Symmetries in Quantum Physics

Symmetries in Quantum Physics

This Page Intentionally Left Blank

Symmetries in Quantum Physics U. Fano Department of Physics and James Franck Institute University of Chicago Chicago, Illinois

A. R. P. Rau Department of Physics and Astronomy Louisiana State University Baton Rouge, Louisiana

Academic Press San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-flee paper. ( ~ Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495

United Kingdom Editionpublished by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Fano, Ugo. Symmetries in quantum physics / edited by U. Fano, A. R. P. Rau. p. cm. Includes index. ISBN 0-12-248455-X (alk. paper) 1. Quantum theory--Mathematics. 2. Symmetry (Physics)-Methodology. 3. Mathematical physics. I. Fano, Ugo. II. Rau, A. R. P. (A. Ravi P.) QC174.17.M35S96 1996 530.1 '2--dc20 96-2004 CIP

PRINTED IN THE UNITED STATES OF AMERICA 96 97 98 99 00 01 BC 9 8 7 6 5

4

3

2

1

Contents Preface 1

xiii

Introduction

1

T r a n s f o r m a t i o n Theories: Klein's and Dirac's

........

3

1.1

S y m m e t r y and the Selection of Variables . . . . . . . . . . .

1.2

Algebraic Elements . . . . . . . . . . . . . . . . . . . . . . .

11

1.2.1

Vectors, tensors, and related quantities . . . . . . . .

11

1.2.2

A d d i t i o n and direct p r o d u c t of tensorial sets

1.2.3

Linear t r a n s f o r m a t i o n

1.1.1

5

E x a m p l e s of tensorial equations . . . . . . . . . . . .

5

....

.................

14 14

1.3

R e d u c t i o n P r o c e d u r e and Irreducible Tensorial Sets . . . . .

1.4

F u r t h e r Aspects of R e d u c t i o n

.................

20

1.4.1

R e d u c t i o n procedures

.................

21

1.4.2

Labeling of set elements . . . . . . . . . . . . . . . .

1.3.1

1.5

An analytical example: R e d u c t i o n of tensors

Block diagonalization of the reduction

1.4.4

Phase n o r m a l i z a t i o n

1.4.5

G r o u p theory . . . . . . . . . . . . . . . . . . . . . .

1.4.6

R e d u c t i o n as an expansion into eigenfunctions

1.5.1

........

23

..................

23 24 . . .

.....................

25 26

A l t e r n a t i v e sets of c o m m u t i n g invariant operators

1.6

18

22

1.4.3

S t r u c t u r e of the Book

....

17

........................

27

Quaternions ............................

28

Problems

29

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

PART A STATE R E P R E S E N T A T I V E S A N D r- T R A N S F O R M A T I O N S: THEIR

2

CONSTRUCTION

Infinitesimal Rotations

AND

PROPERTIES

and Angular

Momentum

33

2.1

Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.2

Analytical Example: Infinitesimal T r a n s f o r m a t i o n of Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . .

38

The Angular M o m e n t u m Matrices of Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Phase normalization . . . . . . . . . . . . . . . . . . 2.3.2 Definition of a s t a n d a r d base . . . . . . . . . . . . .

41 43 44

T h e F u n d a m e n t a l Representation . . . . . . . . . . . . . . . 2.4.1 Significance of half-integer j . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 47 49

2.3

2.4

F r a m e R e v e r s a l a n d Complex Conjugation 3.1

51

Analytical Representation and Implications of Frame Reversal . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.2

3.1.1 Explicit form of the m a t r i x U . . . . . . . . . . . . . 3.1.2 Properties of the m a t r i x U . . . . . . . . . . . . . . Contragredience and the Construction of Invariants . . . . .

58 59 62

3.3

3.2.1 Contragredient tensorial sets . . . . . . . . . . . . . 3.2.2 Invariant products . . . . . . . . . . . . . . . . . . . 3.2.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . Cartesian Base for Integer j > 1 . . . . . . . . . . . . . . .

64 64 67 68

3.3.1 C a r t e s i a n - t o - s t a n d a r d t r a n s f o r m a t i o n . . . . . . . . 3.3.2 Phase normalization of spherical harmonics . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 72 74

Standard r-Transformation Matrices a n d T h e i r Applications 4.1 Explicit Form and Properties . . . . . . . . . . . . . . . . . 4.1.1 Spinor m e t h o d . . . . . . . . . . . . . . . . . . . . .

