In Figure 8.12.

In a circularllarge·e orbit. an electron orbits not only the nucleus but aho all

Figure 8.1 2 Small·( �13te\ approach

clo,er 10 the nucleu\ than do large·(

�Ulle\.

�-o::;::;? roC3f1M1 ' 100'10:' Sm;al ( :rIpl

n: IT

states orbit a smaller positive charge and thus have a smaller flegat/t'e potential

• 0)

" (t

tnergy than they would without the other electrons. They are in a relatively high energy state. On the other hand. an ellipticallsmall-t orbit of the sarne 3\'erage radius-that is. the <;ame II-brings the electron closer to the nucleus. v.hich. therefore, is not screened as effectively, resulting in a larger negative potential energy. Of COUf'ie. an elliptical orbit also e'l:tends Jarther from the origin than does a circular one. but the screening "cloud" is clo.')e to the

ra.cLI. U'� •

� clh,tlcill mo.l

electrons closer to the nucleus than itself. We say that the inner electron\ screen the outer ones from the nucleus. In effect. electrons in circularllarge-(

It:

3d It

-

2J

310

Ch.pter 8

Spi(lll(ld AfOmic Physics

ordering TABLE B.2 SubshelJ

and capacilY

owering effect at small radii dom inates" A / h , nucleus. so the energy-l oUgh / " the ot be solved exact y, apprOX imation techni mullielecrron atom cann Verif y II, a the claim: orb clse 86.) its. cer nal Ex uo ula mp Co (See Exercises 48 and a/so provide the suppOrt for an im Approximation techniques . p onan t rul e states In an alom, they do so according to of rhumb: As electrons fiJI 'he / ' / ' y ower energ In cases of lowest 11 bemg value ofII + f. the stafe of " + , h gIven ' " / f belng 11 as a ue va In the nth , I.. sh We refer to electrons wll a / ell an d being in the I1t-subsheU. Thus , UJe subset wilh a given value of t as S she / + e = 2) fills before the 2p sUbshell (n + I fills firs•• and the 2s subshell , t ::::: 3 + ( IS subshell also ), but II IS ' / which in turn fills before the Owe I. r n (n e 4) + ll = fills subshe before 4s the bu. ' ( he 2p), The 3p is nex� n ::::: 5) Table 8.2 illustrates [his schedule. as well as the number of e/eclrOn s abIe fo' fit into each subshell-the product of the number of aIJOwed m v al and /fir values, 2(2f + 1). . subsheHs fill, however, the complex electron inter As t ons cause ; subshells 10 change. For instance. if only the ordering of th S, 2s 2p subshells are full. the 4s is of lower energy than the , and ' and m ". , � 'he nexi subshell to fill. Bur If subshells are filled out to n = 4 Or m ore, .he 3d ' IS of lower energy than the 4s. In fact. as more and more electrons orb' th J{ � fhe tendency is for inner-shell energies 1101 to imermingle A 10 . r a IS lower { is sti/1 ,a lower energy Wilhin a he energy, whereas J S g . . Ure 8. / 3 schematIcally depIcts thIS progressIve change in subshell ordenn g. f d (I oes nor reflect the lowering of 01/ energies with Z, a point we will d ' . �oon.)

For given

s energ�' is lower f�r lJIore-elliplical/lo �ve��;

eqUal OWeS! e. the

(n

ouler

inner

3s

(n

3

•

s

ues

�� �

ISCUSS

Figure 8.13 The energy ordering of eiec(fon Slales varies with Z.

+e

'he

�

3d

�

3d

nUcleus , lOWer

8.4 Mulbdectroft "101m. and the Periodic Table

C hemical Behavior: The Periodic Table Nothing i!> as fundamental to chemistry as the periodic table. The periodicity

. that of the elemenls' chemical behaviors, which depend almost entirely on

�e

and binding energies of each element's valence electrons-­ electrons, dangling at the periphery of the electron cloud. bound weaklY mech �i�s and the exclusion principle play the central role. Without Quantum pnnclple. the ground states of all atoms would have all electrons the in the Is state. They would have compact. spherically sym­ bOund very tightly metric probability/charge densities, would not produce external electric fields (as asymmetric densities would) to interact with other atoms, and would thus bechemica11y inert. The exclusion principle teUs us that there is room for only twO electrons in the II = 1 level, eight in the n = 2, and so 00, so most ele­ rnenlS have an unfilled shell with electrons "sticking out." A chemical reaction involves exchange or rearrangement of electrons between atoms. Hydrogen has one electron. which is naturally all it has to offer in a chemical reaction. It has one Is valence electron. Having an energy of - 13.6 eV. the energy required to free this electron from its nucleus is 1 3.6 eV. lei us now look at the other elements. As we do, keep in mind that each lime we imagine adding another electron, we also add a proton to the nucleus. We study chemical behaviors of neutral atoms, not weird cases of many elec­ trons orbiting one proton. Section 7.8 on bydrogenlike atoms showed that protons to the nucleus lowers the energy of a single orbiting electron. E,'en though things are more complex in multielectron atoms, each proton's atuaction to orbiting electrons still tends to lower the energies of aU levels. Figure 8.14 depicts this trend and guides the following discussion. Note that it is schematic only and not intended to be quantitatively accurate. Helium in ils ground energy state has its two electrons in the 1s level, with spins opposite to satisfy the exclusion principle. Its valence would be 2. except thal mese two complete the n = I shell and are at relatively low energy. Being in the same spatial state. they aren't very effectively screened from the nucleus by one another. and they feel an attractive force to a nuclear charge twice that of hydrogen's. Ignoring their mutual repulsion. each would behave as a lone electron orbiting a nucleus of Z = 2. with an energy of -54.4 eV (see Example 7.8). Their repulsion raises this only to about -40 eV. Com­ pared with hydrogen, then. helium's electrons are very tightly bound and thus exrremely reluctant to engage in chemical reactions. Hehum is accordingly known as a Doble gas (noble and aloof), and its valence IS said to be O. By convention, we indicate the occupation of an element's atomic levels via its ground-state electronic configuration. These are shown in the margm for hydrogen and heltum The super<.eripts indicate the number of electrons in �e ' each subshell. Beyond helium. addlliona! electrons must go into the next-lowest energ} bels. Electron" in n = 2 are considerably less tightJy bound than n = I elec­ trons. Not only are the) farther from the nucleus, but they are also \Creened by the Is electrons from the full nuclear charge of +u. Consequently, it I' the n = 2 eleclron� that panicipate In chemical reactions-the l s electrons are nor part of the valence. Fif'tt h Z =- 3 lithium. with a single 2s \alence electron. number

exclusion

adding

311

Jh•' qwtt -.'iJ\('. 1(,,' ,hi' hlgh-�nerg) elel'lTon 1\ ea".1 � u chm 1 .. � Ihum. the .!1 \uthhd l l\ lul l . l\ppt)�ItC 'PI" .. i.l.IIO\\llOg )' it.ht. \\. the three 111 , and I mum fOr t 11\ from 1 � III Z · 10. the �p filh. "lIh . \\ 0 m "u., I 'allie, , Th"," ,a)enee 01 elcmcnI\ be . VkilD rv.Jm (or ,, ('IC\.'tron Ond } hel l Uln pugrr rrom 1 (Of '/ - .' IlttuuJIl h,l � for Z =' 10 nCOn. 1m

11

�ru..

_

• • r 'tate:..., a quc\llon Iln,es: Nu\\ • � that the \\ I[h the lilhng IIf h re .I:' r�\\111 rent '{)IWdJ 't.lIe,-Jln�rent m, \,.lIuc:-. \\ ilh in e to ,\; ur l.IIl ... the \U b\�11 energy 1\ Independent of 111 tk,.. "klccd 1J4.1 1hc) fill" BCI,."lU\C one 'al " . U e 1\ ) I > " b liN e Ihen h C 2p e1t!clron In Z h��h ." aIlt.lth(r hJ he "\:h

•

thtrc

'\1

� 5 bo

i."LIfT'C'IJlI\ln \10\.'(' llthcr ('Iel'u·on ... are added, beg lnn l "g , roll. SUI \\ nh pul\l\ c energy bet \vecn c.ubon \; fk)(l....t In SnUun 8. :!. the re ' 6 eleetra l ' n� I Wh Ich require, anu,,)mmctnc. � 10" I' �I�I(' I/J(JIWJ th'l .. t thelf tl't\l Ihe U \. • I n l u e\ 1.... � � the antl\YIllIll ' ' ( 1 0 $('(1100 t:d .. i.:UV ''' ,fIIlcrt'OI f\lr tc:<1'llO' dh ct nc � pal ' lal \Iru ' e a . con'cqu aligned nce he 10 'Pill' kno til<' wn a H , In tum m.lulre, , s. und' S "'I, 7 nllrogen. Each of ito;; three "'II fhe IftnJ 'llnllnllC ' v.llh 1 - 'J e Ie c l rolh ha.\ a, are all l i hgned. l.hIT('n.'ol m, \alue, and their 'pin, \, the n • � ,hdl near. completion. Ihe outer electron,. beIng all . t by one anothe r: meamv a abou tht '.mlt r,IJlu,. iJ.n: not ,c�cned mUi.:h hiJ.e th '\e-e"t ey hu\, they become incrca,ing l ,In (\I.'r 1J.ll! one electron (0 complete the ()Ih...·r hand. fl)r "hlch 7 == its I ow �el1ergy 1/ • � ,hell. "0 il prefer. to capture a weakly bound electron of a orher n . . atom rJlhcr than !!l\e up rul) of I'" Ill.my ughtly bound OUler e lectron s L'ke SOdium, � it I llul1nnt' I' quilt' chcmll'ally active. bUI in the opposite sen'\e e tak r i valence of be d 10 sa . I thu!> ( " Lih. and ewi s nllher Ih;m the gi\Cf e 2-8 gen. t\\ l\ t'iecU'On.. �hon of a complete shell. i") said 10 be valence . There IS - 2 and one of +6. nOr betw no f'C.1I dl,tinclIon bel\\oeen a valence of e - I and . 7. Valcnce.. ll.I'C' 'pecitied '\imply according 10 whether the atom u:ua Iy [end� If a sodiu ll1 at to n.,\'CI\C or donmc electrons in a chemical reaction.) Om and Ou , atom Will seile �odium ' · flnne Jwm are broughl logelhcr, the nuonne . ' s danglin g. its g more completin tighlly bound \\ (" tlh . bound valence electron. n = 2 sh " . ell. \lthl)ugh pulling the electron from sodium requires energy. the "hoie n O i uonne'" " """ 2. shell is deep, and the net result is a l ower�enero eoy SIale. A n , Ion' . Ie bond ,.. fomled. in which the ions are held IOgether by electl"O!>t3tic attr t ac IOn. CompJcling the comparison. Z == I a neon differs by just one eleclr n ,odium and fluorine. but it'\ behavior is vaslly different from bOlh lIS � 2 I

•

8.9,

,

_

8.1-1.

ahrupt

' I

9.

�

�� e � ox�. �

f N

N

•

from

e

. fI sh l! I' full. It has no valence electrons to offer. as does sodium, and no hOles t i f 11, as

. (k'IC, Iluonne. It thus joins helium as a zero�valence noble gas. only gru�glng ly partil'lp.llmg in chemical reactions with other elements. A\ Z increases beyond .

10.

Ihe

v.�ence (now excludmg both fulJ "

==

3s and Ihen the 3p slales fill, .

and

�

1 and II = 2 shells) Increases exaCI IY as n.

"

'

.'

"

1,. "

---

,

'­ ,

-

'­ ,

'" "

--- '

,

"

•

�

,

+

*

•

1\

1\,

2

-

-

+'

..'

>.

*

L, 1

Bo

B

4

5

N

c

7

6

o

F

No

N,

M.

8

9

'0

II

/2

does during the filling of the /I = 2 levels. And just as the chemical behaviors of Z = 1 1 sodium and Z = 3 lithium are similar--each having one electron beyond a lined shell-so too arc the behaviors of Z = 1 2 magnesium and Z = 4 beryl�

limn, each with two valence electrons. This is the periodicity of the periodic table! Shown in Figure 8.15, its first column is valence 1 , containing Z = 1 . containing Z = 4 and Z = 3. and Z = 1 1 : the �econd column is valence Z = 12. The eighth and laM column does lIot correspond to a filled /I = 3 shell.

2,

The 3p Siales are rull, but the 3d states remain. Furthcnnore. as Table 8.2 indi­ cates, the 3d states fill only after the 4s. However, the energy jump from the 3p to

tbe4s level is large. Thus although Z

=:

1 8 argon does not have a full II

\1 is relatively tightly bound, so in common with Z

=

=

3 shell,

to and Z = 2, it behaves as

a noble gas of valence O. By these same arguments, Z = 17 chlorine, in the sev­ enth column, is chemically similar 10 Z = 9 nuorine. Both arc valence - I .

Si

13

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.

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J!

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·

• • •

· • .! -:

! � 1 • � -

,

•

�


,

I�

1 ,

· . · '

f

. \

I�

� ;..

;

,

j

•

>

, •

•

·

i.

. "

•

.II ;: •

.

- , ,. � •

.

•

, � J � � ·

"

• =

:

"

,

-

;.

•

,

· ·

�

314

-

. ! . -:� , : � : j

•

" .-

�

•

-

• 0

•

.. '"

,

•

• t '

• • < • , .

·

•

�

<

1 • • , .

. ,

•

" "

·

• "

• •

• •

•

•

•

• "

," ::::

�p ..ub..hcll,

\her the

the 4� filh, Becau-.c of the large cncl)t)- Jump

l£I\lll\OO, thl" begll" a nc" \cqucnce " Ith vaience:o. I and 2, Z - IQ and /. - �O f1'Icrc I" a break. however, betlA cen "alence.. '2 and �---t'Ic!t"ecn the c\lmple�

lran.-.ition illl(l lli the �� ..ub,hell and the ..tart of the �r-"here " C' lind the rklUfnb. "hlch lIl\ol\e filhng 01 d ...ub..helh after the ..hell 01 hl�hcr II ha.. ;tlrt.&J> begun to fill. As the ten �d ..tate.. fill. the ete\:lrOntc contiguratlon Joe .. al\\ 3)'" \ary co",�i...tcntty \\Ith Z. The fin-t deviation i .... Z - 24 chromIUm, 10 �hl(h the 3d level lIlcrea,c\ by 2 mer Z = 23. while the 4\ Ic\el lo..e.. 011\ cI((trOn. The energy leveh are relativel), clo,c, and ,lIch "exceptIOnal'" beha\ · uncommon. Although there are vanou" argument... to e'(.plam de",,\' Illr I' f uoo" rom con,i"lcnl tilling. the force\ m muilielectron atom, are complu;oltcd. IiIld ground-state configurations ultimately re\t on e�pcnmenl By Z 30. the 3d hao;; filled completely. beyond \\htch the 4fJ tilh tn a w a y, addmg a third period of elements of \':llenee 3 to valem;c 8, A.. I\lcnt con' doc' comple ion of the 3p, completion of the 4p level-but neIther the 4d nor the
[ll,lt

not

t

Jump to the 5$ i.. large. The ne:(t row in the periodic table, begmning with Z = 37 nlbidlUm. j, \CI)' similar to the previous. The 5s fills: then a lower-II ..ub"hell. the 4(/; then the Sp. The row beginning with Z = 55 ce"ium exhibits an expan\lon \\milar to the one accommodatmg the transition elements, caused by an merea,e III the number of lower-II subshells filling ··Iate." The 65 fills. then 11\ turn the: 4/.

6p. The elements involved in hanides, Z = 57 -70. al.,o known as

5d. and are

filling the 4/ subsheU are the 1 4

lant·

the rare earths (a misnomer. as many

qUite abundant in nature). The row beginning with Z =

87

francium nllf­

the previous row, The 75 fills. then the 5f. comprising the 14 actinides. 89-102. then the 6d. The 7p fi lls nex.t. adding yet another valence 3 to \alence 8 period. though not all of these elements have yet been identified.

ron

Z=

The periodicity of electron energies is also apparent in the ex.penmentally

determined first ioni,tation energies shown in Figure 8 . 1 6. The flTst Ionization energy is the energy needed to remove the first electron from the atom. Local maxima occur at the noble gases-He. Ne. Ar. Kr, Xe. and Rn-in which elec­

trons are tightly bound. In the elements immediately after the noble gases-LI . Figure 8.16 Fir!.t Ionization energies of the elements, 25 �

;.

�

� �

H,

N,

20 F

-

� 15 u c

• 2

i!! ti:

I

K,

CI

2

�

A<

Xc

S,

10 5

: LI 0

' Rb

10

20

30

50

40

z

60

70

80

90

t e sin gle valenCe el d to remove h e eed n ec. a ene 'eY d fr-u.e \ sily slolen by an e lectron hungry toIll. C s. b. i ea - a F. CI • Br. i. and At h ve 'hel. Na. K. R """"� s e s p" ou, y. a g r e .. ,he noble f uoo drOP" y'--,ei' be,on: ng room or one mOre 'pl..., ha\-'I ir the la _ --' irn � und. and

I It

menl\

_ J ,ly faJfly ugh tile their apPt .

n' eiecuo

EI,

�

.h

bo

oximate orbit radii in a hyd oge h ke for the appr " IZ , 'a _ ll , charge of + Ze. A larger nUCle Chatge lear obtain" nuc n b 7.8 ' , ' '/! . or i method of defi n ng alo Sec:1i on erimental eIeeuun mlC I h an l o ,e '. By o ne exp ""h'c In . m c alo orbits). the radii of nuonne and - g eleclJOn that d" clou In b t" I or I i .. a " pull� an \\ihen , m and 0.180 om. As nOled earlY In Ih1\ ull la�k Ie I d"ffi . 0.050 o y (a tivel pec radiu!. re� nol exactly hydrogenli ke. due t0 quoled as. lron alOrn!> are. t' c are liele ITlU l lum od. matIOns. Esti mate a � ns 1n ood baSIS lor approXi n g _lon . w3\e funcllo a vide pro y . �ec .t ro ns 'In sod'IUrn and fluonne ec t �, bUI lhe I e n nce ctlo and vale lera n" by the . eIec'ron In harge "see . °0 IS 0.053 nm . nuclear c EXA M

3 PL E 8.

: � !u

_

_

•

•

_

_

etTec1i\'C Bohr radlu� results. The .he II I �� dISCU

The \';llence

2 l>tates.

II == fluorine art in in n S "" V " e (;U I e 4(0.053 0.050 nm =

valence electrOn Sodium'!> lone

Zeff = 4. 2

nm)/Z,"

has 11

::=

3.

3 nm)/Z,"

9(0.05 0.180 nm =

theless about the same size as Z =:. flourlne is never . 9 (Note that _ of fluonne s " = 2 valence eleCllons each . l e ma l l s e 0 ou, Acco,dmg 1 ned . We would expect the interveni ng I hydrogen.) so 4.8 are scree 4.2. y nl o of e t the �even II = 2 electrons should would see a charg hly twO charges. b � roug en scre fI as SIX others orbiting in the same range I electrons to e extent. Each som o t other al�o screen each about 3 IS reasonable. screening of tive effec an letely scree ned .by all lower·1I elecirons. II. of radii. so comp were ons == 3 electr ' elliptical orbit s·state, however, IS a highly If sodium's II only I . An ' of Z . tive . should see an effec nce InsIde the lOner clouds. increasing liS prese le iderab trOn cons gl\lng this elec 2 2"1 etfccti\e Z. approximately 3 / = 2.25 times an 11 == 2 is s radIU rbit II � 3 ? of fluonne because sodium's In hydrogen.. an than three times that rodlus I� more Um sodi effectively .,maller charge. The an rodius. higher n. also orbits es beJng In a beSid tron. valence elec . Z Fluanne has

_ _

.

:::;

�

,

8.5

Characte ristic

X-Ra ys

lowest·energy states u P IO completely fiJI the . alOms. electrons In ground.state . d vaI ence e Iectfons can Jump around am ng xClte . um energy. E n some maxim olve no more Ihan le s of ecwnward jumps inv Do . els lev ed fill un higher ns in the ult ra violel nge i ndicales, producing photo 6 8.1 ure Fig as S, tronvoll glh" In Section · much shorter wavelen made to emit be can ms ato r, Howeve

� r: 3.3:

Figure 8.17 Characteristic X-rays are produced when an inner-shell hole is made.

All Jl;i!"eter.ucd

°

o

... ..

". °

b

o

.,/'

•

•

n=1

..... eje ..:tmg an

..... orb lt , illg electron

and U¥king a hole

° 0 . whl�h i� filled b} eled tpn orbiting 0,�....!. an In a higtj"er shell " o ,/ creaullg an )V'ray photon.

we learned that electrons smashed into a target produce a

continuous spectrum

of X-rays via bremsstrahlung-but something else also happens. As Figure 8.17 illustrates schematically, if an accelerated electron knocks an orbiting

electron out of an atom's

illller shell, a "hole" remains. An electron in a higher

energy state can then jump down into the hole. producing a phmon. Of course.

this leaves anmher hole, so the process can repeat. Inner-shell energies are

often thousands of electronvolts. and photons produced in transitions between

them are therefore in the energy range of X-rays. 103_1OS eV, with corre­

sponding wavelengths of

Figure 8.18 An X-ray specmtm. Characteristic X-rays

�

1 0 nm to lQ-2 nm.

Because atomic energy levels are quantized, only certain X-rays can be emit­

ted by a given element, and each element is different, so these

characteristic

X-rays serve as tingerprints by which we can distinguish the elements.

Figure 8 . 1 8 depicts the X-ray spectrum for a particular target elemenl. Super­

Wavelength

imposed on the continuous bremsstrahlung spectrum are some of the target's

characteristic X-rays. We use distinctive notation to refer to the X-ray photons

generated in inner-shell transitions. The II = I shell we refer to as the K-shell.

the n = 2 as the L-shell, the II = 3 as (he M-shell. and so on. These letters are A subscript also used (0 indicate the shell at which a LTansition

tenninates.

Figure 8.19

Ka

X-rays.

advancing from cr on through the Greek alphabet designates how many shells higher the transition began. An cr LTansition begins one shell higher. a {3 two sheJls higher, and so fonh. Thus. a transition beginning at the L-sheLi

(n

•

= 2)

•

and ending at the K-shell (II = I ) produces a Ka line. An LfJ line is from the N­

= 4)

within

a given shell are not down to the L-shell. (Transitions shell (n energetic enough to be part of an element's X-ray specrrum.) Inner-shell elec­ tronic structure is largely IOdependent of the behavior of the vaJence electrons,

with their temperamental periodicities. so the variation of characteristic X-ray

wavelengths with Z j.:, relatively smooth. Fig ure 8.19 illustrate!l. the smooth variation of Ka lines.

•

•

.'

50

, tiO

z

EXA MPL E e.'

. otbInng a cha1geofZarr given in Section 7 8 as neodymmm. 1bt efdPS tor ll1 � b Of . K X·ray Z = 60 gt

_

zt,,;t fSDIIJ* lht W � the

�1I�J

r ON

�

�Jl)

The pboCon t

1\

b;fTOII

otMf clouds, and thai bef re o 5. � ilboul halfoftM L-�hell clouds add 10 the scree 15 o

[55'

he - -IJ6. eV 22 A _

y minus us fimal.

A

_

-

-

59'J

'

.,

/-

3.7 1 X 1a' ,V

=

:z

3 .7

1X

Iran\j.

/ ()4 eV

O.033 nm

between 0.033 om and 0.034 om Wave1e 1 eod)'mJum KK X·-ys n_ 8,, for many elemenlS using simil 1. engths wavel ar e ) e�lJm:lI apprO)(IJ:n 59 Excn:ise h a· f F' dilla 0 e I 8. fils Igure model 9 1 Ihe t w well Actus n

,<>

" ,

Iron) and �ho"s Jus

h0

ore

,I.

IJ

.

ways. Those produced by a "'-J . . X-mys are used many ,_ CharaclensllC IO \V 1l X-ray source for crystallography. Con common • are ·aJ ren rna verse rarger ly. ,amples somewhat unknown are also very usefu tho� coaxed from l. B y X-mys With tabulated values and measUring matchmg chamCteostic re,ali.V , e s elemental compositio . learn much about the. sample n and \4-e , leS, mtenslt con. . . electron IS not the only way to exci te chara . centralJon. An accelerated cIensh. . ' l c ic art p ) es are (alpha also comm X.l'3ys. Prolons and helium nuclei only use . . d ' X-ray emiSS . as partie "" e-mduced ion (PIX and Ihe leehnique is then known '" . E) . . techmque IS liS sensitiVity, this due to a low O N J advantage of The pnm_ e r back . contl The X-rays. uous ung brem bremsstrahl sstrahl ung of ground level spec . of the charged particles and has [rum is caused by decelel'3l1on nothing 10 · . do · · with the specific larger. Given comparab'e kmellc energies, a proton ' w· . Ith a much larger mass than an electron, has a correspondmgly s maller a ecel era· " . " lion. so il emirs less "braking rad latlon. _

�

lJ D V A N C E D

8.6

The Spin-Orbit Interaction

In this secrion and several that follow, we will study Some of the Way" . In which the atom's various angular momenta interact We start with the sim . P �t case, returmng to the one-electron hydrogen atom. In an atom, the electron, with its intrinsic magnetic dIpole marnenl . . " essentiaJJy an orbiting magnet, so it should interact with any magnetic fi . ,� . le present Our simple solutIOn to the hydrogen atom assumed thal lhe only POlen· U·aJ energy In · Ihe alom IS · due to Ihe Cou'am b attracuon · belween nucle us and electron. This is certainly oversimplified if there is any magnetic field pr ese . nt. ' We need look no further than the alom uself. MOVing charge!ii produc fr e ma g. nerie fields. Because il orbits, the electron produces a magnetic field . , and om lts

�

fel�

-

-

1240 eV nm

t

Illog

the tkdron'lj Inllial energ

I l:,mi 1240eV ' nm forhcglve�

I

Assume- 1;�tV� ---.... ;:

In

orb.lS a charge of 59, due 10 screening by .

X-:II�COCIrnbution� from

K-shell eJcdrUl'l and

tJOA. llortMb "

.

�

t\,() The Spin.()fbit IntcncbOl'l

rehiCh ....

Figure '.2.0

pictured in Figure 8.20. the lIucleus is in orbit. producing a tield in !">pe'tive. the electron's intrinsic magnetic dipole moment has an orientatIOn energy.

The t"lectron feels a

netic field. dut" to the ··ortnting" in the ...arne dirt("-tion a!o l..

A detailed study of this interaction is beyond the scope of the text. But

319

mal·

proton.

'th fairly simple considerations. we gain a good qualitative unden;tanding

.... ld a reasonable idea of how much the interaction might alter the atomic 'Y levels. Strictly speaking, this "new" interaction invalidates our naive an "

e�'b . . . h drOgen atom solution of Chapter 7 . However, we WIll 'iee that the mterac· ergy is small, so the effect can be considered a minor perturbation on an li n en ption. eS'ienlially correct descri

�

The orientation energy

of a dipole

IJ.. in a magnetic fIeld 8 is

U=

-

• - - - -- -

'" What the prolon '>tt!o

f.L ' B .

Here the dipole moment i s the electron's intrinsic ""s' related to i t... spin. and

c"4 '"'

due to orbital motion-hence the name spin�orbit interaction. Thus the fIeld is we write

u

=

-""S ' Bdue lo L

8 (due 10 prolon "orbit"')

(8-23)

e

To obtain a rough idea of B. we assume that the electron "sees" the prOlan

I'

orbiting il in a circle. as in Figure 8.20. The field at the center of a current loop is given by 8 = J.LoIl2r, and the current J we relate to as follows:

L

Here we have

assumed that

whether the motion

is

viewed

as proton

orbiting

electron or vice versa. the orbit radius and speed are the same. Therefore, by multiplying and dividing by the electron mass, we relate I to the electron's

orbital angular momentum. A s Figure 8.20 shows, the B felt by the electron is in the same direction as L, by the usual right-hand rules, so we have

(8-24)

Finally.

inserting this and equation (8-8) into (8-23), the orientation energy

U= -

-p.. S · Bdue toL

(

e

- --S me

)( .

I'oe

41TTI'I..r3

)

L -

-

is

(8-25)

The energy is high/positive when the two angular momenta are aligned and low/negative when antialigned. This makes sense, for if S were aligned with L. Jls would be opposite L and thus B . and we know that a dipole's highest energy is when it is opposite the field. As we might expect from Section 8 . 1 . the spin has only two possible components along the internal field. However, the values,

�S -�': t:J - - - ­

_ _

�

_

_

What the electron <,eec;

figure ',2' The electron's lWO possi­ ble orienl8tion energies.

fls 1 ...

of B and L D,,,,,•••

" 1Ui!h--(lIo;q.,,), stIle � \lJlJ'tNte H. S pilmlld h' I,

� ... 1

D�lJon

ofB ,"" L

:t

y., do not correspond to perfect alignment Figure

(see also Figure

8.8).

8.21

illustrates the JlOi.nl

To give us an order-of-magnitude idea of the energy involved in the sp ln orbit mteraction. the product of ILs- and B will suffice. The dipole momenl o " 'b'lIlghtforward.

S

� I's,

'- ]O'''''C"IKrgy �L.llc: po., p"'.alkl 1,\ B S or",',11!; I.

S,

1.6 1 1 1 m, 9. 1 10-3 2 (1 .055 10-34 A 9.3 B (8-24). C. 0 2 n n2ao 4 2p Ve( + I )� V� ,

, I

'"

X 1O-19C

� m, 2 �

X

X 10-24

�

X

kg

J.s

)

. m2

Determining via itself a rough approximation, requires reasonable assumptions about r and First, we note that a state with L = should have no orientation energy at aIL There is no magnetic field if no charge has any consistent circulating motion. Thus. the interaction comes into play only When € is I or larger, which. m tum. requires to be or larger. Choosing the sim_ and r � � . . pleSl case, a Slale. L � ( �

B�

�

(47T

0 1' '

4",n,( 4ao)'

V� r.: V

10-1 T·mJA)(1.6 1O-19C) 2( 1 47T(9.11 10 31 ) 0.0529 10 9m)' 0.28 X

X

�

0

X

kg (4 X

X

•

.055

X

10-34 ) J.s

T

Considering the size of the particles involved, this is a strong field, thousands of limes Earth's field at its surface. Now completing our calculation,

Electron energies in hydrogen are on the order of electronvolts, so We see that the spin-orbit energy is very small. Nonetheless. the interaction is clearly discernible in the fine structure of hydrogen's spectral lines. Energy levels predicted in ignorance of spin to be degenerate are split according to the orientation of L and S. States in which L and S are aligned are of slightly higher energy than those in which they are antialigned, so photons produced in transitions to or from these states are of slightly different energies.

rJ

D

V

A

N

e

E

D

8 . 7 Adding Angular Momenta Angular momentum vectors do not add in an arbitrary way, but rather by strict quantum-mechanical rules. In thi� section, we discuss these rules mostly in the context of adding S and L. but the same addirion rules apply in all cases in

1,1 Ad4ift1 Aqul.. """'*-D

U1

various angular momenta interact within a system-spin angular FI!illure 8.22 Quantum-mechanical rule" allow L and S to add only 81 momentum with orbital angular momentum, spin with spin. and orbital with certain orienWion". orbital. The angular momenta S and L in an atom are not independent. The whole spin-orbit interaction is that spin and orbital angular momenta idea of the through a magnetic field internal to the atom. One result of other each affect of S and L I� that if quantization is probed with a weak external oupling th;� c weaker than the mtemal. so as not to overwhelm the internal much a.fieldnot e do observe independent quantization of each. but rather ion-w interact sum: the total angular momentum J . of the "ector

which

J%L+S

(8-26)

The individual magnitudes L and S are not altered. Both follow the usual fann: + I ) fj and S = Vs(s + I ) h. But the vectors L and S may add L :::

v'e( e

in only certain ways-the angle e in Figure 8.22 can take on only certain val­ ues. We have some reason to expect this. Recall from the discussion of the Stem-Gerlach experiment that if spin could somehow be ignored. L could make only certain angles--corre�ponding to 2€ + 1 allowed values of In c ­ lVith the :-axis of the external B-field. We now know that orbital motion pro­ duces an illternal B-field parallel to L---an "internal ;:-axis"-so it is certainly plausible that the intrinsic angular momentum S should make only certain angles with L. Indeed. we might expect 2s + I such angles. corresponding to the 2s + I allowed values of n1r And this is exactly what we find. The magnitude J and the ;;:-component J: ofthe total angular momentum are restricted to the following values:

}

=

J

,

:::

Vj(j + 1 ) m) fi,

/II)

h.

j %

\f - s\.\f - s\ + 1. . . . . ( + s -

= -i. -) + I , . . . ,) - I . j

U + s (8-27) (8-28)

Each allowed) gives a different magnitude ), which in tum implies a differ­

ent angle (J between L and S. The number of )-values is the number of values from to l + s in integral steps. which is indeed 2s + I , the same as

Ie

sl

+ s. Of course, in applying this rule to hydrogen with its lone elec· tron. s is so j may take on only the two values f + and { correspond­ ing respectively to L and S being aligned and antialigned. In multielectron Figure 8.23 The maximum and mini­ atoms. discussed in Section 8.9. total spin can be greater. and) may take on mum '>UrIb of two vector;. more values. In some sem,e, equation (8-27) resembles standard rules for vector addi­ tion. As Figure 8.23 demonstrates. gi\'en vectors A and B. the magmtude of the \"ectorsum C = A + B cannot be greater than A + B nor les, than - B\. How­ , e\'er, the angular momentum magnilUdes L S. and J aren't simply proponional B 10 the values ( , s, and ). Each is proportional to Vi (i + I ), where i stands for B l. s. orj. Let us see what this implies.

from - s to

1,

�

�.

\A

lit} allowed �) (8-27) j<.; ( T'ht ma\ lmum \aJut

I' lar momc:nlUJn Ing (l)(aJ angu

J...' = \

I l l. and S tI'trr paralld. l.

(/

S)'

+

o

+5

(I'

=

.,

Pond,

I )�

Ih� nlJgnHude of the vector sum would

_

+

'(I + ) ( I + , +

+ s. and (he cOO'es

VI +(1+I)A + vls(s +

I +

.r'

I)h

be

+ , + 2 vir's' + e's + s'e + es)/"

S. and we cOlcl llde Ihat rule (8-27) forbids L and S an e:'(erCl.�e 10 show thai Jmin > I L h' PL' l'\,ldly pJrillll'l. II I' lef! a<.; S1 B) lO'peLuon, 111);1\ < L

+

�

_

which be ex.c,ly opposi,e. (An excep,·IO ,
.

Sunuuanzmg, Ihe alOlll '-", inlemal B-field causes L and S to be " lOcked wgelher" al one of cenam aI/owed angle�, corre!;;pondi ng to the 2s + I val ue !! 01 J allowed by (S-27), l f we p robe angular momenlum quan lizat' n y a p ply . I n,!,! a weak nfffllaI B-field. if i!l quanlizafion of J: thai we obse � Or alo rn � II IIh a g l\ en al lowed value of J. we would find J: 10 be one of the + 1 v al. The Slem-Geriach exp ue� .i/lO'\ed by e� unlion (8-28), erime t of Sec. , lion 8 . ' I� a <,pecl31 case: ( =: O. ImplylOg lhal j =: S = ,1 and cOrrfiecfl y gI.Vl. ng .Iu ...( Iwo componenh,) FIgure 8,24 I·1 IUS(rafeS the relarionshi S Or the cas e Given ( Jnd s, Ihe magnitudes L and S are ji xe , b ( ;g . .f = ur there ar

�Nofe.:

i,

1

�2:'

�

Frgu� 8,24 Possibilities for totnl angular momentum J for case t = 2

- - - - - - - - -.;.:.. ; ,---::: JS=(' - , j)II

_ . _ _ _

,

_

L

� -

-

:-

----.-----

.

_ -

- -

-

-

•

fI +�. 2

+ lJ,,

- - - - - ­ - - -

- -

-

- -

-

___

-

L = ---

-

S -

I

e

-

,.

h(3)-

-

-

--

+ !,, - - - - - - - - -

-

2

-

_1" 2

-

�

�

- - - - - -- -

. . -�

L and S aligned j =: ( + s ; � ,

----

- _

_-

-

_

--

_

-

- - - -

-

-

- � /I l

L and S antiaJig ned

j = ( - .I = J ,

) I

:1

"Jc� bet wecn L and ,.ne'Iolc t '

_

S:

an aligned ",lie ""here )

t + J l"llJl
point In an mfinite num ed. l m ber of dircClion� . "bmll ihal and JI, uncenolll. Vectors L and S J ng avl have �illlllar freedom , 1 Ic 10 1I\ �, unit about J lhc � a . "nlll li\f dlqu" ...

•

,t

�"

d Quantum Numbers G. n0 field re ....enI� quantlza . llo �.... .. Ierna ] f J•. rel

Gi\t'n

d

no

qu

med to m II may ..cen . !

antUIll nu mbers of no w lUI III, arc relevance. I ndIeed ' . , with the Ittul l/ll . a . . orblll. ll !cr ct ion coupling L and S tnto a tOlal J. we reg ard II, (,j, and", us �pHl' "good," qmmtu •

the ,�levaI."!.

m num bers. I I ·IS or J reassunn · g, however. IhtU Ihc the sam is es stat e w lei of I her lIIe and m mu . or J. and 11/ are mlly re wu� /Ill d Ie" gllin the cllse e = 2. S = 'j. IgnOri 1 / COIl�1'dcr • ng any COupling of S and . • HlilL L. , uld take on five (I.e II co .. 2e + I ) values, and IIIr ( for each, there would L. be ' " v',lues of m,., for a totaI f 0 len stales. T e cou . · pO��Ibl"" plll h · lg .III FIg , ure 8.24 S ) . . ' Uowed m values for J = 2 and four for 2. which (11'0 als i o S ��� add to ten. If no a " I B- eld is pre sent, the latter view xtcn I is Correct_the �ta �trollg e tes foml ording [0 j. Bu t a strong enough extern al field can overwhel fO"p� acc o"pli no 0f IIIlf m and I · ·lnSIC · anguIar momentum ., to the alom 's ·IIIlernaI break fhe c case, both L and S are illdepellde mly q�antizcd. Each pre B fieId. In thai cess s : com � ponent along [he external fieldl;::-axIS \\ Ith a fixed . and m and "'s aga lll e ind icators of the system 's state-goo beCome levant d quantum number s. Sec ­ dis es cus s these different behaviors (ion 8.8 urthcr in an external field.

I

n

_

�

, IX

•

.

�

EX A M PLE 8 . 5

It/cnll·f Y the dilferent towl angular momentum States possible for the case e

SOLUTION

=

3, r

=

�.

•

Ifwe .Igllore any coupling of L and S, the number of States is simply lhe prod uct of values of II/f and Ills: (2e + . I) X 2 = 14. Accordm g to (8.27). , valu twO The cs. se and the correspondlllg ed " values of "� arc J. . ow wed [he number of allo •

IS I.

. lmDY.

.

7 =:: £ + s = 2

5 - C - �. = ­

lmln -

2

M OOI,d. the number of Slates should be the same whether or not the spin-orb . n i leraction is a faclor, and ag:un il we have 14, The aClual phY . . are Sical properties JDlDY. 1 ml"

= J2 (� + I)II = V63h = J� (� +--;) 2

2

2

2

2

II

'=

V3511 2

J

,

J�

�

7 5 3 I + -11 + -fI +-11 +-fI -2 ' - 2 ' -2 ' - 2

5

3

2

2

±- fl. ±- li, ±-J fI 2

R.r.tiv;ty .nd Hydrogen Atom Energy levels B(X'au.� 'rin l' reIJli\ l\I'l" in origin. Ihe ..pin-orbit interaction may be \ iev.� �IJII\J 'II'; elk,:1 II t.le\{fO)"' \ome or lhc "ymmerry in Ihe hydrogen (110f t! oUIJ rNWI'"' the' JegenerJt.')". The energy '\ higher when L nnd S are o:lign� Ho" ever. Ihe SPin. ( + I) th.tll v.hcn Ihc) ;Ire llntiah,gned (j = ( IJ l'rt"l" ,"I(radltlll i\ nOI Ihe only re/atn;,lic cause of discrepanc)' berween Iht and tho.�e we actually observe. AnOlht"r ndJ\( (n(�J k\('h of Ch..lpler Jm(Jh �" Iht' elliplil'il) of orbi". Rec.,dl fhal ti:tr il gi\t�n II. II "i1\"c� function of ( a... cOl11parnlnel) energy-If' orb'l IS more elliptical. f ('ncrg) and greater rJdlal k" noJI/(lIlJI -\hl) m Chapler 7. \\Oe nOled Ihnl energy berng Independent of ( is "acciden_ wi." .1 comcqucm.:e of Ihe simplicity of Ihe Coulomb pOlential energy, and a.u

- �).

7

.n�/{Iller . �

Ih;JI dn) m;uhemilflcal allernrion of rhe problem de.!.!roy." Ihis degeneracy. Although IOU Im ohcd 10 quunlify here, Ihe more complicated re/mivislic dependcnn.' ot energy on -"peed i .. ,ueh an alleration. and we find Ihal 0 mare

('1I/pII{" orbi! i� of lower energy than u large- ( orbit of rhe same II. Unlike [he ..pm-orbit inlcraclion, thi<. reklli\'i.�fic lilclor "fleels even the 1/ = I , ( = 0

\t,lIe. " Ion i .. of lower energy (han we would otherwise expect. Of course, the h\O ent'cl.�-the ..pro-orbit inleraclion and Ihe dependence of energy On t"Uiptlcil}"�oe\i�t. Taking bolh inlO
Rpur. '.25 R,'IJII\ L\IIl' �rl1ll1n8 \1/ hIIn'lC" .I,,,n ('nt'I�I\'" I1U('.I,I.\ ("\JUC'IJleJ)

U --"T""-'-v'" . .. I., lpJ,.,. ldv.-

11.. Ii'. HI I. -

(.!s.1i'�

,

c\pet."red nonreJallviqicaU), and (2) the deviation depends on Ihe iOlal angular mOlllenlum. The 'mailer Ihe "alile ofJ, the lower Ihe energy. Consequenlly, 10 'pent) \Iale� of differenl energy. we incorporate into the spectroscopic nOla_

3d .

lion of" Chapter 7 ;1 ,ubscript for Ihe rolal angu la r momentum quan t um num_ Perj. A ..Iale in "hich L and S are ontialigned (j = e - s = 2 =

i-�

(I

.ldv: 'laic, aJld il is slighlly lower energy than

�

�)

the 3d512 slale, i n which

the)" :m- aligned.

Figure 8.25 itiu!.trates the relativi stic splilling of hydrogen atom energy lev­

eb.. Stale;; Ihal by nonrelativistic theol)' should be degenerate

are

sh i fted down­

3p

\\ard and "plil according to the 1OL.1.I angular momentum. NOle Ihat the 3d3/2 and JI'-�r.. �Iatc�, of equalj. are of equal energy. The ellipliciry effect argues thai a

�

� �=e

\Ime--Ihe more elliptic-should be of lower energy. BUI the 3P3f2 state has L and S aligned (j = = I + = f + s , while in the 3dJ12. Slate. they are

(

�=2-

)

3pv� should be higher. As il turns out, Ihe two effects balance. Still, the figure nntilligned j =

-

s). So the spin-orbit effect argues thal the

renect� each effect independently. For LS-aligned Siales. larger e corresponds 10 higher energy. and for a given 8111ialigned ones.

e,

aligned states are higher energy Ihan

.

Figure 8.25 greatly exaggerates the spl itting The spacing of the nonrela­

I,

(ld

---I

li, islic energies is correct. but the slopes showing how the true energies devi­

are are all much Sleeper than they should be. specifically by the constant value 1/0-1. CaJled the fine-structure constant, a is defined

a s

"

41feJoc

==

137

as

(8-29)

1'\\1'" combination of factors often appenl"- in a rell\l\,\ I<1,\it treatment 0'- th� h"dn..'Igen atom. and it .. ....quare.

-5 X \0

'.

'" a f,m mea,tlf'C ,,-,[ the ,",\,\1;\11\''''

I�)"' nonrelatl'\' lsltC expectntion . (Ex.erci<;,e 69 �ugge"'b ho\'. to lint\. thi .. I'.\d�'r In ,he ",pln-orbit energy. who'c value of about \0 !> time' h)drogcn "t\lm

(oe�\e, i.. not cOIncIdentaL) The relali\'lslIc ..plltting of thc h)'dn')gcn _nom enl'f'g) \e,'el.. ,.. obviou,ly ..man. but It doe"n', es.cape our 1ll)\\CC. Prcc"c "'pc\,:­ \to'copi.c equipment can discern the ,light wavelength dlffcrencc .. of photon,

pf\1duced i n transitions expected nOllrelativl"ticaHy to be of the ...ame energy.

ror Ill",tance, n transition from the 3d"l;n state to the 2P Ir:! yidd... .\ pholun of ..hghtly higher energy than does a transition from the 3slr:! ...\tIte 10 the 21',,�:

fine structure

of hydrogen's spectral line ... wa.. one of the Explanation of the �reat early triumphs of relativistic quantun"! mechanic!>.

Even line structure doesn't tell the whole story of hydrogen's !>.pectra\

hnes. There are several smaller effects. The Lamb shift. an interaction

between electromagnetic vacuum fluctuations and the electron . ,piit" ..tate� of

a given

/I

and j according to the value of e. Another interaction. between the

ele<;tron and the proton's small magnetic moment-responsible for the so-called hyperfllle structure-splits each ('1 . j,

C)

level into a doublet. higher energy

when electron and proton spins are aligned. lower when anti aligned. Compre­ hensive Exercise 8S discusses a famous aspect of the latter effect: the ,o-called 2 1 c m line. We leave further study of these topics to a higher-level course.

8 . 8 External M a gnetic Fields and the Z-Axis In the simple hydrogen atom of Chapter 7, energy depended only on the prin­ ciple quantum number

II.

t n Sections

8.6 and 8.7, we

learned that o;pin and

other relativistic factors complicate matters. Energy also depends on total angular momentum. the quantum number j. Still. without an ex.ternal influence

such as an experimental observation, there is no preferred direction in space, and thus no dependence on mc' f11s' or nip A z-componenl is meaningless with­ out a l-axis. But what happens if we introduce an external influence? Given the atom's magnetic dipole moments, an ex.ternal magnetic field i� an obvious innuence to consider. States that are otherwise degenerate ...hould split into different energies depending on the orientation of the dipoles in thut

field. As it turns out. it is easiest to analyze cases at opposite extremes: where

the external field i s considerably stronger than the atom's internal field and where it is considerably weaker. We use hydrogen as an example in this sec­

tion. but we wilt see i n Section 8.9 that the results are applicable to valence electrons in multielectron atoms as well.

Weak Field: The Zeeman Effect

magnetic Ln a weak external B-field, S and L are coupled by the atom's internal cou­ field into a total angular momentum J. Consequenuy, IJ.s and IJ.L are also of JA-J pled into a total magnetic dipole moment ILJ' and different orientations

rJ

D

V

A

N

e

E

D

' . J relaljonships mental and energ ies. Funda ' fylng these ene have different quanti to l WIth 81: cruCH ' '[ mal .'> D difference a faclor-of-rw •

(8- 1 )

rgles.

.

-\'c ,... 31

_

1-' L

-

e

2m,

_ _

e

1-' -

L

=

S

--8

1-'L + 1-'S

1-'J

=

e -2m e

(L

(8-30)

III

,

I'J to relationship relates fundamen ral gouS analo No sum of PL and JlS' =

-

(8 8)

+

J.

I t is merely the vee lOr

28)

(8-3 1 )

J) IS

ships" Because L + S (eq ual to . trates [�e relali on Figure 8.26 i llus . Opposite), PJ and are t no l co ' IS i 25 (to which PJ ' obviously not L + le about (he z-axis. the external fieldl bu t lL ang d fixe a al ;1; J (hal precesses J ari sin g from t1 axis through Other ' precession about an h" an additional L ks S into a . fixed an gle w ith ' wh ile I J ld. The arom s inremal field loc . I fie . tW irl ing the capti ve JJ.. arOund about H un a as s ces pre a lowmg them to stronger than lhe ext erna l field being much they do. The in/ema icult analysis. From equ ation ' 2).Th erwise be a very diff , har would oth .. the ld to al fie ion st port e pro h. frequency is e �now that precessional io n ces lhe lha pre s ter fas ch mu is J ut abo abo P and L. ut S. J .'pinning of of JLJ in the ext erna 'eld the orientation energy for . we so d. fiel al ern the ext " tWI(I 5-lts component opposite J. ' average vaI ue as ]( . pIy use Its \Jm

J,

J,

J

mear. It �

sfl as l ���Phfies We � �fi7J

.

h . w·t ly about Be:o;r' s slow esse J� prec J J , field rnal Figure 8.26 In a weak exte J. t u abo dly rapi precess fixed/quantized. while L and S J prece��es ,Jowl)' about th� ::.-axis/Bnl

I I

- - .-

-

-

- _

.1. .1.

-

_

-

-

-

-

----

--

- -

:..

--

- - - - - -

-

-

----

-

---

" (quantized)

,

-

-

-

,

8

..,.

....

;

�

' \

�

,

\

L and S. locke� toget h r prece.,.. rapidly abou[ / .

I1Ja\ig

\

� ...

'

I1s

"

\.. (opposite S) •.. .

...

..... . . . ........ .. . . ... .. .. .... ..... ..... . . .

,

'

.

..

.

. .. .

cau�ing J.J. fi e ', iLL J .

+ J..L )

a u the t 10 WJ�1 r �idl)' abQ t OppO:.rfe dJre�·lion. with .

.

steady component

J.1.Jovg·

s

tt M EUemal Maanet1c Pields

Even SO, several steps are needed before we obtain a usable e",prc..�ion. From Figure 8.26 we see that I-lJavg

Angle a is between ILJ a�d the iJ-}icos a. and we may wnte

= I-lJ cos 0'

direction opposite J, so that �J

.

(- J)

�

= - --

J

NoW

inserting relationship (8-3 1 ) and J

= 2m, e

J,LJllvg

L + S,

=

(L + 2S) ' (L + S)

=

J

e L' + 2S' + 3L ' S

2me

J

= (L + S) . (L + S) = L2 + S'2 + 2 L S, we can put L . S i� terms of the magnitudes j, L, and S, and thus in terms ofi, e, and s. Straightforward algebra. left as an exercise, then gives Using j2

.

},,' = 2m, e

!L

-

3j(j + I ) - qt + I ) + s(s + I ) h 2Vj(j + I )

(8-32)

Now returning to Figure 8.26, the angle between this average dipole so the orientation energy is moment and the external field is 180 -

8,

U = -JJ.Jr.lVte'!.\ cos( 1 80 8) = ILJavg But cos 8 Using J, = J cos 8 and J Vj(j + 1 )r., we obtain -

=

and inserting (8-32), we have

(8-33) where we define the Lande g�factor as 3j(j gLande :;;:

+ I)

-

t( t + I ) + s(s + I ) 2j(j + I )

(8-34)

Note Ihat the coefficiellt of Bel(l in orientation energy (8-33) is exactly where we expect to find the .:-component of a dipole moment, and it resembles the

and the Z-Axis IZ'7

.55 '

_1 _* 11aJoat In I

,

/)

IUt*

WI

A8

""'.

t.'1 II ,:OYen bot h

c&! • C'�

l!io J . and in Ihto

..c

�

Ihe I h)Jnlg�n atom " ,thoUI JI . wotth RlknMUII IMr II as Imp.. M�"" hI e' ,pm-orhll ml�ml,:( h 1('\',\ oj t lon If'fI"U llnaltun ..hilt tali." anl(IICI.:tlUnl lht ..., mc:rd) addmg thelf cnC mall. .. If< ('fT(":I� lhe� ,,"hen Bu. (wlds r. and t.1Cf1UlI

II" to lho!;t "I Itat Impie:' h,drvBcn

ilWnJ 81\("

I:___ .,tenYf (..kf • f.au r.,w +-

ttk.'

, _

.:",�

I.:UfT('d

cnc:rg l(" .

g...... _,�, �". "', fiR,."

( 8-16 1

,,"cAl. ('\":om,,1 ma!!nclll' lielJ ,phh urhcrn" c degenerate atomIC' '1411" Inlll � + J ..1011(" d�IUl lh(' Il'n).Jit.'lJ Il'\t'! .lcctmJlI1g [0 fill' and lhe dcogr« ,,( ",pl,lrmg Ikpcnlh. Ihnlu� h gl-.k' on J. I , ilnd I 8) the argu_

In liumrtW).

OJ

':nnduJ1.' rhal ll rhe: lil.'lLl "ere /lonuflIforlll. If \\ ou ld (:�C'n dtfJC'lt'nI h,(n," vn \Iah.·.. l,l Jl llt"rL'nl ItIt ..() J SI1.'m-Gcr/al.: h-l) pe e.ltpen. 1 1\'"1' mill 2J + I \cp.tr.lIe nne.... BUI c\ en a rlk"nl \hlUld ttrtoU. l1nt �;,tm ,,1 1 pn·,enn' Lnlmn . .-\, \\ (" nun ..ce. II can \pl U und,lm1 ('\I('mal ti('IJ m;JLe, " , ITW'nI1i llf Sc\'bon K I .

OM

\%to

spt'\'lrul lln(' mhl man)

EXAMPLE 8 . 6

� (.:I) 8n:Ju� J -

�

10 (he I J1 'I.lle!, fII) J, rt'lnL:teJ 10 � l· Dillercnl dipole C . . n \'1T1('nI.Jlll)O.. lhu� \ (\\o enemie ,p I the le\eJ mill ·c \ · · SlmllarJ)'. '"

�1lIl

�

any 01 the Illur ,alu",.

-+:. �

IOr) - .1, . m - J

anJ � j. gl\ ml,! four energie...

_

1M fiN \\e l·dkuJJI(' li'

L.ano.It fl1r (he! '''' 0 �ta(e,.

+ I)

,li'l..oto.:k. " '' . '''

_

( +

l ' ,]

.,

I) -

1(I

" _�

(:!

'

+

I)

+ I)

= 2

+ !(j

+ I)

�

•

-

3

"' ll\ \h� �� '1.\\1.·.. " ,\01) t\\(Hhmh th.II \'\ the \ ' It;:­ �"\Ir(' 1\ ll ,\lu"'lnllc.. Ih\!' relnl"c 'p.\\,:i0>;t N,)\\ u'ln� l� �(,\,

"g...,.. '.27 \" " \\CIl),. mlllncU": neld. th<-t'C' ill'( \I' Ul\l\"'W �I\ll ") h

\Ii..: � thai th� "phttln..

It110,,1'\\1\\ ,

[I ,"

,

I

.'

o , o

.oJ

•. "..

.

,

:

I

,

'" ,

The ,pl itting III the I \ In i.. 5.8 x lO b eV. and tn the 2PV2' ,.. 3 9 X \0 II eV Gi\t�n the c'(lemal field', weaknc!;.. relmi\'c \0 the imernal, II i..I\' , \urrri,mg Ihal 1hc..e shift.. arc ..mallcr lhan the spin·orbit energy of SCl.:liull �.6

\d lh,mg (S·36), with E , " l and E1p,,� representing the Lero-fleld energies, a tran�ilion from a 2111(2 <,tate \0 a \SIn. Male would in\'ohe an cncrg)'

1 •

t :"

difference of

,

,

or eight possible tnmi>ition.. from the 2p,n 10 the IS l n_ the ..election ntle

forbid!> two. indicated in Figure 8.27. differences can be written as

A.. the

AE = (E,p, , - E" ,)

+

reader may verify. Ihe energy

:

B 2 .., " ",

There are six equally spaced photon energies. (d) We see that the spacing

,II

2

B = 2m, ", 3 =-

�;\}

is

( 1 .6 X 10 " C)( 1 .055 X 10'34 J . , )

2(9. 1 1 X 10 31 kg)

3 . 1 X 1O-15 J = 1.9

X

(0,05 T)

2 3

IO-6 eV

te) The splitting is very small. The transition energy is still essentially

- 1 3.6 eV

- 1 3,6eV

2'

I'

= 10, 2 eV = 1.6 X 10'" J

•

t

or L"

_ 00' • ... .. � ...,..

10 dIt local met¥) I Ui:al.lU1 !l,I X 10 thel �J 0 � . n... &k' ,.., oI lbt COftJY lhift Thh tiN nj1h. ''tltIme .... the In ,,, ..tUft lM , Z ;I( 10 cauuDl - umi ,00 , 1), ",I I� l'\lem,11 fiC'/J ,PIII LfII'ID tcI'IC'J II .t-.JUl U': MI ' iCC' \0(\.1 II alacIuC Z " ;I( 10 ' nm .pan •

IDIO au

� � tJ*.'aJ

Effect Strong Field: The Paschen-Back

Jcgcnt'rah.' !C",t" . ' mud (';Ncr if BC"\"t " ...fron rwlrnn& lhe' r.phllln' �If 8-g Ii field (l\e-rn helm" Ihe in!COlaI on the' onkr of ,mil 1�1,j \ll ,tnln ti Id hl�l�lht'r. and each , ... then qua mal woukf oc.hc',.... L\c �'()l.lrk I• •,nd S Olt/f"(j 0 luptnJcnd) ."Jong R ," as tkpldl'd In hgul\� i'\ 2X U...lOg (8<�O), the

IItWt. �MrgJt'

a«

I,

"'t ' 8.. �,

1, -

IA" ' 8", •

-

(

PJ� 8,;\,

and

(

p\. H.;" -

r .,,,, - ,

e In ,

)

"

�e

neola,

B."

)

S� B1."" \1 ;

Alrh\lugh iJ '[nm� field l'olO break Ihe "pm-orbII couphno eo'

aII) ,mall enough th.ll the o\erall energlc"

are

£IUl. IOJ e\1Cm41 hdd

•

E{crv field

+

+

(,

Ll (qUaJlUlNl

2m, "'t '/8'"

e 11/ l"

I1IJ/1Bect'

'I'. eIT lIeCI , ... ...(" •

1m )h8

Figu,.. .'.28 A ,ln1ng c.\rem"l licltl force, borh I. dnu S ro pre\'e�, tnUcpcllut'ntly nbour 8.

-'-

e

� .I U\u_ once agam obtained b y merely

dddln!! Iht'"C' penUWJIIOn, Ill ihe lero-field energie.\.

•

1

s

to}"

(8-37)

I I

1"hc.' f(,IJcr I' l'nnlUr.lt!l'J tll compare thl'" re,uit .... lth the cllrrt'p-'ndmg -lidd o:,uit (::l-�M "nd to lUlIlp.1re Flt!ure, 8 �b (Ull! �t�� Th" ,hl,u ld

bclr d;lf1t�

'II

\I.

h). be'llk',

"

l , nO t , the- glXxi quantum num�("<" it ,trunt! fie-Id. ,

� .... ,.j)..lkIJ lW"C. hut "'l ami ". III

are j

and nil m

8.9 Excitation Spectra

\I. e- h.'arne-o about characteristic X-ra)',- '(1Cctrul l'I1lI" ion, In 'l'i;ti(ln �.S. .•u'�'J �) holl'... "um..le-r" the v"lll.'ll!;c electron:>. Here \\c ta)..e it 1(0).. at pho, . :,In t'ml" ."" due 10 the e,(CltatlOn 01 vale-nce electron, above their ground

.

.

\,tJI�

\' we ).. no\\. III under�tand the richnes� of spectrnl lines. we need to ).. now

1\1.1\\ thc \';\rious angular momenta interact. We have MudlCd the !,pin-orhil Interaction Ul the one-electron hydrogen ntom, In tnu h ielcctron mOIll" the f1\JlI) ..pill ;Jnd orbit al angular momenta give ri);e to more campIe\. Interac­

bllll' To begin, let

1I"

cons ider total sp in.

Total Spin

In Seclion 8.7. the quantum rule!' arc thc same when adding any angu­ e nt _ on l,1I" m � �. Ju�t as L and S can add to a total angular momentum of magnitllde /j(j + 1 )11, whcrc j may take on any integral value between ( + s and )0 \ I -- II. twO cqual-ma nitude s in angular momenta can add 10 a tolal "pin Sr of sr �'T + f)fI, where .'iT takes on integral values betwecn magnitude Sr = ,\ ;- J and Is - ...-1. For two spin� panicles. sr can be only I or O. corresponding !ll Sr =- -iiI! and Sr = O. The z-componenl Sr: obeys the usual quanlization roles, so for sr = I , there arc three possible z-components. +1" O. -11, known as ,1 triplet of spin states. Naturally. for �'T = O. the z�component can only be O. and thi, loner i� I.-nown as a singlet state. Figure 8.29 depicts how the two spins add to form the triplet states. In all three, the angle between the spins is the same. They are as aligned as the \' noted

�

Figure 8.29 Formation of Iriplet states.

- -��-�,�, --...P:. - - - - - - - - '

,' Sr, "" - n.

�

� _ _ �

�

•••

.. 0". •

..

I

!

:

�j}

It

lJ

Tnpl
II

11 - H

,� 1l1Olt'lnlrt.' 'I

Si ng le t dnt i 'Y ffi mClr

ic

0

'�T

�\\t(' the l'llfTt:'r.lndcnu'\ �1\\ l.!en (hi'! repn!\Cnlatlon of lhe triplet 'UUes

and th.1t In Ftg un- K�9 E'll'h ,ho\,' \1ne

,In , e

\\Ith both ...pin ... up. Olle with t'klt� dl1" " ' Jnd \)nc 'lith 0rp'-l \lh: :.co nponenl� - Recall aho t c clai m lna de III S,,'\.'1I\1I\ �, � IIl I.: vnnCI.:IJ(\f\ \\ I lh I tund .. rule: An alltlli.ymmelnc &pat ial &UlI e. \\hll- h hl\\L'!' the repuhl\'c l'nerg) between electron, in an atom inl I . . P. . uh�nL'd 'pm, Thl' lull argument ,\ thol an anuli.YlllmeLnc SplIIW/ Slate ' , where pJJ1 I\ lc labtl, ll" the \\a\'e functIons o/rme give, a sign chaneo ne. WI· 1 1 '\ \ Jrrm!.! ,. ' mmctnc fIIulliplIrlU"/l' ,laiC only I f the SPill slate �I\I!' the re�llln..oJ anll .) i� ') mnll'ln\.' Illl) " gn \:h
�

�

:

LS Coupling

hlr hghtC'r atom,. the u�u,ll mleraction among the various angular momenla is a.. f�)lIo\\ ... The , alence cle�trons' orbital angular momenta interact to form a qll.U lI lll'd Mia orbital angular momentum Lf the spins interact 10 form a l/lI,m tiu'd lotu) "£110 angular momentum S]'; then these (WO engage 10 an Over·

I

all "pilH.)rbll interaction. giVIng a quantized "grand total" angular momentum

Jr- Ea..'h angular momentum addition follows the standard rules we have dis­ \..'u"eJ. Thl" lind of IOteraction is J... nown as LS coupling-the "tinaJ step" is

coupling of L

�md S.

AI ih root is the tendency for the electron repulsion to

(IJII.'tf\'< �·\)rfd.lu\m

�\{l

III Ihl' "palhll ..1.lIt·... IIldulhng. 01 (.·llUr.e. f - \Jlue" .

\\,

hh:h

mull/I'e/nlt'I,' "Iat� Inu,t � (mtl") mnU:lnl', lIl'm,lnd, nlNl.·, n!. tk,..IlI'1.' the l ll -,·\\\t'\'ll Ihl' ,PUI ,tate" Onl) "after" the,e IwO C(lrrel'llion, .,1\'11:" J 1"("1;1 t

a,,:I) \\ co'lk o\cr.III "pin·llrhlt mterat'tlon l'ome Inti) rl;l� nl�' \·h;lfJ.�I\·r llf e\ciled energ) Ic\cb

III mon\w,tlt'nt .1I(lm ..

... nlm:h the

the predominant effect of Ih� 1I11ler·..hcI1 ck�,:lnln' ... ..u111.' J.' 10 h) Jl\l�l·n. lur n Ihe \llknee l'ieelron trum uhno�, ull ihe p'-hiti\e dl'lr�\" III tlll' WllrJ� I\l ",:I\:( nut thc dlee" ol true IllUllle!cclron II1tcraclion', let u, e\lImOIl,leu, T\l ..on 11'4' ,he ..11111'1<..1 ca"t' of IIlteraction between twO "lIencc I.'ll'ctrtln, ;l hdulill

,lh11l1 1O Yo hich one clt'ctron ' " (',\i.',led \\ hlle the other remain' in the I , (Ru ...out �lf the 1 ,\ wuuld Ul.l..e more energy tllUn libemtll1g one ("tlt/lf,ft'Il'Il' ril l! /Hllh ,\)!I.I/Lng the :llom,) SUPPOM! the cxcitcli 11.'\1.'1 " the 2{). By dIe ungular

J�

1ll\1l1lCnll111l adlhtlon rllle ... the quantulll number (1" for the 10\;11 orhltut angular \lWI1\i.'rllUlI.l may he lInywhere from

It:!

-

( ,1

to

«( 1

+

i l ) in integral �tep',

B.'t'.lu'�· t 1� 0 for the 1.1 and I for the 2(1. (rc;w he only I . Th\" �pin .. couple 10 11,"\1 a wtnt 'Pill .1 r of 0 or I-the ..inglet (nnlinligned) anti triplet (I\\igncd)

I

,ute- di'cu....eu abo,>'c. Finally. Lr and 5r couple via a �pin-orhjt 1I1tCfilCIioll angular momentum J l' wherejr may take on any vuluc betwecn hl fllflll II towl

....

.\rl and «( T + (r) in integral ,tep�. For !ir = D. this i" the single vulue f " .. 1 . while for .� I' - I . it Tllay be O. I , or 2. Thus, we ..ee that the excited

hH.:h '

II

-

dt\"lron hns open to II many different 2p states. The conventional nOiation i.. �L\c:n ill the margin, where

( r fol lows

the same lettered scheme

11"

ror ..ingle

c1c,,:trun,. except uppcrca..e. Thus. the «('T ' sr.jr) = ( 1 . 0, I ) state i, thc ..inglC:I 21P I qate, the ( I . t . I ) state ,.. the triplet 2JPt state. and sO on In this IItllBtion. the grrllllllJ ,tate i .. the ..inglet 1 '50, There is no triplet 1 '51 �tatc . for

the \pin ,tate mu..t be anti ..ymmetric when two electrons are in the same 'PII-

na! !ltatc, Table 8,3 ..ummari/es how LS coupling governs stalCS in heliUm

.... ilh one electron excited. In general. different LS coupled states are of dilTerent energy, Of the many factON competing 10 determine lowest energy. the major ones are the follow­ LIl,: ( I ) The preference for an antisymmetric spatial state favors a symmetric

'pill �lale-3 larger $7' (2) The reduction in electrostatic repulsion afforded by baving multiple ( '* 0 electrons orbit in the same direction, "pas\lng olle

farger Cr (3) The spin· (A complete analysis can be found in any

another" less often than if coltnlerrevolving. favors a

orbit interaction favors a .H!wllerj.,.. dedicated atomic physics tex!.)

With so many states split in so many ways. it should not be surpri�ing that

\pectrn can be quite rich. Figure 8.30, though not

10 scale, illustrate.. the

\\ealth of spectral lines in excited helium just through " =

are forbidden by selection rules:

IljT = 0, ± I

3. Some transitions

UT = 0 -f+jr = 0 )

As usual. these are based on angular momentum conservation-the emitted

pholon has unit spin-and the ability of the atom in transition to oscillate as an electric dipole (sec Section 7. to). Because ilsT mUSI be 0, the lines naturnlly divide into those among the ST = 0 singlet states and those among the Sf = I triplet state... Interestingly, from the triplet 2351 Slate, a jump to the

f...h II,ele tron

C. (05C)PIC

'Iotal

2

31po

0

h�P

3'P

o

,

2

3'P ,

2

3'D ,

2

3'D ,

3

3'D ,

2

example of a metastable slale. is a good allowed. This not her than electric is ground stale r only by means o slate can occu und gro s. which ar invariably Transitions to the atOmic olh'-Ion . such as inter ration gene n dipole pholO ratton of lasers, and this particular al to the ope slateS are centr ble Metasta slow. helium-neon Jaser. in the common ited explO is m one in heliu nt is subjec ted to a weak external mag_ that If an eleme II is walth noung d. spill accordmg to m)T by the are further ennche es lin al clr �pe its netic field. to (8- 35 ) ma be used directly. with t. equallOnS (8-33) . leadlflg nomenclature fro Zeeman effect In fac mls (, s. and j. Somewhat ng laci rep Jr and (1' s1" "normal" Zeeman effect, Ie physics persists. I n the the early days of atomic ular mo me ntum alone, e �>componenl of orbital ang ell, are o;plit according 10 th Before spin levels, as noted in Section g"mg an odd number of e was ineXPlic mpl even number, as in Exa understood, sphtung into an Zeeman effect. We now know that the "nonnal" and known as the anomalous . nt aloms whose tOlal spin is O. behavIors merely IOvoh·e muillvale

�

�

�

8.1 8.6,

�

:

;'��

Figure 8.30

) 'So

2 ISO

Helium'� rich �pectrum.

3 1P

l

2 1P,

31

02

) lSI

3'P

3'D

\

-2 eV 2lS1 -2 1 eV

We won't delve deeper into multielectron-atom excitation spectra. eJlcept

10 identify an important alternative to LS coupling. It involves a spin-orbit coupling between eacll electron's S and L, yielding a quantized J for each, followed by a coupling of all these individual J vectors into a quantized grand 10lal angular momentum JT" As the final step is coupling of individual J vec10�. rather than 5T and Lp this kind of interaction is known

as jj-coupling.

it

predominates in high-Z atoms, where the intense magnetic field of the highly

charged "orbiting"' nucleus favors an "immediate" spin-orbit interaction for each electron. Left until last is a J coupling that reduces the relatively weak electron repulsion.

PROGRESS A N O A P P LI C AT I O N S Controlling Quantum Coherence in Spin

It is

impossible to men.tate the role of spin in modem techno­

logies. and the developing field of quantum computing is an excellent example of its utility. Most schemes for building the

basic unilS of a quantum computer rely in some way on �pin-of electrons or of whole atoms or of nuclei. Many exploit both spin and the convenience of the quantum dot formed in a semiconducting �olid. In quantum computing,

can perturb the confmed electron�. A recent ad..-ance by a team at Harvard essemiall)' neutralized these effe<:� by using two nearby dOl5 and twO electrons (Petta.. et aL.

Science. 30 September 2005, 21&0--21&4). In this o;,)�tem. the two qubit states, which can combinelo;,upefllO'£ in an infinite number of wa)'1O. are lho\e in which the electron spins are opposite and in which the)' are aligneJ.-tM Singlet and (middle) triplet �tates of Section �.9. To

however. preventing random external factors from altering

demonstrate the protection of the pure ,;,inglet state. the lWO

the quaotum state i.. cru�ial. .llld in '>orne semiconductor

electrons are begun in one dOl in the � ':>patial ...tate.

quantum dots, the nuclear ,pin.. of the surrounding atoms

requiring, of cour..e. oppo.,ite ..pin... Via external c¢ntro\s-

the r'llh:nll�1 "IICJ} J\ .lIlctn/. ;and oot' d«trolI " nullied Mlin w,,,,,... , , !.l.lt k�� b\ ;t NIner through "hilh "';ill' fUIKII'1f!1 un f"'S§/lunnd .h JC'pl([eJ In Figli/e 8 .ll flIto �mlu,ndud"f nudC:1Il .IlliXl the \.ep;!f3Il-d

'r'"�

t'le\:lnm\ I.h"��rlll.t. aJJmll .l roll\J<.m amuullt !')f Inpl':l

�I..I<· ,...l. Iull<'" and dC"Il'� inS Ill.. ptc\/uu,/)' coht:n:nt

I I I

I I I I I I I , I I I I I I

lIlM:kl " 011(' In onl) llho/U! 10 m Uo",,\cr. b) a (/(lt1 and ,,"cll·/lm...J �rrh'dll(l" (>1 nlt'mal mlL3�t:" � md,"iduaJ c/t'<:'m,,," \Plll\ .If!' �"�rreJ, d k .. :I,Ir!y ;j\(,l'lIgrng out Ihe "'�o,rm,"'I,.-' toel""cn Ihc Lit}" and R'lumm!' Iht pair 10 a l l ey \m�kl '�I(' ii' mll�h .II a nllcro\<xond laler

tt .l n l rure

Plilsing Ent.ngl.d 'n'orm.tion to Photon Spin

In the

mJtk. ellher ow..omc 'mghl (esull. wilh equal probabihty. Thc 810m IS .. Iwo-�[ale qubu (aJrgned or anlH1Jrgned de.:lron and nudear �rlll<; ) !lnd �o is the photon (spm )IJIt'�l. ilod these qubll\ are enlaogled The (,Jlperimem dele.:led photon �pm $1
Important ad\'ance

,)mmeln":
�t":h{ln IL', the '[;:I'." 01 ["0 PJnlde� 1Irt' ('nlanglrel We

Jvn'/ IoI) "hl.:h II III It
lind, line l(l he

111 11.

then /ht' ocher i\

' m /I .

(The panicle�

r.m Ix I�lr uparl "hell tht' ml·,,�uremellt i� carried out. but if (kleln'l rcpre,elll rn,t
�II <:lIu'e /lnd C'n"l'l " lale', Numcmu� entllngJed �)'srems III photon, alone ,111(/ atmtl\ alune ha\c been achievo:-(/ in

III pJrtH;/cl hJ\ now :11<;0 been

IUn emll\ a phIlion OIl II dee" lfe�, Tho t'\luaJJ)' likely

olniiahl!nment ill the cadmIUm ekdron'� final statt' spin

anJ lhe.- ,pin 01 Ihe cadmIum nucJeu\ (the hyperfine

II1lerudlllll referreJ III 111 Seeltoll !In Like eleclrons.

phllWn\ hal'': JU\t 1\1'0 poI�lble lpm llatt'\. and angular

momt'nlUlII con\':l'Iallon dem;mth thaI the photon's spin be

Olle \alue fllr r/el'lmn-nudeu\ alltialignmelll and the Olner value for Illignment. Ho"ever. unttl a mea\uremenl is

Howcl'er. in thiS age of increaloing conlrol over m..tetial propenles and mIcroscopIC SlfUcrurcs. the possibility of

success was the de\'elopmen t of glan[ magnclOresistUflce (GMR) de\'ices. The mO�1 f;lmou�. Ihe s pin

valve. is the

"e)-e" by whIch modem person;)1 compUler; read Ihe

I I MJf(h 2004, 15J-1571. :I !orngle cadmIUm

Irolll,lIIolI' � r<",,,ble. dcpcndmg on the alignmt'nl or

possess Spill, bul elecllic currenl ordlnanly nows in a Slate ofrandom Spill oricntation. �o ..pm is of lillie consequence.

knOlln as spintfOnics, short for �pin electronics. An early

Jem.'n'lr.tJcd. III1J If mJ) one dolY be e.�pIOlted 10 '(lmmUm�'atc quantum mf(lrmolliun fmm one place 10 "nulh..., In a linnel"ll) 01 MichIgan elpt'nment (Blino\'. ct ai , ' .. f. ,ll/fr,

Today's electronic devices rely on eleclrons

exploitmg this distinct trait ha� led 10 [he field of study

f'(,'cnt ),·a(\. but ellt,rngJcrneni between Ihc�e 11'.'0 \ery dlllcrenl kll1d,

Spintronics

(and holes 111 semIConductors). and the vital characteristic e�ploiled IS Ihelrcharge. As we know. electrons also

infonn:lIion Mored on Iheir hard disk,. In the spin valve. t"O layers of melal qualifYing as ferromagnetic (possessing

a 13f!!e. permanent '"temal magnellC field) s ..ndwich a

nonmagnetic conducting layer If the outer layers' magnetizations ate aligned. eleclJOns of one �pin

otienlatioo-intnnsic bar magnets that they are-now easily between them. but if antiahgned. electron� flow poorly. A paSSing data bit (magnetic re!:ion) 00 the \pinning disk temporanly changes the magneliUltion direction of one layer. cau"ng an unusually prominent change in resistance. hence the word magnetore�l�tance. In the late 1990�. this �pin-ba"ed technology ushered in a considerable increa�e

111

mfonnalion "lOrage density over

i� predecessor (which. emplOYing feebler non-lopin-based

Flgur. 8.31 Two eh:�tmn.) WIth oppo�ile �pin� in separate

tioK The du\tcl'\ repre�enl \cmlc(mduCIOr nuclei. which

magnetoresistance. was not "giant") While this benefit has already made liS mark. more applications are foreseen for spintronics. Control of one cir­

perturb tile electron \flln�

cuit by another. the essence of ampJificw itching.

is a pnmary element in essentIally all of our cXI\ling elec­

lfOnic�. and conlIol of spin-polarized currem In an operating circuit is a key goal In splntronic.). An earl}' proposal. the Datla-Das spin field effect transistor (SFET).

-

\loa,>

to sand­

wich a semiconductor between ferromagnch who!te magne­

tilations arc aligned. as tllustrated m Figurt It 32. A potemtal

differtn,e would coax electron.. from the �(lun:e ferromag­ ,

net y,.hose \ptns are aligned WIth lb m.l�netualtfln. W,thout

1 I,

Summary

337

Figure 8.32 A proposed spintronic device. where !tpin preces..ion in an external mag· netic field controls electron flow from one ferromagnet to another.

Source ferromagnet > •

-

1

Externally applied magnetic field

1

1

:� ���� Semconductor i

would arrive aligned with external influences. these electrons accepted. be However. an applied and ferromagnet the drain

ing its polarization, particularly at interfaces between

materials, are major hurdles,

Chapter 8 Summary

For an electron in an atom. the quantum numbers n. (, and m(

are said to specify the spatial state. while m, 'ipecifi� the "Pin slate. up or down.

Fundamental particles possess intrinsic angular momentum. spin. that is inherently quantized, The measure of this property is the value of s. a characteristic of a given kind of par­

called

c particle's intrinsic angular momentum S is

ti le. The relationship between this numerical value and the

I)h

For a multi particle system. the probability density must be

unchanged if any two particle labels are e�changed. Solutions of the SchrOdinger equation not violating this requirement are symmetric or antisymmetric combination'i of products of individual-particle states. For two particles. these are Symmeuic

(8-6)

quantized according 10 the value of a new quantum number. ms'

"" m/l

. = -so -s + I. . . . . s - l . s

"'

The electron is �aid to be a �pin-{ particle. because s and it follows that S i"

\111. 111

(8-21 )

Antj�ymmetri..: (8-121

The :-eomponent of a particle'� intrinsic angular momentum is

S.

-

spin current are being punued, and there is no shortage of

ye:t to be made workable, Other methods of controlling

Vs(s +

-

innovalive proposals. Practical problems loom, however. Efficiently injecting a spin-polarized curren! into a semicon­ ductor (the dominant fabric of modem circuits) and preserv­

magnetic field in the semiconducting region would impose a controllable precessional frequency and would thus control the degree of acceptance at the drain. Alas, such a device has

S=

1

Drain

ferromagnet

Although both give symmetric probabilit)" den<;itie, "hen

(8-9)

squared. they are of different character. Panicles .ill: man!' likely

=

in the antisymmetric.

t

= ; t and S. = ± {Ii. l ne·x­

tricably related to "pin i� intrin..i� ma�;etic dipOle mo�enl. and the relationship for the electron is J.ls "" - (f'lme)S. With S;: and

thus Jls: restricted to t....o . value__. the electron is mU�'h like a bar

magnet wilh only two p"� ,[hie orientation.. alon� a �i,en axi'i.

10 be found close together in

the symmetri..: ..lale and far apart

A system of indistinguishable particle<;

\10 ill

fonn a multi­

particle state of definite eJtchange symmerry--symmetric \ersus

antisymmetric-depending on the particle �pin. Panicles _hose spin

s is an integer fonn symmetric multipartide �tates .md are half-inlt'graJ form

known as bo:.ons. Particle\
antisymmetric multipanic1e �la.l:e<, and are kno\1. n as fennioCl5.

the

apparJtu�. I'.e can ,"hoo�c 10 block one or the"

8« .. UIt 1M 1IlI11-,}'mmcftll; comh....,"...n of1\100 \tillh \10Ith

lame

lull

Iod

01 quan.um numi:rc'n ,. l..kntlcaJI}, O. 00 '\100

Ind/l"nIUI�hl(" lC'nruvoi mI) (I!:cupy !he' <.ame: in
partKie \l;Ile: f:tnlrll.h ate (e:muon,. \0 no 11'.0 c:lC:ClIOn� in /b("

7.

" the: udUiwn PO"",pl("

8. Imagine IwO IOdl�tlOguishable panicles thai sharC' an

umt &(Clm rna)' pt"\('�\ Ilk- wnc <.<:1 01qU.llltum number" Thl�

l'u.:Il}, \Iollen multlple: Inl("rp.utldC' forrC" IlrC' imohc:d. NC"'C'r·

9.

tftcoku. untk,...unJlnl 01' mlk:h alomlc beha"ior 10110\10.. from

Of)("'pu1I�'1C' qU;lIl1um mtch.:tnlC\ and

Ilpphcatlon of me eRlu­

ont \oIuh

tn,

-

1-

'" +1

n. f. "" - I . O. 0, one \oI,m ",. W1d Th(" 8�JlI("( me number of elC'Ctron�, the

hlghC'( mu�t I"Il' the �'.'ue� of many

n

numbtr of .alenee eleelton) --weakl), bound ouler C'lecllon�­ 'hemi'lIl

re3clions

"ilh omer a.oms

I I . The "f""Jdlu) of an atom" i\ a dcblllable quantilY. Why?

deprnd, IOII/natdy on Ihe number 1I110"ed in thr vanous shrlb

12.

IlIIJ sub,he"II.. 'n-lInd (·v,1lu("s)_

Would you guess it to be more or le�, than twice as hard? Why?

13,

Conceptual Questions

Concisely. why i� me table pet/odic"

I". A good elC'clron thief need� a trap at low energy 10 cnlicc il.\ prey. A poor electron shepherd will have at

I. A dipole huhour angular momenlUm can stmply rolate

lea." '>Orne of its nod. dangling ou l al high energy.

10 aJIl!n \oIllh Ihe field (mough il would oscillale unles!> It could \h�d ene�y). One \oIuh angular momenlum

Con�ider to"� 2 and 5 in the periodic table. Why

can.not \'-by?

should fluorinC'. in row 2. be more reactll-e than iodine. m row 5, while lithium, rn row 2, j� les.. reaclile" than

Dot-\ cln;uhlling charge reqUite btJlh angular momen­ tum and magnelU: moment? Consilkr po�jlj\"e and neg­ afl�(" chali�� �Imultaneoud) circulatrng and

rubidium, 10 row 5? 15. Discuss wh:1I is nght or wrong about the follOWing �Ialement Noble ga ...es corre..pond to full shells.

cQuOlen.:u-culalrng_ J, Summanlt the" connC"f;lion be,ween angular momen.um

4_

SolvI08 (or aUcmpllDg 10 �ohe!) a 4-electron problem is not twice a� hurd II� �olving a 2-electron problem

ondlcales advanced questions

2.

�son or a

rermlon IS milependent of Z, Instead depending emirel),

A� individual atom\. would the�e behave: as bosons or as femlion�? Mlghl a ga� of either behave as a gas of bosons? Explnin.

'h allnlcll\t IllfCC' by IIlle""("nIn8 10l'.er.energy eleclron�. The III

ilnlJ\ymmetric, or neither? Explain. Whe"me"r a neutral whole a�om behales as a

10. In nature. lithium e.\i�(s i n (\010 i�OIopes: hlhium_6, With three neulron\ in liS nucku�, and lithium-7, with rOur

Ihe}' OCCUP). ConsC"quendy,

dec'ron\ may I"Il' I�r rom f t/lt' nucleus and SCrel'ned from

availllble 10 plll1l(1pJle

the simple"sl tenns.

on the number of neutrons In Its nudeu,. Why? What,li It lIbout th,\ number that detenmnes whether the alom IS a bown or a femlion'.'

'Ion pnn.:lplC' Onl}' t\o\o t1("\;lIOn� rna) occupy the 1()\I.�t· eMIJ)" \pJllal \1011("

III

("�p«l their mullrpanJ(le" ;l,p..1tI1l1 Slale 10 be symme1nc,

Tht' \d\rOOln�t'r '"'Iua"on c.lIInO( be whC'J

probk'mall�

pie?" E'plain

attraction All other things ocing equal, would you

Comple:'C' ch..,.;.C'nlahun 01 lhC' multld«lIOn alom I�

•

other f rarot

�lUd)". but aJ�o lIs\ume Ihat neither is del'iated too from the [("nter of the ,hannel_ Your tri�nil lI\h " Wh), i!o there an e:l;clusion princi.

16. Lithium is chemically re:Ktile. Whal if e1ectron� were

�

�

quantl/JltOn ilnd me Stem·Gerlach e�penmenl.

spin- IOSlead of spln- . What \'alue oJ Z would result

Compare and con.rasI UlC' angular mornen'um and mag­

in an element reacli�e in roughly the ,ame way as lithium? What ir eJectron� were in�tead spin- / ?

netic moment R'lated 10 orbtlaf mOl ion wilh those Ihal are In/rltlS/e

17.

5. The neutron comprises mulhple chal"8ed quarks. Can a partlclC' thai IS electrically neutral bul really composed momenl? Explatn your answer hydrogC'n ((

- 0) Stem-Gerlach appdl1ltu� of Figure

is aligne"d with

!he fi,-.,t bul totaled 90c about Ihe X-lUis,

in\tead of the";:. Whal would }'ou see emerging al me

rn Figure 8.16, to remove one of helium's 2�.6 eV of energy. h it\ energy \oIhen

electrons require�

/9.

orbiting -24.6 eV? Why or why no(' Early on, the lanthanide.. were found 10 be quile uncooperative when auempl\ were mJ.de to chemically

8 3, you place a \ccond �uch apparalus whose channel so that II!> B-field line� point roughly in the )'-direction

would be Z ror the first nobl; gas?

18. As indicated

of charged con�tituenl� hal'e a magnelic dipole 6, SupPlhe thai at the channeJ'� outgoing end in the

What i f electrons were ,prn-� in,tead o f \pin-t What •

separate them from one another_ One (ea�on can be

20.

seen

10

Figure 8.16. Explain.

Figure 8 16 <;hows that the n

.:

III

Ihe Z

end of your added apparalus? Consider the behavior of

trend

the \pin-up and spin·down beam� <ep.m!tely. Assume

Ihere a drop a\ Z gOC's from " to

thilt when Ihe�e beam, � �eparaled in !he first

10 boron?

III

�

J In 10 filling of

2 �hell (lithium to neon), there i, an upward element�' first Ionization energie,. Why i�

5, {mill beryllium

ZI. ro."

"" lr je\'ch fill, from

Z

""

5 to IO. lh� elemcnb'

fir.l lonilattOn energies tend 10 inc��, BUI. &.!> Ftr:urc 8.16 dearl)' mdicates. It j... slightly f'O,Hf'r to f'(n'lo,e an dectron from Z = 8 oxygen than from l .. 1 mtrogen What might explain thi�') of charactcri�tic X-ray.. start al lithIUm 12. Huge table... en or helium? hydrog not Why mind liS limiting cases of l and 2 mentioned g in Searin zj. ,n SectIOn 8.8, how would you descnhe the �ignlficance of me Lande g-factor

21.

2.&. 1l1e IOlal-lopin singlet state for two ele�trons has one

that the frequency .t ,..-hich an elecuon'lo intrim.ic moment would pn:cct.!o in I macnetic field ,\ gi..-en b) Cd " �8I,"�. Cakulllle thi" f� for a field of 1 .0 T. 28. In the Stem-Gerlach ell.periment. ho'w m\Kh would . hydrogen atom emanating from is 500 K oven (KE � n be dene(ted In tra\leJing I m through a m..gneti( field who-.e rate of change i1> IO T/m" 29. What angle� might the mtrin..i\; angular momentum \'tttor make with the =:-axis for a lkuteron'l

Win up and one down. but one of the tnplet �tales dOC!>,

too. What i!o the difference"

Shu.....

ma@netlc dipole

llca

(See Table � 1 . 1 30.

e'll.lemal magnetic field of 1 0 T What i.. lhe energy dif-

feren(e bet....een the <;pm-up and �pln-dO\llo'T1 "tate,," 31. The ..ubatomic omega partlde ha What .. "pin ""

s

Exercises

mak.e .....lth the :.-axis?

25. The electron is known to have a radius no larger than 10 '11 m I f actually produced by circulaung mass. its

32. Is Lntnn..ic angular momentum "real" angular momen­

mlnnS'C angular momentum of roughly h would imply

H�ry h,gh speed. e\'en if all thaI mass were as far from

(al U..ing simply rp (from Ir X pi) angular momentum of a mass at radius r. obtain the for

the axiS as po....lble.

a rough value of p and show that It would imply highly

relati\'istic speed. (b) At such speed... £ "'" ymc2 and p "" ynill combine to gi\e £ OIl pc (JUSt as for the speedy photon). Ho.... does thi.. energy compare with the known internal energy of the electron?

26. III c\a.ssi(al elccuomagnetism. the \imple ..t magnetic

dipok is a circular current loop. whICh beha\'e!o In a mag­ neuc field Just as an electric dipole � in an electric field. Both e'perience torques and thus have orientation energie:.. - p . E and - ,... 8 (a) lbe lksignation "onen­ •

lauon energy" can be misleading. Of the four cases <;ho n. .... 10 Figure

�.

angle.. might ib intrin�IC angular momentum ,'ector

section 8.1

\

A hydrogen atom in ,t .. ground ..tate: i.. !>ubjected to an

8.�. in ....hich . would work ha\'e to be done to

Ili,holft reorienting it? Briefly explain, (b) In the magnetiC case. using B and JI.

move the dipole horizontally

for Ute magnitude.. of the field and the dipole moment, rbpecliwly. how much won. would be reqUired to rome

the dipole a dis\an(e tLt to the left? te) Having shown that I rate of change of the "Qnentation energy" can gi\'e a force. now con..ider equation (8-4) Assuming that B and po are general. write

- ,... 8 in component form. lbcn. •

llotlOg Ihat ,... i.. nOi a I'unction of pD\ltion. take the nega·

u\'e gradient. {d) l't.w.. relerring to the s�cific magnetic field pictured in Figure

13.3. ....hu;h tenn of your part (c)

result can be dl'>Carded Lmmedlately"? (e) Assuming that po, and fJ., vnry periodLcally at a high rate due to preces­ sion about the :-3..-.i... ....hat

cI-.e rna)' � di<;carded as a\'er­

aging 10 0" (f) Finall�. .a.rguc that what )'01,1 ha\e left reduces 10 equatIOn \8-51

tum? The famous Eins�in-de Haas dIed demon�trate.. It.

Although it actually reqUire.. rather mvol\led

technique.. and high precision. con..idf:r a ..Imphfied

ca�e. Suppo-.e you h"\'e a cylinder 2 cm in diameter hanging motlonle!io!o from a thread cunne.;\ed at the ...el)' center of its circular top, A reprt'oCntame atom In the cylinder h3.!o atomic mas.. 60 and one electron frtt to re\pond to an external field. Imtiall). spin orientation" are as hlo.ely to be up as down. but a �lrong magnetic field 10 the upward direction i.. wddenly applied. cauv 109 the magnetic momen'" of all free electron!> to align wLth the. field. (a} Viewed from abo\'e. Which wa)

would the cylinder rotate? (b) What would be the 101t1a! rotation rate?

33. Figure 8.3 �hows the Stem-Gerlach apparatu\. It re'Oea!..

that sPJn- particles ha\le just two po'<;lble spm statC$. A..sume that when the-.e twO bums are .-.eparated in�lde the channel (though .,tlll ncar Ih c.enterline), ,,"'e can

�

rudy. �Q\I> a s.tt� choose to block one or the other fOf .. ond 'iuch apparatu\ is added a.fter the fJ.r";l, Their chan­

nel<. are aligned, bot the o;econd one IS rotated about the x-axis by an angle
.. of a proba­ tal

blhty amphtude. which we square to gi"e a probabibty. Aher the rUSt apparatu!io "pm-up heam � through

the <;ccond appar.nu,. the prohability amplitude , co..(cfIl2l Tad. + ...in\,f,/21

1."... where the: arrow

mdicale the two po.. ible fmdmgs

for sptn In the SCCCIDd

apparatu... (a) What 1� the probotbility of

ftndin, \be

particle spm up m the '-C\;ond, appatatu'i'll Of finding U

�pln dO\\on1 �ue thallhc-.e probabililics trl:lU �

indl\iduall)' fOf ��ntilti\e \-alues of til and lbll lbrn

length? (c) Di\cu\s Ihe rrlalion�hjp belween lour

um I also tenSJbk .bl8) conflaJlJnB 11m spin

�btlJf)' .unphfUIk ,uth • lpalia/ probahdJI} ampl.,udr. �lt4:h as .po(.f) .. At' �. argue thai although

answl."o in (a) and (b), 38. Two particle\ in

numbc-r of laJuC',. ,1 .\ �liJ/ JU'lltiC'd 10 refer '0 il a\ 8

2

'nlimle-'I..... ,)\,...m

39_

Section •. 2 . J.I. �hm� ,h;ll lhe \}mmelri�' and anli\}mmelri(' combin;)­ Il(1n' (If IH· / 91 and (S·:!O) are \olulion, of Ihe t\.\lO­

partJde S�'hr6dmger e�uJllUn (H· I .l) of/he ,arne energy

,1;l1C', lhl." pm�abi/iIY (hal both panIcle, \.\Iould be found

1

calculate the same probabihly rallO. bUI assuming that their multipatlicle wave function is either symmelric Or

In IhC' Idl ,idl." of the 00' ( I .e. , �tlleen 0 ruld L).

antlsymmetric. Commenl on your results.

Section

repre�enled as fimle well... and invesligale when Ihe

exclusion principle must be taken iOio account In the aceomp3nying figure. diagram (3) show� the four

Ihe 'quatC o( an)' individual-part.c1e lunclion. wch as

lowest-energy wave functions for a double finite well that representS aloms close logelher. To yield the low­

but for the

pmduci o( diffnrm individual-particle funclion�. �uch J�

4t;(.1 )4t,,'(x), I',e gel O.

8.3

40. Here we consider adding two electrons 10 Iwo "atoms."

" (lrthllOomla/"' Th3f i�. if we integnJle o,'er all �pace

�:(t)4t.. (,t) or 4t;.(.t)4t".(t}. we gel I.

Suppose you have two panicles OCcupying Ihese I\VO slates. (a) If d.stingUishable, an acceptable \\-ave function would be Ae-brl l/20x2e-bxll/2. Calculate

Ihm8s.) (b) Suppose now Ihal {he particles are indislin_ gUishable, Using the ;t symbol 10 reduce your work,

.161. ('.;l/culale for bolh the \}mmetrK and JJlti�ymmetric

bUI il ., not nonnJJiled. (a) In applying quantum

;., O't'· .... . .

2

nm\liJnr i, Ihe ,arne J' in E\ample 8.2 (�ee E�ercise

mechanin. lIe u,ually deal wilh quantum �tate� thaI are

The wale funcllons for the ground and first excited \Ial� o( a simple hannonic oscillator are A e-brl/2 and

bililY for bolh being found over all values of x, and.t . (This kind of nonnalizing-a...·we-go will slreamJine

md"Idual·pJrtKle 'Me\. Gilen Ihal lhe nonnali.aalion

W,*(.I,II/J,*.(.I�) !" W",(l,)WIl(X2).

(Not�: smlm7T(L - x,VLJ = ( - I )',,+ 1 sin (m1T.rlL).}

the probability Ih3[ bOlh particles would be on the pmilile side of the origin and divide by Ihe lotal prOba.

i1\ ",.. f 1 , )1#... (,,!,. the un,)mme/ri/ed produ,·, (8-17), 35. �II p.trtu;!c, in J box ocxup), Ihe II I and fI ' ,., 2

est energy. the firsl eleClron added to this system must

Th" happem 10 be rrue (or all

have wave funelion A and is .. hared equally between

tht: '>'�Iem\ '" which we have obtained or tOlbulaled sel� of WJve lunction, (e.g.. the particle in a box. the har­

the atoms. The second would also have function A and

monK o'cIIIJtor. and the h}drogen atom). As�uming

be equally shared. bUI it would ha....e 10 be opposite

that th" hold,. whal nlulliplic3tive comlanl would nor­

spin. A third would have (unction B. Now consider

mali/e the ,ymmelric and anlisymmelric functions?

atoms far apatl. As diagram (b) �how\. Ihe bumps do

(bl What lalueA gi\'e� the \"('(:Ior V

nO[ extend much beyond the alom..-they don't

1

A(i z

�I L r-� c

"j

=

B

..

I

II !

(. )

y)

unit

/0 J / � IJ TI II I

B

.

(el Make a �'mllar <:okelch of PA (x,. Xl)· (d) Repeal pa"� (b) and (e) bUI for point� on the line .12 = L - x,.

"",,).'1.1(' .\)\Iem." �hd(' 1M �pa"./ ('a.\(' ,\ an

"J\e h,"�'liun' i\

bo.\ have a Iota I energy 5 7r2f,�/2I'11L2

(a) Which Slale� are occupied'! (b) Make a �kelch 0/ p.�(x,. x2) veous -1", for points along the line .1 :OC r,

Iht .mlfntnnc'�\ (If fit gJ\"'� th... 'pin ('a!>C' an mtinite

36. \cn/}' Ih.Jl lhC' normalilatlon eon�tant gnen in E\ample � Z j, nlrrCCI for both ,}'mmelric and anri\)'mmetric \loJle, and .\ mdl."pendenr of II and 1/' �17. The gener.. 1 fonn for �ymmcrric and anli,ymmetric

11

I

(b)

overlap--and functions A and B approach equal energy. as do function� C and O. Wave functions A and B in diagram (b) describe essentially identical shapes in the right well. while being opposite in the left well. Because they are of equal energy. sums or differences of A and B are nOW a valid alternative. An electron ina sum or difference would have the same energy as in either alone. so it would be just as " happy" in A, B. A + B, or A - B. Argue that in this spread-out situation. electrons can be put in one atom without violati ng the exclusion principle. no matter what Slates electrons occupy in the orner atom. 41. What is Ihe minimum possible energy for five (noninleracting) spino! particles of mass m in a one­ dimensional box of length L? What if the particles were spin- I ? What if the particles were spin- ? 41:. Slater Determinant: A convenient and compact way of expressing multipanicle sillIes of lIntisymmetric character for many fermions is the Slater determinant:

44.

45. Exercise 44 gives an anti�ymmetrk multipanic\e state for two particle� In a box wilh oppo<.ite spin!.. Another antisymmetrlc !>Iate with spins opposite and the --.arne quantum numbers i�

!

0/1" (xl)m,1

iJI",(x!)m,\ iJi",(xJ)m,\ 1I<",(lN)m,l

o/I�,("I )ml•

1I<"1( x1)m,. 1I<"1(xl)m,2 1I<"1(x",)m'2

o/I",(.ll lm,3

1I<",(x2)m,l 1/1" (x3)md

,

IJ!",(:lN)m.l

The Slater determinant i, introduced in Exercise 4'2. Show that if ..late.. /I and /I' of the mfinile well are occupied. with the panicle in !otate /I being spin up and the one In /I' being "pm down. then the Sialer determinant yicld� the antisymmetric mu!tlpanicle �tate'

1/I..)xl)msN

1I<"�(x2)mJN 11<" (x3)m,N "

J

IJ!.. :l.v)m,.'1

It is based on the fact that for N fermions there must be different individual-particle states. or sets of quantum numbers. The ith state has spatial quantum numbers (which might be "" f" and mCi) represented simply by 'Ii' and spin quantum number nI�.. Were it occupied by the jth particle, the state would be 1/1" A column cor­ responds to a given state and a roW to a given particle. For instance. the fir!;! column corresponds to individual­ particle stale ��,('9m� 1' where ) progresses (through the rows) from particle 1 10 panicle N. The first row corre­ sponds to particle I, which successively occupies all individual-particle Males (progressing through the

N

Refer 10 these �tates a' I and n. We have tended to characterize exchange ..yrnmetry as to whether the state's !>ign changes when we swap panicle Labels, but we could achieve the same result by instead swap­ ping the particles' slates. specifically the II and /I' in equation (S-2'l). In thi!> exercise. we look at swapping only parts of the state-spatial or spin. (a) What is the exchange symmetry-!>ymmetric (unchanged). antisymmetric (switching !>ign). or neither--of multi­ particle states t and It with re!>pect to swapping spalial slates alone? (bl Answ'er the same question. but with respect to swapping spin state<Jarrow,:> alone. (c) Show that the algebraic sum of slates I and U rna) be written

(X)nlJj.

columns).

What property of determinants ensures that the multiparticle state is 0 if any two individual­ particle Slales are identical? (b) What property of detenninants ensures that switch­ ing the label� on any two particles switches the sign of the mu!tiparticle state? (a)

43. The Slater determinant i!o introduced in Exercise 42.

thOt if state, /I and n' of the infinite well are occupied and both �pin\ are up, the Slater determinant yields the amisymmetrio.: multipanicle state:

Show

where the left arrow in any couple reprC!.enb the spin of particle I and the right arrow that of panicle '2 (d) Answer the same questions as in pans tal and (b). but for thi!> algebraic sum. (e) Is the sum of ..talh I and II �till anli�ymmelric if we swap the panide..total-spatial plus spin-states".' (f) If the two parti­ cles repel each other. would an)' of the three multipar­ tide states-I. 11. and the sum-be preferred" Explain. 46. Exercise 4S refer" to state... I and II and puts their algebraic sum in a !>imple fonn. (a) Put the algebra!.: difference between these state:> in a "imilarl) §imp!e form. (b) Repeat part.. \dl and lel of Exerci-..e 45 but for the algebraic difference. "Upy ... 47. A lithium atom h3.!o three electrons. � roo.: indi\idual-panicle "tate" corre...ponding to th
= ( 1 . 0, 0,

+�). ( t . O. O. -�). and , ,:!.O O. +it l stn1

'" 0.0 (r," , "1.1.1 0 Ir}'l . and "2 O. 0 I�)' to represenr lM Ind.VlduaJ·pattKle u.teJ when occupied by

�d�J, apply the 51<1lel detemufWJ' w!.CuSliCd in fucn",w 42 10 lind.an e.lpreuuJII frY JIB antJ�ymmetn' mulbJllld lh e· � Your answn- �kf be sums of It'nm like

Section 8.4

48, Tu Ime�"gale thC' dillm Ihill /owC'r ( impliC'5 101l,el

single elecutln in a one-electron atom of Z = 3. Find the C'nerg)' and orbit 11ldius. (b) The other n = I electron ' being in !he \ame spatial statc. must have the same C'nerg)' and 11ldiu�, but we must �CCOUnl for the repulsion befween Ihe�e electrons. Assuming they are roughly on e om/I diameter apart, what repulsive energy would they �hare, and If each cI:um� half this energy. what would be­ the energies of Ihese Iwo electrons? (c) Approximately whllt charge does lithium's lone valence electron orbit, lmd what radius and energy would It ha�·e? (d) Is /1 reasonable to dismiss the role of the" = I ele<:trons in chemical reactions? (e) The actual energies of lithium's electrons are about -98 eV (twice. ofoourse) and -5.4 eV. How good is the model? (0 Why should the model's prediction for the valence clew·on's energy differ in the direction it does from thc actual value? 55. Consider Z = 19 POl3ssiulll. As a rough approximation' as�ume that ellch of its n = I electrons orbits 1 9 pro. on- that is, on average, half its IOns and half /HI e Jecl.r I electron. Assume that each of its If = 2 fellow " = elecltons orbits 19 protons, two Is electrons, and half ofthc seven other n = 2 electrons. Continue the process, assuming that electrons at each n orbit a COITC'_ �pondingly reduced positive charge. (AI each II, an elec­ tron also orbits some of the electron clouds of higher n. but we ignore this in our rough approximation.) (a) Cal­ culate in ferms of 00 fhe orbit radii of hydrogenlike atoms of these effective Z. (b) The radius of potassium is oflen quoted at around 0.22 nm. In view of this, are your 11 = I through " = 3 radii reasonable? (c) Aboul how many more protons would have to be " unscreened" to Ihe n = 4 electron to agree with the quoted radius of potas�ium7 Considering the shape of its orbit, should pofassium s " = oJ electron orbit entirely outside all the IOlVer-n electrons?

C'nel){)', l:on\lder II \lmplC' l'll�e: lilhium. which h/L.\ two " .. / efectron' dnd a lonc " :! "/lienee e1�·lron. (al l-j"t lind Ihe JPprO�lmale orbit r3dlU�, in tenm of flU' ulltn n • J electron orblling thret' protons. (Refer to SedillO 7.S.) (bl A��umJflg the n = I electrons ,hlefdkancC'1 nut two 0' the pmlOn� in lithlUm'oS nueleu\, find IhC' orbil r.ldiu� of an" 2 electron orbiling jJ nel char,!:C' ofJU" +t'. (e) Argue that "thlUm" Hilence C'1t"-"lron ,hould n·rtainly ha�e InM.er energ) 10 a 2J ,rate Ihan in iJ 2p �tOlle. (Refer to Figure 7.I�U 49. Wnle the efec:t({lfllL' contiguralion� for pho�phoru,. ger­ manium. and l'e�/um, SO. Element 1 17 ha' nc\er �en found. but what would we e'pc..·1 if, "Olience 10 be') 5._ Were il ro follow the \Iandilrd p<.lfIem, what would be Ihe e!ednlOlc confi.l!ur.llilln of element 1 19? 52. The radiu, of !:c\jum j\ mU.l!hly 0.26 MI_ (a) From this. e�timllte the elTectlve charge ifS \aience electron orbits. (b) Given Ihe nalure of the electron'� orbit, is this etle!:{ive nudeilr!:harge rci/,onable? (c) Compare this efJ�·llve Z wilh Ihnl obtained for �odium in Ex4tmple 8..1 Arc the \'111ue\ ill odd" with the evidellce giv'C'n in Figure 8.16 1hal il tllkes bs energy 10 rt'll'love an eleclron from ee�jum lhtm from sodium? Section 8.5 Expl...in 53. Con,ider row -* (lfthe periodic rable. The lrend is 56. Rank the following lines according to increasing wave_ length: Ka, Kp' and La' Explain your answer. Ihat tht' -*� oSub'hell fills. then the 3d, then the 4p. 57. The Ka line in copper is a very common one to use in (3) Judging by adherence 10 and deviation from this X-ray crystallography. To produce it, electron s are lrend. M.hOlI might be �aid of the energy difference accelerated through 3 potenlial difference and smashed between Ihe -*1 and 3d relaliye 10 Ihat betlVeen the 3d and -*p_� (b) h Ihi� al\o true of roM. S·) (d Are into a copper target. Section 7.8 gives the energies in a hydrogenlike alom as Z 2( - 13.6 eV/112). Making Ihe Ihe�e obser\'atlon� In qualilalrve agreement with Figure 8.1J? Ex-plain reasonable approximation that an II = I electron in 54. A Simple Model: The multielectron alom is unsolyable, copper orbits the nucleus and haljo f il:. fellow n = I bul iimp/e models go II long way, Section 7.8 gi\'es electron, being unaffected by the roughly spherical energie\ and orbil radii forone-electronlhydrogenlike cloud of other electrons around it eSllmale the minimum accelerating potenlial needed to make a hole atom�. Let U� \t'e how u\efuJ the�e are by considering lithium. (a) Treat one of lilhium's n = I electrons as a in copper's K shell. ""

'

.

of electrons accelerated in an X-ray sa. Using a beam

3p�12 ----,--

tube, we wish to knock an electron out of the K shell ora given element in a target. Section 7.8 gives the

3P /2 I

energies in a hydrogenlike atom as Z2 ( - 1 3 6 eVjn1). Assume that for fairly high Z. a K-shell electron can .

589.0 nm

be treated as orbiting the nucleus alone. (a) A typical kY. In accelerating potential in an X-ray tube is roughly how high a could a hole in the K-shell be

50

Z

589.6 nm

produced? (b) Could a hole be produced in elements

,, ---'--''--

of higher Z! 59. E�timate Ka characteristic X-ray wavelengths: A hole ha� already been produced in the n = I shell, and an 11

= 2 electron is poised to jump in. Assume that

the electron behaves as though in a " hydrogenlike" 310m (see Section 7.8), with energy given by

�ff( - 1 3.6 eV/n2). Before the jump. it orbits Z protons,

one remaining II == I electron. and (on average) half its seven fellow II = 2 electrons. for a ZetT of Z - 4.5. After the jump, it orbits Z protons and half of its fellow 11 = I of - 0.5. Obtain a formula for II), electron, for a

Z.

Z�ff Z

Compare the predictions of this model with the versus eo::perimental data given in Figure 8.19 and Table 8.4.

TABLE 8.4 Ka X-rays Z

A I(nm I )

20 (Cal

3.0

30 (Zn)

7.0

40 eZr)

1 2.7

50(Sn)

20.3

60 (Nd)

29.9

70(Yb)

4 1 .8

Section 8.6

60. The well-known sodium doublet is two yellow

spectral lines of very close wavelength. 5 89_0 nm and 589.6 mn. Ie i, cau.sed by splitling of the 3p energy level. due to the spin-orbit intel1lction. In ils ground slate. sodium's single valence electron is in the 3s le�el. It may be e:ll.citcd to the neXI higher level. the 3p. then emit a photon a� it drops back to Ihe 3s. However. the 3p is actually two levels. in which L and S are aligned and anlialigned_ (In the notation of Section 8.7 these are. re<,pectively. the 3P3n and the 3Ptn.) Because the tNn�itlOn� �Iart from slighlly dif­ ferent inittal energie� yet have identical final energies

---,-f---

(the 3_f h3\11ng no orbital angular momenlum 10 lead to spin-orbit interaction). there are two differenl wavelengths possible for the emilled photon. Calcu­ late the difference In energy between the two photons. From this, obtain a rough value of the average strength of the internal magnetic field experienced by sodium's valence electron. 61. All other things being equal. should the spin-orbit inter­

action be a larger or smaller effect in hydrogen as n increases? Justify your answer.

Section 8.7 62. Identify the different total angular momentum states

U. m} allowed a 3d electron in a hydrogen atom.

s,

L and S cannot be exactly opposite: mat is, show mat at its minimum possible value. for which) ::- e - me magnitude J of me total angular momentum is strictly greater than the differ­ ence IL - 51 between the magnitudes of the orbital and intrinsic angular momentum vectors. 64. Assuming that the spin-orbit interaction is not over­ whelmed by an external magnetic field. what is the minimum angle the towf angular momentum �ector J may make with the z-axis in a 3d state of hydrogen? 65. What is the angle between L and S in a (a) 2pJt2 and (b) 2Pln state of hydrogen? 66. Whether adding �pins to get tala! spin, spin and orbit to gel Iota! angular momentum. or tola! angular momenta 10 gel a "grand total"' angular momentum,addition rules are always the same: Given 11 = hv' )J U] + I ) and 63. Show that unless C

12

)2 U2 :: ft v'

+

=:

s.

I ) . where 1 is an angular momentum

(orbital. spin. or rotal) and) a quantum number. the

)-,!..ir + I ). where)T may take on any ftv' i i integral steps: and l t -hi and)] + i� n for each value of J.,.. JT, = ffl -rfi. where ffl)Tmay lake J total is 1T =

value between

on any of 'l)T + I ��ible values in integraJ steps from -iT'o +i1' Since separately there wQl.l.ld bt 1j, + I

Me Cheptet . Spin and AtomIC Pby�1C1 pouible vaI� for mIl and 2)l + I (or m l' the total j number of SlIIeS should be (2il .. I )(2j� + I). Prove II; daat 1S. show that the sum of the 2), + I value\ for mIT over .U lht allowed vaI� for Jr i� 12h + I )(2h + I ).

po....ible �pin-orbit energies in the 3(1 state (see Exerci..e 60). Dctemline the splitting of the s00· lurn doublet (the energy difference between the I.... photon\). How does it compare with the line SO pIIttlng or pan (d), and why·l

(No": Here we prove IR general what we venficd in

67.

�.)

EumpJe S.S for the spc'Cial��i l :' 3, i: = The general rule for addIng angular momenta I" gi\en in bef(l� 66 When adding anguillf" momenta with

= �, (a) whal are the po..sible value� of the qua-ntum number}r and the total angular momen­

add two, then add the Ihin!.

�

69. The hydrogen ..pin-orbit interJ.(tion energy given in equation 01·25) i.. (IJ.nf"2/4rrm;,-') S ' L. U!oing a rea..onable. \-alue for

- \"

r in term� 01" 00 and the rtlation\hips

\'Tft';- )

and L .= I '" �how that this energy 1\ proportional to a typil:al h)'drosen atom energy by the factor u:, v.hert a i.. the fine \truClUrt constant.

S

70. Angular m()menta J t and J1 interact so that the)" obe�

the �tnct quantum mechanil:ru rule� for angular momentum addition. If il = I and h = , what angles Ix'lWetn J I and Jz are ullowed�

�

Section 8,8 71. Repeat Eumple 8 6, but a"ume that the upper slate is

76,

netic,; field were now IRtroduced (weak enough that it doe!> not dhlurb the ,pin-orbit coupling). into how many diftcrent energie� would each or these \tates be split1 73. The angles b(tween S and I's and between L and ilL are 180� What is the angle between J and IlJ in a 2P3!2 'itate of hydrogen·! i 74. The Zeeman effect occurs in �odium just as n hydrogen-sodium" lone ls \·aJence electron beha\·cs much a� hydrogen'.. 1.\. Suppose sodium atoms are Immersed In a 0.\ T magnetic field. (a) Into how many le\els is the 3Pln level split? (h) J)etennine the energy �pacing between these Siaies. (c) Into hO\\ many line� is the 3Pln to 3s112 spectral line split by the field? (d) J)e�cribe quantitatively the spacing of these lines. (e) The ..odium doublet (589.0 nm and 589.6 nm) is two spcctral line.., 3Pm -+ 3sln and 3Plf2 -+ 35112' which are �plit according to the two different

equation thai precede" it. A hydrogen alom is �ubjected to a magnetic field strong enough 10 completely overwhelm the spin-orbit coupling. Into how many levels would the 2p level spI· II, . and what would be the �paclOg between them?

B

Section 8.9 What is the angle between the spins in a triplet state1 18. In its ground state, carbon's 2p electrons [nteraCl lo pro­ duceir = O. Given Hund's rule, what does this say about the total orbiw/ angular momentum of these electrons? 79. In it:. ground �tate, nitrogen's 2p electrons interact to pro­ duce h = Given Hund's rule, how might the orbital angulllT momenta of these three electrons combine? BO. A beam of identical atoms in their ground state is sent through a Stcm-Ger13ch apparatus and splits into thr� lines. Identiry possible sets I s7'£rJ of their total spin and 100al orbital angular momentum? Ignore possibili_

77.

�.

ties in which sr is 2 or higher. 81. (a) Show Ihal, taking inlo account thc possible

:-componenls of J, there are a total of 1 2 LS-coupled states corresponding 10 Is2p in Table 8.3. (b) Show Ihat this i� the same number of state') available to two

the 2P lfJ rather than the 2P3n. 72. The "pin-orbit interaction ,pi its the hydrogen 41state

into many. (a) Idenlify these �tates and rank them in order of IOcreasing energy (b) Ir a weak external mag­

,

=

i, = 2 and h

tum JI' (bl huw many dIfferent �tate� are po\sible, and te) what are the VI' mj ,) value� for ench orthe� stale�"l 68. UIoin, the !leneral rule for addmg angular momenta dis­ cu\\ed in Se�llon 1t.7 and further in Exerci\t 66, find the allowed ,·aluelo ofJT for thru spin- reonion'>. First

= L! + 51 + 2 L 5 to eliminate L · S " �,ll,, L = Vi(1 + l )fi. S Vs(s + 1j� J= V iU + I)", obtain equation (8-32) from th�and

7S. Using!!

82,

electrons occupying the Is and 2p ir LS coupling were ignored. A.. i.. done for helium in Table 8.3, determine for a carbon alom the various states allowed according to LS

coupling. The coupling is between carbon's two 2p elec­ trons (its filled 2s subshelt not participating), one or

which always remains 1 0 the 2p slate. Consider cases in which the other is as high as the 3d level. (NOTe: Whell both electrons are in the 2p, the exclusion principle re�tricts the number of states. The only allowed states are th<>"t in which STand (T are both e ...·en or both odd.)

Comprehensive Exercises 83. Relati"i�ti(: effects are rather )mall in the hydrogen atom, but not so m higher-Z atom�. Estimate at what value ofZ relativistic effec� might alter energies by about a percent and whether it applie!> equally to all orbiting electrons or to some more than others. For this crude guess, it

i� acceptable to combine quantum-mech,lnical results

you have leamed, related to energy, angular momen­ tum, and/or probable radii, with some classical relalionship�. �. la) E�timate the repulsive energy between helium's electrons. Do this by comparing the energy required to remove the first electron, 24.6 eV (see Figure 8.16), then the <,econd from the remaining hydrogenlike atom, to the energy required to remove both if there were no repulsion-that i.., if both electrons behaved simply as though in a hydrogcnlikc atom. (b) How far apan would tWO electrons have to be to produce such a repulsive energy? (c) How does this distance compare with the approximatc orbit radius in the Z = 2 hydrogenlike atom? 85. The 21 em Line: One ofthe most imponant windows to the mysteries of the cosmos is the 21 em line. With it. astronomers map hydrogen throughout the universe. An imponant trail is that it involves a highly forbidden If'J.nsition that is, accordingly, quite long-lived. But it is also an excellent example of the coupling of angular momenta. Hydrogen's ground state has no spin-orbit interaction-for f = 0, there is no orbit. However, the proton and electron magnetic moments do interact. Consider the following simple model. (a) The proton sees i15elf surrounded by a spherically symmetric cloud of Is electron, which has an intrinsic magnetic dipole moment/spin that. of course, has a direction. For the purpose of investigating its effect on the proton, treat this dispersed magnetic moment as behaving effectively like a single loop of current whose radius is ao' then find the magnetic field at the middle of the loop in teons of e, II, mI" �o' and ao. (b) The proton sits right in the middle of the electron's magnetic moment. Like the electron. the proton is a spin-� particle, with only two possible orientation� in a magnetic field. Noting. how­ ever, that its spin and magnetic moment arc parallel rather than opposite. would the interaction energy be lower with the prOlan's "pin aligned or antialigned with that of the electron? (c) For the proton, gp is 5.6. Obtain a rough value for the energy difference between the two orientations. (d) What would be the wavelength of a photon that carries away thi!> energy difference? Comp utational Exercises 86. Screening and Energy Dependence on ( :

in Section 8.4. we claimed that the effects of screening by inner electrons were such that for outer electrons. a lower ( would have lower energy. given the same fl. Here we investigate the claim. We begin with a form of the SchrOdinger equation already adapted to the task. Computational Exer(;\<;e 7.92 lays out the steps that

tram.rorm the hydrogen atom radial equiltion (7-31) to the following: -

d'f(x)

d..-c2

--

+

[1(1 '

+

, .r

I)

'-

2

}

)(

-

x) = Ef(x)

--

The function.Jtr} 1<; actually rR(r). and the uansfonn8· tion give.. u.. a syMem of units where )( = 1 is one Bohr radiu!> and an energy E = - I is the hydrogen ground-Mate energ.y -13.6 eV. Computational Exercise 7.92 solve!> lhe equation numerically and verifle<; lhat phy..ically acceptable solutions ocm whon (E. f) = ( - 1 . 0). ( -1. 0). ( -J. I ). H.o). (-\. I ). and (-\. 2)"h" is. tho Is. 2s. 2p, 3$, 3p. and 3d. Notably, energy depends only on n. A common way of anaeking multielectton atoms is to assume that a given electron orbits the nucleus plus a rather inert cloud of other electrons. The second term in brackets in the above differential equation is the electro.static potential energy term. To handle a diffuse charge cloud, we need only insert a Z(xJ in this term, accounting for the potential energy '"feIC by an electron at a certain distance from the origin. Actually. we in'>en it in a fonn obtained later in Exercise 7.92 that further adap15 the differential equation to a numerical solution.

.,, { f(l;

j(x + �x) = 2f{x) - f(x

I)

_

�x) +

2Z;X)

_

},)

E

We study lithium's lone n = 2 valence electron. To a good approximation, the effective Z produced by i15 three nuclear protons and the roughly fixed cloud from ib two = I electrons is given by /I

Z(x) =

7.5exp(- 126x) - 5.5 exp(-1.1 1x) .... I

(a) Plot Z(x) from.r = 0 to 10. Does it make sen;;e at the limits':' (b) Lithium's valence electron is not allo�ed in the full Is level. whose twO electrons have energies of roughly - 1 00 eV, or about -7 in the units orlhi!> exerci<;e. Following the procedures laid out in Exercise 7.91 and u�ing Figure 7.15, wtU,.ch plots the SqUL1� of rR(r). a guide. try values of E and t until you find phy�ically acceptable �olutions whose <;hape is what e'\pect of 2s and 2p !itate'!>. (Plotting out to x = 15 j, �umcient.) Do these n = 2 !>tate'!> have equal energy? Can you c'

we

9 Stati stical Mech a n i cs Chapter Outline A Simple Thermodynamic System Entropy and Temperature

9.1

(;l

9.2

The Bolltmann Distribution

9.3

Classical Averages

9.4

Quantum Distributions

9.5

The Quantum Gas

9.6

Massless Bosons: The Photon Gas

9.7

The Laser

9.8

Specific Heats

0 9 .9

T

hink of it! We're surrounded by countless air molecules. whose velocities we cannot know. much less controL If they all moved

e direction. we would be in lrouble. Why don't they? Whenever in the sam particles free to move about. the answer is always the there are countless samc_probabililies. In statistical mechanic.., OUf concem is making predictions about properties and behaviors in systems where the number of particles is huge­ I�pically Avogadro's number. We cannot be absolutely certain about things.

simply because it is impossible to know what each particle is doing. Fat:ed with incomplete knowledge, we make statistical predit:tions based on the applicable law� of mec/wl1ics--dassit:al or quantum. Net:essarily. our predk­ will be observed is the average of all the things that might possi­ tion of what bly ocwr. Air pressure. for instance. a smooth effet:1 at:t:ording to human senseS. we find 10 be pcrfet:tly explained as an average of fort:es exerted in iso­

lated mit:roscopit: t:olli�ions.

Though it may seem ironk, sint:e increasing the number of particles

would appear \0 increase the amount of infonnation that we dOli '/ know, statis­ tical mechanit:s i� able to make suct:essful predictions. indeed becl/llse the

number of particles is large. Averages over very large numbers are mut:h more

precise than those over �mall numbers. The average force per unit area exerted on our skin by the t:ountle�" air molet:ules in a room is so extraordinarily pre­ dicl3ble as to seem con�lanl. If rooms contained "only" thousands of pan.icles

exerting the same average fort:e per area by t:oltiding more energetkally. the effect would be a terrifying <,cries of sporadit: blows. Consider another exam­ ple: an experiment in v.hich a lone volunteer is placed in the t:enter of a long hallway. What will be the location of the " particle" later? Obviously. a predic­ tion is almo�t certain to be wrong. There are many things the subjet:t might

choose to do: walk one way. run the other. stop to think. But if Avogadro's

number of volunteer..

were ..omehow available and released in the hallway.

347

"a. ..

.

. 11 " " • predirrinn oflhe awl'd/l' location

of the ··particle..·· at a later time .....o Jd i... they didn't decide 10" "tid, ntly-Iha! b" independe �. If lhcy IDO'o'ed . they wherc were prel'I'-Cly rele \Cd be' d l u wo locallon ..IUje .. ...'---dJe U" .veras!: . ... gl\en \peed 3!!. Just a IDUI)' would go one way down the h.111 at a ndently. there might be ..o down the other. Even if they didn·t. move mdepe accurate pred iction . all e pmdul: l il l s WIly to model the correlauon and I" large enough that predic. � e c l i an p of A sY51C'm in whi�h the number . s),stem. It is only In ynamic thermod a nons bc\:omc "ell preciS(' IS known a� pre....ure. te, of " rtlc\ propc ..perae such macroscopIC systems that the a\crag ule The molec o ng. s f aa ln ha\e real meani IUI'C'. and density (01 concentrdtion) . ' . the photons wllhm a bed of glow-_ wire. g ctm u d on c 3 In ons electr the a room. ll: \)"stcm. Two colliding electron, 109 coa.h.�ach constllUtes a thennodynam

'

WOUI� !;

do nol. or "modem." phys ics in the Statistical mechanic .. i.. not nonc!:I....ical. echan ics aTe. Rather. it is a m um quant and ity relativ � S('nS(' thai 'pecial r e;. classical as we ll as quan he t o many to � distinct area of physic.. thaI applie it now ?" is that we will study y Wh lum. The answer to the obvlOu" que�lioli ire it. A gas laser is a requ that cs phy�i soon encountel branche.. of modem ductor is a thermody_ n o ic m e s . and 3 thermodynamic system of ga' molecules . lattice namic system of atom .. bound In a solid d mo<,tly with u ndersta ndi ng energy erne conc be Will In this ,hapter. ""e on specifies the fmctlon of a system 's pani­ dislributions. An energy di�tribull \Iate. General ly speaki ng. the higher the cles that will be in a gis'en energ)' that stale. To will be the nu mbe r of particl e s i n energ:y of a ..tatc. the smaller . let us begi n ever how s n io ct ble predi ..oow how large numbef"'; lead to relia d o nam ic sys­ m r y he t ion In one of the simplest with another klnd of distribut

_

"

tems imaginable

.

.

9.1 A Simple Thermod yn am ic System

.

epicted i n Figure 9 1 . by an imaginary li ne as d . Consider a box. divided in half e back and forth. An mov to free are that ic1e� part t n i o p N and containing d iv ided into right and ecules in a room that is example would be the air mol e d istribution we a!) the "two-sided room." Th left halves. which we refer to one: the number of r d istribution . bUI a simpler consider here is nol an energ. e bother wit h so "triv­ reader may wonder why w particles on a given side. The on average half the particles should be iar a case. The obvious answer IS that. but the ease with an",wcr that is i mportant her e. on each side. It isn't the Simple lOn are negligi­ ibut i!)tr d d iation", from the expecte whil'h we can show that dev sible "'states" pO!o. C idea is this: There are onl y two ble when N is large. The baSI g on the left-and the mg on the right side and bein of a given air molecule-be But If the room contained er of thel)e states is probability of being in ellh are o n each side-the ld not be �afe 10 say that half only four molecules. it wou It would be safe to say one side is. i n fact. 12,S 'k probability of all being on c number of molecules. contained a truly macroscop' this. however. if the room ilities, and ns, distributions re.!lt upon probab Being statistical predict io .. arranging {e.g ething bers of ways of dOing som probabilities rest upon num

.

Figu,.. 9.1 A two-sided room.

!.

Q t A. Simple Thtrmudymumc System

' 0

WZ

of ple� in 5 ba.\.!i.ets). For our two�sided room. the number of ways. li ap olecules so that NR are on the right side IS given by the :' (. IIl$lng N air

�

Flgu... 9.2 Number 01' way" of distributing four partide... on two s.ides..

l roefficlent: bioornia

Note: The reader who

349

\��" \

(9- 1 l

N NR

is not reasonably familiar with probability and stalis�

4

0

;iO is strongly encouraged to read Appendix J. which discusses the above for�

along with :several other basics of probability theory.) For instance. if N were 4, there would be only one way of obtaining no molecules on the right (NR = O}--that is all four on the left. as shown at the top of Figure 9.2. mula,

,id\!

Equation (9-1)

agrees:

.

0 _- ( 4 ) - � _- I

wJ

0

�

O! 4!

The number of ways of obtaining one molecule on the ri ght (NR be four. since any of the particles might be on that side,

IV!1 �

(4 ) I

=

� l!

3!

�

=

1 ) should

4

And there are six. ways o f obtaining two on one side.

Wl

=

(4) 2

4! =

2'2'

� 6

G) @ 0 : 0 @ : 0 o 0 0 :0 o :0 (j) :0 0,0 0 : 0 0' 0

Now a point fundamental to statistical mechanics: All ways of arranging specific particles on given sides are equally probable. ! But a state in which the distribution of particles is uniform. regardless of which specitic particles are on which side, is the most probable. simply because there is a greater /lumber of ways of obtaining it. To see how large numbers lead to a precise average. consider cases of n creas ing N. For the case N = the numbers of ways of obtaining the five i distributions with different NR are plotted in Figure 9.3(a). The total number of ways is 1 + + 6 + + I = 1 6 and is approximately the area under the molecules. Of the curve. Figure 9.3(b) plots numbers of ways of arranging � I possible distributions, the maximum still occurs at N = N, but it is milch R larger I O I ! rather than 6. Even more important, the distributions in which NR N constitute a smallerfraction of the total number of ways of dis­ is far uibuting molecules-again. the area under the curve. In other words. the

4.

4

-from �

4

40

�

�

curve is peaked more sharply at NR = N than in the N = 4 case. The trend continues for N = 400. Only a tiny fraction of the total number of ways is far from the NR = � N distribution, so the probability of fmding a significantly nonunifonn distribution i� very small. For the macroscopic case of 1013. shown i n Figure 9.3(d). the total number of ways occupies a minuscule region

l Eacb \II,,} occur<; "Ith a probability of I 1 IN i x

� x

= 1

bcCilU:IC lhe pmhabllny of a

!!i��n particle beingon a gi\en �Hk

I�! and Ib£

lOla! probability i\ tb.: produ(l, �lncc p;artic1c� by l!Mumptlon. mcwt Indc:pendrolly

.. _ 01" ", ..... .."" ,... '.3

cle\ on two sides of a rOOm_ Number of \1iap of di\(ributing parti

variation as loW numbtr of

paJ1id('� inCre3M!\ from " to

\ .4 X

JOl l

IO�3.

•

•

� "

� •

•

•

N "'

0 •

, •

4{)

(.1 1.0 )( 10" N = 400

,<)

around

NR = �N. True, there is. a possibilit)-' of the air in a roobm b�I. ng pre­ ll IS so extremely unlikely as to .

i

. bUi e d smiss dom·mantiy on one side ed, al vidu tic h indi par le Eac has conclu\ion: robability We have reached our its IWO possible " slales," left or r. g t, a nd the of being in either of of ed have hal f in each state if I· t IS a truly inde l wil s icle part ther_ distriblltion of , tem sys ic am modyn _

t

�

states Microstates and Macro

:

::

ning SOm . mp ta t provides a good framework for lear The two-sided room or n tin cial dis cru the ction b ween d min in p kee us s help th terminology that · c. A hown ·m F'l�ure 9.3( d), in the copl cros I � e t h d . an . f microscopIc , == lon in whi ch ways of obtammg the dlstnbut x ur 103 are e ther . case re state is the sta of e t microstate. The m icro sys­ Each "way" is a differen the states of the in 1' , of dge wle kno ic cop ros mic �ldual tern, given complete m. the individual-particl e slates are roo ed -sid two panicles. In the s 9"4 ' classical cases in general, as discus ed in SectIOn " Ieft" and '·right." (For . re · e. Igno h her we C h For qua . y, w I mpass '.IeIow the Slate may also enco .) um st ant ate a qu lly ura nat is rticle siale cases. the individual-pa tion is unknowable N such microscopic informa o Realistically. however. · lecules crossing the room and know W hlch mo of k trac p kee real observer could (J!2 microstates correspondin g to side, The 1 03 x I R ones are on which me that it is plausible 10 kn0W M assu We ble. sha gui . would be unerly indistin . of molecules on the two sides, but this COn­ n, rallO cent con or r, least the numbe know the macrostate of a syste y the macrostate. To stitutes knowledge of onl exacl microscopic states that do not depend on the is 10 know the properties

��

2� N 22N. \ N ==

�

Simply mum

N == !

,N

;

9.2 Ea""PY .... .,..",.... 311

('\d)' tndlvidual

partic!t:: the overall properties of number. energy. and volume

wei the average local properties of pressure. temperature. and particle concen­ macroslale is that in which the pat_ = uauon. In the two-sided room, the

N

NR !N

centratIon I� three (nnes as large on the left side as on the nght-a ratio tide con = '" case, four microstates correspond to thj, macrostatc. The lll" � 10 - In the \. ":= 0 macrostate ha<; a vacuum on the right. There is only one way to do thi,; macro'tah! = o y one micfO:,tute corresponds to this maCfO<;tate. The a:-. equal particle conccnl.rutions on the two sides and IS the one ob!ainable 1 0

!J

1

N

h

NR � N

N

1 = 4, and IOJ> 0.:' for = "reatest number of microscopic ways: 6 for tht . . of number greatest the to corresponds which of all microstates, II use 1()23. Beca equally probable, it is the most probable macrostale. The mo\( are assumed probable macrostate is what we know as the equilibrium state.:?

Equilibrium If left alone. thermodynamic systems eventually reach equilibrium, a macro­ ,{;lIe whose macroscopic properties of pressure, temperature, and particle concentr'dtion do nO[ change with lime. Begun in a IIonequilibrlllm macrostale, one for which the number or ways/microstates is relatively small, a l.hennody­ namic �ystem will pass through other macrostates, its macroscopic properties \arying in space and time, while inexorably approaching its unchanging eqUI­ librium state. In the two-sided room, for example. if air were released on one 'iide and allowed to undergo free ex.pansion, both the micro�tate and the macrostate would change-individual molecules would move from one side

10 the other and an observer would certainly detect the macroscopic wind. But

the air eventually altains a macrostate of unchanging uniform distribution, because it is extremely unlikely that it would not sometime later. and for all lime thereafter. be in one of the huge preponderance of microMales corre­ sponding to that most probable macrostate. When this equilibrium point i"

reached. the microstate still changes-molecules still move about the roolll­ but the macrostate does not. And if the system's macrostate is the same for­ ever. so are i13 macroscopic properties. This unchanging. overwhelmingly most probable macrostate defines the equilibrium state.

In a more general system, where individual particles have available to ellergy states, we might initially distribute the total

them many different

energy in a strange way. perhaps putting half the particles i n their ground slateS and half in their 37th quantum levels. But, in time, the system would assume its most probable state, with a predictable fraction of the particles in each individual-particle Mate. It is such equilibrium energy distribution ... that

:'-fl.t word ""1 ntarly alwilY� rtft.... 'o �

wa},," lU'l: ,hu, redundanl but ""' often u'l'd for

mlcro�lalt. E�pre" i<m, ,uch w; '"micros.cop,e

clanl},. Th� �mbiguOlh Icrm milt. \\ol'>cn applied

u,ual1y reftr; 10 W m.k:I'Q'oUIC. When apl'llc
(0 a �Y" cm.

m
we study beginning in Section 9.3.

P"IMidt,·

9.2 Entropy and Temperature

[!J

Everyone has some idea of what temperature means, but it may be somewhat disturbing when it suddenly appears

ill an equation. Because temperature does

appear in all the energy distributions we consider, it is helpful to see ho", the idea becomes a quantifiable propeny. It begins with entropy.

P

·,ralt': \\oC

T

",ould Lno", !he m,cl'(Kutt:

o

N

A

L

' II, ' 'IP ' • , 1

ynamics says thai

the dis. Ord thermod er--th no oecoad law of . at is . not decrea.<e, will m In 'y,l< Con ,soiaIed tr ast 10 opy '" ID .... ' , ,

the dISOrdered S/ale ' other .... , ..... oIpbyucs, 1his law has a reason : A rno,... "451...... IS a ' .

. systems t nvanably �o� p .." JJQIe. md tbermodynanuc move towar d ",0,.. ' case of the Iw()-sided roo",. for , n sIan ce' Ih bit ..... ID the N = 4 e N�.'UOt the N. = 0, and ii ,' ,, "' than ble proba IIIIXC Ore d'I __ is , , SOrd R • 2 .

�

•

ertd cules on each Side 1\ less ordered than 0 e n a stale with two mole Wh i Ch I authat disorder increases with prObab ilito n Give . e de y i on , and p '" on , rOba bir " oscopIC ways, we may give the concept 0f ' with number 0f micr disord er a ition by relating it to the nu mber of Way Og. ical quantitative defin s 0f O btai e we choos to ns, reaso defin n l e en Iropy ''ng the macrostate. For severa S as , pro ber of the num s: of m way nth loga ral Po,. tional to the natu

I 'l}

S

.. ka In W

(9'21

Boltzmann cORManl. Because the equil ibrium m acrOM ate C0,. e!Ot number 0f microscopic ways (9-2) say responds to the greal s t h . at . . . m. 'P." jJ mdt/mum af fqu,hbrlll loganthm? FirM. numbers o f nalUral Why the choice of WayS te nd to rs to the Nth power (e .g !N) , So , _ � .• 2 Proportional to numerical facto thOIS de . fi n H 10n to the number of panicles in th gi\'e� an entropy proportional e syste m. C n In ! the = ) o N 0"13. NR == N with macrOStat �ider the two·�ided room 2 e, \It "2. Il lS · 0 1 l IoX 7 no � = k8 In W = 1.: cOinc _ idenco Ih 10: >( • and 0 , al lhe , , faCtO -r of IS the nt 1!I3me con!)ta orde ann r o f "'ag . Boltzm multlpl�mg the nltude a s N. \e. The total number of ways of Second. it makes entropy additi arranglll · gt w h e I f e a ' separat uct nu o mbers of ay independent sy�tems IS the prod S, \V )( \Va . ; the logs. of so the sum the total en o and the log of a product is S PY I . then t�� so let d 'itate a comp ely ion. efimt this ordere th �um. Finally. by at It c n a be entropy. as in the totally Or dered N obtained in just one way has zero It ::::: 0 m. macrmtate of the two-sided roo . Relating entropy to number of ways IS fairly easy in the tw()-Slded room I n ' case'\'. It ,IS, 0f course. more difficult, Nevenh more campIex, reai,ISUC elesS, If' "' could somehow be calculated, the change in entropy according to ( 9·2) WOuld b.,

where

18 is the

erllro

•

S

�

:.S = s, - S,

= ka (In IV, - In

IV,)

= ka I n 3 IV,

(9'3)

Fortunately, there is another way to calculate entropy changes f: that I�. . .In ll1(ht . reader is probably already cases, easier to use and with which the l ar. Bef ore �h we sec how it fits in. lei us iR\'estigate the link between entropy an tem peratUre.

Temperature

. In thermodynamics, temperature IS given a clear, logical defin it"Ian. Identified alike as the measure of and scientist the by layperson ability t0 transf e, Iherfi ' , mal energy. temperature 's de millon naturally involves the p heno menon of heal transfer.

1,).2 Entropy and 1e� 3S3 Imagine two thennodyoamic systems of fixed volume, perhap!i two rigid ((J(Itainer; of gas. The two systems cannot exchange mechanical energy by . doing work on one another. but they are in . thermal contact:' as in Figure 9.4. w that then/wi energy can flow via microscopic collisions. Suppo!;e heat is flowing slowly from sy�tem I to system 2. (We might say that sy"'tem I i� higher temperature. but this is not yet part of our vocabulary.) Slow heat now betWeen the two allows each individual system time to assume a new cquilib­ nLim state and entropy at each step of the process. Now consider a shan time interval during which an infinitesimal amount of heat flows from system 1 10 ,y,tcm 2. The tolal change in entropy is the sum of the changes:

dStoml = dS I

+

figure 9.4 �·o 'y'lem.-. in "thennal conllU.:l'·

Heat flo""

dS-z.

Energy exchange alone causes the entropy change. so dS may be expressed as ta5fclE) dE. The partial derivative keeps other things constant that might affect

Ihe entropy.)

dS"",

�

(�) dE, aE 2

+

(\ as aE

) I dE,

System 2 gains the energy lost by system I . so we musl have dE t

= -dE2 :

Now if systems I and 2 are far from being in equilibrium with one another. the entrOpy change of the tOlal lwo-container system must be nonzero as it moves toward its maximum. Thus. iJSlaE would have to be differellf for system I than for system 2. But suppose that only an infinitesimal amount of heat remains to be transferred from 1 to 2 for the total system to attain equilibrium. Think of the entrOpy of the total system as a function of this small energy. If transfer­ ring it in one "direction" cau!>e!> the entropy to increase to its maximum value, further energy transfcr in the same direction must cause it to decrease. so dS ", would have to be 0 at this point. (A process carried out infinitesimally close to equilibrium is known as a "reversible" process. Entropy does not change in reversible processes.) Accordingly, iJSfrJE would have 10 be the same for both. Thus, systems I and 2 will be in equilibrium with one another if and only if aSfrJE is the same for both. This is exactly rhe properry we ascribe to rempera/ure. For several reasons.4 we define temperature as inversely propor­ tional \0 iJSJaE.

,,

as aE

lin IIcncrol. entmpy depend, nO! only on a ,ys· tem·\ int.mn! energy bm aho on ;1> �otume and lhe numbers of different Lim!> of parudlOSl

chem,cal �p«ie.. h migh' be in'� b} hc.llm8 at co..,....n ",tum. (.1£ > o. .1\. = 01. b� fltt

upiUlS'oo r.1£ = O • .1V > 0). 0< in a chrmical ndtllt, tilt 'oJu","" DOl Iht

reacUOfI that chanse.

Internal «nelllY

- � -

T

(9-4)

A glance at equations (9-2) and (9-4) raises an obvious question: How on Earth do we take a derivative wilh respect to E of a number of ways? We

'In hr..1 Inn,fer hom ')� 1 10 sy>lell\ 2. tht

d;",nkr of<"'; tlw ...

lhe dJ
iJS!�E > 1o<S,/�E1. A dcfimu"fl in>o1\,ng � reciprocal prop<".ly a""P" to 'j,tern I \be I""e. ,·atue (If r lbe definition at.. ag..-a widl tht ,deal ,.. • • to...

_ al4lJ

,

"j" II ...'

'MJII'l lCtUalIy aced the answer, and it i'i unfortunately rather invoh;ed (stf: Appeodil HJ. buI il is wonhwhile 10 proceed a bit beyond equation (9-4) to a relalionship litely famiJiar and show a clear connection to " numbers of waylJo" in ll ieasa one simple case.

Equation (94),

wi

th its parliaJ deri\'ative, relates temperature to a change

dE.

dE

in enuopy on the assumption of con'itanl volume. In �uch a zero-work case. it

would be valid to write dS

'"

(1/7)

But because

heal dQ added to the (,y\tem. we might as well

dQ.

would then equal the

write dS =

( 1/1)

That lhi�

relationship is correct even when two system� are allowed to exchange energy

via

..

oct follows from a �trai8htforward generalizalion (see Exercise

24) or

the earlier thermal-contact argument". Integrating gives a rather famous result:

.6.S � Sf - Si

=

'dQ

JI T

(9-5)

Note that wrule a sy.stem's entropy change depends only on its injtial and final 'lale.., this way ofcaJculating it requires a process in which the system is at each

,�tep infinite.simally c10� to equilibrium with its sun'oundings-a so,caJled fC\>'ersible process. Now let us look at a simple application.

EXAMPLE 9.1

An ideal ga� containing I02J molecules is initially confined to one side of a room. then e\pands to fill lhe whole room. lis inremal energy does nol change. (a) Calcu, hue.ii' entropy change u�ing equation (9,3). Slirling',., approximation, J!�

\ '21TJJJe-J• will be helpful. It is correcl within w% for J > 100. becoming even mort' accurate as J increase�. (b) Entropy/disorder should change by the same amount no mailer how the conMant,energy expansion from i nitial 10 final stale occu.... A�sume thai the air is kept alway� close to equilibrium by doing work

against a stow,moving piston. while heat is conslllntly added 10 main tai n Ihe internal energy. Calculate its entropy change using equat ion (9,5).

"

TlO,..,

(a) Equation (9,3) tells us that 6S = kll In(WfIW;). In Ollr si mple Iwo-sided room,

NV[ON)!J[UN)! J. Thus,S

there is only one way of having all particles on one side, and Ihe number of

way,., of having �N on each side is

·H.I�I'" nolfTUW'C'd � I.jIleUiOO IO .,mr',' -r- n (In !1M: n"" (Jf !IM: kll ... ..� an: Jr/itI'IIK lilt 1n111�1

rntI'OI'Y '0 be lIm'_ "e .. -ouldn·, ontlnJrily Ulln!

of

!1M: d1\Otlkr (I' ,u ,'oofux'd 1<'1 h.ul • room a.. ��ro,

bul Ihr dt/Mt# In enll\'J'_lldl,,>lIkl�' the \lImc �<'I

''''1 ''' � an' ,'On,,,lcnl In I,n""ng !he partKlc,'

Wl('lt4IIgifll iMnnal ener¥)'

I

Because

N is

so large. lhe approximation

is valid.

lioo

Distribu �.3 The 8alumann

Now u�ing a properly of logarithm",

For N :::::: lOB, the second tenn in parentheses is negligible: the factor 2"­ dommate:; the ratio of the numbers of ways. Thus.

In�erting numerical values, 6S � 10"( 1 .38 X 10-" J/K)ln 2

�

0.96 J/K

(b) Because the heat added equals the work done, we may replace dQ with dW = PdV. Using PV = NkB T, this becomes dW = (NkBTN) (IV. Thu",

I

,

j

The result is the same! While this, of course, does not prove that (9-3) and (9.5) are equivalent, they nevertheless are . It is interesting to reflect that by calculating an entropy change through such familiar properties as heat and temperature in equation (9-5), we are. by its equivalence to (9-3), detennining a ratio of numbers of ways, final to initial.

The example reiterates a fundamental idea. Unconfined air expands and its entropy increases because the state in which the air is evenly distributed is a

vastly more probable state than one in which it is unevenly distributed-the ratio of numbers of ways goes as 2N! Thus, it would be extremely unlikely for

the system to move away from equilibrium. The second law of thermodynam­ ic� boldly makes it official: It lViI/ not move away from equilibrium.

9.3 The Boltzmann Distribution

One of the most important and useful predictions of statistical mechanics is how energy will be distributed among a large number of identical panicles that are

able to exchange energy and thus reach a common equilibrium state (but are oth­

erwise weakly interacting. with no explicit correlation in their behaviors). In the following sections. we study several such eqUilibrium energy distributions

and their applications. Each gives the number of particles in a state of a given = 1 states. how many in their 11 = 2, and so on. AJthough the word equilibrium may suggest uniformity. it is the most energy: that is, how many will be in their 11

disordered state microscopically. Energy is constantly being exchanged ran­

domly between particles: some will have less than the average energy. others more. It is one of the wond.ers of physics that knowing only whether the pani­ cles are bosons. fermion\o, or "classically distinguishable" particles, we can say

355

... a.,.. . ........ ... ..... ..

g energy,

e\,hangin Agute ,.5 Harmoni<: O-\CdiaiM

The lI�iIlJtllf', �h;lrc and e"\:han�e cncrg�. one iocr ea'.n g as another dc..:rea.�n. •

I'

/

. "-I 'r \ .., � . ....

4A u.\;) \-------J JAwo \------1 2/tWj)

)

,

\-----/ 0 ,\--1

Once again. the simple reason is th how man) \\'tll be In each �tate. at aver . age wh�n ap�roaches Av?gadro's umbe predictable bc:ha\o'ior. become qUite n r . We begm with the Boltzmann dlstnbutlon , a specIal case that ap . . Phe . s · when particle" are so spread out that the 10d·15tlOguish ability of iden .c h . d ' a] d A goo exampIe IS a gas und bo"oo" or femuon� may be Ignore . e r Ord·I a"" n-I . I , are so far apa h I enllca conditIOns. The gas molecules, t ough ·d rt as to nOt ' "hare the !;ame space. Their wave functions don t significantly overlap . , they can be distingui�hed (see Section

�

8.2).

A rigorous derivation of the Boltzmann distribution from principles of Sta_ II\IICS is fairly involved, Since it reveals as clearly as possible how a d why energy is distribuled as il is, let us first consider a simple special case , Fi u . re 9 S . I bul separateIy I·dentlr,lable (i.e dl·st· . . I eotlca represents a system 0f N ·d ' " tngulsh_ . able) harmonic oscillators, exchangmg energy 1ll some unspecified wa {) ne o�cillator jumps to a higher level as another drops to a lower. We cho se the . OSCIllator . because Its · energy depends "In a SImple way on a harmonic . gl Sin e (see SectIon 5.7). We simplify thing quantum number: E = ,, + 2 s funher by Shifting the potential energy (as is always allowed) by . . 0 so that the he It . h OSCI·11ator In Its II th ground state energy .IS 0. The energy 0f l I energY Ievel . IS thu!>

�

.

(

�

.

1)hwo

(n, = 0,

-�liw

1 , 2, . . .

)

(9'6)

and the total energy of the N oscillators is N

E = L n/1wo = Mfiwo i=l

M,

where i=l

(9,7)

Note that the integer the sum of the quantum numbers of all oscl·11 ators .IS . ' . directly proportIOnal to Ihe lotal energy.

.

UboP

OislJib 9,.\ � Boltzmann

the probability of a given panicle/oscillator being In a given To find slale, we must use some elementary probability and stati!<;tlc' (..ee J). An axiom of statistics is thai. given numerous equally likely pm,·

cncrg��'(

� I· ,· e�. the ,.W' blll

probability that a particular subset will occur is the number of . . . set might occur dIVided by th e tolal number of ways of obtammg . the sub �sibilities. The probability of obtaining subset a is thus

���

P

o

;l. pp

� �

� � �

numb er of ways of obtaining subset a ­ � � � � � � � � =� total number of ways of obtaining all possibililie�

11·,ng this to

probability that particle i will possess energy our case, the .

�( il� quantum number

. will be flr- IS

number of ways energy can be distributed with II, fixed

PII, =

total number of ways energy can be distributed

£/I,

(9-8)

Here we are invoking the same postulate of statistical mechanics used in the [\\'o-sided room of Section 9. 1 : All microscopic ways of distributing a given assumed equally likely, so the probability depends merely on total energy are /llIlI/bers of ways/microstates. relative Now, the number of ways N integers-the particles' quantum numbers­

can be added (0 give the

integer M (see Exercise 34) is

number of ways N integers can add to M

(M + N - I ) ' = M!(N I ) ! M + - I =

( ; )

This would be the total number of ways the energy could be distributed among the N quantum numbers. The smaller number of ways energy can be distrib­ uted with IIj fixed is the number of ways the

can be added (0 give

M-

"i'

N - I other quantum numbers

which is proportional to the

energy. This would be the same expression, except with and M IIj replacing M. Thus, the ratio in (9-8) becomes -

,=

Po

(M

-

n.l + (N - I ) (M - n, )

l

remaillder of the

N- I

replacing

) j (M + MN - I )

N

(9-9)

Suppose now that N = 10 and M = 50. That is. there are ten particles! oscillators. and the sum of their quantum numbers is 50. The average energy per panicle is

E Mfiwo E=-= = 5f1w0 N N _

--

(9-10)

357

_ a.,. ... ,

Iw..,.' . .. .... . ... 1 .. .

..... t.' ProbebtblJcs of a gh'cn rucillaipr being in ih "j Slate. and Boltzmann _"I)'. Eu,,;1 probabilities (9-9)

•

o 17�

Bolt7.mann probabilily (9- 1 7 )

O.IXI

J

:>..

rf.

O_11� 0 1'" N ·- 1 0

001�

AI "" 50

O,O_�O O,O!5

I----�--=::::oe ::: ;o...

o

...... .. ...... . .. . � . .......... ..-I"__

�

"

40

30

50

An a..erage Q\cillalOr occupies its fifth energy level. The probability that parti­ cle

i ha, Ihe

avecage energy-that 1// =

(

quanlum numbeJ'\ add 10 45-would be

� -s

= =

45 + 9 - 1

�

5

and Ihe remaining nine panicles'

)/(

8.86 X 10'/1.26

X

50 + 1 0 - I

10 "

50 =

0.Q705

)

The probabilities given by (9-9) for all values of 1/, are shown in Figure 9.6. The

cune drops ..harpl)' as 1/, increases, which leads to an important conclusion: Varying the energy ofJU�I one particle causes a sharp change in the number of ways of distributing the remaining energy among the

other particles. The greatest freedom to distribute the remaining energy occurs when that one panicle has the least energy. Therefore, Ihe more probable state for a given particle, the state i n which the number of ways of distributing the energy among all particles i s greatest. i s one of lower energy. Equation (9·9) is cumbersome and limited to the special case of harmonic oscillators. In a system of infinite wells. for instance,

squared, and the expression

En

is propoilional to n

would be entirely different. However, it may be

_�hown that in the limit of large systems of distinguishable particles. converge to the Boltzmann probability.6

n

p

all cases

b�

"'I'btdtn.m;;,n and lboJc,;,f lilt forth.:,>!mng

"qlllltum l diSU'lN..l"" Ii delennt 10 App:I).l,\

H. � II U. be,1 Itl _ tao. Lilt �h'hty 1\ flSty/ til'll :and t>eeause KICID, the den, al,OO,

IOfdIIU .Iantics dIt,tllmdll'iuc, 1Uld dJtlcrrOl.�'.

This is the probability that in a large system at temperature particle will be in stale n of energy

T.

an individual

En' where 11 stands for the set of quantum

numbers necessary to specify the individual·particle state (e.g._ n, e. Probability drops exponentially with energy.

me_ fils)'

II i� important to note that the Bolt7mann probability and a1l the di!>tribu­ n�)'et to come can be applied to <,ubsy<;tems of luger s}'!>tems. In small !iub­ ,1�lemS, there might be significant fluctuation... but the prediction slill hold... ";'\ JI'crage, becoming more precise as the subsyst�m grows. The large remain­ Ikf0f theovel"',dl sy..tem IS what we call a reservOir. The temperature. appear­ in2 1n all the distributions we discuss. is a well-defined propert}' of the O\'erall �ro�opiC system. (See Section 9.2 and Appendix H.) While the exponential in (9- 1 1 ) obviously drops off qualitatively like thtcxact probllbilities of Figure 9.6, a quantitative comparison might be more C(ln"incing. Doing so requires a diversion. but a very important onc. for it 'pOtlights one of the most basic tasks in statistical mechanics: fmding an JI�rage. We begin by nOling that the probability summed over all individual­ particle states must be 1 . II

=

"

"

This step enables us to eliminate the multiplicative constant A and write the Boltzmann probability as P(E"l

I

I

e-E,/J:BT

�

2: e-E,/J:BT ,

(9-12)

As always. the average of a quantity

is the sum of the quantity's possible val­ ues times the probabilities for each. So the average energy of a particle is E

== I

E,l{EII)

(9-13)

A quotient like this is a

very common form in which to express an average. returning to the comparison of equations (9-9) and (9-1 1), if we Now insert equation (9-6) into (9-13), we have7

(9-14)

The sums can then be carried out rairly easily (see Exercise 30), yielding (9-1 5)

. ...

'The avera,e energy,� the same for any paruck. <,() we omit the �u�pt j on n. Abo rule the panicle <:anna{ hase _lhon W loud ennnn cannot e�ceed M--\he exponential probability drop. to eMcnlialL� 0 befon: tlu� point ... readIed. and the ttln_cni.,n.."e ofc�u:nd!ng the WIn 10 >nfin't� mtroduces mo el'Tl)r.

tong

.

...... -

agai n late r, bUt fO ' rn ,ound I n the Bol tzm temperature, , o a nn p II proVlItes the link between ro ba " b and N of the exact probabil ity (9-9) ili h . We ". (9- 1 1), and the intege" M . me"' ' e Th ree. ge a e avera ene g , ene 'lp e l. averag rgy I n ensure that the ' th e . IN M' nW()I , 10) . (9 . SubSll "luallon c tUti ' In . ng this t approach is gIVen tOr ':!
own right, we'll discuss Thi result. imponant in its .

'

kaT

hwo

=

In{ )

+ N M)

(9- 1

/

6)

ing this and equation (9-6) b ac k I' n to (See Exercise, 3 ) and 32.) Insen II. ( 9-1 2 , in of term M ty abili , prob N, and mann ) then give, the Boltz P{ E, )

N e- I,( I + N/M) M+N ..

=

(9-1 7

As ,ho"n in Figure 9.6, the exponential Boltzman n probabil ity ag ree s

)

qUite . well WIth the «act probabilities of (9-9). The slight deviation IS d Ue On IY to the fact that the N we have considered IS so small. I n a reali stic acro sCOpic hmit. the mes�y faclOrials and smooth Boltzmann functio n coin Im C d e preC . "ely (,ee Exerci" 33). In the case depicted in Figure 9.6, where MIN = 5, equatio (91 6) gi Ves Figure 9.7 shows the Boltzma = hw,)ln{ l .2 ) " n prObabilit l = y ploned ,e"us energy for three different cases: 5' I , and . 5. � . l e rn � perature IS related to average particle energy, and lliw is the fi Irst en er gy above the zero·energy ground stale. Therefore, the curves repre sent ase C s in which the average particle energy is less than, comparable to, and g reater w than the tiT'" excited-state energy. Sensibly. at the low temperature e ee th at it is quite unlikely for a given particle to be in any energy level' h i er than the ground state. while at the high temperature, occupation of the g ound St ate is nOI much more likely than occupation of the second, the third a n d So fo nh. o

5hwo'

kaT

•

:

ko T/fzwo

o

•

Figure 9.7

� ;

VanaLion of Boltzmann probability from kBT < hwo to k

B T > liw o·

1.0 ,

Boltzmann probability P(£�) VS. E�

� 0,6 ;;

J

_

OA

0.2

o

2

3

7

8

9

"

10

Q ) The BoIWNIM Distnbutioa

\I.e are inlere�lcd nOI \0 much an Ihe probtlbi/il)' of oc('uP)'tni! a n ,late a� In the mUllber of particles ell.:pecled to occup)' that ..tatc. rderred the occupalion number. A (This c;ymbol IS chosen with care_ It i.. a bcr. but we must not confu�e It With the 'y'tem·... total number of pani­ a quantum number. often represented with an II,) The relanor "11h e,� ..." \.· . . a ..tmple one: The number ell.:pected 10 a gl\en ..tate I... Ju...t the OllP,hip is of occupymg that ...Iale limes the total number of particle... N. Thu... protut-ihty 8olt.wmnn (occupation number) di�tribution follows directly from (9-1 1 ) :

()ftc:n

SI':

:.n t/I(

N(E,,)Bolu

(9·18)

represents the full set of quantum numbers needed to \pecify Here again. /I ly. given that summing probability (9-1 1 ) gives I . summing Logical ,tate, )

� 9.18) �.1

lI\u<;1 give N. Though we omit the jU"tification. the principle of Section ..u ll applies: A� the numbers occupying the various states become large. precise. A thermodynamic system started with any odd dis­ dle) al�o become would progress inexorably toward a vastly most probable energy Ulbution of eqUilibrium macrostate in which occupation number drops exponentially with energy (see Exercise 40). Before going any further with the Boltzmann distribution. we call atten­ tion to a distinction often overlooked but crucial to applying all energy di'itrib­ ulion�- From their derivations (Appendix H). we find that they give a number of energy E, not the number of a given energy E. of particles in a gil'llfI stale The\C twO things sound dangerously similar. but here is how they differ' If !here are multiple states of the same energy (degeneracy. discussed in Sections 1.2 and 7.6). then the number of particles of energy En IS the number in a gi\-en state of thaI energy-the occupation number .N(En}-time!o. the number of�tateS with that energy. This principle is so important in statistical mechan­

(

)(

ic� as to merit memorization!

number Of

particles of energy E

=

N. number of state of energy

)( ) number Of

particles in a given

£

X

states of

(9-19)

energy E

EXAM PLE 9 . 2

Con�ider a sample of hydrogen atoms al a temperature of 300 K. (Ordinarily. hydro­ gen uists as diatomic molecules. We consider individual atoms, for the sake of sim­

plicity.) (a) What IS the rallO of the number of atoms i n n = 2 energy levels to tho�e

in n =- 11 (b) AI what temperalure would the ratio be to?

(3) Hydrogen's degenerJC) (ignoring small effects like the spin-orbit Inleraction) IS

2111-the II! duc to ditlcrent ( and m, values (see Table 7.5) and the 2 due to the lone electron's, two allowed \'alue., of �pin. Ins' Thus. the n = 2 le\'ei has eight

rr"

d,

r

Jl

n

16'

.. 0,..

,

.rr 'In

... _ willi !be same eoerJY. Equation (9-18) says that the number of

__

-.lar lDlllllllCe. dat sc* III, (, III( .m, )

(

- {2, l . +1. +D. is

Ji(E1.1." •. ·f} = NA exp -

IbviIlJ die same ena'lY. £2' the

I

I

""

number with energy £,

� .

--_._--._-

number with energy 1;'.

.

""

.'f(E2 ) X 8 .1-i(E.) X 2

The hydrogen atom encr!!ie� are

.ta T- (IJ8 X 10

I I

-

p, O. 0, - 0 the {2, I. - I . + !}, and all

1 states hI"e the \ame number.

- 13.6 cVln2, so that

:!lJlK}(300K) = 4.14 X

number �ith energy E2

number With energy E[

_ _ .

10- 2 1

"" 4t'-I0 2i(I0159

ill

J =

O.025g eV

1 0- 171

AI room lem�raturt, the probability of finding even a single hydrogen atom in an ex(iled state i� pra(IJcally zero.8

(b) Insening in (9-20). 0. 1

= 4t'-IO hV/laT

=-t

T

iii

32.000 K

The Sun's surface i� aboul 6000 K. giving a ralio of about 10-8. We conclude thai except al e:
'NoIr _ .....,. UW;� but tilt tllCfJ)' dlf· fInrII:c ... moll D 6tv. MIPI tilt DmrIIxt .,. ....... -zy lit pakt? Ser E.u:rt"lW )6

f

=

� E, .K(E,)

,'--.2

_ _

� X(E,) ,

.

(9-20)

and al 300 K,

,

ka T

Similarly. there are two n I state� . . f l . O. O. + Jl and { I . O. 0, -il. of energy E. So [he rallo of the number of aIorm in n • 2 Siales 10 tho� in n ... I �tates is ocher "

I

)

£, • • • . • ,

(9-21)

9 3 The � Di.er

�

� says that e a�erage energy is the energy E" or a given stale times Ibe JIIDbU or particles In that (j ale. summed over all states and then divided by

�

� toW number or partlclec; an all c;tatec;.

From ' Sum to an Integral Oftt'n an statistical mechanics. the s.pacing of the quantum levels. is much

!JIIiIlcr than typical panicle energies. This is cenainly the case cla�sically.

"here discrete quantum levels are unresolvable. but is also true in many quan. [11m mermodynamic systems. If quantum levels are Indeed closely spaced. a stales may be replaced by an Integral. As justification. consider the �um o\o'er probability at energy ratiO of the Boltzmann

En to

its value at the ne�t higher

tIIcrgy. E"+I·

thc

En is much smaller than k8T-our definition of "clo..,cly spaccd"­ tiO is nearly I . In a sum, such as in equation (9·13). the 'iummand would

�

If E

+

t

-

\at)' quite slowly with tiltlcenor.

'1,

so replacing the sum by an integral ..,hould antroduce

However. a quantum number is rarely the mOSt convenient Integration Energy is a logicaJ alternative. but here we must be -+ variable (i,e.,

an

dn).

careful. Consider a simple case: Suppose we wish to calculate an average In the numera. energy in a collection of hydrogen atoms via equation

(9·21).

lor. me sum writlen out would begin

Z £,X(E,) = L, E, NAe-E.I'.T El.o.o,+� NA e xp( E I .O,o,+�/kBT) + El,o.o,_� NA e xp(-E 1.0,o._�/k8T) + £,.o.o.+j NA ex p( E,.o.o.+jlksT) + E,.o.o._j NA e xp(-E2.0.o.-j/ksT) ,

=

-

-

(9-22)

Smce the energy actually depends only on quantum number n. this sum over slaltS.

{ 1 .O,O, +�}. { I .O.O. - �}.

�um over

{2.0.0, +�} . . . . . can be written instead as a

energies (a vital point!). E,. E2. . . . . as rollows:

E,NAe-E,lk,T x 2 + E2 NA e-4kaT x 8 + . . . L E" NA e-E,,/AaT X (number or states of energy En) =

E.

(number of states of energy

To proceed to an Integral. we would replace the discrete

En)]

En

(9-23)

by a continuous

ymable E, but. as we see. we .... 1111 need to include the number of states at the

£ dtf(eIeIIbaJ numbtr of lale... '0\ Ithln range df. l)j' -

df

difkIeItial number or \lllt' \10 Ittun_rJn�e dli CIk tNt den\u) \l[ ...

tatc' ha\ ,hffi('

,If.' of £ •

df:

D(£)

Il�IOI'I' "one mer energy,"

o\croge (9·21 ) I n the limit 01 (ij... ;!;!) Ihn\ugh (Q·1�)' l'bUlIt., applYing

energy d�ly �-cd

It\(I.. OO:l)I1lt'�

S(f)IJ-( F)IIE: If. = f JX(E)IJ(f)lIf.

19-26)

an o....erage \13 Integration. sa)\ that IlIne\ the number of particlell In a IS ill enert.Y at ..tMe.. per energy range at thai tht lvera \ Ihe number at imt t S, Y. cntrJ \ummed o"'cr all energies. lhtn stitt of thai c (If energ�. !lo " ran� i mt l lX£). rDtrgy. of partlcie!>. IOllil number complicated and depend� di\ided b) lht tale� can be den,IUC\ of .. 1ining \er. consider the simple" '-"'lUIII) dctem t example. ho\\e srod)'. A'io a fi.... under m ng \pm. Wh�re do ''ie start? What on the syste oscillatOr;.. Ignon of systcm Ihe qU:lnI1 7:l1l0n condilion �pe. cast. oW IN!'I) The an..v.er I� gy? ener 10 s

Thl'lo rorm. wh.ch

our nllll.lel ai' ptl" ltt\e energy.

sc....'c... a...

E.

dE.

ed slate .. \ , relMtS lllo l the' system:

"al l(

Q.l lbe BoIwnann Di.mbuttotl

}fl..' man) allowed .\itate!Jquantum numbel"> ,,� II

changes by I , E changes by

hwo'

are

there in an energ) range dE.�'

,,"0 the ratio of the number " I" ,tale,

�i)\� per change an energy is just l/hwo' More formally, differential number of state, dE D(E)

1

hwo

\9-27)

� ­

[

This density of states calculntton mny seem too cute and easy to be true, oot it is COrTect. Because others later will be quite a bit more involved, it I� helpful to consider briefly a case only slightly different. What if our �y�tem were, say, a collection of particles in a box? In this case,

£

would

J �ain be proportional to a single quantum number rI. but squal'('{1 (cf. Sec­

u�n 5.5). Contrary

to the oscillator'S equally spaced energies, the levels

g�1 farther apart as " increases. Shouldn't a " density of states" decrease as tnergy increases? It is left as an exercise to show. by the same procedure OL'

above, that if E

Increases.

0::

,,2,

then D(E)

0::

£ - In.. It does decrease as energy

We close the section with a basic question: What would be the average

fntrgy in a system of oscillators if the temperature were high enough that the quantum levels could be considered " closely spaced" ? This is the limit for \\hich we have tailored (9-26). Inserting (9- t 8) and (9-27),

Several things cancel top and bottom: the multiplicative constant

A,

the total

number of particles N, and. perhaps surprisingly, the density of states. While D(E) always cancels this way when it doesn't actually depend on E, we are

careful to put it where it belongs, for in several cases to come, it will depend on £. Now integration (see Exercise 38) over all energies from 0 to inlinity gives a very simple resuh: (9-28)

I> our switch to integration valid? We obtained equation (9-15) via summation:

It is left as an exercise to verify that in the limit of closely spaced energy levels,

""0 «

k.T, equation (9-15) becomes (9-28). It fits'

Ity 01

• ,".

.�

'1,1'

161

9.4

Classical Averages

applicable only when Ind· The Bo ltzmann distriburion. is l'ioll ngu ' I \ha b There . are quant u m Ignored be i/j, may s particle thefTn Od identical or Y n arr Je ul�hab dISllRg . are bUI llc}' the Bol t m , lems in which the particles z a . n '1\. n d l')l . ribu t' dl'ilnbulJon. because i l i� the is often said to be a. cJasSICaJ IOn c OrrecI . one cal . quantum mecham where beh l i hml the avi n systems in ors c Oov u a erge . 1 0 ICaJ the class . . Classical averages naturally Involve not summation Over IIo Wed quantu inuum of "Ckl%ical ')tar eS slates bur integralioD over the cont III th at . · n coordm ates. d In eartesHl . dYdZdv-'dv IS, Pan;. x cle positions and "cIOCl. tles. .\dl'. y En become.... E(x )' IS 0 l' infinitesimal elemenl. and Ihe energ 1 ..t .• 1;:), kinetic energy depends on velocity and pOlenilai ener� ie's e P nd on PO . 5itio .. I ·' raIher than energies We d n Because we Inlegrale over Sla 0 ' nOI . need a me uely assu mfin den. , man y stale sil} of slaies. bUl lhe fact that we lses a n i nt nly principle lells U � er. eSling and subde Question. The uncerta s t ;� P QSlilo n e known simullan be! ously a cannOI with nd velocily/momenlum abmI u� . pre I. . of a contmuu m of C si . speak to o sense no makes such n 5 so " reaJI)" tates. Tt, . . . e e an r I S between pOSIllOn and velo ' mescapable granulanI)". a spacing Ctty Val ues · I eSIi. e h fhat e\-·en can be Shown th H N · Istmcr. d· can be declared lndy at the re ' , IS a o ne. mechanical states an to-one correspondence between quantum d Ih ese . . . "dI.SlJ .. o. guishable pos!llon-velcx:uy !!otates. so that aCcount ing for the g ra nUlar i "h, would just introduce a proportionality constant Ihat in calc ulat'1 09 an ave r ag e would cancel top and bottom to common quile calcula I!I it te averages 0 In c1a�sical systems. f qUa nt llies for speed, instanc or n e, auo Fonu nalel other than energy-loc We nee d no a model for all averages. To a! new appararu!!o. Equ31ion (9·26) erage an b . . ar I. · I wllh · [rary quantity the expOnenti pur It In the Integra Ju_st . al Boll llllan o probabilily factor. (Note the analogy to expectation value i n qua I u m mecha n. )/2 ic,. where knowing the probabIlity density, /",(x , We find t�e a . verage . of · th e Inlegral with the prob ' . any funellon 0f x simply by pUlling H 10 ab Iltty') . . . ..I .. OC followmg example Jllustrates the pomt.

.

.�

� �v\

becalJ�

�

e\

�

,

i\

Q.

.

EXAMPLE 9 . 3

I J ,

-

By appro�imaling lhe 3lmo.)phere 3!. a colu mn of da\sica/ pan icles of ass a( caitu/ale the avera e � IcmperalUre T in a uniform gravitational field g he ght oIn f an . air molecule above Earth \ �urface

g.

St)lU"I�N

Probability depend� on energy, and in this example. we have

£

=

KE +

, U.,.,..,. .'-

=

I

(

111 ' -2 � '* , .

". + � ', .

+

yo, )

+

mgv. •

I

'

2 nn·-

+

mg),

We may use (9·26) willi two simple change�. We average ), rather than E • nOled. there is no need for a density ofSlaies.

and, �

9.4 Clusical Averages

v=

"nit constant

NA

367

h' NA exp[ -(�ml,2 + mgY)/kBT] dtdydzdl',dl')dl': JNA exp[ -(j 1/1\.2 + mgY)/kBT] dxdydzdl',dl\dl':

cancel<. top and bottom. leaving the all-important exponential

Boltzmann factor. Furthermore, integrations over all variables but

J sepamte and bottom. Carrying out the remaining inlegrations, we obtain and top el canc

l�'e-(m�>'/(�T)d)' y= 100e-(mg>/k8T)dy

mg

Tht: result rnake� sense qualitatively. At a higher temperature, panicles are more energetic and so should rise to greater heights. Conversely, if either fI/ or g were larger. we wouldn't ellpeci particles to rise as high. Inserting the mass of an N2 mol­ 1 ecule. m = 4.7 X 0-26 kg, and assuming a unifonn temperature of273 K (although 1101 of uniform temperature, nor is it all nitrogen). the result i� Ihan the elevation of Mount Everest-a fair approllimate value. 8,2�. a bit less

the aunosphere is

The Maxwell Speed Distribution The distribution of speeds in an ideal gas is

an

important topic in classical

thermodynamics. Known as the Maxwell speed distribution, it is just one of the many faces of the Boltzmann distribution. We begin by considering lhe average of a yet unspecified property fly)

that depends only on the gas particle·s speed, v. Examples would be the speed ibelf and kinetic energy. The energy is assumed to be solely translational

kinetic energy. dependent, of course, on 11.9

We now average, as ill Example over the volume V cancel,

9.3.

but in this case the spalia/ integrations

J j(1I)e-!m,;J./k8T J e-�m,.2/k8T

dvxd\l>d\'�

dl'..dl' dl' J :

Sincefty) depends only on speed. rather than on velocity, we further simplify things by using nor rectangular velocity components bUI spherical polar coor­ dinates:

I'x' l'y.

1'_ --) V. e. c/>. By analogy with dV = ,2 sin 8 dr d8 dcjJ in spmia/

�ngular integrals

coordinates (Se�tion 7.4), the "volume" element in I'elocirv coordinates is tN, dIll dl': --)

v2 sin 8 dl' dH dcjJ. With this replacement. the cancel. and integration o....er speed l' alone remains:

"PoJ)al"""� ,deal JUC' 1\;I,'e ,n!emal degree. of f-oom_ rotalJON-t and vibnlJJ>lUJ. wh>eh dqIend ... "'.-�r,>m< in tb<: loau On ,..,J.alf.... {'O
_

°eli '

Nt 'Me'

I,

/

v;;:;2(.ts T/m)J

'The deoominIror is a srandard Gaus!.cian inregral. whose value worh our to be l.

Rearranging a bit. we amve at

(9·29)

1[,.)

J1[,,) (dPld")d,,.

J1[,,) dP

P

= = where Viewing (9·29) a' 15 probability. we see that the quantilY in brackets is a probabil i ty per unit Speed.

pe l')Ma�...."tll

=

liP dl'

�

=

l( m )]/2 -

1r

.--

k• T

r ,2. 1,2e -iml /*8T

NIV.

Often. the number per unit \lolume per unit speed is of greater inlerest. which The follows simply by multiplying the probability per uni[ speed by result is the Mu,..·ell speed distribution.

(9·30) Figure 9.8 plols [he diMribution. A'i a reference. it indicates vrms' the rms speed. This is a \lery useful quanllly, clo�ely relaled 10 the a\lerage kinetic

energy in a gas, and is an excellen[ example of a .!.peed-dependenr quanrity one

�

-the square vnns • root of the mean of the �uare of the speed-hence. rm'i. In Exercise 4 1 , it is shown 10 be V3kBT/m. may calculate from the distribulion. II is defined as

Because Ihe plot of

n(l')�bweJl Increases from

0 before decreasing. i[

mighl be tempting 10 conclude Ihal probability does not strictly decrease with energy. On the contrary. the probability of occupation of a does drop exponentially with energy. but the

,..... '.1 Mu...u 'P'Od di""bu.,,,.

number

given slalelveiOCit)" of slates-velocity

\leclors-at a given speed increases as the square of the speed. Consider an

analogy: The set of position \'ector� of a fixed magnitude de\cribes a sphere of radius

r centered at the origin.

Doubling

r would quadruple the area of the

sphere. In effect. the "number of posHion \·e<:to,,·· of magnitude

r increases as

the area of a sphere, proportional to ,2. The same may be said of velocity. The .2 "number of \lelocity ve<:tors" of fixed speed v increase.!. a\ ) It is the three.

J

dimensional \lelocity element

dl·.d�')d�·.. = \.2

...in

8 d\· d8 deb

that inrroducelo

this factor. which should nOI be surprising. for this differential element accounts for the " number of stales"

o 0$ J.O 1.5 2.0 2.5 lO ,/JU.1I.

in our cl3s\lcal \umlintcpral. AlthOUgh the

number of slates/velocities increa.�es with speed, causing thl' Maxwell dislrib. ution 10 increase initially. the exponentially decaying probability does eventu. ally wiD out, bringing die distribulion to O.

9.S Quantum Distributions illator example of Section 9.3, we said that the number of way!) t' the len·osc "st ibuting energy with one particle fixed at energy E was the number of ", IIfdI r t J\' � the o her particles' quantum numbers could add up to account for the maining energy. We implicitly assumed that switching the labels of \\ �'Iem's re ,) If twO particles yielded a differem way of distributing energy. For instance, == 5 and nil = 2 for particles j and k was one way, then ") = 2 and II� == 5 B different way. However, if the particles are indistinguishable. then different ways, and if numbers of ways are different quantum­ these are 1I0t y echanicall than classically, then probabilities will also be different. From ese ideas, perhaps seeming of little practical interest, arise many starlling consequences. and far-reaching Chapter 8 that there are two very different types of indistin­ from l Recal uishable particles: bosons (integral spin) and fermions (half-integral spin). how and why the distributions of energy in systems of v � e noW investigate differ from the Boltzmann distribution and from each other. w should O these t distinctions most easily, we consider a simple case: four parti­ To see the basic cles, a, b, c, and d, bound by a harmonic oscillator potential energy. It is convenie nt to define the ground-state energy as 0 and the spacing between

�;r ��

�

levels as SE.

("i

�

0. \ , 2, . . . )

Now suppose that the system has total energy 28£, that is,

Given this constraint, how may the energy be distributed among the particles? Table 9 . 1 shows all possible ways/microstates for distinguishable particles.

We see that the smallest quantum number is most common; the largest. least.

TABLE

9.1 Ways of distributing energy U£ among dJstinguishable particles a, b. c, d

Probable

Probability

number or

P

(#/40)

particles (P x 4)

•

0.1

0.'

cd

12

0.3

1.2

.b

2'

0.6

2.4

Torals

40

1.0

4.0

Number of

times n

1

appears

Ways

• "

b

e

d

I

ab

0

bed aed abd nbc e(1

.,.

ad

be

bd

be

ad

bd ue

(#)

IJI a.,..

9

• • _ 1 ·1 ... .. ' _ .

,-"'.... 9.10 Tht Ihrre tiL,lritllilion, for 0!C'i1� al l"o,\o. tnICnnc=Jliltc=. and high

trmpn:alUt"e, S

�·IEJ \ " E

(hIrmomc

�

O'>Clllaton)

S,*-bn�[cm Bolttmilnn

kIlT = � t:

kilT = 6-

kilT = 5&

.1

.-\[ Jow r, [he Bu'\C

Em\lo:rn j,

Illfhtl [h.m Ihl" Alliltma"n HI

Iht' �nlulIJ �t.l[e

2

�hlle [h�' f'o:rmi-OlrJ�'

, ,

,

nl.'\<:r n�'�'�'J,

\

..

I

I

r.

-\[ high the: three di,trihulil'"' clIlOcill...

,-\11 ,'i 3rt: « I

j

--. - - - - �::

.

2.

o E

Summarizing:

"'&oft t.1 t

A'ttftSt l�llIillor c=nergy

�"UJ kmpenlluft It,.. tht

o(poruclcs.

Ihrtt type'

1001

...

I �,

'0'

O,�li

1,0& I.BT

1.�f,.

2.M.

At high temperarures. panicle.� are .'tpread out sparsely over individual-panicle states. With occupation numbers much less than I. II is irrelevanl whether the panicles are bosons. fermions. or classically distinguishable. and the quanlUm distributions thus agree wilh lIle da�sical. At low tempenuureli. bosons tend to congregate in the lowest-energy individual-panicle state, while fermions tend to fill states. one panicle per state. up to some maximum energy. PIOb of average particle energy. .;;hown in Figure 9. 1 J . are also iJluminal_ jog. (The ta.;;k of obtaining them is left 10 Computational Exercise 94.) At high temperatures. ks T » 60, bolh boson and ferotion cases are essentially plots of a linear relationship E = ka T. the same as the classical result (9-28). The panicles are spread so diffusely among slates [hat indistinguishability is unimportant. Even so. the energy is somewhal lower for bosons, which tend 10 Congregate at lower levels more than would be expected classically, and is

!Of!1C'\hat higher for fermion�. which llvoid already-occupied lower energy

�

sdk�. At lower temperatures. the a\' �ge energy for OO\on\ approache,

O.

. femllon.. approache.. 2 f... the logical average when equally hilt thai for 1C',el� fill unifonnly from 0 10 a maximum of 6. Th<: �mooth curve" In Figure 9. 1 1 rely on the assumption that Inlegration

:"'-rd

' Iid-thal ksT i.\. comparable to 6. which i<; a fairly high quantum le\el I ,'ll

I'tr" low temperature... where kBT is comparable to the 'pacing IIwn But at t'l:I"o\et'n indi\'ldual levels, ..ummatlOn would be required. The bo.\.on ca�e is ularly "en"'!live: the minus \ign In the denominator of (9-32) cause", a patti" One result is thai the vaS! majority nounccd increase III N(E) near E rro . til' particle.\. In a boson gas may drop mio the ground stale. leadtng 10 "supcr­

� O.

nuid" behavior. We di ..cllss this amazing phenomenon in Section 9.6, where

e the electron gas, one of the most important/amion "y"'tem ....

\\C' 1I1�o inlrOduc

BUllel us first take a look at an aspect of the Fermi-Dirac distribution that ha<; oOl.hing to do with the specific system.

The Fermi Energy

As Figure 9.10 :.hows. at low temperature. che Fenni-Dirac occupation number is

nearly l out to a certain energy. where it quickly drops 10 nearly O. This energy is known as the Fermi energy EF• Strictly speaking. the Fermi energy is defined as

t.

the energy at which the occupation number is The convenience of lhis defini. B tion is that the constant in (9-33) may be written in terms of Ep no matter what thes)'!item might be. Given that .N"(E)FD

=

� at E = Ep we have

= -

2

:::::::.

Thus. (9-34)

Figurll 9.12

Fennion oscillators in Ihe loweS! pos�ible energy stale.

In this form. it is easy to see that in the limit T -+ 0, the Fermi-Dirac distribu­

lion is a "step function": I for E

< EF and 0 for E > EF. Figure 9 . 1 2 illustrates

this all-or-nothing filting of states at T = 0 in the oscillator case. Since it is really just another way of writing B, which is, in general. tem­ perature dependent. the Fermi energy does vary with temperature. bUI its value at T = 0 is most important. Figure 9. 1 3 for the oscillator case shows why. At T '" O. the occupation number drops 10 zero abruptly at EFO' When k BT is not o but still small compared with EFO' the plot passes through at nearly the

1

'i31l1e energy-the

Fermi energy is essentially constant. Moreover. it deviates step function nearly symmetrically about the Fenni energy. Only from a true becomes com arable to EFO is the bulge significantly depleted and when kBT o t passes through the pl ;: at significantly lower energy. When we speak of a

�

Fermi energy. we are usually referring to very low temperature. so it is justifi· able to refer to "the" Fermi energy as if il were constant.

Epu - (j,

t · ·

\ \

' ' '

· ''

·

·

>,

·

\

\

'

•

!

! : /!

(

j

and disuibullOO 11 r - 0 The ftnni·Dirac ,." ,.... }-emll rft'l!(i}' t.. \lIUe ..... ltmpcnturer.. the � kJr 1oW r 0 mil( ,ink from ,u ..r d l EfO

E '!!

,

,,\

r � I,t

II

9,6

",I lOw T ,.�� Ptila

I).fa e....

o ,� L

0

'.

_

--

,

hNU,h H AI wne " urn T: t" c••

" AI

f. F1

� f

-

f

•

Tne Ouantum

Gas

n\ to a \ituation more reaI'I�h. c ''-A_ um dl.,tnbUlio UI4l\ . '. the quant annl .... no n we do, we enter an area 0f ph)"" \\e mmon ongln. A" ilco III c J c f"":llIatl\f' J,ltrol..:! of Ihe mo" aeli'e In r'''ar h ' mvohmg . h., b<en un' 1'I9!h Ihe Ih". ,,"" , .. wQUId fiInd completely baffimg. I ph)".IC\"t a ·\:la.....ac3. are f t-o.:hilWlf" that . h N masSive particles �.I" one In ",hie ider Clln, \t 'IIo e "The " ". tern ... (The massless h .10 a cI�...\Cal g a u\e" mo\et.: are a:. \J \ " In thf'.'( dllfl(n'Il an: nol eon...denng ordinary gase \ Ho."er, " 9.1. un Se
: ::;�"

�

,pe)ed ,

��

.

;;rn : ::;

';""

���'

9,6 Thc> QuantumOu

371

F9" 9.14 Cl;Mlcal di�ting\li�hable particles at high T. and t�o different lo\\,·T tdA1L\'f'. (Adapted from Nobel lecture by WOlfgang Ketlerle. Rei'. M(l
().",\lbt:r �OO2.)

(b) Ferrnions tilling lowest levels ullowed

1,1SpKad,oul particles 81 h1gh tcmperature

(c) Bosons all m the lowest po�,\blc level

So far as calculations go, we will mostly consider circumstances where quantum states are closely spaced and integration is valid, so we will need the density of states for our new system. In Section 7.2. the allowed energies in a 30 infinite well nre shown to be

(9-35) Computing the density of states is fairly simple for one-dimensional oscilla­ !QT!i. because E is directly proponionai to the lone quantum number fl . How do \\e determine the number of quantum states in an energy imerval dE now? This is one of those junctures in physics where we learn that the best reson is a trick: Picture the set (/Ix' liy• Ii�) as coordinates in three-dimensional "quantum-number space." depicted in Figure 9.15, with each point represent­ ing a quantum state and a IIlIif volume thus enclosing one state. The radial "distance" II from the origin in this space would be given by /I =

Vn2or + 112y + 112�

FIgure

,,'10' ' n 2mL2

--

Allowed stales in Ii 3D

case, 5.

Thus.

E�

9.1 5

infinite well. The shell encloses all stllles in range dn al '"radius" ,,-in this

I

,

(9-36)

0'

Sensibly, the " radius" increases with energy. Now, if the energy increases by dE. the radius increases by a corresponding dll, so an energy interval of dE encloses a " volume" in quantum-number space of i41Tn2 dll (i.e .. a one-eighth ;tCtion of a spherical shell of thickness dll). But a volume in quantum-number Ipace is a /llImber of quantum :.tates, so we may write differential number of states in range dE

=

i41Tn2 dn

•

.'

.�

l (n,. ",. n,)

• •

!

•

:

•

"'"

(3. 1. 2) ".

1M

051.

' 11 -_ . .. '.. "'1IId'_.,

D(E) .: difftmltial number of S(atC!i In rang e dE

- == �4lT1I2 d"

dE

dE (9'3) )

or course, D(£) should tit I function of E. and what '"e h ave d . e pends ' 11.1 show thai. usmg relatlOn hip ' \eft ..,, 111 t'ltrtiK 1$ (9-�6 . � II may be eliminlled. kaviog

D(E)

=

m'\ �L� - £1 .2 , 'Jr- �_\ Vi

) betwee n

m:��

. . . ' IS more conventent 10 express this in te •U'Iy. II •l F 'lU rms ' le-ngth L but of the volume V = L . We mu�t also .

account (Of the multiple allowed spm slate�.

1

. A. good sAaI1Ulg

D(E) =

(2� + I ) 1II� 2V " . '" ". !. v2

a��

II. I t

t"

(9'3 8)

of the box's . ply ' by 2$ + Sid. l 10

£112

. would be the familiar avera CalcUlation ge

(9'39 )

.

. "le e an p . quantum distribut ,. · U nerg),. K: ' UolortunaIC y. the parameter B 10 ions (9-32 D( ) and ( depe'IMb on the sy�m. and_ the complexity oft� formulas and £) p 9,33) venllo of r of B and. therefore. for the quUfl a cO \ , o d ...fonn cU:lllation ' tum gas t . , . 1"10\1,' �.: \0IullOn IS re\-'e<.UIng. Exercise 72 OutI' lu r'lu. t'o'tll an appn:lmate 1nes the . _ Steps: .. er.; 0 the particle density /V. Th O\ thai gwt T) a.o;. a !iCn�r. In p e result

£( T)

E(

£

..

)1

2 '

r[1

.,

,,' !' Vi

N (N)I +

(2s + 1)(2"mk, T) '/2 II

�

...

]

re

•

is

(9-40)

In contralil lO our oscillators, system, in which volume was not even a COn SI'deralioR. the 8\1erage energy In a quantum gas depends on both tern�

N/V�and

. 1e� together-increase , NIV. Squeelc Ihe partie particle denSity while for bosons it increases, energy sign) average fermion (bonom T is not too low and when only Note that this approximation is good

_re a t rd

��reasthee�.

e Partl_ de density nol too high. in (9-40) is sensible, being the f .... "ous .".... .. LeI U!i look.closer. The firsl term aVer. seem, would It then. particle. that ,icai gas ... a. the effeelS age energy of a c\ of the second lenn. Actually, we c an con_ quantum indistinguishability must arise in . . relying on ideas already introduced. F irst, as deplc linn this in two ways, each ted . ' . . . behaVior should prevail if in Figwe 9.14(a), we clatm thai clasSiCal particle separation!>. Wavelength des ha,'e shoff wavelengths and large n gas panicle is V3kBT/m. Therefo a of .. speed (Section 9.4) nn. , nd the d a spee

depen�­

�

s is roughly the cube root of the volume I f3. Thus, the second tenn in equation (9-40) is per particle--thal is, d IX (V!N) s are much smaller when �articJes' wavelen proportional to {Ald)3. It is small (the claSSical case), becommg large when than interparticle separations overlap. cles' wave functions begin to

TIlt linear distanee d between particle

�th

parti­

The 'tcond path i.. more roundabout but take� us; through a re�uh that h;'l�

""') application". It is ba<;ed on ho\\ diffu ..el) the particle" are 'pread l)ut IItIllng the Io!nergy levels. To begm. we calculate the Fe-nui energ) at T co o. \Ul�t1) 'peaking. the Fermi energy applies only to a ga., of fennioo,. for l)ol), dIC� c«) the exclu,ion prinCiple. But for now we vie\\ it 'Imply a, It \)nc­ pu1lde-�r�statlo! qaodard. I f t e energy of. n. syste.m. no matter �hut kind of p.trUd¢. IS much greater than Ihls Sla�dard. il l:; unhk�ly that multiple pan �cle, behavlo� should prevml. The slart11lg pomt IS un Itlti;lbit any ,laIc. and classical �'rre ssion for the total number of particles:

�

(9·4 1 )

(Remember: Number

= number per state X states per dE X dE.) A gre EF' Thus. we simply integrate the number or stntes per energy O(t:) from 0 to £F and solve for EF·

N=

j,fF o

(2s +

I )mlf2

7T2 ttl V2

V 1/2 dE £

=

(2s

+ I )ml/2

7T2 ,,1 V2

( i)

V 2 _E�,2 3

or

(9-42) This important result says that even lit zero remperature. the greater the den­ 'ilt)· of fermions, Ihe higher will be lheir energies. We will have more to say �bout this shorlly, but let us first show how it figures into (9�40). If we rearrange (9-42) a bit. we have

Thu s. we see thUl the second term in (9AO) is proportional to

(Er!ksn3n.

If kBT » £p only the classical first term survives. The particles are spread

t,Q diffusely among states that quantum indistinguishability can be ignored.

Summarizing, a gas will behave classically so long as the "quantum term"

in (9-40) is small. which can be expressed two ways:

W' « I

or

N

-

�,

V (/Ilk. T)'/2

«

I

Classical limit

(9-43)

When quantum effects do begin to arise, the average energy in a boson gas i� lower than in a classical one (as was true for oscillators in Figure 9-1 1). Being

related to average energy. the pressure is also lower. Conversely, the average

energy and pressure in a fermion gas are higher than in a classical gas.

.,. Ck;:1Ii '

�.. ......

.

EXAM�LE 9.' S� IbIf dtt ftJrro,ea moItaIkI In 1M illf around u .. need nOI be trealed a!t

awWbIop·rud"

indisn:

EqCMCKIII ('9-4)} dwMdd prtnu the' aru �«. I f a.lf 1.11'le, indeed behave cI
-

.'

.lp l (nJ,(s Now

-

nl :l.

1I/1 2(k8 T}5'2

IMMlIIj: 80'l--nllmfcn\ ,han"----df jjlml.)�pheric pres�ure, a plausible room

1(JI!fICD1W'C or lOO K, and tho;> n1{IIC'l"u/ar ma_�, of diatomic nitrogen. we arri..C' at

The nal.o 1\ rmu:h less than I. <.1\ it would dearly abo be for oxygen and for any other Itntp('r.lIUrn nalUrall) {1I:�urring. '0 {he ..:rilerion for classical behavior is mel.

'."'" " I�LIn' In IltOf( pO�I1I"1f ,.,,1' tile lhe "n,Ie· .. tom J"llCttlllaJ fncql" 10 lhoil 0(, finllt ",,,II.

(

From Iht

c_\.1mpk.

"e

�ee

(hac our Jlffio\phere is "classical:' but In the

man) impnrtanl bound ,,),tern, where qunnlum indistinguishability cannot be rgJlOr(J, (ttlf _�O infinite well model )Ield::. excellent predictions. One appJica­ Iwn an\C'\ /0 nhRlolvgy. By treating them as .�ystems of gravitationally bound

rermlt)n\ and USing re,ult (9�2). we may predict the radii of so-called White d�arI' staJ"i and n�urmn stars. The conc/usion\ are �tartling: A typical radius Ii" a n('ulnm .�Iar who,� ma�s is somewhat larger than our Sun's is only about /(I lorn' betn...es 89 and 90 lay out the calculations step-by-step. Another \'1:1)' l'ommon applll'iuion is thai of the electrostatically bound conduction ('1c�l'lt\lnS 1 0 a melal,

Conduction Electron Energy Levels In melals.

of each alom\ valence electrons are shared among a1l atoms, moving es...n .e tially freely in three dimensions through the solid, They are some

known a� conduction electrons and are a good example of a fennion gas. This "ga.s" is confined in a potentiru well " hose origm. illustrated in Figure 9.16, is

the." (}('t effect of the potential energies of the positive ions spaced regularly tMlUghoul the solid. The top diagram shows the potential energy of an electron ----

. .

I

- ..

10 the presence of a single positive ion-a simple Coulomb potential energy.

Thre< ions. of course, make [he potential energy more complex, but as more i(ln�

are

added, the net polential energy begins to assume the form of a finile

well, In a cf}'slaUine solid. the density of positive ions is so great that a finite well becomes an excelierH approximation. Often the even simpler infinite well model v.e have developed is valid, Accepting thist le[ us investigate whether the t'/eclron gas in a rypical metal must indeed be treated as a quantum gas.

\J.� The Quantum Gas

EX� MPLE 9.S .... \�,1,I1ft( . (hal

th� !;onduclLon e1�utln� in a pi�e of ..jher beha\�' ,1\ a fenni'\11 ga', ,,1ltI �a.:h awm I;onlributmg one eleetron Calculate the ...-ermi e"erg). tht'n Ct'mpare that might be ex�ted classically at room telll�ralUre (The den· It ... tID the en('r�n' 1 . 1 x 10 kglm ) IO.5 .. l \lh'('r \J� \lr .

,

lq-42)

give.. the Fermi energy in term.. of prope::rtie.. ea..ily dctcnmncd. f4uation mlli. \ and ..pm of electron\. but .....e mu" abo detcn11lne the numbc.'r of We know the �'I.lIldUdi\ln electrons per unit volume. At one conduction dectron per ;ltom, ,hi .. ., t\jul\'a1entl) the number of silver 1I10m\ per unit volume.

N �'

/

mass volume mass/atom

£F • =

10.5

10.1 kg/Ill'

107.9 u X \.66 X 10.21 kgiu

7T2 ,,2 9. 1 1

X

X \O-�I kg

8.8 X 10-19 J

( 3

27Tv2

-- 5.86

:::. 5.5 eV

X

= 5.86 X I02� m l

lj)l8 m

Were we to apply clas�ical physics to the conduction elcctron gil), the il\'crage T_ Al II room temperature of 300 K, this is only IlOOul 0.(,)4 eV, energy would be and \\-c have shown that even al ..ero temperature, electrons in �ilver IllUSI fill �Iales up to 5.5 eV! Here we have an excellent example of T « EF, whcre (9-40) approximation, ils " quantum term" much larger than Its clas�ical awful an liQuid be first lenn, Because k8T at room temperature is only about Er:' the occupation number distribution would resemble the extreme nonclassical T = 0 plot of Figure 9.13. To electrons in a metal, room temperature i� very cold. Temperatures hundreds of times higher would be needed before they could spread out among higher energy ICI-ds, and because such temperatures would vaporize mosl metals, we conclude thaI conduction electron behavior is a decidedly quantum-mechanical liffair.

�kB

kB

rlxi

That the exclusion principle rorces conduction electrons to fill up slates

h

unusually high by classical standards has a famous consequence, one that puz­ zled p ysicists terribly before quantum mechanics mmured. The electrons' free­ dom to move about in II melal was obvious, and if behaving as a classical gas, that

Figure 9.17 Electron energies in a

"cold" metal.

freedom should increase the heat capacity of metals relative to nonmetals. But the

,

electrons' contribution is, in fact, very small, and the reason is that most of them,

I

sandwiched between states already occupied by others, simply cannot gain energy from a modest increa.se in temperature. Only those few at the top near the Fenni energy, with some "room" to spread oul, can gain energy. Thus. the large majority ofelectrons contribute nothing to the energy storage. (See Section 9.9.) Other properties of metals also come into clearer focus. Work function, the minimum energy required to remove an electron from a metal. is si mply the energy difference between the Fermi energy and the top of the finite well n i which the conduction electrons are bound. pictured in Figure 9.17.

.--

---,

�

I

V'I I

j

I

,

:

j E,

379

I II, Iv. Ml'nc' tl«tnm\ '1 m' I

Ih

8ivu'� II a IIt'I ne3d'I\!! .:har!!e •.....'--

ft .t 'u �,�\'

In '�k'liIl '

,�d raisinS 1'/k'/�) k\'eb there

\

i

O· •

t

.;,

,

+ + +

•

.'

while lowering k\'cl� In

m':;�1

2,

prudll�'lng a .:on;lc t t potential

Metals in contact

MoreO\:er. it is the difference in the work functions of different metals that is re'iponsible for the phenomenon of contact potential, the potential difference that dc\eJops when different kinds of metal come into contact. Suppose that

;

two kinds of melal are electrically isolated and at the same electric potential arbttrarily design:ued V = 0, and that the work function of metal I is greale than that of melal 2. as shown in Figure 9.18. When they are put in conlact, the mo�t energetic electrons in metal 2-lhat is. those cP2 below the top of their

finite well-are of greater energy than the most energetic in metal I . Seeking a tower energy state. eleclrons from metal 2 move to metal I . This effectively shifh Ihe energy levels of metal I upward, through an increase in electrostatic repuI$ion. while correspondingly Shifting the energy levels in melal 2 down­ ward. The lransfer of electrons Slaps when the two Fermi energies are at the

same level. to At this point, metal

I

has a net negative charge, and the potential

difference is V, - V, = (, - ,)I(-e) = (, - , )le. While contact potential between dissimilar metals can have obvious effecis-such as promoting corrosion-it alone does not make a battery. An equilibrium is, after all. eSlablished in which electrons cross the junction in equal numbers both ways. with no net transport. In facl. a voltmeter, which actually measures [he " electrochemical" potential difference, would read O. However. when combined with nonuniform heating....i....<J fferenl temperatures at different places-calling into play another property in electron transport, a "bauery" does result. The dependences on the various properties are too imol\'cd to discuss here, but they produce a reliable temperature-dependent potential difference. A bimetallic device that exploits the effect to measure temperature is known as a Ihermocouple. and it can operate over temperature u'Onl�' " mlnUle I'no:lnon M l� ':Ooou,IJon dO:ClR'n, JK..-d lO lran,ter h) ,hilt (neli)' levd� by ... clectronvolh n«tled, 5« h.:n:I'" bQ for e 1M I JIJ1 onJcr·')I-IThI{!nilud� e,lInu.l(,

ranges too extreme for conventional thermometers. The conduction electron quantum gas is obviously important. and its cen­ tral role in modem electronics is discussed further in Chapter 10. BUI other systems of fennions should exhibit the same low-temperature characteristics.

�.(> The- Quantum Oar.

E\'tO whole

'11

aloms. so long as they are fermions, should till state.. one-by-one

()'(1m the bottom. See Progress and Applications for a discu� ..ion of recent t:(perimental verification that this i<; indeed true.

The Bose Gas and Bose-Einstein Condensation Inltre,!

in the low-temperature behavior of quantum gases has mulliplied in recent years. since the achievement in the mid-1990s of the usly �normo Bose-Einstein conden!)ate. At very low temperatures, the minus sign (Jr.t true

in me denominator makes the Bose-Einstein distribution (9-32) very large near

E ::< O. The physical consequence in a boson gas is a decidedly nonsmooth dropping of panicles into the lowest-energy quantum state-and amazing behaviorl Helium-4 was the first example. As noted. under normal conditions. gases are so diffuse that it is irrelevan whe er the particles are bo.sons r . ((noions. However. the hehum-4 atom IS a spm-O boson: the total Spill of Its

�

�

?

nucleus is 0 and its two I s electrons also add to 0 spin. (By contrast. the �elium-3 atom. with an odd number of spin- fermions in its nucleus. is a

�

renoion. In the present discussion. "helium" means helium-4, the vastly more abundant form.) Reluctant to solidify, helium atoms move as a fluid down to

lery low temperatures. At some point. of course, interparticle forces become large. and the fluid is no longer a simple gas as we have assumed. but ,

becomes a liquid. Nevertheless. at very low temperatures, about 10%-a considerable fraction!--of its atoms "condense" into the same lowest-energy

Figure 9.'9 Superfluid liquid helium

individual particle state. U there are cases where quantum effects should be perceptible. this certainly qualifies. -

lndeed. liquid helium ex.hibits so-called superfluid characteristics: negli­

creeping out of its quarters. (Courtesy of A. Leitner Rensselaer Polytechnic lnslitute.)

viscosity and near-perfect thermal conductivity. Both are consequences D oible ex.treme order that reigns in a system where many of the particles move the of

essentially as one. Where motion is disordered, viscosity and resistance to

beat flow follow, but ordered motion implies streamlined collective behaviors.

In liquid helium. the transition to the unusually ordered state occurs at about 2.2 K. Ex.ercise 63 obtnins an order-of-magnitude prediction of this tempera­

lUTe from the criteria of (9-43).

Superfluid properties are responsible for a remarkable phenomenon.

shown in Figure 9.19. If. at very low temperature, a bowl is lowered into liquid

�elium and then raised above the surface, the liquid will crawl up the �ides, seemingly defying gravity. and spill over the edge umil the bowl is empty. The

mechanism is capillarity combined with negligible viscosity and perfect ther­

mal conductivity. Helium adheres to the inside surface of the bowl. creeping in

a film up the side, similar to ordinary water in a capillary tube. In an ordinary

liquid, a height limit is reached, due to temperature variations between liquid

and bowl. When the bowl is hotter. the film evaporates; when colder, vapor condenses back to the liquid below. This, however. cannot occur in liquid helium, with its near-perfect thermal conductivity. Additional helium moves upward in the film. unimpeded by viscosity. to grasp the bowl at ever greater heights until the top edge I!. reached. Because the interaction.. between particles are so strong in liquid helium-4, it doesn't demonstrate Bose-Einstein condensation as simply as we might

.

805O n5: The 9.7 Massless

Photon Gas

gil.. of photon

... we n (,. but 10 ..lud) a a I\e'\\ 'P'.t. art �m ' 'l uppI� i�n beUII (9·)9) ale " of '>I Ic ty ,... dc!n"l abl g a � e m u quan t The , for . stilet. . . �'ty of 9 mllssh't' pa . condition l ·35) for clc !<; T\.. rtl blll11lln an enerIY qu · \C bI.Cd on '. '1 .• C an.u1 given m equation (C IKIP' II I' _ . . 10 Ap penulx -IOl. nv !;U � .'" dc I def\.'i .

�

•

ity of

wd . . _,,
,.. ,n ,,00' of

en,,!,Y· bu' frequency

ot'

.,Intt.''' In ber ntial num differe-------- �

e UMR, E • Itj. w

df

,,-on,"crt thi!> III

r iliffcrenual numbe _ _ _ _

-

tlf 87TV - -- ::::: --I'

range

c-'

the u�ual number per unit

1.'11' ,tate' in range

--;J£!h .

D(E)

�

tiE

=

energy

87TV 7(E/h )'

811'\' - £' -

h' c'

important way. Whereas f u differ.. in another applicati on ' ' " ,. �Qn fi 'The: of ma,,'" obJecls 'S con ned 0 some • fi"d number ,ch , Id wh ,\>OUI " l n '1<" containi.ng iquid heli� � phOIOR..., A bowl ' 't the ca.'iC for volume. \hi.!> isn It reneets them. All ob'je � ' . ...; we mighI say . , ('llrb helium alom d ab nn mit pho.ons. The poi", 's that lO a not e and absorbi ng g em,lI,n n.ly y . "'" how.,,,, .re cons.a t.. ·farehund; rather. 'II is determined b ' . I ..Cho�O . . . ISO l}l;T nUffi the phOlons and .he Surphoton gas. rium between the . oR� lead 10 equilib of photons havin gas whllC'\'er condili of a e 10 thlOk may seem ·mang . g, can <en"nly Imag_ rounding mailer. It we Su' brium w,m onaller. equ,h '" . ng bei .empc"rure and one IS honer than the other. wh_ . O'ieral1 slate where . . ' num quthb matter or vice \'ersa. If Ine a none . from photons 10 t' IS transfened prob.ble/d'SOrdered a moS! energy on s'YetaJ solaoed. ' are omeTW S. ' bu' "Ii Y lhey exchange e resuII. . ,arm temperature must ! Un! How do we use the B sl8te 0 ' 11 difficulty: seem to lead 10 But this would we have said 'ha e for fIXed, r N ,s no' buoOon if Ihe numbe ? E,ns"in dislri '0101 number of part,des S e is relaoed '0 .he B .er panme JUS! aceep. e dislrlbution', question. For now, we re,olu"on of me Append" H for • conelus,on, B ; \ . bappily simple use the lools we have developed. we art ready to Disnibution in band.

:

��

�

IXA MPLE 9 . 6

-__ ('Mil) In Itl< fllon ot dC\;lh)ma�llC'h': r.-J.ah\'I\ ptrmea,c,1l11 maRcr. ...at'f C'OOt;Ufb; \)'.•..-.Ilal,ng .:haq:I,.'\t partldc' H\Wo 10k) t� CMlJlC''' �\I t�

•....-e .

�II

nkbahun and

� compare• (a) hr.t .:akulatc \1\(- 1�\lal r"'*'1\ cnc�� in a wlurtl(' \'. \"aIlIl\I II . � 'atf'. I'll !I\\l,J \1\CN"-t' III '�Ill�lk \loCO ",.11 hcl"k:chlfth tend w "� (' ! k nNll �� \11 a m"�'''''':llrh: n'l/rm Id man) panldc.., II "lllfllc" qu.mlll), : , a ",m,,"" qu.tnllt) 1 � � t h... the cncrJ� Of 1I.,C'ra�c cnerg� \11 II pan!..:\(: f�� mlt1ul \101" \'It" ",dul

�*':.

1.� II

(I))

�

ro' \'

d\ t ' I� 'If"

" ..uml" that an ideal I1lUIl.lImn;c gJ.. at

STP enJ K,

II,.. lhe

1 atml '\ in cl\luhhnum

lUth cll'.:tn..lnlagnetu: riu.!lal,on Find the ratio 01 the phlllOn intemal energ)' III the itlttmlll l.mctu.: cncrg) UIII�H�. (If the gil' ,OlUTlO N

,11 -\' nlltccl carlier, the clCIlOIlHnalQr 11\

(9·26) i, the tOI;\! number of rli\nL':Ic, Ih:

Dumer.lll1t is the tot,,1 energy. and to calculate II. ... , e need Nlf,) and D(f:lcqlJ.llIl)n.. (')-J2) nnd (9-44), n:..pecti"ely-wilh 8 I

jE

I

_ _ _ _

1 ,1- .IIT

I

811"� E2 dE II' c-'

We allo" lur all pos\lble energie<; from 0 to infinity, '0 we may u\e the mtei!ral

1.'" " Tt

pro\lded If we mal..e the definitIon: f.

UpIIOi""

-

811"\'

h" c

()

I ("l -

I

l8Tt, Thus,

, (k,T,)-k,Tdx

(9 46) We sec that the energy in a photon gas grows as tempemture to theJounll power As a rule, we e�pect ll\'t!m8e particle energies to be proportional to Tlo the IiTht power. and this is indeed the case for the photon gas, The total grows so quil:l..ly becau..e the /IumlJer of particle!Jphotons grows as to the third (�e EJ(cn:i� 74).

T

Ibl For electrOmagnetic radiation,

The kinetic energy of the nom., : is N times the average energy of the ideal gas law,

UI1'lJ!lCf

=

N X

�k8 T = �PV = �(1.013

X

The ralio is

6.1 X IO--() :( 1.013 X 10')

=

4 X

105 Pal

� k8T

U\lOg

X V

10- 1 1

We conclude that under ordinary conditions, the fraction of a thermodynamic �yslem's total energy 10 the form of electromagnetic radiation is very small.

Rad iation Blackbody ht: Lig S and link between ternpera ynamiC ture ,he crucial thermod provides s nic e of the electromagn . na,ur cha me the "lc 'ical ous ly,

v, statis was mostly a m all "on. Pre er of ,emperalUre 'ic rod.. ne ag l1l a gIven of ..... u v... 1..r ec' .. originated e obJ as law n an me and man r m e ly a fro -Boltz coming ,he Stefan radiation's .Inten ,he e, at rodia,ion Opl . h , si'y eXat observa"on on for rimental i' governs heat transfer via observati ause Xpe e bec ortan' n' of the . very imp cation. Accordln " s,ateme · 1 Justifi tlca gly, The law is 'f'. ked theore aC on I . " ds ' . on, ye depen Exercise 78), was one of sta oati (see rod wS follo agnetic the law electrOm evemenlS. m which achl fro ), y -46 earl (9 . celebra,ed equation in,egrand in (9-45). It IS 'he ics' more xamine ,he han mec Ie' us e us'ical t, photons igh Who those 50 of ins gy ene ther r_ ener fur oonagnellc To gain alai eleclr , ,he 10 on around E. contribu'i range dE all sm a gies lie in QuanlUm

.

•

.

terms off, ess this in pr eex r d f: hJ. we may e df arou n Using E % uency rang freq the pholOns in

10 the

/if

dUphotOn

==

3

. giving the contributlon due

8"V

-;hf/k8T _17 d/

(9-47)

energy density or blackbody s spectral ' is is Planck' th 9.20, ure learly, the total pho,on energY has C C ndix . Ploued in fig · ed in Appe ency at whic h thIS m3XJmum f W, also deriv thai the requ . radiation la wn sho be . This is the essence of Wien's , and " can . aI ,.,e ,emperature. u a maxImum 10 n o rti uy . wrec ., _, propo nally put forth with0ut theoretica

l occurs IS , was origi an-Bollzmann rtio a1 S,ef propo reflec ' The like 'ed , ation.) law, which 79 for the deriv (See Exercise on. mum at a frequenc 0 3 illz; the fica,i maxi justi e has a curv K 300 o Itzmann law is The 2 dence of in Figure 9.20. The T' depen THz 6 d a a,47 un are . urve) K case (,he 4600 K curve, gy in the 4600 ener 'alai the r o f au', alsO bome

; ;'i �

the S,efan-B

gy radiated by tromagnetic ener Elec 9.20 F;gure lures. dlfferenl,empera

er the c

an objec' versus frequency " ,wo

4600 K ,

2300 K 238 THz 476 THz (1260 om) (630 nm)

f

es that in the 2300 K case. Extrapolating to still lower temperatures. we why something at near room temperature, such as a human body. c3ll easily see

. tim <S 'l�

ely liule electromagnetic radiation. and essentiaJly all of it at wave­ emits relativ

r than the 400-700 nm visible range. "Red hot"' is very hot! lenQths longe o II is worth noting that our derivation of equation (9-47) here, which treals netic radiation as indistinguishable boson particles, differs con<;id· electromag el1lbly from that in Appendix C. There the radiation is treated as standing IrOves (albeit groups of photons). which are distinguishable, justifying use of Boltzmann distribution.

me

9 .8 The Laser

scientific discoveries have attracted more attention than the laser. Few modem Eye surgery,

communication, nuclear fusion. information storage and retrieval,

product bar: code readers, pre�isio� ali�n�ent, and �olography are b�t a fe� of . The heart of Its utilIty IS Its productlon of coherent light. Light its applications.

when it travels in only one direction, is of a single wavelength, and is coherent meaning there are definite wave fronts where the oscillating elec­ is in phase, tromagnetic field is at a maximum at a given instant-depicted in Figure 9.21. By contrast. light from an incandescem lightbulb diverges in all directions. contains a broad spectrum of wavelengths, and comprises disjoint pieces of waves produced by countless independent oscillating sources in the hot fila­ ment. While ordinary light can be focused with lenses or mirrors into a near­ unidirectional beam, and gas vapor lights may produce reasonably well-defined wavelengths, a laser does these things bener automatica1ly. Above all, the '"in phase" nature of its light cannOl be well duplicated with ordinary sources. How is all this accomplished? Light from ordinary sources is produced chiefly by spontaneous emis· sion. one of the photon-atom interactions shown in Figure 9.22. Suppose the light-producing material is a gas. such as in a fluorescent bulb or sodium street­

Figure

9.21 Coherent versus incoher-

ent light.

light. An electron in a higher atomic energy level drops to a lower level in a short but unpredictable time. producing a photon. Transitions between the same levels in the material's many atoms produce photons that are of the same wave­

length but uncoordinated, moving in arbitrary directions and not in phase with

each olher. Another possible photon-atom interaction occurs when an electron

is promoted to a higher atomic energy level by absorbing a photon of precisely the correct wavelength. A third interaction is stimulated emission. An electron

Unidi�tionaJ Monochromatic Nor in phase

Not coherent

n i ane energy level. if intluenced by a passing photon whose energy is precisely me difference between that level and a lower one. may be induced to drop to the lower

level. In essence. the first photon stimulates the emission of a second photon of precisely the same wavelength. Moreover. the second photon moves n i the same direction as and to phase with the first; they are coherent. The word laser stands for "light amplification by the stimulated emission of radiation."