75 76 78

vii

4.1.2

Algebraic approach . . . . . . . . . . . . . . . . . . .

80

4.1.3

First order differential system . . . . . . . . . . . . .

81

4.1.4

Second order differential equation . . . . . . . . . . .

82

4.1.5

S y m m e t r i e s of the s t a n d a r d r - t r a n s f o r m a t i o n s . . . .

82

4.1.6

Integrals . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.1.7

r - T r a n s f o r m a t i o n s in the Cartesian frame

4.2

Macroscopic Applications

4.3

Applications to Q u a n t u m Physics . . . . . . . . . . . . . . .

...................

85 88

Particle transmission t h r o u g h a Stern-Gerlach magnet .........................

88

4.3.2

Angular distribution of a particle in orbital motion . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.3.3

R o t a t i o n a l eigenfunctions and eigenvalues for s y m m e t r i c - t o p polyatomic molecules and heteronuclear diatomics . . . . . . . . . . . . . .

90

Spinor and vector harmonics

92

.............

C o o r d i n a t e Inversion and Parity Eigenfunctions . . . . . . .

93

Problems

97

............................

5 Reduction of Direct Products (Addition of Angular Momenta) 5.1

85

4.3.1

4.3.4 4.4

......

99

Structure and Properties of the Reducing Matrix

......

....................

100

5.1.1

Spinor approach

5.1.2

Normalization . . . . . . . . . . . . . . . . . . . . . .

102 104

5.1.3

Recurrence relations

107

5.1.4

Symmetries . . . . . . . . . . . . . . . . . . . . . . .

107

5.1.5

..................

Reduction in the Cartesian frame . . . . . . . . . . .

109

5.2

Reduction of r - T r a n s f o r m a t i o n P r o d u c t s . . . . . . . . . . .

110

5.3

Irreducible P r o d u c t Sets . . . . . . . . . . . . . . . . . . . .

112

5.3.1

Special cases

112

5.3.2

Symmetry ........................

113

5.3.3

P r o d u c t s of contragredient sets . . . . . . . . . . . .

113

5.3.4

Wave-mechanical examples

114

5.3.5

Multiple products

...................

115

5.3.6

Coupling diagrams . . . . . . . . . . . . . . . . . . .

116

5.4

......................

..............

S y m m e t r i z a t i o n of Wigner Coefficients: Invariant Triple P r o d u c t and 3-j Coefficients Problems

............................

........

119 123

viii

PART B TENSORIAL

ASPECTS

Tensorial Sets of Q u a n t u m

OF QUANTUM

PHYSICS

Operators

127

6.1

The Liouville Representation of Q u a n t u m Mechanics . . . .

6.2

Q u a n t u m Mechanics of Particles with Spin ~1 . . . . . . . .

128 130

6.2.1

132

6.3

Two-Level Systems . . . . . . . . . . . . . . . . . . . . . . .

Base sets of matrices and operators . . . . . . . . . .

135

6.3.1

A t o m in a radiation field

6.3.2

Light polarization and Stokes p a r a m e t e r s

...............

136 ......

137

6.3.3 6.4

Further applications of two-level systems: Occupation, creation, and annihilation operators . . 1 Wigner-Eckart T h e o r e m . . . . . Particles with Spin j > 3"

138 140

6.4.1

Density m a t r i x

140

6.4.2

Multipole expansion of operators G . . . . . . . . . .

6.4.3

Physical implications of the triangular relation k _< 2j 145

.....................

143

6.5

Systems with 2j + 1 Levels . . . . . . . . . . . . . . . . . . .

146

6.6

Transfer of Angular M o m e n t u m . . . . . . . . . . . . . . . .

148

6.7

Calculation of Matrix Elements . . . . . . . . . . . . . . . .

150

Problems

154

............................

Recoupling Transformations: 6-j and 9-j Coefficients

157

7.1

160 161

7.2

7.3

7.4

T r a n s f o r m a t i o n Matrices and Their Analysis . . . . . . . . . ?.1.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2

Group properties . . . . . . . . . . . . . . . . . . . .