To understand how we exploit stimulated emission to produce a coherent light source of mall)' photons. we need to consider the exchange and balance of energy between matter and the electromagnetic radiation with which it interacts. As we do so. an lRtriguing conclusion arises.

• , ,

•

Unidirectional \ 1onochromatic In phase

Coberent

Figure 9.22 Th different photo .... . n-atorn Interaclions

Spolllatle(lU� emi�\lon

Absorplion

Stimulaled emi��jon r is a gas of atoms and restrict OUf aHem,· . . . �� IWO of their energy SlafeS, as In Figure 9. 22. transitions between just The I and the upper as level 2 . It is reasonabl level as to refer we e til lower Siale n. ,po ' should be proportional to the nu mb [he rate of spontaneous emissio R n r level, N2· We use uppe the from n dow drop [0 ASpon for thi s of aloms poised t. stan con ality 1ion propOr

Le' us assume mat the mane

�

Rspan

- Aspan N2

proportional to the num ber . Similarly. Ihe rate of absorption should be of par". . d , promote b e to b ut pOIsed It · are must that also depen des in the lower level d e t f · h · h Just ng h are at t t 0 photons energ y nearby of to prom on the number ote them. This energy would naturally be £2 - £' == dE. For the number of pho . ­ . [Ons of this energy. Ie! us lemporanly use the SImple symbol Y(.6E).

R'b'

= B.b, N, Y(IlE)

for the proportionality constant. It was Einstein who fiJrSt . charted the cour.�e we follow. and the coeffiCIents are known universally as the Einstein A and B coefficients. Finally, the rate of stimulared emission sh ouid depend on [he number at the upper level, available for emission, and aga . m an the number of photons ofjust the right energy. Here we use

Bab

"

R,"m

= B,"m N, Y(IlE)

Now. a'>sumjng equilibrium between atoms and photons' the rate at Whlch · . . atoms make upward transHlons from level I 10 level 2 must equal the d ward transition rale his is known as the "principle of detailed balance." .. ply put, the two emISSIon rates must add to the absorption rate.

�

�:�

9.8 � l.&ter

A\pon N2

+ B�um N2 Y(a£) = Bab\ NI Y(,1E)

Solving for Y(.l.£), Y(IiE)

�

N,

N2

Babs

'[he ga5 atoms. being diffuse. obey the Boltzmann distribution. �o the ratio NI ' \'2 is e-EII*BTje-EJkBT = etlE/ksT. Thus. Y( liE)

�

s j --'Bs '-:::--= A po Bab

,'Elk.T

stim

- --

B,I»

Here is where the "intriguing conclusion" arises. We defined the quantity of photons of energy 6E, but we know what this must be! uE) H as the number energy density (9-47), with d£ (i.e., the photon spectral is the photon gas the atomic energies £1 and £2) replacing hi in that fonnula. energy; not one of The t1Ei*oT is in place where it needs to be, and the only way the re�t of the denominator can be correct is if Bs1im = Babs' This means that if the numbers the two levels happened to be equal, a stray photon of the of atoms occupying as likely to induce absorption as stimulated emission! just is ncy proper freque don't want these rates equal. We want stimulated we laser, a in Of course. emi��ion 10 predominate, but how? As the reader might have guessed, simply making N2 » N1 would do it-Rwm would be much larger than Rab�-bul such a situation is certainly not !he way things are in equilibrium, where number drops with energy. (Note: We a.>sumed equilibrium to prove that B\lim = Bab�' but they are indeed constanle,. so they are equal even when the situation is far from equilibrium.) In fact, such an overpopulation of higher levels is crucial 10 laser operation and is known as a population inversion. Being unnatural, it must be established by some e�temal means. Bul so long as it exists. one phOlon of the proper frequency. perhaps the result of spontaneous emission, may become two, which become four, and so on, very quickly resulting in a large number of coherent photons. Before investigating how a population inversion might be established. let us take a look at the basic elements of a laser, shown in Figure 9.23. Because spontaneous emissions could iniliate separate "outbreaks" of coherence in dif­ ferent directions, the medium ie, tuned to amplify the coherent light injust one, Parallel mirrors are placed at the medium's two ends. Photons not parallel to !he axis have relatively few opponunilies 10 induce stimulated emission. while those parallel renect back and fonh. providing many opponunities. The length along the axis is tuned 10 a standing-wave condition L nAn, so that waves moving in opposite directiom constructively interfere. In the re<;ulting rtSOnant cavity. the radiation parallel to the axis is greatly amplified. and off­ a�is radialion ie, dimini ..hed. In fact. this resonance is so selective in ib tuning !hal the variation in the Ia.<;er wavelength is typically several orders of magni­ tude narrower than the natural " line width" of the spontaneous atomic tran"l­ lion thai gets the prllcess e,tarted. (Accordingly, simultaneous output of multiple closely spaced wavelengths within the line width is possible; �ee

It

I

=

Figure 9.23

Basic componen15 of

a laser

Mlrrof

f Pun::'ing •

<e�

,put)

317

,.". , z.. nat >JO,inal l.ucr. Wllh tptr.1 n... l llube \urrouDd'ml R$CIIWJ' Qlur, of ruby

h(' n ..i� so., B)' m.tkmg lIle mm"Or 81 one end only partially refiecling. a frac _

III.n vf Ctlhc:-renl lighl i� allowed 10 exil lhe cavity.

There are \('\"Cral "a.p of addmg [he exlemaJ energy needed [0 prom olr d('dflln� (0 the higher le\·els. One j" optical pumping. used mosl often I n �."d·\I�Ie' Ja...el'. ",·here Ihe Ja�r medIUm j). a rransparent solid. The medi u

fra:

1\ \uhJe<:lcd 10 it \·el")· mleno,e lighl of broad spectrum. of which al least a (Ion /\ of the required frequenc)" Allhough much of the inpur energy is diss I_ pated J\ hear. a \;gmticanr amounr becomes coherenl Jigh!. Usually. 0PI/ca l pumping m a \olid·,rJle laser I), pul\ed 10 allow lime for cooling. Figures 9.24

and 9.25 ,how '"0 example, of optical pumping,

The firsl is the Original rub) lil'>er. who�e optical pump h a nashlUbe wound around a ruby crystal. the "las­ Ing" m;uerial. Figure 9.25 shows a one-pass laser amplifier, i n which a bank of high-inten'otty lump" " charge\ up" neodymium glass plales just prior 10 a lase r beam" enrry. When Ihe beam enlers. atoms in the glass deexcile and a much

more p<merful beam emerge).. (The beam i\ polarized. and light is passed

Wllht1Ul renection by orienting Ihe plates at Brewster's angle.)

AnOlher means of energ} inpur is elecrric discharge pumping. used in ga, lu\cr,. Arom, are rai\ed 10 higher energy levels as lhe discharge ionizes 'oml� alOm\; and produces generally violenl mol ion. Because excess hear dissi� piuion /\

Ie, ... of a problem in a gas,

U\ oppo,ed to pulsed, operation.

il is u.suaJJy possible [0 sustain continuous

•

.-\ VC\lflg problem remnins: Optical pumping can increase the population

of a higher energy level onl} when the number of electrons ar the lower level

Figure 9.25 A one-pas, IUlIer amplifier. An lRU is a replaceable unil of neodymium

gl.MplllfC'

Flashlamp cassatte

Slab cassette lRUs (2)

Q � l'h< L.a,<,t

::) llOd the point where the numbers in the twa Ic\'e\<;

ho" Lnten...e. it cannot increase a higher lc\'cr... popu\a· ater. No matter are equal. for ,\I thi ..

1� g

Oln

.

8

I · stmlU · lated eml'iSlon. · (Rememlrption I'" na more probable than IS po.1LIlt ab�( ' ::::;: Jb B . ) Nor would this stimulated emission be u'ieful. 'iinec It

t-er

,Ulll

. · herent phOIons from t he optlca · 1 pumpmg �ource . ed by the mco ld be induc alone is also incapable of e'itablishing a population Im cr­ dbcharge �

\\��tricThcn slate. a�hj,elllsslable E�

how can it be done'? The problem is sohed by u<;ing a mcdium with . . . Most downward atomic transItIons occur 'ipontancou'ily 8 S. A few. occurring via relatively inefficient mcchani ...m'i u 1 0 . t abo 3 Section 7.10), take much longer. typically 10- s. The upper level 11l 1'uch

..)\In.

\\,�t'

!' Ir:lll s ition is known as a metastable-nearly stable-state. 1 To see how this solves the problem. consider the simplilied model in

n collcction of atoms with only three energy level!': the ground figure 9.26an excited metastable state E,. and a yet higher. nonllletastable state �wt, £I ' . £1' from which electrons drop preferentially to E2 rather II�an fully down to . E2 - EI ), so that essentIally all atoms arc TIle temperature IS low (kaT r he ground state. Very intense optical pumping is now applied. so intense

<

{

ut

that a point of equal absorption and stimulated emission is reached between all le\'cls: NI

Nz

=:

=:

N). Then the pumping is slopped. After about

10-8

s, all

declfOn� will have dropped from the "fast" E] Slate. but more to E2 than to £\ .

For Ihe next _10- 3 S . N2 i s greater than NI ' and the situation il) ripe for a cas­

cade of stimulated emissions between £2 and £1' In practice. pumping need 1101 be stopped. and output is continuous as long as pumping keeps ahead of

the ,timulated emission.

which would otherwise quickly deplete £2'

What we have just described is the

efficiency is

the

four-level laser.

three-level laser. A way of increa-.ing 9.27, in which another

depicted in Fi gure

"fast" level is employed, above the ground stale but below the metastable one. leI us call this the near-ground state. Its energy is E ' and we redefine the

I

ground state as Eo. The benefit of this arrangement is that in the first -10-8 s after pumping. all excited slates but the metastable one depopulate. Thus,

rather than dropping to a nonempty ground state, electrons in the metastable

�lale may drop to the essentially empty near-ground state. The four-level laser

is more efficient

simply because the ground state is not the terminal state for

Figure 9.26 Energy transitions in Ihe three-level laser. E,

E,

------

- --­ :--'\

• • • • • • • • • • • • • • • • ,

Metastable

r\1l moms in their �ound 'Iate�

\

'

Slime electron, in all highl'r leveh

../

"

Within _10 1I �. £� .' depopulate,. but nOI metastable £�

.

E,

II . ' .. " ' " II " '. II '" II.. ..

,

.. .. ..

Equilibrium

u

• • • • • • • •

After pumping

...

....

j

\

1alion in\er).ion.

wht:rC �>

�

!

I'

• • • • • • • •

Population inver�ion

389

FIg"'" 9.27

FOU1_le�'el laser.

•

--E, --

1 1 JJ

epopuJale�.

\

og a large _,<:

"

"

IJlion 1n\'er..ioIJ,

MeWlable

>N 1•

/'

"

"

•

--E, --

E,,---­ Equilibrium

-'

..

. e •• • • I

Afterpumpmg

Population iove nion

, IIY Complere population inversion between £. . ion ' . A "mUll '.;mulated emisS 2 . '0 P"'"P the ground state so ncar e mp, havmg y as , d WI'[hour '" nd E is obrame " s.. lasers oflen employ even more tha I n fou, aser. In prac,'c, ' . in the three-IeveI I . evel Jaser slllI applies. (The se four-l he [ f 0 IS bas the ( mic b u on_ energy leveIs. �eren A pop . . " n IO t uIatlo ' Section 10.8, IS d'lu ' cussed rn . ver_ . dUClOf d,ade Iaser. dls ' . has some interesting tWists, meIudmg a complet II ely sian is SIi/1 needed, bul . ) . msm cha me , g . dj(ferenl pumpm Is are employed, only cenam substances mee, No maIler how many Ieve . . op" ation. As noted earher, the first laser r I ase . reqUlred for . na the cnle uSed a Three-level and optically pumped, ruby ruby crystal as the med·/Urn. lasers . . I nm rIgh . Four-level neodymIUm glass lasers. used m produce laser ( nm infrared light · n expenments see Seclion I I . 7), produce 1060 IiUSlO The . · producmg red-orange mOSI common gas Iaser Is Ihe helium-neon, 632.8 n . m . liShes ' populalion inverSIOn m neon by a roundabout Iighi, It estab" pat h . . fi " rom electric discharge 10 metastable hehum states, Energy passes rs , . then via . " . ' helium-neon coII'lSI'ons 10 neon stales at essentIally the same energies as those . . I he helium, followed by slimulale� emlsslon to lower neon Slates. in -,?riven by . "g 'pp/icalions, excllmg advances are cOnstanlly rapl'dly expandI.. bemg . . . made in finding new laser materials with improved capabilitIes. •

69�.3

Cl D

V A

N

C

E

D

9.9

Specific Heats

The ability of a substance to store energy internally is of great concern many fields. At hean, it is a macroscopic avera i ge of microscop ic energy age. and thus it is an excellent example of the stor ulility of stalistical mechanics. A basis for understanding specific heat s is the equiparti tion theore When derived using classical stalistical mechanics, if is lh. as fallows:

�

A PMticlf in a lhennodynamic system af equiJibri um will have. O average, fiBT ofenergy for each n . Independent varIa ble. or degree medom. 00 which its energy depends of quadralically.

9.9

SJ!C'=ific Keats

of independent variables and quadratic dependence are a one-dimenE.�amPies 1 2 1 2 · , which has 2 degrees of freedom; an atom lator, E = oscil . aI 1100 . aI _ 1 2 I ") I ") . gas, £ + whIch has 3; and an . a mooatol1llC Ide ';hich has I . We won't rotaling about a single fixed axis, E =

� � 1

2:mv + "2 KX - zmvx + 2mvy

.

ect

\"e the theorem;

2mv:-, �/w2,

rather, we will investigate the circumstances in which quan­

mechanics limits the theorem's application. As we shall see, low tempera­ . problem. FIrst, though, let us see why the theorem is so important. res are the IU According to the equipartition theorem, for each degree of freedom,

Eone pat1icle =

_

"2 ksT 1

per degree of freedom

energy is just the average energy times the number, the Since the total internal of a mole of particles (Avogadro's number N inlemal energy

Uper mole = N where we have used R

(he amount of

E

A "2 kST = "2 RT 1

1

A) would be

per degree of freedom

NA ks· Although specific heat is usually defined as

heat required

to warm a

unit

mass per degree of temperature,

R

measured in Jlkg' K, we will study the related but more reveahng molar heat capacity, an energy per mole per degree. in J/mol · K. At constant volume,

when no mechanical work is involved, this is defined as

a Upe r mo(e

Cv � --"a"=T=

Thus, we see the utility of the equipartition theorem: Yielding the coefficient relating

U and T, it is the route to heat capacity.

Gases

�

The equipartition theorem says that the internal energy per mole of a monatomic gas is RT (degrees of freedom: x y vz.), so the molar heal 3 capacity is 2: R. A diatomic gas molecule, however, may also possess energy

v ,v ,

associated with rotation about the two axes perpendicular to the axis through latter the x-axis, this adds 2 degrees of freedom: w., and

the atoms. Calling the

w.. the angular velocities about y and z. (There is no rotational energy about the x-axis. because all the mass lies along that axis.) A diatomic molecule

vibrate along its x-axis, adding the 2 degrees of freedom of a one­ and xrelative' According to classical statistical mechanics, then, a gas of diatomic molecules, with 7 degrees of freedom. should have U Diatomic ydrogen·... molar heat capacity does approach but on ly at at room temperature. It more than 1000 K! As Figure 9-28 shows. it is doesn't appear to store energy in vibrational motion. The reason is that. so far as quantum oscillator states are concerned. room temperature is very cold.

may also

dimensional oscillator: "rel3lille

r mole = �RT andCv = �R. h

�R

� R,

NA kg

8 31 � J/mal· K

391

capacity with tem peraru " 'drogen's molar heal re. (M 1 VOle . Vanauon 0fh ) .' reached.) l PIateau IS lJona vibra ' the before . s I e ocl3 . dlss cule The H1 mole

FigU'" 9 28 .

..... .. ...

7

V'Ibrat iOl

,

, ,

J

2

ROlalioflllJ

---:l./--T_ ranslatio na 800

[40 200

80

1400 2000

T(K)

' the energy jump to the rlfst exc ' kB T is much less than liwo . ited sta That IS, ' brauona ' te' I speCific ' VI e t h determme [0 wish heat if we n .

Conseque tly,

contn' b u. energy, even though the osclllaror average ' the Bo ltlm tIOn by caJculating an . n ' "on IS diffuse gas 0f rno leeu les, mtegralr i nval'I d. d'IStn'bution is correct for a W e = ka T, and II clearly won t do, U integrated to obtain (9-28), E per mole Would to Cv WOllId be the co " be R T.' and the vibrational contnbu[Jon nstam R ' 11 lar rat particu tempe any at ure. in" "kick ly Equ ation ( 9.would not sudden 15) , , IS ' the correct rou te'. obtained via summation .

_

.

.

'

9.29(a) and noted in Section 9.3, at high tem As shown in Fioure c perature . ' where ka T » /iwo' this vlbrauonal energy approaches the expected r Inear ' h the classical increase with temperature kB T· Th'IS agrees Wit eqU ipart'alO , n I , theorem: 2 degrees of freedom gIves 2 X ,ka T, Bu t at low tem pe . ratUfe ' kB T « hwo' the average energy Increases only very slowly becau c se lew of the molecules would be energetic enough to occupy any state othe r than the ground. .

.

•

.

Let us now see how this affects the heat capacity. For a mole of OSC I· 1la' tors, the mlema ' I energy due soIeIy to VI'brauona I motJOn ' would b e

fzwo

Differentiation then gives the vibrational contribution to Cv.

Cv. Vlb. '

'

lal

=

f'I9U'" 9.29 Average energy and molar heal capacity of a syslem of oscillator.. at Icm­ pcratu�l> where koT - "wo· R

O.5hwo

1.0"Wu

1.5"wo

2.0"w(l

(b)

,.)

Figure 9.29(b), essentially the slope of 9.29(a). plots Cv. vib: Its resemblance to the vibrational jump in Figure 9.28 is conspicuous. From 0, it jumps to practically

O.3tULIO· At around this temperature, translational kinetic energies great enough to begin an interchange with vibrational energies. Given �hould be 14 \JIat Wo is 8.28 X 1 0 radls for the hydrogen molecule, the corresponding temper­ ature works out to be 1900 K, which is in good agreement with Figure 9.28. The other jump in hydrogen's heat capacity. around 200 K, is due to the beginnings of energy exchange with rotarional quantum states. The basic idea is the same. Unless the temperature is at least comparable to the jump to the first excited rotational state, rotational Slates cannot participate in energy e:
R at kBT

=::

occurs at a lower temperature.

In summary, we see that the spacing between quantum energy levels must ta k be en into account when applying the equipartition theorem to predict heat capacity. At low temperatures, certain degrees of freedom are "frozen out,"

effectively storing no energy.

Solids

In a solid, atoms can't move freely. They oscillate about fixed locations and do not have rOlational degrees of freedom. They may be thought of as being held

in place by "springs:' due to interatomic forces in the three spatial dimensions. Because a one-dimensional oscillator has 2 degrees of freedom. we should e:
�

capacity 3R. tn a metal, however. the conduction electrons do not oscillate with

the individual atoms, but move freely about the material. They constitute a gas

coexisting with the solid. and with no vibrational or rotational energies (being "'monatomic" ), they should add a factor of R to the molar heat capacity of the

�

�lid. Classically. then. the molar heat capacities of solids should obey Insulator: Cv Conductor· Cv

= =

3R

4.5R

Classical expectation

classical arguments bre* dolV n at nsing thai surp be t m temperature. the mal ar h roo shOUldn' il at n --.01 eat eve By _ _ particular. from those of insulators-and ures. In much at per er lV 't ditT e loW tem ctors don ' a metal fill the IOWek · 10 s! ene ectrons of condu I e es . rgy 0 "iti � c """ ...... on. At T dIOary temperatures. . the reas IS SO E · But at or '""y have .'_ au m",lmum F a to Example 9 .5). A modest . up (ct 0 be ible te",. .. ht as wel l states p" · that It ntlg packed IOta essentiall the s E i o", n eetr tha .. arne . have th e e much Ie F litt le with te very would stili e; chang rature. gy perature average ener if their zero linearly With from and . ase < els how_ ncre energy lev small . ( It I is very electrons should ' y n l the acit b e mUch that their heat cap II we accept , h 84 showS.) statistical m ec al classic aOl cise cs preExer how well rn ever. as showS just ' 9.3 room te ' e near pera . Tabl ture. FO . crystaI h ne soI Ids . r of a factor . ' for cen.,n Ie the pred" ' tlil. . clty 1cllo S . 3R n cap. faI iS fO near t very hea " s r diet ductors. It . . atlOg atoms find room mperatu rs and can vlbr se re who both insulato . those specifically many solids. � aves of the atoms in a soljd beh at 101V tOO cold. heal capacity the how te analysis as or stiga same Vib the y inve rating To to appl B be tempted n or rotatlon translatio . we might free ure for ut the perat . tem allowance kly. A1th U h h ugh without tho wea s. very cule ct t ey do gas mole molecule; intera They a Simple. Gas so time. 't a at isn SSe on nllally situation one collisi do so only are independe energy. they motions e nt lonal [n a xchang e se vlbral . partidO' who . ate th pendent. separ . The mOllon of · Y mterde . free and ons are strong atontlC VIbrati ver. aff ect all which owe ose of h around I . ons soI'd nd It. the moti all those arou aoalyze t we h cts e do affe Col/e How one ctive elike f""hion . on. in a wav them. and so atoms? motions of the valent approaches, the same two used to rent but equi diffe twO ' the stan di ng-w There are ave bound in some region' ' ic radiation agnet trom in A density ppendlX analyze elec energy C , . and k s spectral to denve Planc . an t l es betw sim'l I great are approach . used There een 0f Secuon 9,7, 4i approach tic "vibrations" bound in a the boson ga omagne . electr I 0b�ect and . . s bound 'm a soI·d treats VI.brallOn , the OSCIllations as ing-wave approach stand the case. cavity. In either

BT

��

chanK'

•

.

:!

I

"

ed solids

es of select TABLE 9,3 Properti

(g/mol)

e" (J/mol ' K)

3.5 t

6.9

24.2

AI

0.90

27,0

24.3

P

0.77

3 1 .0

23.8

S

0.71

32. t

22.6

Fe

0.45

55.8

25.1

Ag

0.23

t07.9

24.8

Solid

c.. (JIg ' tel

1I

m

3R = 24 9

•

9.9 Specific HeIb

395

e standing waves of different energies and uses the Boltzmann distingui\habl The boson gas approach, on the other hand, treats the energy as a ion. ibut di�tr tinguishable boson particles. In the case of electromagnetic radia­ .u of indis are, of course. photons. We now assert that mechanical particles the may also be treated as a gas of free bosons, known a.. solid a 10 I'lbrations quantum of vibration-has zero spin, and it may phonon-a pbonons. A tr3wl from one point in a solid to another. carrying an energy Iif. The reader suspicious about phonons. Are they real particles? The answer is no. may be same sense as photons or electrons. They have no existence apart the nol in of propagation. However, whether one takes the standing­ medium e from th boson gas approach, the predictions come out the same, the or oach appr wave and this is strong evidence that vibrations ill a solid may be treated as com.

�on,

prising (In integra/ mill/ber ofdiscrete phollons rather than diffuse wal'es. The Debye Model

Whichever approach is taken. the steps to finding the heal capacity of a crys. talline solid coincide at many points and are generally referred to as the IXbye model. Because it most closely parallels the treatment of photons in Section 9.7, we use the boson gas approach and begin with the internal energy a phonon gas. analogous to the pholon expression (9-45). To adapt this equation to phonons. we need only modify it to account for three principal

of

differences.

Maximum Phonon Energy While photons of arbitrarily short wavelength are possible. if improbable. in a cavity, there is an absolute lower limit on the wavelength of vibrations in a crystal. Figure 9.30 illustrates the reason. ln a vibrational wave, adjacent atoms oscillate at the same frequency but are at dif. ferent poinlS in their cycles. A long wavelength involves a small phase differ­ ence between adjacent atoms, as shown in Figure 9.30(a). Figure 9.30(b) shoWS a short wavelength, in which adjacent atoms are one-half cycle and thus wavelength apart. Let us refer to this as A in . As suggested in Figure m 9JO(c}, it may be shown that for any wavelength shaner than A m n, there is a l wavelength longer than Amm for which the atoms would be at the same points in their cycles. (The proof is left as an exercise.) Thus. it makes no sense to speak of a wave for which A IS less than the spacing between adjacent atoms, � because it is physically ideltical to one for which A is greater than the spac­ ing-the sine waves coincide at all discrete atoms. Accordingly, the minimum wavelength is 2a. twice the atomic spacing. A minimum wavelength implies ' The integra­ a maximum frequency and thus a maximum phonon energy E

one·half

�

tion in (9·45) must be terminated at Ernv;.

in

Propagation Speed All photons move at

I]

4

m;u:

c in a vacuum. Vibrational wa\'e.'o solid. however. not only don't travel at c. they don·, all travel at the same speed. But we ao;...ume that as an approximation they do move at the same speed-the speed of !\oound I', characteristic of the material. We replace a

(by 1'5'

1 B� the .."" ...n"l> do _ ...,.Jlcide ..
396

C"'pter 9

SUlli'tkat Mechanic�

Iy In a CI')')lal. on .30 ' Ftgure .

ter than 24 m vibrational wavelenglhs grea ake se

nst.

(.)

A ",�

(b)

Figure

9,31

�A = A",,, l A =A ",,"

Atoms in a crystalline

solid.

« )

Longitudinal Oscillation Density of states (9·44), obtained in App endix

C

�

includes a factor of2 for light's twO independent transverse polarizatio ns, In . solid, longitudinal vibration is also allowed,. A�sumlllg equal. speeds for trans. verse and longitudinal waves, the net effect IS Simply to multiply (9-44 ) by �. Taking the three faclOrs inlO account. equation (9-45) becomes

(9-48) Figure 9,32 A cube of allowed wave

numbers and a spherical octant of equal

"volume,"

Still, we have yet to link Emax to the minimum wavelength

2a.

In one dimension, atomic spacing (I would be [he length of the solid

divided by number,

2LIN.

LIN, and the minimum phonon

wavelength would thus be

In three dimensions, determining the minimum wtl.\.Idength is a bit more

involved. Consider a crystalline solid with atoms at the d)mers of cubes, as in

, Thus, the spacing Figure 9.3 1 . The number along on� side is not N but i\ JI3 , !�'ng one axis" would between atoms is LIN , and the mlllJmum wavelength "

be

2LIN!lJ,

Wavelength isn't a vec tor, but there i s som;

is: wave number, (Remember: p = lik is a vector fOI

Using

k = 2rrlA,

the corresponding

maximum wave

would be rrNJ13/L. In three dimensions, the wave-nu,�, anywhere within a cube in wave-number space, depk'

has one comer at 0 and is of side length rrNiI3lL. h.... would obviously depend on its direction (it would be

'ng related to it thaI

if P = hlA = hk.) �ber component k vector may

poin�

Figure 9.32, that

(!Inurn magnitude :test when pointing

r IO"ard we fa ,,\(1h.l

rJle."

here

corner). But by finding a -,phencal �. tant that enclOSc:s the __ we obtain a reasonable average m

(::�1�t.

all.lmum---the ''--'" '-".us" of the !.phe -

. k be mu." EquatlOg the vOlume., of the CUbe '"' Quid \ and

(�:)' i(j �k:,,,)

,p

"

(6W})'"

k "" =

=

�

k

_

only relate to E. For a simple plane wave k . w' need , ::= 'Tr/� ' with OUf assump We no kl21r. Thus, hrfA, tion that aII , E � 111 = In' waVe '" s move

IhJspI eed e ,arn

�O

.

v�•

we have

Emax

hV,km" _

=

Beforen. '

. o definiti

- liv

27T

( )1/3 3

al the

N

_

47TV

�

inse rting this in (9·48), we streamline things. Wit . h the follo wing . To at

Em" k.

_ -

( )1/3

hv, 3N k. 4rrV

-

_

(9·49)

k gy divided by the Boltzmann

constant (in J/K) . g an ener . . of s umt'i ' T0 hu Bein . as t he De bye temperature and nown IS . It re de atu pen d ... on proper· telllper ' I y and vS' Equation . I . N'M (9·48) th u'�. become� . 0fthe given materia s tie Upho"o"

=

J.'.TD 0

E

I

eE/'.T

.

9N

_

I

2 (k.TDl, E dE

(9·50)

this by redefining the dummy variable of integration. We simplify

Elk.T �

x

. Therefore

' and using = t is, A Finally, assuming I '1101, tha atil r: 1 energy of a solid the intemal vibr

N

-: .,01,

Alas, after 01 Figure 9.33(a) ,he

molar internal elle we also see Ihat "h

temperature. This (1 which increase� .� :

"

=

9R

N

T'

{"F

Ti, Jo

e'

x'

_

k8NA 5' R. we obtain for

I dx

(9-5 \ )

work. the integral cannot be done in closed form.

•

re,ult of a numerical evaluation. As e.pected, die

!'proaches the classical 3RT 81 high _penlll.le but temperature for a solid is judged relative to its

Debye

, stnso. for To is proportional to the speed of 1IOUIId va' strength of interatomic forces, aDd stnJo&er ''IpriDp'' _

"- _

1ft

,

i JRI· rI

.1 I

"

d('�

l"

lr

JIlT

..

I

r

L,

o

O-STo

I.OTo

I Ofo

(b)

f.f

� �. molar roer lump" between fmph ,,, . -. ra!ure. II npproachc!t the classical 3R b rempe high AI . • U heJl l'.:Ip;J�II) · ally pred,cted hea t c ' ". drops \(�I) /0"' T Planed with IhC Iheorel/c I(l 0 ap Quantum level.... Figure 9.33(b) sho

acJlY a re al cry"ilallme solids of dittere ever ... of nr . D mea,ured heal b e ' Ye , the appro.",mal l ons: we ve made along th e Way lc:mJX'ralure�. Ewn given _ nd tranwer!le waves, geOmetry. depen umhlml 'peed for longlludmal � den t enl ,... excellent and greatly strengthens 3\craging. and '0. on-agreem [he c . ase " solid as a phonOn I energy 10 a cry....r3I Ime 0as for lfearlng rhe \Ibrauonn c:. ("'. If lC . _ solido;; 1iuch as glasses. In which the atoms r Iheol)' talh for amorphOu\ a e nOI parrem. Liquids are similarly difticu ll lo analyze.) • atrJnged in J rec:uJar al

l'apaci fie�

•

EXAMPLE 9 . 7

��O� aboUI O.73R,

The molar heal (apa(il), of diilmond ill 293 K i � 6.0 lImol . K

(.:I) An.:onJing 10 Ihe Debye model. at whal lemperalu re would Ihe

II}' be 2.5R'l (b) What

I (.) I I I I

I I

I I

is Ihe "culOff' (maximum) phonon frequency

-­

3r �eal

capac. Or diamon d?

From Figure 9J3(b). we �ee thaI il heal capacity of O . 73R ocCurs al arou nd r =: 0.16TD. Therefore.

7:

D

3

293 K 0. ) 6

_ ) 830 K -

. The \-ajue 2.5R occurs al around 0.51 To = 0.51 (J 830 K) =:!: 930 K. Which hap_ pen� 10 agree "ell w/lh e.\penmenl.

(bl Using (9-49). ihe maximum phonon energy i:,

E�, = IsTD = (138 X

( Jmal

/O-" I/K) ( 1 830 K)

£trl1IX

= -h =

3.8 X lOll Hz

= 2, < .

x

_

1 0 20 I

,IOGRES5 A N D APPLICATIONS A SEC: Many cting as One

Studied for mo�t of n �'('ntur)', hquid helium. with its two components,uperfluid and normal-and strong interactions is an imperfect Bo�e-Ein�lcin condenSaie (BEC). BUI in 1995. after yean. ofstow progress. the quest for the first true BEC ended (Nobel Pri 7C, 200 1 . Cornell, Ketterlc & Wieman). . and a huge upsurge In BEC research began. The first ,ucces� WIIS in II dilute gm. of several thousand rubidium-87 �WOl�. Rubldium-8? has 37 electrons. 37 protons, and �O neutrons-an even number of fermions-making the �oncc[ion (I boson. (Section 8.3 discusses composite particles.) Magnetic traps conlined the aloms. while laser cooling and simple evaporative cooling. the mechanism that cools the human body, combined to lower the temperature to ie'S than 0, I I-'K, A major hurdle was keeping the atoms in a gaseous stale where evaporative cooling is efficient. o,crcoming their tendency to bond. Figure 9.34 documents the achievement. The spike shows a large preponderance of alOms in the same state, Even ils shape confirmed the WCCC5S. "Nonnal" atOms would spread symmetrically after the nonunifonn confining fields were switched off, but the condensed atoms undergo dispersion (see Section 6.4) as a lingle wave funclion, which "remembers" that field and thuS �preads at different rales in different directions. True

BECs hu\e no" been produr.:cd gases of bo�ons.

III

numerou.. ditJerenl

Whol. Atoms as Fermions The celebrity of the low­ temperature quantum gih that began in 1995 has been magnified by numerous rehllcd breakthroughs In 1999. researchers at the National ln�titute of Standards and Technology (NIST) produced the lir�t system of whole atoms behaving a.. a degenerate gas of fermions. The term degclle/'ale here means that the particles' wavelengths are long and overlapping. !IS 111 Figure 9.14(b), so they should exhibit quantum behavior. For II fermion, this means a tendency toward lilling state�. one per p!lrticJc. to 11 maximum level-the Fermi energy. Atom,> containing an odd number of fcmlion� should themsclve .. be fermions and so should lill state... this way, but demon,�trating this required surmounting not only the same challenges as in the BEC but also a challenge special to fermion�. The goal was reached in a gas of potas�ium-.tO ( 1 9 electron!>, 1 9 protons, and 2 1 ncutrons). As in the BEC case, lhe atom!> were held in magneto-optical lrUps (MOTs). and final evaporative cooling lowered the �ample to about 0.3 11K. The "new" hurdle was the reluctance of these fermions to collide! Evaporative cooling requires collisions to reesablish t

Figure 9.34 Number of rubidium atoms. plotted vertically. versus velocity in two dimensions before (left), near (middle). and al achievement of an essentially pure BEC (right).

_ 0.... . ....... ....,

"

• • ,1'" j_ _ "'� .. _ "' _ "'" .... ID 1Ift ,...1n IIXOfdlnJ 10 ipfn �"PD" heR ,......, dar aJIJIbUwd �r.J UId IJICrlmIC qular mot_IlIUm of all d'tc 110m', romIilvrllCZ .ad "''0 fcrnaJoM .. the' UIhr tpUJ mae �', '"'touch," (Aligned tpI_ Impfln an anll.Yntrnr'U'M.: palw 1Ialr. "hKh I� 0 _bm .,...It locaboM COInc..Jr-...a« S«tton 8.2 ) A ,k-vtr r.ohn.... trtpprd IWO ,&<1(5 ot dift'ftmI 'lipIDS. .. hleh "an coil•• br(
Jto,('15

Fetmlons '-'ring to Form Bosonl

Yel ;tnulhC'r pl.l)C'r In

lhe t'pandln, ul.....,lld Jo1� drama 1\ 3 ') J.lem of' femuon,

rdut'tltm, tlt'tw\ ll'ln InII.Illl0nall) allntwJtcd III t-�10\

up:rtlUIJII)' a"d cuncknut"m tl! it §mglc \talC' Tn tht nalLlra) qUllll(ln ,," MI bu ,tit".. ..10 ,'ermwn, hll\c ("n&lm 10, &\ boMlru"" Ihe IIls\\cr 1\ pamng lSc1.:til!n �,J)_

ThaI . pealf or (tnmOIl� �y btha\C u, a t-lM)n

1\ IOlc�\ting

C'notI,h, bul ptmalb t\C'n Oltlre Impmt.ml l\ ho\\ the palrin� ,,'(Jill(" .haul.

rh.) 11.7

mulll r�rlctl qUC'Sllim to 11«11' and J he')l lhal t'IC:ln on man) olher e" llto)? artJ' l''­

•

ft'�h IUpk:

('\1m CI)\tnolol,) Il) \uptn:twJUdl\ It, While

Slngle-Atom laM' A\ 1�l>C'" attract new applicahon\ Ibtu ch.tra..'IC'rUtic."s come under clo..er <'Cnlllh}- On� mU h. tlUdK-d Inll of the cumenliona l la\er i:; the flut:l uallon m I h phCItllO pnxfudlCln rolle. 01 no ..igniiicance in many appli�-�Ii(ln\. \ut:h "nm-.('·' can l)(he quile a problem Ih othcN. In lhe pWRlI'lhg field of quantum compuling. for In,c.ance. thc abiht) 01 eenain �imple syslem\ to m anipulate '!uanlum mhlnnation i\ hindered b)' fluct uations in phOlon llUlpul In an Ironic IWI\I, given thai large num ben. u�uall)' Impl} \mall nuduallom. n uclua tlom. In laser light can be \UPPf'('\\l"lI by Jl'ar'lHinR the nu mber of alom� i nvolved. Slarting Ih the late 1 980�, numerous experiments produced la...:r operation viaJu\t a few atoms. and in 2003. the logical limIt \\a\ reached Ih .... hich a smgle mdefa tiga ble cesiu m al()m ",-,� the whole weight of photon produclion (McKeever. Boca, BooLer. Buck & Kim ble. Nature. 1 8 Scptem�r :!O(n). A \JX'cial optical trap imposes the "sirong

�

wuphng:" reqUired when \pontaneou., misfires cannot be wicralcJ, r-;alur.tlly. outpUI is low and t his single�atom . laser, JepiCled in Figure 9.35, differs in several funda_

mental "3)\ rrom the com.'entional laser (an inversion in a pOpulillitlh 01' one?) bUI il\ imponance is the prech,ion wilh ....hll;h

il can produ(;e photom. In the field of quantum

u)plograph)'. the abi lity to produce 3 single pholon on

demand 1\ veT)' imponanl. The Mngle·atom laser i� one of \(\cral mules that rna)' provide the solution. High Power from Wiggling Electrons

An e ntire ly

PlUIn, o{ tl« IronS 10 Suptn:.·llnJlI�'li\II) had lleen ,ludieJ

diflerenl ....3)' . of producing coherent light. and one al the

(,'r iJ«I1Jc!'io, the tiN r\ltkm:c {ll !ennllln alom pairing \\01'"

(FEL). In this device, who�e components are �hown in

lhe JISI.'(lVCl) In

1�7: tNtlhtl Ptllc,

19%, Lee. O..hemtl &

Ridlanhunl ll, supcrtluldll) 10 lI'!uiJ helium,), ('-Ilmpri\ing

! c]a,:lmn,. � pmhm,. and I n('Ulnm, The �ut't\equenl

thtortlllill e\planatlon INobel Priu. 2(XH. uggcU) was

0PJxNle end ," Icons of power. is the Figure 9.36. Ihe

"

lasing medium" consiSls of a relativistic

beam of free eleclron� sent Ihrough a �eries of magnets . which cause the eleclron o;

Ihll rcJilll\cl� dl'i'W11 .Illm.. formed p
InI(roM;lulfI. \(1) mlk'h h� that tlt'1\\«n el«ltlln� in

!mpen:onduClmg mcuJhc elemcnh. bul l O\ol nng a more ,"'Omplc.,

rclall\ e mOIIt\ft. A qu.tlitatl\"("ly iliffcrtnl poUring

mcdwmm �mcd 10 be .. pJ4)' "hen. follO\\i ng the 1995 \\llershtd. c\ptnrntnh finl pnlduccd BEC", in 8�� of ('"mon atoms. Hen:. the Interaction between aloms wa..

rree�elect('on laser

10 "wiggle" along

Figur. 9.35 In a ,ingle-atom laser. a lone ce�iurn atom (black dol) trapped between mirrors MI an d M.. is pumped

h

by e,'emal \Ource H3• yielding very precise p oton OUiPUI

in tran,itlon g. Source n.. drives a roundabout relum 10 the s!antng le\'cl.

Mort range and sirong enough to pnldlK'r' a 81s of pauN

il),",

t"o--Itom moJ«uJt5. which ,,'OI1dcnsc a.� boron.� and we� lam �n 10 e \hl M the hlllnwk lraJl of 80!0C'-Einstein l'Ondrn\3I1on almtdy s«n In liquid hC'lium-3: supc:rt1uidity. A bridge btl,,«n the m«han.'ms was formed when il WM mol' n pos\ittlc. b) manipulaling w eJllC'ma! ml$:nc:tic fields. tll ""lune" 1M Interaction 10 an inlermediale stalC'.

The "ori. continues. purred by the btli�f that a btner undt�Wlding uf femuon pairin, mighl lead 10 rt\Olulionary o1ppilnllons. such u a room-lC'mperalure supel't'(lnducIt1r.

their paths.

l'� il'

il,

4'

4

3

Pro,reu and Apphcatiom

Figure 9.36 F!l:e-electron laser: Electrons '\\"iggled" b) a coherenl, high-power light.

sc:ne� Ill" magnel\ pn.ldu� Output

Electron accelerator

�' ,

Total r efle ctor

All accelerating (i.e., wiggling) charges emit electro­ magnetic radiation. and in the FEL, the inlerplay with the

numerical approximations. But then we run into another quirk of nonlinear differential equations: Extremely small

electromagnetic fields causes the beam of wiggling electrons to bunch up in a periodic way at the laser output

extremely large differences later. What actually happens

\\aI'elength and generate coherenl light along the axis of the

may thus appear completely unpredictable. (The

beam. The chief disadvantage of the FEL is that it is not a

atmosphere's behllvior is governed by nonlinear nuid

tabletop device but a massive facility relying on a high­ energy electron accelerator, explaining why FELs tend to be found at large universities and other research laboratories. Their great advantages are very high power, typically more

changes in the conditions at one point mlly re�ult in

dynamics, and it is often mused that the weather at one place may hinge on the earlier flap of a butterfly's wing on the other side of the world.) A common example is a nonlinear (non-Hooke's law)

than I kW and in some cases much higher, and easily

oscillator driven by a periodic force. For certain values of

wavelength, which may range from far infrared to

driving force. However, if the amplitude is changed even

ultraviolet and beyond. These capabilities make the FEL

slightly, although the (friving period has not changed. the

tunable wavelength. often by a factor of 3 about the design

the force's amplitude, the oscillator's period is that of the

allr:l.ctive for applications in biology, medicine. nuclear

system oscillates at a period larger by a factor of 2. Further

fusion, and materials science. among others.

change may reveal Ilnother abrupt doubling of the period. At some point, a small change may produce aperiodic:

Chaos

While we have concenlrated on thermodynamic

behavior! It may seem random: it may seem like noise; but

systems in which the behavior is predictable. the scienlific

it isn't, for it is governed by a mathematical relationship--­

community has only fairly recently recognized in many �ystems a special kind of unpredictability: chaos. It arises

however intractable. Such deterministic but seentingly

in nearly every physical and biological science. in many

unpredictable behavior is what we call chaos and is found in mechanical oscillators, electrical circuits, fluid now,

seemingly unrelated situations. The unifying thread is that

optics. chemistry, population sy�tems, biological de\'elop­

these situations all involve a phenomenon governed by a

ment. heart fibrillation, the wellther. and on and on. Much

differential equation that is nonlinear. The differential

work is now being done on controlling the onset of chao!>

equations we encounter in this text are linear, depending on a function and its derivatives to only the first power.

of sense from it in cases where it

Dependence on any other power makes a differential

a natural characteristic of the system under study. Chaos

equation nonlinear. Quite often there is no mathematical

is a huge and expanding field. The interested reader is

technique able to sol�'e it. so we must fall back on

encouraged to consult the literature and the Internet.

in cases where it is unwanted or on extracting some kind ,'>

either uncontrollable or

40'

n. ..... II a device tJw produces coht�nt liabt

r:::�:�

.. .. _ If ,.,.. . ..... .. ' tilL .... . ... . poIIII ... _ , , g ... " .. ... tI . " 'c ..-., A " ., ' .,.. 11 .. ... .. _ . .. _ _ ... _ of .. " _ ... $ ' n, ..,· ' . ... " ui.. . ..,. ... . " '\ ..... ..., .. .. ndd .. ... " '11 ., .... .. __ ...., .. , 4 _ _ IWD .. .. _ bE p ' •d

d e

,$ ,

"

(9:2)

(9:1) 10 . ......._ _ .. � .... IIUIIIber 01 .. , . ' '. • . .,. ...... ..... ...... ...... .,. ... .. .. � . ..._ ... ....- ooIy .. .... _ oI ..... n 71 e! t til � ., 1M ."nn diIb'itu­ _ _ ... _Bh ' , ... _ .... _Dn:.

X(£!.. -

1

1

(Ila - ...,." " '''' ( '"

•

(9:) 1 )

...,..'

1

...,."

_

1

+ 1

(9:321

(9:)))

' -':y ill all .. .. s t" .... .. rI .. .. 0 � ... 1M ,$..... ....... COGYGae '" Ibe "dIasicar' 7= indhti,..tisbehiil ty r .. . .".1- •• • ' ... .. ...... ..... . &IlDCb .., .. .... . .t 1000 " 1 AlII 1·. ...... _01"" _ _ _ af .... . . _ ... 11oc ... . ... Mu... .. « n I'_ di8Ir .. II low ...... 1IIdIor hiP , , _ _ _ .... oII p111i: ...... .... .. .. 7'11 • d . .. .... __..., ........ .. wIIi)c Ihr: _ a.. ... ..... _ occ. ' ..• .... oI l � ,.u. .. .. 1M _ _, • _ Il1o """"'"'"' • ..

.

_ .. ......., .. 0, C '

en

I

I

-.

.. . . ..... c6II iIMiIhc a _ GWa' .. if .. ..... ...... .,a I pMidc

I _ II _ _ .. V, ... _ _ _ ... • ....... .... .... -- - - - "" "' . ...... ...., of _ � _ afr ' _ .. _ ... . .... .... ....... . ... .,-. I _' " I ....... _ � .. ... I • ..... iz . ..... . ..... ... .. .. , rI _ I , ' "' .... - . _ "" .. ... .""'ip ' I ac 7 , 7 .. ..,. " • • •, . .. .. 01 .... . : .. .... ... af- , .... . ... .... .. 7 z

...

Vi.I

. m 7';'" A aoncquilibrium population i .. Ifj W in lite Ii....-producin' medium. in Whit.: ... ... 01 cIecaoas in . hiJher..enelJY §Iate is greater than IbIA . � s&IIe. Ir the: hi,her Slale IS meta......ble. e ....., 10 It by ulemal optical pumping or electric diKt\arat .. JR'CDted � quickly returning lind fNlolabhshing eqw, ...... A pauln, photon wtKKc energy equal." the dlff� r and lower state!> Ihen precipilales Stimul ..... � •• OM PI which the newly produced photon i" cohertn widI die fint. With . population in\"erSlon, the process quick! t � '-. two. two prodUCing four. and , � '"""'"' . prodUCIn, SO CIIt---Ud . powerful coherent hcam re,uh�.

.....

�

•

:

IndICates advanced questions

Conceptual Questions

I. Coo§ider I �ystem of two identical objocb headmg

SU"light IOWan! e4U.:h other. What would qualify and what would di�qualify Ihe �y.)lem iI!> a IhnmodYllamlc syslftn. and how. if al all. ,",ould thi\ relate to Ihe e1a.�, tieity of the collision·� 1 WhaI informalion would )"ou need In order to \ptclfy the macrostale of the air In a room'" What Informallon would you need to \pe<:ify the Imumlate·' 3. Gi\"en an arbitrary thermooynanHc �y\lern. which is laraer: the number of po'�lble macrmlolle... or the num. berof pouible miCl"O\late�. or i� Il lffipo\slble to �a)"'1 EApilin your answer. (For most �y\tem\. both arc ill/in'''' but il is �Iill po��ible to an\wcr the que�tLon ) f. A scientifically untrained but curiOU\ friend ask!>. "When I walk inlo a room. i\ there a chance thai al/ the air will be on the otheT �Ide r· ·. Ho,", do you ans,",er thl\ queStiOll'� So Derend 01 refule the following claim: An energy dislnz bulion. such as the BoltJ.mann dl\tnblilmn. spcofies Ibe microstate of a thermodynamif.: \)i'tem. .. Not IUtplisingly. m a col/tttion 01 O\I.:illaIOr... as In other thermodynamic sy\tems. rai�tnj! the tcmperature cawn panicles' energle, 10 Inc�a'>C. Why shouldn'tl point be reached where there are mme panicles in som(! biab eacray state than in a lower ener)!), ,tate'" (The fundamental idea, not a formula that ml�ht arise from it. ill be object.) 7. Wbea woukl a density of !>t.ale� be nc.:Jcd: In a "um ova lIlIeS? in a sum o\"er (!nergie\? in an Integral over COCIJies? in In integral o\"er stale\"l .. Suppose we hive a system 01 identu.:al particles IIIIIWiDa injusl ODe dimcn�lon and lor \\ hu.:h the energy quudizItion relationship is £ - bnYl \\. here b is a

�\'I/l'I..ntlind tl an Integer quantum number. DI'r.:u�\>

17. Cla" I�all). what woulJ he the a\er,.. energy of. par.

\\hether the den�lt)' or ..tale, ,h(lUld be Independent or

tide

f, llfl ln(rca!
whllo: mtatln, aOO\Jt th� r·a,-,,')

fun�tlOn 01 E_

l8.

t. 8� ,on�ldering It� COIl'tJluenh. delermille the dlmen­ '>Ion� {e,g,. length, dislance o�'cr I1me. et(.) of the \knonllnator III equallon (9-26). Why i.. Ihe re..ult

III 11 )) 'h�m

ot p;1f\ldn I� to fl\t)\le In the n-·plane

At hl1l-h Icm�rature. the a,ell1l1': energy

�ll

a da,".:al

one·dlll1en\ional o�dl1at\lr 1\ k�1. and lor an Itom in monJlllmK Ideal 18'. It 1\

;.l"r

Il

[:.,phun the difference,

u"n8 the cqulpanmon themtm

o,.tn\lble1

10. �re are more pennutallon\ or partlr.:ie labeh when \\\0 p:lJ\lcle� h:me energy 0 and twO have energy

I

than

\\hen three particles ha...e energy 0 and one has energ)'

Exercises Section

�. (The total energle� are Ihe �ame_1 From thl� observa­ til," alone argue thaI the Boltzmann di.!-tnbution should be lower than the Bo�e-Eill\tein al the lo\\.e�t energy

II.

level. The Fen11i energy in a quantum gtl\ depends inversely on the volume, Bn�ing your ..n\wer on Simple Chapter 5--IYpe quantum mechaniCS (not such quaint

nollon� as 'queezing clasSical panicles of finite volume into I container tOO smali). eltpltlln why. n, A. block hal> a cavity Inside, occupied by a photon gas. Bnelly explain what the char:lCten�tIC\ of thiS gas \hould have to do with Ihe temperature of the block.

13.

Con�lder a gas of atom" that might serve al> a la�r medium bUI that is

lit

eqUilibrium, With no population

m,>ersions. A phOlOn gas coe)(i<;ts With the atoms. Would a pholon whose energy IS precisely the difference between two atomic energy states be more likely to be ab!oOrbed or to induce a stimulated emission or neither" We expel.:! that III eqUilibrium the numben. of atoms at different levels and the number of photons of:1 gIVen energy should be

stabf�. h your answer

compatible?

1"'-

What is special about a metastable Stale, and why is it so useful III a laser? Why wouldn't a nonmetastable �tate at the same energy work1

15. In a certain de<;ign of helium-neon la.,er, the chamber contatning these gases ha� a perfect mirror at one end. as usual, but only a window at the other Beyond the window i... a region of free air space and then the second mirror. which is partially reflecting. allowing the beam to eXIt. The resonant cavity belween the mirrors thus has. a region free of the helium-neon gas-the "lasing material"-in which you can in�rt something. If you in�rt a ,heet of clear pla..tic at any orientation in this region between the milTor.. the la!.er beam disappears, If the same .!-heet i � placed I n the beam outside the par­ tiany reflecting mirror, the beam passes through it. regardless of the oncnlalion Why? 16. Suppose that in Figure 9 27, the level labeled £1 ' rather \han the one labeled £., were meta�table. Might the material �till

functio �!i. a la,er1 Explain_

19,

9.1

C"n\idcr the tWQ-\idcd l'Ol'm \al WhKh 1\ more hk.ely to ha"e an Imba1an..:e of fhe particle { i e.. Nl( 11

�N + 5):

N ., 20 or a mom With N - 6()'l (Nme: The . room ....Ith

total number or wa)� of d'�lnhutmg panl.:le�. the ,urn (If WZ_ from 0 to N, I� 2"_) �bl Whll.:h " more likely 10 have an imbalance of 5� (i,e"

NMo -

;N -+ O,05N)" (c) An

..,-erage·"lle room i\ qUite likely to have a trillion more air molecule, on one �Ide than on the other Why may \\.e \ay that preci..ely hall .....111 be on each \Ide')

20. A t.....o·�lded room contam\ \1)( pMtlde,.

11.

b. c. d. e.

and /' with tWO on the lelt and four on the right_

(il) De"l.:ribe the m;\cro�tnte Imcro�tnte..

(bl ldentlfy the pm.\lble

, (NOll': With only '1)( pJrtu.:les, thil> isn·t a

thennl)dynamic �)'stem, but the �eneral idea Stl\l

applie\o, and the number of combinatIOns is tractable.) 21, Con..ider a room divided by imaginary line.. imo three equal parts, Sketl.:h a Iwo-a,.i� pill! of the number of way, of arrn.ngin� particle, vel'ou, N.,I and N .... for the cao;;e � n.. �� Note that Nm,Jd1t: " not llldcpendent, being of N � I(}"

Nlch

Nn&ht

and - Nlcfr Your axe.!- 'hauld be COUM N Nn¥M' and the number of way' �hould be repre.,cnted by den\lty of Shading. (A form for number. of wa), applu,:­ able to a three-Sided room 1\ given

Appendix J. but

the queslton can be an,wered Without II.}

22. The Stirling approximation. JI

•

III

Virr J't tt2 e-J , I�

very handy when dealing with number.. larger than nbout

100, Con�ider the following ratio: the number 01

ways N particles can be e,'enly divided between two halvc\ of a room to the number 01' way<; they can be dIVIded with � on the right and 4� on the left_

(8) Show, using the Stlrhng appro�imation. that the ratio i.. approximately

(.In-l61lf>j5)\' for large N.

(b) E�plain how thi\ fih with the claim that a\'er;tge behaviors become more predictable in large sy"'\cm,

Section 9.2

23, The entropy of an ideal monnt(lmir.: ga, i, 13I2Wks In E +

NkS In V - N.ls In N. to within an additive \:omtant

Show that lhi\ implil!l> the com:r.:t reiation.ship bet\I.'ccn

internal energy £ and tcm�rature.

... ChIp.."

SiIbec"CII MIll_ ."

K. The dIOpOm show, .... .y"""" lhaI mlY ..cIwIg< _ tbmnoJ and .. e d lU llcaI lt,

....onductm' .. puII.Oft. Btau.. boIh £ and I'mol chlnp. we ,OnSJdtr 1M entropy of eacb sy«m to tit a (Ull(1lon of bach' S (E. VI. Considenn, the e..\change of lhmnal entfJ)' unl y. we lIIuN In SectIon 9.2 lhat II ..... J"h.5OOlble to define 111 1.1 d..SIa£. ln the mOlt �n· craJ �ar.e. PIT ,s abo dcfinN as �mcthlOg. tal Why huuJd pr6\Urf \:"URlC" IOltl ria),. and II.' \Iohal mlghl PIT ht equated.' IN()I�: Ch«k In 5tt whether the unil\ make �n.'it.) Ihl (Jlven thl\ rcl,UlllO'Jup, oJIow that dS " dQIT tRelTlC'mt'C1 the fl",1 h&w llf Ihennod)'nOtmll" .)

I_ I

28. In a large! ..ystem of di\tinguishable harmonic oscilla.

ttlf', how high doe.. me temperature have to be for !he probabiht)" or occupying the ground slate to be le\\ than 29. In a large ,,},tem of di�[inguishable harmonic oscillator how high doe� the temperature have 10 be for the pmba tile number of particles occupying the ground state [0 hi; Ie" thlfl j '] .30. ObtaIn equatIon (9- 15) fro m (9-14). Ma�e use or the followmg sums, correct when �tl < I :

�?

I

�

2:X' � I -x 11,,0 � x-:: :"-" 2:1Lt" � -'( I x) ' " 0

31. Show that "'Iualian (9- 16) foUows iTom (9-15) and (9- 1 01 ' temperature and M and .v given m (9-16) and that between E and n in (9-6), obtain equation (9-17) from (9- 12). The first sum gi\en In Exercise 30 will be useful. 33. Show that in the Iim.it of large numbers. the exaCI prob­ abdity of equation (9-9) becomes the Boltzmann proba­ bility of (9-17). Use the fact that KU(K - k)! .= }(A, which holds when k « K. 34. The euct probabilities of equation (9-9) rest on the claim that the number of ways of adding N distinct non­ negative :ntege� to give a total of M is (M + N - 1 ) [/ IM!(N - I )!J. One way to prove it involves the follow­ ing trid. It represents two ways thaI N distinct integers can add 10 M-9 and 5. respectively. in this special case. 32. Usi ng the relationship between

\(10 K. I� bndl)' put in conlBd · wllh a ·'hut' ubjC1.·I. I"l - .IUO K. amJ 60 J of heal flow, Inlm :he hU1 111'IJ�l ln :he \:"old (mc The objecb are then K'polI3ICd. Ihen temperature, ha\in8 l."han�ed negligibly duc 10 1MI' large 'IIC\ la) What are the \:"hange� in cntrup)" t'll ('a�'h ol'ljt'd and the ,y\(cm a\ a "hQle'� tt'll Knuwlng \ml) Ihal lhc,c (lbjel."1\ Mt' in l'ontact Jnd lit the 81\en temperature" \\ hat i\ the ral io or the proba­ hlhtlC�' (Ir thclr being Jilund In the ,ec{lnd (final) state hI thell �lf Ih�lr 1'I<-mg hlund in th� first (inilial) �Iale? Whott ..lIlt' uu, re\ult �uggc�l"1

1..4. '\ "'l:uld" IIt'lJed. 1".

x 2

Section 9.3 l6. We tl4:t.N the eUl."t pnlbabilili(� of equatIOn (9-9) on

thc ,,'Iaim thal lhe numNr uf way� of adding :V di�tinci ntlnnesath e inlegerv'qullnlum number;; to give a total + uf .w IS l'\l .'11 - 1 WIM'tN - 1 )�I. Verify this claim (al for the:" CJl'oC' ,\' :- 1. M = � and (hI for the case

.'v .5 .'tI =- �. 27. Four wSllllSui '>hable hannoni.... osciIJato� a. b. (". and d �

•

rna) el.(hanl!c energy. The energie\ allowed particle a are

EoJ = "./u,,1I' those allowed particle b are Eb = "t.'IWO' and �l ('n. Ctln...idC', an overall state (mll�tate) in which the totlll energ)' IS 3Awo. One possible microstate would have' partide. ,,(. "d) = (0. 0, O. 3). .. miCTO.\tales. (b) What i� the probability (8) UM tIll p u s \ihle thai 11; given particle will be in its n = 0 stale? (c) Answer pan (tI) fOI all other pos...ible values of n. (d) Piol the prut'B.bllif)· \'enU-\ ". !O

X

X

X

X

X

X X

X X

The X'� represent the total of the integers. M--each row has 5. The 1's represent "dividers" between the distinct integers. of which chere will of course be N - l--each row has 8. The first row says that "l is 3 (three X 's before the divider between it and "2)' "2 is 0 (no X's between its left divider with n l and its right divider with "3)' n) is I . 114 through n6 are 0, 117 is I , and "g and 1/9 are O. The second row says that "2 is 2. "6 is I , "9 is 2, and all other n are O. Further rows could account for all pos­ sible ways that the integers can add to M. Argue that. properly applied, {he binomial coefficient (discussed in Appendix J) can be invoked to give the correct total number of ways for any N and M. 35. Con..ider a simple thermodynarrtic system in which par­ ticles can occupy only two states: a lower state. whose energy we define as O. and an upper stale. energy E".

la) Carry oUl lhc �um (with only two �tates in tegration i, ccrt.1inly not '>'alid) giving Ihe average particle energy ,

t, and plol

your reo,uh Ib a junction of temperature.

(b) E).pl.1in qualiillively t why it �hould behave Q\ it doe�

(e) Thi' ,y,lem ean be u,ed os a model of paramagne­

\I�m, where

mdividuRI atoms' magnetIc momenl\ can

tIther be aligned or unlialigned with an eXlemal mag­ netir field. gi, mg a low or high energy. respecti\cly_ Describe how

the averilge alignment or antialignment

depends on lemperature. Does it mllke .,ense·)

36. EXllmple 9_2 ohtains a ratio or Ihe number of particles expected in the /I = 2 �tute to Ihat in the ground �tllte. Rather than the II

=

(al t' �mg eq\lllt'��n t�'�l, \'ah:\llat� thl: prohahililie or n, �ing n. 1 . 2. •md , __ (b) HIl.... Illany p'lrtl..:\e,. Nil' w�lulJ he e'J'lC"Cted in ea..h level '! Rllund N..:h \tllh� neare�1 mte�
2 Slate, consider arbitrary 11.

(a) Show that the mlio is

number of energy E ---:---;---''--0''' = 1I2e-13_6�V(I_" ll/kaT number of en ergy

E\

Note that hydrogen atom energies are E " - 1 3.6 eV/,,2. (b) What is the limit of this ratio as /I becomes vcry large? Can it exceed I ? If so. under what condition(s)? (e) In Example 9.2, we found that even .11 the tempera­ ture of the Su o s surface (-6000 K). the ratio for It = 2 is only 10-8. For wh at value of 11 would the r'Jtio be Om? (d) I� it re3listic that the number of 3toms with high It could be gre3ter th3n the number with low /I? =

'

31. Consider a system of one-dimensional spinless panicles in a box (see Section 5_5) somehow exchanging energy. Through steps similar to those giving equ3tion (9-27), show that

38, By c3rrying out the integration suggested just before

eqUillion (9-28), show th31 the 3verage energy of a one­ dimensional oscillator in the limit kaT » fi.wo is kaT. 39. Show that in the li mit /two « kaT, equation (9-15)

the number 01 wuy� illr YI,ur di\tribution,

(d) Culculute the outllher 01" ways il" thcre wel'e' (I pacti-

cle� III tl

I , and none hig.her_ NOle

saml.' IOtal energy.

(e) Find ,ll le:\\1 one other t1i�tributi\ln In whi..:h the

I I osciHlltors �hare the ,atlle energy. and calculate Iht' number 01" W.lYS. (I) What do your finding' �ugge �1 ! "

Section 9.4 41. Show that the nn' �peed of a ga, molecule_ defined a" \'rm. -

_. 3kV9 \ '., i� given by "\.17: BT'm.

42. (a) C3\cu131e the avcf'.Ige ,peed of a ga� molecule LO a classical ideal gas. (b) What is the ,erage velocity of ;l gas molecule') 43. (a) Using the Maxwell speed di:<;tri.bution. detennine the most probable \peed of a p3rtic\e of ma.ss m in a ga:<; at temperalure T (b) How doe" thi.. compare with \'rm.," Explain. 44. Determi ne the relative probability of a g3S molecule being within a small range of �peed!o. around 2\'nn, 10 being i n the same range of speeds around "rm' 45. Figure 9.8 cannot do justice to values at the very high­ speed end of the plot. Thi, eJ<.erci�e inve�tigates how sm311 it really geh. However. although integrating the Maxwell speed diwibution over the filii range of <,peed� from 0 to LIlfinily can be carried out (the 'u-called G3ussi3n integrah of Appendix K), over any re\tri..:ted range, it is one 01 those imcgmls that_ unfonunately_

becomes (9-28).

4(1, We claim that the famous exponentiaJ decrease of prob­

cannot be done in do�ed fonn. U�ing a computational aid of your choice, �how that the fraction 01 molecule... moving faster than 2\'rm� ii - 10-.1; faster than 6\'m>§ "

ability with energy is natural. the vllStly most probable and disordered state given the constraints on total energy and number 01 particles. It should be a Slate of m3X.i­ mum entropy! The proof involves m3them31ical tech­

-10-23; and faster than 10\"nn� i" - \0'1",4. where " !'IIt'

nique� beyond the scope of the text, but finding suppon i� good exercise and not difficult. Consider a system of I I osci1l3tors sharing a total energy of just 5�wo. In the symbols of Section 9.3. N = I I and M = 5.

0. 5 ill It

Ihal thi� abo hl\� the

from Exerci�e 41, ....

V 3kaT··m

IExercise �8 uses the�e

value� in an intere�ting application.)

46_

For a room

3.0 m tall. by roughly what percent dlle� tht:

probability of an air molecule bl::ing found at !he ceiling

tillS, bul even /j�ring them all won'l take forever_ if's fewer than lOll)

dttrer hum m. 01.,. cquaJ.apeed moIeC"uk bring fouod

at W rJoor1 � my wanMioo lit CtmperaNrr from /bJJ to ",bn,. 47. To obuin eM MUIWII .peed dI�bibulion. 1IoC' �umed •

(d) Suddenly. a fantastic change! All six volUmes at'l! \·olume X-It'S impossible 10 [ell them apan. For each of the three distributions described in parts (a), (b), and (c). how many differem (distinguish_

u,uJorm fCmpentuR'. • tinC'tlc-onJ,. energy of

l. ;IS

I M( � ... .: ,;I. and W(' I.uumcd dllIt lIo'e +

wll/!ed to lind thC' .� ot aD arb.D'aI)' function of \

AIoo, W WI,.. WC' .>blaJned. probabduy per unil Ipc't'd, '"'.J. 1.1 Anum.os I umform U'mperalure and

.ntM,.,,.ufE -

....um'n' \IoC' WI�

�m(\� I: '" 'D + mgy aod 10

(c) compare?

find rtlt ntragC' of an arbitraty

IUIk.·Uun of)', ob'.1D a probabili,y per unil height, P(.I'l,

,\ ItIC' dC'm,'}, 01' Ik almo\phC'f(" � N1 al an altitude of HOO m (abou' .1000 II) lhan 31 5ea level'I (c) Whar of rhC' ()� in 'hC' il(mo�phC'lT�

018. A I1llrtide \ubJ«l to;l p!aoel\ gravllatlOnaJ pull has II.

;;: �m\,J

In"'! nJ«han'\'IIJ e-�rgy 8,ven by Emoch,I"'C.r C;Mm,.'r. where m h 'he- panicle'\ mll5S. M the plane['s m.h\. tint.! G the grll\llll:llionul conslant 6,67 X 10· I [

N m!/l..B� " mil)' c.'>capC' ii' il.\ C'nergy i� Um-rhlll is. if

ih JlI.,\il'lC' KE 1\ cquul In magnitude 10 [he negative PE

hc)ld,"� 11 10 thC' �urfll(e. Suppose the particle is a gas

mc'le�:ule to iln atmo"phere, (a) TC'mpc:rarures n i Earth's

�Im{hphcn' lIlil}' R!'arh /000 K RefC'mng to the \-alues

�

lI t lnc:d in hen:ise.j.'i and gIVen that R

Earth

= 6. H .

X

W " In ilnd �'f:Mlll c 5.Q8)( J()2� kg. should � f:"nh ..hie' 1\1 ··hold on" 10 hydrogC'n (1 g/mol)? 10

Mmgcn (28 glmol)" (l"'III�: An uppc:r Iimil on [he

numl"ot-r (If nm/t'\:u!(', ,n Earth'� IItmosphere is about

lO.h I (bl The-ffltlOn\ mass i� 0.0/23 times Earth's. ils

rmhu, 0.16 tifTle\ Earth \, and its \urface temperatures

K. Shoold if

�

floor is proportional to a shelf's height, how do

the 10131 energies of distributions (a), (b), and

...

fbI A'5umlnB II rC'fflpc'llIfure of 300 K, how much less

ri� III .170

able) ways are there now? (el /frhe energy you expend 10 liff a volume from the

Ilble 10 hold on [0

(f) Use these ideas to argue thaI the relalive probabili_ ties of occupying the lowest energy states

silO/lid

be higher for hosons than for classically distin. guishable particles.

(g) Combine these ideas with a famous principle [0 argue thaI [he relative probabilities of Occupying (he lowest states should be lower for fermions than for classically distinguishable panicles

51. There is a simple argumem, practically by inspection. that distributions (9-3 1),

(9-32). and (9-33) should

agree whenever occupation number is much less than Provide the argument.

52. Equ3tion (9-27) gives the density of stales for a system of oscillators bul ignores spin. The result. simply one s[ale per energy change of fuuo between levels. is incor_

reel if particles are allowed different spin Siaies al each level. but modification 10 include spin i s easy. From Chapler 8, we know that a particle of spin s is allowed

2s + I spin orientations, so the number of states at each

level is simply mulliplied by this factor. Thus,

D(E)

"

(2' + l )/�wo.

(a) Using this density of Slates, the definition NfllurJ(b +

I)

SKtion 9,5 49. \eritJ.' Or.ll ihe rrobllbihtie� shown in Table 9.1 for four

N"

Ji\lingui,\h3hlt' oscillators �hnring energy 2liE agree

the floor, and si� volumes oran encyclopedia. A.

0, E, and

B. C,

F.

(a) l.i�t all the way� )'OU can arrange Ihe volumes with 6\·e Oil the floor and one on the Sixlh/fOP she/f.

I ABCDE, -, -, -, -, -, FJ. Li,t all Ore \�"l1ys ),ou can arrange them wilh four on One way might be

(bl

II\(' floor and IwO on the third she/f. fc) Sho\lo

Ihllf l�re are many more ways, relative to p&1S (3) and (b), to arrange the sUI: volumes with

IWO on [he floor and two each on the firs! and sec­ ond ..helves. (There are SeVeral ways to answer

-

5>, and

�

J N(E)D(£)d£ o

14'llh Ihe ("'1.1\("1 prob3bililies gil·en by equalion (9-9).

50, You ho\"e ..i, shelves, one above the olher and all above

I.

calculate the parameter B in the Boltzmann distri­ bution

(9·3 1) and show

that the distribution can

thus be rewritten as

.N(E)SoIU = (b) Argue that if kaT » 0, the occupalion number is much less than

I for all E.

53. Using densily of Slates D(E)

(2l·

+ I )/hwo' which

generalizes equation (9-27) to account =

for multiple

•• 1$$ .UOWN ,pin ,tat�' t�e� Exen.:I'>(: 52), the doeflnition

�l21 +

Section '.6

I ) - l\ and

N-

II1II

SL ShO'\\' tI\at, u"inl equation (9-)6), density rJI ....

J .'·(E)D{E)d"� �

(9-381 follows from (9-)7), Dcn,ilY of Mates (9·)9) don not dcpead. 01'1 N,_ total number of particles in the sy'tem; neUher docs the .... ,il), of ,late, in equation (9·21). Why bOt7 60_ For ;l panide in l one-dimensional (to) box. Ell is pro­ ponil1nal tll a ,inglc qUlntum number n. Let .. UmpIify thing.. by ignoring the proportionality facwr: Ell ,.1. For I :\D 00:\, f... " , " = , the 20 and + " +

59.

•

",he f\lr B In di,tribution\ (9-32) and (9-33\---<ardul u..e III :!: ",ill ,ut )'our ",ori.. b)' about half. Then plug Nt,:i.. In and ,hoI'; that for a ,)'..t�m 01 �Imple harmonic

"".1101101"1. th� dl\tributlon\ become

"!

. ,

"�

_

:

00\ i, I-alrly ob\·ioulo. (II l'he table �s . start on

.__,ElaE •

� .I,'

and

,-..,.I.T

S(E)m -

YlJU will need the lallowing integml:

I

' 1"[ 1 ' 1/8).

J��{ Br:t. 1 r

'

d:

SJ. E,erci�e 52 gives the Bohzmann distribUlion for the

'pecial cll�e of �imple harmOniC O�CillIl10rs, �xpre.o.sed

NhwrJ(2s + I ), Exercise Interm�of the eomtant ti. 53 g'�'e� the Bo\e-Ein<;lcm and Fermi-Dirac distribu-1I0n\ 10 that case_ Con�ider a temperatu� low enough that we might expect multiple parllcle'i 10 crowd into

!

_IO\\er energy Mates: til T "" f, How many osc:illalon would be e.\pecled in a �tale oflhe 1000est energy. E : m Consider all th�e"'"'""-Cla'i�ical1y distinguishable. boson, and fennion o�cillator\-and comment on the diffe�nee�.

55. Exercise 52 gl\'e, the Boltzmann distribullon for the

'pecial ca..e ol "lmple harmonic oscillators. exp�ssed In lenm of the con'lanl t, .

Nllwc/(2s +

I ) , and Exer­ CL"e 53 give!> the two quanlum di�tributions in that case.

a\:cuunllng for allowed lola-t«. Complcte the table, 5top-­ ping after thc 10th Male (ItalC, no!: encray) for all tbrcc ca�,. (hI Fmd the number of Itates per e.nc:rgy differ­ ence I-or the first fivc stalc, and \he lal1 five ltales for all three cases. For in\tance. for the fin.t live in the l D cll,e. i\ " 5 stales per energy diff�rcncc of 24, or .5/24. (c) Overlooking Ihe obviou..ly crude aspects of this accounting. docs thc " density of slatcs" !.eCm to mcrea-.e WIth energy. decrca!oe with energy. or !.uy about the ,ame')

20

ID Sta tt!

£

3D £

Statt!

.....

£

I

I

1.1.1

3

2

4

2.1.1

•

3

9

4

I.

,

"

Show Ihat both quantum di�tributions converge to the Boilzmann in the limit ksT » 60. 56. {a) From equation (9-34) lind the Fermi-Dirac distribu­

tion given In Exerci ..e 53, obtain an expression for E.,(n. the Fermi temperature for a collection of

fermion meillator" (b) Show that fro = 6>. (c) Plot Ef(T) vmu, koTff. from 0 to koTIf> '" 1.5. (d) By what

percent doe, the F�rml energy drop from ilS maximum

T'" 0 value when toT ri'C\ to 25% of fi.?

57. E'(en;:i�e 54 calculat�' the three oscillator distributions'

£ = 0 \'alue, m the \pecial ca,c where "sT is

l&. Using

a very common appro'(lmatlon techniquc. show that in

';;',4,� and I . for

the more general low·temperature limit, k T « 6, the occupation numhl:l' occome f.lksT.

tht di..llnguishable. bo�tln. and fermion cases. �spec­

ti\ely_ Comment on th�!'C re,ulls. (Not�: Although we a.\sume that "sT
\'hw/( If +

1), we aha sllll

assulm Ihat le\eh IlR cJo�ly <;paced-that is,

koT » IIWO' )

61. Calculate the Fenni encrgy for copper, which ha.s a den­ sity of 8,9 X

1O� kg/m' and one conduction electron

per 810m, h room Icmperature "cold"? 62. Copper ha!O one conduction clcl;tron per atom and a den"lty 01 8.9 "

IOJ kg/ml

By the criteria of equation

(9-43). ,how thal al room temperature (300 K). the con­ duction elCi.:tron gas mu�t be trcaled as a quantum gas of imli�1ingui�hab'e particle,_ 63. Obtam an order-of-magOitude \'a1ue for the tempcratUff at w hlth helium mIght begIn til e:\hlbit quantum! ,uperfluid behavlor_ Sec equahon (943). tHelium',

specific gra"'lty " about 0. 1 2.)

.... Dctermme the den'lty 01 Mate!> Ot£) Cor a 2D infinite ....ell

(ignOring ,pin) 10 which

E

""

.."

-

In'.

II&. Humpat 9 .. 1n\o'Cllilaied one cnceriOl'l fw quantum IndtllUIJUlthoIbthlY "" b mCI'eftR: 10 "�ril' nitro­ JeD Heft "'" IIIWS1I,aIC me other, (.) CakulalC' lht � � -.WtnI IIllfDJO'J moJ«ub n i lhC' '" Auurrle' a t.crnpcntun u' }OO K. a prn.�Uf'C' Q' I "Im, .aad AI, 1I'uI, ., If'" nllNJftl. 'hI Calrolak lhC' �"l"('­ kn6rh oI. '),J'U/ ftlrroacn molecule.- In the." . ,el H,,� cb )'ou, rnulb �1.uC' III E\amplt Y,.,,· 66. Tht r� ...., ... 't .. tklilX'd b) IF '" ¥obr:rc 1:, .. lilt fermi C'lX'lJ),. llJeo h:mu C'rJ('rgy for e(lndue-

;""1'

11011 C'lt!,;lhlm If) It-d,um

Fc-mu wl,,'II),

a'll \\fhA,

,�

J I C'\". (a) ('aku/alt' Iht'

\looM bt tht �.nrkng'h of an

C'l«uon "'ilh Ih,� �dil,;it)", 10:1 If each ,odium alom

�-onlnbult� nne l'nntJUl'!i,ln dC't:tnm to Ihe d�crron ga.�

10. The maumum wavelength light that will eject electro

dit

fmm melal I via the photoelectric etreci is 410 nm. f:n� II)ct;ll2, it i� ::!80 nm. What wou ld be Ihe potelll;al J'eren.:e ihhe:,e: IWO metals were put in contact? 11. Copper hJ� J den�ity 018.9 X J03 kglm\ and no photo­ electrom tlfe ejected from If If Ihe wavelength of Ihe incidenl light i� greater than 275 n m (in the ultraviolet r.rngeJ. How deep i� Ihe well in which i ts conduclioll electron�---Olle per atom-are bound? 12, Deri\'3t10n of equation (9-W): Our modd for calculat, ing E i\ �quation (9-26). whose denominalor is the 10lal number of panicles N and whose numerator is the total energy of th� system. which we here call U IOIaI' , StJtl wilh the denominator:

lind �I-d'um 'I'"m, all' \p.!Icc-d ruu8hl} 0..17 nm lipan. i.\

it n«tUllry, h)' tht \'rileria of' t'qualielO \9-.$]), 10 treitt 1/1" l'nntJUl'II,Ifl d(·..-rnm !ld\ a\ il ll'I/JI/tU!II gli�',1 67. Til .IIl'nin CtJllllliun (9-41), we l'akulilled a inIal numher ot krmillm N;" a funl'lInn ,1( E.'f. as�uminF r ", 0, �tltninJil \llih Ct{ualfnn 1\)·4/ t. Bul nOIt' that (\)--l)) IS the Jcnl)mlf)ltl,!r u( nur nwtJd f(l( (alculM,ne ,1\ erage pani­ d... ...n('f¥Y. C'\Iualmn (9·101 h� numenllor " the 10131 (Ib oppu..w hI a\ela,!!... panidej cnerg)'. I4hil'h l4e'll caU {llIuJ he'.... I In mhcr l4'onk the Iiltal �)\tt'OI c:nctg)' U,\ 1� II\COrasc: panlde �'ne�_\ E tlm(, the /(J/al number oJ' P.V1,�·Ic:, V ) (ukuI.de {'"U/ iI' ;J fun�:tioll ofEf ilIId u�e Ihl� ... �ht'\l Ihotl lltt.· min' mum IT .. 0) ene(8), oJ' a gll� .., �pJII � 'ennJtln� "iii) hI,' nnllen

t>7 ";Jkulalt'� Ih�' minimum total energy in a !(omliom ;tIlIj i\ applicable 10 conduc­ li un l'/cl'tnm\ In lI IlWI,./, The (II't'IlI,i:e panicle energy i� the Mal encrg)' Ji\'iJl"1.l "y the numner of panides N.

68. E\cl\:i,,�

-j

N=

o

In�ert the quantum gas densilY of Sillies and an expres_ sion for the distribution. usi ng j: to distinguish the

Bose-Ein�lein from the Fenni-Dirnc. Then change vari­ J IIble�: £ ::: yl, and factor B,,+\ /(sT OUI of the denomina, lor, In the integrand will be 3 faclor

C)- I

== I :!:: c, a .sum of two inlcgrals results each of Gaussian form. The integral thus becomes two terms in powers of liB. Repeat the process, but i nstead lind nn expression for UlOldl III tenn� of liB, usi ng

U.�ing (I

+=

�Y" em ,II' \PIII

.�hl)n' th.." hc: ..\cruge panidc energy E ofII conduclion

cl«" nm al 1,1\4' tl'l1lpt'rJIUre IT iit 0) is (J/S)£r' This (vIm I' ,'llm'...niC:nI. br:-in,l! rulher �impJe, and il Clln eas­ ily t'IC' put ," Icrms Ilf N. I', and m l'ta equation (9-42). 69. Thi\ pmfJlem imc�ti�ltle\ what frJclion of the anlilable chtUJc mu�1 he Iran,terreJ from one wnductor to unulbr:-r ItI pn,Ju,� a IYPlcal c(lntao.:t potential. (a) AS:l rough "rpn):{imafion, freal lhe I.:OOOUl.:tors as 10 em x 10,'m .;qUiin:' plale, 2 I.'m 3pat1--3 parallel-plale capae­ ;1II(-!j(1 thai q 0= n'. where C ::: £0 (0.01 m�IO.02 ml. How mUl'h dwgt' musl be translerred from one plate

hi lht other 10 pmdu,'t' II polential dift'erentt or 2 v·, (bl Approx;mllitly ",'hal fTUl.'lion would Ibis be or the lotal number of �'ondul'lion electrons in a 100 g piece of copper. whi"h hi&!> one conduclion e1cclron per alOm?

/ Jf(£)O(£)d£ <�

U"", =

/ 00

o

£Jf(£)O(£)d£

Divide your expression for UIO by thai for N. bOlh tn 'd] temlS of liB. Now JIB can safely be e li minaled by

using the lowest-order expression for N in terms of liB.

Section 9.7 73. Heat capacity (at constant volume) i, defined as aUIdT.

(a) Using a result derived in Example 9.6, obtai n an e"pression for the heal capacity per unit volume, in JIK · 1113, of a phOlOn gas. (b) Whal is it� value at 300 K? 74. (a) Show Ihat the number of pholon� per u ni t volume in a photon gas of temperalUre T i l> approximately (2 X )07 K-Jm-3)T3. (Note: /000 x2(e-" - I ) I dx == 2.40.)

Ib)('omblne Ihi!> wilh a re\ult derived In Example 9.6 1<' ,hll\\" that the average photon energy In a C8\i ty

temperature Ti, gi"en by £

•

�Ith�r "de \If t� ccnlr"l value. 1M relOOnant cavity is C"!.;Jdly Ml cm Illng. til HI'W many wavelengths fil the-­ 'tamhng-wa\C' l"IIndltltln', (til II" only a singlc wave· I�ngth were dC'In=t.i. \I,\)uhl �'hanglng the length of the ca\Ity help? Expl.ltn

at

2.7 kaT

1�. l'�lng the re\ult of part (8) in bxerci!>e N. detennine tilt" number of photons per unit

"olumc in outer space. temperuture-the so-called cosmic background \IIho�

trrnperature-i� 2.7 K 16- "The temperature of our Sun"s surface i� -6000 K. (I) At what wa\'e\cngth i� the spectral emls<..ion of the Sun a ma:(imum" (Refer to Exercise 79 .) (b) Is there

Section 9.9

\omething conspIcuous about this wa"elength?

11. At \\"hat wa"elength does the human body emit lhe

ml.'(imum electromagnctic radiation? Use Wien's law from Exercise 79 and assume a skin temperature of

70 F

18. The electromagnetic imensity thennaHy radiated by a body of temperature T is given by I '"'

uT4

where

u

- 5.67 x 1 0 8 W/m2 . K4

This is known as the Stefan-Boltzmann law. Show that this law follows from equallon (9-46). (Note: Intensity. or power per unit area. is the product of the energy per unit volume and distance per umt time. But because intensity is a flow in a given direction away from the blackbody. the correct speed IS not c. For radiation moving unifonnly in all directions. the avera e

�

component of velocity in a givcn direction is <4c.) 19. According to Wien's law. the wave length "max at \IIhich the thermal

emission of electromagnetic energy

from a body of temperature T is maximum obeys

Show that this law follows from equation (9-47). To do

this. use!=- d}.. to reexpress (9-47) in terms of " rather than/. then oblain an expression that. when solved. would yield the wavelength at which this function is maximum. The transcendental equation cannot be solved

exactly. so it is enough to �how that }.. = (2.898 X

10-3

m K)/Tsol\"es it to a reasonable degree of precision.

81. Someho\\; you ha..c ;t two-dimen'ianal ..al id . a !theet of atom, in a 'quare latt ice . �ach atom li nkcd to its four clo�est neighbor-. by four ,pring\ tmented along the twO perpendicular axe,. (al What would you expect the molar heflt capacity 10 be at "ery 10\\ temperature and at "cry tugh temperature" (hi What quantity would dClermlne. roughly, the line between low and high"

82_ The Ochye temperature of copper i, 345 K (a) Esti­ mate 115 molar heat (aparity at 100 K. u\ing the plot In Figure 9.33(b). (b) Determine Its corresponding 'pe­ cifle heat lind comp� it With the eltperimental \"alue of 0.254 Jig K 83. From equation (9-51). !ohow that the !>pecific heat (per mole) of a ery!>talline ,olid "aries a!o t� for T « To' 84_ The electrons' contribution to the heat capacity of a metal is small and goes to 0 Ii!. T --+ O. We might

try to calculate it via the total internal energy, U = fE N(E)D(F:)dE. but It i.. one of those integrals impos�ible to do in closed fonn. and SlE)ro is the cul­

prit. Still. we can explam why the heat capacity !>hould go to 7ero and obtain a rough ..nlue. (a) Starting with ..N'(E)FD eltpres\ed as in equation (9-34). !:Iho..... that the slope d.N"(E)ro'd£ at £ = £F is - 1 I(4kaT). (b) Based

on part (a). the accompanying figure is a good approxi­

normal ga!>, all molecule!>. on

""O when T is small In a mation to N(E) ..

such a.. air. when T is raised a lillie.

average. gain a little energy. proportional to kaT. Thus. the mternal energy U increases linearly with T. and the heat capacity. aUlaT_ is roughly constant. Argue on the basis of the figure that in this fermion gas. :b the tem­ perature increase.. from 0 to a small value T. while some particles gain energy of roughly kaT. not all do. and the numtxr doing so is ale;o roughly prop!..lttionaJ

Section 9.8

80. The fact that a la!;.er\ resonant cavity so effectively

to T What effect does this ha\'e on the heat capa�it)·"

(e) Viewing the total energy increao::.e as simply

.lU

::;:::

sharpens the wavelength can lead to the output of

(number of particle, .....ho'e energy increases) X

�veral closely spaced la�r wavelengths. called

(energy change per particle) and a'>Summg the density

longitudinal modes Here we see how. Suppose the

of states ie; ,imply

'ipontaneous emission �rving as the seed for stimulated emission b of wavelength 633 nm. but

particle energies. e;how that the heat capacity unJer

somewhat

tional to (ksRIEF)T (TI)'ing to be more preci,e i not

fUllY, WIth a line width of roughly 0.001 nm

:1

con..tant D o,'er the entire range of

the� lowe,t-temperature conditions should be: propor­

410 � ' _ IUIly _while, rar Ill< �') .,.,..,.", 'I IUbjcd 10 .everal corrediocu from etfC(li we

,port.)

di,tributeJ unifonnly throughout a sphere of radiu\

7 x 10 ' I � m?

(bl How Joe' this result fit with Exercise 87'>

89. When a !>tar ha� nearly burned u p its internal fuel, it

may become a "hile dwarf. It is crushed under Its O\Iin

1

..

enonnou� gravitillional forces to the point at which the

e,clu�ion principle for the electrons becomes a factor. A smaller �ize would decrease the gravitational poten_

0

lial energy. bul as'mming the electrons to be packed into the lowest energy <;tates consistent with the eXclusion principle. '\queezing" the potential well necessarily

Increases the energies of all the electrons (by shortening their wavelengths). If gravitntion and the electron

exclusion principle are the only factors, there is a mini. mum total energy and corresponding equilibrium rJ.diuS.

85. In b.ef(l\C l�. It 'Impl� Iw\)-\Ialc �)'Iem is \Iuditd Auurne Ih,u the pan,dc, ilrt di,lini!u\,hable. Deter­ mme the molar ,pc�lfl": heal C\I of Ihi' material and plot II \'("U\ r hplam qUillililtiV"cl) wh) it �hould �hll\'e a II dl)('\

• 86. Pnwt thai ror any ,mC' junction ,.n(h I/J) orlAn\e­ length ,hunC'f lhllll 1tI, '" hen: a '\ the al(lmu: �paclOg. there

.\

+

a 'Inc lunction .... .Ih II ....a\elenglh longer than

� lha' ha\ the ..arne \alue' al lhe poiOb 1 = a, 2n, 30, and \" on. I,\o'ott: II " pf\mably ea,ier 10 work with

wave number than 'AIm 'AoI\denglh. We ..eel 10 �how thai for ('\ltl) 'o\oaH� number <�rro'rr than nl" there is an �ul· ..alC'nl (tne le\, than 11'/11.)

(a) Treat the electrons in a white dwarf as a quantum gas, The minimum energy allowed by {he exclu_

3 (3,,'f,')liJ

sion principle (see Exercise 67) is

Ucleclmn\ = 10

--

m;'2V

NS/)

Note that as the volume V is decreased, the energy does increase. For a neulral star, the number of electrons, N. equals the number of protons. Assum. ing that proton� account for half of the white

M

(neulrons accounting for the other dwarf's mas'; halO, show thaI the minimum electron energy may

be wrillen

Comprehensive Exercises 87. Nudear lJen\lty 1\ approl.imately 101 7 kglm3, (a) Treat­ Lng them 01\ a g� of fcrmicms bound together by the tnonclecll'mldbC) "'trong inlemucleon auraction," calculate E�, for !he neutron, in lead-206 (82 protons

and

I�" neutron.. ). tb) Treating !hem the woe way,

�hal \Iowld Ef be fl)f !he pmton�'l Ie) In fael. the ener­ gie!o of !he Il\O
the Fenru energ)'. are �..entialll equal in lead-106. What tw been lefi oul of parb ta} or (b) that mighl account for

lhi�"l 88. From elementary eleclrmtatic!>, me lotal electrostatiC potential energy In 8 !>phere of uniform charge Q and

where R is the star's radius.

M

(b) The gravitational potential energy of a sphere of mass and radius R is given by

3 GM�

----

5

R

Taking both factors into account show that the minimum total energy occurs when

radius R 110 g i\'en by

3

I

Q'

U = ---5 "1I'eo R

(c) Evaluate this radius for a star who�e mass is equal

tal What would be the energy per charge in a lead nudeu!> if it could be uenled as 82 protons

(d) White dwarfs are comparable to the size of Earth. Does the value in part (c) agree?

to that of our Sun, -2 x 1030 kg.

MeftUeS

I

E,.:n:l�e 89 discusses me energy bal:mce in a 'Ahite tendency to contract due to graVItational 90, u"urf The balanced by a kind of incomprt"s�ibility of I;; on attrod1 due to the exclu'iion principle. \ e1ectrOll t�O;: lal M3Itcr cont(lins protons and neutrons, which are also fennions. Why do the electrons become a hin­ drance to compression before the protons and neu­ trons do? (b) Stars several times our Sun'!> mllSS have sufficient gra\itutional potential energy to collapse further than a white dwarf; they are able to force essen­ tially 1111 thcir mailer to become neutrons (fonned when electrons and prOtOM combine). When they cool off, (ln energy balance is reached similar to thai in the white dwarf bUI with the neutrons filling the role of the incompressible fermions. The result is a ncutron star. Repeal the process of Exercise 89, bul assume a body consisting solely of neu­ tron�. Show mat the equilibrium radius is given by _

R -

( )

3h2 31T2 '" 2G 2m�M

-

--

(e) Show thnl the radius of a neUlron star whose mass

is twice that of our Sun is only about

1 0 km.

Com putational Exercises 91. Referring to Exercise 40, write a compuler program to find (III possible distributions (i.e.. sets of No, NI • N2• Nl, etc.) in which I I oscillators share an energy of SlIwo' Calcukltc the number of ways in which each dis­ tribution c:m be obtained, (lnd comment on your results. 92. Defining temperature through a derivative of a log of a number of ways-equations (9-2) and (9-3}-may be hard to embrace. but it's built inlO the Boltzmann dislri­ bution and all others. We can at least verify it in the oscillator case. Equation (9-16) comes from equating Ihe average energy Mllw,jN of a collection of oscilla­ tors to the average predicted by the Boltzmann distribu­ tion. According to equation (9-4), the temperature it gives should be (OSlaE) I . Is II? Rell�onable precision demands a fairly large syslem. Assume the number of pl1l1icles N is 10.000 and the total energy E is 49.000 hwO' It's ea�ie�t to choose energy units in which !two . I . so that. according to equation (9-7). E and M ure both 49.000. (a) Calculate the number of particles, N", at each level by multiplying N by the exact proba­ bilitie� of equation (9-q).-u�ing n in�tead of n,. since

411

our interest is th� lotal nUn'lber al l gIVen level. nor: a �pecitic particl�. Note mal wilh liwo • I . n i� the level'" energy Round each N,. to the nearnl integer. then determine the total energy simply hy multiplying Nn by II-the number limes the ent'rgy at t'8ch level­ and summing over all level�. It mighl �eem that the total should be 49.000. Due to rounding, If won't be. but it should differ by le"s man I q.. Entropy is fairly easy to calculate here. The' number of way� of reanang­ ing particle labels when N panicle� are distcibuted in different boxesl1evels is N!tn,,(N")1 (See Appendix J for the proof.) Use this to caleulflte me entropy via equation (9-2). (b) Repeat part (a), changing only M. making it 5 1 ,000 instead of 49,000. (c) Calculate the quotient: change in entropy over change m energy. Bcaring in mind that we consider a system of N "" I 0.000 and M lli! 50,000. how does it compare with equation (9-16)1 93. Write a computer program to delermme the probable numbers of particles occupying o�illator levels for dis­ tinguishable. boson, and fermion particles. as plotted in Figure 9.9. but with the following changes: Assume six particles. and for the tOlal energy, assume first mat it is ME (the lowest possible for fermions) and men 1 2oE. Comment on your results. Here is some help: Call the particles' energies e( I). e(2). e(3). and so on. If the total energy were. say, 6, let e( I) range from 0 to 6. e(2) from e( I) to 6, e(3) from e(2) to 6. and so forth. By making each particle's energy greater than or equal to the previ­ ous. we exclude distributions that differ only by a per­ mutation of particle labels.. This is perfect for bosons and fermions, which dOli have labels, and it saves a /01 of computational time. Run through all the possibilities. If the sum of all particles' energies isn'l 6. go to me next. If it is, count the number of particles in each [evel and save this se, wim II "serial number" for later retrieval. But if the number at any level is greater man 2, designate the set as unacceptable for fennions. After­ ward, for bosons and fermions. add me numbers at each level for all acceptable sets. men divide by the number of sets. thus giving average numbers at each level. The "trick" for the distinguishable case is to multiply me numbers at each level for each acceptable set by the number of permutations of particle labels. 6!tnll(N,,)!. where n runs from 0 to 6 (and later 12), and afterward divide not by me number of sets but by the sum of all these numbers of pennutations. 94. Exercise 53 gives the Bose-Einstein and Fermi-Dirac dislributions for the special case of simple harmonic oscillators. expressed in tenns 01 a constant c.. (a) With equation (9-26) as a guide. write a formal ellpression­ not aClually anempting to carry out inlegrah.--fol the "

B o n d i n g: M o l ec u l e s a n d S o l i ds Chapter Outline \0.1

When Atoms Come Together

10.2

10.3

Molecules ROlation and Vibration

10.4

Crystalline Solids

10.6

Conductors, Insulators, and Semiconductors

\0.7 10.8 10.9

Semiconductor Devices

10.5

Energy Bands and Electrical Conduction Semiconductor Theory Superconductivity

10.1 0 Fullerene�

T

�

he lopics we study in l is ch�pler are lhe foundation of a large . pan of sCience lodaY-1n partIcular, novel materials and modern Our focus is how atoms bond together to fonn molecules and electronics. what characteristics and behaviors arise when they do. As in the o , lids and ectron atoms, it is impossible to analyze multiatom bonding multiel of

�

a se �olely from first principles-to solve the Schrodinger equation. On the other �and, physics would not have advanced far had physicists t::lckled only those could be solved w ith pencil an� pape . T�us, it is of great problems that � . . we learn to combllle first pnnclples thai wIth SImple models and rtance impo qualitative argument», if found to agree with the experimental evidence.

10.1 When Atoms Come Together

Underlying much of thi� chapter is the idea that when isolated atoms approach

one another, electrons are influenced by more than "their own" atom. Our con·

cern is no longer isolated atom wave functions, but those of a more complex multiatom potential energy. Even so, the resulting muhiatom states are often recognizable as closely related to, or combinations of, isolated atom states. Fortunately. we can grasp the main ideas in a simple, one-dimensional system. Consider the pos!>ible Quantum states of a single electron in the pres­ ence of two "atoms," represented by one-dimensional finite wells of width L

and separation a. We choose the finite well as a model because it is capable of holding an electron bound, as is an atom, and it allows wave functions to wan­

der through the classically forbidden region, as the unrealistic intinite well would not. We must allow for mingling of wave functions among atoms.

413

stales. which molecular . pairs of ve states. ha . lS" d atOm" P " atOIl _ of "olate ' 10c O' wO " T ' differe .. ,0. � rns and p." e: 10 su rg nve lion co

at I"ge 5Op",.

(e) (b)

L

(ol

energies for the fOUf 10West nctions and fu e . ' II. They somewhat resem· OWS wav tion IS sma 0.1(a) sh ' ose separa wh / Figure 1 s ' g e fi'ntte well. (The .op light a sm s/atom 0 f en Ons w . func.t. states in .hree, the ligh. red below" gy wave energy below " has . est �ner red low ur dark distorted antmode. ble the fo odes, the ) only one badly ur antln one has f o has red a larger separalton, A. dark gy. red one ener m ow O'to . a' h'tgh and . and the b m palfS has .wo, for are essen.ially equal, and 'hei, palf eS . of .he lower these s.at , r eve How rg,eS nt in the right atom and 0ppo. y coincide. ), the ene uall I(b virt 10. ilar. also become closer. At Figure very sim . er pUlr have tions are of the upp func eS gt e r ene wav converges 10 a single energy. atom. The each pair left (c), the . . l 1O site in ' equal energy. The low·energy paIr on. Figure . approach rati epa oer s . pairs ,.ill iaro algebraically added and sub· ce that eS tha" ' f cOinetden stat no .wo . is 11 to It ds, the two In o.her wo separauon ted '·alOms." at large 0'0 isola .he of converges 1 atomic teS s of .he two " � I s.a combination r ' are the ea lin .ed trac combinattons of the different states that are stateS are nverges to co r molecular pai wells, there would be three upper there .hree larly. the es. Were states. Simi . stat erge to three m ato ration would conv 2 lSola.ed. large sepa a. ich tWO " � · ncome together, theU stateS, wh n N atoms olecular general. whe n � I m In related m es. N olecula of sta. r I band, � or mn a se', isolated.ato ne 'o form and b' b t , and molecular com so te . her N,slll m sta.es form anot ISola.ed.a.o iC stateS m ato 2 energy of the isolated-atom state n � around .he . ster stateS; their clu rgies Figure 1 0.2 Illustrates band's ene n deereases. n ratio give sepa A on. atomic wells!atoms to produce apart as .he states of N and spread of atomic ning e atoms nbi the cOt and do belong to separat ' . electrOns can schematically e WhIl ntrate in thIS chapter cular bands. e 26), we conce N.state mole (see E"rcis apart far e. atoms are by the whol when those are shared that s state iatom on trUe mul.

I

.

:

'

I

1 0. 2

Mol ecul es

energy than the se(>'state is of lower . the molecular hen w s molecule eleetrons IS crucial. As they Atoms form atoms' valence the of The behavior lower e nergy may resuIt. But araled atoms. . atom orbits, a + ated· separ molec le, two ve to their u possible case: the H rearrange, relao st imple the s g iderin � conS e rgy may be en 1 by t01ll begut e Th . ? e re 1 0.3 how . W depicted in Figu tron, elec I � P us the neg· 0 .y : eUC ener "ts I.�n protons and one on's energyelectr the ( I ) ( " ) he positive twO parts: ' 'ded mto ' dIVI wt·th the protons-and " . potential energy il shares . ... t , onS1°der f!.Pil the electron s ns attve, attractive proto two . ' energy shared by the tial repulSive poten "

C

10.2 Molecuta.

FI9"r. 10.2 At small separations, each atomic state becomes N-�tate band.

FlguAi 10.3 A simple molecule.":t+­ an electron shared by two protons a dis­ tance a apan.

an

At large atomic separation. elec::trons occupy 11 '" I and

�;:��r

11

'" 2

molecular

Slates

�;:�'�r

��... '-}<. t{

�:e�ular

State.<;

11

415

'" 2 states at the ,arne

ene�les In eacll atom

t..•

. . • .

.

1 11 = 2 atomic Slates

\,\ {� =:.::;;?: . . \ .

.. '

....

.

1\

'"

I atomic states

Atomic separation (oi····l

,

are

close. tile molecular states spread into a set. or band.

v

. but wilen atoms

a

energy. If the proton separation were very large. the lowest possible energy ha e the electron simply bound to one proton or the other, with an . ....ould

eV. Putting it somewhere between the distant protons would energy of - 1 3.6 ield a free electron of zero energy. At the other extreme, were the protons

y

(used, we would essentially have helium, but with only one electron, which is also easily analyzed. As shown in Example 7.8, the electron'S energy would be -�.4 eV. Thus, we expect the electron's lowest-possible energy to decrease

with decreasing proton separation. However. the positive electrostatic potential

energy shared by the protons increases as their separation decreases. At large separations. it is zero. and at negligible separation, it is arbitrarily large and positive. Altogether. the

-

foral energy

I

is - 1 3.6 eV at large proton separation

and arbitrarily large at negligible separation. Now the important question: Is

there a minimum, less than

13.6 eV? I f the protons' repulsive energy were to

increase faster than the electron's energy decreases, there would be no advan­

tage to fonning a molecular bond.

Since it is an experimental fact that

form a stable molecule. the answer must be yes. Of course, we might

equation.

be

H2+ does

interested in proving this via the SchrOdinger

As always, we would

begin

with the potential energy. Given the

positions shown in Figure 10.3, the electron's potential energy is I U(r) = 41Teo

( Ir -- �axl Ir - � axl) "

+

+

i'

-54.4

Unfortunately, even for thi ... simplest of cases, the ScbrOdinger equation is solv­

able only by numerical tl!chniques. When such a solution is carried out. we find that the electron's energy does vary from

- 13.6 eV for large a to

eV for

_ rmfllIDum f\';lIt:

tHeb\llTl doe<. IIQI ha.\e infinite erw:tgy. ",� _ will see III CbaplQ II. protOrl.'- JauaJl� aar/IIt1 va *

rnolec::ulc altai!\! a

urort& fOla: ",ben If! *

tIll� PCIll' i.i

hcd.

D\ld� bIA Ibf:, H."

� JOg bdCln':

• ,... pul\- jvC energy is added. there is a min' ImU prolon-' _ rn V. and II occur, al a proton separar e .0 3 1 6 (l . I ' lO ii . ,'alue! \ ., ll · of cnttgy expenmenl. 10 [he flllotl OltnI "11h a�re e in I nm. !'epara1ion of 0. 1 1 nm is a s O. J onl v a f ncrion? A .. "a\l� U .1. ru .... O J boIII "hal atom, so [ he eieel.ran I' n H + sho A.nd en drog h me 2 uld ' of Bohr roIdlU'. nd ( hat the groun d i th '" KC' the lOns. Indeed. we fi s ale is be an e�td b) Do .p �tnlO�J) jnnu between the prolons. boul lhe midpoint a C h rn e and tn . \",rn . Ih I " llfx. -J e region ' ',en fUfli."Uon den�lly IS In between ,1to •roe ... . " >em Probabl'I IIY/ch" Ib r 0 . Ihc:' rna)!)nl) with the inlervenmg eIectron cloud is the protons ,hare both n roo l 'lIo : r: n Thl>' JH energy. bond''\ lower of Ihc!' molecular H,- mole ''ould , of course, yield the neutral ron , . d elee r _ a ..ef,;on cUJe . , Adding drasllcally complicate . s thi� . rhe a n POl ectro I enll"aJ tiel but Ju..t as fOf I�U ground state has both electro ... however. the gues ns . . . . hl in o ' m Ihe encl1!J. A<;, we , eo ,aI slale. This ' eoeroyC mlnlml z mg Sh aflng of a . �pall , paIf of "line lo,,"cst energ) . known as a covalent bond. Its Strength oppo..lIe spin is ot n5 (o elcc[ ,"a/ence ',' ' stare being centrally located . ron pal s ' palial and loU dcri \e� Jrom the elect s speaking, both atoms in H, . . Loosely , . -ons I e l l \ lay cl ' . ' ' d 10 both pO S l _ illuade aun behavior somewhat like helIum-more stab ' so we see le and 10 IWO eIeCtron "" . d atoms would be. Taking the separare than ' l\e react , idea fur le�" chem ically also attain grealer slability by sharing ' s should l en elem J e . a paIr lenc . " ,. the:T, \a 0 9) molecule. Fluofl ne (2 diatomiC bonded II) en I cova ' �1 W or eIel (()n5 In a I! . becomes somewhat neonllke (2 =: . ., �hell ' .. 111 the n ::= J 0) by eleclIOn , "e\en ' . F�, WI'th each atom donating an electron [0 an equally , iC dl310m fom11l1g shar ed � . and 1., ,orm m a correspon .' 8r,. ' CI, cule s mole ding w ir. The cO"atenr . ay. pa, atoms must share Ions In the n == 2 shell, oxygen . eIecr two ele With ani), "1'\ . . . c" molecule 0., . SimIlarly, three pairs are sha tnlO palf'i [0 rarm Ihe stable ' •. red In ' unusually Inert for a non-nobl e gas N�, Thl\ In'pIe shanng makes nitrogen ·

Ibe and that \'-hen

_

�

to"IS

::::

_

•

•

'

•

Bonding and Antibonding States U�eful lhough it is, the portrayal of c.ovalent bonding as at0m.s acquiring nOble ga, eharacler by sharing specifiC pmrs of valence electrons IS overS imp lified First, Ihe quanlUm states of a molecule are !nh�relltly different from those of jsoJaled noble gas atoms. Second. "non bond 109 electrons may also be shared Consider Figure 10.4. which depicts states of an electron bound to tw . prolOns. Much like [he finite well "atoms" of Fig�re 1 0. 1 , if the proton s are f apart, then there are two atomic Is stares, and addll1g or SUbtracti ng them giv es IWO equal-energy molecular stales thai are even or odd function s of posi tion. Al lhe �mall 'ieparalion in a real molecule, however, the electro n's wave fun tion j" affected by both protons al once. While qualitatively similar to t ,ums and differences of atomic states, the molecular states are the truly vali ones, and they no longer have equal energy. The even one is given the sYm b I ITls and is known as a sigma bonding orbital. II is of low energy beca u e both p�tons are close to the bulk of the ele tron clou� . 10 the center. � The odd one, given the symbol and called a sigma anhbondin , g orbital, IS . ' Or . higher energy-higher even than the a[omlc state-because the wave funetla ' node keeps the electron away from the cent n ral region. Despite the node, v.a\'e function is also "shared" by both [hiS protons.

;

� ; :

_

u*ls

10.2 Molecules 417 FIgure 10.4 Reducing lllomic separation yields high- and low­ energy molecular Is states. <,...,

�

.J. 1.....ft

Mole<:ular

• �

•

Figure 10.5 formation uf I.t bonwng

and antibonding slales. Plus and minus �fer to 'ign� of the wave function at an 1n�lanl ;n lime, nOI to charge.

•

•

• Isolated atom

1<

At lMge 11rtltlln wpllration. atomic lind nlllle.:ular ,\lUe' nrc equi\ alent pai",. e,l�h being ,um� !lnd I,I\\lerencc' or the other.

•

•

•

•

Is atomic

+ "

At ,mall ,epamt!on, the odd

0 +) I,

+ . + ,+

h

llOtibondillg qll.te i� higher ...n...�y

1<

Atom;c stutes

. . . and the even bondIng

• +1

•

u·1s • + •

ai, Molecular stutes

stute is IOlVcr cncrgy.

Figure 10.5 illustrates in a different way how

arise from

+

I s molecular wave functions

ones. The plus and minus signs do not represent the sign

of a charge, but rather that of a wave function. Remember thai all wave func­ tions conlain an oscillatory lemporal part and thus change sign periodically.

The even u Is state results from the isolated-atom states combining in phase:

in the odd u Is, they combine a half cycle out of phase. Of course, plus and minus are irrelevant to the probability/charge density, the square of the net ·

wa\"e function.

In the spirit of Figure 10.2, the energy-level diagram of Figure 10.6 shows on-dependent link between the Is atomic states and the corre­ separati the sponding molecular states. It is the u I s molecular state that is occupied by the

Figure 10.6 Atomic Is energy levels form bonding and antibonding molecu­ lar Is levels at small separation.

lone electron in H2+ and by the spins-opposite pair in neutral H2. Figure 10.7

rurther illustrates the point. A molecular bond forms because it is of lower

energy

than an isolated-atom Is state. On the other hand, Figure 10.7 also

shoWS that although the a* Is state is of higher energy than the atomic I s state, i! is still possible for an "antibonding" state to be occupied in a

bound mole­

+ z cule. While helium doesn't form an Hez molecule, the ion Hez is bound. Two electrons in the a Is slate and one in the a* Is state give a lower average energy than three electrons in atomic

figure 10.7 Filling of the

ai,

Is states (i.e., isolated He and

He+).

Molecular Is

,---

levels

Atomic level�

Is

Decreasing atomic

separation

Is levels in molecular hydrogen and helium.

� i

� i

H" (Does not fonn molecule)

Atomic Is level

lAt very low t�mpo:rnture. H� does form a ··dmu:r."

IlII c�tremely

weakly bound pair of atoms.

d States Bond Types and Hybri

..... ,•.• ".., �w2p _ ....

aii in atoms with occup·Ied 1'1 - ") ,tatc' e\hibll richer del Ni1turall�'. mole<.:ular -e In m a high-energy u* 2s and low nerg for .. tale .. 21 ffil( IcH:ls. '\to rstand how 2p alomlc sa .. for the Is. To unde a. t "-'3} l ts arne .. h l C'\�nua •' the, ' . ' f l(ln L\ help ul. ef hgrt,� (ll{1lhu,c. a bri I. 's three equal-energy 2p spall""-I nden(e� of hydrogen The an�ular depe

" ,

.(-.J

".lIC' are gi\cn in Table

Y,U ls

7.3 a�

tV,!,l,"! L

ex: sin

i 9 e:!:..

ivalent states depicted in F'g""' d 10 yield the three equ They may be combine a.. follov. ..:

iJ!�:

I/I'!p - 1/I�.1 .+ 1 •

1/121',

FIg�

'0.9 l'orn\
ilf\d J.nllhondmg �tdte'.

.\

�

.�

•

.. ".. "..

· t�· r

.,.�.

[.

"'.

"'.

, •

•

;:p,

",.

' .l.

J

I.. \

•

lp,

All,lml':

2{>1

tJ�.

,

,lal(S

t')

•

•

\,)

r

1f-2p; ,

, '",

2j>,

'\J

.. Mllk"ular fJ'!p, �Ialc'

!\I\>Il11� 2p, ,'''10

•

)

..... 1 . -. .

\.

•

l

')

•

,

1f2{'.

\1(lI(\;uliIJ

tr!P" ,Lilt_

�

=""

1/12,['0

IX cos 8

+ 1/t2,l ,-1

41H+ 1 -

108 .

412.1.

ex:. sin

I ex sin

8 cos cP

(10, 1 )

(J sin
as the 1/12p,; they are JUSt oriented I/J:!p. are the �me shapes The �tate<; 1jI:!p. and to combine this way is to form IowtU� for the state� differently. The impe directions. different other atoms along energy hond!> with spa ial states, ":,,hich form six mol­ ms, there are six 2p . For rwo identica1 ato h the X-axiS arbitrarily 'bo s en ICts these slateS, Wit dep 9 10. ure Fig �ular state,. fer to the wave functi re nus . d rru an s plu \'is. (Again, ru. the molecular 3. 0 and -sin 8 when

�

�;. ;

l

Molecular energy levels in nitrogen, oxygen, and fluorine. FI9.-. 10.1 0

_ -::_ u"2p

_u"2p , ___

-

,

,

�U2P ,

I,t

,

,

".2p

.2p

"

Figur. 10.11 In an ionic bond, an elec­

0,

tron is transferred. and the resultmg ions altrllCI electrostatically.

...·\"al�ncl'

Single 6.1

Being symmetric, molecules comprising two identical atoms have no dipole moment. In bonds between atoms of different elements, how­ tric elec ever. electrons are not shared equally, resulting in polar covalent bonds. For instance. when HF forms. hydrogen's

11 =

I and fluorine's "

,f ekdron Separated atoms

..

= 2 electrons are

f

.0

Nearly filled

,hared asymmetrically. The fluorine end has a significant excess of negative,

n

leaving an equal positive excess at the hydrogen end. The asymmetry is most pronounced when a compact atom that is one

2 ,hell

electron short of noble gas structure meets a large, spongy atom with a lone valence electron far from its nucleus. In an extreme case like CsF. depicted in Figure 10. 1 1 . the fluorine does not share cesium's valence electron; instead, it wholly appropriates it. The fluorine ion assumes a stable, spherically symmet­ ric neon structure. and the cesium ion is left with a stable xenon structure. Such a bond is not covalent. but is known as an ionic bond. Il is true that energy must be expended to produce the positive ion by detaching its valence electron. but the final overall energy is lowered as this electron completes the low-energy outer shell of the negative ion and the ions draw together by elec­

trostatic attraction. In general, molecular bonds exhibit characteristics some­ where between purely covalent and purely ionic. Atomic states may hybridize in many ways to form bonds ofthe lowest pos­ sible energy. Let us consider an important example that leads to the tetrahedral geometry common in carbon compounds. When surrounded by electron-hungry aloms. carbon often shares all four of its and !Ji2. .- I may form the !Ji2p/ !Ji2

1j!�.I.+ t·

1

n = 2

P/

electrons. Just as the

and !Ji2

p(

!Ji2.1.0'

all four of carbon's n = 2

stales may combine to form four equal-energy atomic states that stick out toward vertices of a tetrahedron. as !>ohown i n Figure 10.12. (As we know, atomic s and p states are not of equal energy, but external influences, such as nearby atoms. can

Figure 10.12 Carbon's spl tetrahedral

lobes.

• t . .... ..' _ 0•

, .. ,....

b P�e

10.13 U1o", ' one example of felrahedral bondi ng:

diamond a ly t covalent bond� four atom each othe ? rs. Each carron solid in which o nd IS �ame molecular spatial state. the bo . of elec[TOn� sharing Ihe ...ir a r. . n d'lng . . eo g t binalion of WI lobes (rom the pm1Iclp'lIUl alo�s, as Illus rated i p' m. n Ig . u 10. 14. Because the lobe is along the /rile coonecllOg the atoms, each I' re S a bonds per alom covalent (our ()" makes diarn bond. The presence of olld a ' · yery · 'Irong cr}"�la"ine slructure. A fyplca1 comp Ol/lll In Ihe lelrahedral C On fig ura. titln i� melhane (CH... ). in which each lobe shares a hydrogen electrO n and . . . One of Ihe carbon",; four. Allhough Ihe �hanng IS unequal. wIth the hyd r gen b ei ° . the 0 J ng of lobes at Ihe 9 5 tetrah ..lighll)" po.�ili\C:. Ihe arrangemenl dral a n gle (\CC' E."(en·i.\t )1) is ..ymmetric and leaves methane with no eleCl . nc d'Ip ole momenl. A� Figure 10.15 shows, elemenl.� near carbon in the period·IC table o . . . ften a'�ume a slmdar geomeuy. In ammoma (NH)). one of nitrogen' fi e IV n => 2 . eJeclron.� lakes the place of lhe "missing" hydrogen electron o WI t {he proto . mis,ing, lhis lobe has a negalive charge' giYing amm aho . . - . e1e n ctrjc dIpole moment and accountmg (or many of its properties as a Solv enl. step (urlher removed. water (H,O) has oxygen's Iwo On e extra 11 =:: 2 eJec trons �plilcing hydrogen electrons at two lobes. Naturally, it 100 h as an el dIpole moment. The angles between the nitrogen-hydr eclric ogen bond . s lO affl.In and ox�gen.hy�rogen bonds m · water are found to onia be 107.30 and 1 04 50 respectIvely. whIch are very close 10 the tetrahedral . ' angle (Th . . . . ar rhey equa I I.� · due to ,\ome mJ}[fng WIth the 900 p.{. py' p: ' are no, hybTJ·d·IZatlon. ' )

,.... '0.1J Tht cl')�tlIJhnr ,IN;IUIt' ofdl.mond

Figure

:

�

Figu,.. 10.1" The spJ carbon·ca rbon bond.

.:.. .l. .. .1

Anubonding

.!.... _ ... ...spJ... -+( "

lsolaltd

Prepared

for bonding

Carllon afom

�

-

'

, , , ...,

Bonding

)+.

.fp.... .1

Prepared ....

P P ....

.... __

for bonding

Carbon alom

p .. .... .

10;0",." '""

10.2 MoIeeUica

Flgu...

10.15 SImilar electron orbits. H

H

Ammonia

Methane

Water

EXAMPLE 1 0 . 1

Another hybridization common in carbon bonding is the sp2, which combines only three of its /I = 2 electrons. It can be represented as follows:

One way of demonstrating angles between lobes is to regard the 2P and 2p\, states . as \'eclOf'S along x and y. whose lengths are the coefficients of the states. (3) Why do

we not make a similar assignment for the 2s state? (b) Find the angles between the three states. (c) Relative to these three lobes, where does the remaining 2p state

2sp2

\tick out?

I ION ta) Like all s-states, the 2s is spherically symmetric. 1l has no direction.

AB

(b) We find the angle between vectors by the definition of the dot product, cos 8. Ignoring the s-slate contribution, the squares of the A B= .

lengths of our "vectors" are

and the dot products are

In all cases, A B = .

-

I and

AB is 2. Therefore, cos (}

=

-

1/2, so

0 is 1200•

Thus, the three lobes are evenly spaced in the xy-plane. This is the geometry that gives graphite and benzene their planar hexagonal patterns. Each lobe forms a q-bond with an adjacent atom. Section 10.10 and Progress and Applications

discuss important examples of Sp2 hybridization in carbon and other elements.

I (e)

The lobes of the 2P: state lie along z and are therefore perpendicular to the other three.

421

1 0. 3 Rotation and Vibration In adliJli{ln Itl the different energy levels irs electrons may occupy, a mOlecule rna)' ..lore energ) in the rolalional and vi brational motions of its atoms. Not only are Lhe\t (orm.\ of energy storage important 10 Ihe molecule's heal capac_ Ity, but the fael mal these: energies are quantized also leads to spectral "finger_ pnnl"" mal aid In derennining identity and concentration. Lei us consider a \Imp/e Cil5c:.

Diatomic Molecules ,.,� 10.16 P\�t'nll"" ('/ler$) 10 a ,hollom. nllll('�'u'c,

Figure 10.16 depicts the diatomic molecule pOlenlial energy. A given mole_ cule has a part icular equilibrium separation between its aloms, labeled as a. If rhe aloms-nuclei plus nonvalence electrons-are pulled farther apart, the potential energy increases due '0 a Jess fayorable arrangement of the yalence electron cloud between them, and it ultimately leyels off as all forces dimin_ ish. If the aloms are pushed IOgether, not only does nuclear repulsion increase, 001 also the ooovalence electrons begin to occupy the same space and are

forced 10 higher energies by the exclusion principle. (Atomic motion doesn't bump electrons 10 djfTerenl quantum .�lates; rather, i l modifies the existing clouds.) We see that a bound Slate is possible, and from

Chapter 5, we know

Ihal bound particles must oscillate. Allhough i t may seem that we know very hlile about this potential energy, if the oscillatory motion is small, we can explain quite a bit based 00 only IWO characteristics: its equilibrium separation {I and

its " spnng constant

"

K.

Near a local nunimum, any smooth potential energy "'ooks" like an ideal spring-a parabola. It is left as an exercise to show that for s mall oscillations about its equilibrium separation, the potential energy of Fig ure 1 0. 1 6 can be replaced by

fig"". 10.17 A 'Iimpliti«l di,'lomic mlll('�:ule, t� 10 nllafe about jl� ('enler

(If ma�, and III \'i�ntfe along the mole(uJar a'J�

/

where xr == x -

a (i.e. , the amount by which the actual atomic separation x dif·

fers from a) and

K

The constant vaJue

5

d2U(X)/ dr'

"

( 1 0-3)

Ua is irreleyant to energy changes, our ma in focus. Thus,

we model a diatomic mol ecule as in Figure J 0.17: two masses an average dis­

tance of a apart. free [0 rotate as a whole and to oscillate along the ir axis as lhough connected by an ideal spring. Chapter 5 treats vibrational energ ies, and Chapter 7 tre at s rotation. Exerc i se 7.88), it should nOI

Although we omit the quantitative details (see

be surprising that when both are present, (he molecular energy levels depend

·

ion

10.1 Rutatlon and Vibrat

quantum numbers " and lin tWO

C, related to vibrational and rotational

p tively. The vibrational part in particular is identical to the one­ energy, res ec n sional case.

dime

1110ugh arising d�lron orbital

from a

similar

differential equation, the e here relates 110' to

motion, as in hydrogen, but to rotation of the molecule. Like

:

�

�

however, d i ff rent rotational orientations are allo ed, go erned by hydrogen. :':f. The mass In equallon ( 1 0-4) quanlUm number obeymg me = 0, :!: I ,

a

[, that of

the equivalent

. . . •

single particle. It is given by

1' =

111 1 "'2

/Il l

+

( 1 0-5)

"'1

WId is known as the reduced mass. Were one particle of overwhelming mass. the reduced mass would be simply that of the other ("'2 » nil � IJ. = 111 \ ). This

moles sense. With

an essentially infinite mass at one end of a spring. the only

thing moving. vibralionally or rotationally, would be the small particle at the other end. In general. however. the reduced mass is less than both 111 1 and 1112, EXA M P L E 1 0 . 2

The diatomic r-

HD molecule comprises an ordinary hydrogen atom and a deuterium atom, a fonn of hydrogen in which a neutron joins the proton in the nucleus. Their

1 .007 u and 2.0 1 3 u, and the bond length is 0.074 nm. Using the Boltzmann distribution (Section 9.3), delermine the temperature al which the ratio of molecules in ( :< I rOlational states to those with no rotational energy would be 1/10. Jumps to

masses are

higher vibratiQlwl levels requi re more energy and thus occur at higher temperature (as we soon see). so assume thai all molecules are in their ground vibrational states. SOLUTION

In many ways, this is similar to Ex.ample 9.2. Taking into account the three different rOlational orientations (values of lilt) allowed for e = I and just one for e = 0, the

ralio is number with energy E

o. 1 -;-;-:-= --;: ""number with energy Eo.o

=

3 X

e-Eo,l/keT e EOJJ/keT

= 3e-(EIJ.I-Eo.o)/keT

To calculate energies, we need the reduced mass. Using ( 10-5), I.L = As noted,

( 1 .007 u)(2.013 u) 1 .007 u

+ 2.013 u

= 0.67 1 u

the reduced mass is less than either individual mass. change, the energy difference given by (1 0-4) is

With

no vibrational

423

- 0sq Ie ..... � '1 ,. q "", ... _

(UlS5 )( HI }.I} · s)1 07" X 10 .;�)} '- (0.671 X 1.66 )( 1 0'27 001 1 coV '" 1.82 X JO-!I J •

kS}(0

,10

-

[�

]

J 1.8.:! x 10 oZlP ( I J8 X 10 B J K)T

... )e l

-

10.1 e'pJains h)'drogen\ rotational heal capacu 10.18. (The vibrational Jump j.. dJ\cu\\ed in Seclion

� ��

Eumple

in Fi,1Itt'

.

.

should began 10 I:onrnoole 10 energ)' 'IOrage aO) \OOn as leveIS

mp, �hov.1l .) RotatlO"

abO\'e

the ground S1ale b«OOK jk'ces!!.ible. and the choice of a one-tenth proba bil" ',Y I� a . ' I.. fAIr JDQSUIt'. Ordinary H} ha� the ,arne bond length as HD, ,",oug h II , " itS a somevrhar snWler mfucC'd rna" (-0.5 u), so we \hould expect the JUm , i to p Figure 10.18 dearl beg n wmewiJal llbc.WI: oW K. a,\ y demonMraleo;;,

Sped" If tht \'ibralional and nll.allOnaJ energie<, 01 molecule!' are mdeed res triCted . to . dlsc�lt raJue.., we mlghl e'(peel 10 'ee evidence in th ' only (,:ertaln e hghl lh . �)' emil and ab!iorb Spa-rraJ dala 'hould Iherefore be a good lest of OU mOdel. CleMly, if some of lhe molecule's nbratlOnal/rOlallonal ene r gy IS gl\!en 10 I photon, n or t or bolh mu\1 decrea�e, bUl lhe pos�lble Iransiliom, are

�

restnCled

Fit"'" 10, " Variation (If hydrogen'� molar heat capacity with lemperatu re, (NQ1e : 1k H! molet'ule th�SOClale, b(fore Ihe \'lbrlltlonal plateau i\ reached,) , •

./ -

-.t.C"tioDa/ nrC"Qi +:

80

140 200

TIKI

800 1400 2000

Vibraboll.il

1;1\ \,(\(\:t\OO ru\e� t�e Section 7.10). Fin.t, because !.he photon is spin- l , th(: S)lS­

�11\" total angular momentum before emis<;ion can equal Ihat after\liaru onl)' it­ t ill( the molecule changes by I, leading to the selection rule

�f

:!: I. Second.

e\(o(ltOrnagnetic radiation i!; emitted or absorbed most effec:ti\'l"\y by a chou--ge .}I-.uitll.lil on when it oscillatc!; as an electric dipole. Although charge denSity in

• *'li(lIIl1n' �la,e produces no elcctromagnetic radiation, it docs o:.cillate in a

In.jn\ilion between vibrational states. It may be shown that a charged ham\onic �i\\ator oscillate!; as an electric dipole only in tran!'oition, between energy -.(J.IC' whose quantum numberli. differ by

\(,.:h 1<;

�'1 = :t t. (The rule is valid only

I . so the selection rule for \'ibra\i.onal

long as lhe o<;cillation energy i, low (nOUgh to make the parabolic approximation a good one.) To lo<;e vibrational energy, the molecule must, of cour<;e, jump to Ihc nelll lower II. but the vibrational energy spacing is usuany <;0 much larger than \h(. rotational <;pacing that the energy decreases whether ( decre��e� or increa!'oc!>. By �ublracting the molecule's final energy from ill'. initial for both case!> .1f := -=- \ (�ee Exercise 40), we obtain from equation (10-4) thc pos<;ible ener­ so

.

g.ie� of the emitted photon.

\ . 2. 3, . .

( 1 0-6\

.

�

ced equa\\y on either �ide of Photons will be ob�erved whose energies are spa The energy fi itself i� con�picuously the vibrational jump

f"t�.

absent because angular-momentum conservation requires that any transition in,'olve a (unit) change i n e. The same formula gives the energie� of photons

absorbed by the molecule.

Guided by equation ( l 0-4). Figure 10.19 plots molecular vibrational level�

for II

= 0 and '1 = I , and the much closer-spaced rotational levels f = 0, 1 , 2, and 3. It also indicates the transitions allowed in photon emission and absorp­

tion, which obey equation (10-6). Only six transitions occur between thc states �hown, each differing from the next by

\

h2/1J.t12. The "hole"

in the middle i� due

\0 the forbidden 6.f = 0 transition. We say that the photon energies constitute a I'ibration-rotation band. and Figure 10.20 shows how well the theory agrees with experiment. The figure plots intensity of light absorbed by diatomic He! versus frequency (Le.,

Ephotoih).

Each spike represents

missilJg

transmitted

light. because photons of thut frequency are readily absorbed. The spacing isn't quite equal. As e increases, the interatomic spring must stretch slightly farther to

hold the atoms in their mutual orbit, and a larger separation Q implies a slightly .maller rotational energy than otherwise expected. Thus, af

are somewhat less energetic than the formula suggests, and

= + \ absorptions af = - I absorp­

lions are more energetic. Still, the spacing is fairly regular and the hole in the middle quite apparent. Note that each spike is split because chlorine in nature comprises two isotopes of slightly different mass, giving two slightly different selS of data.

Our simple model seems to fit the experimental curve well qualitatively.

Let us see what the quantitative side can tell us.

G6 C/NpW 'o

l and So/ids

.. � 80ndlOg: M0I-ule"

energy levels, and

photon-produ 1 rOlafional . C .ng uonil1 and Figure 1 O. .9 Vibra = 1 .. W l ru es, selectIOn = I and af by [hem, res- meted .

lt3lIsilions berween

�£yjb ==

£1.0

-

_

.:!:: I .

J

E J.o + 68£ ,

J 2 1 0

£1,0 + 38£ ,

E ',o + liEr £1,0

n-I

Eo.o J

£0.0 + 68£,

2 1 o

Eo.O + 38£,

n

Eo.o + liE, £0,0

0

Figure 10.20 Vibration/rotation absorption spectrum of Hel. The " hole" in the middle is due (Q the forbidden IJ.f == 0 transition.

8.0

8.2

8.4

8.6

Frequency (X

8.8

J 00 Hz)

9.0

9.2

EXAMPLE 1 0. 3 From Ihe data given in Figure

10.20. delerrn.jne (a) the approximate bond

(b) the effective force constant in an Hc/ molecule.

length and

SOLUTION (a) From the tenth line to the left of the hole to the tenth to the right (20 steps) i a frequency range of 1.2 X lO/3 Hz, giving an average spacing of 6.0 X 10 1 J Thus, the photons represented by these lines differ in energy by (6.63 X 10-34

� J.:)

" iU7

10.3 Rorabon and VibrabO

(6.0 X 1011 Hz) = 4.0 X 10-22 J = 0.0025 eV. This must equal the energy spacing hZlJUl2 indicated in equation ( 1 0·6). To find IJ.. we use data from the periodiC table (Figure 8.10), which is averaged over naturally occurring isotopes:

"'H

= l.OI u mel == 35,5 u

( 1 .01 u)(35.5 u ) 1.0 1

u

+

35.5 u

= 0.982 u = 1 .63 x 10-27 kg

Thus,

4.0

X 1O-22J =

( 1 .055 X 1O-34 J ' 5)2 ( 1 .63 x 10-27 kg)a1

Q = t.3 X 1O-10m

=;0

(b) The hole falls at 8.65 X 1013 Hz. or a photon energy hJof (6.63 X 10-34 J . s){8.65 X 1013 Hz) = 5.73 x 10-20 j = 0.358 eV

According to equation ( 10-6), this feature corresponds to the vibrational energy spacing of hv;;;;.. As noted earlier. we see that it is much larger than the rotational spacing. Plugging in. 5.73 X 10-20 J = { 1 .055

x

10-34 1 _ S)

J

K

1 .63 x 1 0 27 kg

� K = 482 N/m

Thus far we have discussed only the molecule's rotational and vibrational

electrons

change states (e.g., energies. Generally speaking, transitions where the Balmer series in hydrogen) involve much greater energy differences (> l eV) than those involving molecular vibrational transitions (>0.1 eV), which are in

energetic than molecular rotational energy differences « 0.1 eV). not to proper scale, Figure 10.21 illustrates the relationships. The curves represent two different electron states-perhaps the ground state and the firsl excited state. When an electron is in an excited state, the atoms are still bound to each other but are farther apart on average, and the interatomic potential energy well i" wider and shallower. Even so, for each electronic stale, there are closely spaced vibrational levels and even more closely spaced rotational levels. Thus, while we might expect a molecule to produce a simple speclr3.1 line when an electron jumps from one of its allowed energies 10 another, vibrational and rotational energy changes break each line into many, yielding a much richer structure. Note that symmetric molecules. such as H2 and N1, lack an electric dipole moment, so they do not produce spectra due to pure vibration-rotation transitions. However, the distinctive structure of Ihe':.e levels is superimposed on l)pectral lines arising from transitions between elec­ trOn states. (The oscillating electron cloud provides the necessary dipole moment in these case�.) tum more

Although

Fig'" 10.21 �lol«ular \lbratlOnal and rotational le\-els ror

1....0 . different dettrun k\d\.

HI�hcr

ela:lf\ln If:o\el \'Ibr.ltllllUl

le,d' II

I I I

l :; ' = ) " :� ��' ,

Lower ek.;lrun

:2

, 10 � tol 11;' ,

D_

ll

ROlallonal e�e \ J

Db....

1 0.4 Crystalline Solids lei u..

now

lum

from mol�ule... the bondmg of a relati\'ely 'imall number of

atom.. into \OmewhJI larger unil", to ca<;e� in which a huge number of atoms bond 10 form ,trur.:lure.. of mar.:ro-.r.:opic \ize_ This lakes us into the field knOWn lb

.

condeMtd matter. b)' far the mo,t active area of physics research today

Conden':>Cd matter Indude.. ,tudies of 'iohd!l, hqulds. and other syslems in which the beha\'ioT'i of man)' alom' together must be understood (such as Base­

Ein,tein conden..ate". di-.r.:u."c,ed in SectIon 9_6). In the following sections, we restritt our attention 10 cl)'�talhne wILd... All elements and compound:. form :.olids at sufficiently low temperature .. a cr)stal lattice, in which certain alom� and/or high pre\,ure mml often a

1-a,:t-(o!1lltmi lubK II�d

.

are found at 'per.:ific location, in a mlcro"copic unit thai is repealed identically countle.... time, in all dimen,ion"_ (Some materials form amorphous solids. in whICh anglc� and bond length.. are \0 Irregular thai the location of one atom i� cs..enlially unrelated to that of another only a few atoms away. Familiar exam­

Side �itw HC'O\Itln;d dOMCSl l'3'l.etl 4hi:p>

0 •

----

'

.

.

n

0

-;;

c.n

Simple ,.;ubic .. ,)

ple� are gl:c>, and rubber,I TypIcal atomic "pacinI! ID a cT}lotal is 0.25 nm to

0.5 nm_ or 5 to 10 Bohr radiI Geometrkal con'ldemtlon, ,how that In three dlmen�lOn.. there are only 14 pcMible lattice 1)pe5. or wa),,, in which atom'l can be arranged in a regular geomelric pattern. (Exc:rCl..e �5 gi\'elt a 'Imple example of"geomelrical con�ld­ c:ratil,m" in two dimen,]Iln�.) Figure 10.22 "how .. �\'eral of the rnO:.t common. The bod)'-ctnltred cubic I' a repetItIOn of cube'i wllh atom... at the comers and (enler. It ilt found in the ,olid form, of the periodic table-s fir-Holumn ele_ .. in a number of tntn..ltion metal", iuch a." ITOn. chromium. and menl'. a., well 3. tung,ten The face-centered cubic i.. aha a repet itive cubic sUllcture. but ....11lI alom, at the center. of the t:ub.t:/II("tl lD,tead of al the center of the cube. The:

10.4 Cryltalline Solidi ept helium) solidify in this structure, as do many transit.ion ele­ DObie gases (exc g copper, silver. and gOld. The hexagonal close:st.packecl includin . ments ,tr\lc(Ure has a six-sided symmetry and shares with the face-centered cubic the of resulting in the smallest volume per atom for a lanice of idenbcal distinction found throughout the periodic table­ �pheres. Eleme ts wi Ihi st cture

�

�

� �

ar�

helium. magnesIUm, ZinC, tttanlum, osmIUm, and many rare earths. Compounds same crystaUine structures as elements. For instance, sodium chlo­ form in the ride is a face-centered cubic structure in which a sodium-chlorine pair replaces

the individual atom. In facl, the sodium and chlorine ions each i.ndependently ntered cubic arrangement. because the atoms of one are a fixed form a face-ce those of the other. Cesium chloride. though resembling the from displacement

bOdy·centered cubic, is an example of a

simple

cubic structure, fonned by

each ionic type independently. Besides their lattice geometry. crystalline solids are often categorized according to how the valence electrons are bound in the solid. Four categories are generally recognized. Covalent Solid

ill a covalent solid. such as diamond, each atom shares covalent bonds with those surrounding it. resulting in an unbroken network of strong honds. Such

solids are relatively hard. due to the inherent strength of the covalent bond. and have high melting points. They are

poor

electrical conductors because an

\a1ence elecLrons are locked into bonds between adjacent atoms. The crystal lat­ uce assumes a geometry determined by the directionality of the covalent bonds. In the case of diamond, it is face-centered cubic. Ionic Solid When atoms with nearly filled shells meet atoms with weakly bound valence electrons. the fanner may seize electrons from the latter, producing an

ionic

solid. The solid is held together by the strong electrostatic attraction between the

ions. Thus, ionic solids are relatively hard. with high melting points. Because the transfer of electrons leaves both positive and negative ions with noble gas electronic structure. electrons are not free to respond to electric fields, so ionic

solids are poor electrical conductors. The ionic bond lacks directionality. and

the lattice geometry is therefore determined by whatever arrangement leads to

the lowest electrostatic energy. which depends on the relative sizes of the ions. in any case, lowest energy results when signs alternate-that is, the closest neighbor of an ion of one sign is an ion of the other. Because ionic solids depend do not fonn ionic solids. on asymmetry between atoms.

Metallic Solid

elements

Except for noble gases. al l element!; have valence electrons. But in most cases. no arrangement of

covalent bond!; can join all of a given atom's valence elec­

trons to those of the atoms surrounding it. An element or compound with "leftover" valence electron ... forms a

metallic solid.

When bound to their i,oiated atoms. valence electrons have relatively high

energy. In a metallic sohll. however. the atomic valence states mix. much as in a diatomic covalent bond but I'nl.:ompassing all the atoms in the crystal.

This

.a9

to • . ne. ... beoor1 bY potassium .I"ne. b) ,hl"nne alo and b) �.. put . 7 IOIid (anned -.w ttl. I;iod _ _ 11CA1I 'LE

.

__

.... ,,1

__

qedL _

.

--':�. - I. When -valence .7'1 " dIIorine

+. "'II '711 "-

logeth". chlorine will . . ........ _tng complete ....hell po"'llwe and

l('IM. 1be: kIIU �houlJ tht-n f\1fffi an K)ftl,;' ",>W1IunH.'hloridc whd. In..thid­ � � IC'MkI d\k1rmO not potassi um .;u fonn an hlflll; dtd. I.' ilJrrul('al ar('l4n ..III .,. dIIf(' tk(trOOi �)mmttrk.lIl. I). i\"�,iWQ h.n. illJ� 'J t�1ron In. phm

.#. I)� ,..tilt. "'-' 11 �11I n..-c ItnJ 1,1 � o:�·\akntl) Wllh mullirk .1then

� Il In a 1.11tK'('. \\ Ilh lC'fI" \'ef C'lt,,:trvn.s, I"<�Ium !I.hl'OUhi li..l(ll\ . mtlallto: ..01 Cb�. hl�' n\JI...tl� '" l'I'urc 10 10. )I,!k\Uld h>rm w\II.lt'nl mo.\�ul«, IlnIJ .. .r.,:,uld fonn II rt\l.IC'\,·ul.u ,,'hJ

Ju!4 • (,. mJI\ IJu,\! Ill\lle(ulc,.

. bondlnt: 11\ \" h," ,,:In b(o \""'1111'1('\

tJtlA. \'OI1It>lIlill1�m �ll the 'MUlti' t )I'IC '

.;I.nJ l '

N.'l m" dn\,:(', Ih
p«" 1:'I:"..... \'(."n ;U,llll'. and t�)n\'t,., 111 \"oOll"uund, ilft' llnt'n panl) ('\\.lknl, I} 1111\1\" .\ $\xxI (' \umple of the f.:vmrle\.u) \,1" bI."'IniJing III "lill" I' Ill( pt\)

�H.

\,·l\J.n$(' 111 f.:h,U1t!.:lcr tulU1ng the Ictra\ alcnt e!CIlli,.'Oh:

\,:.trb!.m (l

•

('I).

_,Ill \Z . loll. �cnn;tmum V • :\2). and IIIl (I - :'iO). AlthllU�h III U·'t', aU ,..... "..l1k" ('\h.tllt the- ".'," alent lluullunll . \tNcture, ek�tn('111 f.:OIl\IUdl\ 1\) rt"I\('I) ' ...� pf\'�

h\lm thllmond.

an In\ultuur, t�) ttO,

U lUll I:lllldul,.'IIlf

�II), ell'\:tl\ln' art Ol)t tree Ill lll(I\C about thc ,olu.l 11\ lllJmoml. tlut thl'Y ' }-hm ('.111 thl \ he e'(phllllcd') The ,,' me" hat un,.ul"l� I n8 If frcrt hHJll ,0 In un .v."tf I' tlw.t e,en (,ll"alent and

10011..' o,;olid, ttlU'1 often he t.re.lted ;1\ thl1u�h (III • \"llko.:c cll'1,;tn.1n, are 'hared by all atoms, Just (I' III (I Illt!tlllhl..' ,ohd. DlIlI..'r·

III tlr..:ln.:al l'n.1pcrtu!, are then ba:.cd on Ihe fact Ihat the elcctn.mlc 'tilte, d\\ ..... 1\'(111 an unbrol..l·n continuum of cnergie� from "hich the clt'�·tmn' lila) "cruralI.' intu energy b..nd�,

�h,:)

10.5 Energy Bands and Electrical Conduction Ii 111 01 a metal', \alenee t'lectron� are rclea�ed to the solid as n whole, why do ", find that the number of nmdllclioll electrons per atom· -tho�e IIctually fl'(e t..' rt"f'l,nd to an eloclrie ficld--often doel; not equnl lhe number of I'alflll'l' clcl" lfl'lb pe-r atom? And if even nonmetals mu�t be U'Ca!cd as thou!!h they frel'ly

-rw'C ehxtrons throughout the ..olid. how can they be eleclrical IIlSul:ltor'? The .I!l\\\ers re..t upon the fnct that the well into which the valence electron' ure �uu�ted is nOl ll liimp le well. It restricts lhe electrons to energy bands. In referenct! to lhe double finite well of Figure 10.1, we notcd lhat the wave fu»,uons resemble states of a !>ingle large well. but they also fonn pair; centered (JI the tnergie-'! of the IndividulIl "atoms." Figure 10.23 demon<,ll1Itc\ the '!(une

l Electrons are attracled to each positive ion in a idea.., In a fOUT-atom "cry�al." model this attraction with a simple finite well. giving low poten '1'I..taI. and we � � (I\(rgy at each ion (center of a well) and high potential energy aI the point<, �'un them. (The functiom and energies shown in Figure 10.23 follow from aurnerical solution of the Schr6dinger equation. See Computational Excrci<;e 80,) With four wens. the \tate\ naturally divide into four bands. To <,ee why, .;tllhider the Slate at the top of the " = 2 band. The wave function. with eight lIIuood(s, is ofJu�t the right wavelength to be zero at points halfway belween tbt: "positive ions," where the periodic ionic potential energy is maximum. nll� �tate's potential energy (expectation value) is thus relatively low, It.. total

�rgy. lOcluding kinetic. differ; IiUle from what it would be WilllOllf any peri­ odie \'llriation due to lhe po�itive ions. represented by the single large well on

Iru"'ur� for u nK>nJ

I"'I"-"",,d 2�V(' 1m ""dall''':' � II1U1,furm,IIW". .I>arnlnm". Il>e II"""""... . cI'"_'en_.,"II.IJ) morl"n" ,..,e.

'" bre�" "t"" '"'0 . " '""I" 00;' polOnti ; od" pori "" ..1.

In ' "y, om "," ,;o,k."

'0.23

"""'..-- p" � �.

n

r 11 = 3

.

,

f- .

·

'\\

••

..

•

\

\

,

dX

,

I

,

,

,

,t

I ;

. ,

\

, ,

•

,

,

Stutes

j

well in one IMge ionil" the UI .,.;/hlJ d'

\

pOlenti�L

.

�--.-- '

f----- ' .

Large well

432

, , ,

./

,

,

,

,

,

,

,

,

, , , , ,, ,

.

,,

,

,

,

,

,

,

,

,

II

,

2

. .,' ...

�

Energies i a band

cluster around the

corresponding

,mgle-alom , '" "ergy. .

,

•

...

'.

'"

n

,, , , ., , . ., ,, ·

·

,

=

band

,

,

. ,

,,

,

, ,

,

,

/

• ,

,

. h aminO
potential energy is high ha e relmively lv high IO al energy.

; ,• ,

,,

\

where the ionic

"

,

". . " '. ".

"

,

,

-------

band

,

,

='.�' '.

band

.. ., .

'

.

4-atom crystal

Single atom

kmt'IiC

... ...e\en (Mlme1olo'hal mi...-.hapenl an\t­ 1btkft. � Re'(l-Iower wa\'e tunctlon ha �, 'Otl' &I\'C\ It a energy. 'I�k'!n.m. "iomewhat lower. but 1\ alMl

.

.»-t'" it the wrong wa\clength to be lCo) between all Ion... Ito< ptltt'J1tial (fItlt) " higher. Thu.... tt"i lolal energy .... nOI much lower than the top-of-the-

tIJI'd ,tatc. The nc,l-lower \Iate . ..i, anlinode.... \\0 lower 'till In kinetic 00\ biJbCf again to potenttal. and 0;,0 again 1\ only ...lightly lower In total energ)' 'tbC wttom-of-Ihe-band \tate. with ft\e anti node!".. 1.. nearly at 11, mal/mum at

all lotCratomic pointo;" Thu.... whIle of !\Itill lower kinetic energ)" It I' 01 \'cry h potential. again re,ultmg m only a \light decrea..e In the tOlal The ..itua­

DiS I tWO ,hange"i dra<.,hcally at the ne)lll-iower energy ...tate. the top of Ihe " 10 IOn .. . a .. ituation m returno;, which nodes fall e'(aclly between the baJlJ. f('lf II

\

Be\lde!i. a ..omewhat lower kinetic energy. it has a much lower potential. '0 the 1\\laI energy drop.. greatly. There is an energy gap. or band gap

On the right In Figure 10.23 are plotted the wa\'e functions and energie, of a 'lI\�le "atom" of the ...ame width a ... In the four-atom cf)'!i.ta\ Note the ..imilaritie... �\\\een Ihel,C wave function... and the lop-of-the-band function... in the cry'tal.

There are three anlinodes per atom at the top of the J1 !be II oro 2. and one at the top of the

J1 =

=

3 band, two at the top of

I . Not COincidentally. the band..' energie,

�bJ\ler around the ...ingle-atom energie.... Thus. we see that each band ;\; related to

an indi,idual-atom state, Figure 10.24. which

plots energy \ersus wa\'e number. further illu\trate...

Ihe effect of the ionic potential In one large well of width L. the allowed ,,3.\e number� k would be multiples or ""IL. and the energy would be purely

I

10.24 The periodIc ionic potential group� the large­ \\ell ..tate� into band\;.

Figure

t

-

I

\

'----'·U ---"-',,,�

,

\

-�, '

",-fl,I ._.-

,

-

,

,,

, ,,

,

. ..

0'

. .

)

, ---,

II

I

-1

.

,

-,

, ,

,

';"'� 7 8 9

2 3 4 S b

kly

10 I I "

.".., .... ...... . _ . + I o«un sharply a' t _ Il1rla. ..... 1.." &lad ,. 0l.l."W' "hc-n II - nA� UI" .t • nrr/a. .., 'o p.

al •*.... __ RlU iltNun ilIlJllh, .... hen=- lIk' potentI -U . .... wIIeftat bonom·of-lhc-band \IJi('\ � large thc� ..

�Lf�'��� , lo� PE ..

•

PE

•

.,. " "•' ll

PE

•

Low PE

lIK� Ilw, ,I.U'" hJ'

Nnll " h." r, \.Otlmxl�, ref alllm an..! Ihe!'\" .\f\' '\ ahlin' \l,('r· � tot.t1 ()[ n.\ .Ulhmxl...... The tlll' ll.- Nnd " + I "11\ h." I.' eJl.+ IJ.' .ntin\'I�" '\1 the rtllmt'fr {'If 'I.\le, Irom lh� h'l' \ll' \.ll\l- Nnd I\t Ihe lht \h7\t " \ . Our -.cel'nd. condU'll'n I' lhu,

tf lhe t�ll' �l"

:"Ili

I I

F ..ll'h oJmt 1.:\''''''1' or '" 'till"'_

larg\.', a bOlnd I' e''>enti.dl) Il n"" mulml lll l.'ne�le, \\ hl'n the, 'pn."td alxne: anti bell", the " ng.le·.lllll\l enefl:I(,' 1' In , Ii r ( tlI :!b. The I.'\ICOI 01' 'pl'e;'ldiog. dl'pcmb on ho\\ IlllKh \" ...rhl('l Ih('� _ r.,' octWel'n the " i\\C functton' llr WUdatll .lIom, (SCI." l\�.un Flg.Ull· the blind " idth Jcpend, 1101 00 Ihc tllll1l/)e'" \II mom, hut IOJ ) (lln,...quentl), Ulcl\'lI'mg l\' a decrcnse" (See E\erci�(' ·un At \el) Clll.�. JWll l ic \"(/ (f\ thc I ,ep.traltlltl', the repuhlon of the !X" III'''' 1011'. Ig.nored III our f\lUl'-atl\ll1 _ ' .1 kad, tll (\ nlpld me In 0\ emil enel)l.) The cry,tu\ 1'(10'" " Ith (I 'P,IClng (k,-;aU..c

\"

'0

",li lt" 'Illall,

.

:10.1

.",11,11. , II ",tll�h the 0\ cr;.11l C"'!I)l.Y

•

i� milll\llUIll.

In 'ull\mill')', from the m(IC"rlI.\c(lpic cry "tal an"e, a continuu1ll \11' cncr�ic\, }QI UIt mlcmm'pu· Ionic potential energy divide\ the continuum Will tt,llId, ftlJ!� III the ioui\ Iduill�alOm \1.L1e,. Figure 10.27 plOh (lnowell Cl\cr�IC' ,cr· 5'1' ",,1.\( number III (I one�dimen"ion(\1 cry",al where i, large and k may he he r electron\ may move In e direclion. The il u\ gencr.l1 f'Jrabulk \Ign. r tIthe of tb:Irc' ,ho,*, ' that a!lowed energie\ are the vioual continuum 01' kinetic cncr· J1t' ,!\iI.ilable 10 "n electron roaming freely .., a m[\cro�coplc inllnitc well, A!t!r.:m E\Ccplion\ ari�e only near k "" IITr/a, where the alignment of wave fundlon node\ anrJ 3nlinode' with the ionic potential lead� 10 a Jump from an ldIu,ually low energy at the lOp of one band to an unusually high energ.y ut the bo.->tIllO\\lf the nc'\t. Thu, Inr, we hU\lc di,cU',sed only Ihe nature of Slales QI'ailabll' 10 an elec­ tnln In a c£),\\(11 We have not udllressed the question of how many electrons might be pre..ent to fill those Mutes, and it is this point thaI go"ern� whether n materi,l! will be a conductor or an insulator,

N

I

Electrical Conduction Before we study the rclation..hip between energy bands and conductivity. it i\ \\orthwhilc 10 review aspect!> of electrical conduction Ihat may be under�tQO(.I

from a classical point

01 view and 10 identify those that require u quulltllm�

mechanical approach. Figure 10.26 Beh;\\i\lr of allowed electron SillIes as atomic spacing varie\. MInimum N alomlc

�hlle)

B�d width

AtomiC 'P' ing

•

, "1 ... .. � . •_ .. Ck;tl l' •

BOlIld, and �ar' 10 a one­ ,... 10.27

di�1 crystal

. .

..

. .....

.

I

) .....

-'1,-----,1f�r¥) 1!4p , II c

2 bud

!

" -.

:�

are free 10 move aboUi i n ondu o piclure. electron, Cl ,. In It\( da....IL:al ions_ This i s a th nec ti s ary , , Ingrt_ io", " Ith the po" colh, "flee 001 th,) ma) electrical current is proportional to e applied anding why the ndcN 10 dlent Were the'e no call'ISlo u , ns, asing function of time, being an IOcre . l'idd ralher than < constantly would ncce erate, field c electri proant to a con�l electron.. ..ubJed represent a retard'Ing force ons Colhsl nt. " Increa>;lOg curre - In dUl,;\Og B ..h:adily ously attmn lermmal speed in whIch a bal· nc tanta m .. �t >; almo . e..-.en�e, electron and the decelerating effect 0[ acceleratlOg field the n bet\\ee anee I " reached , ' clllh",on... Speed ; moves randomly with an avera typical electron ;: CI�...ica\ly. 11 with a positive i n , !I IS ) "t ", After a co\lisio the tempcralu chaneten,lIe o[ - n as another. With no externa e ectr.ic _ onc d-lrecllO monng 10 But when an e,ectric a.. IiLel) to be the electrons is zero. \'elocit)' of all . . . a compOnem field, the B\'erage collisions between the tLme . electrons gam, 10 the nt, pre!\C I' time field average belween colus define 1" as Ihe . lle the field. Let given electron of \'elOClt) oppo� a may have time. in inSiant n time AI any but an average eleClrOn Ii,ions. or collisio lime, [ . span 0 - g [or an arb-Itraty 1eraun eee a ' 1 ree 'i been 1". It WIII thus have acquired an addeIighl shift toward ,elocities o ' m thennal mouon l P� mueh [a-"" rando velocity, The resulting current densi y ) IS drift a� n .... . kno site the field i!> •

I

,

1 I

_

_

'

found Il!; folloVo-'!i:

j•

charge

charge

time . area

-=

distance · area

distance time

:::

e X number distance volume

time

field.

the

It" I'J '" the number of free charge carrier:. per unit \'olume. We "ee that ..tic 3 current den,it)'. is proportional to the cause, an electric which I' (/ftc!. . .. , . y l constant IS . own c i l . 1m d " y 0\. the as t e con pr oportlontu lt h t v u \, law The reciprocal of ils resistivity p--and is given the symbol u.

::rial-the

j = uE

( 10-7)

vity decreases as the collision time 't decreases. One way ecrease the collision time is to increase the temperature, for faslcr-mo\'ing

(I � �ical1y. conducti

��

c1es collide more frequently with obstructions around them. Thu\, the conductors decreases as temperature increases. ductivil)' of o l Cl Much of the classical view of electrical conductivity is valid quantum 'banically. Collisions are still viewed as the origin of electrical resistance.

:e � �

er. we find that in metals. though Ohm's law still holds. the collision larger than can be explained by a theory in which electron\ may m ' e is much llide with all positive ions. The quantum-mechanical explanation i" that near edges of bands) the states allowed an electron in a periodic crys( Ctpt 1. are essentially those of a free particle. The electron is a wave. upon which the \'

posith'e ions have little effect. It is not the regular array of positive ions that up�elS the otherwise free electrons. Rather. deviarions from regularity perturb

the electron wave and thus determine resistance. The most important devia­

U(lnS at ordinary temperatures are vibrations of the positive ions. These increase with temperature. contributing to an increase in resistance. (The facl thai conductivity in metals tends to vary as T to the first power is further evi­

dence that ionic motioll. not mere presence, is key. See Comprehensive Exer­ cise 77.) At temperatures below about 10 K, vibrational motion is so

diminished that the predominant sources of collisions are microscopic lattice

m i perfections. These may be either point defects, due to atomic vacancies or lmpurities. or more widespread disruptions of crystal regularity. In any case. the quantum wells at certain locations are altered, and the electron wave is perturbed. Unlike ionic vibrations. the abundance of lattice imperfections is largely independent of temperature, and so is the resistance lhey cause. In all solids. electrons fiU bands from low to high energy in accord with \hI! exclusion principle. In a conductor, the topmost band is only partially filled: this band is where lhe conduction electrons reside. (Lower full bands do DOl participate in conduction, as we see in Section 10.6.) To a good approxi­

mation. the conduction electrons behave as though moving freely inside a macroscopic well (the entire crystal) with no ionic potential. Their energies are those of an electron

gas (Section 9.6), and the energy of the highest·occupied

siale. measured from the bottom of the band, is the Fermi energy, EF" As

depicted in Figure 10.28, with no external electric field, as many electrons

" I.... .,, �

..... .. Os- to

'l\idl

ickf pllCftYl (k�(� 1nOl'C" da.1J,l(tl Ji ...,nf':Itn •

;S �

\

IIra:.J Jt.Ift'.

""'"

,

,

.... .t

.

.

'

,{.., > 0

0

hlled

....-- Eleclric field

tJt,..-!n( ',tId · 0

/

direclion as in the opposile direction· Theke{ffe in" � in onc Ct of � ukl � mel\ �''1 enta toward Ihose in wbich 10 shi fl clectron mom I' field mal e\le =: P/h) ,;tn the mo menta of all the conduction elecl fl�iI� [he tield. Although I� ()pn ' r tOns only among e higheSI�energ W't' ,h ttt , � Ii« mal IC I_� a net Shill .

e

�

1'.

Y stales. . £f' where the momentum IS 10 the direct'IOn belo" y l h l , t g i ,["Ie' ' of '" fnlm . l.Ie � where I" I IS O�posl le. Classically or qu ' above l:.F'
d

_

�I

EXAMPLE 1 0. 5

(a) Whal is the approx imale atoml.c sep J The dCllSllY ohilver is /0.5 X /OJsilve!;,g/m3. a, . Ex 5 lated in 9 r, le calcu amp Fernli energy in IS 5 5 eV ( I ralion'l (b) Tht' the mOre average themwl energy of classical particles at r o' than 200 rlnles O m .tempera, velociry corresponding Ihis energy, and use it 10 the ine I lUre). Dtlerm estimate Ihe ,

•

--

10

I I I I I , I I

I

t.:()llj.�loll time If elecuons collided with every lalfice ion. (c) Compare the COnd . UCIt\" ' . . u� prrdlCred by thiS model with rhe actual room temperature val ue of 6.3 X 107 n I . m . , Sih'er ha_� one conduction electron per alom.

SOLUT,Q....

lal From the periodic lable. we find lhe alOmic mass of silver fO be 1 07.9 u, The

number of aroms per unit volume is therefore 10.5 X 10' kg/m' ( 1 07.9 u.'alom)( 1.66 X 10-27 kg/u)

� 5 .86 X 1 0211 alom/m3

The volume per atom is the reciprocal of this value, and Ihe cu be roo t 0f t " he ' . 1 ume IS appro:omately the linear separation, \-0

(5.86 X I02B alom/m3)- 11J = 2.57 X 10-10 m ==- 0. 26 nm

/

�' ":" \

1 2(55 x 1_6 X 1O- QJ) 9. 1 1 X 10 ., \ kg

1.39 X lot' Ill/\

..0 lhal

T �

2.57 X 10-10 III 1.39 X l Ob m/s

\d Arplying equntion ( 1 0·7),

u =

( 1 .6 x IO- 19 C)2 (5.86 X 1028 electrons/tn' ) ( 1 .85 X 10 9.1 1 X I 0 3 1 kg

- 3.0 x IO� n-I

•

m- I

This predicted conductivity is less than U100 of the nctual vnlue. All our assumptions are reasonable. except one. We conclude thai an electron pnssc\ a great many !Quice ions before sutTering a collision.

10.6 Conductors, Insulators, and Semiconductors

i

The fundamental difference between an insulator and a conductor is whether the hghest band occupied by electrons is completely or only partially full. We intro­

duced bands in the context of a simple finite well model, but a more realistic

pOtential energy wouldn't change the most important conclusions. Figure 10.29

depicts an isolated atom with a Coulomb-type well, and the result when many

atoms are brought together. The main difference is that the atoms are still effec­ ti\ely far apart for low energy states, so these states do not form a band. Higher slUteS, where the key valence electrons reside, are again broken into bands by the ionic potential.

Figur. 10.29 When atoms form a crystal , low-lying nonvalance levels still belong to each atom, while higher levels become bands.

lSingle atom

o

I+ o

\Crystal

o

t--

.. QQI to ......

...... _ w..

,... 10.30

l.lthiwn !l. pou1io1l1) tilled hand. and beryllium',

fu.l.looc"

I

rt

. ....... for roQdudloll)

.........

--

Bc:rylhum

� romP

Band Occupation and Conductivity 8) the �lmrlc�" 1 ilJl!umcnl. Inhium (7 ""' �) ..hould be a conductor and beryl h um

IL

N

4) an in,ulator "' A� we �n()w. energy band.. comprise N spafial states. Or

'0'-11.\(" IUlh:lllm,. where

i, tbe number or atoms i n the crystal. Thererore II1duJlng 'pm, (';teh Nnd \:3n hold 2N elecITon<;. In a solid of N atoms. mOno­ \alence electron,. It follows that because the 2s \alCnI blhlum wvuld ha\(' ,Iatt in the lithium afom 1 \ onl ) half full. the 2s band in solid lithium is only half full Thu,. 11\ ,hllwn III Figure 10.30. (here is still half a band of unfilled ('fl("� !t'H:h 3\ailahle to 1 /\ 2.� electron... When an external electric field 1\ apphw. the) art free 10 enter the,e 'laieS and thu.. participate i n electrical COn­ dU(tlllfl A bel)lhum alom. on the other hand. has a full 2s level, so the solid, wIIh �.\ \'alc:n,e clectrom, should have a full 2,r band. Assuming a signjficant

F1gure 10.31 (hC'rl.lp fila) l'h.mg(' ful l IUld rmrcy banJ� mtuoi'le p;1rtralh fi11«1 baruI

N

rn('r"g) gap 10 thc ne'(,l-hlgher band, electrons in beryllium should be unable to lreel) ahs(\fh energy (\f to l'hange Siaies in any way i n response to an external lidrJ B('rylhom �hould be an lIl�ulator. BCQlhum " a conductor. It is the last assumption that is incorrect. Given the r.:lll"e ,paCing of 'iome atomic energy Jeveh, it should not be surprising that the comspondmg bands mighl overlap, leaving no gap. As depicled schemati_ f.:all) m Figure 10.31. we find thal lhe 2s and 2p bands overlap significantly in hl-ryllium. Accordingly, the sptem's ground state will have electrons occupy­ mg thl.' lo�er energies of the "J.p band rather than filling the 2s. Thus, beryl_

hum', el�tron� al ..o ha\'e a continuum of unfilled Slates available just " above" them, maling beryllium a conduclor.

Relaling atomic ..tales (II, r. m() to bands in a solid can be rather compli­ (ated. Fortunatel), we can grasp the essentials of conductivity without this \.'omphr.:8uon b) ..imply regarding overlapping bands as one band and consid­ ering how electron.. fiJi the..e combined bands. The highest-energy band that would be completely full at zero remperalure is defined as the valence band. and Ihe bandjust above il is the conduction band. Figure 10.32 i l lustrates the feature.. lhal di\lingUl�h imulators. conductors, and semiconductors. (Plotted -now 1ft' Itr "� , .... 1 PI MIIrrftl,. luxe � ad IWlh,m fi:onro oookt"vIat 'lOlKt.tW ..... OAf,. ., ,....,,. abcI\"C' aba.u :� 1lfII).

alongside the band� is the Fenni-Oirac distribution. As discussed i n Sectjon 9.5. at T = O. the number of electrons per state drops abruptly to zero at EF. whde at low nonzero temperature. it falls off less sleeply.) A material will be a

� 1\.1 1" u..:tor \\ill h,l"" ,.,.:"",.,,-

10.32 a,lnd tilling f,)r .in in ...ui'll\lf. a l;oll\.Iu�·wr. and a 'l'IllKOndu\"IM <11 1('1'\1 I.IIIJ n.ln/ClIl tl'm�r:lt\ll(' ...om(' eledron, In the conduction haml.

1 0 0 -0-0 11 0 0 - - _

-

'

\n\ulalur

'-

. .

C.lnduclor v

T

- -

C\lfl
V.lkn,-(ban�h

SemiCllndliClof

I

0

t 10 { \' 0 11'1'111.11"1

'--.

�\)ndtll'lor if the band occupied by the highest-energy eleclron� is nol full,

\rutking It. by definition. the conduction bnnd. Even lit l.era temperJlure. electron, will be free to ab�orb even a small amount of energy from nil in mi\ band ('lemal field, so there will always be a respomic. A material will be an in-,ula­

lof it" eieclTons happen to fill '>tates just to the top of a band that IS separated from the band above-the empty conduction band-by a large gap. The ncxt­ bigher allowed energy is �o high that ordinary electric fields simply cann(.1t bump electrons up there. The electrons cannot occupy any state.. they were not

alread) occupying. so there is essentially no response to the field. Even at

fJtber high temperatures, the number of electrons energetic enough to occupy 3. \tute In the conduction band is negligible. The term semiconductor is applied to materials that are insulators at zero temperature but in which the gap between the valence and conduction bands ii'. ,mall. By somewhat arbitrary definition, an insulator with a band gap smaller than about 2 eV is considered a semiconductor. With such a small gap, the tem­

�ra1Ure does not have to be very high to produce significant occupation of the conduction band. Semiconducting clements commonly used in electronic devices arc sili­

con and, to a lesser extent, germanium. Both are "covalent" solids with the

�ame lattice geometry as diamond (Figure 10.13). Nevertheless. they are prop­

erly treated via band theory, as though all atoms share all valence electrons. The relevant states are the four tetrahedral states. Four spatial states and two ,pins could form a band of 8N state!:i, which four valence electrons per atom would fill halfway. BUl this band splits in two, just as the bonding and anti­ bonding atomic stales split in the carbon-carbon bond of Figure 10.14. The lower band of 4N state!:i is the full valence band. and the higher is the empty conduction band. The bands are separated by an energy gap of 1 . 1 eV in sili­

con and 0.7 eV in germanium. Diamond shares the same band structure in it!:i n :::

2 states, but its 5.4 eV gap qualifies it as an insulator.

The Conductivity Gap

By applying some statistical physics from Chapter 9, let us investigate more closely the link between temperature and conductivity in insulators and

<.emiconductors.

-\

\

When 1 ,.

(I,

0 10 0 ;\ 0 -

l\'ll,lm:l"r

--y"

1

:.

tl

S"n"�llm\u�I,\r _ _ _ _ ----.1 .

f j �:=::=:::::���. " · "1 ,,

_ eI 57,11 I

'o' . I

.... ... .... oIe1ft;tmns al a ,I\'.:'n tnt'r 1P�

••

t lie -- CllllJY .

T . o and I

�)

.I�·tllm l.tfb otr .t>nIptJ)'.

10

1ID.lt'\:trm, 1'lI.':Up)

.

"""'

£ oIIoy. .. ""'-

:finJ

tht nUlllhcr ('\ClteJ Ill lh(' contiucl1(\n b..md at T higher than 1-", bolwm �lf thlll t'land. " hll'h I'

iF,.lr

.,.

0, \\« il\h,'�r"t'"' lr\\1l\

{k(lIlI,e S(£\n fnlb 00" quickly, (''(tcnding the upper limit tn

infllHty Hltru­ The mtcgration i<,; left 1I<,; nn c\cn:i<,;c.) Givcn thllt till' l!ilp fi'l' CLWlah11"' ami ,emIC(lnduClnr:- i, nn Ihe order of I tV JIlJ Ih;.\1 kil' I' i.lhmll �(\' at f'l"XIUl IClllpcr:llurc, thL' �(\ctor f' E'""i2l"T will he llll!tc 'mall In 1111 rele­ . ,aoI (llnJ.itlon,. We may therelorc u,e the upproxlmatlon In( I + t�) t" lor , «. I fqu,lIioll ( 10·1 0) bccomc�

�'(' nl1 cITlK

lbu', the r:.ltio of electron:- excited at T ':::> 0 to the tOlal number in the valence bifid at T 0 IS

( 1 0- 1 1 )

=

Equtllion ( 1 0- 1 1 ) is a very useful relationship, Band!'. tend to be of order eY, we ,ce that the 10 eV III width (see Ex.ercise 50), and with k Tmonl =o muluplicative factor is roughly of order 1 0 �, However, it IS the ex.ponential factor that dominates. (A realistically varying density of state!'. would change

lo

only the less-important multiplicative factor.) The difference in conductivity

bttween conduclOrs and insulators is one of the mo�t stark contra<;t� in nature and was baffling to early phy ... icists. The energy gap is the key. Given a typical 5 eV band gap for an insulator, the exponential factor at room tempcrallirc

41! Because Nv is "only" of order 1021. we should expect nt1 at all in the conduction band. and thus no current in re!'.pon,e to an electron� tltemal electric field. For a semiconductor, on the other hand, with a gap or 1 eV, the exponential factor i... roughly 10-S-higher by 30 orders of magni­

'II-ould be -10

rude from a mere factor of 5 in

£sap.

EXAMPLE 1 0 _6 Near room temperature. an IOcrease of 1 K decreases the conductivity of copper by only about 0.4%. For a particular semiconductor, it incre(Jses the conductivity by 4.8'1:. Estimate the semil.:onductor's band gap.

I

SOLUTION

ly to an increase in char e conduc!i�iIY sole g ' artie ", increa..ed we need only Lin < d a fa110 band. of ''''"ctIon condu lilt ( 10- 1 I d LOla I i�. ttto..e t" lle

\

If ....e anribute !he !hal

at tM: \....0

lempcOlIUT'e'-

both �ide� Multiplying

Thus. given a

d \hell laking logs b) T,fT/ an

be 300 K, we arrive at and laking TJ \0 4.8% increase

"(� )

I

gives

1 (48 301 K .

'"

!K

Esap

J/K) (301 K)(300 K) 2(1 .38 X lO-B

=lo

Egap = \.09

9 X 10-1 J = 0.68 eV

gennanium. Our band gap estim", ." \0 be thaI for d happens quote The 4.81f q"'to good

ucto r 1 0.7 Semicond

Theory

. uc itors, and inductors we study in lntrod resistors, capac ' of its cin:: u WIthout a means ' of h.-. arn 1ty. I utt d p mIte The simple, I ' m are 0f , ' and magnells . syslems, Ion g nClty s d Istan addres elect ce ublic tory calion-p onic communi I to amplifi . eclr ' R elated IcatlO ' . , e. n n, bl IS lmposSI ficatio . -wouId be , O. teleVI.Slon change ' radi to the t s thing ane. ate of ' the ab'Il'Ity 0f one leleph " " hing; that IS, 'Lgltal computing thal Perf h d I e 0 electronic sWitc eart 'S the h ' If to on, Wh'ICh I amplifier Ihat knows no mlddie another from 0 is merely an swilCh (A , "lion ' onductors so inextricably reIated? vades our civili ' and sernlc mcs ' " are eIectro why ' And est route to the basic devlces . ground") rS are the Simpl " ducto icon thaI sem ental deVice, which we d"". The answer is The most fundam " . hing" " " rna fl SWitc and on ow Ihrough which curr,OI diod" of amphficatl !h, is nexl ,
"

'

r

i:�;

to.7 SmUl:onductor l'lIcory .as

Figur. 10.35 1nennal excitation creat·

�oles tliC identification of effective electron behavior with the motion of po..itive much deeper than the simple notion of " conventional current." In charges is circuits. we often treat negative charges moving opposite an elec� uctory i[luod as a conventional current of positive charges moving in the field Held uic direction. For most purposes. the two are indistinguishable. Nevertheless. it is easily demonstrated that the charge carriers in copper are not positive. The

,iJllpiesl way involves the Hall effect, in which positive-carrier and negative­

behave differently in an external magnetic field. Holes, on the carrier currents behave precisely as positive charges whether the field is electric or , hand other all practical purposes, positive charge carriers do exisl. Let us For magnetic. and see why they are found in semiconductors bUI not in holes investigate ordinary conductors. At zero temperature, the full valence band and empty conduction band leave n semiconductor's electrons no freedom to respond to an eleclric fteld.just like an insulator. When thennal excitation at nonzero T promotes an electron across !he gap, a "hole" at the top of the valence band remains. as depicted in Figure 10.35. Not only is there now an electron free to move in the conduction band. but the empty state it leaves behind also represents a certain freedom in

'1 in the hole is in state n. Now a valence band electron band-the in state n' may ce valen state n, creating a vacancy in state n'. The hole has changed from state intO jump Thus, while valence electron.s are not fr ee---c o nstrained to move /1 10 state n'. only into the one unoccupied state-that Ilnoccllpied state is as free to move in the otherwise full valence band as would be an electron in an empty band. The special nature of holes relies on two further ingredients. First, a lower energy state results when a hole is higher in the band. Figure 10.36 illustrates the useful analogy of an air bubble rising through water. The system's energy decreases whether we view it as the sinking of the water (sea of electrons) or

ing a pair of charge carrien-o Conduction band

___. Condu.:tion electron _____

Energy gap

-��--:

Thmnal excitation

----- Valence hole

Valence band

the valence band. Suppose the promoted electron had occupied state

Figure 10.36 Holes floal.

•

the rising of a bubble (hole). As an electron tends to sink to the lowest energy

Higher energy

state open, a hole tends to "float" as high as possible. In moving among states in a band, a hole's energy varies in the opposite sense from that of an electron. The second ingredient is effective mass. By definition, this is the ratio not the nef force on an electron to its acceleration but of the external force to of the acceleration. This is a more useful quantity than mass, because it is the exter­ nal force that we control via the applied field. For the derivation showing how effective mass relates to electron energy, refer to A Closer Look: Effective Mass at the end of this section. We find that

mdf

( d2E)-'

= r,} , -

dk -

E

( 1 0- J 2)

Although this result is not inlUitively obvious. a special case is reassuring. A free particle has only kinetic energy, so = /i2k!/2m. It is left as an exercise to \'erify that in this zero-force case, the effective mass is just m. To show how equation (10-12) applies in our system of interest. Figure 10.37 plots energy versus wave number for electrons in a crystalline solid. As noted in connection

E�e'.we

"a:::.

Lower energy

,... 10.17 Energ) \C"U� V.;l\o! Dumber for elec­ onc-JlfnC(hIOnal cl),lal. An electron s

IroDS in .

df«u\-e ""'"" dcpenJ} on �hal ,>(ale ,

,

"

II occupies '

""'

f·�e·de.;lron ':Ul"\C.

""�� £.

and '"a

l

,

_

.�

_ _ _ __

l,--.,... . . ' ,,,,._ ,

.

o ,..".. to.lI In I4 lCmu;ondudl>( III I U, tIofh h"le� .md t'1«Jrlln, wn mbule III �'urrenl In the J,rt...III'n lli' an aPJIItc:'\t t'lft:lri.: lidJ

.

,.

Al.t�(1m

111•

.\1 tIouum, "I tlalld�.

" J:rdk " larger. ..,.., - 'I' ''' _ _ m

, ,

'

"

�." AI wp, of tlanlh, ..,�

., .__ .� �

J�f)dl..l ilnd men' are

negaI1H'_

,

�lIh Figure 10.27, In most region.., it is a plot of a free particle in a macro­ �:()pit.: well, wilh £ = II�A.2n.m and lhus an effective mass of ffl. The positive ion, ha\-t little efleel on eleclrOns at lhese energies. Only near the band gaps is

the curvaturr ditferent, At lhe bottom.. of the bands, the effective mass is less

than m (the cunature d1.£JdA.2 is larger). At the tops of the bands, it is negative.S Here the force» mtemai to the crystal are such that !:he electron'S acceleration is

-

-

(lppo�ire the eXlernal force. (See Comprehensive Exercise 78 for an instructive

mechanical analog of negative effective mass,) NO\I, lei us put the ingredienLI; together. In an ordinary conductor like cop­ per, electron' fill )Ialts to around the middle of a band. where they are essen­ tially free panicles of ma.s� III. In

n semiconductor,

however, electrons fill states

1\) the lOp of the valence band, where the effective mass is negative. Because hole energy vane.. in the opposite sense, we might imagine Figure 10.37 flipped tlrc!rl': flC'IJ anJ nln�nl

-�

up..ide down, with opposite curvature. Therefore, at !:he lOp of a band, and only there, a hole has positive effective mass, and thus it moves i n the direction of the e'm'mal field. Figure 10.38 illustrates how conduction electrons and valence hole, respond to a field.

Doping The semiconductors we have considered thus far are referred to as intrinsic semKooduCIOr5. In these pure materials. the number of electrons thermally

'....,. dYt t".... 10.)1 .. mt'lr\._ pt ... ..... 1CI.*' IJ, Thf dftcfne .... 0/.. eko.·, .-0. 111 --" .. oCIIt II 1IdbdIr' Thf t\pI.aMbon gf ,., �-.:"-ioo IS tI\M Jbt p.niculIrty . u. " dill ...

elements-we produce f¥lrinsic semiconductors. These have a large majority

tttoq inttmIJ 'QrttS lw,_ tilt dkl and tilt tloo.-aoa lit MIdI. staltcon"oe) lilt nlrnW r"n.� l{l

greater concentration of charge carriers than the pure intrinsic semiconductor.

dIr 1Otid • • ",hilID (""!urI). thl: t"io:I.uvn "ooid .... -.er)' 1nIIW'�.

e,ciled to the conduction band necessarily equals the number of holes left behmd in the valence band. By doping-addition of small amounts of impurity of eilher conduction electrons or valence holes and, i n either case, a much In the following discussion, we assume the intrinsic semiconductor to be the

1 , 1 !,.,,\){\\I1m :1'".'"

r1l \ ;\ mum�

tn:>e: a covalent lattice of telrovaJent atom'. ,ueb u, ..ih,.:on or

.\0 o·type e:tlrin<;ic semiconductor is produced b)' intcNpef'ing a ..mall Ol 1 n of impurit) ntOI\l\ of a pentavalent element. ..uch tl'i pho..phon.... or : M : Typical doping would have one of every lOS atoms orlhe inlrim.ic male­

�

�m' _

by an impurity. The valence band in lhe intnm,ic ,emiconduClor ,..

.1l ftplal.:ed n ll!lely full, and each impurity atom adds an extra electron for which Ihert.' " '(ll1lp

�_

�

\

roo m. as only four of its electrons can share in the solid's Ictmhcdml

..

tme­

Thi.. doe$; not automatically put the extra electron in the conduction band. t\l.1WI.'\'er. for each impurity atom also has an extra posirive charge. The net effect thaI each impurity atom cremes u new electron state slightly beloll' the C{IIl­ �

�Udi("ln band, known as a donor

state.

This may be thought of �s II MatI! in

extra electron is bound to the extra positive charge at the impurity site, \\hich the ,1.110 the energy needed to free it is the amount by which the donor state ';e�

conduction band. As shown in Figure 10.39, a typical value is only �low the Accordingly, even fairly low temperatures readlly excite eieclrOn'\ from 0.05 eV. to the conduction band. Because the impurity atoms are l>parse and donor states

l..olated, donor states do not fonn a band, so the earlier arguments about hole� s, du not apply. Vacated donor states don't behave as free positive charge carrier but the electrons vacating these states do become free negative charge camel'\ in

the: conduction band. They are far more abundant than any eleclron-hole pairs that might be created by thennal excitation across the en/ire gap (see Exercise

b�). Therefore, in an n-type (negative) semiconductor, the vast majority of

charge carriers are conduction band electrons.

By complementary arguments, a p-type (positive) extrinsic semiconductor

adds impurity atoms of a trivalent element, such as aluminum or gallium.

Although it seems that the missing valence electrons should produce holes in the ,-alence band, the missing positive charges in the atoms again alter the structure,

pushing these unfilled states slightly above the valence band. They are known as acceptor states and are illustrated in Figure 10.40. (Using a system of seven

finite weils, Computational Exercise 81

shows how an impurity can be modeled

as �imply an increase or a decrease in the depth of a well, producing a donor or

I

Figure 10.39 At

T = 0, extra electrons in an n-type semicon­

ductor occupy donor states. At T > D. they easily become conduction electrons.

_

I

.V)'-------.t -O.05 eV

..

Donor ,tates (full)

T0

Conduction electrons

! !

!

field T>O

Electric

441 (hapN' to Bondm8: Molf'f;ure� 4IId

Solids

Figure 10.40 AI T = O. a p-rype semiconductor has empty acceplor slates. which al

T > 0 are filled by electrons from the

valence band. freeing holes.

-r ev)Acceplors!ales(empty) o

0

0

. " ¢

Electric field __

J-O.OSeV Valence holes

T>O

T=O

acceptor state.) Thermal excitation readily bumps electrons from the valence

.

band to acceptor states, leaving freely moving holes.

Again there is always some thermal excitation across the entire valence_ to-conduction gap, creating electron-hole pa irs in both n-type and p-rype semiconductors. Both thus harbor charge carriers of the "wrong" sign (i.e.. valence holes in n-type and conduction electrons in p-type). known as

minority carriers. But far more abundant are the majority carriers: valence holes in p-rype and conduction electrons in n-type. What makes doped semi·

a

conductors unique is the ease with which we c n vary the sign and abundance

.

of the carriers via impurity type and concentration. As we see in the next sec·

tion this is central to modem electronics.

A Closer look

Effective Mass

a particle moving Ihrough a medium. and

Section 6.4 shows Ihal the velocilY of a panicle. called group

\Ie/DeilY. i� related to its mailer wave properties UJand k by

I'

.

paruclc

: 1'

8"'"P

dw dk

=-

As demonstrated in Example 6.3. Ihe dispersion relation rnat gives (J) as a function of

k

accounts for rne effects of a

medium on a particle moving rnrough it. Thus. we may write the acceleration of a particle in any medium. including a cry.�ralljne solid,

d dl

as

0 = - 1'

.

parodc

( )

d dW dl dk

=- -

J2w dk dk2 dl

= --

Effective mass is defined as the ralio of rne exlemsl force to this acceleralion. Ignoring losses due to frictional forces. it is the external force alone that does rne net work on

the force times

the panicle's velocity is (he work per unit time (power). �

dE dr

-

Using E = ItW, we then have F" ,' _ " " " � -.

=

dw dw dk � h -- = dr dk dt

11 -

or, canceling vpMI,cle on both sides,

FUI

=

dk h -;;;

dk

Ill'parllC . ,c dt

1 0 . 8 S emiconductor Devices

how doped semiconductors may be exploited to produce twO \..et U� nOW study important circuit elements. most e of th

The Diode

applications, we wish to allow current to flow only one way. The ill many

most basic is a rectifier. which converts the sinusoidal\y varying (alternating) current supplied to homes into the direct current needed by most eleclronic circuitry. Actually building a device that does this automatically, known as a requires some ingenuity. The primitive solution used evacuated glass tubes housing bulky electrodes and heaters. The advent of doped semiconducbrought a revolution. Asymmetric current flow can be achieved through a

diode, tors

simple physica l asymmetry : charge carriers of different sign in different regions. We produce a diode by joining an n-type semiconductor to a p-type �emiconductor. The area of contact is known as a p-n junction. As shown in figure lOA 1 , if we apply a potential difference with the p-type at higher potential. known as a forward-biased condition, free holes in the p-type region and free electrons in the n-type flow toward the junction. When they meet there, as we discuss below. they annihilate, or recombine. Current flows

continuously as electrons are added at the low-potential side and holes are added (valence electrons removed) at the high potential side. On the other

hand. if the n-type is at the higher potential-a reverse-biased condition­

both holes and electrons move away from the junction. A region devoid of free charge carriers quickly fonns, and current stops almost instantly. Energy-level diagrams help clarify the behaviors . Figure IOA2(a) shows an

unbiased condition. If we simply join a p-type and an n-type, without an exter­ nal potential difference, electrons "high" in the n-type conduction band cross to the p-type side, raising all electron energies there by repulsion. while corre­

spondingly lowering them in the n-type. Few electrons need actually cross

before an equilibrium is reached in which the highest occupied levels-the

figure 10.41 The motion of the charge carriers in forward and reverse biased p-n junctions.

Hole" added

Electron-hule r&ombin"ll\)O

If e e ® e ,'� 0- 0 e '-0

...

..

'

.

'

..

EI':"lron� JJ!.kd

.....

...

n-type

p-type

Electron noYo

«.tl

"---'lN1fIr-I 1 \�-­ +

Forward bia)'

Free !.:harge depleted

Holes removed

.

'

+

e e e

,.......,

p-type

<:) <:) <:)

Electron� removed

n-type

Elet.:tron now (momenlJJ) \'-

Rever.;e bias

...

, ,::'

.

bias, the highe"t-energy , :1 th noe�lemal e .... J()(k, (8) ..

ec lto ,l/1J� ln 8)(hi 1}lX' ( efl) aod n'ly� (nghl) regIons. (b) An e.'( l rnall II.!. l Rgon 10.42 8 e _

.

wne l('\od In eg In the bands. (e) A forward bias eve lIT II tht Pe� [he dogl n 1M:rbal S I_t }' 4L� e( • �tnt .. IIpf'licd lT ' lll. and cUJ'K'nt n�

IA'

-'f

�h " �

e

Ef

(I) l'nbla.o;cd

�

ce'

~ +

".:

Conduction electro

ombin",

eelroo E'n ow

'"

o{j =�,:�n -

'

Vah!'n�hole,

...

(c) For. ard bias

(b) Reverse bIllS

side_� are equal. Because electrons fill states to

apprOX_ . on both PennI. energl.esand . to the acceptor levels in the l n-type the in levels r P'type e doDo . l u . , irnale y ,. resulllOg m a charactenstlc dogleg equalize. thai ' e i rll in ,. • ene'b • these: are .U � . 'e " h cr. T IC ot U to ... he Immediate ' . -I,nee bands from onc SIde ,

'

.

'

'

•

"g.IOn .... conductIOn .IlJld " . Carriers the deplelion. zone, 'has no free charge ill · 100 . . It IS 01. the JunctIOn. c"led discussed In ExerCise 68. is (the tunnel .. . diode. an excep neI thro"gh Yo I·de 10 iUn electrons 10 the n-type no longer have reason to cross r (IOn . . ), and conduction lor . I . S·Iml. ,arly, valen band h'll conducuon (he) would ha\'e 10 "climb" the ce hOles . e band valenc m the n_type lower the to ive" "d to refuse type . he t � In . ' · r F'Igure 10. Slares lor Simp IICily, 42(b) sho Omirring donor and acceptor Ws potentia . I on the p-I!pe s �.dethat a reverse bias-applying a lo�e� exaggerate s p-type energies still higher the dogleg in the energy bands, shlftrng the reIat·'Ve ' h potentm ' hIg , I IS ' I energy for a ne to the n-type. (Remember: Low potenlla gat,.ve . . would very much ers Il e to cross electron.) Note that minorit), ��rn the junc_ . lion in n reverse-biased condition. ConductIOn electrons In the p-type \VO uId . readily .slide downward to the n-type cand uctlon band, while valence hoi es I.n the n-Iype would float upward to the p-Iype valence band ' . But '." normal diode operation, minority camers are scarce, so the reverse CUrr ent IS negligible. Forward-biasing the diode, as shown in Figure 10.42(c), shifts the n-type conduction band upward relative to thai of the p-type. Conduction electrOns , · near I read·, he Juncllon I use, across as SOon as the I y wander, or d·ff UneVenness .1\ eI·Immate ' the same physical · '· same, and once m d. Hoies d0 me region, the two recomb·me; h I at IS. ' ' I e conductlOn eIec[rons Jump · h down to fill the v aIenCe . of course, conserved, and holes. EnelEY IS. the energy lost In the ju mp across the gap can go to heal (some d·Iooes ·mcorporate Iarge heat-sinks) or to Iight In < ' .. . . how h . laC\, thIS tranSitIOn IS I e LED (I·IghI-emittmg ' diode) produces its ligh t, •

.

�

10 M Stoml�ond","tor Dewices

41'

nil "'"c �e (hBI the phOlon ener�y equal .. Ihe gap ene�y. We aho ..ee Ihal the

amount by which the external potentmJ must ..hlft the bands energy is the out. Thus, a diode of I eV band gap wQuld need a forward blm. of ,.p ('\,n them . 0 ng on, before turni \'

REA L.W O

R L D EXAMPLE T H E PHOTOD IODE

of c:
k�'

dt(tric:d energy of conduction electrons and valence holes be<:ome� light energy 01 pholon�. What if we deliberately shine light 111 II diode?

.pplying the Physics Con<;ider an unbiased diode whose band gilp is 1 . 0 eV. Suppose that photon� wh(l�e. exceeds the band gap strike all regions-p-I),PC, n-type. and the Junc­ energy slightly the behavior of electrons and holes. Will a potential diO"erence Describe tion (a)

de\'elop7 (b) At approximately what wavelength would this device-bccome effective?

.urtON tal Ju�t as an mom can emit or absorb a photon of the proper wavelength, with an electron Jumping down or up in the process, the transition in which electrons and holes recombine and give off n photon can be driven the other way. In

whatever region it may strike, a photon Clln create an electron in the conduction band and a hole in the valence, as shown in Figure 10.43. A conduction elec­ tTOn in the p-type, where it is a minority carrier. would be delighted to "slide"

down to the lower conduction band in the n-type, as would conduction elec­ trons created in the depletion zone around the junction. Similarly, holes created in either the depletion zone or n-type valence would happily float up to the p-type valence. If the light were switched off, the diode would, of course,

return to its unbiased state. But as long as it shines, the light drives negative charges 10 the n-type and positive ones to the p-type, thus establishing a potential difference. The device becomes a banery!

(b) The photon energy must be at least the gap energy. A �

he

Egap

-

�

=,;-,-:,-= 1 eV 1240eV ' nm

1240 nm

In reference to Figure 10.41, one might ask why, if pairs of charge carriers "dis­ " appear at the junction in a forward-bias case, they cannot appear in the reverse-bias

Figure 10.43 Photons cre;ue electron­

hole pairs in an unbiased diode. Con­ duction electron� slide down to the

n+type, and valence holes float up to the p side.

case and thereby also allow current to flow. As we see in this example. an energy

inpul would

be needed for each pair created-in fact. quite a bit of energy, far

greater than kilT. What we have here is the essence of the photovoltaic cell, which turns light into electricity. Naturally, many refinements have been applied. A com­ mon one adds a layer of inlrinsic semiconductor between the n- and p-type regions. widening the depiction lone where electron-hole pairs are sorted quickly and con­ tributing to improved efficiency in high-frequency applications. Another. U'.ed in

solar cells. stacks semil-onductors of various band gaps, each "picking off" the lI'a\'elengths it mosl effl"ctively absorbs and producing a more efficient O\:eraU con­ version of natural light

n-type

.

deserve.. mention: rhe diode JSII"'� 0n 0f Ihe diode . �r. (1'L in Section 9.8. ) As In the ".... di<.("u., sed L�� , , . nIV'(1Iuon ...... vr I \t'f < a . o( U;" tta.'i recombmillion I S the SOurce of ligtu . �. '"e hole o(,/f(trO In a in p u SHcn I� energy . holes at the JunellOn r prOdu IOn� of electrom and ra( ·t'nl on," . · , �....e C e Jl(llk ...-� tile c.: er la ,.:,· 00. ,,<; . Iypes. Dlher .. a re c In so • nan t ulallon ,mer _ aVIt )' mJ UII l . _ _ � IS pop _ . w3\.e/ength; In a diode laser, it Can de\;red rhe y if pl am III fo A«\kd llel 10 each other. (Sem IC rhe end\ of rhe cry<;,al para ' On""'d �lmpJJ b) l./ea\ mg duccomplete lJufficienlly renec lion.) gi\'t�� x d e In e B ecaUSe _ 'n re(radJ\; 10('10 h I� " than Jigh,. ." m bmatlon energy. In forms olher sI Ico . - \e� up Ih retO ft) n � It)!;1 0111.:0 I S ... . A marenal prodUCing photons wil er .. h OICe for diode la U fl\lC.)r ch£'. compound lanice tri lI('n ' arsenide II formed of I!> ga/lI um hIgher etlicu:nq V valent symmetric about col U 'ion� ("(combjn O th mn I llenl a[om�. nl.J\:i of the i.lnd IY � t . been used. Diode V and II-VI) have also lasers t ' . fI · wble (borh IU lC. l yp pen . I j .c,· c ---elecrrical power 10 hghl power-grea ter th lve e I ell ies en (aUy h. � ' . code \Cllllners. I ac;er POlnlers. 'en and :1 n 0 -, bar .ound m CD plal and are f 51 of olher iJpplkOJlionc;.

-

ppli4;80 r OJ AIM"he

.

-

-

_

_

-

--

be

-

'

_

.

.

�

.

_

_

_

_

�

.

50�·

The Transistor

. . e up a whole ne\\ world. the transistor is actual ly a fairl Alrhough openm� Y ' Iscu�sed , If we Jam d d' alrea y , hlH'e we . in I 'de a s 0 succes ' pIe e"len�/On f slOn . ' refi mements semlconduCIOf'" wI!h certaJn c tt r imic ' ,. g . III a I lem a We haVe fhJ'e(o a or npn. Because the two are Com plemen , (raml'iIOr. II can be either a pnp ary , npn. we re\lricl our attenllon to Ihe ' . npn tranSISlor IS calJed the ThC P-,).pe center region of an base and the collector. (\Vhl/e and emiUer Seemmgly inre are Ihe n-type regl'am rchange , . . fWO are biased difteren(ly and are of dlfferent size and . c laner .:lble. rh� dOPl.flg ' Ic\e/, IJiJonng Ihem 10 the res ctl e roI es.) In operation, as depicle d in , Figure 1O.M. rhe emitler·base dIOde IS forward blas.ed. so cond Uction elee. rron� now from the emiuer into the base. They are dlsc Uraged from ree om_ , brninu wilh holes fhere by making me base of small SIze and of ver y r �n , • , ' r O ' doping. The beaulY of the fransrslor IS h at lh ese cO,nd uct ron elec trons . are minorit), earners ill the base and the col/eclor-base dIode rs reverse biased. . , electrons do recombin Comequenr/y while a small tract/on of Ihe e and . pass oul me base, the rest are qUIte happy o sh e down to [he collector's conduc. . lion band. Here is I,he essence of amplificatron: The fraction that flow s OUt th col/eclor (evenluaJ/y refuming (0 ,the emiller through the utput circ uit) i pml'ortional to "lid greater t WI/ that ,WhiCh now JOO s OUt the ba\c (returning through rhe mpul c rc url ), FOr herr J 948 diSCOvery � of the tra . n. "htor. John Bardeen. Walfer Branam, and Wrlham Shockley Were awar ded rhe 1956 Nobel Prize in physics. Figure 10.45 shows rhe rudiments of a transistor ampli fier. The inpu t per· -� haps II microphone, lIl ,uuuuce s mmule voI tage vanatlons in the emitler_b e hi(I.�, with proportional variations in the rate at which electrons flow into ( e ba,e from the emiller, A liny fraction of this current leaves the base ' and cIr u. Jates as a varying curreO( in the input (emitter-base) � circuit; the overw heJ m lng rem.1.inder constitutes a proportional varying curre nt in the outp ut . . (emI tter_ collecror) CircUit.

.

.

� .�

.

•

�

. t),plco/�y

.

.

.

�"n�s �

,

51ft).

•

•

?

�

�

!

.

.

:

•

�

t(l'" Supen:onductrvif)

FiSJu",

.tS3

10.45 Th� d�m�nl'" or • lran­

".,hlf umph lit'r

11.'''-'

·

r

···········

-

\ ,

\, \'

:\, ...

. .... .............. .....

........

.

.

.....".

...........

.

...,.,.,.: ;>'¥ .

•..• ..

. .. .... . ..... .. .... .. . . ,..' .... '

.

.

.

Output

" ""

Ullllk, imn 1�_ 1I

re':�lnlbirl.llll>n �lIrTent tltlWI1l� Ilut the t.'I'�. ;Ill
::::::: ,'"

•.

. •

th� mu�h lar!!er f.. nUll in� uur the �'\lllt'dN

Figure 10.46 A MQSFET i.. well suited to i ntegrated circuit .., The g;lIe,

The bipolar transistor discllssed above is just one of many semiconductor amplification devices. Another is the field effect transistor (FET), one type of which is the MOSFET, depicted in Figure 10.46 and discus!.ed in Exercio,;e 71

,eparated from the scmi..:onduclOr by an rn�ul;:urng layer, (;ontrob �Iectron now from the ,ource 10 the doolin,

ib electrical properties and ease of fabrication make it one of the mOSt common ,witching devices in integrated circuits-the heart of today's ubiquitou"

Dr.!in

microchip electronics. A silicon chip can be modified so that doping is arbitrar­

(-wJ[c�lUnI

"

ily n-type or p-type in successive minuscule regions, creating transistors smaller

than I f.lm2 in area. By appropriate doping level in single n-type or p-type

regions. either resistors or highly conducting ;'wires" can be incorporated­

hence

,

Gate (-ba,e)

I p

S()urc�' I emitter)

"

- l lJ.m

Ihe name integrated circuil. By the early 2000s, circuits of a billion

rransislOrs could be "written" into a single chip. Figure

10.47 exhibits a typical

IJ'Jnsistor density. Densities continue to multiply every few years.

1 0. 9 Superconductivity

Some materials at low temperature lose all electrical resistance. Figure 10.48 shows resistivity versus temperature for tin and copper. While copper retaino,;

resistance down 10 the lowest measurable temperatures. tin's resistivity plum­ meb to zero at ils

critical temperature Te' a characteristic of the material. In superconductor. About 40% of the

such a stale. the material is known as a

natural elemental metals are known 10 become superconductors at low temper­ ature. In this section. we discuss features common to superconducting materi­

als and the mechanism by which superconductivity occurs. With no resistance. an electric current established in a superconducting material should persist indefinitely. without an applied voltage. Indeed, many e�periments have been done testing this very prediction. one indicating a min­ imum time for significant current decay of 100,000 years!

Figure 1 0.47 A modern computer processor. The 'mall chip in the center. only 143 mm2 in area. coma;ns about 300 million transislOr�.

Fig,". to."9 When II drops �Io� Ib cnllcal lemper.uure. a �upen:(lndUdOr c'l:peh magnelic field line!>.

,.,..,. 10.... Copprr .. h u� tw rh', (nf:.1 tnl tm«, .hrlc lin be('omcs 01 uptn:ondtKlor

i . 2 .. -

f

:

o

In

,.

8"" . O. T <

T�

,"

Copptr

0'

o

o r�

--r-- f

III

I.�

1(J

2.�

frmrc-l'lIluR' ( 11: )

, 8�u fI: O. T < Tc

, ,

, •

.W

Surface ./

� (l]l

, ,

Another feature of superconductors is that they exclude magnetic field If a material is below irs enrical temperature

lmes. as depicted in Figure 10.49.

h.e.. In a superconductlng stare) in a region of zero magnetic field. and an external magnerlc field is m£rOduced. then the field lines will curve around the

object. ne..'er passing through. We mighl expeci Ihis on (he basis of Faraday's law. Any change 10 magnetic nux wough a cross section would induce an emf,

and if lhere is no resistance. an arbItrarily large surface current i s possible, able

10 produce an Internal field that cancels the external. However, it can be shown that even perfect conductIvIty would allow an external field to creep slOWly from the surface info the material. Moreover, a complete lack of resistance is In�ufficient ro explain crirical temperature i.�

another behavior. If a superconducting material above Us initio/f)' permeated by a magnetic field and {/Iell cooled,

the magnetic field lines will be expelled as it drop� below it�

en tical

tempera­

lure. The resuh is rhe same-a superconductmg \ltate thai excludes magnetic field line.�. Because the external field is not varied, we would not expect

Cur­

rents to be induced that mighl cancel that field, but surface currents do arise, and Ihey persist indefinitely.

To e�plain magnetic effects, we must allnbure to superconducrors not only perfect conductivity but also perfect diamagneli�m. Thi\ being the case, a superconductor always repels a magnet. A vivid example is �hown in Figure

10.50, In which a smaH magnet hovers above a disk of superconducting mate­ rial. LiqUId nitrogen

(77 K) surroundmg the dIsk keeps It below irs critical

temperature, so that the disk pushes away on the magnet and il!; field. ExcJu­ \ion of magnetic field lines by a superconductor is known a� the

Meissner

erred. For all superconducung malerials, there is a limit on the litrength of the external field lIley can exclude. If lIle field is too strong, liuperconductivity is destroyed, and lIle field penetrates the material in the normal way. The field .\rrengrh a( which chis occurs is known as the crificaJ fieJd Be II iii temperarure dependent. being highest at T = 0 and dropping to zero at lhe crilh tl lemperature. It i.� narural (0 wonder what uses we make of superconductoT\. Thi� inevitably

y 10.9 Superconductivit

magnet (above). a To exclude the field lines of the permanent . . (below) becomes an opposmg magnet. Note the vapor dl:.k ng ucti ['COnd needed 10 keep the ceramic YBCQ nitrogen, below its critical id liqu the

Figure 10.50

;: .

1�(l1peI1lIUre.

leads to the reason why so much emphasis is placed on their magnetic properties. According to Ampere's law, electrical current is always accompanied by mag­ nelic fields, and if there is a limit on the magnetic field that a material can bear, then there will be a limit on its ability to carry current. Let us now study two types of superconductors and their important magnetic properties.

Type-I and Type-II Superconductors The discovery of superconductivity won the 1 913 Nobel Prize for Heike

Kamerlingh-Onnes. The first elements found to exhibit the effect are now known as Type-l superconductors. Figure 10.51 (a) shows how the critical field B varies with temperature. Type-l superconductors are characterized by a , sharp transition as the external magnetic field increases, from field-excluding superconducting state to field-penetrating normal (nonsuperconducting) state. Figure IO.5I(b) illustrates how the net and induced fields vary with the exter­ nal fteld at a given T < Tc' Up to Bc' the net field inside is zero. The external field is perfectly canceled by the field due to currents induced at the sample's surface. Beyond Bc' the material is normal. and the external field passes through freely. Superconducting elements tend to be Type L Their critical fields and temperatures are relatively low. typically 0.01-0.1 T and 1-9 K, so their usefulness is limited. Type-II superconductor:-, are distinguished by their tendency to sponta­ neously form microscopic vortices. Depicted in Figure 10.52, vortices are superconducting regions of circulating current surrounding "tubes," known as

455

Figure 10.51 A Typc::.( ...upercondw.:tor. (a) The critical l"ield

Be decreases as

T _ T,.. (bl When It i\ ..uperconducling. the induced field perfectly opposes

the e\temal-the net field in...ide i., O.

./

Super· f.cconduc =: iS�"" ,,"Normal ­ ing

Norm..tl �tate

;::

;

"

Sq... . . .....

B,

-

T,

OK

TemperalUTe Ib)

I')

normal cores, in \\hich the material is in the nonsuperconducting state and through which field Jines pa�s. As the external field increases. vortices become rnore dense, as in Figure 10.53. allowing more field lines to pass through. Figure 1051{a) �how" that the behavior of vortices is tied to the existence of two critical fields. At external field strengths below 8�' 1 ' a Type U behaves jUst like a 'TYpe I. Above BC ] ' the vortex state begins. Field lines begin to pass

through normal cores while the surrounding regions remain superconducting. Increasing flux penetration in the vortex slate is demonstrated clearly i n Figure IO.5..t(b). By the time Bd is reached. alJ of the external field lines pass through . so that Binduced = 0 and Bnet = BelltCmal' and the entire sample is nonnal . Metallic compounds and alloys tend to be Type·1I superconductors. They typically have critical temperatures more than twice as high. in the 20 K range. and critical fields 2-3

orders of II/agllitllde

higher than Type I. giving

them much wider practical application.

Figure

10.52 Magndic field hne..

1'iI\.. ing IhnlUgh ,(mICe, in a type·"

\upcn:ondul·lor.

Figure 10.53 Vortices become more dense as field !.t:rength jncrea.�s.

(Both repre�ent Bd < Bemma.] < Be2· See

Figure 10.54.)

�onnal core\

o

c res

Weaker field

Stronger field

10.9

\I�ld penetrate through an i.ncreasing density of vortices. until at Bc2 1he malerial i

Fi

uft!

10.54 In a Type-II superconductor. an external field increasing from Bd

normal.

� '".I'''''-:' r'V ondllctin� ( 511

Vortelt state

r.

, ,

"

,' : I

,

..

,

I

Melt ---+- Normal "

Temperature

(.)

(b)

BCS Theory Although there are obvious differences between Type·l and Type.1J supercon­ ductors. the mechanism by which superconductivity occurs is the same for both: the ordered motion of the electrons in pairs, known as Cooper pairs. The motions of all pairs are so strongly correlated and ordered that they respond to an external eleclric field as one; thus, they are unable to scatter individually in ways that ordinarily result in electrical resistance. Superconductivity via Cooper pairs depends imimalely on interactions between the conduction electrons and the lattice of positive ions, and there are tWO important clues. First. a given material's critical temperature varies with the average mass of the positive ions according to Tc a:; M� 112. This is known as the isotope effect. Atoms of the same element but differing numbers of neu­ trons in the nucleus. and thus differing masses, are referred to as different iso­ topes. That the atomic mass affects Tc certainly suggests that the positive ions are somehow involved in the process. Second, elements that superconduct at low temperatures usually make poor conductors at room temperature. and vice versa. This suggests that strong interacrions/coUisions between electrons and the lattice--the bane of conductivity at room temperature-are a requiremem for superconductivity at low temperature. The logical next question is how the electron-lattice interaction can form a Cooper pair. At low temperatures. two electrons in a solid can experience a net attraction, mediated by a phonon. (This quantum of vibration is in£rOduced in Section 9.10.) As shown in Figure 10.55, an electron moving through a solid causes a local distortion of the lattice, attracting the positive ions and thus creat­ ing a region of unusually high positive charge concentration. This distortion pulls on the electron as the two go their separate ways through the lattice. We say that the electron has emitted a phonon. When the phonon/distortion approaches another electron. the po...itive charge concentration pulls on it-the second elec­ tron absorbs the phonon The net effect is that the electrons experience equal

M

SU�ty Q7

451

ChoP'"'

attraction between elec_mediated propased by one eIeetran Fig rmation cau defo conveys a hat ee on-t alu -a phon trOns. A I . of \'ibration toward cle pulled J ' are P! h . e electrons gates as a ...]ectron. Th anDther 10 ce for each other-

10

A Phonon U"'- 10.55

. .. . .. .. .. .. ;::: ·

t

::: ·

·

.. ..

..

..

·

..

..

:::: ..

.

'

•

Initial momenta

pairs ha\'e

All Cooper Figure 10.56 cur· m: zero when entu the ...ame mom ero otherwise. rent i� l�ro; nonz

.. ..1' !_· "..

..

.

r-- -- ...

..

.. ..

..

.. ..

; - - .. .. .. ..

..

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.. .. .. .. .. ... ...

.

.

.

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.

. . . . . .. . Phonon emilled

..

..

.. .. .. ..... .. .. -r- " .. .. .. .. .. .. .. .. .. .. .. ..

..

.

-·

F

..

..

Electron current

..

..

..

..

..

r

..

�- - -- " .

..

.. ..

..

propagation

.

•

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..

.. .. .. ..

..

..

..

..

.. ..

..

..

- t. - - - -·

l

•

·

Final

Phonon absorbed

Phonon

•

momenta

particles sharing an er, just as would ard one anoth tow · anges attract·IOn IS . , th·IS ph onon-medlated . Olomenrum ch er cond.Iuons prop . ' the r . Unde cmg a bond. Even e. S produ 0, forc the ion, .... ... ! h u«ctive coulomb repulS . Iy ectron separallon is fa me electrOnS' . I e e th than d ir an V oer , strOno . ally I 0-3 e . small, ryplc . a · mueh iarger t han atonuc spacing In energy IS ·IS IS bindino th o Because f..L m . 1 of r orde r. . large, on the overlap one anothe Cooper pall's many not U5t possible binding great ves vity invol lattice, a superconducti of n anatio The expl also correlations In the system as a electrons, but of pairs een rer�, Bardeen, Cooper, and . lscove . attraction betw theory, after Its BCS as n their work of 1 5 yeam _ whole. II is know Nobel Pnze for ed the 1972 . receiv . who common to Type-I and iOrs r Schrieffe . y all behaV sses essenl1a][ mpa M · enco elssner effect , the eXIS . . . he lier. The theory J ct conducllv l1y, t pelle rs: ucto rcond few at a of supe look its more Let us Type-I1 re, and so on. al temperatu cr itic a of tence s. important point all state of lowest energy Occurs wh n theory, an over BCS to According near Ep fonn opposite-SPin electrons, those conduction tic energe . momentum . As illustrated In the most . havmg . the same net . . all poors . , With palfS net zero per has momentu Coo m., current. each poor the stale of zero . nonzero The is pairs Figure 10.56, in resu all lI to common the momentum flows, nt curre ir a pa experienee when Although members of . a very ordered stale. . any case IS all pairs have � of mass 1Il the centers of 'e momenlum. relalil their · in h · changes T engage 10 t e attraction at all . electrons must . momentum. 10 . clsely the same . reIalive momentum. There is no such freechange theIr to m freedo have some se electron5 that filled up to Ep so tho states completely . on ctr ele the in e dom rth . y states slightly abov EF· Neve el SS must aClually occup ' O fonn Cooper poors nly elee t that is lower in energy. stale l overal an in the attraction resulis state. Raising "deeper" eleetron Ihe supereondUeling in e pal tiei par E, r nea

�

No current

.. .. .. ..

.

•

..

;

,

.

.

t

� -'- - -'- !. . 0

.

,.-

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J .. O ,

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.

)

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: ���

. . 10.9 SupercondtKbY1ty

EF would increase the energy more than the weak attraction � .;tales near

coll1d lower it Although lanice vibrations (i.e., phonons) are actually an integral part of tht superconducting state, electrical resistance is zero because ralldom vibrations and lattice imperfections effectively cannot scatter electrons. To disrupt the collective, ordered motion would require breaking apart Cooper pairs. BCS ltteory shows that at zero temperature. the binding energy of a pair is 3.5.1.:sTc' and. as noted above. a typical value is 10-3 eY. Small lhough this may be, it is slill larger than the thermal energy available in a collision with the lattice at low temperature, so such collisions essentially cannot occur! Thus. the lowest­ state of current flow persists. The minimum jump from the energy ordered state to an excited normal state is somewhat similar to the ducling ;;upercon nds in a semiconductor, and it is also referred to as a gap. b jump betw�e� � The distinctIOn between Type-I and Type-II superconductors lies in the material's mean free path, the average distance its electrons travel before undergoing a collisiol1. A material in which the mean free path is large will have Cooper pairs interlocked in a coherent, orderly way over a vast region. Thus. they lend to assume states homogeneous throughout the sample. com­ pletely excluding magnetic fields. Type-I superconductors fit this description. Type-Il superconductors, on the other hand. have a relatively short mean free palh, so order can be much shorter range, allowing for a microscopic network of flux-carrying normal cores within an otherwise superconducting region. Another interesting conclusion of BCS theory is nux quantization. The magnetic flux passing through any region encircled by a supercurrent, whether in a microscopic vortex or a macroscopic ring, must be an integer limes 1T file. High-T, Superconductors

The year 1986 ushered in an extended field day for experimental physicists

and a challenge for theoretical physicists. A new variety of superconductor

was discovered: the high.Tc superconductor. The materials were compounds of copper and oxygen with other elements, and they qualified as ceramics. The discovery was astonishing because ceramics are excellent at ordi­ nary temperatures. The first one found was L�_xBaxCu04' for which Bednorz and Muller won the 1987 Nobel Prize in physics. The x means that the com­ pound is basically L�Cu04 but with barium atoms replacing lanthanum atoms at random locations. The most startling thing was that the critical tem­ perature, 30 K. was more than 5 K higher than that of any metallic material discovered in seven prior decades of research. A frenzied hunt ensued. A mile­ stone was reached in 1987 when YBCO (YB�Cu307) was found to have a crit­ ical temperature of 92 K, exceeding the 77 K boiling point of liquid nitrogen. Previous work on superconductivity required the materials to be cooled with liquid helium, a preciou\ commodity. The new material could be cooled using relatively plentiful liquid mlrogen. The early 1990s brought the record to around 135 K. followed by a stall of about a dozen years, until in 2006. a copper-oxide compound was reported with a critical temperature near l50 K. These high-Tc superconductors. generally referred to as cuprates, possess the Type-II traits of fmming vortices and having two critical field values.

insulators

.-

---

oUt

. Howe\-'er. (he me(;nani"m of wperconduclivily diffe. .. i n significant ways fro m B the ,Iandard es bcnu\ ior of the lYpe I and lYpe II. For one Ihing. high r -

I

,upen:onductor. do nOI. a, u rule. ('.'(h,bll the isotope effecl. Electron pairs SIlI pia} a role. but the Interaction with the bulk m;lterial is different. The fact tnat

all high-Tt ,upen:ondut:tor. ha\e alternaling planes of copper and oXygen

alorm na\ IN 10 an under,tandmg of the basic eleclTon/Jauice i nteraction. but a comprehen�i\'e theoc), ha, yet to emerge. OWing

(0 the great com:enience of using liquid nitrogen as a coolant

�

high-Tt superconductor. ha\c spawned an industry i n superconducling wire and cables. and numerous other uses have been found. However. much of the early fe\t!r has subsided. as promise has dimmed that these Iypes of male. rials will produce a room-temperalUre superconductor-the Holy Grail of

..uperconducti\'ity.