7.1.3

Factorization of transformations

162 ...........

163

Symmetrized Recoupling: 6-j and 9-j Coefficients . . . . . .

168

7.2.1

6-j coefficients

.....................

170

7.2.2

9-j coefficients

.....................

173

7.2.3

Alternative perspectives

................

Products of Operators

..................... .....................

7.3.1

Unit operators

7.3.2

General operator

7.3.3

Commutators ......................

7.3.4

SchrSdinger equation for a (2j + 1)-level s y s t e m . . .

....................

Combining Operators of Different Systems . . . . . . . . . .

175 176 176 179 180 181 183

ix

7.5

Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . .

186

7.5.1

Interaction m a t r i x elements . . . . . . . . . . . . . .

187

7.5.2

Projection of operators

189

7.5.3

Correlations . . . . . . . . . . . . . . . . . . . . . . .

Problems

................

191

............................

195

Partially Filled Shells of Atoms or Nuclei

199

8.1

200

8.2

Qualitative Discussion 8.1.1

Two-particle states . . . . . . . . . . . . . . . . . . .

8.1.2

States of three or more equivalent particles

.....

202

8.1.3 Q u a n t u m numbers for many-particle states Shell-wide T r e a t m e n t . . . . . . . . . . . . . . . . . . . . . .

.....

204 208

8.2.1 8.2.2 8.3

.....................

202

Triple tensors and their matrices . . . . . . . . . . . Coefficients of fractional parentage . . . . . . . . . .

Algebra of Triple Tensors and Its Applications 8.3.1 I n t e r p r e t a t i o n of X(kq kok~) . . . . . . . . . . . . . . .

.......

210 213 214 215

8.3.2

Quasi-spin and seniority . . . . . . . . . . . . . . . .

217

8.3.3

Quasiparticles for the f shell

218

.............

8.3.4

D e t e r m i n a t i o n of fractional parentage

8.3.5

O p e r a t o r matrices

........

...................

220 224

PART C SYMMETRIES

OF H I G H E R D I M E N S I O N S

Discrete Transformations of Coordinates 9.1 Point S y m m e t r y Operations and Their Groups 9.2

9.3

229 .......

230

Characters of Group Representations and Their Applications 235 9.2.1 Abelian groups . . . . . . . . . . . . . . . . . . . . . 235 9.2.2

Non-Abelian groups

9.2.3

Characters of the rotation group SO(3)

..................

9.2.4

Reduction of representations

9.2.5

Reduction of set products . . . . . . . . . . . . . . .

S y m m e t r i e s of Molecules and Crystals

237 .......

............. ............

239 240 242 243

9.3.1

S y m m e t r y combinations . . . . . . . . . . . . . . . .

9.3.2

Vibrational motions

9.3.3

Molecular rotations . . . . . . . . . . . . . . . . . . .

245

9.3.4

Stability analysis of nuclear positions . . . . . . . . .

246

Problems

244

..................

245

............................

250

10 R o t a t i o n Groups in Higher Dimensions: Multiparticle P r o b l e m s

251

10.1 Four-Dimensional Rotations: The Coulomb-Kepler P r o b l e m 10.1.1 Spherical and parabolic representations 10.1.2 Rotations in four dimensions

252

.......

253

.............

10.1.3 Hydrogen a t o m in m o m e n t u m space

255

.........

258

10.1.4 Alternative subgroups of SO(4): The hydrogen a t o m in external fields

....................

259

10.1.5 Clebsch-Gordan coefficients for products of SO(4) 1 0 . 2 0 r t h o g o n a l Groups in Higher Dimensions

.

..........

261 263

10.2.1 Hypersphere in D dimensions . . . . . . . . . . . . .

264

10.2.2 Hyperspherical coordinates for multiparticle systems . . . . . . . . . . . . . . . . . . . . . . . . .

267

10.2.3 Transformation between alternative schemes . . . . .

271

10.3 Further Developments

.....................

272

10.3.1 Invariance and noninvariance groups . . . . . . . . .

272

10.3.2 "Dynamical symmetries" for a t o m s and nuclei

. . .