Uses

One current use of superconducting materials i� in superconducting magnet.s

for panicle accelerators. Because current flows without resistance. liule heat is genermed and ImJe power wasted. Strong magnetic fields can be Sustained

while incurring onl), the expense of maintaining the coolant. necessary Simply

10 pre�'enl the material from rising above its critical temperature. Obviously. with a room-temperature superconductor, even this expense could be elimi. nated. Other proposed uses of room-temperature superconductors include computers. .... here the ever-decreasing size of microchip circuitry has led

to

an increasing heal diSSipation problem: power lines. where resistance heal­ ing wastes power and mandates high4\,oltagenow-culTeot transrnis.'\ion and unwieldy [ramfonners; and rai/ transponation, i n which friction can be practl· cally eliminated by e�ploiting Ihe repulsion between magnets and super­ conductors to suspend the vehicle without physical contact. If workable. \ery-high-Iemperature �upefconductors can be found. ulies w i l l multiply enOf­ mou�I)'. bringing a revolution in technology.

1 0. 1 0 Fullerenes Nothing could be more fining to discuss now than fullerenes. perhaps the Illost acti\·ely studied novel materials in science today. To enumerate their characteristics is to recount e.�sentially every topic i n this chapter-and a host of others.

1985 With the discovery of the buckyball.6 now desig­ nated C60. a hoI/a", spherical molecule about I n m i n diameter and composed The story began in

entirely of carbon atoms arranged in an interesting pattern of pentagons and helagons. As shown in Figure

10.57. each carbon atom bond\ to three others.

In Example 10.1. we noted that a common hybridization in carbon i s st? which produces a planar. three-lobed pattern and allows u bond\ i n three directions. � DmJn �""'" oIIId/MUttrlfl dt\dcp�d 'IUD' tho: oll.;en�uOl'l thdt thco.c lll1Ktl1R'> �\Cm·

tiled drnnc-�lured l;Iu,rdlllp popUI.ullN II) BucJo..IIlIll>kr ··Bud.)- Fulfer

Graphite fonns i n flal sheets of regular hexagon') in thi\ pattern. Although bondmg between sheets is weak-allowing

them to slide ca...lly along each very strong. due

olher and making graphite slippery-the sheets themselve\ an;

10.10 Fullcrenc\

461

Figure 10.57 The buckyball-a single molecule of 60 carbon atoms.

sp2

bond. FulJerenes share this trait. Curl. KrOlO. and strength of the to the great Nobel Prize in chemistry for their discovery of the 1996 awarded smalley were fullerenes. ." •yballs and similar ellipsoidal structures. such as C70 and C240. are

Bud

studied in many fields. Although practical applications have been slow to appear. some of the proposed uses of buckyballs are as molecular ball bearings efficient microscopic strainers. Molecular solids of buckyballs have been

and doped with impurities to produce conductors and even superconductors. though.

their critical temperatures are much lower than in the high-Tc cuprales. (A famo s claim of 1 17 K became infamous when it was later revealed to be a

as yet,

u

case of gross scientific misconducl.) Buckyballs with various atoms. ions. or small molecules trapped inside-so-called endohedral fullerenes-are studied for their intereMing chemical and electronic properties. They have also been capablc of <;lightly accelerating radioactive decay ('lee Chapter I I .

s�own

Progress and Application�). T�e carbon nanotube. CNT. has emerged as even more versatile than the

buckybaJ! and is seemingly able to fill just about every role we might imagine

in modem technology. The prototype CNT is a graphite sheet (more properly as graphenc) rolled Into a cylinder and topped with a fullerene half

known

sphere. The ten<;ile strength of a CNT-the pressure at which pulling its ends

would fracture it-is typically

1 00 limes higher than steel. while its density is

much less. Accordingly. CNTs are expected to take on an important role as a high-strength structural element. Among the first uses have been sports equip­ ment and protective clothing made of CNT yam. A delightful characteri ..tic of CNTs is that the graphite sheeLS can form cylinders in different .....ay ... .....hich determine the conducting behavior of the

material. Figure IO.5� ..how .. a CNT in which the geometry changes from the so-called armchair config:uration at the top. where adjacent hexagons line up vertically. to the ligLag �onfiguration at the bottom. where they line up in

Figure 1 0.58 A carbon nanOlube in which the configuration changes from armchair at top to zig7.3g at the bonom. The top is metallic and the bottom is \emiconducling.

70 _ ___ .. SoIOI' ... OAO7_

« '''0

�� :

O be,ween ,he o "n l, mtigUfJl lO « An, O fi g ' A rmchair c n IOn" ,II< '1hpcj
�,n, on ,II<

.

6.3 d,' ,c , "« ,, ,he ,u Secllo n ere" ,n, n· a' gr< 01 an: oa'ure ' 01 '"1St of C tary ub< men nano' Jila e Th ' � _ (" ,",,0<" .. <111 "" on make ,hem ef) "I held henlleal " ab,hl) '''II d e D. 5" an 01al J'play. FE hetr m
r;\. lhe1l

""n n
,ha VaCIIIIII/ Scieflce and -necllI,ol (llIrnal ', " .0",1 . The ""odie oon. " "I . J ( E Ds may soon be M c' dl CNT·F bUn 9) r
oj

.

-" rec si"or. In IBM pro. - '" ,he IM Ol,,-,he ,ran ...,n'cndad, . ligh' and ra waves in muc "' \.no" A' . e same way a detect ligh' "nn .. ,ho\\ n '0 n t><,'C w J"S g. et al .. Applied PI. iCS LeIters. 27 " I CN1\ ha' no ,,,,es (Wan clIes rad,o re< na ,en Iha' an ..

� ti,

III d'lameter. �ening a ro"nla1Cly 3 "Ill ' bundle appr p.:d ,ha nc· , ,,, n anode (no' Shown). Ihe hole '0 a 9 eNT, '" ugh 5 'hro . 10. "" " Fill. ,"], ElectrO"V P. ode> . ,ourc'. or ,,,,h 01 ,ueh ""h an 'I"m'" ' n,,"n.1 ,rr:ty' dln" . 1".,. pn" ",m cf

, . D'

10.60 fh� I.!c'lgn l\t I , t)rl\·,J.1

\"opI('lllbt-r :!�. :2607-26(9). If the untenna/CNT length I" not th(' l't.'n'-'Ct

flgUfW

":11.l1uple of the light \\u\eienglh, or if it... oncnlallan i, pcrpcmhculilf tfl thl..' !Iptt pvloU1z:l!ion. lhe reception become... meFricient. Rounding out their

l Onln { -.:\'lk�·I\'f)

n',lIm\'. Ci' , ,n,, ",uperconduct, though ugrun 3t fairly 10'\ tempernture...

\1.10\ of the upphcations nDled above employ �o-called I\lultl\\ulled

�

('T�. d p,\�ted In Figure

10.6 t . of which combination" are emile"'''. Ol,co\cr­

ICC. I\ml the I�,ol prope-rtieo;. and u�c" of CNTs are multipIY'ing :1I a dizzying p..

1\'Jlkr j.. cncour.\gcJ to "carch the Internet as the knowledge unfold .. Fmally. It I'" worth mentIOning a few variations on the theme. J\I"t ll'" ..lIi­ .,)n ca.n be replaced by quadruvalent germanium in some semiconductor lIpph­

�'alloo� tlnd by trh'alentlpentavalent ( I l l .V) mixtures in others.

llunOlubc ... hllve

betn fabricated from silicon and from boron-nitride (I11.V). The latter �holiitl ,(.lnd up betlcr to high temperatures and exhibit superior resistance 10 oxidu­ \I(\Il lhan CNT�. but it is too early to say whO! role either of these newcomers \\ill e\cntually play. Figure

10.61 A mul!iwallcd carbon nanotube.

eN l !l' r It'll'll.! ("" 1It'd tt....''''''I\'r).

I\", CNT l' -

--'--

SI..'lln;e

\-cIlIIUt'rl

-----'-

Gale (-�)

II brra1� fhousunds 01 Cooper pair;, The liberaled elec[ron�

UgItt o.t.ction and fMf'l)' G.pt PhoIOn dc't«tlon i\ "Ital In IMn), arul of K1C1JCl' A\UOOOm)' I� �ps the tint cump,," Whllt fbi: IbIllun of Idcsropn 10 rotM'f

I

art pn'mul�d "UP"ard" acrOl.\ the gap. Ihen [unnel through

h,ht (and .ltb£t part) of Ibt dectroaugnmc Ip«truml ha\e

the m�ulah.lr 10 the Olher �upen:onduclor. resu ltmg in II

1TI(',..urJNe charge tramfer (Tunnelmg of e lC(lrons Mill m

Cooprr p.:II.... I� ,upprr�..ed b)' an e1ltema/ magnelJc field,) by the STJ is that b.:\:au..c thc= number of elecrrons promoted is proponlona/ to UlPlhlr.;fIVUY "",)' he 11t'0'� IQ the JlClt m, JUDlp. The th<' photon coetg), It can discriminate different wa\elengths Itadt't loday, 1 1I'tfrIc1llk" n a,hafll.'t: Intf e#ly phu'u,r.aplll� ra:h:niquc , I lhe ("('0. OJ �hatJC coupled ikYl(c, twnJ on of Ilk:ldenl hght. A di�dvantage oflhe STJ deleelor is the ('\tn:me 1010.- lernper.lfures needed To prevent thennally ,ftc Inl\'Y 111i�"On w;mrrondul1or. In II �,"jlle ceo pi\('l. c'(lh:d ('ledIQn-hole pairs. CCDs in the most sensitive cal:h .ITII JOS pholon e:u'Ue an clcxtwn ,KRl\\ the gilp appl"auon, mu�t be kept cold wilh liquid mtrogen. 77 K. but leUi� ICftC)' (an &PJIfllOlCh I(lU'. JO the \ I Iilk). The the l'ntl..:al lemperatures of the superconducting melals used ..undUl.'lIon tle,·ltuIlJ .re tr-.IPpeJ In .ln dC'i:tw�taUe potenllal III STJ.. an: '>0 low Ihal liqu id helium cooling is requir'W ",ell LTt'11c:J t'I" L'fw¥td tU('mOlI e/c.:tmdt'S, "hlk the OPP Uely LhouJed IloI.:'1 11l' repell.:J Irvrn lhc 1o\dl. �nlIO' It'C(JIllllJO;laon ThC' nllmtocr Ilfd(drvn\ N.w light·Emitting Oevices The ligh'-emil ting diode krlJmulll'I06 In ea:h PJ\t/ kf'l\SS II �D gnd I� n:aJ ha� been al'l)und sin(e the 1 96()" Howe\'er. progress toward I mmcdl.ltcly If'ef the t'r"l\ur� B('siok� tie,", a "{lrlh(ll"\(' \hC)ller �a\elenglhs was rather slow It wn� not until the III I\trlll](.lm�, thIS U Iht' I} r<' Ilf light ,(n,,'t in luda) "' mid-l990s that bright blue LED.) became available. The JIgH.l1 \llft)('rJ\ ,.\ pftlnll'lng alt"mallie Illthl' reo for OIaln difficult}' wa� fabricating a semiconductor that �"ITl( ilpl'h�'alt('n� 1\ Ihe �ur'l'l\:l'nJudJng tunnel Jundlon produl'ed photon� efficiently and had the appropriate large t s r11 Jeled," . OI:PKt,·,J In h)!tlle HI ��. In thl\ deli(e, �liII band gap. SU!.:(e�scame with gallium-nitride, a 111- V • •1O In lu la1mg h:'&i,IO '{'par-JII" 1"0 In l!cleh'I'nlC. 'nl �mi(onduclot. While the vi...ible ,pectrum wa... thus IUpcn.onJu rlol'l. and a ,h)!hl ptltl'nll,:II 11iZl' i, bPphl'J covered. the LED has recently gone well into the ultra· .nl"<1 tht entm: 'fa...l ·h in ;l \CmiwnJudor. eleXlron, in a \ ICllel. By inno\atile technique.... 'lCicnti.)l.'. at Ihe NIT Basic IUJ'IC'f\'Il!hJULI.lr' n'" a ,ap. the enerlD rrljulrrd to fr« Re�an:h Laboratories in J3jXln have fabricated aluminum them IRlm thcu ("otlpn PiI.N. B ut II IS much ,mJII(r. nilride LED, that emit al 2 1 0 nm (Tamya.<.u. et al.. Naturr. mt:l�rtJ '" me\' r.uhcr ,hllO ('\' When II \'I'iblc photon. In 1 8 May �006. 315-328), The achievement is an imponant the.- ,'\' ranFI'. 'tn�I" the ..uJ'lt'rnlOdul"lllr vn the high enelE} }Iep In the quest to replace less-efficient sour.ce�. such as ' ·",dc. I' J.!C>n I pnlm.lll' lin del'fnlll aenl" the gap; ,",tead. \apor lighh, over a range of wavelength�, The researchers \peculnlc thai the new device may be useful in high-density

unpnmd mr1lbll)' O'IiCI IbI: yt.n.

WmJ

oIlighl Mt«tlOll

ha\'t talen ,rat kap& In ttttnt dcadr:s. ;and lura'

Figure 10.62 In ;1O STJ /'UpC'I'I:tlO"

tlUCllOg 'unnd Junetlnn). It phpt{1O l1n:al, a Ct)('�r piur, anll th( (h"'teO �Il'\:fl\'n\ !Wnnd throu�h an ln�ul.ilJng t\unec

an.: 01 the pnrrury ad\'anlage� offcred

infom13tion �lOrage 3nd in ai r 3nd water purification and \lcrilization. Another recent breakthrough with enormous potential

j, the light-tmitting transisror(Feng. et OIl . . Applied Physics

Lmm. 5 Janu3f)' 2�. 1 5 1 - 1 53). In a trami�tor. while rnO\1 of the electrons tha' flow from

Ihe emitter into the

bast �lide down to the collector. a fraction engage III dewon·hole recombination in the base. con...tituting the ba�e·en1J tter input current. Ordin3rily. the recombination

-.

_�\l k

energy ,imply produces heat. but a research team 31 the

Uni\l!n.ity of llhnois hOI:' found a way to divcn it to infrared light. In efleet. the bipolar tran�istor's standard electrical

\._--).

input and output channel s are joi ned by a thi rd channel :

(''JOPCf p;u....

I

light output, proponional to the olher 'wo o

Thj� photon

generation can be controlled at far h igher frequencies than

in an LEO and opens up the revolutionary �l"ibility of

I

repla(;ing the ele(;tron currents that flow throu�h an inte­ grated circuit\ tin) wires with light s ignal\.

s_ �ond...ctin9 MgBz Coming as qUite a !;urpri'OC: when di'lol.:o\'ered in 100' wac. the \uperconduetivity of it w» JOIIl'CMum dlOOnde. MgB� (Nagamatsu. et al.. Natlu�.

_

Flgu,. 10.63 The structure or Mall' CiraphilClib heuBonal planes of boron. with mqne iwn aloma between.

I Man:h 2001. 63--64). With attention focused on the

,upi'les Ilnd their critical

temperatures over 100 K. a T, of

081)' �9 K did not dominate headhnes In the popular press. .. in \'IUt thi\ cn1ll.:al temperature wru. nearl y tWlI.:e as high a III)' Illl-metal wperconductor previously k.nown (though

boflln quahflc.\ as a "metalloid""). Furthermore. the fact that

MgB� exhibited the isotope effect

suggested

me �ulXn:onductl' Ity mechamsm was more

that

hk.e that m

al c.upen.:onductors than In the mysterious

�on"ention

cuprates. On the other hand. MgB2 Wlti found to ha"e a unlqlK prope:ny: two gap e ncrgie... Both theory and C\ptnment \OOn attnbuted

!Jus 10 certain characten!ltlcs

of the MgB2 ,truclUre. As illustrated In Figure 1 0 63. the bOron atom.. form he"t3gonal !ohcets Ii"-e graphite. With IT.bonds In the plane of the sheets and 'IT-bonds

perpendicular. The larger of the two gaps is the re!ouit of

dectrOn� In the u states bemg more strongly coupled to me lamce . The stronger t he coupling between eleetron and phonon. the higher the binding energy of the Cooper pair. The "ea�er coupli ng in Tr states makes the gap energy for tk..e el�trOns only about one-founh that of the CT elec· tron). A particularly i ntere'lmg feature of MgBl is thot

Chapter 1 0 Summary

wholly appropriate� the vnlence electron 01 a ..

is known atoms'

l-alence electrom. Molecules and �olids form whenever the \l3te\ thal lhe electron, occupy re..ull m a lower energy than for Iwtated aloms. The,e many-atom ..tate!. may be thought of as

N atoms

11th lingle·otom state become, ..m N- state band. In molecule'>. a (.:o\"aif;:nl bond occurs when two

bond. the

electrons. from each atom. oc..:uP)· a molecular spatial �Iate centered bcl..-.een the atom� The Clll'rFY I' low because both atoms attract both tiectrons. If the atom .. .n' not Identical. the electrOn� are noc Ihatt.d equally. The e\trenl! W'oC. where a - I ,alence atom

one

I

"alenee atom.

as an ionic bond. and the attraction is ,imply between

negati'e and po)lti"e ion,. 1he character

Bonding between atom� in\"ol\"C�� rearrangement of the

combm3tions of �ingle-J.tom ..tates. When

various other element!; has. produced eXlJ"emely high upper critical field� B,�. even surpassmg tho� ot the cuprate\. The interplay between the two !ouperconducting mode, in MgB� i, still not fully under�tood. but IU. investigation i!o already adding conMderably to our undC'nta.nding of superconductivity. and who cnn sny what might be around the comer.

alloying Ilnd dopmg with

of

n\O'it molecular

bond!. is bet\O.·een purely co,alent and punly iom..:

Pure solids usually form in a regulnr arrangement of atoms known as a crystal latti..:c. of which there are man� geometrical types. Regardle�s of the lalllcc type. bonding i, u ually catego­ rized a� one of four dift"erent kind,: covalenl. ionic. mctalh..:. and molecular. The bondmg in the fiN t�o is a. .. in indl"idual molecules. A metallic solid', \"Illence electrons are �hared by all atoms. They move freel) through the whd. ginng mel
E.".. m �I and often be ft.-d

..

'00'"

iO!Kh.

\'ak",-� dtctron\

betn, t.ow.t noI to rndJ"'lduaJ aaom!! 001 10

1M .obd -' . ..hole Tht poIeIIlUll C'ntl'g)

one' lDk"""*"tJPK

mu'"

pokatiaJ wc-II. hul

IS cs,'ltnuaJl� tJul of

,"11'1 � wpmmJ'O'C'd pen­

""', vardtlon due: 10 �ruWf)' .J*�d �i6\'C' wn� Be..·au� .\"

II to 1MJc. -=b ",.1oI.aI( entJ1'y Nnd is a ,·inu.! �·(.-.nllnuum of .rkl\wd eDnJleJ. Tbt ranI" nf J"ortudikn energlC'� ht-I""ttn

Nndt .. known as . talnd glp

Tkc:'«tncaJ !;Ondu.,:tJ\ll) (1f" wild I� celakd III ftlllllg of

III batKh. The rcpmotl twkJ th.l l\ lu/l III T - 0 " koo\\-1.1 a\ the vaJen.."'C hind II C'1�tton� fill p.1n of tht- lK\t-hi!!�r band.

known Il\ 1M- �·,.ndUtll(.lO ltand, tht- materiill lIoi/l be an electrical wnJu..:wr. lbe hlldle'I·C'm:rg) d«tnlO� may en'er immedi­ lilel)' hl,her ,Iaft.. ..nd Ihu\ �,pood 10 an uttmal dec,ric field. If ('/«Inm, fill \LiIIC'\ (1nly 'l) the lOp l.f the \alencC' band, the malenal ,,"III f'I(' an ,"\ulllll'r Hd\'ing no easily acce�sibk Matt's, W dC'�·tn1n\ hll\'C' nil frrcdom I() te\pt"lnd. An ,"\ul"lor wllh II \mJII \alem.:e·conUudion band gap is ul lnJ II \C'mlc,.nJu�·lllt Common e1til1l1ples are "Iicon and ger· m..murn W" h a \mall band gar. thermal C'\citation can pro­ ml)(t .. \lllmrKAnI numtotr 01' el�trom from the valence band 10 IhC' ulhrrv.I'>O empry nmdu!;l1on band The,.:' re�pond to lin e.\l('m.a1 li('ld JU\I a, do dedltms In II condu�·lOr. The resulting \'iIIlt""e band \lk:.IInu('\ aho !;onlnbule 10 det:lrical condw,;li\,­ Ily Tbr)' art lnllwn '" hole\ and re\pond 10 an external field a.)

wh)" both energic\ in the lower band �hould be roU!!nt)" C'tIual to (hat of the n = I atomic state and why both enc:rgie\ in the upper should roughJy equal thai of the n :! atomic �Iale. =r

2- L'pon what definitions do we base Ihe claim Ihat the 1/1, and �1r \131eS of equalions (10-1) are related I. r

�P,

'

3fld Y JU\t as 412/', I!' to ;:.1

J. Seclion iO.:? discu�ses u-bonds and ,"-bonds for p-�tate� and u-bonds for S- Sla le� , bUI nOl 7T-bonds for f-sUltt\. Why not? 4. or N�. °2, and F2. none has an electric dipole moment but onc docs have a magnetic di pole moment. Which ' one. and \\hy'.' (Refer 10 Figure 10.10.) S. It takes le�, energy to dissocillte a diatomic fluorine mole_

cule than a diatomic oxygen molecule (in fact. less than one-third � much). Why is it easier to dissociate nuorine?. 6. Why is covalent bonding directional. while ionic bonding is not? 7. For the four kinds of crystal binding--covalenl. ionic'

moleculat-how would the density of Lhroughout the solid ? Would it be constant, centered on the atoms. or largest bet\\-een the alom!>? Or woul d it allem3le . with a nCI charge denslly positive at one atom and negative at thc next? 8. Why should magnesium form a melallic solid? PII'toIIl\(' I;h4l',:e �'anie"" The energy necessary 10 break: the ionic bond between a 9. The numbcr"\ of C()nductll,m banu electrons and "alence sod i um ion and a fluorine ion is 4.99 eV. The energy band h(lJe� c.:an he: ,ancJ I�I) by doping. Of Introducing into the to separate the sodium and fluorine ions that necessary /4111\'(' It �m.tlll frJ!;tllln \11' impuril) a,om�. Tri"alent lmpurit) form the ionic NaF crystal is 9.30 eV per ion pair. al�lm� )Idd a r-')pC' '\('mkllndu�lor, in \\hich hdenct holes far E1tplllin the difference qualitatively. fllJlnumhc:r l'ondu!;tllln cI�(n)O�. whik an n-type \emiconductor, a me m l consisting principally of copper alloyed Brass is 10, where �"Undu'"tHln ela:ltlln� prNtlminatt. resulls rrom pentava· wilh a smaller amount of zinc, whose atoms do not len! Impunry alllm\ ImportUllt de\'icts are rrude by assembling the crystal lattice but are i n a regular pattern in allemate \1LnOUS �'Oll1blnatlOn' of n·type and p-Iype �C'mjconductors. Join­ somewhat randomly scattered about. The resistivity of inB ant or e&;h type in II p-n junction yielcb a diode, which higher than thnt of either copper or zinc at room bra�s is aJl()w� e/cclrkld current III pil'>\ (lnly one way. Three together temperature, and It drops much slower as the tempera­ rtlrm a D"lUIsl�t(1r. II baMC clemen! of mplitication. ture is lowered. Whal do these behaviors tell us about Many melah and II growing number of novel materials electrical conductivity in general? /().\e al l te,i\l.lnce belo",,' a malerial-dependent cntical temperaII. Explain the dependence of conductivity on temperature turt. bc�·t'ming �u �n'onducto� CUrrtnl is carried by electrons for conduclOrs and for semiconduclors. tw,'und In C\II,lper pall1t.. the rt,uh of an attraction mediated by 12. In the boron atom, the single 2p electron does not com­ tht cry,tal Jauice. IUld all patrs mO\'e as one. The effeci is bro­ pletely fill any 2p spatial stale, yet .raNd boron is not a ken bJ high ltmpcnuure and/or it \trong: magnetic field. conduclOr. What might explain this? (It may be helpfUl to consider again why beryllium is not an insulator.) 13. The bonding of silicon in molecules and solids is quali­ tatively Ihe same as that of carbon. Silicon atomic stales become molecular states analogous 10 those in Figure Conceptual Questions 10.14, and in a solid, these effectively form the valence I. Thr left diagram in Figure 10.1 might represem a tWO­ and conduction band.f. Which of silicon's atomic stales alom cry�raJ with two ban�. 8a�ing your argument on are Ihe relevan! oncs, and which molecular state corre­ we Linelic energy insick eilher individual wel l, explain sponds to which band? metll1!ic, and

valence electron s vary

decrease the. conductivity of a conductor I" What factor> 1I'i \emper'J.\ure increases? Are these factors also present In II semiconductor. and if so. how can its conductivit), vat)' with temperature in the opposite sense" Il. Based only on me desire to limit m;nonlY carriers. why would silicon be preferable to gennanium as a fabric for doped semiconductors? 16. Why docs the small current flowing through a "versc­ biased diode depend much more strongly on tempera­ ture than on the applied (reverse) voltage? I7· In II concise yet fairly comprehensive way. explain why doped semiconductors are so pervasive in modem technology. 18. It h often said thaI Ihe transistor is the basic element of amplification, yet it supplies no energy of its own. Exactly what is its role in amplification? 19. The "floating magnet trick" is shown in Figure 10.50. If the disk on the bottom were a permanent magnet, rather than a superconductor, the trick wouldn't work. The superconductor does produce lin external field very similar to that of a permanent magnet. What other char­ acteristic is necessary to explain the effect? (Him: What happens when you hold two ordinary magnets so that !hey repel, and then you release one of them?) ZO. The isotope effect says that the critical temperature for superconductivity decreases as the moss of the positive ions increases. Can you argue why it should decrease? 21. What is a Cooper pair, and what role does it play in superconductivity? 22. Describe the similarities and differences between Type-! and Type-II superconductors. 23. In a buckyball, three of the bonds around each hexagon are so-called double bonds. They result from adjacent atoms sharing a state that does nOf panicipate in the Sp2 bonding. Which state is it, and is this extra bond a u-bond or a 1T-bond? Explain. 24. What are some of the properties of fullerenes that make them potentially so useful?

Exercises Section 10.1 25. Fonnulate an argument explaining why the even wave functions in Figure 10.1 should be lower in energy than their odd partners. 26. The accompanying diagrams represent the three lowest energy wave functions for three "atoms." As in all truly molecular states we consider. these states are shared among the atoms. At such large atomic separation. however, the energies are practically equal, so an

electron would be Just al< happ), occupying any combi­ nation. ta) Idenllt)· alg<-hrau: I:ombinalion!o of the states (for imtanc:�. I + 1112 + 1I112) that woullJ place 1M electron In each of the three 810m). (b) Were the atoms closer together. the: e:nergle� 01 \tale� I. 11. and m would ..pread out and an elc:.ctron would occupy the lowest energ) one. Rank them In order ot Increa.<;ing energy as. the atom� draw c10scr together, E�plain )'our reasoning.

II

III

Section 10.2 27. Section 10.2 gives the energy and approximate proton separation of the Hl+ molecule. What is the energy of the electron alone? 28. From the diagram!'; in Figure 7.15 and the qualitative behavior of the wave funclions they represent, argue that a combination of the 2p� (Le., '"t "" 0) and the neg­ ative of the 2s would produce a function that sticks out preferentially in the posirive z direction. This is known as a hybrid sp Slale. 29, Electron affinity is a property ..pecifying the "appetite" of an element for gaining electrons. Elements, such as fluorine and oxygen. that lack only one or two electrons to complete shells can achieve a lower energy state by absorbing an external electron. For instance. in uniting an electron with a neutral chlorine atom, completing its n = 3 shell and forming a CI ion, 3.61 eV of energy is liberated. Suppose an electron is detached from a sodium atom, whose ionization energy is 5.14 eV. then transferred to a (faraway) chlorine atom. (a) Must energy on balance be put in by an external agent, or is some energy actually liberated? If so. how much') (b) The transfer lea\'es the sodium with a positive charge and the chlorine with a negati-,'e. Energy can now be extracted by allowing these ions to draw close.

the P slale thaI 1\ onenled along the �-axis. and thu� )Iick, ('411 in Ihe +: dlll:cI,on (see Exercises 28 and J]) If f"ltUfC' JO.11 1S It true plClurc. Ihen in It coordinale ) �IC'm roIalC<.! counterclockwise abolll the J-:u;i� by Ibt It:'tr.thCJrnJ angle. lobe II should become lobe ,. In lhe 1)("0\ fnlmc. l,\'alue� are unaffecled. bul whal had been uluc\ II1 lhe ,;:x.plant:' become values in the z'x'.plane. .i("(ordmg lOX == .f' cos a + z' Sin a and z "" ' t' co, a - f sin a. v.here a is 109.5u• or cos-

...... a lllDl .laale ... dow mtlll Wy apprukb 10 N!I;OWC!f' ''' � apMded UII ,.., la)" (..-, ThC' actuaJ ..,.,... 01 * .rom. lil a N.c, molecuk is 0.1" nm

He. modi JDMr an nJCtJY " lbt mokculc Iban the

8Ied fltUnJ aIOmI '! .piIi JI" F..urt'11C 29 outJlDh ,.. encIJ)' may k e:tu.ctnf by lrMIfftnllJ all d«tn,Jn from an aJom thai easll), bcs an el«tma 10 OAt with a lalp appetile for e�' Il00_. dlen "" tlwm, the IWO 10 apprtb. tal'

1(-0

(orming an

,Ollie: boocI

{al Sbow rhal lobe /I become� lobe J. NOle thaI since nCllher Ihe 2r ,,'a'e nor Ihe radiol pan of Ihe P Malcs is aITecled by a rorallon. only the angular paris gi\'en 111 equllrion� (10- I ) need be con<;idered. (b) Show chat If lobe II is instead rolared about Ihe �-axi!t by �imply �hifring ,p by =- 1 20°. il becomes lobe... III lind IV.

(I) Con ••, Jepal'llicly lhe �'bC\ of bydrogcn bonding wllh fluorine and wdium bonding �ilh tlul.rine '11 c-kh C:�. I)IIW do.\C musl ltlt ion... approach

10 "bnu cven." wben:: 1M cnerg)' nudcd 10 Inlndc-r ItIt clc�:cron f,(olw«n lhe scparnlcd alom, j.\ tt.I,anctd by tht clCi.·Inl\Iaril: polcnliaJ energy of aU,..ljon? The i(ll1llarion cnergy of h)'drogen is 'Jb rV, rhar o( Ml(jium I� 5, I rY. and the clc.:lron Iffinily IIf l1uvnnC' I� ,l-WeV Of Hf and NaF, IIIK'" �'on,jden:d 10 be an joni( bond and the tJlhcr " ctwalC'nI rond Wbl(h j.� Yrbkb. ilnd ,,'hy" reac:h

(b,

JJ. Tht \'('rlICC'� of. Ittrahairun an:: four \eniccs 0" a cube symmcrncAJly dKt�n ....) rhoar no 1140 are ad)Jttni. Show rhal che' anBIc htlw«n rhe- venil'C'.) o( a teuahtdron 1\

IOY,5"

•

Jl. 8y the "\'fi'wr� I«'hni"uc vf Eumple 10.1. �bow Ih.1I 1M an,les htlW(C'n IlI lohe� (If the hybrid WI "ale� at!: 109,50 JJ. In 5«llon 10,2. we di....:u.\,ed Iwo-Iobed P,. Po' and Po II�es and (our·/ohtd hybrid Jpl slllle�. Another lind of hybrid Ilalt:' Ihal slicks oul in jusl ooe di�'CIion i� rhe .'p. formed (rom � 5in,IC' p .�l8rr and an J 'faIe. Consider an al'b11JU)' combination o( the :!f Siale wilb the 2p, lIlak'. ul Uli repn:!iC'nr Ihis by co.� r ¢tz.o.o + �in 1" 1/11. 1.0' (1bc IriS (1I<,'lon; t:'nsun: nonnalilalion. In carrying OUI dar inrc".l. cross lenns inlt:'gn.lt:' 10 0, leaving

fl�z.o.o j! dV + sin� .,.111/12. 1 .0 FdV, which is I.)

COIl.,.


(b) What value o( T leads 10 the maximum probabililY. whal is the value of the maximum probabiliry. and

whar is the corresponding ralio o( 1/11.0.0 10 I/I'!, I.O? Ie) Using . computer. matt . drnsif)' (shading) plol of the probIIbiliry densif)'�nsif)' ,'Crsus rand 8(or the .,.·vaJuc found in pari (b).

• .N. Referring 10 equations (10-2). lobe I oflhc hybrid sr QIes combines rht spherically symmetric s siale with

S.ction 10.3 35. By e)(panding an arbitrdty Vex) in a power series aboul II local minimum assumed 10 be III x = a. prove thlll the eff'e(li\'e spring comlanl is given by equation ( 1 0·3).

36. The 1I11eralomic polenlial energy in a dialomic mole. cule (Figure 10,16) ha... many fe:uures: II minimum

energy. an equilibrium separalion. a cunarure. and SO on, (a) Upon wh:lf features do the rOlal;ooal energy le\e1s depend? (b) Upon whaf fealure� do the vibra· tiona/ leyels depend? 37. The bond length oflhe N'] molecule is 0.1 1 nm, and its etfecllve .)pring conslanl i� 2.3 X 10] N/m. AI rOOm temperalure. (II) what would be Ihe ratio of molecules wilh rolational quanrum number ( = J 10 those wilh ( = 0 (ar the �ame vibralional level). and (b) whal

38.

would be Ihe ralio of molecule.. with vibrational quan­ lum number " = I 10 Ihose with 1/ = 0 (with Ihe same rol:llional energy)? The bond lenglh oflhe Nz molecule is 0, I I nm. and i(s

effeclj\'e spring constant is 2.3 X JOJ N/m, (a) From the SIze oflhe energy jumps for rotation and vibralion. deleonine whether eilher of Ihese mode.� of energy

slorage should be aclive m 300 K (b) According 10 the t<jUlparrilion theorem. the henl capacity of a diatomic

molecule ;'Ioring energy in rotations bUI not vibrations should be R (3 trans/mional + 2 rOla[ional degrees of

�

freedom). If it is also storing energy in Vibrations. it

1

�hou'd be R (addmg 2 vibrational degrees), Nitrogen's molar heal capacity is 20.8 J/mo! • K III 300 K, Docs Ihi� agree with your findings in part (a)?

J9. The effeclive force comlanl of the molecular "spring"

in Hel is 480 Nlm. and the bond lenglh is 0 . 1 3 nm,

la) [ktermme the energie... of the tWll lll"",e\t-energ)'

together Quanlif)'ing tht: eMfI),·\O'tIIering effect or

\'I\)ratlonal "tate!t,

haVing alternating plu� Itnd mlnulI l,7h",e. can be rather

(\)) For the�e energies.. determine the amplitude of \' ibration

if the atom.., could be treated

ritmical particles,

Ii..

oc;ciHahng

YKi

_

i is ih

UI For the rotational frequency, a!)�ume that lowest nonzero value,

Vi(I +I) h, and that the

momcnt of inertia I i s �l(I'!.

Ie) h It valid to treat the atomic !>cparation as lixed for rotational motion while changing for vibrational"

.10. Starting with equation (10-4). show that if .In is.

-

I as

a photon ilo emittcd by a diatomic molecule in a transi­ lI(ln among rotation-vibration state!>, but Il! can be :! I , then the allowed photon energie!: obey equation ( 1 0-6).

.n. Vibration-rotation speClra are richI For the CO mole­

cule (data are given in E",erci�e 42), roughly how many \'013tional levels would there be between the ground vibrational state and the first excited vibrational state?

�2. The carbon monoltide molecule CO has an effective spring con'itant of 1860 N/m and a bond length of

0.1 13

" one-dimcnsionlll ealcu� \.).\ion is in�tru..:t;\'e, C'(lnSlo.ler an infinite line of point charges alternating betwt'C'n +" and

... the lc) For thc'>C energies, by what per\:entage dot atomiC ..eparalion fluctuate" classical vibrational frequem:y (d) Cl1kulate the_ p- and lhe rotational frequency W w"t- =

10\'01\'«1 h'r a 3D lani�e. Nt

per

be helptul: In(1

Section

,

" I

2 band and the oo\t('m of the II

:III

3 band fur a

W8\'e number have \"" tl)' d,lkrcnt energie).

48. A, we ,ee in Figure, 10 :B. in a one·dimen�ional !.:ry�tal 01 finite well�. top-ol·the·t'land ....Ialc\ do\e!y re\Cmbk inliOltC well state'i, In la!.:t. the I'amou\ partldc-in.a.bOll. energy formula give� u fair \'alue for the energle, of

these ,tates and thus the energies (If the band, to '"hlch they belong,

(8) II for rt to thaI formula you u,e the

number of anllnodc\ to the ",hole W3\e

functlllO. '"hat would you U';C for the box length L? (b) If. in'tead. the n in the fannula ....ere . laken to refer to bdnd n. could you �lIl1 u�e the fomlula? It \0. what would )'1..11,1 u\c foc L?

in the lower-energy electronic ...tate? Explain. What

vibrational energy levels.?

-L{- \)"'11

wells. Explain \\-h� the� 1\\0 \Iah.'\ 01 rou�hl)' equal

(c) Explain why the energ;e� 10 a band do or do not

about the rotational levels?

Ie�'el!." (c) And what effect doe� it have on the

-

one-dimensIOnal "cry�tar' con",lIng 01 !-e\en finite

the same. farther apart, or c\m.er together than lho�

(b) What effect does it has'e on the rotational energy

"{)

10.5

the II

tional levels 10 the excited electronic state to be spaced

on the bond length and force con...tant" Explain.

+

41. Make rough ....kelehe' 01 thl' '"a\e IUnctlon, at the 1\IP uf

�3. From the qualitative shapes. of the interatomic potential energies in Figure 10.2 1 , would you expect the vlbra­

atom (a) What effect. if any. does the replacement have

ion. For \lmr\u':lty. ",,,,urn... that 3 po�ltive charge 1..

at the origin Th... fulloy,.m� puwl'r �nes e'pansion W il l

CO might absorb in vibration-rotation transitions.

from H,! 10 that a deuterium atom replaces a hydrogen

with a uniform

Wh)? (bl Calculate the ele..:tm\tatk potenltal ene'll)'

nm. Determine four wavelengths of light that

"-'. A<> noted in Example 10.2, the HD molecule differs

�(',

....paung bet....«n ad.la..:ent loppoo.ite) charges of Q, la) The cle..:tn\\tatir.: f"'tenltu\ tnefl), pt'r ion i.. the same illr a g!\cn Pl"ltl\'e I(ln a:o; lilt u " \'en Ilegati\'e ion.

49.

depend on the sile of the cry�tal 3., a y,.hole In Figure 1O.24.lhe n

I bilnd end\ at k

'"

4rrlL,

while in Figure 10,27 It end\ at rrla. Are t.he�e ..:ompill1· ble? If �o, how?

50, Assuming an interatomIC 'paCIng of 0.15 nrn. ot'luto a rough \'alue for the width (in eV) 01 the n one-dimensional u)\tal.

=

:! hand to 3

51. The density of copper 1\ 8.9 X 10-' kg/ml. it.. Fermi

energy i� 7,0 eV, and It h..� one condur.:tion el...ctroo per

Section

10.4

atom. At liquid nitrogen temperature

45. 1\Yo-dimensional lattices with three- or four-sided sym­ metries are possible. but there i<; none with a five·sided symmetry. To see why, con.... ider the

following: A piece

of paper can be cut 1010 iden tical equilateral triangle� or '>quares with no exce",. Prove that this is not the case for equilateral pentagons. •

46. Exercise 29 notes thai mlln: energy is required to ionile

(77 KI. it\ 0=<>1'"

Q n . m, E,timat... ho..... far a con· duetion electron would travel before (:olhding and hO\l.' tivlly is. about 4 x 10

many copper ion\ it \\-Quld ra", Section

1 0.6

52. Carbon (diamond) and ,ilk,'n have the same covaknt cry�tal structure, ),et diamond is transparent while sili

�Q(hum than i� retrieved hy adding that electron to an

con is opaque to \i'ible IiFht Argue mill this should be

I�olated chlorine atom. hut the NnCI bond represents a

the case based only (In the �hfferen�'O:

lower energy becau�1! tho! attracting ions draw close

roughly 5 eV for diamond and

In band Il� I eV for ",ii.:on.

5l. In diamond. c.rboa', four full (boodingl l aod p spatial

swet become _ bind, aod the four cmpry (aatibonding) OM. buomt . highcr-eDrrl)' band- COMidering the b'Od In the bIPd gaps of diamond. s.ihrofl. and germa­ nium, e.. plaill wh)' It rrugbt not be SWplloing that "CO\8lent" un beha\ICI II _ Io."onducung melallic wlid. Sot.. S«UOll 106 nolts th.t &-\ aLlie' of mlstancr. ionic Vibrations 'Ive wi) 10 lattice imperfections al around 1 0 K. A IYplCal 'pring r;on5l&n1 ber..«n alom� n i a wild is of (lfIkr of mIImtude 10:' N/m, and typical

'PI"IRI IS nomimall)' 10' 10 m. Eslimale h()\\.' much lhe vlbnatlnl atom,' localion� might deviate, as a fraction of their nominal \eparation, at \0 K 55. Carry out the integralion Indicated in equation (10·10), S6. The rc�l\livil)' or ,ih'er i, 1.6 x 10-8 n m at room tcmperalurc (:lOO K). while that of �ilicon i� about 10 n · m hi) Sho\\ thllt th... dil>parit)' fonows, at leaslto a mUlh onkr of malnllutJe. from the approximate I eV '

band tlap In ...lieon, fbi What would you expect for the n",lm-temperature rr)l�ti\'it) of diamond. which has a

band gap uf about � eV?

57. Show Ihal for a n'lOm-lemperatuTC lot'miconduclor with a band gap 01 1 ev' a temperalure ri�e of 4 K would rai� the �'ondul,'II\'lt)' b) about 30�,

58. The femu \elocit)

�'F i� defined b)' EF =

"real" mtlal copper. will have true positive charge

carriers and eqoal numbe� of negative ones, even at lero temperature. 61. Verify using equation ( 1 0-12) that the effective mass 0r a free particle is m.

62. (a) Compare equalion (10-1 1 ) evaluated at room lem_ pcralure for a �ilicon band gap of 1 . 1 eV and for a t vn cal donor-slate/conduction band gap of 0.05 eV.

' ",.

(b) A.,suming only one impurilY atom for every lOS silicon atoms. do your results suggest that majority car_ riers, bumpe up from donor levels. should outnumber minority camers created by thermal excitation across the whole 1 . 1 eV gap? (The calculation ignores the dif­ ference in density of stales between donor levels and

�

bands, which actually strenglhens the argument.)

63. As n crude approximation, an impurity pentavalem atom in II (tetravalent) silicon lattice can be treated as a one-eleclron atom, in which the extra electron orbits a

net positive charge of I. Because Ihis "atom" is nOI in free space, however. the permitivit y of fre e space. eo' . . muSI be replaced by Keo' where K IS the dIelectric con_ stant of the surrounding materia\. The hydrogen atom ground-state energies would thus become

! ml'f:, where

£, 1\ the fenni tneq:y. The Fenni energy for sih'er is

5 5 eV (a) Calculate the Ftnni velocity. (b) What

\\(luld be the wavelength of an electron with this veloc­ ily'', (el Hu.... dot� Ihi� �'omparr \\'Ith lhe lattice spacing of

OA\ nm') Doc\ the order of magnilude make sense?

59. For II small tempernture change dT. II materiaJ's resis­

LJ\'II}' trtciprocal of conduclivily) will change linearly

aCl;onimg I�'

plaT)

Given K = 12 for silicon. how much energy is needed to free a donor electron in its ground state? (Actually. the effective mass of the donor electron is less than m ' e so this prediction is somewhat high.) Section 10.8

oco

Pu + dp

:::: po(1 + adT)

The frnctional change in resi�tivity. a, also known as the temperature coefficient, is thus

64, The photons emitted by an LED arise from the energy given up in electron-hole recombinations across the energy gap. How large should the energy gap be to give photons at the red end of the visible spectrum

(700 nm)? 65, When electrons cross from the n-type to the p-type to equalize the Fenni energy on both sides in an unbiased diode. they leave the n-type side with an excess of Estimalt 0 for �iliron al room temperature. Assume a

band gap of I I eV

Section 10.7 60. A semimetal {e,g., anumony. bl�mothl is a materiaJ in which electrons would fill states to me top of a band­

the valence band--eJla:ept for the fact that the lOp of this

positive charge and give the p-type side an excess of negative, Charge layers oppose one another on either side of the depletion zone, producing. in essence. a capacitor, which harbors the so-called built-in electric field, The crossing of the electrons to equalize the Fenni energy produces the dogleg in the bands of

roughly £&IIl' and the corresponding potential differ­

ence is

£gap/e, The depletion zone in a typical diode is

band overlaps very slightly with the bonom of the nut­

I ).1m wide. and the band gap is 1.0 ev' How large is

hi,tler band, Explain why such a material, unlike the

the built-in electric field?

66- In a diode laser. electrons dropping from the conduction band, across lIle gap. and ;nlo the valence band produce the photons that add to the coherent light. The ZnTe Iase:r has a band gap of 2.25 cv. About what wavelength laser light would you expect it 10 produce? connected in series with a resistor and a ,1' An LED is power supply of variable voltage. The supply vohage struts from -5 V and increases linearly with time to a muimum of +5 V. The figure shows the supply volt­ age. the voltage across the diode, and the current in the circuit as functions of time. The LED does not glow for the whole time shown. (8) Explain the plots, and iden­ tify Ihe period when the LED glows. (b) Approximately what color is the light produced by the LED?

R

p-t}'pe bands constitutes a potential barrier (see S«tion 6.2), but in the tunnel diode. it is so thin that significant tunneling OCCUl'S. The CUlttnt versus voltage pial shows that unlikt a normal diode. significant CWTeIlt begins to flow as soon afa thtre is an applied voltage-bcfore the bias voltage is

E,.It!.

It then dt!Cn'4US (so-called nega· tive resistance). be fore again increasing in the norma) way. Explai n this behavior.

Tunnehng

region

n·type

\ "�=:_:,"';�\,.=_-�..., --E:-:::� =: -

LED

.6. V = O

t -

h

\.

h' - h h h'

-

Increasing.6.V ----+

Supply t5

Supply

+.

+3

LED

+2

� +1 � 0

Normal diode

,

",

Tunnel diode .... . ..

.

..

0 -1 >

-2

Voltage (forward bias)

-3 -.

-5

6

'<

g

u

2

,

,

10

,

15

I(S)

,

20

•

3

,

,

5

5

,

"

0

0

69.

Volumes have been written on transistor biasing, but

Figure I 0.45 gets at the main idea. Suppose that Vile = 1.5 V and that the " input" produces its own voltage. 0.1 V sin(wt). The total resistance is 10 kfl in the input loop, which goes clockwise from the emitter through the various components to the base. then back to the eminer through the base-eminer diode. This diode is forward biased. with the base at all times 0.7 V higher than the eminer. Suppose also that

68. The diagram shows the energy bands of a tunnel diode as the potential difference is increased. In this device,

high·i mpurity atom density causes the occupied donor and unoccupied acceptor levels to spread into impurity bands, which overlap, respectively, the n-type conduc­ tion and the p-type valence bands. In all unbiased diodes, the depletion zone between the n·rype and

Vce =

12 V and that the "out­

put" is 350 n. Now. given that for every 201 electrons entering the eminer. I passes out the base and 200 out the collector. calculate the maximum and minimum in the sinusoidally varying (a) current in the base-eminer (input) circuit. (b) power delivered by the "input," (c) power delivered to the "output," and (d) power deliv· ered by Vceo (e) What does most of the work?

for a large inpu! vollage. Bearing in mind th8! tile vollage across !he forw�rd-biased base-emiUe r . diode ofa bIpolar IranSJ<;lor IS aJways aboul

71. .5bccb an ...,..�potilioa diqn:m. complemen­ ..,. 10 AJIft 10....... IbowiDJ vaJaxe hok motion and conducdoa eJeccron ,.mcipelioo in an openIing pnp

while Ihe inpul currenl I!> proportional 10 Ihe ot. · urpul · curren!. wauld you say thal the bIpolar trdJIsist� . Impedance? Whal abou · ha.� low or hIgh ,"pul t lhe MOSFET?

E

"'""'-.

71. In fDIII)' tlnds of iDrqmrd cireu.ts. lht preferred elemen' of amplifiubonlswirc:h.ing is DOl the bipolar rrarwlror diJcuued in lilt cbqler. tut the MOSFET ImetaJ Olide st'miconduc"ror field effecl rransistor). The

12. The diagram shows a bridge rectifier circuil. A sinu, soidal inpul vollage is fed inro four identical diode each represented by the Siandard diode circuit sym , . The symbol mdicales the direction of conventional ,"'

attOmpMyillJ diagram shcnu one in i� "nonnally off"

�

llale: CondUcUon ei«rrons cannol n()\\i from the n-Iype

1DWft. which is 8II.logou§ to the emilter. "ovet lhe

­ renl flow Ihrough the di�de. The plots show input and oUlput vollages versus lIme. Nare thai the OUlput volt. age is sirictly in onc direction. Explain (a) how this . cuil produces t1Je unidireclionuJ output voltage it d lJ"· and (b) what fearures in the output plOI indicate that band gap of Ihe diodes is aboul half an electronvolt, (I e

bump" in the p·lype region 10 the n-type drald. analo­ ,ous 10 the colleclor. (An npn arrangemenl is shown. bul JUSI as for 1M bipolar uansiSlor, a pnp would yield the complementary device.) The importan. difference is thai rather than a ditttl electrical conlac" O the p-Iype

�

th

reBion, 15 in the base of.he bipolar. the cenler lead. the

r'

pie. lS a conductor bonded [0 the p-rype region bur

mighl seem rhat about one volt is correct, bUI conSide how many diodes are on and in series at any given inSlan!. In fac!. allhough nol the usual habit. it mighl be

separated by a thin Insulaling layer.

more accurate 10 plol Ihe outpUI voltage shifted upward

O.t,t

d

-

(o.qpo)

I

1 -)/

)'//1

-

"""""

c....".)

relative (O the inpu!.)

�

Diode

(a) Explain how applying a bias 10 the gate can cause

lhis device 10 rum on. Should the gate bias vohage be positive or nrgativt (relative to Ihe source)? Why is

lhe conb'Ol mechanism referred to as ''field effect" ?

(b) The MOSFET IS often said 10 be a ''unipolar''

device becau� "alence holes (conduction electrons in the pnp device) do nOI play the m i portanl role thai they do in the bipolar. Explain. Would you

t'J[PCC' a significanl cWTtnl through Ihe gale due 10

electron-bole recombination in the p-rype region?

Why or why nol?

(c) A low·inpul-impedance device is one in which there arc large oscillations

./f!

in n i put cWTtnl for small oscil­

lations in the inpul voltage. Correspondingly, a high­

inpul-impedance device has a small inpul current

-2

Section 10.9 13. The diagram shows an idealization of the "floating mag­ nel trick" of Figure

10.50. Before

i t is cooled, the super­ conducting disk on the bollom supports the small

permanent magnet simply by conlaCl. After cooling, the

magnet floats. Make a sketch, showing what new

magnetic fields arise. Where are the currents that

produce lht!m?

78. A string wrapped around a hub of radius R pulls with

force FT on an object that roll.. wilhoUI slipping along hori7ontal raih on "wheeh" of radiu .. r < R. AMoume a mass m and rotational inertia (a) Prove that the ratio of

l.

FT to the object"� accelerAtion i� negative. (NQ'�: This

Permanent magnet

object can't roll without shpping. unle"iis theTe is fric­

tion.) You can do thi"ii by actually calculating the accel­ eration from the translational and rotational second laws of motion. but it i., pos!-ible to an..wer this part without such a "real" calculation. (b) Verify that FT limes the

speed at which the �tring moves in the direction of FT

(i.e., the power delivered by FT) equals the rate at which the translational ttnd rotational kinetic energies increase,

That ...,

FT does a\l the work in thb system. while the

"internal" force does none. (c) Briefly di�uss how parts

4 The critical temperature of lead is 7.2 K . What is the

7 .

binding energy of its Cooper pairs at zero temperature?

(a) and (b) cone.,pond to behaviors when an external electric field i!- applied to a semiconductor.

5. The magnetic field at the surt'ace of a long wire of

7

radius R and carrying a current I is IJr1I21fR. How large

a current could a 0.1 mm diameter niobium wire carry without exceeding its 0.2 T critical field? 16. In Chapter 4. we learned that the uncertainty principle is a powert'ul tool. Here we use it to estimate the size of

- . . . . . .. .. .. t--..

a Cooper pair from its binding energy. Due to their

Side view

phonon-borne attraction, each electron in a pair (if not the pair's center of mass) has changing momentum and kinetic energy. Simple differentiation will relate uncer­

tainty in kinetic energy to uncertainty in momentum,

and a rough numerical measure of the uncertainty in the

79. The accompanying diagram shows resistivity (recipro­

cal of conductivity) data for four solid materials from

77 K to 273 K. scaled so that the maximum value plot­ ted for each material is I . Two are metals. one of which

kinetic energy is the Cooper-pair binding energy.

undergoes a transition between ordered and disordered

Obtain a rough estimate of the physical extent of the

spins in this temperature range. Speculate as to which

electron's (unknown!) wave function. In addition to the binding energy, you will need to know the Fermi energy. (As noted in Section 10.9, each electron in the pair has an energy of about EF.) Use 10-3 eV and

9.4 eV, respectively, values appropriate for indium.

Comp rehensiv e Exercises 17. If electrical conductivity were determined by the mere

plots correspond to these two metals and what the other two materials might be. Explain your reasoning.

a.• a.' 0.7

static presence of positive ions. rather than by their

a.•

motion. the collision time would be inversely propor­

0.5

tional to the electron's average speed. If. however, it were dominated by the motion of the ions. it should be inversely proportional to the " area" presented by a jiggling ion. which is. in tum. proportional to the square of its amplitude as an oscilJator. Argue that only the latter view gives the correct temperature dependence in conductors of U

;x

.,- 1 .

Use the equipartition theorem

(usua!ly covered in introductory thermodynamics and

also discussed in Section 9.9).

0.4 0.3

0.2 0.1

o +-��---�--��200 I SO 100 so 250 300 Temperature (K)

... a$

,, _, . '.. ... :: .. . -

Comput.tional &.,ciMS

.... ..... . 1_ . .. " .. _ 0 18

C $ p"

&

dAIII odien"(L·J C'h.v1gr XI ' Xl' and X, '0 1,2. 2.4• • J 6- resp«t1W'ly. Mhi�'h gwe., 'Ot'\en �U
.-

fl --n.- S. ... ... * ' orr. II t. ...... . icII ....... .. • .. . ... ... . J ClKJp. " _ _ .-colrl. * j

•• III 2 1 'b .... �l ... ., '4'*-'· • • .., ....".. "' .&. �, . 3iH.&.J

(dl Coa,ider tht


wavelength f ml\�hapC'n Ihough II rna)' �). POlenfial entI'Iy depends on the ampli lUde of the wave' fUOl'hOQ in rht' ....alls. bttausC' Ihe,e are the only plilcC'\ w�re

.... .. _ aoIy I/{sJ .... .. _ - , . O' .. .... 4kr) .... ..... odd « fteD ,.... 01 J OUr ..-- � will ..... . onHi 7 ..... crylGJ . II _ "' _ _ A llaiMr ( ....... .. on1N) """ ..., .. _ .... I/{s, ..... b) -.. • .....,.. ..u

OW' V(.rl i� nonuro Of a "01\(' fundion were alwaJ\ zero ",'hcne\'er L'(l) I\- nonlero. the expeclar'on valuC' of U would � lem,) Comp.lfC' rhe appro�lmate LnetIC

enelJ'ie� of Ih('\-(' IwO \la lC"

their potC'ntlal ('nregic\-

,... .... . I Mr)'1IMn: AClplta. UIIII-width qion

_ .. x, __ • • 1, ....., fu """"" NU. x. " . ( . ... 1)/2

+ ..,aI1'u - (x -

"'''21·(8 - w�

1I. _ _ X bJ ... .... II1. dIio """""' d. IOd d by .. .. In. • • ' Now .. caa ..sne . UhUblllS L'O • • , .... .. .. ..... ....... OI

.... . . 0. U(.)

•

I . O' X J . X:-aad x).

Vo .-tr. 0. 0) .Vt(.c, X" 0) tw(J', Xl' 0) IW(., Xl' 0)

Til ....... oI.,� UlAl alJows '" 10 po. � for ,..,. .( "",, 1" Yow II it odd or evea). buC Il lbo me&ni .... til tffect, 0lIl' aytIII .... tbreI lIkW welli II nqltive .(. 10 d .. IIWII .... ('"* Many funtIcr IIUIbcs lit possible ..... .. tr.rwmt provided EurtIW 80. for eu.mplt. (an ...... "InCU ...,.. ..... Ibowl", the effect on bud .... n. .... olalOmlc.. aJIO UIlly be varied.) t . " .... Dr1inr tl(.I) .ilh X I. X:' and X1 .... 10 I. 2. '" J. mpectJ�ly. This .i'fi unit·widlh

.. ..

....... 1IpIIWd by wall. of Iift'O widlb-chaI is, �

.. weD 7 1M1lb " For U". VIC 20. UId for olx, Lilt

0.001 ,a) AlUowiDa '" C1uIpea 5 JWcItliAts OD c:boo\.

.. �) ... �Ar). tal bocb odd UId fMIl functioDs II _ aiol .... 01 E bJ ...... . II �l m.lbples f1I .1r lid pkIaiDa: *' rauJb fTooI, • • OIO.t - S. FiBd , .. aDowed CIICIpCI. NoR tbal lbt indicaIof 01 havi", ...... . aDowed eDaJY IS _ Rip 01 � diverJillJ .....1 tIiI l'bI loweII eDeIJY cw cwo will take Ihe mosI walt. EacepI for ... Iblft 1I1O need 10 CJ.teed thtce � fiaIns- Ibl Widl . IOd ' " " . . ..6... .. l . plftidt-ia+boII. aapa .. simply ffflL2. How do ,.. -.- �. and why do some qree beUer

lun..:lioO\ ol ltlc lOp ofthr fint ...J\e .

Nrtd and lhr bOl'rt1m 01 rhe \Cl:ond. KmefJC energy drpend� ()n momC'nlum. and thu� on the funcllon ",

'1.

.

then dl�cuss quanlIl311\rl},

Argue thal lhere �hould be a

rel.mel) l.llJe energ) Jump from one Molle 10 the olller

Define Uff) With XI' Xl' 2 4. and 3.6. J'C''Ipecli\-'ely. which '1\C'� MId XI \('1 10 1,2, . iC\en cquall)' \p.u;ed l,I,ell\ �paf(ued by walls of�ldlh

Donor and ACttp.or LneJs:

O,� For Uo- U'-I! 20. and for ,lx. u-.e 0,001 (a) (No,,:

Th,\ p.il11 can be \-klpped If al\o working Eurtise 80.) follOWing th� Chapler 5 gUldehnes o n choosing 4t1O)

md ;(.l.II, le�1 bolh odd and e\'en funclions al dltrettnl

mal value, 01 £ by find109 1/1 al all multiples of.lx and

ploumg the re�u"s from \' '"" 0 10.t

::

5. Find 1 4

aUOl,I,ed energi�� NOle thai th� mdicalor o f ha\Ing

pA.,o;ed an allowed energy i� Ihe nip of Ihe di\erg:ing

l�e-r laJl, Thr 10we\1 energy or IWO Will lake the mosl work. Except for Ihese, there i� no need 10 exceed three: significant figures. Aflerward. make U ,>cUller plol of

fit \'enus n. wheR: n gOt'\ from I

10

14. (b) Now

replace IWO alom\ with impurity aloms as follows: For or Ihe well al 2.4, change from 0 either th� l,I,e/l al

1.2

B

100.1 (NOIt': Changrng one weI! automatically changr�

!he cotre'lpondm& well al negative � We avoid Chang.

ing lhe outermosl well� �imply becau� it doesn 't work

a\ nicC'ly,) TIus pUiS the boltom of Ihe aheR:d well al O.IUo' or 2 unlh. AgaJn find 1 4 energies, and make

another !>Caller plot (c) Repem pan (b). but chOOSing

B In lhe " Impuri ly" alom. pUlling irs bollom 31 -2 unih. Cd) Discu!.s how Ihe impurilies added in pan�

- 0, I for

(h) and fc) cotre\pond to atom, I,I,ho\e ...alence differs from thaI of lhe intrin�ic atom!., (e) If each inlrinsic

110m comes with two eleelron\. and Ihe impurilies

come wilh one and three, R:specII\'ely. which slat�s would be filled in part:. (b) and (c)? Remember Ihal

there are IwO �pin Slate,. (n Discu,.. the overal l result

of adding Ihe impuritie�,

N u c l e a r Phys i cs

11

Chapter Outline 1 1. 1

@

I \ .2 1 1 .3

I tA

Basic Structure Binding Nuclear Models Nuclear Magnetic Resonance and MRI

1 1 .5 Radioactivity 1 1 .6 The Radioactive Decay Law 1 1 .7 Nuclear Reuctions

. n the preceding chapters, we tended to study ever-larger objects with photons and electrons. one-electron moved to We began � tron atoms, then to molecules and solids. In some sense, n the multielec

I

n a[o l.n" ',nue the trend would lead to other disciplines-chemi�try. engineering, to co and �o on. We now head the olh�r way. .

contam nuclei, studying the "atom" usually means Although all atoms of its orbiting electrons. The nuclei are assumed to behavior on the fIX'using nuclei . But are not always inert. For one thing. they can s n tiaU y inert

b< e se

' and such nuclear reactions release spontaneous1Y or otherwtse, ra f menl, grrnous amounts of energy compared to chemical reactions. In nuclear 0 ttmg e1ectrons t hat are 0f I'title concern. It IS true that they ,"h �ics, it is the orb" ,

P } bound to the nuclei, but compared with the energies within the nucleus,

�ctron �

bindi ng energies are usually negligible. The nucleus is the focus. hard problem t? solve. While much has been learned, it is nfortunalely, cs contmues to be an active area of research. auClear physi , ' cannot be overstated. h ex.plalOs the tance 0f nucIear phYSlcs impor The

�

and nuclear weapons; nuclear magnetic resonance perotion of nuclear reactors produce images of the body's interior; and radioactivily­ routinely used to

�I �

ergelic particles emanating from nuclei-is both a useful diagnostic tool, via n

dioactive tracing and dating. and a mounting disposal problem. as we con­ artificially produce more radioactive materia1s. Moreover, tinue to unearth and

the study of nuclear physics is an essential step in our quest to unravel the fun­ damental structure of the physical universe.

11,1

Basic Structure

All nuclei consist of protons and neutrons, known collectively as nucleons. Masses are given in Table I I . \ . Introducing important symbols, Figure 1 1 . 1

depicts a helium atom-two electrons orbiting a nucleus of two protons and twO neutrons. We use the symbol N for the number of neutrons and Z, the

475

-

-

-

-

�

-

-

-

-

-

-

-

-

- -

�-

- �-�-

.., , , , ... -

I 6126J17 )( '0 .17 ...,

"

•

••

I 6'49273 x 10 11 ..,

o

If •

•

-

9.1(9)( 10 u k,

•

....

hr.

.....

-

� '=>

J.(J07276 u

I 00tIM.'I u

� "'AA X /II � u

lor rbt number of prolons. The ..ymbol A. caJled the IRI5s

II &he lOCal number of nucleons.

At we found in C'bIpIer 8. an element's chemical behavior depend, onl}

a. die IIUIIIber of eJecuons miring il.\ nudeu.�. equal 10 Z. But nucle, of .he .... demenI rarely beha� alike jf they have different numbe" of neulmn,.

ftar ...-.xe. while tbt nucleus of ordinary hydrogen 1\ "" mply a prolon. about 0.01$" 01 hydroaen IIOmS in naluft' have nuclei con ..i<;ling of n prolon plu\ a

aeuIrOn. Known IS deulerium. or hta'o'), hydrogen. rhi, alom io; roughly [\liice a.�

-rve . ordiaary hydrogen. Deuterium participale\ in chemical reaclion.')­ 01 orbiting electrons-precisely IJke ordinary hydrogen. mak.

M •......., .

ill cbemicaI .,...a:ion or me fWD problematic. However. il participates in """"' �gemenls of nucleon!'.-in va'itly different ways. Nuclei with the same number of prolom. bUI different numbel"\ of neutron�

ft _ 10 be IIafepeJ of the same element. Thu\, we \Dy thai Ihere are two ....lIy occurring isotopes or hydrogen-ordmary hydrogen and deulenum. dlpic:1ed in Fi,we 1 1 .2. Both art stablt. remaming unchanged mdefinitely. HycIropa his . third isotope. tritium. with one prolon and two neutrono; in Ihe DUt'1eus. Whik it can be produced artificially. it o. e�senllally absent In nature because it i

OI1lIa". meaning Ihal it is subject 10 a "'pontaneous nuclear

reaction thaI changes it 10 a different element, known as radioactive decay, We often use the convention elemenl-A to refer 10 di fferent jr,otopes. but hydrogen

i I specill case. Although we could speak of hydrogen- I . hydrogen-2. and bydroaen-J. we usually use the name.� hydrogen (undef'>lOod 10 be A deutaium. ond lritium.

=

I).

Helium his two naruraJly occurring isotopes; helium-3 (two protons. one

neutron) IDd Ihe more

abundant helium-'" (two protom. two neutrons),

..... 11.2 Isocopes of hydrogen. ,

,,

.- - -

,

,

•

•

, , ,

,

,

,

,

,

0

, ,

,

'-'

�, 5_

,

,,

.. - - .

'.

d1 . -- -

Deulerium: Stable

, ,

,

,

,

.. _ . -

Ql

,

TritIum Un\lable

•

,

,

It 1 But&: StNc:1\II'I 477

I/'I.l� ("I' protonl\, six. neutrons.), the most abundant carbon tsoto� an JJl'l I' the hasis of a standard measure of mass. The atomic mass unit U 1\ C

� ��O. �

to be one·l\\elfth the mass of the carbon·!:! atom, including ele"tn\n.. t'1ement� have a host of naturally occurring isotopes. For e"ample, hn �1an)'. �uch as aluminum. have only one. Some ha\'e none; they aren't \0 nature. Nonetheless, they may be produced artificiall)" or the) may \1:1-' necllngly from the radioactive decay of other nuclei. While no l'iotOPC� duced· in nature arc �lable-at least, none has yet been artificially pro 11'1" �,'('"I and uranium·238 instance, For unstable. are nature in prese/lt I'"otopt:!� undergo ld U , silver than nature in abundant more do �llnl.2J2 are both .... cti\'e decuy, but they will be around a long time, for their half-lives (see lJ l( ,l r ll ons of years. St.;U(lU 1 1 .6) are billi ,

. oJ

1.660!,')Q I( '0 to k.'l

�

_

tkl� Size

'{b(' fir'! ex.periments ,uJJ hi� colleagues (Section 7.3), fired

to probe the nucleus" carried out by Ernest Rutherford Figure 1 1.3 Probing for Ihe radiu!.i of Ihe nucleus. Q' particles toward a thin foil of gold. icle is the nucleus remaining when helium's electrons are stripped n (t part A Gold nucleus a pllr\II:le The vast majority of a panicles passed through the foil undctlected. missed! A few bounced backward. Evidently, these had encountered a ct. heavy, positively charged object-the nucleus. The size of the a ilm p was determined by increasing the a particle's speed. In shorl, the idea � udellS -I ! 0, hat an a particle approaching a nucleus loses kinetic energy to electrostatic t energy, and a head-on trajectory" as in Figure 1 1 .3, would enable it to closest. AI high enough initial speed, ils closest approach would r03ch JPp btt':tch the nuclear surface, and the scattering would change in an obvious \\a)'. for the force would no longer be simply electrostatic repUlsion. Scattering experiments indicate that nuclei are roughly spherical. with radii obeying the approximate relation

�\\TtIt)av.' )

�ential

r = A II]

X Ro

(I I-I)

where A is the mass number and Ro is 1.2

X 10-15 m. A femtometer (fm)" size. Typical atomic dimensions. nuclear of measure good a thus to- S m, is defined by electron orbits, are 5 orders of magnitude larger. If lhe atom with its electron cloud were the size of a football stadium, its nucleus would be a bil �maller Ihan a ladybug. The dependence of radius on A in equation 0 1 - 1 ) is conspicuous. The nuclear volume is I

( 1 1-2)

Because neutrons and protons have nearly equal mass. the fact that the volume is proportional 10 the number of nucleons means that it is proportional to the lotal mass. All nuclei have about the same density!

-- � -

--

�

--

-� -

Scale of nuclear dimenSions: l fm = lO- l� m

mass

A X mRlldeon

volume

A X }1rR5

p = -- =

,...:c.:.:,-'-'-':::'-i'� =- }1r(1.2 X 1.67 X 10- 27 kg

JO Z5m)l

( 1 1·3) '"

lOl? kg/m'

Conslatll density, in tum, suggests that nucleons behave as incompressible objects packed as close together as possible. Although this view is oversimpli.

fied, experimental evidence confirms thai nucleons have a strongly repulsive hard core. The volume allowed each nucleon in a close-packed arrangement

�7TRij.

So would be the 10lal volume divided by the total number A-that is, we see [hat the effective radius of an individual nucleon is Ro' or about I fm.

1 1 . 2 Binding If il is indeed true that the nucleus contains many prolOns close together, then il is natural to wonder what prevents it from exploding. To this there is a sim. pIe, if smug, answer: Another force is at play, an attractive force thaI holds Ihe nucleons together. It cellainly isn't gravitation. Simple calculations show that this attraction is woefully inadequate to hold protons so close together against their electrostatic repulsion. The "new" force is apLly named the strong force. Just as all particles endowed with mass attract one another gravitationally and those with charge artract or repel electromagnetically, all nucleons possess a property mal causes lhem to attract one another via the strong force. The strong force is one of nature's fundamental forces: gravitational, electromagnetic·weak, and strong. Actually, what we refer to in this chapler as the s[J"ong force is only an aspect of the strong force, for which we often use the

1 2 discusses all the funda+ 1 1.2 compares the relative strengths of the fun.

descriplive renn internucleon att,action. (Chapter mental forces together.) Table

damental forces, based on assumptions of typical charge, mass, separation, and so on. The strong force is not as easily characterized as the more-familiar

forces. For one thing, it does not diminish gradually with panicle separation, as do lhe

I/r2 forces. It is strong only for nucleon

separations less than about 2 fm; that is, only when nucleons are not much farther apart than the effective nucleon diameter. AI larger separations, the strong force is negligible. As we shall see, this plays an important role in the stability of the nucleus. Electrosta.

r.-alE 11.2 F.""

Forces compared ReLative

S......

.....

......

-I fm

"""","",- . -10-2 _10-6

Long: (X 11,2 _IO-J 1m

G...... . ...

Long: (X J/,2

Wak

_10-39

tic repulsion, allhough weaker than the stroog force at close range, becomes a significant destabilizing effect in larger nuclei, where protons are too far apan to anract via the strong force. (In no ordinary ci rcumstance is gravitation sig­

nificant. Weird, unearthly cases, such as neutron slars and black holes, are dis­ cussed elsewhere in me text.) It might seem reasonable to claim that because the nucleus would other+ wise explode, there must be another force hidden from our senses, but where is me proof? Much of our evidence comes from scattering experiments in which one nucleon is fired at anomer in all possible combinations: proton­ proton, proton-neutron, neutron-neutron. The details of how they scalier

ation about the nature of the force they share. Figure 1 1 .4 "t U!I mform

potential energy shared by a nucleon at the origin and B theat strong-force �piC distance r, and it illustrates some of what we have arbitrary other force is attractive, though with an incompressible hard �.a The strong short ranged and strong. We also find that whether it is between an

�

nd it is olOns. two neutrons, or a proton and a neutron, the attraction is very c'0 r identical . it does, .h�wever, deJ>e:nd on e orientation of the nucleon not �urpnsmgly, there IS no simple formula for this force, .. ins, Perhaps indeed has yet to be fully characterized. Nonetheless, we can explain quite a bit from what we do know about it. ore

�

::a,.,y :h.ich

Sta bility : A Theoretical

Model

engaging in an extremely strong attraction, we might lmag­ With all nucleons could bring together any combination and have a stable nucleus. vast majority would be unstable, disintegrating in a time span the owever. [allging from an instant to an eon. A few would be stable. The following discussion of nuclear stabiliry may seem rather s.peculative, forces in the nucleus have defied formulation of a compre­ but the complex htnsive theory. so we must resort [0 a model-well-informed but simplified guesswork. [n a later section, we discuss well-established models thai refine we one we begin here, bUI it is good to gain a quahtative understanding of the \arious factors fust. As we do. we won't speculate whether any particular combination of nucleons should actually be stable; we will simply argue ....hether a given factor should tend to make the combination more stable or le�5 stable. A nucleus is more stable when its constiruems are bound in a state of lower energy, requiring a greater expenditure of energy to extract a repre­ q:ntali\'e nucleon, Energy is all-imponant.

'ne that we

�

Two-Nucleon Nuclei

The simplest possible combinations of nucleons are two protons (p-p), two

oeulIOns (n-n), and a proton-neutron pair (p-n), Whkh should be most stable? . Let us look at the factors The short·range strong attraction shouJd create some. thing like a narrow but deep potential energy well. and if the force doesn't dcare" whether the particles are protons or neutrons. the weU should be the \.llIlC for all three combinations. This condition is no guarantee that any would hold together-wells can be too shallow to have any bound state.!. (see Exer­ cise 5.38)-but at least they should all be the same. On the other hand, the p-p combination ""ouJd mclude Coulomb repUlsion between the proton!>. which v.ould rai� the energ), and this argues that it should be somev.hat less stable than the other IWO. .. are ValId. but the ltuth is that onJy the p-n. known � a These argument dtUteron. form' a bound nudeu'i (that of the stable deuterium alom dio;cussed earlier). 1be n-n and p-p don't stick together at aJl. Why" The internucleon aruaclion i, .,pin-dependent. and II i... stronger ""hen spins are aligned. We will :e 1.5 Ie.. 'loigOlficant In larger nuclei. but here it call.. . • pendcn e find thaI milo d thai Ii crucial in all nuclei: the exclu�ion principle. factor r e h t o an play into

Fig.". 1 1 .4 The basic elements of the internuclean (strong force) poIe11tia1 energy-a strong. mort-range anractioD with a repulsive hard core. Sudeons Iolrongly atUaCI

lpo!cntia] energy drops). but only when close e/1Ough to "'toUch.�

ur,}

f -'7'!!'...-----" H-+ . rn -U

.

•

•••

:,0

.............

�p!

When roo dole, die.. han:! can::s repel.

,

,.... f1.l

The dnlrcroo'UtUlJ'Oll

Mtd profua buund In. well rnullIIJI

" -+�-r

from IIIftt w.:tt� poIftIuaJ cntlJ)"

Proton�

WId llC'utrons

art" I-pln

i f(mlion�, '0 they cannOI occupy

the same

�.)\I("m. The' lo�(�1 energ) po\�ible for Iwo nucleons in a "dl �Id haH' bolh in Ihe ground �piliioll �Iate with their ..pins aligned (the !OWe In lhc

�

mentation of slronger allm:l.on I, 001 ,ut:h a stolle is forbidden 10 the p-p and thC' eu-Iu�i(ln prinl,:iple. h Ihi' indeed why Ihe p-n forms:, but the P-p

I)-I) by

and n n do n,,(') E'p'-'nmental endence \lenfie.. thaI the deuteron i.� ani)' ... no exciled bound Slates, jusl the one �ly bc.-.und: 11\ potenllal well ha �mund slate, .\IO(f'O\er. the neulmn olnd proron ,pins art' aligned (its total SPin -

I� II. If til< p-n cannot �md any other way-with '>p,ns opposite or widJ one nudeon in a higher 'patial \tJte-lhen the other two should IIor bind at alL The enC'Iog) in the deuleron i� depir.:led in Figure 1 1 .5. (The repulsi\le nucleon

core, are i�nored.) Th'o nudeom occupy the lone bound slate in a well that ari\e\ from their mutual altrJclion.

fIsIu,. 11.6 RwumS tntfll} ret

Arbitrary Nucleon Number

nuclcvn JuC' II' Ihe ,'rpng mh.'mud("(ln

We no� com.ider nuclei compri�ing any number of nucleons. addressing in rum the efted� of the Ihree main faCIO,...,: the strong force, Coulomb repUlsion,

"lIr:atlion poly fill: �mJ"C'sl nudt.

t1a\e' lew Ix>nu. (Itt nudl'on, In 1.l11!C"

ilnd the e:(clu,ion prinCIple.

nucld. many lIurk.-ons J� !umlYnutd

Strong Intemucleon AUraction A two-nue/eon nucleus has only one bond. but a three·nucleon nudeu� ha.� three. Thus, the fonner has half a bond per

nlJ/1ron and the lalfer one bond per nucleon, A four·nucleon nucleus would ha\e m bond" or one·and.a-half bond� per nuc/eon. so a representative nudeon would be harder to extract than in a three·nucleon nucleus. This mcrea\ing [tend n i bond� per nucleon does not conlinue indefinitely. however. Nudeon� attract, bul they also have a hard core thai causes the nue/eus to

l .,

grow like a collection of hard spheres stuck logether. The intemucleon aUrae· lion i\ \0 short range that a given nucleon attraclS only Ihose immediately surrounding it. �o each surrounded nucleon will have Ihe same maximum number of bonds. (We ,ay that the force salurates.) Of course, nucleons al the surface are not completely surrounded. but as the nuclear sphere grows, the number .11 the surface is a diminishing fraction of the lotal. The ratio is essen­

F/vu1"9 11.7 [pu/mnb rrpul\ion faj'e, prI" "n eneq!ll,·'.

lially area/volume =

41Tr 2/� 1I"r3 oc

IJr. Thus, the average number of bonds

per nuc/eon should inilially increase fairly rapidly as the nucleus grows, then gradually approach a con�tanl, as more nucleons become surrounded. Figure

Cllulomh

J ,

I

1.6 show� the trend.

Rather than bonds per nucleon. however. we now speak

of binding energy per nucleon, Binding energy is the energy rnal would be required lO pull all nucleons apart. and binding energy per nucleon is the frac·

E,

E,

-I---,;�·l

-t--�:

Scurrun

T

;--

lion required for a representative nucleon.

I:

E,

Coulomb Repulsion All pairs of protons in the nucleus repel. In effect, this positi\e potential energy shift� all proton energies closer 10 rhe top o f the finite well in 'II. hich all nucleons are bound. as depicted ID Figure I I .7 for the case of helium-4. In such small nuclei. all nucleons are

"

touchi ng:· \0 there is slill a

,(rong net attraction between all pail"!. But in large nuclei, paiJ"\ of protons can be too far apart to attract via the strong force, and these pairs add to the nucleu, an uncompensated nel repulsion. Thus, while under control in small

Figur. 11.8 Binding energy per nucleon due to both the �trong inlemudeon attraction and Coulomb repulsion. t.ur�e nudei hone proltln\ thai do nol

Small nude. ha\c:

allriltl.'� \

Ie..... NInth pc:r nudeun

remove lrom lin

2

A

Coulomb repulsion should be an increasingly destabilizing factor in The energy needed to extract an "average" nucleon becomes pro­ ones. larger Combining this with our previous arguments. we should smaller. gressively expect the binding energy per nucleon to vary as tn Figure 1 t .8. Accordingly. �omewhere between the extremes, there should be a mass number more !ltable than all others. for which the nucleons are the most lightly bound possible. nuclei,

The Exclusion Principle To this poim. it would seem that the beM way to produce a stable nucleus is to build it of essentially all neutrons, for only pro­ tons repel. This argument overlooks the fact that protons and neutrons tn the same nucleus must obey the exclusion principle, each independently. Putting Coulomb repulsion temporarily aside. the most important consequence of the e'(c\usion principle is that for a given A, the state of lowest energy would have equal numbers of protons and neutrons. Why? Suppose that Z does equal N. as illustrated on the left in Figure 1 1 .9. The lowest energy state consistent with the exclusion principle would have aU the Figur. 1 1 .9 I gnoring Coulomb repulsion. the exclusion princi­ ple argues that for a given number of nucleons. energy should have N = Z. Slanmg wIth equal

•

numben. of nCUlron� and

f: E,

' higher energy level_ _ _ _

\

-

•

E,

E,

_

./

E)

�7

Neutron

d conveni ng one to

an

the other force� a nucleon to

pruto"",

E,

__

the lowest

Ez

E,

Proton

.

--=- : - - - -

- .�--=..-..:. :=.=..;..- =� --

..... ft. to In .... nucki. .heo Coulomb�ltJOCI becomes l.I,,"ti·

e.I. dw JowaI tnftJY IAouJd

�N>Z

Anllmn cnerJ} Ic\-e!, filled. c&:h �ilh an 0ppo"He-�p,"s pair. up 10 some maxi_ mum cnergy_ Comlde-nng only !he inlemudeon altraction, protons �ould occup) 110 lde-nllf.:aJ ""cll and would fill le\eI� 10 the same maximum energy. NtM. keeping A that a dtcreas(' b)' J in Ihe prolan number would mean an 1f)(:Tea.\e b} 1 In the neutron number. Becau�e all lower energy levels are filled,

ncubun wQUld ha\e 10 occupy an energy levcl higher than the previ00\1) occupied prolOn leH!I. PJld me 10lal energy would increase. The argumenl applic\ )1.1\1 � well If we chan!!e a neutron 10 a proton. Thus. Ihe slate Z :::: N is

lhc

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....,-1-

�

of 10....('\1 energy (If elmer top level initially had only One neutron or prOlan, we might ha\'C' to changc a I.C':cond particle before the energy increased.) UIU_\ now faclor in Coulomb repulsion. In �mall nuclei, where essentially

all nucleons lauch, all prolon repulsions are overwhelmed by the strong attrac_ tion. so Z :!' N should be me more �table Slate. In lurge nuclei, uncompensated Coulomb repul�ion, may shif! me prolon energies upward enough to align wim

different neutron 1C'\'els. as depicted in Figure 1 1 .10. The lowest energy slate­ with me lap.. oflhe proton and neutron levels roughly equal-should then ha\'e

more neutrons than protons. A balance is struck between having eXira neutrons \toe as nonrepuhl\e "glue" and forcing them into higher-energy states by the

e).du�ion principle. O\craJl. our modcl �uggests mat a represenlative nucleon should be most ti!!htly bound-in a 10we�1 energy stale-in a nucleus of intermediate mass number, but il is nOI a function of A alone. The bindlOg energy per nucleon

should ha\-e a rtla/h'e maximum at Z a N for small A and al N > Z for large A.

Stability: The Experimenta l Truth The \alidilY of an)' theory rests on ils agreement with experimental evidence_ Obtaining evidence to support our model of binding might seem 10 inVOlve pulling a nucleus apart while measuring forces. bUI it is much simpler than that. All we need to know are nuclear masses. If we were to pull a stationary nucleus apart into separate stationary nucleons, we would have to expend our own energy. which would increase the system's inlernal energy and therefore It� ma�s. The mass of the parts must be greater than that of the whole bound nucleus. Figure 1 1 , 1 / bears this oul for our simple!!t nucleus.

deuteron mass = 2.013553 u proton mass

,.... t t.11 A dtUlC'ron ""C'IJlh) IC'" lhIn the sum of liS parh

+ neutron mass =

mas� of pans - mass of whole

=

1.007276 u + 1 .008665 u = 2.015941 u 2.015941 u - 2.013553 u = 0.002388 u

We o..ee lhal we must do work to pull the deuteron apart. For an arbitrary nucleus. lhe energy required to pull all the nucleons apart-Ihe total binding 2 OIJ�.H u

energy BE-is lhe difference between the final and Ihe initial energ ies which ,

Proton

Nrutroa

1.007276 1.1

1.008665 1.1

o

<)

follows !limply from [he mas� difference.

( I 1 -4)

I I 1 BmdiftJ 4U

,.t.ppl)'lng this to the deuteron, we obtain BE

� (0.002388 u X 1 .661 X 10 " kglu) (3 X 1 0' m/,)' 1 = 3.57 X 1 0- ] J = 2.22 MeV

The atomic nucleus isn't special in this regard. The \anle argument... hold for electron-proton system that we know as the hydrogen atom. Gh:en bound the must be added to pull the atom apart. equation ( 1 1-4) ,how... that the eV Iblt 13.6 oltOm should be less massive than the sum of the electron and proton masses by ' 4 X 10-)5 kg. The important difference is in thefractiolla/ mass change. It i... 2 small for the hydrogen atom: (2.4 X 1 0-)5 kg)/( l .67 X 10- 1 kg) � 10 ·8. For !he deuteron. it is (0.002388 u)/(2.013553 u) iil 1 0- 3. Although h,% may not sound like much. as a mass-energy conversion It is huge. The 5 orde", of magnitude difference between electronic and nuclear binding energies is typi­ cal. and the simple reason is that nuclear bmding involves much stronger

��ry

forces.

Our theoretical model looked at binding energy per nucleon. so to judge it� merit. we need to divide total binding energy by total number. For the deuteron. this is (2.22 MeV)/2, or 1 . I 1 MeV per nucleon. For arbitrary nuclei. equation ( 1 1 -4) would suffice for BE, but because tables invariably list atomiC rather than nuclear masses. we will find it convenient to use an alternative form. ( 1 1-5)

Here. the mass of the parts has the neutron number N multiplying the neutron mass mn• as it should. The proton number Z. on the other hand. multiplies not the proton mass m ' but the hydrogen atomic mass mHo Therefore, this term is p tOO large by the mass of Z electrons. However, M�x stands for the atomic mass ofthe isotope of element X with Z protons (and electrons) and A nucleons. and thus subtracts the proper number of electron masses. (Actually, this form is not perfectly accurate, for it Ignores electro" binding energies, but they are rtf)' small in comparison. See Exercise 18.) The hydrogen mass is given in the margin. along with a conversion factor very handy in nuclear physics. EXAM P L E 1 1 . 1

Calculate the binding energy per nucleon of an iron-56 nucleus. SOLUTICIN

Fromthe atomic mas!> data of Appendix I. we find the mass of iron-56:

The neutron number N i.. A

�:Fe =

Z - 56 - 26 = 30. Using ( 1 1 -5),

BE = (26 x I 007825 u =


55.934939 u

u�

+ 30 x 1 .008665 u - 55.934939 u)e:!

B\tldmg enerqy, at

rn.

,

u

',J

3te('! from masses

1 .00782c;. t.:

)( c'

931.5 MeV

4M a.- ' 1

Nudar PII�

more (han half a nucl�on ma.u' � cOn\�l'\ion fal:lor rna) be u-.cd 10 c�pres.') r in MeV/u. Tht iron nud�u\ j, l�.., ma"I\� than the \um of lis parh by

to.52�-I61 u) x 931.5 MeV/u

=

492.3 MeV

Thtrt art 56 nucleons. so the binding encrg} per nucleon is

BE nudeon

'""

-192.3 MeV 56

=

8.79 MeV

Figure I Ll 2 plots BFlnucJeon for the naturally occulTing isotopes. We �ee how it validales our model by considering different views. Figure 1 1 . 1 3 is a lOp view that shows the localion, of naturally occulTing isotopes i n the ZN. plane. The) describe a path known as the curve of stability. As our model

predicted. for "mall A, the grea(e�t stability occurs at N =

Z.

but it tends

to\\,aro N > Z as A increases. Figure 1 1 , 1 4 is a cross section of Figure 1 1 . 1 2 taken roughly along the curve of stabihty. Because BElnucleon varies with both N and Z, no two-axe!>, plot can represent it for all isotopes. However, Figure I Ll'" �how.� the general trend with nuclear size. and it looks conspicu_ ousl) similar to Figure I l.S. From the deuteron's I , I I MeV, i t rises to a maxi­ mum at aboul A = 60 of S.79 MeV. shared by iron-56, iron-58, and nickel-62. Of all elements. these have the most stable. lightly bound nuclei. Thereafter, the BE/nucleon plot falls slowly to 7.57 MeV for uranium-238. owing to the growth in Coulomb repulsion between prOlons too distant to engage in the mtemucleon attraction.

Figure 11.12 Binding energy per nucleon versus Z and N.

"

-140

'-120 -100 -80

> •

"

- 60

] • 0

-40

, 0 , " .,

-20 T

'0

,

40

Z

60

I

80

"

F....re 1 1.1 3 Naturally occurring nuclei cluster alOllJ the curve of stability. <:un,0I \tabililY

"

+

t

+ t +

20

80 z

We stress binding energy per nucleoli because of its imponance to releas­ ing nuclear energy. Binding energy alone increases rather consistently with A. For instance. it is 2.22 MeV for the deuteron. 492 MeV for iron-56. and

1802 MeV for uraOlum-238. This occurs because the number of nucleons increases. and each nucleon 1� bound by some net attraction to its nucleus. However. if we wished to find a minimum-energy state for a fixed amount of nuclear matter, we would not group nucleons into nuclei of the largest A. Gi\'en J total nucleon\ di\ided IOto nuclei of A nucleons each. the total binding energy would be the number of nuclei JtA limes the binding energy of each

Ducleus: (lIA) ' BE or l · (BElA I. We see that for fixed l. nuclei with the high·

est binding energy per nllcleon would give the highest total binding energy­ the lowest energy �late-and would enable us to extract the greatest amount of energy for other purpo�e... . We will return to these ideas in Section 1 1.7.

- � 11

Nucle&J Ph)'\a

., ..

FIIUN "

Banding energy per nUI:I�on ver.u� A "F,

,

7 >

� •

c

S

" • c

-

W '"

• l

4 )

2

,

rH, :

, " , ,

lO

"0

100

200

A

1 1 .3 Nuclear Models We now discuss two of the most useful theoretical models of the nucleus. Each has its emphasis. Although they agree In some ways. each rests upon unique assumptions and simplifications and makes its own predictIOns. Until and unless the nucleus is "completely underMood," there will always be rOOm for complementary models.

The liquid Drop Model The force belween the molecules In a drop of water is strongly repulsive at very small separations but attrnctive at greater separations. In fact. the polentiaJ energy quaJitatively resembl� Figure 1 1 .4. This explains why water is e5senllaUy incompressible while nonetheless exhibiting a reluctance 10 be pulled apart. To move a water molecule to the surface, we must pull It away from those that would otherwise attract It on aJl side�. This takes energy. Thus. the lowest energy state of a water droplet is one in which the surface area b minimum-a sphere. Similarities between the energle, In a liqUid drop and those in the nucleus gi"'e the liquid drop model Its name In both cases, mterior particles are attracted to all panicle� m the Immediate vicimty. Were all particles interior. the total bmding energy would be directly proportional to the number of pani­ cles. Particles at the surface. however. "hare fewer bonds With others. which reduces the total binding energy. The centerpiece of the liquid drop model 1s an expre.....ion for nuclear bind­ ing energies. To know blndmg energy IS to know mass, so It may also be said that the model yields nuclear ma�se�. We have already dl\Cu)o�d the go...eming factoT\. and the quantitati...e expression we now obtain Iii a sum of teons arising from each.

1 1 .3 _ _

.,

Volume Term nsider only the internucleaR attraction and ignore the fact that not all nucleons are surrounded in the nuclear interior. then the total binding energy shOUld be directly proportional to the number of nucleons. Thus. we begin our sum with the tenn shown in the margin. If we co

BE = c,A +

c

While the proportionality constant 1 has been theoretically predicted, the ItfJIIS yet 10 come include similar constants, and all the constants are usually deter­ boSe" so as to give the best overall agreement with experimentally

Figure 1 1 .15 Ifj neutrons become pr0-

tons. !he cl1CtiY ;..,... by

c

mined value�. The question might then arise: Is this really a theory. or just a

experimental findings? The answer is that it is a model of a way of codifying p le x system by experimental knowledge. The known binding "ery com

�

/' 1/1 1// /1

guided

energies per nucleon describe a certain shape as a function of Z and N, and as Figure 1 1 . 1 2 shows, it is not a simple shape. For most functions with arbitrary it would simply be impossible to fit the + + constants (e.g. ,

c]z2

c2N

�

c3).

known shape. That the functional form of the liquid drop model can be made to do so as well as it does is strong evidence that the assumptions upon which it is based are valid. Furthermore.

Jill -rl

we would expect for a good model. it may be used to predict binding energies of isotopes yet unknown.

Surface Term

"

sllrface

-

reduced

by a factor proportional 10 the sur· the total binding energy should be face area. which is in tum proportional to volume to the two-thirds power. We know that the volume is proportional to the number of nucleons, SO we obtain the leon given IR the margin .

Coulomb Term

reducing

the binding coulomb repulsion raises the overall potential energy. energy. The electrostatic potential energy shared by two protons separated by

e2/r. In a nucleus of Z protons, there are Z(Z -

I ) pairs.

and typical prolon separation i� of the order of the nuclear radius. which equa­ tion ( 1 1 - 1 ) tells us is proportional to A Ifl. Thus. we arrive at the tenn shown in the margm. Asymmetry Term

Coulomb repulsion aside. a lower-energy state occurs when N =: Z. due to the exclusion pnnciple. To quantify this effect. we first note that neutron and pro­

Ion energy levels due to the internucleon attraction alone should be nearly iden­

are filled to equal heIghts. E", and that the levels are !\lIIgbly equally spaced. oE apart. "-, depIcted in Figure 1 1 .15. lfj neutrons are tical. Suppo
to protons (or 'Ice ,·ersa), the top neutrons must go from neutron

level EF to proton level EF + (jI2If>E. The others must make an equal jump. if me energy of each of j particle.. increases by (jl2�E. the total energy increases

by (fn't8E. We can ea."lly ex.pre"�J

10 terms ofZ and

N. 1 f we stan. with Z = N.

men gain j proton.., and lo� J neutrons. Z - N must now be 2). giving j= - N). Thu,. an a>ymmelry between Z and N should decrease the

l(Z

�

�.

as

nucleons is lower. They should be easier to The bindmg energy of extract. because fewer other nucleons are around to attract them. Accordingly,

r is proportional to

!PIE.

lIZ �

A' J

')

./

-

\ 1 .:\ Nudc&T

f"IIUf11 1 1 .16 Compari�on of Ine �emiempiri.cal binding energy formula ami aCluill 8f)nude�ln. , ,

7 , •

�

I

�l ,

) •

,

�

,I

� '4 � ''d : , o pm � �'I ----------------

.

Allual

, , ) ,

50

150

100

200

A

Ihe�c effects are often appended to the semiempiricai binding energy formula.

but in most cases (helium being a notable exception) inlroduce only small cor­ !telions. Their explanation really belongs to the shell model.

The Shell Model

An indication of the complexities of the strong force is that the two most use­

ful nuclear models are so vastly differem. Except for the asymmetry term's

use of the exclusion principle. the liquid drop model is classical, while the

,hell model is thoroughly quantum mechanical. In the liquid drop model, essentially classical particles interact via strong interparticle forces, and total binding energy is predicted. In the shell model, particles are treated as occupy­ ing quantum states in an average potential weB that largely ignores specific interparticle forces. The shell model's unique successes are predictions of nuclear angular momenta and the tendencies for both N and Z to be magic numbers or at least even numbers. The shell model is to the nucleus what the treatment of Chapters 7 and 8 is to the atom. As electrons in an atom orbit in a potential well resulting from their attraction to the nucleus, the nucleons in the shell model "orbiC in a potential well that is the net effect of all the forces they experience. After our having repeatedly pictured the nucleus as hard spheres stuck together, this may seem absurd. Wouldn't collisions preclude the possibility of a given nucleon roaming about the nucleus? Actually, the idea that motion might be rather free quantum mechanically when classical physics would suggest oth­ erwise is nothing new. As we saw in Chapter 10, electrons in a conducting solid move rather freely despite the countless positive ions around them, and their wave functions !-.pread throughout the volume of the solid. Nucleon \lave functions must also permeate their enclosure. Moreover, if they assume

Mode\!!. ...

... 0,.P.. "

""-- � • lo'4c t-cnerg) o\crall ,tatc con,i�tent with the e,c1u�ion pri nci ple. the) \O\O·I�1d fill all mdi\ idual .. tate, to a ma.,imum level-the Fermi energY-and Ihl' "'·uuld prt\·ent mo,' nucleon, Irom c hanging states at all. They could not undc-rg.o the momentum I.:hange, characteri\lic of a collision. Only nucleons nt"aJ the ''lop'' h.n e much freedom. The real challenge in the shell model of ·

the nudeu, i, that the fon.:e, are far more complex than the Coulomb forct: npcrienl.·ed b) eltttron� in the atom. The form of the potential energy is rather 'peculati\e. A ,tatting as\umption IS that the nucleons orbit indepen_ dently of each other in one large ,phcrically sy mmetri c potential well. A rounded finih! wel l ha.. been most widely used. Let us see if an even Simpler I.:hoice j, u..eful. EXAMPLE 1 1 . 3 Nudei in ucited �1i.ltt:' can lower their energy by gamma emis�;on, in which a pho­

Ion carrie� away the e·u:-e,� energy. As a rough approximation, assume that the nu("leon� beha'e a\ though trapped III an infinite well. Make a reasonable estimate or lhe enell!) of the photon emitted as a nucleon makes a transition from one energy Ie,.:1 10 another. SOL�TION

The enell!Y

Ic\·.:1\ in the 10 inlini{e well are

A c:ubit infinite well might be more realistic, but this would merely replace ,,1 by II!, + nJ + n! . .'

and \\ould not change our conclu�ion. For L, we use a typical nUclear

diameter. Suppose A

:0

lOll Using equation ( I I-I),

2r-= 2(IOO) tl3 X 1.2 X I O- I s m a The ma,� of!l nudeon is about

E :o

1 .67 X 10-27 kg. Thus,

( 1.055 X 10-34 J . s )21T:!n2

2( 1.67 X 10

.�7 �

IO- 1 4 m

kg) ( 10

"

' =

In ) �

(3. 3 X 10-13 J)w

">

a

(2 MeV )n2

We might expeC! energy jumps to be of severa! MeV. Although this is cenainly a j� no coincidence that typical gamma energies are on the

crude appro,imation, il

order of I MeV.

The shell model was proposed when il was found that nuclear stability e\hiblfed some periodicity. Careful inspection of Figure 1 1 . 1 3 reveals a dis­ proportionate number of nuclei with either N or Z or both equal to one of the

magic numbers: 2, 8, 20, 28, 50. 82, 126. Apparently, great stability accom­ panies Ihese values. Because il was known that atoms of atomic number 2, 10,

lB. 36. 54, . . . are unu\uaHy 'ilabJe because they possess complete shells or subshells. it was natural to atlribure magic numbers to similar behavior in

allowance for a strong nuclear spin-orbit lUteraClion. thl' hypoth­ nuclei. After

("i� "as confirmed.

Anomer success of the <;hell model j<., thai when expanded ..lightly to specific interparticle forces, It ex.plains the pairing effect-the ten­ allo\\ for oey for Z or N to be simply even numbers. This is also apparent in Figure

1�.13

d

in the clustering of stable nuclei along lines of even N and Z. Finally. the

,hell model predicts nuclear angular momenta, which agree \"'ell with ex.pencntal observation. This should not be too surprising. for we know that solu­

�

�

ation for a spherically ..ymmetric potential energy on of the Schr dinger eq� quantized angular momentum slate!.. ields (orce) Y I�entral

1 1 . 4 Nuclear Magnetic Resonance and M RI

� fmsging

l!l

An"ular momentum is quantized in the nucleus, and so is its ineviwble COOl­ l on: magnetic dipole moment. (See Section 8.1.) Magnetic resonance (MRI), a popular term for imaging by nuclear magnetic resonance,

exploits this quantized magnetic moment to produce images of the body's interior. The most important requirements in producing such an image nre gening

whatever one uses for a probe into the body, interacting in different ways with different features once inside. and detecting the interaction from the outside­

a\1 without damage to the tissues. Light fails as a probe because its penetration of the body is very short ranged. The requirements are met reasonably well by

X-rays. They pass easily into the body, they are absorbed at different rates as they interact with electron clouds in different regions, and the portion that

passes through unabsorbed is easily detected. In a simple X-ray, the transmit­ ted intensity registers on photographic film and yields an image of average

density. The intensity is different, for instance, where X-rays pass through �kin, muscle, and bone than where they pass through skin and muscle only. However, X-ray photons typically have energies measured in keV-easily enough to tear ions apart-and thus carry with them a degree of biological

hazard.

To produce an image using nuclear magnetic resonance, electromag­ netic radiation is sent into the body to interact not with the electron charge densities, but direclly with the nuclei. The body is immersed in an external magnetic field so that the magnetic moments of the nuclei have quantized ori­ entation energies along that axis. To cause a flip to a higher orientation energy, only photons of a certain frequency will work--only these meet the resonance

condition. (Electron spin resonance, discussed in Section 8. 1 , is very similar,

though it exploits electron spin rather than nuclear spin.) As the nuclei jump back down or relax, the absorbed energy is reradiated in all directions and ,

easily detected. The size of the energy jump is proportional to the strength of the external magnetic field (see Exercise

35).

In MRI studies of the human body, the field

is chosen so that photons in the radio frequency range have the proper energy to excite jumps. Such radiation passes easily into the body; however, with

P

T

o

N

A

L

Fig"" 11.17 A health). lI\ln, human brJ.in. a.� ..cell �y MRI

photon energie, far les\ than chemical bond energies or ionization energies, il P(N!� no biological threal. MRf ha<; further advantages:

I. Because il use') an external magnetic field that is nonuniform, MRI is beller able 10 "look" at '>pecific location� within the body. The reso­ n plane where Ihe field has one value

nanl frequency i� different in

than in a plane where it has another. i

The absorption and relaxation process i, more detailed. I t varies with lime in a way thai depend� on the nature of the surrounding nuclei and alOm�. This lime dependence thus serves as additional information Ihal distingui')hes one malerial or tissue from another.

3. While the most common larget nucleus is hydrogen, the incoming

radiation can be cuned to resonanl frequencies of different nuclei, which can provide an imponanl ahernalive "view:' Figure 1 1. 1 7 shows a nicely detailed image of a human brain. Although i,\ medical applications are best known, nuclear magnetic resonance is used in many areas of scientific research. Its ability (0 "see" the structure of complex molecules is tndispen<;able in chemistry.

1 1 .5 Radioactivity

A nucJeu1\ may be bound yet still in a relatively high energy state, and if it can

allain a lower energy, it will eventually do so. Nuclei fitting this description

are

twned unstable. or radioactive, and the process is known as radioactive decay. Radioactive decay usually lakes the form of the spontaneous emission of a small energetic panicle. as in the gamma decay of Example 1 1 .3. Of course, energy is

maVilintemal ene�y I1ttrea�.. We u-.c relea..-.e
... ,w

Q

" .

a<;.

(nl.

Figure " ,"

•

-(mI - m ,)c2

nll)r

( 1 1 -7)

�.1

lei u, laKe a look at the various fonn<. 01 radioactive decay.

Alpha De cay

We fil'l encountered

Ol decay in Section 6.3. where we noted that It\ c1a\''II­ (a\1)' bafflmg trail'i can be under�tood by 3'iSUnltng that Ihe (Y particle (hehum Rudeu..) quantum-mechanically tunnels out of a large nucleus. But it i\ natural 10 Il'K what maKeS an a particle so effective at lowering the energy of 'iuch a nudeu, Figure 1 1 .14 shows thai hehum is unusually tightly hound for I" \lIe; that is, Its mass is less than the sum of its part\ by an unusually large ..mount The figure also shows thai the heaviest nuclei tend to be less tightly lloUnd than tho'ie of somewhat smaller mass number. Thu�. it shouldn't be ter­ ntll) �urprising that a large nucleus, called a parent nucleus. might split IOta a lighter. more tightly bound one, called a daughter nucleus. plu ... a tightly

bound a particle. Figure 1 1 . 1 8 depicts the process. Because the (Y particle

lale' away two neutrons and two protons. the daughter nucleus In (Y decay ha� t"O lewer of each than the parent. EXAM P L E 1 1 . 4

,...

Ura1llum-238, one of the mo!.t abundant radioactive isotope\ in natun=. attains a lower rnergy state by a decay. What is the daughter nucleus. and how mut:h linetic energy is relealOed? N

Crantum·238 has 92 protons. and 238 - 92 ::; 146 neutrons, so the daughter ha!. 90 protons and 144 neutrons, identifying II a!. Ihoriu01-234. The reaction I!. thus

2:�U

_

2�Th + i He

Obtaintng the masses from Appendix I. the change \0 kinetic energy

;0;

Q = (m. - mf>c2 = 1238.050784 u - (234.043593 u + 4.002603 u)]931.5 MeV/u = 4.27 MeV

Note Ih31. just as we do in calcuhning binding energies, we use atomic rather than

nudrar masses. The "extra" electrons included in the initial mass are subtracted thr fi nal. 92 = 90 + 2.

\0

Soielly speakmg. the a panicle and daughter share the energy released In Q

Alpha .,Jc..;4)'

decay. However. when a massive object splilS into two (nonrelativistic)

p4RII.:lc

p�s. momtnfUm consen'allon tell� u� that the ratio of their kinetic energies

i" !he im'uu of the ratio of their mas�e� Thus. the

the kinetic energy

a

particle gets almost aJl

aeto Decay Electrostatic attr.k:lion cannot [rap an electron in the nucleus (see Exercist

4.73J. yel ","e find thai electrons are

emitted by some nuclei. We conclude thaI

they art 'pontaneou�ly creared, then immediately depan. When created and emillcd in this wal. an eleclron is called known

a�fr decay,

II

fr

particle. and the prOCess is

The phenomenon of fJ decay is certainly curious. If charge is [0 be con. served, what must be happening in the nucleus? We find Ihat after p- decay, the remaining nucleus, agam known as a daughter, has one more proton and

one less neutron than the parent had. In effccl, a neutron in the parent Sponta.

neously changes inlo a prolon and an electron. Charge is conserved, BUi this is not the whole siory. As in

a

decay. if the parenl were 10 decay

inlO only a daughler and ap particle, the tiny p particle would carry off essen­ tially all the a\'ailable kinetic energy Q. TIus would make Figure 1 1 . 1 9 alann.

ing indeed. It shows p particle energie� found in a large number of p decays,

The) vary widely and are often essentially ;,ero. A f3 decay with only two

product panides violates energy conservation! I

Things get C\'en worse. The simplest example of f3 decay i.. neutron decay.

While neutrons inside stable nuclei live forever. a free neutron is unstable.

Because a free prrnon is Significantly less massive than a free neutron, a neu. IrOn may attain a lo",er energy/mass via f3

�

decay into a proton. However, if a

�

�

�pin- neutron were to become only a �pin. proton and a spin- electron, the

final loral spin would be an integer. either I or

The ml'teliou� variation in p pat1ide ene'llie� F"'...... 11.19

O.

Angular momentum could

not be consel'\'ed. Allo.....ance for the creation of photons (�pin- l ) would not �oh'e the problem, for Ihe lotal spin would still be an Integer. All other 13-

decays share the same problem.

Thi� �eeming violation of fundamenlal laws of physics caused much con­ �temalion among early nuclear physicists. Wolfgang Pauli postulated a solu· tion: The decay produces an unseen particle, which carries away angular

i

momentum and \'ariable por1ions of the available kinetic energy. Char e and .. KE" ... ,

=Q

KE. 'UIIIIfiafm CftnD waT lTiwed 10 produce ITa. lilt bile-til; eDffJ) lWIO ... wId 110 /clip be suapl)' dIt IIMnII' of!hi: ... ... but ad! pu1Kk _Id Clil sn III U1JVW)'ta, (racllOn OYUtlc rII
oIdwMW* aIttD

angular momentum could be conserved if it were uncharged and spin-i' (By the rules of angular momentum addition diM:u�sed in Section 8.7, three final �pins of

� can add to the neutron's initial �.) Moreover. the f3 particle does

somelimes acquire a kinetic energy nearly equal to the

Q calculated by assum·

ing a final mass of only daughter and p, so it would seem that the mystery par·

ticle has negligible mass. After much effort,

Pauli's un\een panicle was found,

and Ihe fundamental Jaws �aved. We call 1hi� new kind

of particle

the neulrino. It� discovery lagged its

prediction because Ihe neutrino is elttremely difficult to delect. Having no

nor do they experience the strong force. They interact only by the so-called weak force (discussed In Chapter J 2). which is so feeble thai mo�t neutrinos tra\'chng through space and encountering Earth pas§ right through it. Perhaps it I:..n ', surprising that charge. neutrinos do not mleract electrostatically.

I I ,S RadiO&C� 495

much to learn about neulrinos.

that

Recent experiments suggest we still have uu: neutrino has mass, but the most we can say is that it is very much less than electron. It certainly can be neglected in a calculation of Q. We IMI of the r discussion of the neutrino to Chapter 12, with the following leave furthe

Figure 1 1 .20 fj decay_ A neutron become.. a proton. and an electron and 3ntineutrlno are emitted.

e� neutrino and be to soon decay of kind ssed. Physicists aren't even discu lementary comp :03 different particles. While tception: The particle produced in {3- decay has traditionally been referred as me lJlltineutrino. symbol V, with the term symbol v applied

s

we uphold the tradition, the distinc· . 1 ay. p IOto come never will tion Figure 1 1 .20 depicts a Ir decay_ An imponant question is bow {3- decay aJ10ws a nucleus to decrease its energy. Consider the case of boron- 12. ure if these are

I�B.

Parent

,

*' -

J3

�

particle

Daughter ;;

which has five protons and seven neutrons. If a neutron changes into a proton

on, the daughter nucleus has six protons and six neutrons. Lighter and an electr be more tightly bound when N = Z. and when both N and Z ru:e 10 tend ei nucl so it is reasonable that boron-12 might attain a lower energy via ers, even numb

�- decay.

. . Simply to account for nucleon number and charge. tbe fr particle IS often /3 meaning that it has no nucleons but carries a charge represented as Thus. we express the {3- decay of boron- 1 2 as proton. a of that e it s oppO

�

-

-,

"

58 -

" 6c

+

0 f3-

-I

+ v

The energy released in this reaction is again -6.mc2. However. because there are five electrons in boron- 1 2 and six

in carbon-12, using atomic masses

would include an extra electron in the final mass. We circumvent the difficully by allowing Thus,

this extra electron mass to account for the mass of the /3- particle. /3-

decay: Q =

(mparem - mdaughtcr)£...2

Applying this to the boron- 1 2 decay

( 1 1-8)

and using atomic masses from Appendix

I. we have

Figure 1 1 .21 P+ decay. A proton Q = ( 1 2.014352 u

-

12

u) 93 1 .5 MeV/u

=

13.4 MeV

j

I f3- decay is /3+ decay. Consider N , which has five I neutronsand seven protons. It could become C. with six each. if only it could change a proton to a neutron. This is effectively what happens in f3+ decay. depicted in Figure 1 1 .2 1 . Charge is conserved by the emission of a positively charged electron-the positron we first met in Section 3.5. The neutrino again ensures conservation of angular momentum. Thus, muogen-12 f3+ decays according to Complementary to

�

1 2N 7

The

--

"c + 6

0 + + 1 f3

+

v

atomic mass of the daughter would be one electron mass too small to can­

cel the number of electrons mcluded in the parent atomic mass. so to find the

-

-

becomes a neutron. a positron. and a neutrino.

Parent ,.

./"

� . 13

Daughter

-

particle

+,

,.... " .22 Electron captlm. In whic;h an orbitini eleclron comblne\ with a proton 10 tonn a neutron and neutrino.

energy released, we mu�t include twO electron masses with the final one to cancel the extra parent electron and one for the {3+.

For nitrogen-7, Ihi\ gives

,

Q := (12.018613 u - 1 2 u - 2 X 5.486 X 10-4 u) 931.5 MeV/u =- 16.3 MeV

ft·

Daughter

Figure 11.23 The "din:l:lions"

of a

and f3 decay\, and the decllY �eries of uranium-238 and Ihorium-232. 145

As in all ronns of radioactive decay. the final mass is less than the initial. interestingly, although a free neutron is liable to {3- decay to a proton. , free proton cannot /3+ decay to a neutron, for mass would increase. In a nucleu!t, however, a proton can decay 10 a " more massive" neutron, because factors such as decreased Coulomb repulsion can still result in a lower total mas�intemal energy. Although free proton decay is predicted in unified theo­ ries (Section 12.7), it cannot occur by /3+ decay. Figure 1 1 .22 depicts a third fonn of {3 decay: electron capture. A nucleus with 100 many protons. the type that might tend to P+ decay. may approach N :::: Z another way. It may seize an orbiting electron and effectively conven it and a prolon into a neutron in the nucleus. meanwhile emitting a neutrino Thus. as in /3+ decay, Zdecrea�es by I . and N increases by I. Many nuclei tha; {3+ decay also engage in electron capture. Often. electron capture is "energeti_ cally preferred." in /3+ decay. the mass of [he parent nucleus must be at least one electron mass gremer than the daughter, for a positron is crealed. In elec­ tron capture, however, the electron already exists, so the parent's mass can be as small as one electron mass less than the daughter. Carbon- I I , though able to become boron-II by both processes, " prefers" electron capture approxi_ mately 100 to I.

"c + 6

IIS +

ll- - 5 -°11"'

v

The energy released in electron capture depends simply on the difference in the O1omic maSses of parent and daughter.

140

Electron capture: Q =

130

125

,

I'" 7S

rn, ss_

80

fJ< decay EI«tron (aplure 85

1

90

(m nl - mdaughler)c2 pare

( 1 1·10)

The "extra" electron in the parent atomic mass accounts for the mass of the captured electron. Figure 1 1.23 shows the effect on Z and N of the decays we have discussed. By altering 2 and N differently, they constitute different ways of approaching the curve of stability. Heavy nuclei with too many protons often a decay (Z and N bolh reduced by 2). light nuclei with too many prolons /3+ decay or engage in electron capture (2 reduced by I. N increased by I), and nuclei with 100 many neutron� /3- decay (Z increased by I . N reduced by I ). Also shown in Figure 1 1 .23 are the decay series of thorium-232 and uranium-238, two radioactive but reasonably abundant nawrally occurring

l i S RadiOllCtivity

de:ay need not result in a stable state; many daughter nuclei are i�10peS. A . seh'es rad Ioactive. Thus. one decay can be followed by a serie!> of other a

497

Figure " .24 Gamma decay.

(helTl

d/J decays. Moreover. many unstable isotopes decay in more than one way. may branch. governed by probabilities. Even so. a stable a decay series eventually be reached. Thorium-232 ends up as lead-208, and will s oucleu lead-206. It is true that lead is not one of the most stable as 8 uranium-23 know. the most tightly bound are around A = 60.) But these we clel. (As l stable simply because there is no form of radioactive so peS are s t i l i to le that can take them to a state any lower in energy/mass. Con­ deCay availab are isotopes that appear to be on the curve of stability but that there \·ersely, because there is a form of decay that leads to an even lower mass. l e lit un�tab is potassium-40. discussed in Exercise 43. A goad example

:

* parti:ie y

�uad

Gamma Decay An alom in an excited state may lower its energy by emitting a photon, and so maya nucleus. When emitted from the nucleus. a photon is called a y particle.

and f3 decay, one element becomes another. because Q' and /3 particles carT)' charge. but in y decay. pictured in Figure 1 1 .24. neither Z nor N change. er Gamma decay often accompanies other forms. In many deca.ys. the daught. Its and it may continue to lower ucleus is not left in its lowest energy state. about excite states of nuc\e y much learn We photon. a emitting nergy by c�enstlc . cb a e energies ar gamma Moreover. studying their gamma emissions. [0 a

�

:

�

��

ofagiven isotope-a spectral "fingerprint"-and thus serve to Identify It. (See

Progress and Applications.)

Spontaneous Fission

A somewhat rare form of radioactive decay exhibited by a few of the heavier

nuclei is spontaneous fission, il1u:.trated in Figure 1 1 .25. Unlike other decays. which produce one fairly large daughter and a small particle or two. in this decay. a nucleus fissions (splits) into two nuclei of intennediate mass

number. The driving force behind "pontaneous fission is the reduction of the destabilizing effect of Coulomb repulsion. It is true that fission increases the total surface area. This alone would the overall energy, requiring more

raise

nucleons to be pulled to the surface against their mutual attraction. But split4 tina the whole into two smaller nuclei also decreases the number of protons •

that are too far apart to attract each other, so the number exhibiting a net repul-

sion is reduced. which more than compensates for the increased "surface

energy."

As Figure 1 1 .25 sugge<;t�. <;pontaneous fission also releases neutrons.

Heavy nuclei have many more neutron .. than protons. but intennediate-rnass

nuclei are

most tightly bound .... hen Z and N are more nearly equal. Thus.

Figure " .25 Spontaneous fission. in

which a nucleus splits into two smaller ones. Neutrons are a1so freed.

_parent

.",hen a heavy nucleus brea\...... up, the fragments tend to have too many neutrons. A lower energy re<;ult ... If the ex-cess neutrons are simply freed. This

freeing of mUltiple neulron!
10 t1<;<;;on is a very important factor in the release

of nuclear energy. We will return

tll

&

it in Section 1 1.7.

Fi""ion fragmenlll

C d.

"

....

".udhe W. ........ . .,

� • � ituIope with • aoandiOlClivt material gives the II'AIII:riaJ.

UM.'" 1Ipwnwe. rec:opizahle by the enersy of the particle emittccl Sell dfic nSF ch expIoiu this in DUIDeI'OU! ways. To trace how M>mc IIOnra­

•

dIoKtiw: IMIIS'iah move about or how they slick to one another. radioactive IIOIDI lie

iDIrodIlCed, either as a loose mixture or by being bonded 10 specific

IIIDIcaIJ.r 1fOUPI. 1be subsequenl behavior of the materia] is then easily fol. lowed by the lelhale decay of the radioactive tracer. In one of many biologi.

cal

research

applications• • slerOid molecule is lagged (covalently bondtd) WIth tritium. • /J entitlei'. and then mixed with a protein. The .H�rojd·s abilil), to bind to the protein is clearly indicated by how much trilium ends up stuck to the proIein. Let us now look II another characteristic of radioactive decay that I allO much e..ploited in ICteDlific investigation.

1 1 .6 The Radioactive Decay Law

All forms of radiOKtive decay fundamentally change the parent nucleus, and the nuckus may decay in a particular way only once. A good analogy is a light. bulb. which can bum out only once, after which 11

IS fundamentally different

from a wading lighlbulb. Furthennore. decay!>. are governed by probabililles. It is irnpo5jible 10 know exactly when specific nudel In a sample will decay. One

nucleus may decay righl away and another after

very long time. In a large sample. however. there should be a predictable a\'eroge time. characteristic of n

the panicular decay: it may be long for the a decay of isotope X and shon for the /3. decay of isotope Y. The lightbulb analogy is again helpful. One neVer knows when

8

IighlbuJb will bum out. but in a huge office building with thou­

sand", of IightbuJb X. thert would be a predictable number burning out per day. A rtliable average lifetime would be apparent. In a similar building with thou­ sands of lightbulb Y-a competitor's lower-quaJity unit-a predictable average lifetime would aJso be apparent. though it might be much shorter. No mailer whar value the lifetime might be. if we increase the size of the sample (building). the number decaying (burning OUI) per unit lime should increase proportionally. In OIher word�. the change per unit time i n the number

of nuclei present should be directly proponional to the number presen!.

dN ex N dl

-

We make this an equation by using the symbol A for the proponionality con­ stant.

(Note: In this seclion. N stands for Ihe number of nuclei. rather than the

number of neurrons in a given nucleus.) Becau�e the number of radioactive nuclei that are present is constantly decreasing �uires a minus sign,

dN dl

=

-AN

(dNldl

<

0), the equation

( 1 1·12)

1 1 .6 The I\dOIICIiw 'DIcQ Law

piffercnllal equation.. In which a quantity diminishes at a rate propor­ to 11'0 1O,lanlaneQu, ,·alue are very common in S4:ieru:e. (A. famihar

oooal

(\IJI\PI� I� a

�

ah'�

capacllor dl,charglRg through a �..i,tor. The CUl"1'ent. whll:h is the of the rate of change of the capacitor', charge, i, propor·

dQld,.

QIC.)

The ine"itable �'ult i, an to the voltage acro...... the capaCitor, decay. To ... how thi ... . we \Cpar ale .., ,anahlc nlial �

r

tiN -A d, N =

{nl(graung. tlmc from 0 10 a� the number of nuclei goe.. from

{(�'

•

-A

1.'

tit'

No to N. we hllve

or

Th( number of nuclei falls ex.ponentially with time. It is worth reltcratlRg that �\l3Iion ( I I - I I). on which. ( 1 1 - 1 3) is based. is �a1id only If the number of pMtnl nuclei yet to decay IS large. Only then Will the cx.ponentlal decay be

smooth. If the number is small (the office building is small or most Iightbulb, out). decays will be sporadic, and the number remaining win art already burnt Ilu(IUale considerably about an ex.ponentially decreasing trend The constant

A

is called the decay constant . (It is not a wavelength n

A,

the more rapidly the Equation ( 1 1 - 1 3) says that the larger the value of Equation ( 1 1 1 2) agrees. When N decreases. we speak of decay rate, er numb hol'.'ever. we usually mean an absolute value. R, in decay� per second, Accordingly. ( 1 1 - 1 2) is often written as R

= AN

( 1 1 - 1 4)

We conclude that the rate decreases ex.ponentially with time, just as the num­

ber ofnuclei present does.

A property of ex.ponential functions is that they decrease by the same frac­

non in equal successive intervals of lime. If in time interval I . the number

drop� to half its milial value. in interval 2, it will drop to half of lhi� value. or one-quarter of the initial value; in interval 3, it will drop to half lhi� value. or . one-eighth the initial The time required for the number to fall to half tIS initial

\alue is the characteristic by which decay rates are usually categorized. It is ..:alled the half-life and 1\ given the symbol TI12• Figure 1 1 .26 shows the decrease of

N in terms of half-lives. The connection to the decay con�tant is t is TII2 when N is lNo. Thus,

,uaightforward. By definition.

...

Figin' 11.26 Exponential decay. .,�,

, :/'

.

"

;·\' L

_ _ _ _ _ �

T"ILI 1 1.:1 ......

Jle

.' .

,

,H

��Atl

�Om"

Sc.-IC'I.·IC"d �i1)"'

Ora, ModI'

Hal'·tlr,

P'

A =

12.-' )"r

p ..

Canceling No' taking nUiural logs. and rearranging, we have

.. �

)(

Iflv yr

10 2

( 1 J-15)

­

TII2

Sen.\ibly, Ihe decay comlanl i� large when the half� li fe is small, and vice

,·ersa.

Given Ihe complc�ities of nuc/ear binding and the possibility of mUltiple decay mode�, it i� nOI n �imple mailer to predict half-lives. Many have been mea�ured, howe\er. cO\'ering a range from aboul 10-22 s to 1Or7 yr. Table 1 1 .3 gi\e.\ a few examples. Many more are given in Appendix

I.

EXAMPLE 1 1 . 5

A vessel holds 2 j.l.g oflritium. (a) Whal is ils initi!!1 decay rate? (b) How much lime will elapse before the decay rate falls to I % of ils initial value? S('lUTION (a) Deca} rate is proportional to the number of radioaclive nuclci present. which

we find in [he usual way. Using the atomic mass from Appendix I,

N=

2 X IO'9 kg 3.02 u x 1 .66 X 10-27 kg/u

:::: 4.0 X JO I 7

The decay conslant we find from equation ( 1 1-15), and the half-life from Table 1 1.3.

A

In 2

= --

Now u)ing (I 1 · 14),

12.3 yr

X

I yr :-c � �.3.16 X 107 s

=

1.78 X 1 0-9 s-I

Although the number decaying each second is large, it is still a very small frac­ lion of the tota.l number.

.

Because R = 'AN holds at all time... the decay rate dro� to It;;. when the nurn· , ber of tritium nuclei present does. In�rting N = looNo in equation ( 1 1-\3), • ,-A I N = N. u - , 100 0

1 100

_

1 100

In- = - I

A

-In -

(

=

= 1.7 8 X 1 0-9 .. 1

An altemative route .IS

81.7 yeaN.

(1:' )'

=

2.6 X 109 s = 81. 7 yr

iIi)' which gives n = 6.64. and 6.64 half-h....e.. is

Radioactive Dating

i portan An m

t use of the radioactive decay law is radioactive dating. in which detennine the age of wmething old. such as a fo�sil. a rock. or to pt \\e attem There are \everal method... using different radioactive isotopes. All an artifact. ..t. If rely on assumptions about the abundance of certain i<,otopes in the pa number and the final number of .. ample are known. nuclei in a initial both die a )irople calculation gives lhe lime elap'\ed.

REAL-WORLD E X A M P L E CARBON-14 DATING

.... The most famou.. te�hnique 01

ioacti\"e dating relies on the i,\otope carbon-14. 5730 years. The assumption in this te�hnique is that the abundance of carbon· 14 in the biosphere. the pan of our world inhabited by and inleraclLOg with biological organisms. was the same m the p.tst as it is no\\, rad

which p' decays with a half-life of

Although carbon-14 doe... decay, it is continuously repleni<;hed by the action of

c�mic rays. high-energy particles I"rom <;pace. When theo;e panicle" �trik.e nitro. gen in the upper atmosphere-. they initiate a nuclear reaction that proouces the I�o­

topt, \lj-hieh then �pread", throughout the bio�phere. mamtai ning a con..tant ratio of carbon-I" to 'table carb..m nu..:lel Facto,", ,ul:h � atmo...pheri, nuclear te,ting

and industriahutlon haH' had eft!:l:t",. but the ratio i.. u!>ually taken to be about IJ X IO-u. Carbon· 1 4 dating WO(�!i only for (formerly) Ii\ ing organi ..ms. \\'h.ile a planl or animal !i\'e\. It e .....:han1!c carbon "'Ith the eO\Lronment through Its food

,uppl) and the
an..! (rum that pomt on...llrd,

cea�.. to n..:hange carbon

""Ith

the amount 01" 1.�n- 14 d�---ays

Applying the Pl'tysks

Wb.1l lt the ate of a 1000§il �plc: thai. conLnn.. 6 r of carbon and has a dec:ty rate of

JOdtca)"s peT minute

� ---

.'mm

. 12 ...... .. _ _ 1 .... 01_. 00 3.0\ . \0" ",_ ....... ..., . " ..... ..... ... .... arbon-12. The fraction thai would initially _ _ _I'D

No - ( I 3 )(

10-12)(3.01

X

100J) - 3.9 X

lOll

.. wiD _ ... _ .......

We may ftnd dae pcaenl Dumber via ( 1 1 - 1 4).

:;�

c:; ay � . l _O .

-

6 ( 10

- 3.83 X

lO-l l s - 1 X N

�

N ""

1.31

X

10"

1.31 X lO" _ 3.9 X 10 " l'-P81J(JO-Il\·')1 - , - 2.86 X 1 0" � . 9000 yr To P a lUffic:ieedy hip fJ- ekeay count rate, the lroditlonal method or carbon_I"

dIIiq requira a llrJe amount or carbon, me��d in grams. Retrievmg thl5 much

from Ibe t.c. of, valuable archaeological \pecimen may be out or the question AD iIcreasiqIy common melhod Is« Prol:re�� and Applicahons) determine5 tilt pRSCDI amounl of c;ubon-14 by mu\ �pectro\copy and can get by with a sample

size OD tbt order of, milligram. In any case, carbon-I 4 dating, like all datmg tech. Diques. bas limits. It is most reliable ror lime �cales withm about an order or magnl. _oI"" haIf·hle.

To determine the ales of rocks and other things that do not exchange Cat­ bon with the biosphere. other methods of radioactive dating must be used. The . .... . of I rock is the time thai has pa.�sed �ince it formed a solid. One tech­ nique (or delmnining ages of rocks use� the decay of potassium-40 to argon40. 1be assumptiOD is that when the rock fanned, it contained no argon at all. AIJoD is an inert gas. so il does nol form compounds. and it easily passes of I moileD mass. Thus, any Ar-40 aloms present now should be Ihe result of

OUI

r.dioKIive decly since the solid fonned. A slight difficulty arises In determin_ iDa die IIUIIIber of K40 decays. h is not simply the number of Ar-40 atoms praent. for K-40 also decays ro calcium-40, and the rock mighl wrll have beea ""toaCami.DaIed wilh Ca-40 when il formed. However. decays 10 Ca-40 IDd Ar..tO occur in I fixed ratio. so !he roral number of K-40 decays is directly

proportioDaI lO tbe Dumber of Ar-40 aroms. Adding the number of K·40 alOms decayed IDd the number presenl gives Ihe imtial number, and with the initial number. present number. and half·life. we can calculate the age. Other radioac­

tive iIocqIc!. found in various Iypes of rock and commonly used ror dating are

unaiam-238. decaying (0 1""-206: lIIorium-2J2. decallng to lead·208: and rubidium-B6, decayin, 10 slrOntium·86.

I I 7 Nuclear

1 1 .7 Nucle ar Reactions

lear reaction refers to any occurrence in which nucleons are TIte tenn nuc between nuciei, much as electrons urc exchanged exchanged or . h ged atoms in a chemical reaction. Radioactivity is a form of nuclear reacbut 11 spontaneous one. We now study nuclear reactions that nre mduced

��'eell uon,

· Ie. An exampIe ·IS · h another partlC Ieus WIt b) ..triking a nuc

�

�

but the neu�on estabilizes its nucle�s. hich then bre s � Boron-I O is stable, . change m kmetlc energy .IS again given by equation The pieces. tWO [0 • d·IX I , 11,7). Using Appen

(

1 .008665 u ( 10.01 2937 u +

-

7.016003 u

-

4.002603 u) X 931.5 MeV/u � 2.79 MeV

Once again. kinetic energy increases as mass decreases. In chemistry, we \I'ould call this an exothermic reaction. Of course, endothermic reactions are

possible. If the reactants have kinetic energy before the reaction. it is quite possible for the products to have less kinetic energy and therefore more mass. Mru;s increases in the following reaction, commonly used to produce monoen­ ergetic neutrons:

that Q = - 1 .64 MeV. It is left as an exercise to show Exothermic reactions are of particular interest. To release kinetic energy,

masS must decrease, which means that the products must be more tightly bound than the reactants. There are two essentially opposite ways to do this. Because binding energy per nucleon peaks at around A = 60, a given number

of nucleons would be in a lowest energy state if grouped into sets of 60. In nature. of course, nucleons are not all grouped this way. Therefore, energy can

be released by breaking heavier nuclei apart or by putting lighter nuclei together. These are the processes of fission and fusion.

Fission

Heavy nuclei have less binding energy per nucleon than do those of intermedi­ ate mass number. We can exploit this, as illustrated in Figure 1 1 .27, by break­

ing a heavy nucleus into intermediate-sized pieces in which the nucleons end up in a lower energy state. Lower internal energy means smaller mass and a release of kinetic energy. As noted in connection with spontaneous fission, the driving force is the reduction of Coulomb repulsion. Few isotopes spontaneously fission, and even these few usually decay other ways. Of greater practical use is

induced fission, in which we stimulate release

ReactiOM

S03

_ n � "" • 0. 11 ..... " n

.. DK _

I� YU tmimL * • .... ... ,

•

-

•

-

-�

'.....

IIrt. (GIIbnb

•

�

10 1111 120

�,

"I'" T-" "T'"

.....----.:-

1010 ! ('Ill 1 1ll1 :00 �� 1...,

•

1 1 1 · 1 61

", drpk.Ud 10 fl,un:' I I �S. a l'-!)� nudeu\ ab..orb.. a neutron. becom ing Men), an C'l.dln.l llotlt!C' llf O-�.'6, Ihen ti\\ll)m In one of a ho\t of " 'Y \ Wr dl�u.\..' uranlum·�35. NIl Ihert art other nuclei thai behave \lml­

PO\\lblr fi\\ion

larIy. noc�l}' pJulonlum-,:B�.) The lin.1I �lOclu: energ), 1\ ...hared by the fnamtn� blnum an.J kl)'ptun to the lir.1 rtaCllOn---frecd neulron.,. and ., putidn. Fu�. tht fh'I()O frogmen" lend 10 be radlOaCII\e. and lhC'y

�Itut more C'IYIJ) In Lht form of f3 particle" and neutrino.. Becau3C lht

tnc1J) �ft� IS ratbc'r I.:'ompIKaltd. .a Mmple calculation of

_

Q 1\ nOf possible,

but wt ,.-an tsbmalt Its \aJut. AC,"l,"hng 10 Figure 1 1.17. lrom mas� numbcl\ •

near !40 10 Ihos,t ncar

• •

of the product nuclei). binding

......Id ._ by abo," 200 /drV. •

-• • •

I.)) (. RlUl!h a\croge

attfJ) per nu.:1con inr:1'C'a..�s b)' I,hghtl> le,_' lhan J MeV. Thu.., In a reacllon In whrc:h about 2.w uramum nue/com �nd up near A = 1 20. binding energy All tilt rea:tion\ in ( 1 1-16) ha\'C free neutrons among (he products. As DOkd in Section 1 1 ..5 m C'OIUI«tion With "ponlaneous fission_ the reason IS thai

beIvy nuclei ba\'e • higher ntutron·to--prolon ratio than do hghler ones. The JRICIICe of IffIIItipk neulrons ImOng the prodUCI!O of Induced fission is faleful

for if we colIecI muy nuclei together, a

cbain reaction

....

�

IXlS\lble. Suppo

tbIt eKIl rmction liberlles " product neub'On.... If a neUlTOn I'> reqwred 10 initl. _ . racIioa IDd each reaction produces n neutron!). then each reaction may

� .. �"h<.-r f\"a�:ti,)n,,;

l'a...-h of th.:..." m;'� lIuti,lte II �\l.h('1""_ .11,,1 "ll I'''. "Ith :b ..u;.\.".....I\(' �t'ncraul'n {\.'k,l"'lIIlt " lIIlIC" thl' l'nel):� Ill" thl' rll'\I�IU'" lT "'111� tk<' �'nl'll!) relca"cd III c;ll,:h n,,,i" Il, Ii'll' energy t\'k',N"l1 11l the Jlh �l'nl'h\

�f\'(

..1('1 IHlIIIJ Ix'

L' c "

II ,.,\IUIJ IIKn.::N' n p.mcllti"U)

,\

-

J ' L ;0/1 (.

\ I I 171

Ilh time.

G"en th.lt urt\ll\um-235 llucll'l In "'''I on tnl1h ,\IlJ that (\ ...'....mtillK·\'\I" fl'-

Q;U1l'd. 11 I' re;l\ooable I�l \\llnJl'r " " I . \\l' lI\'Il ' t . \lll1n�11.! ,,,,,\1 1.:1'1,110 1'Coll'tion, III lluIU(\:. It I" une thH\g I,lr multiple neUII\ln" 1\1 l\e fltJtlH'd, Nit qUite anutht!r for all 01 them 10 be (Ib,w/'/lt"d b) \lther IIr;,,\iu1I\ � 1'\ thl pcl Ihl' proce,,, ,. n n ' . .

I1,U(kI 1ht'rt';llter, L1nulLun1-:!35 l1uI,.'lc\ lire e 'lremeiy f;lrelkd III 1I.llUl\', "u the \'bJ1'II."I.' " l .m) n!ill'ti,'n "cHing off another j.., I'Cll1otc, To Cll,ure th"t a ,:htlu, 1'.:.Il' ·

tit.If'I \\ill be ".:11' !>.u!>.taining, Ihe Ih\lllllablt: muterial 1ll1l!>.1 he ,lfTangel! III till' 'f'' "'(' JnJ ...hape-itllU t\ critil'1I1 ns.\cn\bl�, Thl' re.,dl.'l llI..y hil\\! hl.'al\l pI' r � I�'nn ,'rlflwf IIIU\S. It h tnlc Ihat there i ... a mini mum /tIm l tor iI chain 1"\:.ll',

b11fl 1\1 tIC -.('U ...u!>.l:ulHng. but it II; Ihe miTllIllUIl1 only for n eompll'ldy pUl� ',tl11' rk ;iIT.ut��·d 10 a ...phere. Geometry anJ impur;tic" arc fm.·lu..... One l'l the '''"') " of _

( .1. A ma�<; of ur:lmum·:!15 thai \H1Uld �"11lJ; nl.!ulron, '" through the ...urface U'e

1\' \'nlK'ill 1f ammgcd in a "phere would nol be critical if formed mto ;\ !lat !>.hect. Il.' t,here wlmld be too much area from which to lose neutron,. A ...phere !\iI" Ihe

flllnt01Urn �urf(lcc area to volume ratio, However. e\'en for a "phere. there " tI 111LnU1\I1I1I �i/c. occause the rutio incrca�e� :IS the silc of the sphere dccrca"e....

1l11puntlc" come into play becaU',c they may lIb�Ofb free neulron... without Ii..,

...umlng, 1i.) compen!>.ate, the "...!>.embly would have to be larger.

\\'hether an a....cmbly i .. critical " determined by the multiplication con·

\tanl 1. Ihe a\'crage number

of reaction, I;ct ofT by a given I\:l\l..:tlon. \I three

""utRln' were liberated In each reaction and each induced another reaction.

the number of r!.!uctiono; in each generation would be 3 time\ Ihe number in th!.! prt,;ou, generation. and k would be its maximum possible value 01 3. How­

�\I!r. the reactiom, in ( 1 1 - 16) show that some fissions liberate fewer neutrons, Lo,�' tit the surfnce and absorptions by impurities would further reuuce the

multiplication con:-.tanL Thu,. to give the true energy rclea\cd for tI given gen­

�r.ltion, we repl.lce " in equation ( 1 1 - 1 7 ) by k,

'\ �ustained chain reaction require... a multiplication constant of at Icast I . If l > I, the energy relea..ed mcrea\cs exponentially with time. If k

I , succes­

\i\e generations would all release the same amount of energy. giving n steady

chain reaction. If k

<

1 . the energy per generation decreases ex.poncolially with

time. The chain reaction-however il may have started-would die out. An uncontrolled chain reaction is lhe basis of lhe "atomic bomb:'2 Sub­ �'TItical (k

<

I ) parts are brought quickly together to form a \upercritical �...embly (k > I ), and the rapid exponential increase in energy re!'>uits in an e\plosion . releasing

5-6 orders of magnitude more energy per kilogram

txplosives based on chemical reactions.

lhan

lEllr'� III

Ih Iluckar IIge, bl1:'\\'ill� lh� "" kUI ...

IIpa" WI\) populllrly

,

",k=d II'> u� "�phllJl\I! Ih�

nlom." cv�n Ihough onhnM)' ionitllll(ll\ ,phi, un alOIn. tarly h"nln ",(';Iron_ w�rc \lUh \tnn�.J "aloml� hamh......

1l1I.J lh'" nilffie Mud.

Lller

wcapoll' m\(llvlnll lu'Illn of hydrogcn "",lllf'l'. w('rt

lermed

bomh�."

·hy.Jrogcn homb,_' LJu\h lin: '[lude,I<

The Fission RNCtor A. s�ad)' unchangmg #('ntr.lIIOn (If cnerg) require ... a multiplication faCior fj"lon reador. an a\�cmbl) in \\-hleh A j\ capable of I!xceedm8 I 1.\ c(lOlrulled '" 3.. l{l"�p * uf I Confrol rods are made of materials that readily a� nculron.., tt'ma,"in� othern.l\c inert. Boron and cadmium are COm. of J In a

mon!) u-.ed If inuuduced Into a region v.here a chain reaction is occurring,

wnrm' rod, reduL'c the mullrplici!lion factor, Becau\e they can only decrt'llJl!

k.

�hll.:h mu,' be mamtamed al

I. i\ fhsion reactor O1U\( have a pOlemial mul.

Ilpllcation wn\tanl greater than I .

Gneo me clo\c pm'(imil), of ti\�iomng nuclei, i l might seem that If A \omehov. be�an (0 (''(r.:ced I. Ihlng� would gel OUI of control almost instantly.

Ica\ iog no lime for COITCCUVC me4l... ure�. Fonunately. some neutrons come from decal of the ti\�ion produCI'i and therefore lag behind the fission by man)" ..econd� Though Ie�, than I Ilr of the free neutron'), the delayed neutrons

are pU1pO-.e1) relied upon in a reactor to keep k al I . Without these neutrons, the readion would die out: wilh them. It build\ slowly enough 10 permit con. lrol rod, to be readJu\tcd. Uranium':!35 i� the mo't common fuel in today\ fission reactors. yet It IS \el)' difficult to concentrate. N3Iural uranium i'i 99.3% U·238. which absorbs fl(utron, bUf rarel)' th\lon\ Becauc,e they are the same element, the two cannot be ,<,paroted b) ordinary chemical method\_ The allematives

are

tedious, rely.

ing only on the ,maJl ma\� difference between U-235 and U·238. Common

method.\ include (I) deflect ing the ion\ al different radII in a magnetic field_

the cyclotron method: (2) estabh,hing different concentration gradients in a hlgh" peed centrifuge; and (3) exploiting different diffusion rates in a gaseous ..tate through

J porous barrier

In any event. reactor'i do nol use pure U-235.

Some opel'3te "ith only the �mall fraction of U-235 present in natural uranium. while othe..... use uranium III which the component of U-235 has been enriched. Ho"" much the uranium muSI be enriched depend.;, on the effectiveness of the

moderator. Ncutrom produced in fission-"fasl" neulrons-tend

10

kllletic energie\ of aboul [ MeV. but U-235 is more likely 10 absorb

slow, or

�

have

"themlal." neutron,. whose kinetic energy is characteristic of the ambient lem. KE = kBT « 0.1 MeV. I t is the moderator's task to slow the neu­

perature:

tron� and thu� increa�e the likelihood that they will mduce a U-235 fission.

Simple mechanin show� that the beSI way 10 slow a particle is via a head·on colli..ion with an equaJ·ma�� particle. Because protons and neutrons have about the ..arne ma�". ordmary water. with single·proton hydrogen nuclei, would \e(m 10 be a good chOice for a moderator. Unfortunately. hydrogen readily abwrb.. neutron�. becommg \table deuterium. Thus, lhe requirements for a

moderator are small nuclear \ize and no appetite for absorbing neulrans. These

requirements are adequately mel by graphite (carbon). helium, and ';heavy

waler." In which the deulenum alom� already conlain one neutron and do nOl readily abwrb another. Even so. mo�t reacto" In the United Siales gel by with ordinary water. becau\e their fuel i� sufficiently enriched in U-235 10 more than compen�te for hydrogen'I, para\ltlc neulron ab.sorption.

Pluionium·239 behaves very much like U-235. so it i� also useful as a tis· �ion fuel. Although il i� nol found in nature, it may be produced i n a breeder

I I ,7 Nuelear Re.;:tiona

The �I�m�ol'" of a pre'�urized ",aler �aclor cmpJ\\)'1n8 ouckar fi"ltln Ent'tg)' frum fls..ion ht'ats water. Ftgu,. 1 1 .29 ,team generalor, creating ,Icam to run a lurtnoc. h ' h �ntcl' a . "

Control rod�

the presence of U-238 is actually desirable. When U-238 reactor, in which neutron. It briefly becomes uranium-239, then {3 decay� twice. ab"orb� a which then serves as fuel for the reactor. Pu-239, becoming All nuclear reactor.., currently producing commercial power are fi...... ion (as opposed to fusion) reactors. Figure I 1 .29 illustrates one of the most common I}pes. a pressurized water reactor. in which heat from the reactor transfers fir\t to water. then 10 a heat eXChanger in a steam generator. From lhl' point. 11 operates just like a fossil fuel power plant, with the steam dnving a turbme

that runs an electricity generator

Fusion

lIghl nuclei. like heavy ones. are less tightly bound than those of lOtermediate mas.. number. Thus. a'i Figure I I 30 shows. if light nuclei are thrust together 10

form somewhat heavier ones. the total mass should decrease and kinetic

en�rgy should increa..e. Thi... IS fusion. A typical reaction is

iH . 141

20

u

+

IH

I

\.0018 u

-

1

2 He �,0 160 u

+ l'

Q � 5.48 MeV

A\ we see. helium-3 is Ie,.. ma......lve than hydrogen and deuterium combined. De:)pite crowd109 two proton.. mto the same nucleus. thi.. process decrea'ie.. the mlernal energy. becau,c the number of attractive. energy-lowering ..trong bonds triple.,. from one 10 the deuterium to three in the hehum Fusion i.. the mean, by \\-hil:h ..tars generate energy. The mo..1 abundant element in stare; i., hydrogen. Olnd a series of fusion reactions. known a, the

.- " .... _. . .... .... "."-"Ift � tUt c P

-

J

•

L���.,..

•

.. 01(1 eo ..

or-......

.,.

T

or

,..-

T- '1'"

1(1) 1.:0 1".1 100 J ," 1 1m 2.)1 %4t1 A

,. ,n ,..11 • C'Jdr. 1tI!.h If ant\l\f'.It'1l) Iv".:lfIj the lJ�hl l) bound hehum� ........, ID Fipre I I J I , 1M 10.1 (")( t'ic$IO' with IWO pn)ltlfl'i uOIllng 10 fmm "

dr .. C lWl

Q - U -l1 MeV we' know. lhtrt j, Ik' bt'lI.u't�J �I(' uf Iwn pn1IlJn'i, Fur (hi, lu\jon 10 OCCur. a . It) bc..',lnlt' it o\-'urnm plu� it ptNlmn and II neu. JIft*JII musI dna be- ,nd....w mno.

.

..... .. .. TIIt __ ..... .,.. "'- .. -. .. """ ..... _ . dul _ •

� dUll im'oh'h lhe' "'cal. t{'n"t'. the pnx'c,>" h reliul\cly ,low In lhe­

lieU fuak., lhe drUitron L1.lIJl!'lint� with all{llhl'r pmlun ,

I

,

10

form helium-).

Q - ,'i ..J8 f\leV

l : jH + , H - :Hc

.

fa tbt fiaaj *" '.'0 helium-" nudci fu�' 10 toml helium.....

Q = 1:!.9 MeV

o

o

,

... ' 11

,

,.., t

o

.

, •

D

Tbis SIep mu.� bt pm:'Cded by Iwo of step), I and 2. Allo�elher. the process resUs ia four proIOOS (si:4- from sle� I anti :! nunu, 1\\0 from "rep 3) becom­

.... hdium-4 plus positron.s and neutrinos. Me30v. hi/e, 24,7 MeV or energy is rdeaIed. which is further increased by pair anmhilation of the po'i,trons.

While abe poIOn-proCOn cycle" i� lhe primary energy \oun.:e in our Sun. in stars wbcre dae � and Mlium concentration are hi!!h enough. another

!

!

poocess GCaII'S. 1Wo He nuclei fuse h,) form Se. which j<.. un"I,tble and would DIIUI'IIIy decay bad EO rwo helium..... nuclei. However. if enough fast-moving

II' ...... .... .......

nla\,;l..-I 4ft' W'\)UOO,. am.:ltht'r Win hllo(' �fOft' 1M �II)' (;'an ",-,.:ur, ....... P1 U hll� lx'ul\\J �c With this nudeu... begil\.\ the- carbon .:yde, • � n'kllt rtlrlJ "J� (,If "Nmm�" h�'dro�en. 1bc:o slC'P-\ un: &.\ 1'01l,''O\s: '

I

�

I: J..'

t I \. ... ,H -", �N.

'\:, + "

\ - : t>

�:

110 -

t'

I J"I

t

7'

2: ',

� I-, ,

_ .13 0

1

4: .. \'

6:

1-iN

-

I�

... ...

It>\:

+

+�,6

+ \'

:H - \�O

� U - \:c �Hc +

lJ du' qd�, l',ulxln- I :! l' merel)' a c,ltal),t. It i" ne..:c"'\lU) to mllull(' IhC' ",,1t,.'(" '" Nil U I' nol ... ... 00 up. I'or It n:.lppe:i.II' at the ('nd, The n(1 dle�1 I'" th.. , wnc J.' 11\ Ihc- pf"oton proton cydc' f,lur PI"\'h)ll\ becoming hellUm-4 plu.. IWll

I"l'tllnm\ ,Uld '"U neutrino', 8l1t II pn"k:ccd, much !'lI\ler. hC'cll"''''� III !HI P'oli nt �"<' It 1'('\luII",(: n pmton 10 hccomc n neulron befort fus.lon can ol,:l.:ur. -\1 hl�h cn\lu�h tempermurc\ and panicle cOnCelllnlliom.. ('Ic-men" ,It

t\d! hl�her Z thnn carbon mll)' be fonned in ':olan., Beyond ubl.lul :\ - 60. 0 lIud(,1 tlCcome In\ tightly bound. \ dement, beyond the cnrrt'pomilng /. 11ft'

iPIlh Ie,\ ill-ely 10 be fomled by fu...ion. Uo"ever. capture of n�ulron... lre�J III (fb
�;(J(ln (If elemenl' of higher Z. A '"lUlat bUI quicker pl"l'ICe,' III ,upcmo\'ae tlC the primary ,ource of the heavie..t element\ III the Ulll\-'�"'l' fi �lu8hl lll

Whe�J\ li....ion mny occur o;.pontaneou,I)'. there ... no 'ueh thlllg U'" 'llOl\. tlnCOtl' fU\lOn Nuclei are po\iti\'ely chargeJ and mU'i1 be !on:t'J H)gelhcr.

Thc\ mil.)' "ell he capable of fU'..ing ... in the �hon·rilnge inlernucleon (\tIrae· lilJl1: but only If their long-range Coulomb repul'iion can

be

\unnounlcd

FiSure 1 1 .32 depic,,", an unulogy: Imngine trying 10 roll a boulder lip the side

Figure 11.32 Nuclear fusion: over the Coulomb

hurdle. then mto the strong force well.

Coulomb repuhion Strong

illlC'mudeon' Jttr.I"ion

- -;:;;: - Q , - -- --':'; With I:lloll!!h inll:l\ KE. il can �fall in tho.: hoic" tu ;'

mur.:h lowef rn�rI!Y,

,....,.

/

.....

- - - - - - - - -

--

Ef

.... ""'*' I I

".... "'-'

of. ""kano $0 dull II falb 1010 a deep hole ar lhe rop. If the boulder (one

nuclcuJI

IS

DO( headed dirccrly roward lhe cenrer of the volcano (another

nudC'u\) II \utricient -;.pced, It will be deflected to the side or will simply roll

b.:k down Onl} If mltially mOl ing f
hen If ol particle dleab ntar lhe fim�h line by runneling through [he rop of the

c1a.'\K.tII, forbiddtn hilI. the larEe available energy. E( - £.. cannot be

�Itastd ",lhout enough mirlal ene!}!y 10 make tunneling likely.

We conclude rhat high demily and temperature should favor fusion. A

greater den\lry of nudel implies more frequent collisions, and a higher rem­

pel'"dture meam lhal a greater fracllon will have enough energy to surmount the

Coulomb barrier. The condilions are easily met in a star but not so easily On Earth The Ii"r \uccess was rhe hydrogen bomb. The high density and remper_

alUrt "ert produced b, �etung off a fiSSIOn device. or atomic bomb. If prop­

erly arranged near the fi\\iomng marerials, hydrogen Isoropes will fuse tn abundance Pound for pound. fusion e�pJosions are more energetic than fis­

\Ion e�plo\lom for lhe \Imple rea...on rhat. as Figure J J . 1 4 shows. the binding

energy per nudeon mcrea\es more sreeply from light roward intermediate than

from heavy loward inlennediate.

The Fusion Reactor The !!rt3ler their charge. me greater becomes Ihe difficulty in forcing two

nudel do�e enough to fuse. Therefore, 10 achieve significant fusion under

c�lndllio", not \0 impraclical as delonation of a fission device, the following

reaction'l of l = I nuclei are mosr promising. (D Siands for deuterium and T

for tnlium.)

,

2

[0 + , 0 -

',D

{ IT + : H 3

,He

+

Q = 4.0 MeV

I 0"

I

+ 'T 2 e + on I - 'H

Q = 3.3 MeV

( 1 1 - 1 8)

Q = 17.6 MeV

The two D-D reactions occur wilh about equal probabilily.

Even these best hopes. however. require conditions of extreme densily and

temperature that have thus far prevented civilization from explolling fusion as a

viable power source. To gain some idea of the difficulty. let u\ set the average

thermal kinelic energy equal 10 the electrostatic potential energy thai two +e

charges would have when separated by the rough nucleon diameter of 2 fm:

, N o m ( 1 .6 x 10-"C)'

9 x 10 - ,

C

2

x

�

10'" m

T 55

3

= -( I 38 x 1 0 " J/K)T 2 '

6 X 109 K

1 1 .1 �Ud�ilf R�a.:-t")n� e resti mates the t mperature required ' Speeds S ove naturally vary 0ver a and particles can quantum-mechanica1 1y t fltL . unnel t hrough the top range, thus able to COntinue bum" of lhe our Sun IS 109 Its ' fuel at . 'ef ..only" abou bam temperature t would vap0rize 101 K. n this any pan of I eve ' a solid , structure SUI . to avO!' d contammation' . fore, here it the fuel III a heS. T . . fuSIOn reacto r I'd so h I structure as WIthout o t s! Two me lhods have onfined . recei . ve e ved &: the conll nement and inertial C Co most onil nement . n: magnetic artenlio

511

.

'

IOUe

F'19\1re l' .33 MagnetiC confmement (a) Charged panicles in a toroid 1'0110.... helical palhs along field line�. (b) A tokamak adds a spiral field that average::. out drill along the ax.i".

C Confinement Magneti

field, a charged particle is campe1I ed to desc . be a spiral pat h n . and thi s may be ex I ng and about the field lines, W plOited to confime . . . ao nuclei magnetic confinement seh . ost proml smg eme ,s ' ba ed on a The m toroidal n in Figure 1 1 . 33(a) . A g show as , field . f as 0 deutenum 0r tic deuten um· m'gne ' 1 y to produce an ion'lzed heated SUffi Jelent is m plasma of elect . . rons and 1 t11','u . ' . nucle I. These Ions spiral about and along 'he circular e sitiv field r Ines po mSI de " . ver, the field weakens near the outS]' . toroid. Howe de of a toro'd ] , causll1g the the direction of the toroid's . . es to drift along aXIS (OppOsite ways for Charg h escape us d t an es ) confine ' ment. �0 solve this te charg problem the opPosi . . . ' 'mc1 udes another set of coils that use e1 ectr deSign ak omagnetlc mduc. tokam ' Own field perproduce a h uge current 10 the plasma, which produces Its "]on to . toroldaJ field. The the net field to , depict d ' e 10 F'Igure 1 1 .33(b), ular pendic . the torOl'd and averages out drift along the ' clfcles It as axis. als spir confinement alone does not ensure pressure or eless, erth Nev temperature . . to produce slgOlfica�t fusion. Several heating methods have been ough en high squeezes the plasma With additional magnetic fields', ano'her IOJects " tried. One . . partlC 1es, whl�h are unaffected by the fields and th u can neutral erg n s � high·e duected for mosl.e e�tJve heatin�. Unfortunately, other problems ily eas be onl� �o random Col1l�lons scatter Ions, but all heating methods also arise. Not bilize the fueL Trymg to hold a plasma in a magnetic tield while tend to desta can be very frustrating . The harder we squeeze at one place, the "squeezi ng" i� . S out elsewh�re . quicker It S UIl1 magnetic-confinement research facility in the United States A relatively ne", '

magnetic

�

�

�

nal Sp�\ ,)(.al Torus Experiment (NSTX), a collaboration involving is the Natio II I universities and located at the Princeton Plasma Physics severa1 laboratoril... Ie suggests, the plasma in this device is confined in a Laboratory. A s tJ han donutlike, though still with a hole in the middle. h region more sp mflguration. This shape appears to have several advanFigure 1 1 .34 sho i1 tokamak in securing high pressures with fewer instatages over the L:lJ ahead to demonstrate its ultimate potential. # .. 1 bilities, but mOl magnetic confinement is judged is power amplificaA criterion , 1'1 released from fusion to energy needed simply to protion, or the ratio eluding energy needed for confmement. Established vide heating- !)r pt and Japan have achieved values approaching I in tokamak facilitlt rac;:.tical breakeven still appears some ways off. Never recent years. thot. t hurdles has Jed or abilities to surmount many curren theless. optiml'in the ITER. :rger than anything presently in operationto an undertaki n tries worldwide ... the pledged support of many coun This proposed

,.)

Axis

.. Spiral fldd

......

(b)

pattern

,.... 11.'" 1lJt NalltlnaJ Sphcnt.:31 Toru, E"perimcnt and 11\ mlere.'oting field pallern

and j, e'pre<;\ly dedicated to producing. In e!....ence. a first-generation fusion power plant. Depicted in Figure 1 1 .35. It i!. intended to su�tain a power amplifi­ t:Jtion of 5. The plan ca1l... for conf;truction to be completed i n France i n lime for pla,sma testing in �OI6.

Inertial Confinement

Inenial t:onfincment in\!olve� merely deIJvering simultaneous blows to a quan­ tity of fuel from many sides \0 quickly that it has no time to slip away, as �hown in Figure 1 1 .36. The blow can be \truck by high-!.peed heavy ions or by laser pulses. In the forefront of inertial confinement research is the Omega Laser Facility :lt the University of Rochester. Here. a target contai ning deuterium and tritium is struck simultaneously from all sides by 60 beams of 35 1 n m wavelength laser light. In an instant, the target i" vaporized and the fuel trano;formed into a high. ....ma. While "ludie... at this facility continue, with pressure. high·temperature pla an expan<;ion in the works. the approaching leader in mertiaJ confinement is the National Ignition Facility (NIF) at Lawrence Livermore National Labora­

IOI'Y Still under construction, though already i n partial operatIon. NTF. depicted in Figure J 1 .37. will eventually have 192 la'ier beam\ directed at the target. Although direct impact on a deuterium-tntium fuel capsule is possible, the method expected to yield best results i\ Indirect dnve, i n which the beams strike a hea\)'-metal (gold or lead) ve�sel containing the capsule. The metal vessel <;trves as an intermediary. converting the la�er energy to X-ray\. which [hen compress the fuel. Although irs primary goal i\ to demon'ltrate net energy gain in inertial confinement. NIF field of plasmas and fusion.

i!. intended to have a number of applications in the

.....

ed). The 11.J5 ITER (propos

flBI fusion power pbnt1 (NOIe ..... near boaiom

... r.. -.'

..... ', .. la ...... ceII • fig... _ or _ upo .. -, ions mike a quaaIiIy 01 r.I • ca:e fIom lti_ _

Figure 1 1 .37 The NUlionul lgnilion Facility. The insert depicts indirect drive. in

which the ia!.cr beam... Jr� mcid�nt at the ends of a metal vessel. producing X·rays that compress the enclo...ed O-T fuel cap�ule.

.

.,.

Ch..... 1 1

Nuckw Pb7"'.

FISSion and Fusion: A Comparison

A\ soun:e!> or poltntlaJ \(lUrcC'o;; of power. fj,,<;;ion and fusion share certatn char. ik:lC'mllc\ Bccau\C' hoth (" plait the energy a.. '>ocialed with the strong force rad�r than the reJari\eJ) ",cal elcdromagnelic forces of "chemical reactol'\:: lhc:-) do nor uo,c much fuel. Neither requIre" the huge volumes of fuel needed by coO\'C'ntionaJ f�",1 fuel power plan"', Moreover, they do not produce carbon dlO\ldt gao;" an iPC'\ltabJe product of fo..... 1 fuel burning that is suspected of con­ tnburin� to global warming. On the down ..ide. both fission and fusion cause the 'tnK'lure!lo lhat confine the fuel 10 become radioactive. The useful lives of the\(: \lrudure.. are limited. and dhpo\J.J of \ueh "ho," m::l.Ierials is a problem. In other 3.\pecb. ti.....ion and fu\ion differ in very Important ways. First is the nJlure of the fuel. While mlher abundanl, uranium and thorium (3 promising breeder fue/) are mildly mdloactlvc and. of cour'!e, require mining-a costly and ..ometim�.. contenuom process. Breeding of Pu-239 from U-238 makes very efficient U"oC of natural uranium. but plutonium is highly toxic chemically and O1u,t he handled With great care. Deuterium. the future fusion fuel, is chellliCally harmle.., hydrogen and is fairly ea�ily distilled from natural water. As a fuel �un:e. II would be practically mexhaustible. Second. and far more important. are the " combu,tion" products. Fi",ion products lend 10 be fairly heavy ele. menb that are them,el\'e� highly rad ioactive. often with long half-Jives. Dis. po� of thi, wa,te 1'i therefore a 'erious concern. Fusion products. mostly helium ISOIOpe' and tritium. are much less hazardous. Helium is hannless. Tri· tlum ;\ radioaclive. but jl� half-life is relatively short. and it i s nOl chemically pO/�onou!J. (t \, far le,� dirty" than typical fission products. Finally, fusion does nOI invohe a chain reaction. \o,.ith its delicale exponential dependence and Con­ tro"; in"olead. It "bums" II, fuel m a way completely analogous to the chemical fires of comentional power plants. The advantages offered by fusion certainly merit further effort to make it a commercially viable power source. "

REAL·WORLD EXAMPLE E X P L O I T I N G AN A B U N D A N T E N E RGY S O U R C E

I I

Unlike earlier onel>. our final Real·World Example involves physics nOI yeT put into practical applicatIon. However. amid growing concerns aboul reliance on fOSSil fuel,. the �ubJecl is anr3cting increasing scrutiny. If the technical hurdles can be sur­ muunted. the .:nailabililY. relalive clean burning. and power per kilogram of lomor­ roy, " fu,ion fuel rna) amwer the energy needs of our advancing civilization. lei us -.ee h{l\l. Ib energy potential compares with thai of the reignmg king. Applying the Physics

The energy a\'aiJable in a barrel of crude oil. released via ··chemical fire." is roughly BTU. About how many barrels of crude oil would be needed to equal the energy that could be e�lracled from a barrel of seawater by fusion of its deuterium nuclei? (Nole: I barrel 158 L and I BTU = 1 055 J.) 6 million

=

SOL' T,(l�

To begm. we find the approximate number of waler molecules in a barrel. The mol· ecular mass of waler is 1 8 u. For our eMimate. the density of water is a suffiCiently good approximation for seawater.

x

10-l m1) ( lO' kg:m1)(158 - _ mas<.,imolecule 1 8 X 1 .66 X lO·n kgl molecule

dem�IIY ' volume

--

_.-

.. .

- 5.3 X 1 ()27 molewle\

The number of hydrogen atom�. at two per wilter mole<:ule. would be twice (hi, rlier In the chapter and given in Appendl' I. the percent natural ea number A'i noted . um I� 0.015 % Thu\. abUndance of deuten number of deuterium atoms

-

=

2 X 5.3 X 1
x

0.000 1 5

1.6 X I
Equation ( 1 1 -18) gives fu<;ion reaclions of two deuteron" which occur \\IIh roughly tqual probability. \0 each atom '.f share of the energy released should be about half the 3,erage of two Q values. or about 1 . 8 MeV. The energy relea�d by complete fu�ion of the deuterium IS therefore total energy

= =

( 1 .6 X 1()24)( 1.8 X 106 eV)( 1.6 x 10 19 J,·eV )

4.6

X 1011 J

ill 4 x

lOS BTU

The ratio of the energy available from the deuterium to that from bummg the �ame ,otume of crude oil IS (4 X 1(8)/(6 X 1()6) if: 70. About 70 barrels of crude oil "ould equal I barrel of seawater burned by "'fusion fire." Inefficiencies in the standard methods of converting heat energy to electrical energy would reduce the usable power in both processes. But even allowing for effi­ ciency difference:. at other stages. the availability of seawater relative to crude oil makes our result worthy of con�ideration.

PROGRE S S A N D A P P L ICATIONS Akering a Half-life In reference books. we find hundreds ofhalf-!ive.. Ulbulated. ...uggesting that they are not subject to \'1lriation_ For the mo..t pan. half-lives do appear to be impervious 10 external mnuence�. but the poSSibility of ell.ceptions has been ..tudied for some time. Of the various fonns of radioacti....e decay. electron capture might �m most susceptible, al> It depends not only on e....ents within the nucleus but alo;o on the electron orbiting outside. Actually. It IS Important that the electron is not a/nays ou�ide. and as we know (Figure: 7.15). wave functions of s-stale electron� are nonLero at the ongin_ Still. even this fonn of decay ha�. until re.,;ently, appeared largely men. Howe....er. the ubiqultou.. bUt:kyball has now entered the picture. Since the dl,t:O\ery of fullerenes. science journals have been abuzz \\Ith novel way... of getting various atoms or small molecule!> m,uJe theo;e all-carbon cages and then studying the effech. A team of researchers in Japan has succeeded in :.tuffing C60 molecules with Be atoms. which decay by elc\.twn t:'lpture with a half-life of about

�

53 days (Ohtsuki. et al_. Ph.\".fiml R�I·'�"" lLtters. 26 February 2001). While other environments 'howed half­ life variations of no more than 0.2%. inside the C60 it dropped by nearly I�, Idea� of substantially allering half­ lives to aid with radioactive wru.te di�posal are fanciful. but thiS ne\\- result is certainly of interest. shedding light on radioactivity and the unique environment m!;ide the buckyball. Pyroelectric Fusion A.. shown m Section 1 1 _7. if thennal motion alone is to drive fu)oion. seumg the electro�tatic potential energy barrier to the a'erage thennal energy implies a ,·ery high temperature_ However. '>etting It equal to q� \/ showl> that a deuterium nucleu� need onI)· be accelerated through about I MV to tu..e 1,I.Ith another. ev·en unaided by tunneling. Although not an obviously practlt:al route to fusion power, thl� alternative method of �tid..ing .. recently been demonstrated in a deuterons together ha surprisingly convenient way that may find numerous

. 11.38 The eicctric field around a pyroelectric cT)" [1l1 and a ' Fig .... �hdrp lip IIlmll!� deuterium lind dri'(" il lo....ard a deuler3led larget, Vtht'rt nudei,lt 6,,1\," (Xl'Uf'). 101

.<,

L,th,um IllnlalOlt cl)'�tal L'i mm lead �hicld

Dtlllcrlltcd target

Copper mc�h

l imrUni1111 apphcatl\m, A pyR'leledrit C!,)'\laJ i� one that C.1U�' dlilrge 1\) 'epar;lte when 11 1\

hCille<.!, RC!lot"arche......

connection with pyroelectric fusion.

0 + D - � He +

n.

As pre<;ently implemented. the deuterons emerge from an

:11 un.A U'�'J Ihl' cHect to produce d('u[('ron� and to

accelerator with u kinetic energy of 2.8 MeV, giving 6 MeV

accdcnllc them 11.)\10ard l,ther deUicron .. (8 Naranjo, el 81.. .\tlt'lln', 28 Aflnl :!O()�. 1 1 5-- 1 1 7). Figure 1 1,38 iliustrall!S

neutrons. Leaving the helium nuclei behind, the neutrons pa)� through a collimator. producing a narrow beam. then

Ihc app.tratu, Pl)'lli,,: ..:harge from one face 01 me crystal

10 the target. Here they scatter inelastically from nuclei,

\,:11111:\:""..1 at a ,harp IUllg,[cn

tip.

Healing the cry�ta1 by

rai�ing the nuclei to excited states from which they deexcite

IU\I

:!!\ K proJUCt'd an ek�lm: field at the lip greater than 2S GV/rn. \utrlnenl])' high to IllnilC nearby atoms in a �ur­

by gamma decay.

mUlII.ltng �;I' nl lkutcrium. Thi .. field lind thaI of the crystal

gerprint�. di�lJt1gujshjng one element from another, and this

It-.clt' then drme the n:\uhing ion.. awa)' \;olcOIly. imparting

is one of the great pOlential advanlages of the new tech­

"'-melll: t!nergle, In ewe" llf MO keV The ions then struck a \;lrgCt 01 deUICTJIl'l..l erbium and, rutled by tunneling, fused in $"R::;ll numtk:" Thc c\ idcnec W;l, a nux of2A5 MeV

dependent on and potentially obscured by the local electro­

nt.'Ulnln� from the rc;u;tllln

D + D - i He +

II.

which was

As noted in Section 1 1 .5, gamma energies are like fin­

nique. While X-rays respond simply (0 electron density and

MRI images work beM for only certain isotopes (in ways magnetic environment),

NSECT can

lay bare the presence

fut In c".:e�, 01' Ihe b,Kkgn)und le\'t.'1 and proportional to the

of all demenl'>. The only exceptions are hydrogen, whose

r.lthl!r mude\t, the reM!,U'Cher.. antiCipate that larger tips

energies are above 6 MeV (Helium isn't of much concern

ion �urrent, Although the actual energy output in lhis setup

.... ;.\

llr UrTl\y� o/' tip' wmbinl!d with operation m cryogenk

tcmpcmlure, l·,lt1 \�all! the !!flect by a lactoror

1000. This

nucleus has no excited stales, and helium. whose excitation in the body, anyway.) Becnuse neutrons, being unCharged. penetrate the body quite well . the teChnique has the pOlen.

m,ly lead \0 a rehllhk. lli1ndheld neutron ,ouree, and one day

tial for true 3D imaging, lind the ability to distinguish all

nught e,en power thru,ten;

elements would be a great advantage. For reason� not yet

m minimure �JX1cecraft.

fully understood, cancer.'> in the early siages cause changes in ratios of certain trace elements. Thus, by comparing

Medical Imaging: Turning on Gammas with a Neutron Seam Medical imaging ha, seen the estabh,hment ofa

gamma intensities.

numbc:r of new technique, In receO! decades. among lhem

ccrn by a nonimasive scan long before present �canning

NSECT might

identify regions of con­

thl." IT \Can, whn:h employ, X·rays. and MRI. which uses

techniques would notice anything unU
r3dio·lrequen�� e]ectwmagnetic radiation and an e�ternal

ti��ue extraction required by other methods. Of course. it is

magncti� field. All imaging methods have their strengths

natural to ask what hazards the high-energy neutrons them·

and weakne" e'. so the search for ne.... ones continue.'>. A

selves might pose. Inve!>tigation of this aspect continues,

promi,ing technique in early de\elopment is neutron

but the belief is that the radiation dosage per volume would

,timulJted cmi�,ion computed tomography. or NSECf

ultimately be less than that in a standard che,t X-ray.

::!O06. 3�75� �39(1). Thi, method use\ relati\ely high-energy

Be�ides early cancer detection. the technique could provide a vital new .....ay to track movements of metabolic tracers_

(Aoyd Jr.. et al.. Plwllc'i III Medicine and Biology. 23 June neutron,. produo.:ed in the ..arne fusion reaction noted in

both stable and un�table-within Ihe body_

lapt1e r 1 1 Summary

A nuclear ruction ocan when nucleons IR

or exothennic, deprndin, on whether the IftUI tncreua or

11It �"OI'\.,tiruenh of the nUC!c=U.Io-proions and neuuon!io-are a.. nucleon.. The mas,," number" IS lhc num· lfIO'N'I collec1i\,t=ly

�nI.

� nuckon�. and the atomiC num�r Z. which Identifies the

is the number of protom. Nudei of the ..arne alomi/'" _bet bul dillercnt number;, of nculron!> are called isOiopes of

ement. Nuclear rad II an: in the range of femtometcn de ,I\cn d

10 I� m). about 5 ordcl'\ of magOltude lImalier than 1 fro •

orblh In atom,. All nudei have about �i typical of det:tron uggc l ing a do\csi-packed arrangement

tit ....me den�lly· s

of

$

spheres aacomprt\�ible The predommam forces In the Rudeus are the "mng fon:e. attraction between all nudeon\ but With • \(Jtlllg. �hort-range , l't1\tr.11 hard core: and the electrostatic force. II weaker btl! rar·reachmg repul.lon between all prolon�, Along with mort'

cbIDfDd or

ekchanged amon, nuclei. haccions nuty be either endoIhermie

A

tbt (\(Iu�;on prmt.:lple. these forces Cllu\e nuclei of imennedlale '" 60, the ftil" number to be most �table. For nuclei around

r-eal(\' energy per nucleon would hil\'e to be eJ{pentied to !>C'pa. r,ltC the nudeon� FurthemlO�, lI�ht nucleI are mO�1 stable \lben the number. of protons and neutrons are about equal. \lillie In he:wy nuclei, the mO�1 lIghtly bound amlngement ha\

111:« ncuU'OM than prolon". reducing the destabililmg effe!;t 01

Coulomb repuhlOn_ The enel1!ie� bmdmg nudeon, 10 their nudel are about

� \If\le� of magflllude larger than tho�e binding el�lron� to thtlr atom�- Thu" much mon: � than the atom. the nuclcu� ha� , rna" �Ignilicanlly Ie" than that of Ill. �parated p�, Binding

� may be (alculoled directly !rom thIS ma.�s difference.

( 1 1-5)

�s, and the kjDdie

melJ)' �Ieacd I """ by ( 1 1·7)

Partkularly impor1lnl ekolhemuc

raelioln.

8ft' fi

iuI

mass nuclei. In fission , hea"y nuclei are

light nudei are brou,ht l o,etber. In either

pltl. and in fusion.

CaSC'. lbe lou man

energy become!> the kinetic energy of the products. •

tndicates advanced questIons

Conceptual Questions I . Explain 10 your perplexed friend how you can brin, together Iwelve object.. of malts 1.01 and end up with one obJ«'1 of ma�s 12,{Xl

2_ How i\ it Ihal a high binding energy is It low energy?

J, In S�lion 1 1 .2. il j, ltaid thai iron and nickel represcnt maximum \Iability. Chemisu), emphasize ihal helium

is Ihe m(hl ..Iable elemen!'.) How can thew: daims be recont'iled')

4_ Why mi!:hl a flat-bouum tinik v.ell bc= a better appro"i. malion of the porentlal well confining nodeon.. lhan a

Coulomb well. tapering to a lov.-'csl energy in the middle?

S. The \cmiempirical binding energy formula has fout tenm, SUpp!.l\C we h.we a nudeu"i with H I proton and

22 neutwn,_ For each term In the fllrmula. indKale

(without calculatiOn) whClhC'r addins: one mure proton

would cau\e an increa�e or 3 decrea"ie. and c.I[plain why

it should ha\c Ihi\ efle!;1 Foeu, on the: undcrlying idea.

Be(ause of our incomplete lnowledge of the �trong force,

tht nudear "problem" cannot be 'olved e.l[actly, Models

mUM

be u>ed The liqu Id drop model Jnd \hell model ha\'e been par­ tlCuliil'ly succe" fuL Unstable nudei nUJin a lower energy state via radioactive

decay. In .....hich the nucleu\ e1(pels II boo1Id two-proton--twl).neutron partJr.:le: f3 decay. In "hich a neu­ uon ba:o� II proIon. (lr "ice \'el"J. and an electron or po)ltron

decay, Fo�

include

a

IHTratcd; electf\ln capturt:. in which an orbiting electron is seized .mdcombincd "ith a proton. producing a neutron; and y decay. in

\\lIlcll a photon (ame\ away e.l[ce�\ nuclear energy.

RadlOacO,'c decay i� probablJi\lic, Consequently. the num­

ber of radioactlH': nudcl pre,ent In a sample drop, e1(ponen­ wly with lime. The mu,t o;ommon measure of Ihe decay rate 1\

half·life, TIll' or the lime required for the number of nuclei pre­ 'oCTIltO decay 10 hall

ih

"

inlflal 'aluc

1-'1

In 2:

T1'2

6, The \,}" mJJority 01 mC;hured nuclcaI' "plm; are fanly �mall. between lero and nIRC: time

� II. Mure 10 the

point. nuclear �pin doe' nllt uniformly ux:rea$C

With A

Argue that if the ru.�umptl(lfl' in the- llihell model �

"alid. then the\e ob\er'\iltl(,"� are not unc'(pected,

7. U..mg Figure

1 1 1 4 d, a guide. e'(plain why the hca\icst

of natur.lJ1} oCl:urring nudel mi�hl be inclillC'd to cmit

a

panicle, bu t nOI 10 emil OIher "mall groups of

nucleon�

8. Cerl3in nudei wilh hllif-li�e, between dOl)!> dnd . few years are found in nature

Ifl

small abundunfo."eS Ih31

do not chan¥e at J.1l mer man)', many )can, How is Ihis po�\lble" (Him_ Natural uranium and Ihonu m have l't'/)' long half-Ji\e\.)

9, Why dlle' fi,,'lon of he:a,y nudci tcnd 10 produl:e li'ce neutron,')

10. I n bolh

( I I-IS)

and

fusion, which ekploit the vel')' tight bindin, of inrenncdiale­

O-D rcac.:lj�'n" ln clju;Ulun ( 1 1·/8). 1....0

deuleron, fu\C 10 produce tw.I} particles, .. nuckus of

A = 3 and It free nu..:k"tlfl

M.lU d<-xreases because the

517

...... _ fII .. A - 3 _ . _ _ .... = "ps, ...... ' •.- r:l dll ftJID ' cw. Tbe ...... _ fII · " h _ _ IlilL Wily c." 1M 1 _ ...., .. .. . belium-4 IMtCDs ... -.. ehe? WIly _ 1IIUIDpIe por6clcs be , ......

I L 'IIIortG- 131 .. _ _ .. Eanb ...s is .... _ .... _ • • _ Iuol. b __ _ -'Y WOO .........13I. ....,.. _ by ..... nwn­

... r:I prDIDDI lDd DCUInJDI. which means thai it i5 DOl the IdUII "IliaD full. WhaI isotope iI'?

11 An untrained but �'t'C friend CJ.claims. '1br:y say thai nuclell' efterJY can be releucd by sUctin, nuclei ....... ...s by brakina lhcm oporI. Tho' ....·' .. mike _,,,

SuaiJhten

OUI yOW' friend's confusion.

Exereises -. , ,,,

� � many other spheres could be PU1 in

h a ,Iven sphere. calculate !:he maximum conlICt wll

possible number of strong bonds per nucleon for a SUr_ rounded nucleon. (Not,: Because nucleons share a bond. CKh nucleon "owns" only half a given bond.)

RecaUin. mat !he deuteron. �H. has half a bond per Dudeon. how docs your calculated maximum num�r

correspond to Figure 1 1 . I4 �

11. Calculate the binding energy per nucleon of

ICCMetium·98.

13. Ca1culale the binding energy per nucleon of carbon-12. Soctlon " .3

14. Using the �miempirical bindIng energy formula. esti­ mate die ma�s of a europium I S2 atom. 15. According to the stmiemplrical binding energy for­ mula. what �hould be the binding energy per nucleon -

oft«hnetium-98,)

26. Nuclei of !he same mass number but different Z are

1 1 WbII Indion of spII:C is lK'hIalIy OIXUpied by iron aaelei in • ""solkl" piete of iroo"! (The density of iron b 7 11 x 10' q/In'.1 .4. AI, wbat speed would a puticles t..ve to be dim;ted al aoId foil If ICXIW � to contact lOki nuclei? IS. Prom Ihe abuDcIanca and aIOalK' masses given in AIIPeDdi " , or the lwo naaurally OIXurring isotopes of boron. determine the avenae atomic mass of naruraI bcwoa. C� Ibis with the YIIlue given in the peri. _ ...... of Cbop« 8. •6. Determine 1M approllimale ratio of 1M diamettr of a uranium nucleus (A ... 238) 10 thll of a beryllium aucleua fA - 9),

-. , 1 .2

.,. (a) Pew . DUe.. of... - 220. CIIimaIc very rooshly how ...., auc..... WCMlId be .. tbe swface. (b) Irlbe bind iDa eDCIJY of an ilIItrio, nucleon due 10 the inremuckon ...... weft � and if GJJ auclecm � as tiJhtly ·

....... .... _ ....... ....." wouId be �.-" iCIID IIX:OUDI W difl'cmM hindinJ of 1W'f'.:e nucleons, ......,. wbll would be the total biDdiaa eDCrgy?

... 1b taIlO'Ve c.. deca'on &om helium ft:quircs 24.6 eV, .. ftIDOVIDI ill llCCODd labs S4.j eV. Tbe ionization -U 01 � is 13.6 eV. Wbcn applied 10

beIituD-4. by whll pen:eDII&e is cqualion ( 1 1-5) in enor due 10 its ipori:aa of dcctrooic blading eDaJies? It.. Sbow IbIl Ibt enerJ)' requited to remove a neutron from beIiwrH is 20.6 MeV. • I, cluaical. bIrd-spbcrc UllltDptiORI. wbll smallest ...... of A. would mab ODe aucJeoo surrowxIed? Relale

,aur _werlO P'apre l l .14.

11. Iy

known � isobars. Oxygen I S and nitrogen I S arc iso­ -

-

bars. (a) In which of the facto� considered in nuclear

binding (represented by terms in die semiempirical binding energy formula) do these two isobars differ" (b) Which oftbe isobars should be more lightly bound") (c) Is your conclusion m pan (b) supponed by the dc<:ay mode information of Appendix I� Explain. (d) Calculate the bi nding energies of oxygen-15 and nitrogen- I S . By how much do they differ" (e) Repeal par1 (d) but u�e the semiempirical binding energy fonnula rather than the known atomic masses. 27. (a) Calculate the binding energies per nucleon of the isoban (defined in Exercise 26) boron- I 2. carbon-l 2. and nitrogen- 12. (b) In which of the lenns oflhe semi­ empirical binding energy fonnula do these binding energies differ. and how should Ihe�e differences affect the binding energie� per nucleon� (e) Delennine the binding energies per nucleon u!>ing the semiempirical binding ener&)" fonnula. and discuss the resulls. l8. What does the semJempirical binding energy fonnula predici for the ma�s ofa carbon-I 2 alom? Comment on your answer. 19. 'The fint lwo tenns in the �miemplrical binding energy formula deal !.Oldy with the intemucleon atlrllction. (al Calculate lhe ratio of the second lenn 10 the first Ieml for A 20. What does it say about the surface nucleons" (b) Repeat part (a). but for A :::: 220. 30. "The semiempirical bmding energy fonnula predicts the binding energie� of neon·20, iron-56. and uranium-238 within about l 'it Make a "' x 9 table \'\'ho'>C rows are the three i�otope�. who�e fir\! four columns are numeri­ ca! values of the four lenn� in the fonnula, whose next four columns are each of these tenn� di�'ided by A. and ""

_ 119

..ho�C final column IS the sum of the preceding fOUf. Ol�uss In some detail what the tobk reveals. se 19 notes that the energy needed to remove l1. Exerci neutron from hehum-4 is 20.6 MeV (8) Show that the energy required 10 remove a proton IS 19.8 MeV (bl Why do these values disagree with the BFinucleon ...alue shown for helium-4 in Figure I I 14') (Hint_ Con­ SIder nuclei around hehum-4 In the figure.) a semicmpirical blndmg energy pu nucleon for­ lI· Obtain mula- Using this as a guide, txpl:un why the Coulomb force, which is only about dx5 as strong as the mternu­ clean attraction for two protons "in contac!" (cr. Table 1 1.2), would eventually have to become a dominant fac­ lor in large nuclei. Assume that Z, N, and A Increase in rough proportion to one another )3. Equation (9·42) gives the Fermi energy for a collection of identical feonions packed into the lowel!il energiel!i allowed by the exclusion principle, Argue lhat if applied 10 neutrons or protons (ignoring lheir repulSion) in a nucleus. the equation suggests that the Fenni energy IS roughly the same for all nuclei. Making lhe rough approximation lhat lhe spacing between quantum levels is a constant in a given nucleus, argue that this spacing should vary from one nucleus to another in proportion to A-I.

number ahfnN e,cccd� lhc: number antiahped by le"l!i than iii" )�. ..

8

Section 1 1 .5

A, noted In Seclion 1 1 5. carbon· I I dtcay� to boron· I I How do the various fkton Inwlved in nuclear stability argue: that <,u.;h a decay IS fa\·orabk'l 38. Calculate Q for neul.n.Jn decay, 39. How much kinetic energy 1<, �Iea� and whal is the daughter nucleus in the decay of polonium·210� 40. How much kinetic energy is �Iea�ed and what i� the: daughter nucleu.. In the fJ decay of nitrogen· I 3? 41. Oxygen·19 {3� decays. What i5 the: daughler nucleu�, and what may be said of the kinetic energy of the enulted fJ· particle? 42. Polonium·207 may undergo three kinds of radloaclJve decay: fJ"'. and electron capture. But It does nOI fJ decay. Given Its po!>ltion relative to the curve of stabil· Ity. explain why this IS not unupected 43. Potassium-40 (Z "" 19, N 21) is a radioactive I!.OtOPC that IS rare but not unknown in nature. It is particularly interesting in that it lies along the curve of stability yet decays by both fJ'" and f3 -that is. in both directions away from the curve of stability. (a) Identify the daugh. ter nuclei for both decays, (b) Many factors governing nuclear stability discussed in the chapter (e.g., th� in the �miempirical binding energy fonnula. magiC numbers. and even numbers). Identify those that would argue only for fJ . decay. (e) Which factors argue only for fJ-t decay? (d) Which argue for either fJ- or fJ" decay'? 44. Find Q for the decay of berylhum· 10. 45. Calcium41 decays by electron capture. (a) Find Q for the decay. (b) Show that caJcium4l cannot decay by fJ" emis§ion. 46. Using the semiemplrica! binding energy formula to determine whether energy would hkely be released. would you expect the hypothetical nucleus i�x to s nf!utrons. the tol31 binding

37,

n

t

a,

""

are

section 1 1 .4

All target nuclei used in MRI have an odd number of protons or neutrons or both. What does this suggest about nuclear spms? (Not�: 80th the proton and the neutron have gyromagnetic ratios.) 35. In electron spin resonance (Section 8. t). incoming elec· tromagnetic radiation of the proper (resonant) frequency causes the electron's magnetic moment to go from its lower-energy. or "relaxed," orientation, aligned wilh the ex.temal field. 10 Its hlgher-energy antialigned state. MRJ IS analogous. A quantity commonly discussed in MRJ IS lhe ratio of lhe frequency of lhe incoming radia· tion to the external magnetic field. Calculate this ratio for hydrogen. You Will need to adapt equation (8·7) to !.he proton. Note that the proton gyromagnetic rauo, gp' 155.6. 36. MRI relies on only a tmy majority of the nuclear mag· netic moments aligning with the eXlernal field. Can· sider lhe common target nucleus hydrogen. The difference between thl! aligned and antiahgned states of a dipole a magneuc field IS 2JA..B. Equation (8·7) can be used to find JA.. for the proton, provided that the correct mass and gyrOmagnetic rallo (gl? 5.6) mserted USing the Boltzmann dlstnbuuon, show that for a 1.0 T field and a re�onable temperature, the

)4.

10

-

are

Q

arc

=

... " �_� . .. 0.

CIIIIJY would Ktually dlCmur. bplatn wtaM .t wrong e.

�

with the naIVC' atpmcnt. il lht reooll lpced of tht dluJhtef nudeu� when ,.,Ho 0 dcaY''t CTrcal aU mobon Il!o nontd'II\1i�bc.}

_ " .6 51. Given IOJbally 100 a of plutoOJum.239. how much time mUlt pus for the amou.ntto drop 10 1 g? 51. 1l1e IMial decay rate of a wnple of a eertal.n radioac· tlve ilOlopc II 2.00 X 10" � I After haJf an hour. !he decay rate 11 6.42 x 1010,'1 Detennine the halr·life

of the itOlope.

51. Tbc half·hfe TII2 1$ not the a�ngc lifetime Tof 3

radloacll� nudeu!> We find the a"'erage lifetime by multlpJyma t by the probability per umt IInlC P(t} that the nucleu\ will " Iwe" thai long.lhen inlegrating over all IItnC. (a) Show that P(t) \hould � gi\'en by At-AI (Hint. What mu�t be the lotal probability?) (b) Show that T - Tlr../ln 2. 53. Eighty centurie� atter its dealh, what will be the: decay rate: of I g of carbon from the thigh bone of an animal" S4. A fOiU,il specimen ha.\ a carbon· 14 decay rate of 3.0 s' 1 . (a) How many carbon· 14 nuciel are pre�nt? (b) lrthi� number 1\ rn the num�r that must have been present when the animal died. how old i� the fouil') 55. A fossil \pecimen has a 1 4C decay rate of 5.0 s I Ca) How many carbon-14 nuclei are prt!otnt? (b) If the specimen I� 20.000 yean; old. how many carbon·14 nuclei were present when the animal died? (c) How much kinetit: energy (in MeV) is released in each /3 decay, and what is the tOlal amount released in all fJ decays 5in� the animal died') 56. A bone of an animal contains tli mol of carbon when it diei. (a) How many carbon-14 atoms would be left after 200,000 yr') (b) h carbon-14 dating useful to predici the age of such an old bone? Eltplain. 57. Pow.sium-40 has a half-life of 1,26 X 109 yr. decaying 10 calcium-40 and argon-40 in a ratio of 8.54 to I If a rock sample conllllnod no argon when it formed a solid but now contains one argon-40 atom for every pot8ssium-40 atom. how old is the rock'! 58. (a) Oetennine the total amount of energy released in the complete decay of I mg of lritium. (b) According to the law of radiOllctive decay. how much time would this release of energy span? (c) 10 a practical sense. how much time will it span? 59. Gi"'en initially 40 mg of radium-226 (one of the decay product!. of uranium-238), detennine (a) the amounl that will be left after 500 yr, (b) the number of a parti­ cles the radium will have emined during this time. and ecl me amount of kinetic energy thaI will have been

released (d) Find the decay nne of the radium at the end of the 500 yr. 60. Ten milligrams of pure polonium-210 is placed in 500 of water. If no heat is allowed to escape 10 the Sur_ g roundings, how much will the Icmperature rise in I hr? S.ction 1',7 6 1. Determine Q for the reaction

62. Calculate the net amount of energy released in the: deuterium-tritium reaction I 2,0 + J, T - "2He + on 63. In an assembly of fissionable material. the larger the SUr.

fnce area pcr fissioning nucleus (i.e., per unit volume)' more hkely is the escape of valuable neutrons. (a) What is the surface-to-volume ratio of a sphere of radius ro? (b) What is the surface-to-volume ratio of a cube of the same volume? (e) What is the surface_to­ mlume ratio of a sphere of twice the volume? 64. If all the nuclei in a pure sample of uranium-235 were to fission, yielding about 200 MeV eaCh, what is the kinetic energy yield in joules per kilogram of fuel? 65. (a) To release 100 MW of power. approximately how many uranium fiSsions must occur every second? (b) How many kilograms of U·235 would have to fis. sion in I yr? 66, 1\yo deuterons can fuse 10 form different products. Although nol the mosl probable oUicome. one possi. bility is hclium-4 plus a gamma particle (see Concep. tual Question 10). Calculme the net energy released in this process. Consider equal numbers of deuterium and tritium 67. nuclei fusing 10 fonn helium-4 nuciei, as given in equa. tion (1 1-18). (a) What is {he yield in joules of kinetic energy liberated per kilogram of fuel? (b) How does this compare with a typical yield of 106 Jlkg for chemical fuels? 68. For the carbon cycle 10 become eSlablished, helium-4 nuclei must fuse 10 fonn beryllium·a. Calculate Q for this reaction. 69. Ignoring annihilalion energies of the positrons, how much total kinetic energy is released in the silt·step carbon cycle? (There is a quick way to answer this. and a much slower way.) 70. (a) How much energy can be extracted by deuterium fusion from a gallon of sea water? Assume that an average D-O fusion yield is about 2 MeV per atom. the

I!un:"

(b) A modem supertanker can hold 9 X 107 gallons. How many "water tankers" would be needed to supply the energy need:. of greater Los Angeles, consuming electricity at a rate of about 20 GW, for 1 yr? Assume that only 20% of the available energy ac(ually becomes electrical energy. 71· A fusion reaClion used to produce neutron beams ('lee Progress and Applications) is 3 2 I iD + I D - 2 He + on

,

M\uming that Ihe kinetic energy before the fusion is negligible compared with the energy released, calculate the neutron kinetic energy after the fusion.

Com prehensive Exercises 72. The binding energy per nucleon in helium�3 is 2.57

MeV/nucleon. Assuming a nucleon separation of2.5 fm, detennine (a) the gravitational potential energy per nucleOli and (b) the electrostatic energy per proton between the protons. (c) What is the approximate value oflhe intemucleon potential energy per nucleon? (d) Do these results agree qualitatively with Table 1 1.2? 73. For the lightest of nuclei, binding energy per nucleon is not a very reliable gauge of stability. There is no nucleon binding at all for a single prOlon or neutron,

111

)·el one' is "table (so far as We' knOW) and the other is not. (a) Helium-_' and hydrogen-3 (tritium) differ only in the switch of a nucleon. Which has the higher bind­ ing energy ptr nucleon? (b) Helium�3 is stable, while tritium, in fael, da:ays into helium�3. Does this 5Ome­ how violate law,,? 74_ You occupy a (me-dimensional world in which beads­ of mas" 1110 when isolaled-attract each other if and only if in contact. Were the �ads to interact solely by thi� uttr"olclion, it would take energy H to break the con­ tac!. Con�equently, we could extract this much energy by stick.ing two together. However, they also share a repul�ivc forcc, no mattcr what their separation, for which the potcntial energy is U(r) = O.85Halr, where a is a bc"d'<; rJdius and r i" center-to-center separation. The closer the beads, the higher i<; this energy. (a) For one station"ry bead, by how much does the energy dif* fer from "'o(.1? (b) For two �tationary beads in contact by how much doe!. the energy differ from 2mocl? (c) For three beads in contact (in a line, of course, since this world is one-dimen�ional), by how much does the energy differ from 3111F? (d) For four beads in contact, by how much does the energy differ from 4mF1 (e) If you hud 1 2 isolated beads and wi<;hed to extract the most energy by sticking them together (in linear groupings), into sets of what number would you group them? (I) Sets of what number would be suitable fuel for the release of fusion energy? Of fission energy?

1 2 F u n d a mental Particles and I nteracti ons ----------��.' -----

Chapter Outlinf:

1 2_1 122 r:u 12A

HO'o'> Force Act

Anllpartlcle,

Force., and Partide\: How Many"

Particle Pnx!lJC,;lIon and Oe1ectlOO 1 2 � t:>«;ay Mode, and Con'lervation Rules in the Standard Model 1 2 6 Panty. aw¥e ConjugatIOn. and Time R�eBa1 1 2 7 Unified ThC'une.. and Coo.mology

r"f"hc \oColt�h 10 lind the fundamtntal building bloch of nalUrt tw 1 been gwnS nn fur lung tunt_ Ancient ptu� po!olUlaled •

..- die: "'�lfld compn<.ed llmpl) alt.

....;d.e1. earth. and fire. ScicnUrK !'>tudy dl"cnc fabrx It came 10 be acccpu:d that � ,.,or\d ."u JIId' rne.ak\t a mote � of ill""'� of '" t\k:h thC"re ICC'I'I1ed w be douns of dJ ..I Ul c t l)'pn: the "ckID'l't\ ,- \ bnct rC1um 10 &. Impkr \1IC'4' follovoed the dl�O'otty that �l the tkIDCnb �J 10 bit kull of the 1rOlIhC lhrec ··rurxbmenuJ� partKks' pro­ ... DCUIftJfl.\, an,J el«1roo
��

-" �t"al1,. rnc.uth h&, aptn �duced the ou.mbrf of perlJCks,� of bcIJII I� Protun_'l. rrtJlruM., And man) of the QC'Vo parucln ha\.c � \U\k.'"tUIC' and appar 111 be rntn'1) .,:hfferenl Combn�llom of • hmned. tumbtr \.'II mutt-M!U: COO,lllucnl , oovw ca1k.J quarb. In thu c�, ...c di� (UU Iht pmubn, � of the bwklan, bkK:b of nAtUre: and the lMCparablc

� of tXlI'Io the) lntcnc1

1 2. 1 How Forces

Act

abe pft),iQJ. 1UU" cnc- (OftUSU 01 e� panlClc$ alI ltlltr&1 In ,;,nl)' . (tv. dl\llftl.1 ••,,'l.. But Uu� llmp&e outlook l� the: result or

" D moo •

"1It'*,

...- 0I �, IDU
pad! ..... by uch _ 01

� III � ct. ;aI mcct'lal'lln.. .'C lad 10 c;(!nCmmIC Oft r� 1haI "'fIll! pb)uu.I coab.."1 ""II IerNcIft. � foren. U'Id. fnctllOD. " cun· _ nccpcJOa " pa'toU) b &ppraA k> ..... 1broaJb • -':11lCl l..-r. Ul • .... clcaramiapftivn.. _ 5rant ltw force ,.lItIouI Itepa IIUd) of 113'" � CIIftI&, .. mare Ibc- ruSe IbaD dlc "� TCINOa. for Ul\WICC'. " 523

"'Iuallr a L"o.tI"< manife..Wlion of miaO\copic electromagnetic forces. Hu\l, C'�r. we: are I.u�hl th.3' p.utjdC'� do nul e:u�rt forces on one another af a di\lllIlce. Fun.."C' j, rraJ/� com'C'yed b) [) pervR\i\'e field. gr..lviwrional oreJectromagnetic as 1M t:a� may be. We no"- ake r the idea one .,rep funher. The modem , jew i� Ihat a tundameOial force between particles is COn_ \C'yed b� the /t\l'hWlge of a media.ing par.ide whose role is special 10 that fOfCt". The medialmg particle is also k.nown as a field quantum-a particle of field-which giH�' thc Iheory jr� name. quantum field theory. Thus. there are h\.O dJ\o;es of particlc�: lho�e Ihal experience rhe force. and those that are en:nanged 10 con...ey il. (A� we will see laler. Ihis is a bit too restrictive, as the pJrtidc t")(changed is in S(lme cases capable of experiencing the force it COnvep.J Con,ider the following analogy: Physics students Anna and Bob slide along a frictionless frOien pond. Each is unable {O change velocity without cnf:aging in a mutulI! force with .�oll1e other object. As shown in Figure 12.1 , Anna h:}\ J l>nowbaJ! thai ...he can throw to Bob. The exchange of the snOwball ha� the effect of II mutual repuhive force between them. Anna's momentum is Changed by -P,mw.baIJ as she throws the snowball, and Bob's is changed by as he catches it. We might term this the ;'snowball-mediated force"; .• P,Ik, tklU a force between physics students (the particles experiencing the force) medi­ ated by a �nowball (the particle conveying it). The lower diagram in Figure 12.1 represents the net eITect. The exchange of Ihe mediating particle is the cause of rile momentum changes experienced by Ihe particles engaging in the force. Diagrams of this sort are known as Feynman diagrams. We will see them again in Section 12.5. The 'nowball example clarifies how the exchange of a particle might Con­ ,ey a force, bUI as an analogy to the workings of a fundamental force, it has .�e\"er31 shortcomings. One is that it would seem capable of conveying only a replIlsil'e force. Both Anna and Bob experience momenlum changes away from the medimlng particle. To convey an attraction. the throWing of lhe snowball would have to force Anna fOl\'ord the snowball. and (he calching of the snowball would have to cause Bob to move in the direction from which the snowball came. The total momentum after the catch might indeed equal the lotal momentum before the throw, but momentum conservation would seem to be violated during the mediating particle's ex.change. Another deficiency in the analogy is that for a real fundamental force, Ihe mediating particle exists only during its exchange. The " snowball" is created solely for the occasion. disappearing just as mysteriously afterward. This raises concerns about energy conservation. Although energy might be the same at the end as at the begin­ ning. wha! about the interval during which the mediating particle performs Its task? Mediating particles do not obey the usual momentum and energy rules in the way that " rea)" particles do. and for this reason. the particles exchanged to convey force are known as virtual particles. In fact, they differ from the Cor­ responding real particles only in their lack of adherence to those rules and in being undetectable in such a state, which the physics of quanrum fields tells us IS fleeting. In many ways, the phenomenon is like tunneling. in which kinetic energy is negative and the probability of its occurrence drops off rapidly with separation.

..... tl.1 A 'lWI.'C � WdttIb � ..,. n.:1wtJc ola ltlOWbell

•

'

----::-!'Im)�l!dU

...

12.2_'"

Although not rigorous. one way of relating the range of the force to a of the mediating particle is via the energy-time uncertainty principle. poPCl1Y dered for an mterval of ttme �" there IS an Inherent uncertainty ,lE in

coo�i

dlt energy of a system. governed by

!J.,:>£

'"

h

parucle of energy ,lE could appear and disappear without "erifiably up�et­ so long as it existed for no longer than ,lr :::::: h/ �E. ,n \ g energy conservatton the temporary problem with momentum conservation.) By covers iS also a force whose mediating particle has mass must be a shortargument, ge force. If a massive particle must be created, there IS a lower limit on the

; �� :

ount by which the total energy could deviate: the mass/internal energy of This, in turn, implies that the maximum time the particle could lnal particle. Even at the speed of light, the particle could travel at ive is 6.1 �

hlme2 .

\Ilrv

most

ax � cat :::::

h

-

me

h 1 range ::::: -­ em

(12-1)

If lhe mediating particle has mass, the range of the force is lirruted. Conversely, a force whose mediating particle is massless could have infi­ nite range. With no mass, there would be no lower limit on the energy of the mediating particle. Its strictly kinetic energy could be arbitrarily small, so the

time and range could be arbitrarily large. Consider the electromagnetic force. Although it does fall off with distance, it nevertheless reaches infinitely far, so

it would have to be conveyed by massless particles. Indeed, electromagnetic

forces are conveyed by the exchange of massless photons. An electrostatic

repulsion between electrons. for instance, is conveyed when one electron emits a virtual photon that is absorbed by the second electron.

We will return to equation ( 1 2-1) when we discuss specific fundamental

forces in Section 12.3. Here. we merely note that it has been used in several

instances to predict fairly accurately the mediating particle mass from knowl­

edge of the force's range

before the mediating particle was first detected. This

ISone of the most convincing arguments for treating forces as the exchange of particles.

1 2.2 Antiparticles

Before delving further into fundamental interactions between particles, we must understand antiparticles. For each kind of particle, there is an antiparticle

that shares

essentially all the properties of the particle except that it is of

III

_ 0." tI

" "n _ t .., � .. ... i_ _

ClfIPlCiIe dtIqe. Section 14 introduced the positron-me antiparticle of � eJoc:croa.. 1be poUlrOII bas the same mass and spin as an eleclTOn btu is ofpas_ ime cblrJe. An el«Cron and posib'Oll can be crealed in me process of pair production. When they meet. pair annihilation may follow. in which they diuppear and Iheit mass enerv is convened 10 pholon energy. Many OOltr

anap.tides have been found. such as the anliprolon (negalh'ely charged) and

antinturron funcharpd). II mighl seem mal an uncharged neutron has no prop­ erty to disliJlguisb il from irs anlipattide. However. the antineutron's distinct

idenbfy is coofumed by the fael that it does annihilale with the neutron

�

whereas rwo neuuons do nor annihilate. Acrually. as we will see in Sectio

12.3. the neutron is not fundamental-nor is the proton-and its internal struc­

hft disbnguishes particle from antiparticle. (Some uncharged particles. such as the .0. lack such distin,uishing sU'Ucture and are their own antiparticles.)

The convenliona.! symbol for an antiparticle is (he same as for the particle but with an overbar. The antiproton is thus p and the antineutron ii. An alternn_ live convenbon is often used for charged particles. The positron is usually rep. rtsenled ef- rather than e -, and the antiproton is somelimes wrillen p-. Similarly. the JJ. f- and JJ.

-

are anlipartic/es

of one anOlher.

The existence of antiparticles is an experimental faci. bUI there is a theo­

retical basis: relativistic quantum mechanics. Let us pursue jusl enough of this advanced theory to gain some idea of how combining relativity with quantum mechanics might suggest the existence of antiparticles. The SchrOdinger equatIOn for a free particle 1ft one dimension is

A2 a2 2m ax>

---- "'(X, I)

=

Expressed in lenns of operators, where becomes

- jJ' "'(X, I) 2m I

ih

a aI

-

"'(x I)

(12·2)

'

p = -ih(ajax)

and

E = ih(a/at). it

"

= E "'(x, I)

As we know, this equalion is based on energy. In the absence of external potential energies. the kinetic energy of a particle. enerxY E. However. because for kinetic energy.

p2/2m

p212m,

equals its total

p212m is not the relativistically correct expression

= E cannot serve as the basis of a relativistic

replacement for the Schrooinger equation. A logical basis is the relativistically correct expression

(2-28):

(12·3) To obtain a relativistic matter wave equation, we might try inserting the appro­ priate operators and then have each tenn operate on a wave function:

c'jJ' ''' (X, I)

+

m'c' ''' (x, I)

=

£' '''(X,I)

11.2

( 1 2-4)

Klein-Gordon equation, this has been shown to yield correct �nown a� the the behavior of spinless massive particles at all speeds. Par­ about tiolls redic the related Dirac equation. In nonrelativistic quantum obey spin de� with spatial states that are solutions of the SchrOdinger me<:hanics, we work with equation, and we treat spin states separately. In the Dirac equation, spin is d as an integral part of the overall state from the outset. We leave Incorporate equation to a higher-level course. The mOSI important point )lUdy of the Dirac it and the Klein-Gordon equation are based on ( 1 2-3) and thus �ere is that both

�

share a "problem."

The main difficulty with basing a relativistic matter wave equation on equation ( 12-3) is that it involves E squared. If E == p112m, then E is positive. Solutions to the Schrodinger equation are thus always of positive energy. But equation (12-3) is equally well satisfied whether E is positive or negative. There are perfectly valid mathematical solutions of ( 12-4) for which the total energy is negative. The question is how to interpret these mathematical solu­

tions physically.

There is another difficulty with ( 1 2A). It can be shown that the integral

I",'qrdV does not change with time if 'I' is a solution of the ScllrodiTlger equa­

tion. This is reassuring, for we have always interpreted this integral as the sum of me probability density '1'.'1' over all space-the total probability of finding me particle-and we claim that it must always be l . However, the integral may change with time if 'I' is instead a solution of the Klein-Gordon equation. Were we to interpret it in the usual way, the particle might either appear or dis­ appear as time passes! We can fonn a density that doesn't change in time from

solutions of the Klein-Gordon equation (see Exercise IS), but we cannot inter­ pret il as a probability density, for it can be either positive or negative. A conspicuous link between the problems of negative energy and negative density is that the sign of the density is correlated with the sign of the energy. positive-energy solutions correspond to a density of one sign and negative­ energy solutions to a density of the opposite sign. This clue serves as the basis

for the following

interpretation: The new density is

charge density, whose

integral over space must indeed be a constant in time but may be of either pos­

itive or negative sign. A negative-energy solution describes a particle identical

to that described by the positive-energy solution, except that its charge is of me opposite sign-the antiparticle. The idea of negative energy is still troubling. Without going too deeply into relativistic quantum mechanics, it might be said that an antiparticle state is a "hole" in a sea of allowed but usually filled negative-energy slates of the

particle, as depicted in Figure 12.2. A unifonn "Dirac sea" of charged negative­

energy particles would be undetectable. A particle may fall into a hole and thus disappear, but only if the hole already exists-that is, only if an antiparti­

cle is present. In filling the hole, the particle surrenders an energy equal to the

Antiparticles 127

Figww 12.2 ��r a pc.l\iIlH�-encrg)' panicle drops inlO an cmP')' nt8ati�-e-eotrgy 'Iale. or IWO po\ilive-energy panicles annihila�.

tht re\ull ;\ the ..arne

l.wrc1� l:

0

=:::s:::== Hok

;.0 E •

-

q Q

-q

Annihilation

£ . mc�

=

..

_O£ i!! mc2 £

""

0

nr(,l

I � ----__ v __ __ __ __ __ L-

E ;,r m�

E li! m("�

tlifferen(;c between the positive initial and negative final energies-twice the panicle's energy. This lo_�t energy goes to photons. two of whjch are needed to con\crve momenlum (see Exercbe 3.41). Because equal energy would

be

rdeasetl in either view. we see that it is equivaJent to regard the filling of the

hole u\ the annihilation of two particles. each of positive energy me?

1 2. 3

Forces and Particles: How Many?

Efforts to demonstrate the essential equivalence of seemingly diverse forces have been very succes�ful in the past century or so. For instance, electric and magnetk phenomena had been once thought to be completely distinct. but by the early 20th century, they were conclusively shown to be simply different tnce!. of the single electromagnetic interaction. (Exercise 2. 1 1 4 shows how merely changing references frames cau�es E and B to "mix.") Electromagnetism wa.\. in fact. lhe fIrSt force to be successfully married to the quantum-mechanical idea of force being conveyed by the exchange of particles. Known as quantum electrodynanlics. or QED, the theory was worked out independently by Tomonaga. Schwinger. and Feynman in the 1 940s and 1950s (Nobel Prize,

1965). Our present knowledge indicates that aJI forces three distinct fundamental interactions: the

are

manifestations of

graVitational, the electroweak, and

the strong. Four had been accepted until the 1960s, when theoretical advances by Sheldon Glashow, Abdus Salam, and Steven Weinberg, working indepen­ dently, showed that the weak and electromagnetic interactions are really just two alipeclS of the sanle electroweak interaction (Nobel Prize, 1 979). Nevertheless. the two forces

are

still often referred to separately because the " weak part" is

manifest only in cenain reactions among subatomic panicles. It is not revealed in ordinary electromagnetic phenomena. Much research continues to be devoted

10 showing the equivalence of the

three remaining imeractions. A theory in which all interactions appear as

...

11.3 Forca and ,.,ida: How ...."

TAIlI U.1

Fundamental lon:c\ and pan.de.. Gn,llatlon

-

--

,.".., -

MaWcnergy

_ ...... ",.

GrlI�lllln"

_

Ie-' \IP'

10 )9

-

I/rJ

.a o 140. 1)5 •.!. •

-.

1"'

\

< 6 X 10-11

0'

....

1 1m

�I.O

0

QWlrb

......

Parti(lpanL' In gna....tatlon. rICCU1.l....ealr.. .

�Ipant.. In p.m\uaILaR lind eleclIOweak Spin f.kI:lnln. e C"KU\Ilno. \'�

\'"",", p..

,

I

I , I

t.fI(:lItnno.

Charge:

0.5 1 1

-\

< 10 5 .06 <.;

,.ll(tJlnno. I'" buon. 1

Ma�\

, ,

0.2

\.78 X 10-' < 20

0 -\

0

-\ 0

Up. ...

Down. d Strange, .. Charm, c

Bouom. b

Top. I

! !

-, -'0

,

- ' 00

I ,

I

! !

and ,tRlog

- U )(

1 0'

-4.5 )( In:' - 1 Ki) ( 1 0'

merely different aspects of a single fundamental interaction would be the ultimate unified theory. Alas. such a theory has eluded many of the brightest

minds in physics 10 this day. Unified theories are discussed in Section 12.7. Adding to our suspicions that the fundamental forces are somehow related are twO characteristics they share. First, the fundamental particles experienc109 the force are fennions, and they engage in the force by exchangmg mediat­

ing particles that are bosons, Second, for a fermion to engage in a force. it

must pol,�ess the property associated with that force. These properties are

and

mass/energy. for gravitation; charge or weak charge, for the electToweak;

color charge. for the strong interaction. Table 12.1 summarizes the

�tate-of-the-art in fundamental particles and interactions. As we proceed, we .."iII see how the pieces fit together. Let us now study qualitatively the pre�ntly accepted fundamental interactions and introduce the particles that engage in them.

Figure 12.3 The prolon and neutron are not funda.mental. They contain

quarks.