10.3.3 Adjoining an extra degree of freedom . . . . . . . . .

274 276

10.3.4 Alternative reduction schemes for multiparticle systems . . . . . . . . . . . . . . . . . .

277

11 Lorentz Transformations and the Lorentz and Poincar~ Groups

279

11.1 Lorentz T r a n s f o r m a t i o n s . . . . . . . . . . . . . . . . . . . .

281

11.2 Generators and Representations of the Lorentz Group 11.2.1 Four-vectors and the Lorentz metric

. . .

.........

11.2.2 Generators of the proper Lorentz group

283 284

.......

285

11.2.3 Lorentz t r a n s f o r m a t i o n s to r - t r a n s f o r m a t i o n s . . . .

287

11.2.4 Spinor representations . . . . . . . . . . . . . . . . .

289

11.2.5 Neutrino and electron spinor states . . . . . . . . . .

291

11.2.6 E l e c t r o m a g n e t i s m and its q u a n t u m . . . . . . . . . . 11.3 The Inhomogeneous Lorentz (Poincar~) Group

294

.......

11.3.1 Generators and c o m m u t a t i o n relationships

.....

295 296

xi

11.4 Field Representations . . . . . . . . . . . . . . . . . . . . . 11.4.1 Massive systems . . . . . . . . . . . . . . . . . . . . 11.4.2 Representations of massless entities . . . . . . . . . . 12 S y m m e t r i e s o f t h e S c a t t e r i n g C o n t i n u u m 12.1 Symmetries of Radial Eigenfunctions . . . . . . . . . . . . . 12.2 The Full Noninvariance Group of Hydrogen . . . . . . . . . 12.2.1 Alternative decompositions of the noninvariance group . . . . . . . . . . . . . . . . . . 12.3 Dynamics and Symmetry Transformations . . . . . . . . . .

297 298 299 301 302 305 308 310

Bibliography

313

Index

317

This Page Intentionally Left Blank

Preface

This book deals with the role of symmetry in physical theory. The search for symmetries and their exploitation for both practical calculations and basic understanding of physical phenomena have been central to the development of physics, particularly quantum physics. Symmetry considerations emerge first from direct inspection of geometrical or other circumstances. Thus, invariances under rotations and reflections of physical objects allow for classification schemes based upon these symmetries, both for the varied states of the objects and for the transformations connecting different states. The treatment of other kinds of operations, more abstract than rotations and reflections in the three dimensions of physical space, extends the scope of symmetry studies. We aim at stressing such extensions, building upon a thorough analysis of rotational symmetry. The present book grew out of an earlier work developed four decades ago by U. Fano and G. Racah, Irreducible Tensorial Sets. That monograph on the consequences of rotation and reflection symmetry, compact and not developed as a textbook, joined other books on the so-called angular momentum algebra that is central to much of atomic and subatomic physics. It clarified the issues underlying phase standardization and treated angular distributions. Parts A and B of the present book grew out of updates of and elaborations on that earlier monograph, particularly through occasional teaching in the intervening decades at the University of Chicago. The final Part C of the book has been built more recently to deal with extensions to many particles, higher dimensions, and more abstract symmetry elements such as those that involve stretches in radial coordinates and other variables leading to transformations between an infinite number of elements. Sections 1.1 xiii

xiv

Preface

and 1.5 of the first introductory chapter provide details on the organization, as do the introductory pages that open each of the book's three parts. This book is intended both for use as a graduate school text and for independent study and research of particular topics. As a text, the amount of material is probably suited to a two-semester course, although a subset of chapters may be covered in one semester; most of the present Parts A and B were covered at Chicago in one quarter. Subsidiary material, not immediately needed for the main development of the text, has been set in small type. The first nine chapters include problems to serve as homework assignments. References have been placed within square brackets in the text and listed in order at the end of the book. Since we do not intend references to assign credit, original contributions are not always cited; rather, they have been used as pointers to the literature for specific points or for more detailed consultation. Much of the material in this book stems from research supported over the years by the U.S. National Science Foundation, most recently under Grants PHY92-17874 at the University of Chicago and PHY92-10081 at Louisiana State University. The typing of the manuscript was done at the latter institution by Ms. Karla Tuley and Mr. Uday Patil and we thank them, as also Dr. Thomas Heim who helped at the finishing stage. Much help has been received over the years from several colleagues, most notably from Professor J.H. Macek, who co-authored this project for a brief period and whose assistance was especially important for Sections 11.1 and 11.2.