The Strong Force Seither the proton nor the neulron is fundamentaL As depicted in Figure 12.3, each is a combination of more-basic particles, called quarks, that are bound together by a mutual attraction. the strong force. As far as we know, quarks are the fundamental fermion� that engage in the strong force, and there appear to

Prolon

Neutron

-

a.,kr 1 1

Pat .M" PIIIideI IDd �

be six types. As each was identified or proposed. it was given a fanciful name and the names have stuck: up down. strange. charm. lOp. and bottom. . properties are sUllllDlll'ized in lIble 12.1. 1be proton and the neutron comprise ...... quarks each. with quark _I uud and udd. "" pectlvely. From the lab. uIated charges oflbese quarks-+ for the up and for the down-we obtajn = + I and the prqJCI' charges of the proton and neutron: + + + = O. Panicles composed of quarks are known by the generic name badroDs. We discuss other hadrons later. Perhaps the most remarkable propeny of quarks i� that they have noninle_ gral charge. This would appear to dash Qur time·honored belief that any amount of charge is always an integra] muhiple of the "fundamental charge" e = 1.6 X JO- 19 c. But the belief is safe for the time being. for no way has yet been found of isolating a quark. Moreover. there is reason to believe that it will never be done. Because the electro.!.tatlc force falls off as 1/r2. the energy required to pull opposite charge� apart. e..en mfiDitely far. 15 finite. The strong force. 00 the other hand. actually get� stronger With separation. (The force holding nucleons together in the nucleu�. dlscus�ed later m the section. IS an aspect of the strong force that doe.!. decrea.'-C with separation.) Consequently. infinite energy would be required to �parale quarh. Quark confinement is the tenn used to describe the refur,al of quark.\ to be o;eparaled. Certainly the attempt can be made. bul what we m"'ariabl)' find is that Ihe energy expended is sufficient even al vel)' �mall separation�. to create quark-anliquark pairs. We eod up not with separaled quark«., bul '>Impl) With a grealer number of quarks. which fonn multiquark hadron� of JOlegr.d charge. E"'en so. this explanation would seem to raise more que"tlon� than It an...wer,. If quarks can. not be isolated. how do we know they are of nOnlntegral charge') Indeed, how do we know they e\len exi�t? As DOted in Section 7.3. Rutherford dl'ico..ered the atomic nucleus by probing gold foil with high-energy aJpha particle... . In J very Similar way, much of our infonnation about the IRtemaJ o;,tructure of hadrons ha.s COm� from experiments in which they are struck by high-energy electrons. Funda· mental point electrons are �nt in to probe for other fundamental POint pani· cles (i.e., quarks). In deep Inelastic .scattering experiment'!. electrons of energy greater than 20GeV and momentum 20 GeVIc ha"'e been .,mashed IDta nucleons. 1be corresponding wa"'elength of the e1edrvn' I' on the order of 10-16 m. Because this is only about a thl!' nudeon riKhus. the electron should indeed be able to reMlh'e �tru'turaJ del.lIls v.lmlR a nudeon. Tbe'le experiments have provided unml\takahle e...iden!:e that the nudeon COnlaln� . three particles of fractional ,�e. (The upcnment arC' . !nela. ...tic. becau\C' some the eleclTOn's kinetic energy creales additional hadrl)n'.J The mediar:ing boson of the slrong for!:e IS the gJuon Thus. hadl(ln may be thoughl of as quarb bound l08ethcr by C()ntIRUUU' cxch:ange of gluon�. Gluon! are spin-I and mas.o;leu. AccurdlOg 10 equallon ( 1 2 · 1 ). this �oold seem 10 imply an infinite range. whICh lit � Jlh the oh'Cf\alion that the �lrong force actually increases with quark �parallon. HrMe\cr. lhc tfteetl"'e range I� very sbon. As nOled above. attempting III rull lILWr" apan, e"'en \mall dl\­ tances, simply generates mort hadron'!.

-! i j

j

-�

-

The�

i � � -

fa

.

_ a.,IIr "

ne

• • m' ..... .. ...

c..__ of Hodrono

WU IU, quMt, and "It anuqu.vl" a !l'�al number of dillcreni hadron, at'{' ,..sa1*. Man) uf the JIOUlbll' CIlfTlt'lln,Jliofl\ ha\-'e been produced In high. eGCrJY. UIC'1utic coIlIlIoo!j; In \\ hu.:h l.inC'llI: energy b com erted (0 tTle(j In. n d ..n JftN>). Table" 1 2 .2 li�1\ ", along wllh &hell quart r;'OfI'rnl ma."�". iUk.I ()I�r propertIC'_ A II have antiparticle, in whkh C*:h quart .\ �rl",:t"d b) lh anliquarl.. Threc-qunrl.. hadron .. are known .., .."... (from thr GR:'C'l for ··hea....) " J and 'u-o-quark hadrons as mesons Hnlmncthate). The �utnln. for m,tance. i .. a baryon As nOled earlier. color ncuuahty drnwlth fhal II ha\'c one quar/.. of each color. An c'tumple of a rot'Klfl i� tht pIOn, \\hil'h come, in three variclie... The 7r+ is of quark COntent ud. Ih + I ch.illJc i\ ru.:ccl4mlcd for by Ihe . charge of the up quark and Ihe +J1 dl4tJC' ur the anlioown. Ihe antiparticle of Ihe down quark. The up and amid. tl"n I.juut\ mlghl be red ilJld anlireJ, green and Ilnligreen, or blue and anllt'lllH." The anlipartid� of rhe tr' i� the 1T� , of quark cOnlenl lid and there '

.

.

+�

,

The 1TO ha� no declric charye

I

(lilt Il( ctw}'t'

.

.

J[ i.. a mi ;(ture of {he charge.

,mJ nll\lr-nc=urraJ quJ.ri. combinjjlion � uu and dd. "'·rvrn Tallie I� .:!. we \(.'(' Ihat m;lfly hadron' are much heavier than nuele. .

l1l1' GJ\'alt'f ma.,.. OUlpUI rtl.juirt� greater kinetic energy input, so heavier one�

lenJ hI be more recent di\CO\'erie.� u..ing higher-energy acceleralors, From the OI\cril,!!e hfclime dala, "e see [hat all non-nucleon hadrons are fai rly shor1 h\t'J In (,'J"C!' \If \er)' 'hort-Ji\et.l particles. it i, common to specify an energy "ldth rather rhan a lifetime. Acconltng to the uncer1llinlY principle, a pani. dc.'\ en!.'!}!) (an he known 10 no greater precision lhan !J. £ :=!: Il/T, where i is Ihe lIme mlen'al Juring "hich rhe pilrticle e�iMs. The lifetime corresponding III il wiJlh of 50 MeV is about 10 �,\ �.

Intrinsic Properties

� (S�IIOn 8.7). three-quark baryonli �hould be fermions that are either spin.� or \pin.�. a� Table 12,2 renecl�. (Aclually. it isn', quile so easy, See Progress ;nd

All quurk, are ..pin- fennion�. By the rules of angular momentum addition

Apph(l1tion� for a nOle on

lhe

spin crisis.) Simi larly, two-quark mesons are

'pm-O or 'ipin- J b<}.�ons. In general. when we apply the term spin 10 a hadron. we rtall)

mean 101.11 angular momentum,

which includes any orbiral angular

momentum its C'onstiluenls might have.

rei are di lferenl p� i� les. F�r Insl�nce, the 1\0 and 1,° are .both uds and

Table I :? 2 'pm,

shows thal .

'PIl1-� . The dn;tmcllon lies

some hadrons have Ihe same quark conlent and

111

the mtnnslc property called Isospm, I. It is

an angular momentum, as is spin. but

not

ir does have a similarly quantized "third

�.

component" IJ and obeys the same addJ!ion rules-hence, the name. The up

have isospin I = while the strange quark has isospin = O. I n the !O, the isospins of the up and down quarks are aligned, giving a

and do\\n Quark each

I

100aI isospin I = l . while in the .\0, they di�tinclion between the times)

is nOI based on

are

KO

anlialigned, yielding I = O. The

K� and K� (the subscripts refer to shon and long life­

isospin, but on an unusual mixing of the

particle and

TA I L

ut.:c.,J h4J.l\\n.. Commool� pn.,,", -

_

I 12.2

:----

\1..",

�-,.ns

r "

•

1'1«\ Ie:)

..,!,

9-&()

�)

•

\

1

I liN

,..)

I I�'

t�')

I I I fI

{-, (ddsl

1 141 I ' I Ci

S' .1IJ.,1 5 1�'1

.-

lLUl 1 + lu

1 tuooJ! l

tlllW1

� I�l l'· IUU,1 l" ,\J,bl

(�) ::.0,1,1,,1

t'

!' II '


L'�I -

I�.'� .! 11l 12.'2 I �J ! I.'S.'

IJ�" OM7 1:"'1 I 'J

�

1672

-

Slran�

Sptn

I I

.... ,.

(l {l

,

,

I 1 , ,

!

I

I l

,

I

, ,

! !

-I

2

� (l {l (I (I

,

I

I

,

l

-2

-2 - .1

Ufellnw. ,.

1.1,

I . +1 I. I

tor _kim " T) I ;110

l ), l Ol

S O \( IO " ,

t . t1

' -l X III >I ,

n.O

� (\ )( I(l '",

I.

I Ci X

. , �. "1

'

, !. +1

1(1 '" . co ,

� " )( 10

\ ,6 X IO

lid i

,,�

(I

1. + 1

0

1.0

(I

Jl

....

n

•I

•••

n

nll\

..,

n

01 1 \

(,1,1)

,..

n

II ' (uJ)

7.·

II

I..

(u� )

I\.

....�\t.J�. sci I

II.l'i t.J•• k11 t..:

KI)!

.,

,..

.)

�-lO MrV

I. - I

··m MrV

, t1

- IO MeV IO MrV

II , � 2 )( 10

\0;.'

Hea\)

mo..."

I. +!

I· I

I. + 1

,..

I. I

tNJ

(!>II)

;\ 2 )( 10

Me-V

I�t Me-V 'II MeV

� I Me\' ' I MeV 50 MeV

�\II1I.lJ",nR qllJrli �)t.'Ild the �tlange

J,I/I (c d

,' 10l)

II

11. 0

H7 lc\'

("")

"-IN.l

II

n.o

50 lcV

\

Ib antiparticle KO. A d,...cU'.slOn of this miKing may be found m higher-level . le�l\ on particle phy...ic... Yet anolher mlrtTl... ic property given m Table 12.2 i.. strangeness. Sen... , bI). II is the \lrange quark that endow... a particle with slrangenes"" By arbl

tr.tI)' sign chOice. po......e ..-,Ion of one strange quark gives a strangenes\ of

.

I,

of twO, a "trangenes.. - 2; o f an antistrange. a st.rangene!<.\ + 1 . and ..0 forth. We dl'cuss 'tTangenes, further in Section 12.5. For now, we note ...imply that

the hadron.. conlaining only u. d, and s quarks can be grouped mto ..everal

related multiplel' accordmg to ..pm, and that an individual member of a mulli� plet is distingui...hed by its ...trangeness and isospin. (Four such groups are ...et "'part In the lable.)

Strong Force

"';eulml atom" may attract each other electroslatically. Each may have an eleetnC dipole moment (permanent or induced). and if the moment... are properly

onenled. a nel altracllon re\ulls. Tenuous Ihough it may seem. Ihl... attraction is how some substances, notably water. (ann solids (see Section 10.4). In a -,imi­ lar effect. color-neutral particles may auract each other via the ...trong mterac­ lion. It i... this atlTaction thai holds the nucleons together in the nucleuli. Called lhe residual strong force, it 1 -' not a distinct force, but rather a coar...e manifc-,­ talion of the Irue ..tmng interaction. Still. it too can be viewed as being

" "

1 � )( 1t) , .

I.

..... (u .. )

1.0

Ii II x HI

1'1 I

.

1 2 )( III "

1. 0

120 Mr\'

\o;.�I(lh) \o;.�{MI )

2_

10 "

n n

120 MeV

I, '1 i. m., t, tn . , I. I

10 "

I � I Me-V

'"

Jd)

I

I,

••

I. • I

" (du)

..

1.1,

0

''''

.. (du )

-

The Residual

0

-

J20 Mr\'

I . ... 1

0,0

I",'

....

,,11(1,11,1

..-lO MrV

I, - I

111 ..

..

-

M.-

tM«\k'1

I;!OMeV

�. -1 i. -i � ,,

.. · (uJ) "",1,1\1

".

I . ... I

I. ,I I. I

,,-

con....eyed by the exchange 0'- a particle. Color-neutral uud protons and udd

neutrons may exchange a two-Quark, color-neutral combination of u, d, U, and

d quarks-a pion. Although the lightest of hadron�,

which explains why the residual �trong force

i.�

(he pion does have ma.ss.

�hol1 range. In facf. rough

knowledge of the force's range Jed, lhrough equation (12- J), to a rea�onably . accurate prediction of Ihe pion's mass before II was ever detected (see

Exercise 9).

The Electroweak Force

Particles lhat possess charge or weak charge ca n engage in the electroweak force. Besides quarks, this include.� the class of pal1icle� k nown as leptons (from the Greek for "light"), which lack color charge and are therefore blind

10 the strong force, As shown in Table 1 2 . 1 , there are �ix known leptons in three pairs: the electron and electron-neuuino; the muon and mu-neutrino: and Ihe tauon and lau-neutrino. (All 81')0 have antiparticles, bUI whether tho\C of the neutrinos are actually distinct particles is nOI settled. See Section 1 2.6.) Despite their different masses, the elecuon. muon, and tauon are very mUch alike, and one oflhe open Questions in high-energy physics is why we " need"

[0 ha....e all three. Although earlier work had um fied the electromagnetic and weak forces, elecrtoweak theory received a great

boo.�1 from

'I Hoof! and Martinus Veltman around

1970

the contributions of Gerardus

(Nobel Pri/e. 1 999). Their new malhematical framework made eXl.:ellent predictions of Ihe ma<;ses of mul­

tiple mediating particles a decade before the..e particle... were discovered. II is now confirmed thaI the electroweak force i, conveyed by the eX Change of

any of four mediating bosons: the photon and Ihe weak bosons W �, W-,

and Zoo A glance al their proper1ie., quil.:kly reveals the complexity of the elec­ troweak interaction. The photon, exchanged in the " electromagnetic part" of

the interaction. is massless, while the W:! and

ZO partic le ..., exchanged in the

"weak pan," are among the heavie�1 fundamental particJe<; yet delected. If there is indeed only one underlying inleraction. the ..e four are all field quanta

of the same field. Bul how?

The schism is known as a spontaneousl) broken s),mmetry. The asym­

melry is the different behaviors of the elearomagnetic and weak forces, and it occurs at low particle energies. A toy top pro\'ide\ a u-.eful mechanical anal­

ogy: If <;pinning at b..igh rotaIJonal energy, II will be oricnlt!d 'ymmetricall), relative to the ground, but at low energy, II �pontaneou..ly drop' to an asym­

metric repo� on it5. side. The true underlying symmetry of Ihe elel:uoweak interaction becomes apparent only al particle ener�I\.'S mUl.:h higher than those pre�nlly accessible 10 observation A \'alUI! of "hoU I

I

TeV 1<; a common

guess. At very high energies, the IOlrin"'ll: m:.ss vf 11 partklc� becomes irrele vam. The massive particle momenlUrn--enr:rgy re/atlon. + m2(A

£.'

l'"1c2

­

becomes that for a mas!lle.slo piiJ1Jcle, E - Pl. and the beha\'llIrs oj Ihe mediat­ Ing particle!! converge.

!

All ieplon� are Splfl- fenmono;, .'\eutn no are un<..harged ItIld Iherefore do not engage in strictly electromagnetic Iflh:ra<.. llon'i, but all Icplon� engage In

....e». lnteractlon"r-all po'Os.e!.S weak <:hargc. We won't siudy the rclatioMhip btlwecn charge and weal.. charge. except 10 note that the greater strength or tbC clectrorn3J,'!:nctlC furt.:e relative to the weak fon;e is not because weak c/UJrge 1\ Vo'caler than elee-trk charge, but rather because of the mysleriou" rna';'s difference between the mediating bosons. Sonaro man is an impedi­ ment re!opon 'ilble for born the weal ness and the �hort range of the weak force relatin� to the electromagnetic, The lhcone5 of the "lmog and electroweak. forc� together form what ii\ a ..mg symmetry: ..ix quark, �o\\'n as the Standard Model It exhiblli a ple and U� 1ephlO'l ()cl.:urring in three incrca,ingly ma.'ir.ive tieri, or gerwraliom. (harges of and (-e. 0). The hghte ..t generation each cllmpn,mg net. t is the ); V (C, i) and (J.L . v,,,); and the hea'o'\4 c compn'iC the (u. d ) und (e ('\1. 1\ the (l. hI and ( '1 \'r)· I n many ways the Standard Model i, a very har4 rnonlOU'i thc()f)' that e""plaln\ much of our ob-.ervationi, However, the pil:1Ure 1elc A relallycly new " problem"" 1$ neutrino mass, discu\sed in 1<; sull inC(lmp _ SeCtion I :? 5 But the nddle thaI has been center ..tage for d«:ades I the llpon· 1,JJleQU..\ly broken s)mmctry of the electrO';\'c:U force, The IUwwer may he �'ith the one p.JTIiclc c("t"-"Plcuou�ly miuing In the Standard \'1odc:1. the most sought After to high-energy physics: the Higgs boson It ii difficult to over­ ,ute the Importance of the Higgs bosun, for It i.. believed to be 'Ahat IS respon­ Sible for the mar.ses 01 all partldes' In �hort, �pace IS thought to be permeated by a field. called the Hi", field. through \\hich all particles naturally must pass. This fidd covems the behaVIor of panldes passing through il in much Ibe same way lha• • <wlid goo.'cms the beha..ior of conduction electrons pass.­ Ing through IL. prodUCing an etie<:ti"e mass ISection 10.7). At the relative lOW cnerg.e!; of panide� we ,,--an obs.eT"\ie tOOa)'. an �ymmc:tr) In the HiW field leaJ, to a hard way and an ea�y \\'ay to tra,e\ through \1oving me hard .-:ry produces the characlen tic of ma.o;.s. Phol0n'i mm'e the easy way and are mus �sless. \\ hi Ie the W and Z mm-c the hard way. The particle aUoClatN \lith the field IS the Higgs boson. Predl\.:llon.·; vary, but its rna,s I" C:4pected to be In the ,c:ner.1l \ h.·imty of 2no GeV/c2 Enc:rgle<;. ...ailable ut upgraded and new colliJen In the next few years �hould euher confinn Il'i exi,tence and su� the Standard 'hN1el· -or hasten r.carchc-s In new dl!'eC\lon,.

(+�e. -ie)

•

•

The Gravitational Force

Gencnl relarnlly kif u !hat all parucles willi rna\!. or any other rorm or CDeTg} attract ",nc anochcr v.a ,he gravitational rorce otr.·tously. this broad cafCpzation inclu� at. lhe fund.ltnetual particles. (or C\en rDaiiless parti c-I..� mm'e at c and thm. rune Lmetk rnergy, HCMC\'CT. to dale. no quantum aspects of the pa'ltnu_nal fon:e- have been obSt'...."Cd. A nu.jor difficulty ,,5 its �larj-\e ....runes. 8y �Icgy _ ith me two other iateTXtion..\. \\c assume that ttunp .utrxt one another gr .. .. .utionall)' via. lbe e,changc of. field Q1WIrum. which should be � I bccau.\e the fon..-e IS of infinite nnge. �'oreavcr. Ja\Cn grwviry', anr-.&C'tlVC-onty nature the lhcory of gravitation SUgttSlS tha1 d'us mcdt..lt1nt bcrtun lhould be of spIn :' It tw. e\'Cn mn " ven • namecn,itnn Yet ., Iln'er I;trcn detected. We .,11 return to this quantum�ham,alt} mtr.m I�Dt

force _hen "c dlSC'UQ unified theonn,

1M

0 . ..

'2 A .... _ ' ...' ...... ... .... . ...

1 2 ,4 Particle Production and Detection

As noted earllft'. OW' \jc", of \l.hl(.'h panu:le, arc' truly fundamental ha., We will now !)nen), ..uryc), the progre..\ IIId upcnmenw mtan.. thai ha\'t' brought u\ 10 \\here we ate.

d'lanJed qUire a bit mer lht ye......

A Bri.f Chronology

Pemap.. nOi �urpn\ingly, the tirol \ubalomic particle 10 be discovered was the clccrron, in 1 897. PnxIJ' of a compact. po�ilivcly charged nucleu\ came in 1m. followcd in 1919 by Ihe d"cmery of Ihe proton. Despite mounting Ifttere\1 and rapidly ad\ancmg technology. the neulron wa\ nOI discovered until 19.n. This give\ \orne idea of the difficulty of detecting uncharged particle... En�n rno\t of today\ parlicie detector\ detect uncharged panicles tlnl)· indirectly, by anal)'/ing the behavior of the charged particles with Which tht)" inter.k:t. The fiN \oun:e of high energy for particle creation was natural: COsmic rol)\ reaching Earth from \pJce. Co\mic-roy particle\-mostly proton\--oflen

h.&\(' k,netic ene�ie\ greater than I GeV. �o they

arc

energetic enough to pro­

du(.·e other panicle, ""ho\(' ma.\!> energie� are comparable to the nucleon·s ..... 1 GeV. Be,ide� the ptl\ilron, co\mic ray\ ...cned a� the source for the discovery 0. 14 in the 193(h and 1940, of the muon (m :-:: 0. 1 1 GeV). the pion (m.". J4 GcV), ilnd Ihe kilon (m... = 0.49 GeV). After the kaon. or K-meson. a great =

man) (llher hadron� were idenlified. But co...mic rays "ere soon no longer the Ix,t energ) source. The panicle aCCeler310r had arrived. Although not realiled at the t,me. the discovery of the kaon had brought

10

three the number of quarks �ubJecl 10 study. The fir�t hadrons discovered

"ere merel), differem combJOu(iom of the same three quark�; the up. (he down, and the �trange. Indeed, after a decade or �o of finding recurring pat. terns, sirong suspicions aro�e that the known hadrons were composed of a

�maller number of more fundamental panicJe�. The three-quark theory that came to be accepted is due to Murray Gell-Mann. who was awarded the 1969 Nobel Pri/e for his work of 1964 (and by whom the name quark was chosen). The theory was widely embr3ced. but it was not until (he deep inelastic scat­ tenng e'{penmenls (Section 1 2.3) of Friedman, Kendall. and Taylor in 1 967 that the fir..t direct e'tperimenlai evidence of quarks was obtained. Still. other more massi\e Quarks lay undi<;covered. A� the energy available to us has increa<;ed. so has our ability to produce mas...i\'c . particle�. The appearance of much more powerful accelerators since the 1970s led to regular <;uccesses in the production of many theoretically pre­ dicted panicles. Consider a few examples of the progression: I n 1974. the 3.1

(leV JII/I particle. a cc meson, wal, di...covered. adding a founh quark. Discov­

ery of the 1.8 GeV lauon. the third massive lepton (joining the electron and muon). came in 1975. The bottom quark, the fifth, W3.!. idenufied in 1977 via production of the 9.5 GeV Y me...on. a bound bi>. The weak bosons. W:!.

(80 GeY) and z<> (91 GeY). were found in 1983. And In 1 994. 'he sixth quark.

the 180 GeV top. was positively identified.

Sllm� parude ;}c'"derato.... ,....lE 12.3 l.ocaUon

;;;;q'"(

1)..

SloI.ntocd. CA l'SA

lUH�:.u�·�

Accelerators and Detectors Tl.1da)'\ accelerator-. may be categorized m several ways' ( I ) They may have (\100 col liding

beams. or one beam and a stationary targct. A colhdcr I!'. mher.

cntly more difficult to align, but technology has \unnounted this problem, mg ib advantages decl"ive in many applications (see Exerci..e 2�). nlak

,,-(cordingly. most accelerators nowadays are colliders. (2) The accelerated �article\ are oflen protons or electrons. including the p and e-t . but heavy

Illn,

are

also used. (3) The particles may be linearly accelerated. or they

may follow a circular path. In all cases. electric fields speed up the particle ... ",hile turning i\ done by magnetic fields. Data on a few of the many colliding­ beam faCilities are given In Table 12.3. In a synchrotron. the magnetic field

i\ constantly adjusted so that while being "kicked" by electric fields the

panic1e'i S1ill move in a circle of constant radius inside a narrow tube. Figures 1 2.4 through 1 2.7 show features of several of the larger accelerators.

Of course. it does no good to produce a particle if we can't verify it. Reli­ able detector-. are crucial. Of the many types, a few of the more Important are

the scintillation counter. the muhiwire proportional counter. the Cerenkov detector. the sihcon detector. and the calorimeter.

The scintillation counter houses one of many substances known to pro­ duce a flash of light when struck by a charged particle. a common e1tample Figure

12.4 The accelerator chain IH CERN. coveri ng about 70 km2 in Switzerland

and France. The largest black oval i\ the Large Hadron Collider (LHe). Linear accel­ erators (\inacs) are the !;ources for various particles.

, '.

UNAC "'''

.

�7

" ...

p·P. \cl)' high enel1!)'

SPS

p.bar - �""':::::: :: ::: ::::==:::;:::::�b=::::::;:::r· .... . -P
D (("I1Il A

P

LHe -.

\ f·.

PS

_ __ �

. � ......

...

'.

BOOSTER

n.TOh·

.

... . . .

. .

. .. ..... ...•...... . •�. . . ....

>�uckarph)\ic\

P. low energy

12.6 At � Slanford LH�ar AccderaIOr Center(SLAC), eleclrom and positrons Irom II .H,m·long hnear 3Ixeicrator (C.listing Injector) are u�l!d for variou� e,\periments. "fhto BJ.BJt del�tor \tuche� CP \iol:lI!on in the decay of B me'>On� (Seclion 12.6). F!gln

Electron

PEP·Ll nng'

,,"""- 1.<.. energy nng (new)

EI«tTOn�

Figure 12.7 Thc= 1

1�n\llc= dlamc=tc=rTc=\atrQn \luJIt"

({'\\h­

'Ion.. of hlgh-c=nc=rg� protO"� and anllprotl'"'

Targel hilll

Anhl'''ll�'n ,oun.:t"

CDP

BO(hler Lm��

ClId,,,roft·\\;tl!(\n

�.

NeUlrl

Anll,PrOIOIl I'rolon dlfC"IOIl d\�IOIl

Nal cry'HaL The flash is amplified in a photomultipher being 3. thallium-doped Efficiency is high and rc"ponl;e time short A nlUlliwire 3.2). tion tube (Sec a grid of conducto� :u altematlllg electric contain., counter al roportion . . region. When a charged particle enters the region. II wlthm a gllio-filled

�enU3.1

around ii, which then causes a brief current between nearby con­ iOOllCS the ga, ning which conductor<; are conducting at a given time e�tab­ Detcmli ductors. location. and computer analy�is of the .,ignals provide... an h,hes the particle's (,cel1ent picture of the particle's motion. "Proportional" refer.. to the proportion between the energy deposited by the particle and the electnc respon\C of the

')e�"

gas discharge detectors, like the Geiger counter, give only a detector. Simpler or "no" respon...e and do not track a moving panicle. In a Cerenkov

detector. a charged particle enters a medium at a speed greater than light move...

in the medium. (Of course. it never travels faster than light moves through a 'lac­ utJm.) The particle emils electromagnetic radiation called Cerenkov radiation. In much the same way that an object moving through air at supersonic ...peed pro-­ duces a shock cone. Cerenkov radIation fonns a d;�linctive cone whose angle dclelT1lines the particle speed. Silicon detectors are grids of p-n junctions (Sec­

uon

10.8). A charged particle excites electron-hole pairs in proportion to the

energy it deposits in the region through which it passes. and the electrical signals II different points in the grid map out the particle'" path. When a panicle enters

a calorimeter.

it adds energy. which i" then shed in numerous ways--creallng

�howers of other particle�. for instance. Analy�is of these secondary occurrences detelT1lines the incoming panicle's energy.

1 2.5 Decay Modes and Conservation Rules in the Standard Model

Accelerators produce many particles for u s to study. Almost all the ones of InteTt!!t are unstable. Indeed. it might be argued that there are few stable parti­ cles. The J.L and T lepton ... are unstable, and all hadrons but the proton are known to decay. Some partIcles decay so quickly that we can study them only

.. ..oI tI "_,_'" __ '1,_=_

la4Ii«lIy dwouah 1M swtJdtt. lhII rrwll from thtlr �-J). Thu..._ nponIDI lo IIIIdcnIIad dr:ay pm.� D«J,

art

.... ......:: or ___ drpmLhna on .. hcthcr the IOlera..:tlon � of ,....... phoIoGt. ot -.eaL bosom.. �recll'cl� ... From mee ...

.. c

II 1\ 'to')

d!l.' lfin! as nrong. C''ce

�'M thar momentum anJ Clll'tg)

•

In\'(lhl"\ lhc

I.:oo-.er\ alion ""00-

linin &hr � 0UICunt of all pro..�. tntI'!), cunloCr\;l11t1n. f\lr In�.

� dca). tn \\hh:h lmclil.: �ICr}.''Y IRnea-.c\. mu" tIC' �icd h) • MiQ'iC' ID m.tiS.. ll'tL, aillfl(' ought ,ugge<,,( thai uli ma' we

wilt. &b da. I

pwucb \!add be lubk

to

\h:a� III tnt' hght,,·\t \"1ble panicle

In paI1.Ki(

ph) \k' hi-nu. � coo-�alion NI� ('till)\.' IOhl pia).

Ih ..,uru:aU). II hla., often Clune hi the' JtI�nll(ln ul" ... dcnti,'" that one 4U'Ull1t)

or

onatbcT

te(ntS 10

� l;on�"'cJ_ Aflcr 'lanJlng the h!"'t of labonou\

rutin). xne "r IhtSC art \lllkl.dl) pnxlilllueJ con,er\-ell quantltie\ ilnJ theu �on�f\ ..lIon �('('J'lcJ a, II

\t OOC

'I

\\ ..� once

kUk1 of

ON ""

In

1.1" (,\1"

nature. M I\lule, han: �en made

lbou,hl III tlot c\ln-.eneJ. Nt \\.f' Rim loo,", thai it

ent"rg) .nd Ih.i� II mJ� (hi1R�e "hlle total energy I"

.... t

" ..

mertl}

"II con"

d.U4.'W' \ariOU$ dCI.:a),. "e WIll ref....r to the rule, ,ummariled

TliIhlc 12.-4 8('.&1 In mind thut JU\t

Illr " rna', ("on ..enallOn." ,orne of

it'

a pallem that ha� oot \"ff been \'10-

Ihrw rule:!. lit N'I("d UO (wlhlllg: m(ICc than

1.ltd. They UOlkrJ\1 �'\lnIIllUJI feu" c"mcol. In OlaO) C3'1:".

3

con\enatlOn

rule has t'l«n fuund hi he:- ()�"ycd onl) hlr .1 particular I)pe of decay, For lO\tOln...-(', 'Iran�('nc�'i I' (1\0'>1."['\(..1 111 ,{rong anti elcctromagnetlc Inleractlon, kit no. In

....

caL line' (another c\ample 01 the '1,)'n1nlclr)

the eIC(;lro'Wcal

III

mhT.,('{llm),

fh( �tand.nJ nlnwnalllin IJ'"

01' da"ical phy,ic, appear 10 be \ule,

MI11lk"nlum. cn(r�). •\fl�ular momentum, ilnd chilrge are alway .. con�en'cd"

And all tnh:ra..:ll\)fl' "l.'l.'m to Cllll,cnc colm, Another quanlllY apparently con­

"i."'nC'd In all

CJ'l"

1\ blryon number 9UI),01l number

B

I"

+ I for baryons.

I t(lf antlhar)llO\_ Imd 0 1M Ilonbar),on" linlll falrl) f(\·(nll�. lfpton numMf Wih aho belie\'ed 10 be universally \'t.ln-.eneJ_ It 1\ l"\lnllnlln 10 define one number for each generation. The dl',:tf\ln icpllln numlxr Lt ,' � I for the electron and the eleclron neutrino, I hlr their Ilnllp.irtlde'. and 0 for all olher particle�" The muon leplon

numt'<'r L

flo

'AI". 1 1 • �1f1lt {(Iflcn;allllll� N ::� k:.

.,-

_ _

....! ..

-

-

and tau Itpwn number Lr are defined analogou,ly. I n 1998.

7'

_ _ _ _

\� .....,. � 'hF .:-.

.."..

"....tIc-r

1'Iriwabtn-

"....... " . .....

tOt

11.,. I.,., 1.,1

,..

,�

,,,

'o.

Yo

,,,

Y..

-: :-

_ _ _

bptOil

"'7'

_ _ _

_ _ _ _

Stranatn ffi

Paril)

::C

-: ::_

_ _ _ _

Charge

11 ""

IP)

Co!\.iugalion

RenrsaJ

IC)

tTl

)'t,

)'t�

\6

Y6

Yt,

Yc,

Yc,

Ye\

Yo.

't'e,

No

No

No

No

ot'lt.\UWll that l)tl� I) pc l)t n(,O!fmll CIlO ,,:h.lR�(, mhl "'lhl,h('1 \1"l',II,t!tun (mJ lh'I.'u"cti further In Se('lh11l t :!.bt Wh('1\ Ihl' um n('Ulr I�k-J :tUI". Ofl(, lepton num�r 111U'" dl:cfeJ'c ;\, anothl'r ml.·l"1:"il'(" . RC'I,.·C'llt �n..' I.f " 'Ine hight) ultu,ual lll,.'I,.'u)" \ ..ec Pn1grt''' >lud Arl'hl,.·,llh1n,l ,,\,,1 Q1Q(" t thaI C(lll'CC\illilln III lertolt I1\lmlx-r mOl) 11\11 l'oe a UI\I\\'"",I rule \#Ihd�". "Inhl ..t "II pml'e,"c'" ('on"'I:(\I,.' the Ihr\.'(" Il'rh1n "'tm�r' .adcr
rttdtno.'(,

.... J'

F.ynman Diagrams

fht t\NC element of (l FcynnuUl dingmm j, a \"erte, rcpre...ellting. the jnte-rill;· U � ...hl)\\'" Ihl: tlllll llf a fundamental fermion with a mediating OO'on, Figure I : I«(tll'e' hlr emi ..,;on of 1\ bo..on by {t fermion. Thc hllril"�'nlal 'l\.i.. i ... lim�.

-.!1:lOl·lOf! from Icft to righI, and Ihe \crlical axi, ..tamb. In ror al\ ...p.llial dlnlC.'fllolOn... (Feynlllan diagram:-- do not really repn:..ent ...pallal tnoti�ln.) krllnoo' and mediat i ng bo...ons arc repre..clltcd by ..tmtght ltnd Wl\\ Y Iinc.... l\.'rl("\"Il\ely. Only particle .. pm'''c).sing color can engage in the ...Irong force, "0 III-" llOl) ic:rminn... Involved in the "l(ong "ertex. are quar�.,. Note that in thi ... \((tC\. the quar� may c hange color. To conserve color III Ihe t nlCmCllon . the ,lu\)O carrie» a color-nnticolor charge. The electromagnetic vcrtex ...how... Ihal JIl fundamental particle.. can inter:'lct electfomagneticully. ex.cept neutrino.., v.hlt:h arc uncharged . Their strict ly weak inleraction llla\..c� them very ditlicull tOdelec!. The wea\... verIice.. reneet the fact that all fund:'Ullent(li fermion.. may Inlcrad yiil lhe ZO and W"! bosons. In the W= case . the charge of tht: fermion \\ changed. and a trun !.\llulal ion of particles lakes p lace. Thu.., W+ em i...,ion , I (hange" a +):-charge quar\... (u, c, or I) 10 a -]-charge quar\.. (d. s. or b), or a n�utnno 10 the other member of its lepton pair. Emission of a W · hu, the �)ppO�lle effect.

To obtain the ve rt ices for absorption. we would draw the boson as coming vertex rather than leaving it and switch its charge or color charge. Into the

Figure 12.B

111Ic: rule:, I'CrtjllllHli l

Emission venice)' for strong, electromagnetic. and wenk processe.... Time advunces horilOntnlly 10 the right . u bo::.on (w:IVY) and leaves a fermion.

.I,. fermion (straight) elllil�

� Same quark Quark � Gluon

Strong

u.c.ort

quar\;. �quark

"

QUMk

Samcquin(

or � lepton

(e,cept v)

r !epl(ln (cxct!ptll)

Y

Eledromagnelic

-

d.�.orb

W Quark

p�

or It!

Same qUUT\" or

Zo

pton

Weak u,C,ort

d,�,orb

quar�uark

W

C �I'� I'

W

,-

"

".

.. 0.... .

,.

P "

.. ....... ... .... . _

en-allon

\ II rmtt.\lUO llf a ptNti,·C' L·,ln'enes charge, so would the absorplion of a nega.

t"'iC' I RrplaL"mg all parttde... � 1m their antiparticles would give the antiparticle

\("I1I\·C'\

With jU\1 a few other changl!', the vertices can represenl other proces<;e\. h�urC' 11.Q ,hmo, .. 'orne c"(ample ... If the final fermion of an emission VCrtc:c, 1'1 �rlJced "'th 1'\ anlipanidc and the diagram i:o. rotated 90° counterclOck_ "i�. iI lhJgrnm flIt the de\truction of a panicle and an antiparticle results. Re\C'r.aI along the IImC' a", repre�enb Ihe creation of such a pair.

We rerre..ent "hole" pmce'�C'� by putting vertices logether. Figure 12.10 is

the prl.lM;'-pe F(")"nman dia�ram of an interaction between fundamental fermi.

llO...,

One elel·,mn emit, a photon. and the other absorbs it. NOie the similarity

h.l lhC' \nowb.Jll·medialed lorce of Figure 12, I

Figure 11, I I ,how, diagram\ repre\enting various Iypes of spontaneous

del.:ay. Diagram (a) 'ho\\ , the

f3

decay of a free neulron-a ""eM: decay

be\:au,e it tn\"{)h e, the c"(change of a \\oeM: boson. Diagram (b) depicts the wcak It dn:a)'- It, lifetime of 2.6 X 10-8 s is typical for a weak decay. (The

neulron\ 1 .5 mill. lifetime IS unusually long.) Diagrams (c) and (d) show weak

and IL t decay,_ An example of an electromagnetic decay is the decay of the �o in diagrum Ie). Invol'lll1g Ihe stronger part of the electroweak force,

!

l'Ieclromagnetic decays occur in a much shorter time than weak ones, typj. l'aJly 10 t1 s. A gluon-mediated �trong decay is the K*o decay, shown in dia_

gram (0- Involving (I yet 'tronger force, srrong decay lifetimes are typically

onl) IO-- J-' �, Once cre�lIed, a pnrticlc able 10 decay strongly does not survive long. Indeed. ,uch a ,hon,lived entity is often called a resonance rather thun u

particle. It:-. conslilUenh "�tay in tune" for scarcely longer Ihan it takes them to po" one anOlher 31 high speed. Diagrnm� (c), (e). and (f) in Figure 1 2. 1 1 bear OUI Ihe conservation rule

,..... 1'. 10 H«h'On� mlC'l"&C1 by

Q("han,ing II rOOlon.

,� ,

for 'trangeness. While conserved in electromagnetk and strong decays, (e) and (I"), slrangeness is not conserved in weak decay (c). The W emission changes the strange quark to one of another kind. The values given in the ,haded rectangles ilJu,trate consen'ation of other propenies. The final mass is al"a),s less than the initial, as it must be for a spontaneous release of kinetic energy. Baryon and lepton numbers are conserved 10 all of these decays. By

me angular momentum addition rules (Section 8.7), the final angular momenta are able 10

add to the initial. Note that the spins don't add up in the K*O decay

of diagram (f). in wh.ich a spin-I particle becomes two spin less particles. But

the decay is allowed because the kaon and pion depan in a state of unil orbital

angular momentum. Total angular momentum is still conserved.

I!.� Dec..,. Modft _C� ... .. - SS._IoP'P' _

...

•

.

&�.

"

,

,

9)1.

p

•

• •

, w

,

W'

, 1191, i .� I

,

0. 0.0

•

-1,0,0 O.SI.

,

+

'411, 0

o. ,

� ..

(a) n �p , �.

j

0.0. '

� :> Ib)

,

,

•

d

"

d

d

<"

.-. \

0.

, j

140. 0

106,

0,0,0

0, - I . 0

940,

"IT"

-.

+

----

.'

W·

, j

0, 0. 1

,.,

�'.

(d) �' -+ �. t 1'( t

(c)�" � n + 'IT·

�o

� =====

+ 1. 0

I'",

".

, 119�, :i j),O, 1

10

+ ' , 0, 0

�.

,

o. i

CI.

•

, 'i

lie


w

d "

"

,

,

,

1 1 16. 0.0.

, 1

I

0.

:

0.-'.0

0.51.

, j

- 1 , 0, 0

, O. ,

+ 1 .0,0

"

896, I

140.0

0.0.0

0,0,0

O. I

494, 0

0.0.0

0,0. 1 (f)K'O � K' + 'IT

EXAMPLE 1 2 . 1

-

By considering the conservation of charge, energy (i.e mass decrea'ie), angular momenlum, strangeness. and baryon and lepton numbers. mdicate whether each of the proposed decays is po!!�ible. If it is not possible, indicate which rule!! are vio­ .•

lated. If it is possible, indicate whether it is a strong, electromagnetic. or weak decay, a/ld sketch a Feynman diagram. Use data from Tables 12.1 and 12.2.

..

... a..-r " ' .... W

�

II "

\

. ,_ _ . � ... .. I'II'bdn -' .... . au.

f-1'J1IIM' � for

C1wrr .' �.:IftWn-N ' " .In *' dca)s.

de.=.Jy

'.I, Tht- ","00 �� .

1 '8 )( I()' ,\k\" � final " I (If) \k\". .:uJd no Nryorb are

lMl j � l Tht ,nlDa] kpIoo IIumhcn � II •. L,.. l.�) to. O. + J I. and lhtm mU'1 be a W. for iI Z couldn 't •

j

J.

1I:r:nc" rhe r

,

"

• �

I

v --

,

.

1-.

•

J

-

. �- - .

�

\.

•

<::::: '•

(bl Ihls ., {IlfhiJden (til two al."l"OWIh. '-.r.I. allhllugh Ihe lin;)/ ma.... 01 2 x 49.1 \k\ h len lb3n Ihe Ulililll L ' I � \1c\" and no leplum are m\ohed, II hegm.. "'llb a b.Ir)'UIl num!Jl.or (II '" ,md ('nd, y,Ilh nil baf)on.. Se..:ond. angular m.JffiCOIum "lInml( tte .;on-c"o.'d Tho.' milial ..pill I'

•

de:'

are

can',

�plllll$.�, Orhl/al iUl�ul.:lr ffi(lmCnlum.

mae up tbr d.lkl\'n.;(',

�. while Ihe prodUCI pani,

y, hlch

l'om!', i n IOIegl'",1I \teps.

fd SU)L'e Iht � anJ n il\' b.llh baryun� and the loa,'n i.. a me..on, Ih... deca) would ..nnse....(' t'w)(ld own""'r N,I Ic:pt(ln, are Imnhed ,\ finill orbllal angular

•

1

�

rnomcatum mi�hl ilHow ItIt \Ptn partlde III de..:.:l) 10 a ,plO' and a "ptnlc" plOl\. Ik*l"\'ft', ��-aust the final ma" 01 �O MeV + ...9... I\.leV i\ greater than tbt imllJ.l I2_12 �k\, the � .;annlll dn:a) Ih., wa)

• •

I ,JI

'The ioitel;mJ fllud rna....." ! ' an: 1 1 16 MeV and 9]8 MeV + 140 MeV, The \0 and Inlipnlll'n m: Pll(h Ill' /:Idf)'110 number - J, Ilnd the pion ha, bar}'on num�r � . I 0.10 bar)l.lO nurn"., ,, l'lln-.enaj, "The \0 and antiproton are "PIO-l' \\'hile lI\(

pion •• plnl"" Thl\ dIXa) 1\ allll\\,cd and doe.. c",:..:ur_ Becau,e the \0 con­

r.un,;} strdll�e quad.. , \\hlo.:h h a/:l-.enl in the produch, ,(rangenc" " not con· 1 2 . 1 2 ,hm.... the FeynOlan

,('ned, \I) thi, mU\1 t'l" a y,cal. drt:a), Figure

Lhagram, A \\ /:Io,l"m I.'han,!;(', the:' ,Irange qUilrl. 10 a u_

,,,,lie- that Ihe-re i\ n guiding principle in particle phy\ic\ thai says that if a \.k,:,t� nit! I),;\.'ur-If II Violate,

110 con�ervation rule\-Ihen il ,mlSt OCcur. For

I1Mny h"dwn, and lepton.., there are multiple avenue\ of decay that obey all

the- f't.'quired wn..enation rub and therefore do occur. For example. the i ma, de-con ' Hi T - e + \' + ,. nnd the \O via fo _ -p + e + + I' , ,

t

.

t

r

The r..'on,eJ'\ntlOn rule... we hnvc discu\\ed govern not only spontaneou!.

del'.)}' of fundame-ntaJ partjcle� and hadron\ but abo other interactions Ixtwecn them,

EXAMPLE 1 2 . 2 \lJJl) \If I\lo.ia) .... c(llhdo.'f'!i, \nM_\h prt'lIOn, together_ Con,ider the two proposed rtac­

IlflJh �hll'o\n be/(lw. (a) 0..-.=, either \iolate an)' of the con\er\'ation rules in Table

12.4 (e,dudm8 P. C. and T)'.' It \0, \\.tuch rule�" (b) If not. whal minimum kinetic

l"11\''i� mu,1 the Iy,o m i llal particle" ha\e for the reaction 10 OLeur" I: p S

\a)

+ p - p + P + 7f+ + 7f�

l,;",-N

R(';t,,:livn

2: p

+ p-p

+

P

+ 71"+ + 7r'

I \.'On-.el"'le, ch�e becilU\t the final proton and uotipmton cancel.

Enc.-rg)' (an be l'On-.em!d jf the initial l.inctic energy i" high enough. Pions are

'!opink�', ,\1 Ihe t....�l 'rm·1 rartKle, lxh.1'( al\\l all(f the feKU\'n "an �"" nscl"'(

!\OO krtCln... \'r 'Irang( 4u;u-l, llJ'( mh,ht"'\t .""'(\(f. lhe-

lIlltubr O).lRl('ntum

I + 0 + O. '>1.111 t.. (\t.lt ......"'. inl!I"! b.lI)·\,n number '\ I + 1, Ilnd mt" linal I' I sn'
" mIl",. aq;ument..

'........ 12.'1 If aU pmnh l1G an arrow

.1'( re-dta.. n ill pum!", on lht 'lJIPO"lle i.tt \If the \Iflilln. the nltauon dt�li'1fI

i.. un.:han,w

Pant) ,n\en10n

nll,mentum

,b) lbe tnl'rg� 01 the final \Iatc

.t ttit, that lllihe 10111.1.1 b) ..t If'll.. tn I't"a..:tllln 2 t,.. Ih< ma..\ tllerg)' 01 the two rlOn... Thcrchll'(. the Inillal J'f\.'ll.n, mu,1 ha\C' lUlt"tll' energy 01 at lea"

dot"

not dl.n,t the- dll-n;lIun " f "'Iular

,

2 x 140 MeV . 280 �1(\,

1 2 6 Parity, Charge Conjugation,

and Time Reversal

Figure 12.14 (.1) A nudl!u", In..m� I

at the remaining com.erv.llion rulc� of Table 1 2.4. parity Let U' lake a 100"- . . . und tnne conJugatIOn, revers., ·. aI . Thcy toueh upon I'unu.Ullcnurge . h c in\l�r-ion

dc\:;l}' (\:II Th... poxc" " e,",·«1 wllh pari!)' i"e\'cI"ocd.

umt lIt 'pm to f3"

space and lime. while ra;\ing difflcuh question". Parily inversion change.. the "ign of all walial coordinates.: (.\. y. ;:) -+ - I. -_". . ;:). Strong and electromagnetic mteractions. appear identical from it

1 ,dea.' of •• ...

Anllular nl(ll1\enta arc unl:hangcd. hut

,...h'll,: II;e\ arc �lN"hltC

I vertcd view. but the weak mlcmction does not. The Mmple't c,amplc pJOly_m 1,/3 decay. To see thi.. we fir<;t note that If a point particle were orbiting in a about the ;:-axis and we change its coordmate.. according cI(ld.."" '''' direction It would still orbit clockwi..c. a... depu:tcd in to (l, \'. =.) -+ the direction of spin (intrin,ic angular token. arne .. the By Fi5!.urt 12.13. m�menlum) is unaffected by a parity inversion. Now consider the (3+ decay In

tt

_

(-x. -yo -z).

(am' away from the nucleu

S"

p

s·t r"'' "o

Fie.ure 12. 14(a). The neutrino and positron move in oppo.. ite direcllon� and ..

I unit of angular momentum (theIr "'pin!) arc

dligned). What happen.. if we change parity? The spins of the nucleu.. , neu­ mno. and positron are unchanged. and the neutrino and positron move oppo­

••

\ite their previou., directions, as shown in Figure 12. 14(b). If nature were

lal

aly.ays the same under a parity inversion. this process should occur with equal probabi lity-but it doe..n·!. We say that the weak interactIon is not Invariant under a parity inversion. ..ion replaces all spatial coordinates with their opposite, Just as parity inver

10•

s· t

~

s,·t

tt··

s,

..hortly. we expect the process 10 Figure 12. 1 5(a) to be just as unlikely as the

one in Figure 1 2 . 1 4(b). We ..ay that the weak interaction is not invariant under charge conjugal ion. On the other hand, the combined operation of parity inver­ �ion and charge conjugation produces the decay in Figure 1 2 . 1 S(b), which we Ifoufd expect to !lee. With ,-;ome rare exceptions, to be discussed later IR the

s

@

.

t.eClion, the weak iOleraction "ieems to be invariant under the product of the two lran�formations, referred to as a CP transformation (C for charge conjuga­

tt. I.,

tion. P for parity).

--

-

-- -- -

� --' , �

\bl

,

..e::r<;t:d. Ity re'

antihydrogen, become readily available.) However. for reasons discu..sed

--

"

Figure 12.14 under charge conJugation, all particle.. chnnged to antIpartIcle... (b) The:: ..arne proce.... \'iewed with [lar­

Figure 12. 1 5(a) an antiproton emits an electron. Actually doing the laller tlperime nt would requIre antiatoms. (Only recently has even the simple5t one.

.-

l

Figure 12.15 (a) The (3t decay of

charge conjugation replaces all particles with their antiparticles. For exam­ ple. the diagrams of Figure 1 2 . 1 4 would become those of Figure 1 2 . 1 5 . In the �� decay of Figure 1 2 . 14(a), a proton in the nucleus emits a positron. In

j

p .

--�-.

- _.

-

� -=:"""--'-

1 ..

s·t

~

l •

s, t

{bl

p,

__ t , � ", ,,,, _ _ ' ... ... 0..- 1' ''

common "'.1) "I i."\pl.unm, the L,>mnlClr) 01 the \\e.o.. (on:e

.appeal 10 d'k' Iwficit, ,., Ilk' fk"Ulnno Hdll,:II) rrkr\ 10 the onenratlon ot I . a tklinlle !kilo:. piIl'tJC'k., "flO tdarwC' 10 ,1\ m"�nlum [lcdmn.. do nOI hne 'iOO ,11 lCln l dlre..: +: Ilkm .. OltHe m/.. \\.- uh ..pm up. II) \urpo� an i."lmrun "

1\ 10

pto .lnd tlJI,lmcnlllm ..hgo,.'d In iI fr.lme mm 109 III 1000 mI.. In tht ';'z dir«1lon rd,dne hI the lil'l. the ek,:!ron I" 01(1\ mg al 'iOO m/.. III Ihe flt'.I((JJH r dlr(\:lItlfl .ru " 'rIO up, �l thoIl I" ..pm and 1I111menium are anl.allgned. The �I

Ipm and momtnlUm v( tht.' \.lIl1(' t"Ie\:lron �'an � cllher aligned or anriallgned. dependmg lIn tht- 'rJ� 01 n:il'I\'OCC A mtt.. ,le .... P'lnu.:le. on (he Other hand. hllI\L', at c 10 JIl) IrJffit'. 8el"'IU'c II �'anrll)1 Ix' mcn..J..cn. If II .. 'Pin and rnllflienlUll1 Jrt' aJlgnt.'d 10 one Inll1lt.'. Ihe� mu", be ah,gned 10 all (rames. In the \tant!JlJ Model. neulnno' .1n.' mil"Ie". '0 tht.'y \\.-ould h.ne I.Iclimte hellellle,

I"h,' nt'urnno Iwuld 011"01)' h,ne I" 'pm opp(1\Ile II' momentum-referred 10 Idf.h;lmkJ "hu.:h e\pIJIO.. \\h) tht.' dec'l) in figurt' 1 2 . J .f(b) would not

.h

C\4:,'ur Tht' rl('utnnu \\ould halt" an allllpilnll'le the IInllOelllrino--",hlCh IHlulrJ tx- n!!hl-h.mrJed. I\-Ith 11\ 'pm pardllel to 11\ momentulll, In Frgull' 12, l'i, IIl l'thKh panldt', hll\C Oe-cn c;hangcd 10 anllp;tnicle\. the panicle pro­

rJUl't'd

'Min tht" dCl"twlI l\tlulrJ hale 10 Pc the antlOeutrino. Thus. Figurt

1 2 l'ita) \\llulJ nOI bt- oh.."r.cd, am.! Figure 12, 15(h) \l.ould be. Ont" pwNt'm '" IIh the c\planallon

Jfld"

Ihal ncutnno.. apparent I) do ha\�

r\

The ('Ildence -':\1,"('.. Imm n(lulrino o')Cillnlions. Ihe ah.llly of one 1..lOd

tll neutrino lu (·hang.... 1II1{) anlilher Th" phenomenon was li"l documented at

tnc: Su�r KanullJ..ande facility. loe.lIed

Japan and opcrntcd Joinlly

III

l'nlloo SIJle' A, noled earher. the nculnllo mlernCI.. only

"'ii

I't

Ilh lhe

the \l.eaJ.. force.

v.hll'h male, II t''(lrelllel) dltilcult 10 catch ('cc E�ercl ..e -'9). Located I lm

unJt"!J:round III ,hleld II from CO,IOIC ray'. Ihe Supcr KamioJ..Jnde I� es'itn11<111) oj huge \\aler lanJ... J.. ,ho\\11 111 Figure 1 2 . 1 6. Tholl",:lIld.. of CerenJ..o,

Jt"te(.'l{lr.. eJ('h emplo) jng J pholomultipller lUbe. walch (or Ihe light gener.

i1lerJ b) ch.lq!eJ panicle, 'c.ltlt"red by

rnre

neutrino Inh:raclion.., Re\carch al

antllhrr mUllen,e fJr.:ilit). the Sudbury Neutrino Ob�enalory III Canada. has

n'<.'ellll� confirmed mat a long IrJ..�ome ,hon!all In the ob..er\,cd neutrino flu..;

!mm the ..un rd
Llnfl)rfuo>lIc1y. neulrlno 0'C;II3110n, lell u.. onI) Ihe diffl'r(!IIC(!S in neutrino

1Il••\\e,.

nOi 1\ h.1t tho..,e ma..,�e� actually arc. Another factor related 10 neutrinO

ma.,.. h.1' ,great bearing on the c\;planation of parity \'iolation. Con�iderable

doubl ha, ari'en lalely on Ihe queqjon of \Ihether there really is a difference bcl\\een a neUlrino and an anltneulrinO, Ob\ lou..I)'. a (ull under\landing of the

\l.t.'ilJ.. mteractlon, \\.-ill ha\e 10 awail lhe an" ...er, 10 thc..e quc'lions. The neu.

Inno hil.. become one of the mo�1 acll\'e research IOpIC.. In phy')ic...

A� nOied aNne. the \l.eal. tnteraCllon i .. not always mvariant even under

Ihe ('ombmed CP Irnmfom1Jlion The fir.,1

\

IOlalion

wa.. dlsco\'ered in 19&1

b, Jame.. Cronm and Val Fitch (Nobel Pnze. 1980). who found that in 0.3% of I �'tance�. the Kr me..on deca).. \l.eakly m a way Ihal i� nOI mvarian! under

CPo Decade.. of in!en..e ,tudy did little to .. hed light on thi.. enigmatic phenom­

enon, \\'im higher-energy acceleralor\ producing heavier particles. anemion 'hifted to deca).. of 8° (b<1) and 8 0 me�ons. which ..hould behave much the ,arne

a., the K me.,on., but \\illt fewer unrelated decay� 10 ob'ICure Ihe resulls,

FinaJl) in 2001. conc/u')ive evidence of CP \'iolation in these decays was

found. Much ha� yet to be learned, and it is of great importance. for

CP viola­

tion is thought to hold the key to another asymmetry: the apparent imbalance between mailer and antimatter in the universe. Finally, time reversal replaces

1

with

-to

I n effect. to view the time­

rt\'ersed result of a given process would be to view a motion picture of the

process run backward. Almost without exception. microscopic processes are

time-reversal invariant. They obey the same laws of physics whether time

moves forward or backward. A general result of quantum field theory is the CPT theorem, which ..,tate� that under the combined operations of charge con­ Jugation (C), parity tnver�lon (P). and time reversal (T), all interactions are

invariant. Absolutely no e.'(ceptions to this symmetry of nature have ever been found. It follows that i f proce�ses are invariant under the combined operation CP, they must be invariant under T. On the other hand. if a process is not

12.1 Unified �

Another prediction of GUTs is the existence of magnetic monopoles, which should be very massive. Efforts to find these have also failed. On the bright side, GUTs, unlike the Standard Model. predict nonzero neutrino mass, which is now established. The quest to unify the strong and electroweak forces continues. As the yet mysterious asymmetry that bteaks the electroweak into eleCtromagnetic and weak parts is expected to disappear at particle energies abOve about 1 TeV. it is speculated that the strong and electroweak forces might converge at around 1 012 TeY. Note that this is a fantastically high energy-a megajoule (a car at 100 mph) per elementary particle! The prospect of unifying gravity with the other forces was gloomy for a long time. Gravity is a very tough problem. To give some measure of the diffi­ culty. Einstein himself, driven by tantalizing similarities, spent much of his later years attempting to unify gravitation with electromagnetism-without succesS. However, although quantum gravity still lacks experimental evidence such as the graviton. recent theoretical advances offer reasonable hope for the future. A promising direction for the unification of all forces is superstring theory. in which the fundamental particles of the universe are one-dimensional )3 "strings" of el<.traordinarily small size. on the order of 10- m, the so-called Planck length. What makes the theory "super" is that it incorporate!> super­ symmetry, in which each fundamental fermion would have a bosonic super­ Symmetric partner, and vice versa. For the electron. there would be the (boson) �\ectron: for each quark, a (boson) squark.� for the photon, the (fermion) photino: for the graviton, the (fennion) gravitino: and so forth. There would also be multiple Higgs particles. Space would consist of 10 dimensions rather than just 4. with the extra ones " curled up" and largely hidden from view. Alas. no supersymmetric partners have been detected. and there is as yet no experimental verification of superstring theory. A theory that goes beyond a GUT, by also incorporating gravity, is often referred to as a theory of every­ thing (TOE). At what particle energies might gravitation unite with the strong· e1ectroweak force? Who knows!

Back to

Cosmology

Becau<;e gravity is the only fundamental interaction that is both long range and of one ..ign (auractive only), it is by far the most important force in the motion of the things we see in the heavens today. But things would have been quite different in the early moments following a Big Bang. At such an unimaginably high energy den"ity and temperature. one superunified force may have gov­ emed all. Only later. with decreasing density and lower average particle energy, could (bymmetrie!'l CTeep into the picture and the one force manifest iL..elf as many. Before we discuss conditions near the Big Bang. which are na;essarily the most \peculathre. let us take a look. at more recent eventS. The Universe Expands A wate�hed e\-'ent wa.... th� decoupling of light from maner. for it has left us a very reveahng sign. When panicle energies were higher than typical atomic binding energies. maller would have consisted not of neutral atoms. but of a

'-"'eo.-., 149

- �"

.... _.. J ...... . ....... plavnI of electrons and positive ions. These charged particles would havf iDknCCed strongl), with the elecrromagnelic energy penneating the uni\'f�

and an equilibrium would hive e�i�led. However, when (he temperalure Of�

dropped 10 about 3000 K. al which kaT approaches alomic binding matter formed neutral aloms thai the infrequent remaimng Interactions were no longer able 10 keep pholons in equilibrium with the mat. Wliw::ne

enetJleS. so much

ter. 'I is said th.af the photons wert "frozen oul." They embarked upon their random, independent journeys, contributing 10 the universe a blackbody spec­

K

uum characleristic of the 3000 temperature at which they ceased to interact. Such radiation seen today was produced in equilibrium with the plasma very far away. very long ago. But the uniyerse was expanding even then. and thiS

distant plasma would ha\'e been receding from Earth a t a sign ific ant fraction

of c. Accordingly, the radiation received now is redshifled, so that it Corre_

sponds to a blackbody spectrum of much lower temperature. Confinnation of this oocc:nario came in 1964. when A A Penzias and R. W. Wilson (Nobel

Prize. 1978) discovered what we now know as the cosmic microwave back_

around (eMB). Its 2.7 K temperature is famous. and the agreement with the theory of a universe expanding from a Big Bang is excellent.

are many unknowns in the expansion of the universe. We can 10 grasp some of the factors with a simple model: Consider a projectile of mass m launched straight up"-ard from the surface of a planet of mass M. Still, there

begin

Il\ energy could be wrinen as

whert r is it!; distance from the planet's center. If £ is positive. the projectile

escapes, for it has kinetic energy remaming when r is infinite-when the poten­ tial

energy is zero. If E is

negative. then

r

must still be finite when

the kinetic energy drops to zero. The projectile would return. Zero energy is the dividing hne.

(This is the condition of escape

velocity from classical mechan.

ics.) Now suppose the planet explodes, expanding as a sphere of unifonn but decreasing density in such a way that its outer edge is always at the very radius

r(t) of the prOjectile. The projectile's behavior would

Gauss' law of gravitation), but we could replace

be exactly the same (by

M by density times volume.

or. after rearranging a bit,

(12-5)

This fonn is more general than it might seem. It would govern the behavior of

every chunk of the exploding planet, provided no fragment ever overtakes another.

A conspicuously similar expression governing the ex.pansion of the uni­

can be derived from Einstein's equations of general relativity. It is knOwn as the Friedmann equation: verse

( dR/dt )2 = -R-

11Ie quantity

-S"Gp

3

K

-

( 1 2-6)

R2

R(r) is called the scale factor and is a measure of the size of the

K is analogous to the projectile energy. but in general universe. The constant refer to the overall curvature of space. If K > 0, space is posi­ we y. relativit

tively curved, like the surface of a sphere, and the universe cannot spread out = 0, much like a projectile wilh nega­ forever. It must eventually return to K < 0, giving space a negative y If curvature, hke a saddle, expan­ rg n e live e

.

R

�ion can continue indefinitely. The density of matter decreases as the universe expands. An easy way to as I at the present age of the universe, in which account for this is to define

R

PMIR3,

where PM is the maner density now. As case the density of matter is we make this change to the Friedmann equation, we add one other density, P A'

We wilt have more to say about this later. For now, we simply nOle that it is a constant, independent of and thus has the weird property of not dissipating

R,

as the universe expands.

K R2 To put the Friedmann equation in a fmal convenient fonn, we redefine our mass and time units to absorb some constants and set

K' R2

Rpresent

dRldt to I at the present age. = I

(dRldr) present =

1

( 1 2-7) This equation suggests vastly different behaviors depending on the three

parameters OM'

fit\' and K'. As noted in Section 2.8, the universe appears to be

nat, meaning that K ' = O. But even this allows great variation. Suppose we dis­ pense with 0A' lfwe simply evaluate equation ( 1 2-7) at the present age, we see

thaI OM =

I . In the units we have adopted, the matter density for which the

ulliverse is flat-the critical density-is conveniently 1 . It is left as an exercise se if we choose the present time to be I , the actual solu­ \0 show that in this ca

tion is

R(t) = [�(t

1

- !)J2/3.

The universe would expand forever, though at an

ever-diminishing rate, as illustrated by curve A in Figure 12.17. The figure also

plots r alone. This would represent a simple case in which the expansion rate was always

what it is now and the "age of the universe" would be I . Curve A

shows that the actual age would be less, due to the backward gravitational pull. The Friedmann equation has a fairly simple solution in the case just discussed. Although it is more difficult for nonzero

flA,

this case cannot

Friedmann equation

Figllr. 12.17 .>\ cnlKal UOI\t'1">e or ma\� aloot'. CUI"\t' A, Uoooid "" paRd (lll\ l� ",r, It.. brgrnnrng Uoould be Imer than a ron'talll" peN e\pan\IClO III loday\ nile "oold 5ugge..l, A nl\mOlogKal nm"wlI. I:ul"\e B. �'an an:eierale e\pansion B SmJII. oonJ'cro

1

(o_moIO�H;al

t'on,!anl

•

Con_!,anl

,

\Cloclly clp,an\lOn

Cnh(al dell<.ity

J

(nil (o..mologiclil I.:On"',lnI)

,

1

,

,

•

7

be ignored. Thi� weird density i� the cosmological constant mentIoned In Section

2.8. attached to space il�If and undiluted a\ the universe \preads. As just one e'(ample of tile possibilitie.., CUI"\'e B in Figure 1 2. 1 7 �how.. what would happen if the den�ity at present "ere agam critical, but with I q, being due to f} \. The CUI"\'e looks very similar at early Limes, bul n \ cau<.es the univer;e later to expand more rnpidly, In fact it would eventually grow exponentially (<;ce Exercl"e 52). Compu. tauonal Exercise 54 lays out a method of sohrng the Friedmann equation numeri­ cally and looks at se\era.! interesting cases, The value of the cO$moJogicaJ constant i'i unknown. but it could explain a peculiar feature of the early univcN:-innation.

early Times Although there will continue to be II\ely debate about many of the "teps. there is 'iOme consensus on a rough chronology of event.') beginning with the Big Bang. Figure /2./8 iIlu�U
their lone unified interaclion. It ",oold ha\e been unimaginably \mall-()rders of magO/lude smaller than a nucleon. Very MX>n, howe....er, a quiCk. tremendous e.�pan�ion occurred, called inOation. Thi� is the pre\'aihng e�planalion for a "problem" in todoJ's uO/verse. The univel"\C IS actually very homogenous. �uggesllflg an equilibrium. But its parts are much too far apart to have communi­ cated, Jln/�ss they were Once in closer contact before undergoing a drastic expan­ �ion. We aren't sure whal caused innation. bUI a nareup rn Ihe co�mological constant is one possibility. Somewhere in this early period. sJXlIltancou'l symme­ try breaking would occur. FirM. gravity would ...ever ItS relanlln\hip with the

genealogy 1 8 The

_ · 1 2. ll ' fig

of p3nll.:Je�·)

,trong-electroweak force. "lriking out on it.. own at an carly agc. rand unified next rebel would be the 'Itrong. �plitting off from the eleclroweak. Cooling ....ould . continue. but '>0 long a� paJ1ic1e energic'\ remained above about would \till be plcnty of energy to en,ure an abundance of 1 TeV (loJ GeV). there ... of fundamental fennions: the least ma�"'I\'e tier (u, d. e. " e)' re gencmtion all lh e the middle (s. c, l./... I'J.!)' and the heavie...t (b. t. T. " t)' along with their antiparticle.... called the quark-lepton soup. thiS hoi mixture 1\ \ometimc\ I By perhap" 1 0 � ... average energies would have dropped to approxi­ mately I TeY. whereupon the electromagnetic and weak force... would part company. thu .. beginnmg the four-force, asymmetric behavior we ob\erve today. Thi\ 1\ roughly at the Imllt of experimentally attainable energie,. At around 1 - 1 0 I-l' .lntl I 0. 1 GeV. the heavier fermion\ \vould have become

�e

.

lOcreasingly \caree. lor the} are hable to decay to lighter particle\. and avail· able energle\ v.ould hI>! \O\ufficlent to renew the !'upply. Below thi, limit. the univer,e would (.:011'1" mn,tly of nucleons and light lepton\. After \everal minutes. average enc.:rgy v.ould have dropped to -I MeV. Thi\ 1\ roughly the binding energy per nudeon 10 the atomic nucleu'\. \0 nuclei would form. At around a million y�:lfS lnLl. a temperature of 3000 K. a, noted above. atom.., would form. Heawilly bodie, arrive on the !'cene much later

.. a.,., t2

... I " sd ... _ . ' ..... .. ... ... · , · ..

Condusion

AI presenr.

we' hive I fairly ple.t.�ing view of fundamental panicles and mler.

acboDs: Silo qlW'b. six leprons. and

three torees. But many que.�tions remam:

JUSI how does the uymmerry in the e'ectroweak interaction come about'> Will the Hills be found? Dots il tll:pJain mass' 1 . Wi" a theory encompassing all forces be established. and whar will it lell us about the universe? What really

happened right after !he- Big Bang? Was a uniled (orce shared by panic'e� III(ft fundamental than quatb and 'eplons'l Are there other fundamental forces'? &citing discoveries lie ahead.

Producing I au.rtc-GIuon ,..."..

nllJ)' be ih own antlp.utJde. and if 11 1\, the accounlmg fads. MOl"C(wtr. the ncUlnno produced in the "first" decay of a

��lIy Iht world\

mo6I·pow(f"(ul kcdetaror o' ions is lht RelltlH�lic He.a\·)·

Ion Colhdet IRHIClIlI Brooihavcn

National uboralory It

!muhe ,lold iOM tOllelher at II rrcmendou\

100 (i(V per nudeon. Sut;h • (cn"ll:lVu\ Impa.:t of man)' nudrons i\ <;&id 10 "melf' neutrons aDIJ prottln, for an inslMlt inlo I quarl· .Iuon plas.ma. mu,:h hkt wnJlllon.\ in the early uni'·e�. A \ef) IQkR'\lInB phenomclwn olxctvN II! RHIC 1\ JCt qllC'l'Khmg. m which JCh pruduced m faul)" �imple colli\ioo\ appear to be .\mothered by I swann 01 \ubr.equent \.'OlIi�lon� In the funot!\ mas, \1'" quarks and gluon\, Further sNd)" u.oolJ tell U' mut;h abl.lUl the\t t"On\IIluent' (If the nueleo"," RHIC may al�l hdp II\(: tilt spa crisis. I penulent pulLle In hlsh·eners) phY�lc�. Although Ih (hru spin. qu.tl.� contribute to the menll ,pin of the prolon.

j

dnuble beta de(ay coold. in effect. anmhilate With the neutrino from the �econd. lea...ing Ju\1 two eleclrons 10

be

eJ�led from the nudeus, A repan of neutrinoless double

bela decay ha\ been p ublished, but the e\'idence is not yet

(on\idered condu\l\(, If il can be \ubSlantialCd, an actual \alue for neutnno m:bs-\'c"(ingly unavailable from neurrino o\(IIIJ(lon \Iudies-mny follow. a!t weJl u!t \ub\lantial rethin king of the Siandl.rd Model.

Oart.: En.rgy The (erm dark energ) I� heard much 1Mw.;tda)" and refen to the defi(J1 between Ihe entical tkn\lty and lhe mailer density of Ihe universe. The Ufl!\erse

apptan to be Critical. or nat. but mailer dens ity. both

hee Section 2.8). appears to account fOr

uperiment,,1 e\'ldelKe in lOt 198(h �h{)\\.'ed that they

lumlnou� and dark

acrouDl for Ie\\ than a thllll PI.uilble call(hdllel for

only aooul J5� oflhe entical \a/ue, Whal about Ihe olher

mduI, up Ibc Jlfftrtn.:e mdutk orbital mOl:ion,
momenta Ilf the 'pin· I IIlu(IlI'. and the quark·antlquarl pain. appcarrnll llJld dllappunng Within the pMon\ lively interior

A Con...-vation � Hangs in th. B.a!anea A rare fonn 0' �lI\'e J«:ay 1\ double beu deca)'. In a pn:UhM

leaptroa. a nudCI.IS that unnot beta decay. b«au..e the

wuuld·be daughter IS hl'
65ft'· Lillie IS known, bul theones abound, One propo§cd ,andldate IS "quIOtessence." an unseen form of energy that could vary With positron and [Ime. Howe...er. the more

common view IS that dark energ), i!t synonymous With the (o\moI081(a/ con� t an[ . ....hich slill doesn ' ( provide much

help. thiS con�tant wouldn 'I �tretch Ihrnner wllh the

gmwlllg UOl\e",. but would be so methL08 special 10 the

,ery \'acuum of �p�e Ihelr Theorie<. of electromagncllc

\ac.:uum fluCluations ha\e been qUite �ucce�srul in

at OIK'C If the dauShlet ill the end of !he double proce\\ i�

e�plilining Eanhl)' bchll...io...,. But calculations of the

product\J In onhnatl beta J«:a)'. double beta deca) produce�

a bit off-too IlIie by mOf"C than 100 orde..., of magnuude String theory. whICh hopes 10 umfy all Ihe fon.:c\. i� IJ� chlmmg m on this problem. Some ...ersions of \tring Iheory offer a pos�Ible explanation. though IC(Ompanltd by Jegiom of othcr dimensio ns and alternalive

hlblCf tIwr the onSina/ �nt JU'1 1b a neurrino I'

two neurrino�. HIMe\er. one (If the honesl que\t� In high­

energy ph)'\lc\ loday i� ntu/rino/tl! double bela J«:IY, lbc

traditional \iew \11 beta de<.:ay. In harmony with con<;er\'afion of lepcon number. 1\ lhat when a nucleus under):oe\ p­ decay. the dcctron. (If L• • + I. mu\t be Joined by an QIItmeutnno of L,. '" - I But so fM a\ \\-e kno\\-. the neurrino

co\mological constant baM:d in pan on �l miJar a�sumptrons

are

um\e!'SCs Dark energy IS one of the blg8e�1 m)�tenes m modem ph)�I(i today.

IiIptlsr 1 2 Summary The .cud) of fundamental partu:1r:s and their interactions, knuv.n dC ph)'�ic� OT high-energy ph)'�IC�, I' the que\! to undern af U . . J' I UnL\-erse on the most bd"C le...el po\slble. At l;IfId the phY�lcal lc\lons between panlcle� occur In dl�crete events. It\d. mtcr: share a force by c'(changmg II. mediating fundamCntlll fenmons ..... or field quantum. . . .. t panIcle hM an antiparticle wlth the same Each fundamenal but of opposite charge. Antiparticle) particle l1li-" and spin as Ihe relativistic quantum mechaniC). by predlded recognizes three fundamental forc� physics nt. p�se jilt At II. propeny requ,red to engage in Ihal force. and e5 1O\01\" EJ<:h own medlatmg 00500(5). In order of decreasing ..II has its ( ,tfCngLh• Ihey are the strong force, the eleclroweak force, and 1f\CgJ3\;talLonal force. The slrOng force requires the property of color charge. \\hl'h can be red. green, or blue. The fundamental fermions '/Iltlt lhis property, called quarks. come in three pairs: the u and s. and the I and b. The mediatmg hIhon i� the gluon do the c and �Ied individual quarks have ne\'er been found_ The small. "I gluon-bound U01ts are qu rk-a01iquark meM)ns and three· � quMk baryons. known collectively a� hadrons. The proton and oeul1On are baryons of quark conlent uud and udd. respectively. 1l'I ' nucleus, all nucleons attract each other via a c� mani­ ft\tauon of the �trong force k.nown as the residual strong force. Tbc property required for the electroweak force is charge/weak dwge. Beside� quarks. leptons po�sess thi� property. There are thrtt patrs of leptons: (e . �'(). (p. - . \I,"), and (.,.- . \'�). The medi­ aung boson� are the massle�s photon. a�socialed With the elee­ uomagnetic part. and three very massive weak bosons W� and 't. �sotiated with the weak part. The asymmetry belween the ion IS an allrac\i\e f'Io"O parts is not fully under<;tood. GraVitat lora: that pulls on all energy and so is felt by all particles. The presumed mediating boson. the graviton. has not yet been found, Decays and other interactions between particles are gov­ en'Ied by numerous con�rvalion rule�, Some rules are obeyed

�s

In only certam kind� of decays. Many physicists believe that today's three forces will one day be revealed as merely different aspects of one fundamental IllleraClion The unification i� expeded 10 be apparent only at f\treroely high particle energies. Advances m cosmology may clarify the picture. for it 1\ beIJe\'ed that Ihe universe began with I Big Bang. in which all forces were one. :-:ndicates advanced questions

Conceptual QUt.stions 1. Explam to your fnend. v.ho ha.. Just completed a study of classical eleltrtlmagncti.,m. the term field Qllomwll_

lO\:rea-.e \0 p.ltcntial cnerg) a� t.....o atom!!. \0 a thalamI\: mole..:u\e {:ct c1o..er I.. panl) due 10 the (1m: cltttron\ hem& fon;ed \'I) lhc: e-..d . u"l�ln pnnc1rk 10 higher cnergle.. Can you �ue \Ioh) the rc'lduai �tmn, force-the Inlemuc\eon oIl1ra..IIOo ('II ChapteT 1 1 ..hould ha...c a "hard (lire''') 3. Why could It be arguer.! that the tJ.ngc of the ,trong fOl\:c \h
2. �

Exercises Section 12.1 9. From the expenmental evidence that the I-orce bet ....cen

nucleons has a range of about I fm. obtam a rough value (in MeV/ell for the ma�� of the panicle exchanged to convey this force, the pion From the masses of the wea\;. bo!>Of\!o gi\'en 10 Table 12.1. show that the range of the weak pan of the e1e..:­ troweak force should be about 10-' fm_ 11. Someone propose!> the eXlstcnce of a new force who�e range is 10-20 m, We found 10 Chapter 2. that ,," • . About ho.... accelerators tum kinetic energy into rna much energy do you e!.timate an al.:celerator would need to create the mediatlOg particle for l'iuch a force"

10.

Section 12.2 12. As in electron-positron annihilation. ....hen a proton and

antiproton annihilate. twO photon\ can be produced Of what wavelength are these photons� (As�ume that the pair is initially at rest.)

5S5

I . ' ,·on 13. To �how Ihal lhe Klem-Gurdon cqua

M.llul/Un�

ha� ';llid .

Ih<.lt equatIon the lorm C !'undH'>o (lI

Illr ne,allve \a/uo;:' 0I E'. "en I,.

_

( 1 2.�, Ii sau\lictl by a ";I\

•

�

-

-

A,.:tr- ,"" I� a �oJ ulJon 0( both

'4. (.1) Show Ihal '" fl '. "

equatl on�. the Klejn-GII don and the Schroom,gcr llon \olu a j, 611 Ar.l.· C(J� (h) Show 1/J,1I 'V (I, " nger Schrixh the not of of the K/t',n. ordon

b

c'Iu;llJ()n

.

_

hUI

__

_

_

and

�luve (0' Show thaI 1.JI� j� 11 combi natIOn of po -Gordon in e Kl nt'ga!i\-'c enc;gy \oJWlon\ of the

cljualwll hee Exc"':I\C 13J. 2 (d) Compare the IlIne dependence of )"'1 for II'I ilnd 1¥2' IS. For \oIUll()n\ 01 the Klein-Gordon equillion. the quanlJly

jIJ' .

d

ill

1/1

ill'

,

;'/

'I' .

j., Interpreted a\ I.:harge den\!!)'. Show that for a po�jl.\c.cncrg}' planC-W3\C ..olulion (\ee Exercj\e 13). it i .. a re;,1 (."On�t..nt. and

for the negative-energy

�()Iution. it i.. the negali\'c of thill (';omtanl.

mCl'hilnic �. governed by the Schnxh nger equiltion. the [01.11 prohabiJity of finding a partKle doc� nol (,;hnnge ",lIh lime.

16. In nonreJali\i�ti(,; quantum

Section 12.3

�

17. The electron energy mentioned i.n Section 12.3 fOr

�

ineta�tic scauen ng ex nments IS 20 GeV, and � . momenlUm JS gIven as _0 GeVIc. Why SosimP1 ', cOfl\e�ion 'l \\hat energy wOul d electrons 18. Approxi mate ly need to as probes th al Co uld re\'ea useful be to have l features ' 18 m, the approxllnale range a� \mall a� 10

of the

weak forcc" ... 19. S "ppo<,e a force between two particles dec....�c . .... ith .� ' to F = k! . r b. What is the•...limi di\tance accordmg t on b lf . , ' les the energy required to separate th e panic intiInn ' ely ' ? . far is not to be IIlfi nl ie .

Try ing to pul l two quarks apan Would produce mOre quarks i n groups. or �adrons . Supp�se that When the �eparat ioll re:t(,;he� I t m ( th e approxlIllate radius of a nucleon). the lightest hadron. a 1To. is created. (a) Roughly how mUCh. force is involved? (b) Com pare this with the electrostatic force belween two fundam """ charges the same d'1�lance apart. 0oes your result ag ret wjth the �trength<; in Table 1 2 . 1 ? 2 1 . For whidl pani cles does Table 12.2 show both panicl e and antiparticle? Which particles are their Own antipaniCle? 22. Symmetries are compelling in physics. The pro penies of the particle� in Table 12.2 �how So me i nteresting ones. (a) Make II two-dImenSional plot, with strange. ness along the vertical axis and the t hird component of isospin '1 on the hori zont al . Add SpOts representing the .:l . . and the nine other baryons below it in Table 12.2. (b) Arc properties other than strangeness and " corre lated? 20.

(.1) Pf(lVC ii, Bcgin wi lh the time derivative of the total prot-.ilhiljty.

(J

I'

til .

Section 1 2.4

23. To produce new particle.., a..:cekrators often smash two

equal-mass objects together--p H,)ton and proton or electron and po�jtron, The tbr�\hnld energy is the kinetit' energy berore the collision n' .led simply to produce

'I' -(.I. I)'V(X, IJ til

/("'(<.1)

ill ,

� " (u)

� '/" (x. f)

"

af

'/'(x.

f))

dx

Then use tht:' Schri.ldingcr equation to etiminale the partial limt:' dai\atiw, in favor of parti31 spatial derivativt:'s, inlcgmlc hy parts. and ..how that the result i� O. A..sume thai the particle IS reasonably "'el l lo'."ali /cd. so th<11 'P and a'IJIliix are 0 when e\'alualed al --: 00. (b) Does Ihllt procedure lead to the ).ame conclusion if the wave function mUltt obey the Klein-Gordon rather than the Schrildinger equation? Why or why not?

lhe linal particles-their m; So Lr. ' �mal energy alone, with no leftover kinetic energy. r"f ,ider a COlliding-beam ;Ides of mass //I are acceler3lor in which two ini:I' moving at the �ame speed rt'l.· !'>�. IQ the lab. Assume that

the lotal mas� of the (stat,...,}, sion is M. Show th3l lhe th!'.:

24. Exercise 23 discusses the l:l1 ticles of mass III in a coltidir;",

duce a final stalionary ma�

instead a stationary-target Yj'

energy is needed to produc�

.t'

that the threshold energy i� (,\

I

'drticles after the colli-

:('

'nergy is (M - 2m)2. ;d energy for two par;n accelerator to pro-

:he accelerator is

I;)ore initi31 kinetic une final m ass. Show

/. '11 -

2m)c2. The

calculation is greatly streamll)' ! by using the momen­ tum-energy invariant, discus: ,I i n Section

2.10 and

"-_

117

. The main point I� tha\ ,L Exerci..e :!. 1 1 2 me q

(£'
-

P i ,,� a', must be the same not Qnl\ bero '

.

uantlt\

.

COII1\100 bUl aho In an,' et Jnd aft�. r Ihe r eren� hame � h 'IS. to equllte thi\ qUa approac "tIt) The qUtek f-ma�\ frame \\ h .' . Ihe cenler-o ' tlna.l. colli sion In ere .L C I U ' . . _ . Its 10 value bf'f(l� thA , 11· y. r Ona "taIl 1� • a�� 1 J' .. \:0 \10" In the m . me. \\ here Ihe largel l� �lahon" n.- '-' " • fr'.l lab 'OU \\11I -J 1\e'e(j 4 "' � + lII" e to eliminate the rno\ to U�e £- = p'e' IIlg pani· ntum in the lab framc. cle"s mome and 24 g;\C the threshold en,rg. 23 ises ,e� 1or . ; E;(erc 2". of mils", III cnn produce p(lrt,c1c<; a \\ hieh twO gl\Cn . g m be II·d· am I and co In �tationary_targel [lla�s . respectIvely, Evaluate the Iwo for It accelerators. collision ;n which Iwo protons become lh1.' protons ... antiproton. How much mort '. energ,· ·IS neede , '' od onc d . get? for the stationary tar '' ',. lead nucleus :It rest it' roughly 1 0 14 n " d·IQmcter. 26' A . through the laboT:Jlory wllh a kine!"IC energy mg mov If Ihlck would Ihe nucleus be III tl of 600 TeV, how Ie . n.'I llo f mo o ion direct

M

ufrrr the:

o

•

... tll .. r 3K. .'\u ... fI + IT 39. t\' - J.l.� t \

.J7. :::

classical magnetic force formula F ql'B is 27. The correct relativistically. But if a magnetic field is to keep a high·energy chnrged particle mOving in a '=

..., .1 0\.. �'l

4t.

circle, it must satisfy the relativistically correct relationshi

�

between force and centripetal acceleration,

F == 1,.III(I·-lr). (a ) To keep a

I TeV proton in a I km radius circle. as is done at the Tevatron, how strong

must the magnetic field be? (b) How large would the radius have to be for magnets of the :'iame strength to keep a 20 TeV proton in a circle?

42. ITn

-40

•

•

1\ lo"' '" . .. Tr

.... J.l.

.. \.",

't

In the rollo",lnll eun:iab" t_o protOQl art unubed

together In an IIIlttmpt to wavert ILlnetk tMrv lato _

�nd new particles. Irtdkalt ..btther the p� rtIIidioD. l\ posslbit. U not. lndkalt ..hkh rules

an vMMte4. COIIIIdeI'

onl)' thost ror charat. Inaular raomnatum. aDd baryon

number, If the fft(tlon II pcalb\e. ukulalt tbt adDImum kinetic entra,y maulrtd or the coUId'ol pl'OtoDL

Section 1 2 . 5

28, Although 11:11 truly fundamental, the residual strong force shan:'! by nucleons can also be represented by a Feynman l!i:.tgram, with the pion filling the role of mcdia1ini: 'oson. The accompanying diagram repre. sents a

,. .10

... 2'1 + rr

b

h·

by a 17"0 t exchan�i'

between a neutron and a proton mediated '0. Suggest a diagram in which they

neutro:

particle . (We demand that there be a

proton only at the start and finish.)

43,

P + P_p + p + p + p

44. p + p - p + p + n + n 45. p + p - p + p + n + n

46. p + p _ p + K+ 47, p + p _ p + K+ + 11." 48. p + p _ p + I' + Ko

Section 12.6

49. If a neutrino interacted with a quark every lime

lheir

\Cparation was within the 10- 11 m fIIlJlC aenenlly

accepted for the weak force, then the emu IICIioo of . neutron O!" proton Maecn" by a neu1rino would be aD die

o ------ p

29,

Table

L.


aemal rio

eonstittLt I great de:.ll I

,'es the "width" of the K"O meson. lately what is its lifetime? (b) This meson's

I S on the order of I fm. Would its

have time to "bounce around inside" a

order of 10-16 m2. Even at luch sepanIioD, baweYer.

the probability of intenctioa is quite UDIIl. Tbe nucleon appears 10 have 1ft ctfec:tiw Cl'OII sectica �

(I) A_ bow IIIIIIY _ .. ... _ . ... .... sectional .,..? (b) WIw 1I .... �ty neutrino pIIIin. tbn>u&b _ II1II .... .....

only about 10-41 rrl-.

\here in a column IhnIuah _.. _ on will octuaIIy "bin

_ .... 11

n·

.' ...... .. .. , __

If we multipy both sides orequltion (1 2-7) by R2(t) whole equatIon, men caDCC!1 dR(I)/dt, the result is

_ ,&7 •

.., ... JI ... .... ... aiIicaI(O.. - I). � tr 0), "' ''' 01, a_MP:aI a.a.I (II• • 0). - _ (12.'1. be ....... .,. •

'!""'" (1 - Ir"·

, _ _ lI(� oI (, II . ... if_ (Il.n _ . ..........-etecf waivase. . _ _ _ _ 01 _ " _ ....... ...... JIN"iIII "'" ... .., 110 wdb die diI·

Equation (5·35) comes from the Schlixhnger equallon' which says thaI

.... ... . ..... .. ..... 0I _ 1IId Ndi.­

d',,(x)

.. \111)' ...... dw .. oIdw uaiwne. MaDer dmsify is

......, - po.,.._ .. ... ....... -

,,,�. ..... II.. .. die ..... deDaify now. bdidiOll

.....1)'. bowewr. would be propwtioc&I lO IIH'. (Nol oaJy doll the volume incrc&le. bul al50 all wavelengtl!s .. IINIdIed in proportion 10 R. lowerinJ the enetBy "1)' by Ihe un (aelor'.) Thil denJ.iry drops !tiler • die ualvene pows, bul it al50 flOWS III(ft qukkly in ... bKtwIrd bme direction. In adler wonb. IonJ aao. Ibt aiwnc would have been r.hatioa domirwc:d. $bow tbII i( tile function IPed (or matter deMiIY in .,...uon ( 1 2·7) is replaced by one appropri* 10 �. but retainin, lhe &IIUlDption 1M! K' and 0A are boctJ O. Chen the scale r.:tor R would grow as ,Ill 11 SI»ow !.bat cbe ptUCnCe oIa positive cosmological CODICaDt in Friedmann equation (12·7) m.w. as R bcoma very 1Itat. lead 10 an exponential expansion 0I1he urUYCnC. 51 You _ • pomisina: theomicaJ physiciSi who does nOl: bdievc IhaI pviry is • ctistinct fundamc:nral force oot is iIIIIIIId ... 10 Che ocher fon:a by an all-encompusing rellciviltic. quanfUm·mcchan:ical rbeofy. In particular. you do noc believe Nt the universal gravitational CODIIIftC G i. really one of naIU/'e's elire set of funda­ IDItlItaI constants. You believe lba' G can be thrived from more-buic constanl5: the fundamental conSlanl of qaaatwn mechanic•• P1anck's COIlStalll It: the funda· IDIIllII speed limib"l the pro".,ation of any force. the .".ed of IiJhl c: and one othcr--.l fundamenral length I. imponanr CO me cae wlified force. Usina simply dimen­ tioaa1 ....ysit:. . 1iAd . formula for G. [ben an order·of·

...._ .. qloe "" '" - 1enJdI·

Computational Exercises M. HeN we solve die Friodmaa.a equatioa numerically _ loot II some iaIueaIiaa: IpeciaI cues. (Review _ '.10 aacI die Ouidoliaes . ... bqinaia8 of .... Chapter .!i COIDpIIatiooaI E1cttises.) The first step II 10 COIMI1 &be Pried"""n IDqUIlioa co • scco..ordcr od cti&rutiaJ OfIUIlioa lib daoIe in earlier cbaplen.

'

cate I bme derivative or lhc

-----;J;2- =

-

,,"(E - U(z)) 112

Adapting (5·35) 10 our problem thus gives

R{t + j,)

- 2R(t) - R(t - 61) +

(

!/I(x)

J\,

0"

AR I 2R2( 1) - O ( I1r2

This would allow u� 10 plol all values of R(/) from a starting poinl li the presenl age. whose lime we choose IS I We would also like 10 look backward in time, and the modification is straightforward.

R(t - .1t) •

2R(t) - R(I + .6.t) +

( �M

ZR · (,)

�

- n\R(')� '" .

LeI us usc: 0.001 for .6./, Together with our choices in Section 12.7 that both R and ils time rale of change are I at present. we thus have R(I) = I, R(I + Jot) "" I ,001, and R(I - .11) = 0.999. Now we have e\'erything needed to plol R(I) both forward and backward from / = I However, with an R(/) in the denominatot above. a 0 value could cause trouble. So incorporate inlo your program instructions that (I) will assign 0 to any "new" \'alue of R(I) that the equation might cause 10 be less than O. and (2) will check to see if a prtvi(}uJ value j� 0, and if so. simply assign 0 10 the new value. With thest instructions, your computer will never actually try to divide by O. If your universe ever shrinks to O. it will Just stay there. The values we now input and check are thost of OM and llA' Ploltlng from I :: I to t = - 2 and from I :: I to t 18 (except as noted). consider the follOWing values oflllM. ll.�I. and in each case di<;cuss your find· ings. (a) I I . 01 (b) (0,6. OJ (c) CIA. OJ (d) (0. IJ. with an upper limit of 1 = 4 (e) (0.95. 0.05J. wi th an upper limit "'"

of' 8 (f) 10.6. 0.01] (g) IL4. 0.011 (h) 10.5. 21. w;th an upper limit of t 3. =

"'"

Appendix A:

The M i ch e l son-Morl ey Experiment

Michelson-Mort�y expe:i ment was intended to determine the speed at '('he arth in its orbital motion moves through the ether, the medium of light which E presumably permeated space. The experiment was based on agation that proPraet that if light must move through the ether at a fixed speed while the the . · parallel and perpendicular to the moves relative to Earth, round-tnps

ther

e h would take different amounts of time. Measuring the time difference t er the ether's speed relative to Earth. The analysis is identical to e, uld then give ion �ften discussed in cJass�ca1 phys�cs. ��I in a situat . SWimmer, who always SWLms relat.J.ve to water al speed vS' In a a ider Cons ' (me medium) that is flowing at speed vr relative to the ground. An ver erver on a bridge overhead watches the swimmer perform (wo differenl Obs in Figure A . l : ( I ) a round-trip from just below the observer, 10 sks, depicted � upstream a distance of L, then back downstream to the pomt under the bserver, and (2) a round-trip with the same starting and ending points, but in

a pOint

:hich

Ihe swimmer swims a distance L perpendicular to the bank. For both of both round-trips,. (.he swimmer'� velocity r�lative to the ground is the

legs vector sum of Ihe velocities of the sWimmer relative to the river and of the

river relative to the ground. From Figure A I , we see that the speeds resulting

from these four vector additions are

round-trl·p 2'.

II

away,

2

-

..VIv's

-

2r v

Vback. 2

I 2 + v'

= ..VVs

r

In both cases, the time required for a leg of the round-trip is the distance divided by the speed. The total limes are thus round-trip I :

round-trip 2 : Because

crip 2.

Vs > Vv!

-

III =

11.

=

L v,

v,

L Vs + vf

+

L

Vv;

-

V;

+

=

L

VV' ,

-

2Lvs

�,

v' ,

-

=

(A-I)

2 v ,

2L\/V; V; V;

-

V;

(A-2)

If. we see that round-trip I takes more time than round­

Michelson and Morley applied the same analysis 10 light. If Earth in ils

orbital motion about the Sun passes through the ether that permeates space,

lhen the ether moves relative to Earth. Thus, Earth is analogous to the bridge

A-1

-

-

�

-

-- -

-

-

-

-

-

Figwe A.1 Perpendicular round-trips through n moving medlum_

Current "

Vb&
and me ether 10 the river. If there really is a medium for light-an ether-men so the light beam is a light beam would always have to move through it at analogous to the swimmer. Thus, we replace Vs with c and vr with verner. From equations (A-I) and (A-2), the lime difference between round-trips of Jjght

c,

beams on Earth is

AI

=

2 Lc

--,-= ,c? - \ ther

�

2£ Vc 2 ,

C-

-

v;ther

ve2ther

(A-3)

two round-trips. If L and Vs (c) are known and a round-trip time could be measured, either (A-I) or (A-2) alone would suffice to calculate vr (v th r). A far-fetched but useful e c At this point, it may seem a mystery why we bother with

example illustrates one djfficulty: If the speed

vr

of the river current were

I mls but the swimmer swims at Vs = l OO mis. then the round-trip time would

djffer very little from what it would be if the river's speed were O. Consider­

able precision would be needed to distinguish between a current of 1 mls and one of 0 mis-about I part in

104. In the case of light, the "swimmer's" speed

is roughly 10,000 limes faster than the current (Earch's motion). A round-trip of a light beam on Earth would be very short in any case, and the slight lime

difference caused by Earth's mOlion would have been beyond the capabilities of early instruments to measure.

Timing

two different round-trips would not

solve the problem. because both times would still be minuscule. However, having light beams travel two different round-trips is the perfect applicatioD of the Michelson interferometer. By observing

interference

between the beams,

exceptionally small differences may be detected. In an interferometer, depicted in Figure A.2, a light beam is split in two by a half-silvered mirror, the beams take different but equal-length paths, and when they reunile at the half-silvered mirror, the interference between them is observed. 1f the beams travel at the same speed, constructive interference is observed. But if the beams move at different speeds, they will take different

1 : "-

Figure A.2 The Michelson inlerferomeler

' ''

.

Half-silvered

mLrror

Mirror

Beam I

D�'I

__ __ __

;q

_ __ __ L __

l.i.

Interference

incoming light

2L through the apparatus and will have passed through a different number of periods-in general, a different interference condition will prevail. Were there an ether moving relative to Earth, an inter­

times to traverse the distance

"

ferometer would have a "slow arm" and a "fast arm for round-trips of light. Equation (A-3) would give the difference between the two times, from which

we could calculate the number of fringes (cycles of light and dark) that the pattern should be shifted from an equal-times condition.

Of course, we can't observe the initial interference condition and then ask

that the world be slowly brought to a halt as we count fringes until the equal­ times condition is reached. The trick is to begin with the interferometer ori­

enled. as shown in Figure A.2, with one slow

arm

and one fast arm; observe

me initial interference condition; then rotate the interferometer by 90° while

counting fringes. The interference condition would start at some arbitrary state; pass through a symmetric, equal-times condition, in which the ether " then continue to a final inteIierence condition in "flows at 45° to both anns; the fast ann, and vice versa. The number becomes arm slow which the former of fringes counted should correspond to twice the difference in limes between the tWO round-trips. We find the number of fringes that the asymmetry would

shift the interference from the equal-times constructive condition simply by dividing time difference (A-3 ) by the period Tof the light wave. Number of fringes from equal times

(

2L c -

Vc2

T( c2

-

- V;th,,)

\'�ther)

=

( H) ( V;;,,)

2L I -

TC I

_

Ln the original Michelson-Morley experiment, L was approximately 1 1 m d an the wavelength 550 om, corresponding to a period of 1.83 X 10-1 5 S.

---

-

AP pendix

B:

The Lorentz Transfo rmation: P l otti ng Eve nts

tnUl"romullion equations allow us to find cOQnlinute" 01 e\ent' in (derence frame from their coordmates In another. it i.. "h(l po.....,blc I�l different fmmes. which can add much to our IIlsighl. (, enls in d such plots, it is helpful to take n look at a related 101m" To understnn l1n of ll'(es. Consider two reference frames that differ not due to relahve

fht L�lrentl

on:, ph:

�lI

as Illustrated in Figure B . l . simply in that frome F' " rotilted hy (DI,.'ltion. but. the z-axis retaltve to frame F. An arbitrary Iwo-dnuen,u:mal about "' angle has different components in the two frames. A.. ..hown In '''tion \'ector r

�gurt l

8

B 2. these components are related by

( =

)"

=

('COS 8 - y'sin 9

_,.'cos 8

+

x'sin (J

['J [

or

y

=

cos 8 sm

8

-

sin

J[ J

8

co.. 8

"

)' ,

(B- \ )

Let us see what a plot of the x'-axis would be in !eom of x and .\'. Thi,

l\isl\illl points for which y' = O. lnserting y' = 0 in equations (B-1) gives

x = x'cos (J and y = x'sin fJ

�

� x

= x'sin fJ

y = tun (J x

x'cos fJ

Flgl.lr. 8.1 A frame rotation and a vector r. y

y'

x' F

8-1

---- -

-

...

. , , , " , _ _ _ • •

_ _ • • � ,._ T _ _ •

- --

•

·

• •

. -.

-'

. -

·

•

.... h . --... .... ..... , "' .. � . . ""­ .. • •.,..,.. .. 0 ... •

_.

•

-

•

-

•

• • •

-- " -.-CC-,.

� - ...... � I _� � -� ....... --•

.. .

- � -."---.

,

•

_.

•

Go

•

� e.l Space-time axe!> and evenb according 10
,60

, o · ,, 0

/

..

� /

+60 ,

"

/ /

,

,.., , , , , +20

,,"

@I EI

,

,,

, ,

,

,

, ,

+�o

-20

,

,

-20

x=t

"

-40

� 40

60

-60 (.)

I

,

,, ,

,,

,

,

,,

, ,

,

t§

,,

,,

\

This bas slope elv. Unlike the F-F' transformation. both axes In the S-S' tran�· fonnation have positive slope. Clearly. a plot of S and S' should look different from a plot of F and F' . Figure B.3a shows x'· and t'�axes plotted on the xl axes of frame S for ' , :0 ,/j c· The x' - and t -axes have slopes of and respectively. Perhaps m�st important point to bear in mind as we interpret such plots is that lines of constant time must parallel the position axes. This is certainly true when axes make right angles. but it must hold in all cases; otherwise. a given I1OII;:ero lime would intersect the position axis, which defines zero lime. Because the Lorentz transformation from S to S' differs from the S' ·to·S trans· formation only in the sign of v, if we plot x· and r·axes on the x'r' axes of frame 5', both would have negative slope, as shown in Figure B.3b. The focus of Figure 8.3 is the set of events of Example 204, also shown in Figure BA. (�'ote: In Figure B.3, time advances in the upward direction. while in Figure B.4. repeated from Example 2.4. it advances in the downward direction.) The rJtaded regions represent the space covered by Anna's (red) and Bob's (gray) �hlps over the 60 ns interval in which events 1-4 occur. To see how these space·time plots describe the events, consider the following:

'{3

ilie

�.

In Figure B.3a, Anna's ship has a positive velocity relative to Bob, while in S.3b. Bob's has a negative velocity relative to Anna. 2. The sequence of events according to Bob is as follows: events 1 and 3 simultaneously. then event 2, then evenl 4. According to Anna it is event I , then event 2, then events 3 and 4 simultaneously. Figures S.3a 1

and B.3b both demonstrate these sequences.

-

,

,

'

o r

"

,

,,

,

�L , , ,

,,

,,

,,

,,

,, ,,

,

,, ,,

,,

,, ,,

,,

,

,

,,

, ,

, (b)

3. In eiIhct filW't', we see thai evenlS I and 3 occur on the same t :;: 0 ine l bur on different ,. lines, demonslrating relative simu!laneity. S _ im iluly, events 3 and 4 occur on the same t' :;: 0 line bUI on differe nt , lines. 4. In both pIOL�, al Bob's ' =- 0, Anna's ship, bounded by events 1 and 3, filS belwun me left end and the middle of Bob's ship, whose length t. According 10 Bob, Anna'Jii ship is conlracted. Similarly, at Anna's t' :; 0, Bob's ship. bounded by events 3 and 4, is contracted 10 half the length of hers.

IS

,..... ... The pu.In. of IWO ships.,

IC!C'n (rom both frames.

Evtnt (0' i o E\'ent 2 J

E\tnl J

0

[+20

Event J Elloli+§t>

-

[0

V

40,! I 40,l>

C+ii2)

£20 ! IEve20.nt 2!. 2O.l> ) C+.§> gl+2m@j> o g&J�) Event 3 F§t> G§> ill [

HO

Event 4

+60

I

Event 4

301 1+

Ap pe nd ix

c:

P l a n ck's B l a ckb od y Ra d i ati o n Law

is defined as an object from which electro ..\ bI llc kbOciy magnetic radiation he ther ly I mOll' on of ,Its les sole due to t ma ((Il1lJ\1i charges-it reflec •ion 3 . 1 , an b' ts no light . S ect 0 Jec! 0r temper I M l'lotCd in ature

'r

T has an interior cav . . . ity. the cl1 radiatIOn e Xl. t.mg the cav' ity through a 1romagn c h:c Small hole approximate� e . ody qUite weli . Expen'ment eSlabl'IShcd that the electromagnetic ene a plackb rgy " , a blackbody or cavity attalOs a ma)nm by d ,1te um U r! at a certain frequency and e . her 51'de 0 the maxim on Cit ' um. Classical the � wward zero , o�' howe",, ' ass the on um d pti bas on e tha s t electromagnellc Whl.Ch L radiation is strictly a ' " I ' failed miserably. t 'I ndIcated that the energy should be walC, a monotonically req 0 tion uen , d" func e)' lvergl g as the inCreasing frequency approaches infin � : ause the theory a?reed WI expenment at low Bec ', i frequeocies but failed at enC freq ies, the let u d lscrepaocy was tenned the ultraviolet Callas bI)Igh/Ultravio ­ h e Ofcourse, there was no real catastrophe. Classic al wave theory is sim­ Ma x Planck solved the problem. equate. As we outline his work, we will apply some principles of statistical mechanics (the topic of Chapter 9) . unfamiliar, but we Will still dearly see tha ay be the most important point­ theory differs from classical wave w ne e theory. b o Our goal is to calculate the spectral energy density: the elec tromagnetic de a cavity in a narrow frequency range be energy insi tween/and/ + df.

r

.

rr

�

u'O� d

�n:o ::

dU(j)

=

energy in frequency rangeftof+ df

1\<0 factors go into this quantity. First, treating the energy as residing in stand-

, , waves, there will be a certain 109

, "e in the range f to f + "os

df.

b

r

'

llum er 0 standing waves whose � H equen-

dN(j) - number of waves in frequency range/to/ + d/

__

y in a given wave of frequency /mig Second' although the energ ht fluctuate as ' energ is agnet const y ic antly emitted and absorbed, it shoul electrOm d have a . predictable average, which we deSignate as £(f) .

The

total

EU)

1&

average energy expected for a wave of frequency f

energy in the range of frequencies aroundfshould be the product

dU(f)

�

£(f)dN(f)

(e- I )

We consider the two pieces separately.

C-1

-

--

-�

-

:,:

:.,: :..:

-- - - - - - ....:: � ---..:..

..:.. - "":" "'""---

'-

- - - - - -

--�

-

r._. of . W_ of Fl9quency a. • .- • ..., • • .., ...... - -J emil

aad

I

oIIoorb eleo.

s • ...,. .. ..., oi l - 01 I'toqueacy l is I _ of PRlI>o7n ? • d t pi "'nlc' p. .. Ibe importanl factor. known as • • p • •'IIJ II I � 01 _ objec:u-often IIOms or

L

...::

.... .. II _ _ _ yotic oocillitioas of floquellC)' I--o­ _ .. . .... .111 '''''· _ .... 0IICrJJ E i.

1'(£)

oc

.-E/"T

utd T i, the ..mpera..... We obtain the I possib/� value £ times multiplying .... ....,. fJI • •'.... by the pt' •., fil E!? 'p',. tIIIIl value. summing over all values, then dividing by

.... t., II .. ...

en ........1

... ... ..' . wNHIr.

Of CICIIIIIe. . 10111 probIbiIity should be I . bul expressing the a\'crage as a quo.

.... EE3.... . 1O..aid cIeaI.iq

with the proportionality constant in the Boltz

_

_ poWJilily. See AppeDdi. J for further discussion. ' At ... paiIII. the classical theory and Planck s theory diverge. Before pur­ ..... Ibem. we .. ..... PIaock oriainall)' proposed that his new quantization cIIim be IIfPIied DOC eo the elccrromagnetic radiation in the cavity. as we will do. 111: 10 tbe � 0ICi11atiq in the walls. These "resonators," as he called

..... coaId chIqe eneqy oaly in discrete jumps. Einstein later showed that b Jlva the UIUIIlPtion of tbennaI equili rium between walls and radiation In the cavity. such . view is equivalent to the quantization of the radialion, which is our maiD iDIereIt. Now, 10 clarify the differences. let us follow the classical utd _ 1pIII'OICIIa side-by·,ide. ......' lAy.'i rll", 'II.,ufKIor. .., .... .... .. ..,...,.01' 5' h iii dill COMi._ an to infinity. TiI ___ . .......

If/) -

L·

lIP ... ...

. 's

... ... ,..... (C3).

" is an mteger and h I COMtan\.

-"c: . o--

1

,."

.�r�

Carrying OUI !ht summation.)

(C· 2)

•

rMliltion of frequency lis allowed only

thoK energies given by £ ... nhl. where

..-
c.,iII . * ' ,iliw.

'-"1'.

1be new chum I� that rlrcrromagnrtic

7 "EVJ

hi tVl�r

_

I

(C·3)

..... C.I pIotI tbe tworauhs versus ! They agree at low frequency but, ' a.pM!tf.J iacreues. the clusical result remains k8T while Planck s

hc:� O. If ....e accept light's particle nature. we inlerpret Planck', re!l.ull � at frequency f. nhf, compri�!'o n photons each of energy !olkM\: Tahe energy is kaT only If the thermal energy of the charges in the II 1'bt' a\er ge energy walh i... �umcient to create many photons. At high frequencies. where .. energy per photon hI is much grealer than kaT, iI is unhkely that random � energy would create even a Single photon. Thus, we eJ;pcct to find no

�.,

",.. C:.1 Avenp ..., " _ I. cluaicall, ood occadiDI 10 Planck..

pIIOCon energy at arbitrarily high rrequencies.

dN(f): N umber of Waves i n Range f to f t- df

· --� 10 � -� � -(lu.ucal ...", 0.1

J. Ota

I�" O. 0'

same quantum mechanically as clas...ically, \0 It is "The quantity dN(J) is the distinguishing the old from the new. But it is important In it... ftOI. a factor in Planck's triumphant theory would be incomplete without d. l and n (lWn gh , generahty, we assume that the cavity is a cube and that the or With no loss is in the fonn or electromagnetic standing wa'les. Because there eDtrgy inside twists In three dimensions, let us introduce some or the ideas in the art a few xt or one-dimensional waves on a string. conte impler

\

One Dimension

On a string stretching from .k..cribed by

x= 0

W(X, I)

to

x = L,

possible standing wa'les are

= A S.ln- cos 27rft

n'"

(C4)

L

\\here W(x, t) is the string's displacement from the flat equilibrium state. This function has nodes at x = 0 and x = L, has n antinodes, and oscillates with frequency f But what really governs the behavior?

Waves on the string must obey the wave equation for the string. which is

of the fonn

iJ2

iJ2 V' = I) W(x. W(x. l) --, 2

01

-

ox-

(C-S)

This is simply Newton's second law of motion rearranged in a form most

applicable to disturbances on a stretched string. (The proof is gi'len in most m[J"O(iuctory ca1culus�based physics texts.) We verify that (C·4) obeys the sec· ond law by showing that it is a solution of the wave equation (C·S). Ta.king the par tial derivatives on both sides,

The function

W(x, t) it!\elf cancels, leaving (2'1if)2

=

y2(n7rfL}2 . We conclude,

then. that the second lay, of motion is obeyed for aU 'Ialues of x and t so long as

_.•

0.'

., � ....;--

_ _

"

n f< .'2L

"..,. c.z .sr.ndiaI wnet of dJlfaaI,

tip .... . as IJIIIIIlws 01 ...... .

,.,... on I liM. ,

}

4

5

The allowed frequencies are separaled by v/2L. Figure C.2 represents the allowed stanwng waves by points on .!I one-dimensional axis (again, Ii is the

6

��

number of anlinodesJ. There is one allowed standing wave per cbange in fre. Quency of ,.,U, so the number of waves dN(/) per frequency is

" - I

til· yjU

2L

dN(f)

I

df

1'/2L

one dimension

-- - -- = I'

Three Dimensions Whereas mechanical waves obey a wave equalion derived from Newton's sec_

ond law of mOlion. light waves obey onc derived from the fundamental laws

of clecrromagnerism-MaxweJ1's equations. Despilc the waves ' different

(

)

natures, the w(we eqllarions are very similar. Light waves obey'}

" arl W(x, y .::, r) •

=

,

I' �

iJ2

aXl +

a2

iJy2

02 + al2

W(X, y.l.t)

(C-6)

For a cube of side length L, the three-dimensionaJ sianding wave is

lV(x. )" ;;:,1)

=

A sin

n

-T"-sinT sinTcos 27Tft "y'"Y

-rTX

11

TTZ

(C-7)

where nx' n ., and n. are me number of antinodes along lhe three axes. Again, we , find the relationship between frequency and the allowed shapes of lhe waves_

thaI is, the numbers of antinodes-by inserting

lion. (C-6):

[(

n"

y2 �

Again

n 1TX

n)'1TY

n�7Tz

T

"" "y"Y ,'" , - sin T cos 27T!t - sin A sin L L

)'+ (L) + ("'' )'] "

-(21Tf)2 A

sin ' -- sin -- cos -sin L L L ",,, ,

W{x, y, Z, t) cancels, and we obtain f = �n 2L

W(x, y, z, 1) into lhe wave equa­

where

n

111

27Tft =

Vnx2

+

"

n2y + n2 z

(C-8)

We see now that different standing waves may have the same frequency. For

instance, the frequency

(nx_ n n;) )"

=

f = (v/2L) V27

(5, I, I), (I, 5, I), ( I . ], 5),

is shared by four different waves;

and (3. 3, 3). So how do we deter­

mine the number of waves per unit frequency? 'The eq_ IClIIIlly appJ_ l
Like Figure C.2, Figure C.3 represents possible standing waves as separate

points, but here they are points in a three-dimensional "antinode-number space,"

labeled (3. I . 2). for example. is the standing wave willi three anti­ IfJIIeS along the x-axi'll. one a l � � Land two along ::. All potnb "'lth die same radiu!'.. n � + n� + n�. ha\'e the same frequency. and me

1be

point

_ _

..

..

_ber of polnt!Jwaves in a region is jUM the volume of that reglon. Thus. ror ' cbange dn In "radius." df would be (v/2L) dn. and corresponding dN(j) would be the volume of an octant of sphencal shell: a a\'es of w �r the

,"'II',.l X dn. The number of waves per frequency is therefore dNU} df

-- =

i1Tn2 dn (v/2L}dn

= --

three dimensions

(C·9)

L:nlike the case IR one dimension. dN(f)ldfin three dimensions ill an mcreas­ of n and thusf(increasing as the area of a sphere). Obviously. the "..0 . we have used here IS not absolutely precise. for a number of eometric l trick But the graininess of n is not of much can­ not continuous very low frequencies. Now, using (C-S) to put (C-9) in teons of pt at c ctm x fl . we have

function a is �a\'es a e e 1I1Itherthan

function

Finally, because light is transverse. there are two mdependent possible polarizatiOnS of the electric field vector (or spin states) for any given wave. doubles the number of states at any frequency. Includmg this factor of 1. replacing L) by volume and v by Its more common symbol c, we have

\\bich

V

dNU} =

g�Vf' df

(C· 10)

The quotient dN(j)ldfis known as he density of states and is used in Chapter 9 "gas" of photons, which IS important to the operation of a laser. It is 10 study a "'Orm rei rating that while considering the number of states/waves of frequency

t

te f. we have not in any way addressed the q�estion of how much energy would be e�pecled at frequency f That was the POlOt of the previous calculation and is where the classical and Planck's theory differ. dU(f): Spectral Energy Density

We now combine our results to determine dU(j) via equation (C-I). Equations (C-2) and (C·I0) give u!'. the prediction of classical wave theory.

dU(f)

=

"V

g f' df kBT --;s-

classical wave theory

Planck's correct result comes from equations (C-3) and (C-IO).

Oen:s.oly of states EleClromagneto

radiati:>n

p'

.1:. . ;.,"

I e " _Iy :Je:·,1Y

Fit.". C." £nervy pel Wlil frequency.

d..nit'a! won"l Planct.

!

ElIperimentaJ datil, pnd PIllnck's theory

f

dU(1) �,::h/�_ 8"vI2 dif �" fI ',T

_

c'

(C- I I )

1ltt two are planed in Figure C.4. Both initially increase because the number

of Slates dNU) increases asf2. Bul Planck's ultimately falls to 0, because al very higb frequencies. there is nOI enough thermal energy 10 produce even one

pholon-and it is nOI possible to produce less man one. Mosl important, it is Planck's theory thai agrees with the experimental evidence.

App end ix 0:

Solvi ng fo r the Fourier Tra n sfo rm

Here we derive equation (4-22), extracting A(k) from equation (4-2 1),

+ 00

J


_

ly by

A(k) eil" dk

(4-2 1 )

00

[

e-ik'x, then integrate over x from

The crick is to multip AfteT some rearranging on the right side, the result is \ es. ;,·d

L

L L

+00

+,

",(x) e-,k', d> �

A(k)

+'

]

-b to +b on both

e;(k -Kj, d>

dk

(0-1)

lei us concentrate on the integral in brackets.

J "

ei(k -k')xdx

==

ei(k- k')b + e -i(k- k'}b k')

i(k -

�

2 sin[(k - k')bJ k - k'

(0-2)

-,

Note that we have used the relationship sin z =

K).

from the Euler formula (Appendix To obtain our final result, we will let

(e+h - e-iZ)/2i that follows

b approach infinity. but the behavior

of the function on the right side of (0-2) is a bit peculiar. Figure 0.1 shows Ihis fUllction plotted versus k for two different values of In either case, it is just

b.

+

21(k-k') times a sine that oscillates back and fonh between I and I ; but the larger the h, the faster the oscillation. Also note thai as increases, the function beComes more sharply peaked at k

=

b

�

k'. ln fact, by I'Hopital's rule, its value at

b

k k' is 2b, as indicated in the figure, so it becomes infinitely tall there as of the right side of (0-2) over approaches 00. Now, it happens that the all k-the area under the curve-is always 21f, independent of or k' (chang­ ing k' just "slides" it right or left). Thus. the infinitely tall peak at k = k' muSt ==

integral

b

effectively become infinit�imally narrow. At all other regions on the k-axis, it oscillates infinitely rapidly between

+ I and - I . When mUltiplying any reason­

ably smooth but otherwi5c arbitrary function-such as A(k}-il would cause

the product to average 10 zero. [n other words, in equation (0-1), it really doesn't mailer whal A(k) is anywhere bUI aI k = k'-its values elsewhere will be

0-1 -

-

-

....,. � -----� ,-- �

Flg...,. 0.1

• = k'

An arbitrary �mooth function A(k). and one sharply peaked at

A, b becomes large. 2 sin[(k - k')bJ/( k - k') becomes a narrow

but tall peak. with rapidly o�cillatlng "wings."

2b\---'''-... --j .

2

k-k'

Arbill1ll)' function

I�e b -..... '. 2 sln({k-k')b] k-k'

A(k)

\mall h·....··"

,

averaged to zero. II might as well be just the constant value A(k'). which can be side of (0-1) pulled out of the integra1! Therefore. in the limit b -+ co the becomes

+� -00;

A(l) !;m

2 s;n[(k-k')b] k-k'

b.....OO

dk � A(k') !;m

b-OO

� A(k')27T

+00 -;00

right

2 s;n[(k-k' )b] k - k'

dk (D-l)

where we have used the fact, as noted, that the integral is 21T. Having allowed b to go (0 infinity on the right side of (0-1), we do the same on the left side, arriving al

+00 ;-00",(x)e-'"x,,-,

� 27TA(k') or A(k') �

+00 � ; ",(x)e-iKX dx _00

2

(D-4)

Now that the earlier k is gone, we can choose to omit the prime, giving us equation (4-22). Having shown what we sel out to show, it is worthwhile to take a second look at our "peculiar" function. In equation (0-3), it might be a function of any arbitrary variable-for instance. z rather than k-and the function it accompanies in the integral would also be a function of z. Dividing both sides of (D-3) by 21T. we can thus say that

+;00 -00

{

A(,)

Hm

b_oo

s;n [(, _ " )b]

l7(Z - , ) ,

}

d,

�

A(z')

(D-5)

The function in braces has an unusual property-it "sifts" out the value ofA(z) al the point z' only. It does this because it goes to infinity al Z = " but is zero

App.ndi. 0

\

This may sound rather pathological. and indeed it is. but it is , f)'where �IS is in fact the way we represent a particle whose 0 p ySiCS and ,\c! used 1 nonzero only at the point z == z'. ycisel (I1ucn . lOwn pre It is called the IS }u . . . We leave further study of thiS 'tion n tio wen d ne function to a fu

I"'" pirac

delta

....el course. ef-Ie

. n �,g

.

Solving for the Fourier Transform

0-3

Appendi x E:

The M o m e ntum Operator

of the momemum operator rests on the assertion that the derivation

A(k) is to wave number Ie what .p(x) is to position x: a prob­ '!be er traJlSfonn Fo� aJllplitude of finding a value-of either x or k. The symmelr)' in the abli t iy and k appear in equations (4-21) and (4-22) is certainly suggeslive. ,

",ay x

,(.<)

s

J A(k),'"

+� %

d'

A(')

(4-21)

-�

%

1 _ h

J

�(x)e-'" dx

(4-22)

-�

more, much of what Section 4.7 tells us about the Fourier transform rther f" 1" particular. Figure 4.26 shows a Gaussian function that progressively 3gree . plot to a very D� from a spiky function of well-defined x in the bottom chang..... . well-defined wave number/momentum In the top plot. of wave plane d Fourier transform is a spiky function of k. Just as \�(x)12 is the probabil­ finding the particle at x, V\(k)\2 is the probability per wave W d per istance of of finding the particle to have wave number k. It follows thai if

b:se

��mber

..,e write X

J

+� %

xl�(x)I' dx

-00

then we must write

Ii

%

2"

J

+�

klA(k)I' dk

(E- l )

(We take up the 2 17" later.) Because p.- = Iik, to know k is to know p. However, we wish to calculate using q,(x), not its Fourier transform, so we eliminate A(k) by insening (4-2 ) and its complex conjugate into (E-I).

2k

or, rearranging a bit.

Ii

%

� J J J , ,, '(X),+'''' (,,(xViI") dx dx'dk

2

+00+00+00

(E-2)

-00-00-00

E·' Ii

.. ...

. .

... .. " -

-

Lef us iJaItCa- by pms die Ierm in perenrheses involving x.

Assumina: char I/I{x) is well behaved. faJling to 0 at x :> zoo, the fml teon on the

riabl is O. When we reinsert the second term inlO (E·2), we see that k cancels.

or

(E-3) Now the integration o..'er k may be carried QUI, Writing it explicitly as a limit. we have

/ t'+&\l' ••

lim

b_1

"

-lj elk ::: 11m

b_OO

-.

- I ff

Thus. (E·) becomes +00+00

k = -

21T

"" (x')

-00-00

{

k

=

'f""['f'''' '(X'){

-00

_00

'"

--'7x'--::::1: x

' (2 sin[(x' -x)b] _ , ,_ - -'_ ") .
-

;(x' x)

b-I)()

}

2,in[ (x' - X)bJ d"'(x) ' . (-i) -o/ dx b-OO X x dx dx l im

Grouping together the functions of x' and gives

bUI here we must be careful.

'in[

moving the 217" into the integral then

(x:

x)b J ._"" ,,(x - x) lim

-

}

]

dx' (

-i) dop (x) dx dx

(E-4)

JnlegnJs having the fonn of the .:I integration in brackets are discussed Appendix D.

Using equation (0-5), we arrive at , ""

- f k =

-""

d",(x)

'" '(x)( -i) � dx

in

(E-l)

. ••tor 1"" n..-. Appendix E Tb� Momrnlum ....

Now. regarding the factor of 27r in (E-I), for that fOffilula to make sense, the average of I must be I .

J

+00

2".

I

-00

y

�( k) 12 dk

� I

(E-6)

To carry oul this integral, we would follow the same cou rse as before. skip­ the integration ,

g onl

by parts that cancels to the

��'e one-and replaces "'(x) by

(E-S). we would have

�

J

-

k--equalion (E-6) doesn't

j times its derivative. Thus. In place of

+00

"" (x)"'(x)dx

-00

secause this must hold. equation (E-6) must also hold, so we see that equation (E-!) "needs" the 27T to satisfy nonnalization. Finally, multiplying both sides of (E-5) by Ii and usin g p =:: hk. we have

J

+ 00

P �

-00

"" (x)( -iii)

d'�;:)

dx

must applyaLthe(Noopete:ratoHadr -ili{d/dx) .to the wave function rjJ(x) in the probabll­ consIdered the full 'V( , t), l, integr we would have x �plOyed partial derivativeswewith respect to x, and to be genera l, the operator �s usually written in that form.) We conclude that to calculate the expectation value of the momentum, �e

E·3

Appendix F:

T i m e Evo l ution of a G a u ss i a n Wave Pa cket

tafliog from a general wave group---a Sum of quantum-mechanical plane S ' s -ou r goals here are ( \ ) to determine what coefficients of the waves

�'uce

�roup u

that is initi�lly Gaussian in shape and (2) to determ a ine the that Gaussian group. OUf general wave group of ion t l o given in ev is time

·21). �uation (6

J

+ 00

'V(x. /)

�

A(k)eICIa-.,) dk

(6-2 1 )

-00

the Gaussian form we wish it to have at 1 = 0 Equation (6-27) is

'Vex, O) =

C e-(x/2e)2 eiko>"

(6-27)

The �t step is simply to evaluate both sides of (6-21) at 1 = 0, with (6-27)

jJlSeJ1ed on the left.

/2e)l ei*tit = C e-(x

J

�

A(k)erh dk

(F-I)

�

this equation for A(k) is actually carried out in The task of solving

section 4.7. The conclusion is

We conclude that inserting this A(k) into (6-21) ensures the correct form at I ::: 0 and describes,

at least formally, how the Gaussian wave group evolves in

time.

J-00 :J;

+00

'V(x. r)

�

e-"(' - .,)' e'CIa - �) dk

As noted in Section 6.4, w depends on

k

(F-2)

differently for different wave

phenomenon, and. strictly speaking. without knowing that dependence. we can go no further. However. our wave pulse has a reasonably well-defined

F·'

,., '-"'ndU: F

TIme Evolution 0(1 G.UWIII Wave PIckel

wave number/momentum. Being itself a Gaussian, A(t) is a peaked function wim its maximum occurring at to' Accordingly, mos! of the contribution to . integral occurs near .to' and if we replace wet) With a power series expanded about the point *0. we introduce linle error. Thus, we write

th�

w(k)

where

�

w(ko)

+

s(k - ko)

/

dW(k) s a dk k o

aod

+

jD(k - ko)'

DE

d'W(k)

Inserting (F-3) back info (F-2) yields

"' (x, I) =

-

I�e-e2(k

-

dk2

*1» 2 ei(U- ",,(to)I -S(A: -�)I-

(F-3)

/

(F-4)

�

fD(k - toF,) dk

�

A change of variables, Z =: k - ko> simplifies this expression considerably.

+00 J �e-c':2ei(z,r + Y-W(kO)I- SU-�D:ll) dZ +00 ei(.tv-IoI(�)I) / e-rz2 ei(:.x-sv - � Dr') dz v:;; +-0000 �ei(V-W(ko)t) J e-(e1 + fiDI)r e(ix - ist)zdz -00

"'(x, I) = =

=

- 00

Ce

The remaining integral is a standard Gaussian integral, given on the text's inside cover.

Rearranging and using simply Wo for the frequency corresponding to the cen­ we have

tral wave number

'I'(x, I)

=

Note that al

t

[ -(X

ko'

c

VI + iDl/2e' exp 4e'(1 =

0,

- Sf)' + iDI/2e')

]e' o'- W'; (k

I

(F-5)

this function is indeed the Gaussian wave pulse of

(4·23) and (6-27), but now we

see

how it propagates in time. The complex

exponential on the right is its "oscillatory part," including explicit time depen­

ko and %Iko' The

dence. It is a plane wave of the pulse's approx imate wave number frequency Wo, and if moves at the corresponding phase velocity

other exponential is the "Gaussian part," which modulates the plane wave.

Appendix F

TIn\c Evolution of a Gauu.ian Wave Pal:kct

um occurs at the point x = st, so it moves at the speed s, the group I� maxim The Note that what we have is essentially what we see in Figure (light red) move at one speed-the phase crests velocity-and the �i1Iatory

6.20.

\"Iocity.

uffiple Gaussian envelope (dark red) moves at quite another. To calculate the probability density, we note that

+ iOt/2£2

- iOt/2e2

�

I + 0','/4£4

and

I + + iDt/2e2

I

- W'/2<'

�

2 + 02t2/4e4

reader can show that the probability density for wave function From these the

(F-5) works out to be

I'I-'(x, ,)1' �

cz

VI + D2" /4<4

[ -(X

exp

- SI)' 2<'( 1 + D"'/4£4)

]

The oscillatOry part and its phase velocity have vanished, irrelevant to the opagation of the particle. Only the group/particle velocity s remains. pr

F-3

Appendi x G :

The O p e rator fo r L2 tknVC the operator for the square of the angular mo mentum, i.�. we �gin 1"... L r X p, whose components are ""

==

"�

, determine the operators for these quantities by inserting the basic momen­ position operators. Exercise 7.40 car ries OUf the steps for turn and L,. Expressed � I'mates, the resuIt . , IS . In ,pherical polar coo" .

L,

""

, -jft­ ,�

lei us do the same for Lx and Ly.

iII

=Y(-ih2-) - '(-ih2-) oz

oy

(

)

i)' = , -ih2iJx

-

X(-ih.2-) dZ

To e�press these in spherical polar coordinates, we tran sfonn the partial deriv­ n[i�es using standard rules of partial differentia tion.

.

L

<

['

'" - h i y-

iJz

- z, iJy

1 (G·I)

G·, -- -- -- -- -- --

--

1i_,jn .l..!!. 'I' ... lsm t 1 tP 02,

ng

i} �

we obtam

-11 \"

�O"N squaIl... I

�

.. -,.

iJ8

+ cot 8 cos

+ "

¢

iJ (j

- <.,

==

s

"

_

Cot 8 cos Ib.i.. iJ�

i16

••

+

<",'

_

-

,to

t/J

iI't'

+

.t

-II2. iltP +

.. � -"

L2 �

.

'"

"

- - II·

cot tJ

- - + cot (j iI�

+

• - + cot fJ +

(cot� ,

· rlfJ

,


iJ8

. . . · ,fa' l� a' f a' � j r"' + C'\C. 8- j

L1 L;, + L.' ==

d 'l'

- <.t "".

tP ( p ,

"" -,.

•

4J .0 i16

d
:e�0I :) . ""¢ ,�� ¢ .�)] -" �.' � '".:)�"'¢! '",�) " [cosl cOl e sin � � 05 t/>�) � '" '" .�� ..�) + <01' . '" .:�in .:)] . ¢ <'" ¢

Li '

.:)\"(Sin ¢ :(�in tP�)

C 8 08 _ ot cO', rJ,

'

(

+

;,e

I)

"'

i/rJ,2

'

W

)

,

C Ol

j

a' 8-

w-

Ial1 c..c B - - �Ul9- + �8a' it(J iI6 iltbJ

j

_ Ii, a'

iJ4tl

_

)

Appe ndix

H:

E n e rgy D istributions

Our goal here i s to detcnnine, for a thermodynamic system in which

E,

N",

N

parti­

cles share a total energy the number of panicles, expected to occupy individual-particle state 11 whose energy is We stan W]t.h a statemenl simi­

lar to equation (9-9):

N"

particles probability of being in state II

M

=

Ell' W(N - Nil' E - E"N,,) LW(N - N" E - E"N") N,

The numerator in equation (9-9) was the number of ways of disrributing the remaining energy - fI, among the remaining [ oscillators. Here, the numerator is the number of ways of distributing the remaining energy among the remaining panicles not in state n. It naturally has two argu­

N-

E - N"E"

N - N"

ments, but we assume no explicit form. The denominator. as always. is the total number of ways-a sum over all possible values of of the number of ways For now, we assume the restricted. panicles to be indistinguishable; we with

N"

Nil Nil

have not concerned ourselves with which panicles are in stale n. And we leave the limits on the sums un�pecified, deferring until later concerns about what val­ ues is allowed to take on. The average number of particles in state n follows the usual rule:

L NIlIV(N - Nil' E - EIlNIl) -"'L'----W-(-N---/'"-)­ N".-E --E-N - Nil IV Nil

N� average number of = particles in state II

(H-I )

N,

We assume that

Nil N,

« so that the olher particles constitute the of the system.l , or the va�tly larger part Given that is unknown, it reservoir how to proceed, but a power series in the " small" is the key. obvious is not First we take a brief mathematical aside. Suppose we have a function of twO arguments. each of which is a function of x: JTa(x). b(x)J. Consider the

power series expansion.

[

I a(x).

+

b(x)] = 1[0(0). b(O)]

(,u/ia(.,).b(x)] d

)

., <"'0

+

I ( d'

2 dr' /[a(x).b(x)]

)

i! + . . . x=o

'I! would ....m . tha, ,n m:lOy Q>e5 "" - "'. oJ; ...,.,1)' fI,,/Iaq;e, In room',empe""""' hydrogc:u g-. for i",u",,;c. if ,,_ N .. Lhe pvund ""'� .tale. !hen N - N• ....I.. d be nemrly O. for nc:arly all lTl\>lcculc. � In Lheir ground �;� _ ,ro all , • • • • . Bu! ,,,,,1,,, ,,,,,,,0/ eneJt!l ,11.1) I� JW1 of the tySKnI- with .. mcrg) i, e,cl"wlged and ",,,,- �pnf JIIVP"" ert1C' 4efillt" the conuooa � An) chatF ID the number of paru.:1n tII :t pvao oQIc f1IYibD­ _
H-1

.... .. _ .. . .. .. so

To """ ""' ...... .. ...u die follow"" rq>bcemenh

• •

/ - Ia M '1o . _ .\

"

'.11

.... l.

£..

N.

I fi ll

,kJiOttnlh' ('( Q aDd rhr dJo.;r • • ''. it IQp:aL f{f' .. � the urWl tbJtlJ· fhe .. . _ aha tCJlIiNco. triq: Iht atlU1DIre'DU of K Ul t\.jUlllu,m ac}(j hm�;bOrb 01 dw unaD IfwII Tbt � MUpllunz " hmtt I" the Ioguntlun vI I'" The '""-. .. ... � 01 .11' Ilhrra\• 1D\'ohe ... om"'lQalun.ill IJCllln. 'urh • .. A' lId .. hea • b rrpbi.:rd by .' '... .... hen:' « N. \uch r':l�ln IrIId to .., ,¥OJW'fIWIIy _if,h A� : An up.«nttal ti. l�t ...'\'Iur�. nul hlk"iI, lQ IU ap.aJ " 1a fl"-ftI 01 \� and lor a' the IUltM I('ml ,,\.uIJ tit oil) "'-'fuI IJ'J'CUttmiJl1oll 8U1 cbt Ioprithm of an nponenli.t.l H hnc1U' Rcrl.k:ln, In

(H·')

,\.

J

_ (tt Z�

UJ\1111 btm t\� • ,�

• O. the

IWt) rtmobnlng pJ.r1JJ.1 dcm-atl\e<,

� �'" the hdl.\ ard F.. 'k' lhc'>. � pntpt"ttIC' "

f I� U\t.'rall "I)'ICnl, (If' rt':'>('r.

nlU r-nm li:iirulaXlI9-21. \IoC hit,.:" In ,� • 1),1f1. '0 th(o,c leml\ arr pa,,,al ,,k!m. Ml'� of !he ,utr.t11 �)Utm' enfn'p} III (he til'! \10(, gl\t" the 'ymbol R.

"I{n(.� E)� .\

•

u

\1dJou&b we \Iotll not uudy It (I ,.. pfllJ",ttivnal

(H·51

lo the chemical potenlial of

lhr WI'rot T'tIt � propnt)'. ht , .... C'\·('r. !\ qUilt familiar By definition (9-4),

�lnr"( \',E)J n:

•

"(.I I.) .f:

I •

,.T

(H-61

Eocrgy lh<;.tributIOnl., depend on tempera_ hft. and II i, hm' ttw: ICnl(l('f.Jlure tnlel'\ the: pKlure. Sub'ltituling into (H-n Do DOC O\<'f'fcd thl' unp.wunf "'tep.

...... ... ...... ...

,

I

.

...

., .... .. ....... - . .... . .. ...... ....... ... , ,... .. .. •

•

"' •

•

\.' E

..,.... ..

...r ..\ 111& finished

� .... .

... t

J"l-l\\ tr �fl6 u,p.uwc:III and ,1MlUd now IR;\('ft thl.} ba'l ,n o ,H ' )

OW

IN

..e

l

.'trqe num� of

toile' "

rarth:h , .."!I. tn

...: ,\." (.\0;, 1:').-

ell.

kmpoRlDl

r ....Jt.T

,

� W(:\. t.,..

caN. lJ'.'.T

,

" .\, r � ,

t."'....'

" '.

\11 ')

_ thai \\\,\'.. 1::1. "hll.:h I" I1lfje�nd(nt ll' thC' "Ummatilln InJco" ."ft' hili ..lin cckd 11" "(('� .\ 111 ...\) C'"..,I) O\C'rloo\..ed Nt o' .�at I n'Jxullln:t. h'l' WtN. I::l ..

:(tIta1ru .11 the me .....) lillo;tona" and lither ,'8dllr. related hI J''''ln�nlnl � In ttl<- llH-ra1l ,)..Iem That It nted not C'\cn be' \..m w.n 1nt"n Ihllt fM l __[wlt." and Frfflu·Dinu diHrtburlOf1,\

� SHlr," 111\'01\''''/ \\e

upplll'abit «Iordlt

('ff'

t

of rhl

mpllf) the wlUlo.. ahead ".Ih a ..tamlanl malhcmatKal tn�\... That t n

ph III the fllll(.l\1im� 1, lelt "... Ii menial ('eJ'CI\(: Ilvera�C' numbtr of partn:-Ic.. In 'itGtt II

(L

el

()a

N.

- - - In

71

t

,,11&1 'At have thu\ far referred 10 a.... "the average numbtr 01 panll;le.. 11\ ..tate

." is \0 fact the occupation number, Naturally. It ,\ ..ubjC'ct tu nUCluiltloll'. hut

(

, �"mc.. Inl'reillilOgly preclo;;e as numbers grow,

a I L' oN, oa n N •

'''ol,T)

(II �)

Onl) now rnulit we di!ootlnguish between fermion!l and boflon" for tx.)!oon ...

I:N•• where :

-',ll.'

� a geometnc \erie\. Terms get small very quickly, '0 Il Wilt Introduce little N i\ unrt..tncted, The ... um I� of the form

1\ '

0

•

and \\

error to e:(tend the ... um 10 Infinity. enabling us to u\e the well known f(\ult.

!:-o:'"

- I/{I

.1'1£,)8' = -

o

ela

(

0:). Thus.

In

o In( I oa

' L (e N ,0 ,

)

T)N,

O - fJ••

a

aa

= - - In

(I

,

0

• .. . ' ... m m " '..

...

.. ,.

7 H a.., = n

I

NeE.)..

•

C"'+EJtar

_

Fer fermioas. Nil is restricted Co the values 0 and .......

N( E,)FD

=

Willi � . B.

- ...!.. In

iJa

( ± - Tt.) N -0 •

(,-0

EJ••

_

I

I . so the sum has jusl Iwo ...!.. In( I

+ ,-0

aCt

- Eo'••T)

these are expressions (9-32) and (9-33).

The Boltzmann Distribution The foregoing derivation a.IOSUmed indislinguishable particles.

Which panicles

occupied swe " never entered the picture. However, there are N!INII! ways of switching specific particle la�ls between state

n

and the rest of the Slates

without acwaUy changing Nfl' and these are indeed different ways

if(bu1 only

if} IIw ptJrticlu orr distinguishab/�. We may account for the Increased permu­

lations of particle labels simply by including N!lNn! in all sums over Nn. II car­ ries through 10 (H-7), whereupon

N!

cancels lOp and bottom. so thai (H-8)

becomes

-

If we again allow N to extend 10 infinity. we �ee thai the sum is of the " �

form

I t"/m!

M-O

=

r. where z is e-a

[,PaT

Thus.

or

X(E,,)BoItt

=

r'+ E"Ik, /

With r" • B. this is BoJtzmann distribution (9·3 1 ).

the Photon Gas : ... . ..... d I IIIi..... ooIeIy by ... .. ·-- willi wbicb iI i-. 1IIdn!p ... ..1" of the matter by eaau (II lbis _ ..... ... die pboIoa au -'¥Yo

m (H·S), we """"1.. ",,, .

_

_ III .. ". - I ... ... ..... .

AP pe

ndix

I:

Pro p e rt ies of I sotopes

nt Elei1\e

,

�

Symbol

H

I

Helium

2

Litbium lium Beryl

4

Boron

j

carbon

6

NitrOgen

7

,

9 10

"

i2

Oxygen

Auorine ",,"

H, Li Be

B

Atomic Mass

• I

\.007825

2

2.014102

3

3.016049

3

3.016029

4

4.002603

6

6.015121

7

7.016003

7

7.016928

8

8.005305

% Natural

Half-Ufe Abundance'" (lnt:ay Modes)t

99.985

oms \ 2.32 yr (Jr) 1.37

x

10-6

�IOO 7.S 92.5

53.28 d (Ee)

""0.01 fs (20")

9

9.012182

10

10.013534

10

10.012937

19.9

II

1 1 .009305

80.1

100

1.52 Myr (W)

12

12.014352

II

1 1 .01143

12

12

\3

13.003355

14

14.003241

12

12.018613

13

13.005738

14

14.003074

99.63

IS

15.000108

0.37

IS

15.003065

\6

15.9949\5

99.76

17

16.999\3\

0.04

18

17.99916

0.30

19

19.003577

F

19

18.998403

Ne

20

19.992435

90.48

21

20.993843

0.27

22

21.991383

0.25

23

22.989767

24

23.985042

78.99

25

24,98537

10.00

2.

25.982593

1 1.01

c

N

0

Sodium

N,

Magnesium

M,

20.2 ms (Jr) 20.3 min (EC. /3+) 98.9

l.l 5730 yr or ) II.OOms({r) 9.97 min (j3+)

122.2 s (W)

26.9 s (13-) 100

1 00

5

100 26.981539 27 AI Aluminum i3 g half-life-are mutually cxc!u�i"c_ . percenI nalural abundance and being unslable-bavin a n I!aVlng a • Us a u, y. 0 on'um.232. and uranium-234, -23 , and -238 are exceplions. . m--4. th J>ot8SS 1u capture. SF §lands for spontaneous fiSSIOn. s for elecuun st.aIld EC t

\-1

.. .... ",, . f ..... ", .. -

z

27.976927 28.976495

28 29

29.97377

3D

IS 16

3/

p

PhMphOrus

J2 J2

s

Sulfl.lf

JJ 34

"

36 17

CI

Clllorine

"

J7 36

Argon

III

"

40

20

Scandium

So

22

Ti!anium

Ti

24

25 26

27

Cllromium

Manganese

Iron

Cobalt

Mo

F,

Co

0.02 75.77

34.968852

24.23

36.965903 35.967545

0.3]7

37.962732

0.06]

39.962384

99.600

0.01 17

41

40.961825

6.7302

"

v

Vanadium

34.969031 35.96708

9].2581

40

34.99523 39.962591

41

40.962278

42

41.958618

96.941 0.647

43

42.958766

0.135

44

43.95548

2.086

45.953689

0.004

48

47.952533

0.18 7

"

44.95591

100

46

45.952629

8.0

47

46.951764

7.3

"

47,947947

73.8

49

48.947871

5.5

49.944792

5.4

50 23

0.75 4.21

33.967866

38.963707

46

21

3 1.97207 32.97146

39.963999

39

C,

Calcium

14.28 d (fJ-) 95,02

'"

K

pO!lI5Sium

19

30.973762

31.973907

50

49.947161

0.250

51

50,943962

99.750

50

49.946046

"

51.940509

4.345 8].79

53

52.940651

9.50

54

53.938882

2.365

55

54.938047

100

54

53.939612

5.9

56

55.934939

91.72

57

56.935396

2.1

"

57.933277

0.28

"

58.933 198

60

59.93381 9

100

1.26 Gyr (P-. r r , Eq 50 ms

(fJ+)

0.103 Myr (EC)

Atlp·ndta l PIopIdia ol 1L11II'

8Jeldent

t

� �

57.935346 59_930788 60.93 1058 61 .928346 63.927968

63 65

62.939598 64.927793

69.\7

64 66 67 68 70

63.929145 65.926304 66.927 129 67.924846 69.925325

48.6 27.9 4.1 18.8 0.6

'"

G,

69 71

68.92558 70.9247

Gallium

)1

60.108 39.892

G,

70 71

69.92425 7 1 .922079 12.923463

21.24

Germanium

)2

p.rsenic

Jl

Selenium

}I

Bromine

)5

Cu

Zn

Zinc

J1 18

)9

40

Rubidium

Strontium

Yttrium Zirconium

68.077 26.223 1.140 3.634 0.926

30.83

73.921177

76

75.921401

7.44

A,

75

74.921594

S,

74

73.922475

0.89

76

75.919212

77

76.919912

9.36 7.63

78

77.917308

23.77

80

79.91625

49.61

81

81.916698

8.74

79

78.918336

50.69

81

80.916289

49.61

78

77.9204

80

79.91638

81

81 .913482

1 1 .6

83

82.914135

11.5

84

83.9 1 1 507

57.0

86

85.910616

17.3

91

9 1.926270

Rb

85

84.9 11794

72.17

87

86.909187

27.83

S,

84

83.91343

0.56

86

85.909267

9.86

87

86.908884

7.00

88

87.905619

82.58

94

93.915367

Y

89

88.905849

Z,

90

89.904703

51.45

91

90.905644

1 1 .22

92

91 .905039

17.15

B,

-......

(Ilo
27.66 7.72 35.94

73 7'

KrYpton

36

� N.tunI Abu�·

58 60 61 61 64

Ni

II

Copper

Atomk Mus

A

SymbOl

100

0.35 2.25

1.27 min (jr) 100

...

.. '"

_ _ _ _ _ _ _ _ _ _

96

96 97 100

44

Ruthenium

49

50

TIll

9.63

4.2 Myr (J3-)

R"

" 98

97.905287

"

98.905939

1.86 12.7 12.6

100

99.904219

101

100.905582

17.1

102

101.904348

31.6

104

103.905424

18.6

103

102.9055

100

102

101.905634

1.02

104

10).904029

11.14

105

104.905079

22.33

1 06

/05.903478

27.33

108

107.903895

26.46

110

109.905167

1 1.72

107

106.905092

5 1 .839

108.904757

48.J61

106

105.906461

108

107,90418

Cd

5.54

95.907599

109

Ag

S,

99.907477

97,907215

Pd

I,

24.13

98

Palladium

Indium

9.55

91.905406

To

46

Cadmium

16.68

14.6 min (p_)

Rh

48

95.904678

100.910345

Rhodium

Silver

15.92

101

45

47

9.25

94.90584 96.90602

98

TecMClitml

14.84

93.905085

94

"

.jJ

1 00

91.906808

92

M,

Molybdenum

M'Odts)t

2.80

92.906377

9J

Nb

HaII'Lili, (Deca

17.38

93.906314 95.908275

"

42

% NaluraJ Abun d a n c-e·

Aromlt Mass

y ,,-,,:90m:4--'lii,==--.:::��

s.......

--

Z

1.25 0.89

110

109.903005

12.49

III

1 1 0.904182

12.8 24.1 3

112

J I 1.902758

113

1 1 2.9044

"'

1 13,903357

28.73

1 16

1 /5.904754

7.49

113

J 12.904061

115

1 14.90388

12.22

'.3 95.7

112

J J 1.904826

0.97

"'

1 /3.902784

0.65

115

1 / 4,903348

0.36

116

1 1 5.90/747

14.53

1 17

1 16.902956

7.68

118

1 1 7.901609

24.22

119

J 18.90331

120

1 19.9022

8.58 32.59

122

121.90344

4.63

12'

123.905274

5.79

132

1 3 1.917760

40 s

(p-)

_milK .

f;1tJ11oenl t

�

'2' 123

120.903821 122.90421 6

To

120

1 19.904048

122 I2J

121.903504

.24

123.902823

.25

124.904433

126

125.903314

128

127.904463

130

129.906229

125

124.90462

127

126.904473

124

123.905894

126

125.904281

128

127.903531

129

128.90478

130

129.903509

13.

130.905072

21.2

m

13 \ .904144

26.9

'34

133.905395

10.4

136

135.907214

8.9

140

139.921620

c,

133

132.905429

135

134.905885

B,

\30

129.906282

0.106

m

1 3 1 .905042

0.101

\34

133.904486

2.42

135

134.905665

6.593

\36

135.904553

7.85

137

136.905812

1 1 .23

138

137.905232

7 1 .70

141

140.914363

\38

137.907 1 1

\39

138.906347

99.9098

\36

\35.90714

0.19

138

\31.905985

0.25

140

139.905433

88.43

\42

14\.909241

11.13

\41

140.907647

.42

141.907719

143

142.90981

12.18

144

\43.910083

23.80

145

\44.91257

\46

\45.9 1 3 1 1 3

17.\9

.48

147.916889

5.76

ISO

\49.920887

5.64

145

144.912743

Iodine

Xenon

�

cesium

55

Barium

56

51

Lanthanum

58

Cerium

59 6D

61

HaIf·Ufe

(Decay ModeI)t

Sb

1eUurium

53

% Natural

A....,...,.

"

,�

Alomlc Mass

A

symbol

Praseodymium

Neodymium

Promethium

X,

L,

c,

p,

Nd

Pm

51.36 42.64 0.095 2.59

122.90427 1

0.905 4.79 7.12 18.93 31.70 33.87 59.4

100

d (EC)

0.10 0.09 1.91 26.4 4.'

13.6 s (In 100 2.3 Myr (Jr)

18.3 min (Jr) 0.0902

100 27.\3

8.30

5.98 h un

Propertia of llClllDl*

....

1-6 _dlx l Properties of Isotopes

Z

Element

Symbol

62

Samarium

Sm

14H l 1998 146.914895 147.91482 148.9 1 7 1 8 1 149_917273 151919729 153.922206

3.1 15.0 1 1.3 13.8 7.4 26.7 22.7

144

63

Europium

E,

J51 153

150.919847 152.921225

47.8 52,2

64

Gadolinium

Gd

1S2 1S4 155 1S6 157 1S8 160

151. 919786 153.92086 1 154.922618 155.9221 1 8 156.923956 157.924099 159.927049

0.20 2.18 14.80 20.47 15.65 24.84 2L86

65

Terbium

Tb

148 159

147.924140 158.925342

66

Dysprosium

Dy

1S2 156 1S8 160 161 162

1 5 1 .924716 155.925277 157.924403 159.925193 160.92693 161.926795 162.928728

67

Holmium

Ho

152 165

68

Erbium

Er

69

Thulium

Tm

70

Ytterbium

Vb

lutetium

Hafnium

Lu

Hf

163.929171 1 5 1 .931580 164.9303 19

0.06 0.10 2.34

0.14

164 166

165.93029

33.6

167 '"

106.9.n046 167.932368

26.8

1 70

169.93546 1

14.9

169

1611.9342)2 167.I)J11194 )6\).934759

171

170.936,13

2.37 h (Ee, a)

. I ()()

1 6 1 .928775

'"

I . O h (/3+, EC)

18.9 25.5 24.9 28.2

163.92919X

170

Half-tife (Decay Modes)t

1 00

162

1.61 22.95

JOO

0.13 3.05 In

J 7! .93fi378

21.9

17J

1 72.93X208

lfi.! �

174

[ 13.9]�S59

31.8

176

175.94256-1

121

175

I 74.94077

97.4 1

176

1 75.4J42679

�.59 IJ.l62

112

72

% Natural Abundance·

147 148 149 ISO 152 154

163 164

71

Atomic Mass

A

174

113.940044

176

175.941406

5.20fi

177

176.943217

1 8.606

178

177.943696

27.297

min

({J+ , a)

\

\tflmlc \t_

I

----.

'f·J '" r.

w

PM 94S8 1? 179.1Jr4tJ'i.4�

"0

'-1_14146_

181

. 8<'J4:99_

1 110

I 19.946;m

.2

1111 9-+3:12

183

R.

1 84

829M2 10.1509:')1

186

IS' ��3'\"

185

184.9�2Q51

�1

186.9!W'44

184

(hrni m

plaliollm

I,

PI

18' 1}�)8',

18'

186.95514\

188

IX'"' 95586

"

188.'�5\(

,'"'I

l 4,2x '0.7 ISh

3i.,uJ f1::,fIj 0 ')2 1.58 1.6 IJ.3 16_1

L H.96."67

41.0

191

190.96C'M

19)

,,

1929629' '''

62....

'U9 917

I} I .9610:9

I'H

19\ 96]�55

193

1�.964"fl6

196

195.%4926
869

11

1 16 %6<4:

196

19� %"<w:r

198

19' )66143

9.9-

1 09

16.!C�

Ill;

os 16831 ·N.�:

>'ll

� 9'c.:r :' 1 .9�06P

.,;.0, q'I' U,7

100 0_1"1

HID It i8

?'> 86 6,g"

.!oo )�l�:'

.,

13

0 12

�,

191

lC

)t

9IJ.9K!S

26,

I �2

H.Ir·I.If�

ll)ecay M Oda

om ;:

n_1SJS.4�6

"

C

Ilfl29 }� 1 00

lO

'Q8

(WId

!S3.1524!S8

186

89

lnJiUJD

� Natural \buGdannt

I.tad

81 murn

'0'1

N� 9":"oI.4ItI

111

;;')7 9St9�

"

:a·.cr710;

16

l� 'l7.w.t

..,�,

!Ott I}"�!r'� 20" IjI"66:--

'10

J),J 9H41bl

,

� 1 1 Q9Ur"l

"

� I J 'l99M

,00

208 980174

�IO

109 'Il14095

" �.u

3051m.JDfJj I

2� I 51-1

)00

22.6 yr tJJ

I

IO,6 HI/I I 21 ... 1/1 ) 5.oUl/i )

.. ..

.

, PI, .. ........

td -; d�___.,.__ -"�-=D=== ", ��� -; .....-r M.-

A.

211.99 1255

112

2"

..

212

.,

""wine

R,

.....,

..

"

Fru.;ium

"

R.dium

R.

116

lJJ.99SI 76 216.001889

2/0

1/)') 987126

110

110,017570

214

224.020186

3.66 d (Q) 1599 Yr(aJ

229

"

f'ru(a.,.'limum

'"

"

UranIum

U

2J1

9.1

227.02775 229.032980

2.11.03588 23-1.040946 235.043924 238.050784

234

234.042888

2.17

237.048 167

238

238.05094J

2"

239.052157

2"

244.06-1199

Plulonium

""

"

Americium

Am

24J

241061375

..

Curium

Cm

247

247.070347

97

Beflehum

Bk

247

247.070300

98

Californium

Cf

251

251.079580

99

Einsltinium

'"

252

252.082944

94

100

Fermium

Fm

257

257.075099

101

Mendelevium

M'

258

258.098570

Nobelium

No

'"

259. /00931

IOJ

Lawrencium

U

26()

260.105320

104

Rulherfordium

Rf

2"

26J.J08690

105

Oubnium

Db

262

262. 1 / 3 760

102

2!.77 Yr(p -,o: ) 6 . I S h (.8-) 1.04 h (,8-, 7.5 min (fJ-,

2JO,0�8550 232.038054

)

5.76Yr (,B-)

128.031015

231 '"

Np

NtpWnium

226 02.5402

1J.1.(}·H593

:m

)

228.031064

'"

238

55.6 S (Q) 3.8235 d (Ill, �_ 21.8 min (f3_

127

Th

8.1 h (Ee, Q)

22.1.0197B

228

nwnum

145 ffis (Q)

11.1

130 90

102 )'T(Q") 138.38 d (Q) 298 ns (
122.0/757

226

A,,:!Itt/um

J.OO9�

19.9 min (8-)

222

221r "

_ __ __

208,981� 209.9811148 21 r.988S42

210

114

"

__

lJJ.9'i8691

,..

-

' �(Dtto�Y�M�» _A_� � Halt-Lit,

� N.luraI

100

14 Gyr ell') 24.1 d (p-) 32.5 kyr(a)

0.0055 0.720

99.2745

245 kyr(CI') 704 Myr (Q) 4.46 Grr (Q) 4.4 d <.B+.

Ee) 214 Myr (Q)

2.1 1 7 d (.13-) 24. 1 1 kyr (0) 82 Myr (0"

SF)

7.37 kyr ( ) ct !5.6 MYf(ct) IA kyr{ct) O.90kYf{ct) 1.29 yr (ct, Ee) 1 00.5 d (ct, SF) 5 U d (ct) 58 min (ct, EC) 3 min (ct) 65 s (ct) 34 s (SF,

ct)

t

PI EkP't'

� Ii»

1 ,0

Ie'

.

lilfl 8ofU'i

1-Ia>-\l1l1Tl

l e lill n 1elln r.

�

Symbol

A

Atomic Mus

'" Nalural AbuDCb.oc:e.

S.

2.3

263 . 1 1 82

8h

262

262.1231

H,

265

265. 1300

Mt

266

266.1318

......ur. lDecay

MocI-.)t

O.M , \Sf. Q)

O.tO '\Q} 2 m\{al "'3.4 m\ \0)

AP pe ndix

J:

Pro b a b i l ity, M e a n , Sta n d a rd Deviati o n , a n d N u m b e rs of Ways

I lighl the main ideas arising. whenever we speak of probabililie� we h'gh , I fiere most equatIon numbers have an n, b. c, or d. The ad physics. Note that 10 [11

�m

not 10 sneak in more equations, but to emphasize the fact that. .... ,.a50n 1S ' Ih'mgs. we realty deaI Wit ' h ingly endIess ways 0f reexpressmg , Ihe seem

despite t.s. few basic concep ' " ' . . onIY a The first concept is �robablltty Itself A quantity Q IS the focus. �nd. 11 k . various possIble values-Q t _ Q2' Q3' and so on-each wIth liS 1a e on . P2' Pl, and so on. For example, the quantity Q might be the an bility-P 1 � Prob3 . ' · '· . of a pet house cat. AI an arbIl rary InSlant, It may be at a Window, Q I ; iocal1on . ' ' a probabl'I'lIy. e sleeping location. Q2 : or Its f00d d'ISh . Q3' each wIth its fa\'oril P, = 0.065. And there may be some P., = 0.780, 0.040, P = ce _ s.an L ' For ,m where the probability is 0 unless an eXlernal agent intervenes, such locaU'ons ' ' ht have a theory Ihat predicts W mIg the probabl'I"Illes. or he pet washtub. e observation, experimenlal but in either case, one thing i s on may rest i n: The sum of all the probabilities-the total probability of finding the a

�t ce:

cat somewhere in the house-must be I .

L , p,

�

I

(j· 1 aJ

Often probabilitic� are based on a collection of data. such as N exams. where the quantity Q is the Score. or N repetitions of an air-quality experi­ ment. where Q is a fluctuating pollulant leveJ. If value Q; turns up Ni limes out of a tolal of N. then the probability is simply

p,

=

N, N

which filS perfectly.

N, I I L, P. � L , - � - L N � - N � N N " N

On the other hand. Ni is often a theoretical number of ways of obtaining a

particular value. For example, suppose we have an office with four identical

cubicles and four workers. We define Q as the number of workers in cubicle I . As Figure 1.1 shows. of all the possible ways of distributing the workers. one

subset has three worker� i n cubicle I , which we designate Q3' and can be done

J·1

FIgun J.t ,·ubldes.

WilYi Q( arranging

r� �r . r

w()(k�r'i a. b. c. and d in four

Cubide J

J

� [�;-�

(j

::�

,-

(j "" .., , (jj) :(jj)

- - -,- - - -

-

, (j :@� -------

, (jj)

:(jj)� (j : -,-- . -

,

12 ways. Another subse[ has all four [here, designated

Q4' and this can be

done only one way. Wha[ are the probabilities? We need a bit more to go on. The mo.�t common addi[ional assumption is thai any way of arranging specific obJecb In specific loca[ions is equally likely. If [his were [0 apply to the work· ers (0 dubious assumptjon for real coworkers), then all 1 3 ways shown in Figure J.I and all the other ways not shown are equally likely. If we were to do an experiment. each way should tum up as often as any other. Therefore,

Q),

In which fhree workers (independent of their identities) occupy cubicle I , $hould [urn up 12 times as oflen as

Q4' simply because there are 1 2 times as

many ways to do it. II happens Ihat the ,otal number of ways is 256 (thai is,

44), so !.he probabilities

are

121256 and 1/256. Again,

Pi

=

NJN. where N is

now a number of ways.

Mean No matler how probability arises. !.he concept of a mean, or average, is always the same. Suppose a quantity can take on only two values,

Q , = - 60 and Q = 2

+60. and that PI is 213, so !.hat P2 is 1/3. With twice the probability, it is logical

thaI we should give

QJ twice the weighting. A mean of -20 would indeed

be half as far from -60 as from +60 and would follow from the general prescription: To find the mean

Q, multiply each value by its probability and add.

(l-2a)

u.t' u� P,

•

NJN. thi,

a....\ume...

_Q=

a common fonn thai may be familiar�

N" �Q,N, Q .... . ' N = N

'"

(l·2b)

value Qj

limes the number of tllne!Jwa)'\ nu' �)... that the mean is a possible obtained, summed over all values. then dl\llded by the total \\ tit)' dle quan times/ways. number of

Qulle often It is relatIVely easy to obtain something thai is. pmfHJrrionai to

�abililY, but actually expres..mg the proportionality constant i� difficult or �sy. In these cases. II IS common to write the mean differently. Suppose P' i� proportional to the actual probability P: P, = A P,' ,

"bere A is a constant. The total probability must be 1 . so

�'Pr =

I =>

L,A P; = A L, P.' = I � A =

I

� p' ""

,

Thus

(1-2c)

Again. in this form,

P'

need only be proportional to P and the proportionality

Q.

constant disappears. the prob­ Finally, if we wish to average something that is ajunction of ability of obtainingj{Qi) is the same as obtaining QI' and the probabilities stlU add to I , so the mean is

KQ)

=

k,j(Q, )P,

(1-3,)

Standard Deviation Our nex.t concept is a way of quantifying how much the values of Q deviate from the mean. Many recipes are pOSSible, but the most common is the e standard d viation. How do we come to choose it? A given value deviates from the mean by deviation: Qj - Q Some values of

Q i will be above the mean, some below. The deviation is a

function of Qj' which we can average via (J-3a). The result of the average IS logical, but not very helpful.

U

_I.. J

....1.., . . ..... .s..IInf � ... Nambm ol W ...P

I1IQII ofdo'......

1:.( e,

- O)P, = 1:. e, p. - Q ,£, P, = Q - Q I

= 0

.

So Ikt don', '\'cragt the dC'\'ration' Ho"'ever. if we a\'erage i rs squOfl'. rhe sum ,an fant no ntg.O\'C' ".Jut.. and 'he farther tha! 0, vaJues stray from the

mttn. lht larger lht a\tnge �houJd be.

deviation. 1:,(Q; Q)lp, To Yield ..omelhing that ha,� the same dimemiom as Q. we rake [he square mean oflhe ..quare of the

RJOt. Siloing

u� !he root-mean-square deviation.

_

known as the standard devia-

1IC)n, Of the many symbols u.�ed for this important concepl. we choose a della.

vr,(e, - Q)'p, 0-4.) only if Pi i zero whenever Q, - Q is nonzero, Note thai tlllS can be ;\1;1lhemalically expressed. Q, Q � P, = 0 In ocher words. the value IIQ ·

..

:no

0,

:I-

=

Q I� Ihe only one ever obcained, Combined with ilie fael ilia[ 6Q does

'pread a., de\'iacion,� increase. we see Ihal slandard deviation is a very logical definition, 51andard deviation i� often not presented or calculated in form 0-4a), but

1O\leJd in a closely related fonn. whose derivation is a good exercise in sorting

oul the \'anous quanlities we have discussed thus far.

�Q = vr,(Q, - Q )'p, = vr,( Qf - 2Q, Q + Q')p, = V"i, Q�P, - 2Q Li Ql, + Q'2 L/ P,

We have used the fact that

Q is not a function of summnlion index i, but sim­

ply a number thai can be broughl outside a summation, In the second term

Q.

imide the radical. we now recognize the definition of Q, and in the third, a unit lolal probability. In the firs!. we have the mean of Ihe square of

IIQ = VQ' - 2QQ + QI

or

IIQ = VQ' - Q'

(J-4b)

While obscuring the fact that the radical's argument is necessarily nonnegative. !hie; fonn makes a simple point: To ca1culate standard deviation. we need only find the mean of the square.

Q2. and the square of the mean. Q2,

JUSI as equation (1-2b) follows from (J-2a), an alternative fonn for stan­ dard deviation follows from (J-4a).

-

( J 4c )

� Differe

nt Route

deviation can hi! el(p�,�d In a seemingly different form and ..tandard t a. matter of redefinitlon_ If the sum.. are nm over Pl l....!Oible .. rtall) ju ..

�

�5

Q

y but Instead over a.n "tnals"-aU mdi"idual mstam:c!f. of of the quantit we will uc;,c J rather than l-then a sum over I With an N In hlch - 'ut'-- for .... I all '''' " the same as a sum over j alone. For example, If in a �ne!io of SIX is lbe sum - 1..5. and e'( riments we obtain only three values-Q \ = \,2. al� pe t .8. 1.5, 1.2. 1.2. \.8. the following are cqui"alent: 1.2. ce �uen

Q, ­

Q�

�8-ln

Q=

�:"'I Qj N, N

=

Q= With this

=

-'-'----'-----""-;-'----==-=6 1 .2 · 3 + L5 · l + 1 .8 ' 2

1 .2

+

1 .8 + 1.5 + 1 .2 + 1.2 + 1.8

6

redefmition. a\l N, would disappear from sums. and (J·2b) and (J.4c)

\\ould appear as

/l. Q =

Q=

�'Zj (

)( QJ - Q)'

....

N

We avoid this route because it tends to obscure the role of probability and clut­ ters the otherwise seamless transition to continuous quantities.

Continuo us Quantities The number of workers in cubicles is a discrete quantity. its values being nonnegative integers. Except on average, we don't obtain 1 .3 restricted to �'orkers in a cubicle. Some quantities are inherently continuous, meaning that

from one value to the nex.t is an infinitesimal change. An example would be

the locations of a swinging pendulum. In such a case, a sum naturally becomes the probabilities of being at particular point locations must an integral. and

beCome infinitesimal; otherwise. summing over the infmity of locations could not yield a unit total probability. The basic formulas translate in a straightfor­ ward way.

'Z , P, = \ Q = L; Q, P , KQ) = 'Z;!(Q,)P,

AQ " V}; (Q, - Qj'P, ,

--->

--. ---> --->

j dP(Q) = \ Q = j Q dP( Q) KQ) = jJ( Q)dP( Q) /l.Q = Vj(Q - Q)'dP(Q)

Note that equation (JAb) 1� unchanged, as its fonn is independent of whether

a sum or integral is involved.

" - , ,1. '' . ,. .,,1 a J

.. 7 7•5 .... .... " .. ... .. . _ . ... _ .. _

FiuUy. I« «JIIIIDuous varillb� iI i� u:o.ually mo� t:onH'ment 10 lkaJ noc

..uti , dilf'crtati&I probebilil)' bur with a prolNlbUUy de-Mi.y. a probabllil), �r _

Q. __ u follow

,

U(Q) D(QI - -- ,,, .... dP(Q) - D((J)dQ dQ

With thi� delimllon. formullL\

0·1a), {J-1al. (J-Ja). IlI1d (1-4a) become

/D(Q)dQ - 1 0 - /QD(Q)dQ iW) - /M) D((J)dQ � Q - "rrrQ - OrD(Q)dQ

N,'IIe thai In quantum rnt\:h4nit:,. the quanfJIy

(J- I b)

(J-2dl (J-3b)

(J-4d1

Q might be location x and the

prohabiliry drn..il)' comt, from me wa ..,C' function,

D(x) = 'I/I(x)i2.

EXAMPLE J . 1

I

,

I

I I/'

VatU(' CJ, �lImbn of Tunn. .\

I

J

,

,

.

.

"

•

4

7 1

Yo'hat 0Itt W mean and \land.lnJ de\iatlon" SOLlJTION The \Ioa)' in whi,h the data are gncn make� _

Q-

(J-2b) quickest for the mean.

" 1 + 2 ,4 + 3 ' 9 + 4 ' 1 2 + 5 , 9 + 6 ' 4 + 7 " . � _ . • -

40

= 4

hlr the: \,andard dC\'ia!lon, we can u\c (J-k) directly.

.lQ

-

1 0,].1)

/.

2 ,

/(1- '1' 1 " ( 2--- ')"

1-------

\

( J -

+ ( 7 - 4)' 1

+

'0

=

1.30

An...'lIher way i'i equation 0-4b). Thj� requires caku/aling the mean of the square. for whll'h equation (J-Jal i, appropriate. the " funclion" being ju�t the square. Equation

Thus,

i� ex�s�d in

N, IN puIS �J{Q,)N,

term, of probability. but P, •

JlQ)

�

N

II in the form

NoW usin, (J-4b)

�Q

=

\/17.7

- (4)' - \.30

We see that 30 of the 40 "alues are withm t standard �iation of the mean, Standud df:Yiation usually cover-, the majority of \la1UC!i.

EXA MPLE J . 2 "'"""' 1be probability density-probabihty per unit ht:ight- for finding a gi\len object at 1 tw:iJht )' is given by

and applies to all values of)' from 0 to +00, Find the mean and standard de.\liation of " in terms of the constant The following Integral will be useful-

b.

fooo Y"�-b"dy

""

SOLU1 c'...

Apparently our resuh should not depend on naturally using y in place of Q_

m!jb'".. t

A, We can use (l·lb) to detennine it,

Thus,

{Nott: In quantum mechanics, the process of ensuring unit probability i!i called nor· malization.} Now using (1·2d),

The result is sensible, If

b increases, the exponential faUs off faster. and we would

expect the average height to be smaller. Moreover, its dimensions are correct. The

argument of the exponential must be dimensionless. so the dimensions of b must be

one over length.

For the standard deviation. we could use (J.4d) directly, but let uS instead use

(J-4b) after finding the mean of the square via (1·3b). As in Example 1 . 1 . the func·

tion here is just the square,

y' = fo

-

00

y' be-b'dy =

b fo

00

y'e-b'dy =

Now inserting in (J-4b)

t.y =

I

\/(211)')

b(2!/b')

(I/bl' =

= 2/b'

l ib

Not only does this have the correct dimensions of length. but it also happens to equal

the mean. We conclude that most values obtained should be in the fange of heights between zero to twice the mean.

...... J� "'-)"101 JWl.. two cl .. c.- I. two ,.,t". 1If*ft.

FH ', r·n;

-r

;' .. ' f, ,

,

; .. f

/' I

r,

1

-

Factorials-Numbers of Ways

1

Suppose we h.t,-e Nd.ffel'C'nl can. and N md.,iduaJ parking splices. How many d,fft'ft'nl .. a)\ L"iU1 we arrange' Ihe' c� m Ihe space�? Any of the N cars could

be in � tim: 'paL't', and an) Ilf th� I'C'maJnlng N - I could be m the second Thu�. tilhng Ju�t the' tiN IWO ,p"ccs. Ihere are N(N - I ) ways. In the nd c. and �rec �pace�. these six possible Ways \pc.:.aJ ca.'iC' of three J.:aJ"\. O. b. .. _ are �hOwn in Figure 1.2. Conllnumg, any 01 the remaining N - 2 cars might be 'pao.'C'.

In the thiN 'pace, �o to thi� point we would have N(N - I )(N - 2) ways. Of cotJ.r>C. in lhe thfU-car case. N - 2 i� I. and no more ways are added. as there

i_' onl)' Oil(: choJ('c for the la\1 car. For arbitrnry N. the number of ways would 2. then N - J. and so on. until again there is only one car lefl So the 10lal number of ways is N(N - I HN - 2) . . . 3·2· 1 . which is the delinJlion of Nf and j� referred 10 a� "N factorial." Nov.- let us de,ignale a group of NI parking �paces as region i. as depicted be multiplied by N

,..".. J,J A �'''Il .." hln ..hJ�·h

�,"anpffiC'nlll' lotoJn.:h " da:l�

IrrrkW'allI

in Figurt J3. By the aOO\'e arguments, there are Ni! ways of rearranging cars among lhe-.e ..paces alone. without affecting caTS elsewhere. What happens if

�e om.. declare thaI it doc�o't maner where the N, cars are in this region. and we will con�ider it as anI)" one way of arranging all N cars? For (lll)' previous

,in,gle way of parking all car; in all individual �paces. there II'ould hUl'e been N, ' different ....ays . (including Ihal single way) thaI would leave the cars not in

region i exactly where ,hey were before. and we have now declared these N '

�f

fomlerly different ways as jusl one way. Thus. the previous lotal number ....ap. . N'. j, simply divided by N/. the number of rearrangements within region j thlll change nothing out�ide. Repeating the process. we choose

another group of parking spaces from the N - N; not in region i, calling this

ne� group of N �paces region j. and again declaring that rearrangemeOls

l

"ilhin il are irre e\'ant. By the same arguments. we must divide the existing

number of ways by

N ! to obtain the new number of ways. If in the end we N spaces into M regions. the number of ways W to park

1

have broken the line 0

cat\ in !!tpaces. where rearrangement within any given region is not considered J different WIl)·. is

(l-5) Consider the limits. If each space were a "region:' there would be N regions. each with one cat-SO that NI '"" I for all i-and W would be simply N! (divided by I ! 10 the power N). as we expect. At the other extreme, if there were just one region encompassing all

�nsible.

N spaces,

IV would be

1,

which is also

We have used cars and parking spaces as the framework, but the arguments

are general. Formula

among

M

(J-5) gives the number of ways of arranging N objeclS

separnte calegories (regions, boxes. energy levels. etc.), where

N

�

panicles are in calegory I-rearrangements within being irrelevant-N2 i category 2. and so on.

N-

al case i!> just two categories, one \\lth rnmon speci ost cO m objects. in which case we ha\'e II e 'flI ),lith \' other

)C'11\ 10g

the

tl

obJe4.:h.

"::N-"-W � _II!(N -! ,,--,

fI)\

ial coefficient and is so common lhllt It hu,

vn as the binom IS kllO\ r to l. fac . 0wn special symbo ""' 1"1� ItS • en

1><" g"

W = nl(NN'- /1)\ .... (N) Ii

(J-6)

Appe ndix

K:

So me I mportant M ath

In

th

ndix. we louch on several mathematical topics of particular impor-

1"ppe

, ·,n ""

•

modern physIcs.

comp lex Numbers x nUmber ha� ,two parts: A comple

a r eal number, and another real number

, IS defined by ,2 = For example. in X = 3 + ;4, lliliplied by ,iswhere and the imaginary part (which is a real number) is 4. 3, part ;e rtal parts are sometimes referred to �Y the notation Re and 1m. Thus. Re 111i:se tWO Complex numbers Include real numbers a subset­ X = \' 3 and 1m imaginary part is O-and may be the thought of a compact �st is. when ing twice the in formationof a re� number. However, they also I

/,

-I.

4.

==

as

as

y,ay of convey . , bey special rules of arithmetic, based on keepmg i (to the first power) sepa­ 2 t , while j becomes - 1 . Thus. given U = "\ + i 112 and V = v , + i \12-

�e

uv = (Il l + i U2)(VI + i v2) = UIVI + ; UIV2 + i U2Vl + = (1IIV\ - U2V2) + i(ll\v2 + 1l2V\)

;2112 2 "

The twO parts of a complex number may each be functions, for example,

U(x) =

u (x) + i U2(X). \ A function whose argument is complex may often be broken into its real parts. A common example is the exponentiaL Consider first an

and imaginary ix exponential whose argument has no real part: e , where is real. The power series eb = 'L:'o bn/n ! enables us to write x

,

'" =

00

L o

{ ix)n n!

-­

" j1- + 1!

= I + ix +

Using;3 = j2;

=

-i,

;4 =

j3j = 1 , is = j4i = i, and so on, this naturally breaks

into two series.

K-1

K·2 AppendiK K Some Imponant Math

The sums in parentheses are the power series for cosine and sine, and the resuil is an identity used very often in physics, known as the Eu ler form ula.

e'X

Euler formula

cos x

=

i sin x

+

(K-\ )

= (eX, cosx ) + If the argument also has a real part, then e'1 i.<1 = e" 2 i(etl sin x2 ). Useful formulas following directly from the Euler fonnula are cos x

=

ei.
+

sinx =

2

(K-2)

2'

In the same way as a two-dimensional vector, a complell number has a real, positive magnitude defined as the square root of the sum of the squares of its two parts.

Ivl

=

Given a complex number or function obtained by replacing

i by

-

= cos x fix)

�

+ u

X. its complex conjugate, denoted X·, is

i wherever it occurs. Ellamples:

v = ul + fix)

VUT

iU2

i sin x

= e-r e+1.I"

f "'" tanh - I ( 3 - i4) The product of a quantity and its

=

U'

= u l - iu2

= f.(x) = cos x + j sin x = f-(x) = f.

=

= r?e-ix

tanh- I (3 + i4)

conjugate gives the square of the

compJell

quantity's magnitude.

( K-3) Figure K.1 The complex exponential

e'" has unit magnitude and "rotates" in

the real·complex plane at frequency

k.

An important special function i.!> the complex exponent.ial. eib

i sin(kx).

whose behavior i!> repre:o.ented In Figure K I

both its real and imaginary

,IO rio

Jf --:f--f..<::::::::=-- �I •

I

partS

CO!o (iu

is

=

cO'Jb)

+

�iIIatory becau.-.e

are o'>C l llaror). of angular frequeoc)" k. and k I Moreover, the magnttude of � I.. I .

must have dimension'i radlm or ..imply m

Imaginary pan

It

or. equl\-aJently

J".l)fIX} - (costb-) =-== c" 10,:.r) t

I

mlbJJlcos(lr) + • smllr))

Moll b-)

The "'mpln ""I">IlCIltial .... 'rot.l!O' on II>< while mamtalnmg unu lrngth.

I

rc;U.1I11agllw, plane .. fJtqU
..... kinds of differential

....... lInsp' oil_

t .... ..," The euie l Io oolve ... ... .....,. ... -. .... " 1 .,• ... often c""".ned to Ie""'" wi' '1, 11l1li111 "e! I,nt111,_

v.nobl• • by n...... .. ... '1'1 . . ... .. a fwlction uch .. j\ •• y. l). we ....il .. . . ... "1 fW plio . _Ie fUn
. . te
;;;�

fIIIIII dift'erenliaJ equations 10 onIiDary -.> .... a IptZ'. ... by which \he fun<1iOll ! is -'1y ....dzd .. ... _ _
K-4 Appendix K

Some Impol1anl Math

First-Order Linear

df{x)

bf{x)

�

�

=>

fix )

�

A'>'

(K-4)

Solution of a first-order equation yields one arbitrary constant, essentially a constant of integration, which. in the above result, is A. It c�n take on any . techmque of sOlving value while still solving the e� uation. The m�thodlcal this basic differential equation IS to rearrange It to dflf = b dx, then integrate both sides.

Second-Order Linear

d'�;)

bf{x)

{A �

=>

f(x)

�

( VIbIx) + B cos ( VIbIx) 0' Ae+iv1il• + Be-iv1il• b Ae+v'bx + Be-v'bx or A sinh ( v'bx ) + B cosh ( Vbx) b sin

Ax + B

(K-5) < 0 > 0

b = O

A second-order equation yields two arbitrary constants (of integration). Note that the sign of the constant b is crucial, for different signs lead to functions whose behaviors are entirely different. If b is negative, the functionJis oscilla_ tory; if b is positive.! is exponential. growing or decaying or some combina_ tion thereof; and if b is O,fis a straight iine.

Useful Integrals Below is a short list of integrals most often needed in the text.

J . (L ) dx = "2 L = ""4 J . ,(mTX) J . (--L ) sm2

x sm -

mT X

x

-

x'

-

d.x

T x, sm2 n7 X dx

=

( -) ( ) (-- \,

2"-,, sin - X L 4mT

L

Lx 4mT

2mrx sin -L -

2mTX Lx' - -- sin 6 4mT L J

x'

LOS

(2mTX) -L

Appendix K

Some Important Math

K-S

An sw e rs to Sel e cted Exercises

Chaple. 2

99. -48 1"

17. 0.1'''"

ll. 43.75 m 23. ]:ller, 0.8 m 15. 60 m. 2.67 x 10-7 ,

lOS.

n. \'!c = 0.78 1 . 1 .04 x 1 0 -1 s, 24.375 m

19. 24 m. I'le = 0.6 31. 0.0067 s behind )5. 9.8 ps earlier

41. 1549 m

.0. (a) - l OO ns: (b) 1 4 1 ns; (c) I OO ns, zero 45. Bob is 60 yr, Anna is 52 yr

47. (a) 32 y" (b) 32 y,

(a) jumps ahead 1 2 8 days; (b) 250 os; (e) behind by same amounts

53. (a) toward. I'le =: 0.25 ; (b) 687 om; (c) 549 nm

55. yes

:/ T : ,= 3k r y m A, a . 1 5 0m 7 2 5 . c

59. O.385c. c_ ", , " (b) (0, c) c 63. (3) 2

II;:

65. (.) (-0.8c. 0.6c): (b) (O. c)

1016 J. 6 X 1 0 1 6 1 , 1 . 5 X 1017 J 71. 9 10-17 kg X 73. 2.5 . 75. (a) 2.19 X 1 0- 26 kg mls; (b) 3.64 X 10-22 kg . mfs; low 40% low. 10-7% X 3 (e) X

77. c/V'i

79. 1.83 X lOS kg/day 8I . 4.71 X 109 kgls

o

83. -c 2

85. 25.1 MV 87. (3) O.759c; (b) 2.07c; (e) O.948c 89. ue = 0.9997 I c.uc = - 1 .62 X 10-Jc. 20.6 MeV, [7.1 keY

91. "'2 = 6.43 kg. ml

I

0

0

0

0 - �,/C

I I I.

J7. c/V'i

V

0 0 -V/l'

0

0

0

0

I

(.) 4mo' and 5"'0"': (b) 0"'uc and

lIS. (.) A \/8/3 and 5A \/8/ 12: (b) bolh

39. it must; lOP passes through (v/e2)Lo cos 80 earlier

49.

I

=

1.43 kg, 1 .93 X 1 017 J

93. Ca) 1.29moc2; (b) 0.795c; (e) 2.57mo' 3.29mo' expo B

2 0"'0'" VSqA/24".o'

Chaple. 3

17. 3.12 eV. 399nm

19. 82.4 nm

21. 6.42

23. 1.22 29. 8.15

X 1031 phorons per sec X 1()6 mls X 107 mls

31. 130.5°,23.9° 35. 0.00659 nm

41. (b) opposite, 1 . 1 8 X 10 14 m 43. 9.1 X 10-27 kg 45, 60°, 5.73 X 10-3 degrees. visible light

47. 7.38

X 10' 10 m. 600

49, 5 X 10-6 Pa, 6.37

51. je.,E'A/h

x

IO� N

53. 2.91 X 10-12 m

55, imoCl, lmo

Chapter 4 11. 40 13. 0.333 nm 15. 728 mls 17. 2.43 X 10-12 m 19. (a) 6.29 nm; (b)D.147 nm

21. (a) 1.46 X IO-I o m to 1.32 X 10-0 m;

(b) 6.23 X 10-9 m 10 2.43 X 10- ro m 23. 2.2 X 10- 10 m 25. cannot be treated classically 27. Ca) 0.14 mm; (b) 1600; (c) 400 29. (a) s/m: (b) 10 s- 1/2v'b; (c) 5 s'I/2 v'b. (25 S - I)b, 25 s - I

AN-1

AIWI

_ .. _ _

71. 1.61 X 1 0 22 kI an.. ' )S x 1 0 11 J. 1.67 )( 10'l7 kI ) 6 )( 10 li mit

.L G. 4I. #7.

10' "'" 10 " )( 23 (11'1 ' 3 )( 10 . J .. (.)0.01 67'. Cb) 1.0:5 )( 10 J I dtpn - 1.83 )( 10 '\) rtd 51. r < 10 ' m

6.l &r

51 (b)

"

�V'A'.'2M'

5'7. .. (4 ....,)A.. 2 2 x 10' mi. 59. c .) O lI4nm. O ' nm 11. 0 1 '. 6J. 9' nm 65. 67.

...

I

2.

Ct

Cbapc.r S

53.

I

I X 10' �

lJ. (a) 2t1m. (b) 2Lln 31. 'a) .boul 9'1 as large 33. (.) 0.080 and 0.16; (b) 5.8 x 1O-17nnd7.1 X 10-1 4.. (cn,4 x 10 1M and 1.8 x 1O- 1l1 •

'" v ph.ouot

0,0-'3 s

partid�

- B.n«' - 1»

+ c, '11, u.... nh U.

�( �: y �

(b) R => k?L2/(4 + .l:1Ll). T "'" 4/(4 + klL2)

Chapter 7

- lAo A �inkL + B cos kL - F�L B 'I n U.l ... o(FrrL �� QL) _

m ji(�:t.J��(mK)I ·.lh I

21.

O,� om

2J. 0.609. 0.196.0.609 25. 107 'TrZfa2/2mL2, at center 27. -mi'!2rr4iffi2nl 29. n, "'" 6. nr' '''' 3, 1 . 1 X 10-6 m 31. n "" 6. 13,2 cV. lOS K

33. 5.5 X 10-8 eV, 5 X 10-9

37. 150°. 125.)0, 106,8°. 90°, 73.2°, 54.7°, 30°

45. -0.85 cV, magnitude of angular momentum: 0, V 2h.,

V"i2JI, l...component of angular momentum:

-Ii, 0, +11. +211, +311

51. (a) yes; (b) no '3. 0.2 l 2

\lH "'It'll v ICf

_

59, lrm

'7. 0.238 59, ellipse

&J.

61. no 67. 13.5 nm

73. (a) plane wa,'(' (1). Dirac df'lta (I); (b) A =< 1I

,,11-;

Ie) B _ 11 75. tb) no� Ie) 1

3�

,

v2b :

X 10 :!O J 77. (al and (bl do
artutruily high. infinite number of Slaleli slill possible 1 •• 0 323

Il. 0 8661a

hd 0.637

95. '!:b

9'7. (.)

-bV3

ill

77, -1.23 a� cos[ ( 1 .8g eV)t/fi]i. 120ns 79. (a) (�/2 \I'1r)-1 re-r!�o sin8 cos¢, ( l oo61Tr 1 r2e-r/au sin2e cos2cfJ;

(b) angular probabilities: (d) 1#12.1.+l - 1#12.1.-1 85. 0.29 nm

... (bl no 93. (bl

ti;

(c) 71. (a)�; (b) 7S, 0,00069 s, 1.9 X 10-6 m

83. -6.8 eV, 0.106 nm

l15. aA 91. Q

0.30 mls. \'lIroup "'" 0.45 mls 2.79 mls. vJIOUP "'" 1.40 mls. yes

53. zero. � 4. yes 55. (al -33 min; (b) »age of universe

.2 � '\'2, yt'l. v;: I. no I tn(Q_ .. 1 l 4 1 m . )4 lI 4::! X l ()lI m-lx)t'-I\�1"1C 10"J-I)1 X 11 15. 0 b4 om n. U4or 'LJ4 19. 2 )( 1011 35. 0 0.'8 �V

49.

11.

47. 49. 51.

•

45. C + D

(! - 1 '), '" (l - l')'�'

19. (b) O. 0'" M. 90"

_

2". I .. (W1")l 7 1. 7 601 )( 10 If> m.

JI

.7.

45. vpb.w

. IllLI (10 ' 111/ .. .2

.5. 0 00704

43. group 36 em behind. crest 36 em ahead

.on((10 .

lit"";,,,

Cbopltr 6

< I' <

87, (a) 6.2945 X 1()29 kg· ml/s; (c) 3.9 X 1()21 J;

.

+ bv'3 no

(d) 5 X 1012 m and 3.2 x 1030 kg· ml/s or about 5 times

89. En = n2'Trl h1/2ma2, L

=

0

�'

-31i, -2/i '

�.pter8

�-

"

3'1. 1/0

25; (b) J9 ta) 0.

.�

�

g

3? 3p6 3dlO 4s2 4pl; cesium:

Is2 2s2 2p6 3? 3p6 3d1o 4s2

33. � - � sin(2T). - l . l 37. 2 94. 1 .2 x 10 �

4.46 eV lower

,

0.179 eV. O.5:n eV; (b) 0.01 1 nm. 0.019 nm: (e) 8.5%. 14,6%; (d) 5.4 X 101 4 s' 1, 5,4 X lOll s I 41. 33

39. (a)

43. vibrational levels woulll be spaced more dOM:ly. rotational

(a) vp", ::ijih, Yph , �: (b) 20; (c) G, +1), (I. +�),

(i. +�), (1. +�). (1 , -h, (1, -h (I, -�), G, -1), (�. +�), (i, +1), (�, +�), (i, -!), (1, -�). (t -1), (�, +1), (�, +1), (1, -1) . (1, -1). C!. +1), (�, -�)

n. 16r 17. 70.50 83. Z - 20 or higher, and smaller n

levels more closely spaced

51.

0.17 �m.

59.

-700

-0.Q7 K '

63. --!o

eV

65. 10-11 VIm 69. (a) 90

�A. 70 �A: (b) 9 �W. -1 �W; tc) 1 1 3 mW,

69 mW; (d) 220 mW. 168 mW

71. positive bias. bipolar: low-inpul impedance, MOSFET� high-inpul impedance 75.

100 A

Chapter 1 1 13. 3 x \0- 14 15. 10.&11 u 17. - 145, -150£, 21. 6 23. 7.6& MeV/nue

25. &.75 MeV/nue

Cbapter 9 19. (a) 60: (b) 20

25. 8.S1: 0.20 J/K. 6.S2: -0.15 J/K. +0.05 JIK, e3 6X \ 27. (a) O. O. O. 0), (0. 3, o. 0). (0. 0, 3. 0). (0. O. o. 3). .

()2!

(2. 1 . 0. 0).(2. 0. 1 . 0). (2,0.0. 1). ( 1 . 2. 0. 0),(\.0.2.0). (I. o. o. 2). (0. 2. I. 0). (0. 2. O. I). (0. 1. 2. 0), (0. I, o. 2). (0.0. 2. 1). (0,0. 1 . 2). (1. 1. 1 . 0). (1. 1 . 0. 1). (1.0. 1 . 1). (O. I. I, I): (b) O.5

i!c-=c-

29. N""'I'B

43. (a) V2kB T/m; (b) smaller

47. (a) (mg/kBT)e-ms)'/ksT; (b) 8.4% less; (c) 9.6% less 61. 7.0 eV 7 1 X 10-9 m; (b) 2.76 X 10-11 m; (c) classically 69. JO- 1 \ C. 10- \6 71. 11.5 eV 73. (321tsl1/15h3c3)T3, 8. 12 x 10-8 J/K · m3

65. (a) 3.

rl.t.rfl

29. 1.53 eV mUlOl be put in, 0,94 nm,

57. \I kV 59. 3.4 eV/hc (3t2 + 52 - 19.25) 65. (a) 66°; (b) 1450

63. I K

T,

Chapter 10 27. -29.4 eV

4p6 4d1o s? Sp6 6s1 �t ; 'J.Sl 2p6 352 3p6 3dlO 452 4p6 4dlO 41'4 ss2 Sp6 SdlO 1 - ' sr4 6? 6p6 6d10 7s27p6 8s1 53. (al 45 -+ 3d is smaller; (b) yes; (e) yes 55. 0.05400, 0.30°0' 1 .6400, 1 600: (b) yes; ee) -3.8. no

67.

+

89. 7.17 X Ht m

neither;

:

�rt\

....

81. (a) 21.3 MeV; lb) \ 6.5 MeV

(d) antlsymmetric; symmetric; 4S- (a) neither: (b) sum ) yes: (0 (e ,,. \II 1 00(r1) i 1./I \,0.0(r2) J. t.P2.0,O(r3) i I oo(rt) t 1#2.0.0(,r2) f t./I \,o.o(r3) t + !/It'a'o(rl) .1. 1/I2,O,O(f1) i 1/.11.0,O(f3) i !/I\·o'o(r l) .l. 1/11,0.0(r1) i ifr2,o.o(r) i + 1/12'0'0(r\) i 1./11.0.0(r2) i tP1,O,O(r3) .l. iJ!2',O O(rl) i I.o. (rl) .l. l./1 \,o.o(r) i 49. phosphorus: Is 2s 2p6 3s2 3p); germanium: I s2 2�.2 p6 2 _

Rt£,./k8T)l�

85.

aotisymmettic 0.01

. . symmetnc 0.409, antlsymmetric 0.09\

_

\

(a) 2R at high T, 0 allow

81.

"I� \9 �2/2mL2 , 5 :r2fl2/2mLl, � TT2fl2/2mL2 _

\0" m

71. 9.85 x 1O'lI m

•

)So �>mrnetric 0.43 ,

3.9 x

"75.

° 135" '" ",S . 90 'O 15° \05°, \40.8° 1• W " · • " N )J). .spin up. CO S",,,, l)

6,63 MeV/nue. 7.6& MeV/nue. 6.17 MeV/nuc� 6.85 MeV/nuc, 1.25 MeV/nue, 6,28 MeV/nuc 29. (a) 0.42: (b) 0.19 35. 43 MHz!I'

27. (a)

(c)

37. 39.

reduced Coulomb repulsion lead-206. SAl MeV

41. fluorine-19. 4.82 MeV, a maximum iCE, since shared with anlineutrino

43. (a) Ar-40. Ca-40; (b) magic numbers, asymmetry: (c) Coulomb repulsion; (d) even numbers

0.7 \3 MeV 49. 3.94 X lOs mls 51. 18.3 min 53. 5.7 per min 55. 1.30 x lOn, \.47 X 101 3, 0. 1 56 MeV, 0.332 1 57. 4.28 X 109 yr 59. 31.2 mg, 1.1 X 10!9, 1.6 X 107 J, 1.2 X 109 s-1

45.

Credits

ptIolograph Credits

Center for Re�enl\:h \10 C�\mpUllllit)n Oln,J th Apphnm,n, (CERCA)

(tllplfr 1

1...0: w. N. Colley (U. Virgmia) & E. 'Turner (Princeton). I ,\ Tyson (UC Davis), HST. NASA Chlpler 3

J.11: LBNUPhoto Researcher::;; p. 91 (both): Positron

Emi�sion Tomography Depllnment, NIH Clinical Center. �alional lnstilutcs of Hcnlth

4.3: From Claus Johnson. Dietrich Brandt. Stanley Hi�chio. �m } Phys. 42 ( 1 914): 4- 1 1 , fig 8: 4.4: Ernest Orlando

O. Schechtman,

etal., Phys Rei' Let/us. S3 (1984) Chapter 5

5.9: IBM; S.21: Sandia Corporation Chapter 6 6.19: Digital I nstruments; 6.25: Peter Sutter, The Colorado School of Mines Chapler 7 1.20: C. R. Stroud. Jr. . and Michael Noel. University of Rochester Chapter 9

A. Leitner, Rensselaer Polytechnic Institute; 9,2.4: Hughes Research Laboratories; 9.25: National Ignition Facility; 9.34: National Institute of Standards and Technology

9.19:

Chapter 10 IOA7: Intel Corporation; 10.50: Courtesy Author; 10.58:

11.17: ISMlPhotolll�e; 1I.3�: ITER Imemlllionai Fu,,�m Ellergy Org�lnizl\ti\)1\ (IIFEO); 11.37: NII\lOlIal Ignition hll.:ihty Chapter 12

I2.S: CERN; 12.6: Stanford Lint'Olr AccelcrutOf Ccnt(,f', 12.7: Fermilab National Acec\nmlor Lahor
12.16: Kamioka Observatory. leRR (!Ill-lltule illf COl-mie Ray

Cbapler 4

(Awrtllce, Berkeley National Laboratory; 4.5:

Chapter I t

Alain Rochefort. Assistant Professor, Engineering

Physics Department, Nanostructure Group, Center for

Research on Computation and lIS Applications (CERCA); 10,59: 1. Vac. Sci. Technol. B 23(5) Sep/Oct 2005,

p. 1964; 10.62: Alain Rochefort. Assistant Professor,

Engineering Physics Department. Nanostructure Group.

Research), The University of1\)�yo; 12.18: l.awrence Berkeley National Laboratory

Art Credits Chapter 4 4.29: Courtesy of the Royal Swedi�h Academy of Science.. Chapter 5 5.22: Courtesy of N{lIII� magaLine; S.'23: Courtesy of

Dr. L. B. Wang Chapter 6

6.26: Courtesy of Dr. Saw-WllL Hla; 6.27: Courtc�y of Dr. D. F. Phillips Chapter 9 9.14 Counesy of Dr. Wolfgang KCllerle: 9.35: Courtesy

01 Nature maga7ine: 9.36: Courtesy of Or. George R. Neil and Dr. Lia Menninga Chapter I I 1 1 .29: Courtesy of the Tennes�ee Valley Authority; 1 1.34: Counesy of the Princeton Plasma Physics Laboratory: 1 1 .38: Courtesy of Nmure magazine Chapter 1 2 12.4: Courtesy of Kevin McClean. CERN High School Teachers Program; 12.1S: Courtesy of the Particle Data Group at Lawrence Berkeley National Laboratory

CR·'

I n dex

•

,AbIOIuIC fut\lrclpastle'sewht�, 56 ","lmlon, S37-Sl9 �SIIIU.447-448

At'C.dcnlll degeneracy, 258 f,CUIIIdts. 315 ,tJYanCe Lillli Source (AL S). 90-91 � deeay. 201-213, 493-494 Alpha parudt. 207 AAikl'On. Carl O , 85 AJIIIlI lf flUjlltl'lCy, 108 AqIIw momentum IIIdInt. 32()...32S C\!Oen« of quantiu.llOn 289-300 AlIbboodm& Slalu, 416-418 'JdI)drogtn atoms, 216-211 AlbputlCies.525-528 MItI§)'1Mlttric Slate, 303-306 ..wocliltd Legendre functions, lSO AtomIC mus Unit, 477 AtomIC Dumber, 415-476 Atomltspcctn, 2-3. 233-244 ,\lUnulhtl anile, 245 ,\Dmulhal equatIOn, 247, 248-250 ,

8almer. Johann. 239-240 Balmer sene$. 241 BtDd gap. 433 Band occupat1On. conductivity and, 440-441 B&nd width. 435 Bardccn. John, 452 Baryon number. S40 BlI')ons.531 BCS theory, 451-459 geUdec1y. 494-491 Bil Bang, 53, 549. 550 BIlIIiIDB energy, 418-486 B\OOrruaJ coefficient, 349, J·9 Blpo1U transistor. 453 811Ckbody. defined, 74 Blackbody radiauon, 74--75 Planck's law, C·I-Co{) Blickholes. 52-53. 54

Body-c;cntercd cubic, 428 Bohr. Nitls, 122,140 Bohr model of the 110m, 122-124 Bohr radiUi, 12),151 Boltzmann disUlbulion, JS�J66. H-4 BondLng stalCS. 416-418 Bose·Einstein <:oncXnwe (BEC), 399 Bose-Einstein condensation. 381-382 Bose·Einsteln distribution, 37�31t Bose gu, 381 8o�ns, 307. 3O&-309, 37o-371 femllons painng \0 form, 400 massless, 3&2-385 weak. Sl4

Boundary condmonl, 119 Bound $lates, 14\ Brao. W H 102 8T1oU, w L., 102 Bragg law, 101-\02 Brattain, Waller, 452 Breeder rtactQt1, 506-SOO Brtmsslrahlung,19 Bubble chamber, 85 Buckyballs. 460-461

Coheftnt liaht. 1I5 CoibUon lime. 416 Cokx charJe. 5)0 Comp\el. c:onjllple, K·1 CompIcJ. upoaaatW. K·1 Complu. numbcn, K·\-K·1 Compton, Mhur Holly. 10 Compton effect. 10-15 Comptoa .....denl\h. 1),4 Condcnd lnll1a, 411 Cond\lC1ion band. 440 COndIKtion ekdroni, 3'....311. 430 CondIKtivity.437 Cont1ltt poICIltiai. 310 Cooper pam. 45' Copenha&en I�, III

.•

C Calorimeter. 539 Carbon cycle. 509 Carbon nanotubc.s (CNTs). 461-463 Central force angular behavior in. 248-255 Sctu&hnger Equallon for. 245-247 Ceren1r.ov detector, 539 Chain reaction. 504-505 Chaos. 401 Characteristic X·nys, 316-318 Charge cooJugallon. 545 Charge coupled devices (CCDs), 464 Chemical rtlCt100S. 38. 311-312 Classical bound states. 147-149 Classical Galilean relativity. 1l-14 Classicalltmlt. 16. 84, 1 16-17, 253. 377 Classically fomuidenrtgioo, 148 Classical mechanics, 1-2

Cormpondeoee pnnclpk. ll-&4

mtcrowaYC beckpouftd (CMB), 550 CO\molo&itaJ comWl\, 54. 552 Cosmology. 52-54. 549-554 Coulomb repuhion term, 48' Covalent 'bond. 416 Covaleni loOlid,429 CPT�m, 547 CP violation, 456-457 Critical anembly. 505 Crillcal density. 51 Cntical temperature, 453 Cronin. James, 546 Crystal lani,e, 428 Crystalline solids. 42&-43\ Cuprate\, 459-460 Curve of §lability. 484 Cutoff waveien&th. 79

CO\ntK:

o

enel'JY. 554 matter, 54 Dana·Das spin field effect lransiSlOf lSFEl), ll6 DaUghter nucleus, 493 Davisson. Clinton J '. 100 de Broglie .....avelength. IO}-I04 Debye mockl. 39S-398

Dark

Dart.

1N-1

-

"..,.

-

0 .... JII7

- - '" ...., ... -. --_ ....... ..-.. ...... ,,.

D� .:::: .'<;2M.-lJ' *, '

' 1"

� ,""""' J.16 �. � -' J!lI-J!l' f,qIul,bnuIII. HI

.. "" ,.. o..IIIJ crt ... 1M- "", C ,

0., '". _ 4.10 - , .. D.-. ........, 416

.- ,

f\'CIIl llolnlOll. .�Z E1�'banJr .,mmrrry. J06

(jlllU"i,,11 .... 'Jlr p.a<.'krt 1 27-t .l I, 216 lime e",Iul'''" ,,/, F-I F·l Ge'Irr, H�l1t. �.l<}

E.tdU-1lotl pnllC1ple. .101-,109. 4111 �82

El�"lII'1OII talon. 169-112

GrtJ-MoUIl1, Mum,.-. j.1�

E.1.InMIC lInTlICoacfacll.'n. �/I

• f... _ _1rred culIi!.:. 418>-119

F&"lllnIk, 1-1--1-9

Fm,",Dua; clUrnbuaoo. J70-171 krmi tI'ItIJ)'. J1J-)14 Femuoa... J01-J08. J7t)..J7I ,..nl1l of. J08-J09. 400

•

(lamm.a ra)•• �fI

GilUUi�I1 ....�\e 10011. 1 1'1-1 ttl

Et':IP, 167-26I

n-f)-pe, +11�

Do 1111.... .,.,..,... U-w, M-UU

Ii.....,... " 4

" (i3fTln,• .Jttil.l. "91

1i�1Ib. 8

�'�. "" ---418

Dnft ",*ny. 4J6

f-\ilknlX'l. 46O-46J f-1DtIlIl, $a t -j 10

flmOll 1'r.1L:lon, 'II 0-� I J

I:*r fonnuJa, K·Z

.,..,.,... ........ 21'

o-NI ...... ..,. !J4

I'mJUtDC), Qlm&la w;lI\r, 107 1 0K FIVdIU&ll �,oa, _�_�I

n.

� r:fIftnm. 2. J9()..)91

DHA. ..... ....EII �..toh.

-. ­ DappAIr ..r.a 11-- II

F...�re �lIn'Idinge'equ..hoJft. l(JI)...1 1 2

�t, pn,.".w 0l. �'2

..... ...., .. 0kJdr ,-", "'1 DVk .... 1\Nkftt-. [).J DfnI: ....... '11

IIJ- '" 0... .... ....,

Four!II.J1/IIItfIJ,I.OO �59 Fwr·WIC1OR, 51.59 �1«1J'On ,_,H1J. 400

pyrotl«lriI:. jl�-_�16

fi4uibbnll/lt """ drllrlbu.1OIII, _1��

0..-.. �., 4'" � ... ..... NfIM'.... oI. l6I .,..... ........ . ..,U....U . ' D ....., ........ Ie J.. k .. d .

_ no o..p.n... mrrltk'fllt. 110

f'oIIr.lrvd Wcr. ,1",190

- .. ....,

...... '-"' 01. 1.11-.2. ,...., ..... ..JI�J9 �( .... � . _ .._u�.l9 � _Itt_ H'I-H-� f....,- pp. OJ &ao- � U1. I".,(I(). 16�161 � prianpk. /10 rp. bra-

whok aIDIIU as. 399-400

f'i'}1IIJWI diqJvnJ.. ,.,I-54'

Grnrnil rela,,,II). "l-�4

Germer. Usler H • 100

GIIII' rnqnctvrr"�liltKr (OMR) deVICe). J.l6 GI,,1'>aI pmltlOlllllt! )Y'lcm (GPSJ. rd.tlVl!,.­ moJ, 47---49

GI�,I/1. �.l()

..... qumtum l1umbc:� J2.l Got Gran.J ,,",liW thcol)' (GUT), �4R-S49 Orlll'.lil)ll&/ Itn""g. 60, .'5 1 7 GnlltttlonaJ Vrale� 60-6 1 Ground ltate, J 54

.n rnuh,tlr.:tron �I"m. ] I I

On.'llp IrI""'ly. 1'"

ESIdw ..... UJ. ..... 1iifIr..... . 11O-11l ...... AllIM. ,. 6. 76 ..... A -' .�IftcwIp. lltlo

Fit/d tmi'lSiun display (FEDI, 111

H

1iIKft:aI �. t""IY bandI

F�kf quantum. '14

H.alin>n" �.lO. 5_12

.... ..:u-..t.t9 BIktrk .. tIIdYtion.. 112 175

� diIdwJr ,..,..."., Jill BIIc� W"" pulle. 218--220

.. .... ..... WCIc nvn, 109 mall .DItUIn. r. 1

f"ltld tl'feo tmWdoI" (FEn. 45.1

G>rom�tlell' nUlo. 19.'5

f"1dd tmluioo. 1J 1-112

fltlt I«nK'flIft. Jl:'i

H.alf·lrk4�

fiat-stIU!.'ILW �'UfISWI(. 314--31'

H;lU.John L.,. 132

FlnlIt "ell. U8-162 rJ,

lafinilr well. /b2-IM

Fi!lSion. 503-503 rJ flHlOII. '14

HJI(l" 1114 H�n"'h, ThwJore W 132 HIlTllOIlK O\l:IIIIIIOr, 164-169 Ht.\y hydfllgcn. 84--fl�

BIrII:am CIpIft. 496 � co.IIpnfJOIl. ) 1 '

FIHtOn rea.:1On., 506-501

Hel<.entlc:rg. ""'emrr, 1 14

FildJ. Val. S46

Hd'lentlc:rg unanalllty pnociplt. 1 14

"'�. I06-I07

flIIOIfiC'CIlCC teSClClaDCe mntY In/1Sftf

-

......, 01. 1 ....... p'llplftlel. )9, 194-m

lillaraa lpia _ sP'(ESR).M-299

.. c ..... tame, ') f ... .$3'

...... ,J-4J

.. ... , _ lJ6... 1.17

_.36 _ 36

(fRFl), 183-18oI

Fort-n. '2J..nS eIa:Dvweak.5.\4-,),

p;'*-LU5

pncks .... 5]8-535 .scroe,. 529-'34 Forw� cmctition. 449

Fuutier amsImn. 12 ...... '21 _\'UtI, I).r -D-J

HrI,':II)'. S4b Hdlum. 3 1 1 isotopes of, 476--477 Hel.gonal dosesr·p;od..ed Uru.:ture. 429 H'gas b
Q Q Q o

-

IIIf

... _

...

•

I•

"

..

U1

-_".,-!� . ... �

I

2 4'1,1) :==:':141"-'. :-

;aaa. . � :: = :

b.-� � 1. .... r..

• IJ

1 -.]

'1

•

1)9 )\

...�

441 .......... , . . .... 1r.1n, . -- "�"

to

.-- �

- ­

s.-mpukAI biNi. ... ftInJy -

.0&' Sqoon<.- "'_ I.'

•

� «DWII,

a.... •• Tn

--...

II

' ..

bood. .\ "", -, ....... . ,.

......

u..'DII 1InKU, '-19

- .. .., . 7U_ ....... ..., I t..,... ... . ... .... . , .... "1 . ...... . . .... _ 10

"

.

' ...' .. -A1 VI;utiairl. 4.'� DrbQ;d, 04tb

..

..... ..." "' -

"

... .... .....'. R••• \W -. 4lJ ....... . "'.. ..., I 1'19 ...... ,-) it) 9-- 10 _'11) Sot _ 0.-.1 -'11) ,,'.,., ...... O;qth .. I l-- I.

..... , 7 'p••,... .... . �

'7,.... )" 'W '. 0 ' .... .... 'S' 7 � Ml '7 r. , 0IItl)I a1 17 . ' 1m . I, '.. %21

,

7

t.. ... .....

_ Ill

•••• t' - ....

" "s: ...

'7 • ".t " •• .... .... l.1lt. "", )0 I)• • -. 2'71 .,. Q C SF 1)9

.... . '19 � "_ ""' SUaulIan< . ... 10

_ _ .000

",,-.kill. ,,� Cf)',1&lhnr 42�J I

""'"

tDIUC. 429

....... Iic • • �)o

molft.'\llar "')()...&' 1 ___ '9 56 paial fIII1 of WI\� f\&h.1IC1ft. I"S SponaI .... 297 p«W .........,. ). �1 .

,

"toI: .� )f So � of l.irastc", pn'hlhtn. I)

E..aMtaft· f�l1w pmtulatf'

0/• • pccofio: .... -

'''''''1<

2-

'68, )90-)98

2J UI. llI 11'

Sp«ual tDt'fI)' c1rruny. 7+-"S )84, ( I, r�. P«�"'Op1C IIOU:lion., lbO

phID;aI harmonics. :�o

---- -

'" ... lth �-" """""' l
I 2 fbcnnodyaamk 1)'1Ie1D. 14·,'\ I Thom.Ioo. J J . l '\ H.II.'h, 0tnNt.as.. ,).& '0 lRfiGiIe ""elt. l.l)..lJ'

DaJnIn Thermod,

• -, . 199 __ lfi..lOO

_ )0)

.... "' Ill .a.."tioI&.

3t

l'hrff.k,d Wn. )8'J

'h)

1l1l"C'w!J I�IMIOL·). n 1'hmJkll,J w.'C'ieq:dt. n

....... .umbn. 29\ !If1 .. ...

Tuoe-depm;JcJU �bf'Odut't'f IIqIIMlOL 14�-1"�

- lJ<>.JJ'

Ttmc l.hlati�lft. IO. 11

.. _ »t> lP va eIDlWQft, lx:\ • ... 1isuoft. 4"'1 tI'oi.m � ...... Il�

__

)mn�lJ).

Tlmt'-ukkpc:nJtnt 5<:hrtd1naa equation.

� \4

lDS (�tll1. quantum \nl�rltr­ � IkT "'C'\l. �1t��1 t .. ,......,. twW or. 04R4

..... dnOoltion,

1 1 1, 170, J 3-J -&

$IIIIIiIN '1�t. �3� ....., UIth.

l·il

I"�

l8,

_ _I. " .. ,rlK'", ,:\4 .. ....18. n9-·n� . .... .. :

� JIO �1Or). " l-46O ' KIPT... ..,9-160 l)"pt-! and 1)pt-II. 4�S-4S7 � propcnlt'l. 3814382 f*.:,hty, '''6, 547 !iIpr K.amioUndc. SIIptnIrio theory, �" Q

�try• .s4Q s� \lltt. 10)...306

1'ItamocOuple. 380

� Jl lJ iI) .... . ...

� M� ':W ....,.11'1 ,.".... ....... "" \ \tlnI:w»NtIdon 1wId."� V1tt.llt'&NIMtOII tt'lfflY �" oil1 \1It1Ml pIIt.:,"- J!oI 'An\ ", .uo

\\lnKc

lU� Ill '_ H�

hlllll.ltl'l/lI l-n ��It4at. III IU

'J'Ihlll*l. 14'

I�up;"..t r-rt, loW

TraMIllon �Itmtl'lt', 'I �

Tnn\miblOn probabllllY.

...� ..�\.

".,(' lunI.llOrl. II1 )til.

rTln�.\lor. "'2-4'.'

IQI!I-IIN

Tnplet Spin 'lillIe. '.\ 1 -HZ

.J I' "n(' ,1'\1\11'. � 14

W.�C nllmhol',

Thnnchni. 202-207

11111

\I," �f Jl
Il.� "'cd bl.1IOIl�, 'l4

�\Onant, 211

\l,u� fune, '2K, , �

through ""uk bame", 206-2t17

While IIw.rh, �l. 410

Tunnel JunctIOn. 210

.. WlC'n\ 1.11-. Iii

Tumlna pomt, 148

Wlt""l\ R W. ��

1\vln paradox, 24-28

\It\\f� 111"111'11. 7�

Two-noclton nuck1. 479-4140 Type-I supcrcond\k:tl.ln, 4S��6

X

1)tpc- 1I �upcrconductm, 45S-4S6

X'fI�\

dl.lrJt;ltnlll., "6-'Ill

p.-udu�llun of. 7K-11,1

U

mathematical basis or, 1 24-- 1 J I

In three dimensions. 120 Unified theones, 529, 548-549

qo.I,IM W

V.4�(' par1l\"lf R'1.humhIP'o !1In..t.mrnt.1.

Thnnd diode, 210, 4�

energy·tlme, 120

144-145

� tl«fn*. 'II 116

TOII1I "fM. 331�1'2

Unctnainty principle, 1 1 3- 1 20, 173

TtcnprnNre. entropy and, 351-355

1tqICnI pan Of\liBVe function,

Totl' &Il,ular rtlllmtnlun, J. 1� I

Unbiased condition, 449

I

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