Chapter 1

Introduction This book develops physico-mathematical concepts and techniques which exploit invariances and symmetries in the study of physical processes. A simple familiar example of this exploitation is seen in the propagation of waves through a uniform medium which is "invariant under translations." Because of this invariance the wave equation has constant coefficients and plane wave eigenfunctions. Plane waves of different wavelengths belong to different "symmetry species" of the translation group. Products of plane waves with different wavelengths are represented as superpositions of single plane waves, and vice versa. Analogous properties occur in the study of symmetries under many different operations, on variables that may be continuous (for example, angles) or discrete (for example, "yes or no" reflection indices). We deal primarily with symmetries under rotations of coordinate axes, equivalent to--but somewhat more flexible than--the complementary rotations of directions in physical space. However, the rotation symmetries are intimately connected with symmetries under other operations, such as inversions of space coordinates, time reversal, and permutations of particles or other elements. The scope of our treatment is thereby expanded. It extends even further. Because the fundamental representation of space rotations consists of elementary 2 x 2 unimodular transformations, unitary representations of any symmetry may be "embedded" in those of space rotations. In fact, lateness in recognizing the exact role of each relevant symmetry has hindered developments and continues to hamstring their applications. As an example to be considered immediately below in Section 1.1.1, recognition of the isotropy of an elastic medium points immediately

Chapter 1. I n t r o d u c t i o n to there being only two moduli of elasticity that suffice to describe all elastic distortions. Our subject receives its main impetus from quantum physics, even though it actually applies to most branches of physics. The stimulus derives from the widespread occurrence in quantum theory of products of two or more, even numerous, factors--for example, wave functions or tensor operators--each of which exhibits a separate symmetry as a function of coordinate orientation. For instance, the interaction energy of two atomic electrons depends on the symmetry and mutual orientation of their charge distributions, while each charge distribution depends in turn on the product of a wave function and its complex conjugate. Again, when a beam of particles collides with an atomic target, each partial wave of the beam impresses on the target a specific symmetry; this symmetry manifests itself in the angular distribution of the scattered particle and of any subsequent radiation, in the final state of the target residue, and in correlations among these several elements. The volume and kind of such applications are large and keeps expanding. Their treatment involves multiple integrations over angular variables and spin orientations, but the integrations can be bypassed by algebraic procedures whose development and flexible application constitute a main goal of this book. Elements of group theory underlie much of the book. Our subject is in fact presented at times as an application of group theory. It is also often labeled as the quantum theory of angular momentum. We shall indicate the role of group theory and of angular momentum dynamics where appropriate. However, we prefer to view our treatment as an extension of vector and tensor algebra, stressing the geometrical aspect of the combination of symmetry elements in each problem and the rather careful approaches required to handle such combinations effectively. We shall also stress how our treatment fits within the mainstream of classical mathematical physics, relying on expansions into eigenfunctions (an extension of Fourier analysis) so as to separate out the calculation of each expansion coefficient. The interaction of our field with quantum theory has been a two-way process. Quantum physics has provided stimulation, concepts, and notations, which prove relevant even for applications to classical physics. Conversely, the treatment of invariances and of transformations under space rotations provides a pattern that applies to isomorphic symmetries in abstract Hilbert spaces. Some of the flavor of our subject stems from the interweaving of concepts and illustrations drawn from different branches of physics, a mix that consolidates our grasp of theoretical methods.

T r a n s f o r m a t i o n Theories: Klein's a n d D i r a c ' s

3

T r a n s f o r m a t i o n Theories: Klein's and Dirac's As a more pointed introduction, note that the description of any physical-or simply geometrical--process involves choosing a "language," typically a coordinate system or other frame of reference. Alternative languages are interconnected by "transformations," that is, by generally mathematical operations. Successive transformations from one language to another form "groups," meaning that the product of any two group elements amounts to another single element and that the product of two reciprocal elements yields the "unit" element. The concept and theory of mathematical groups, developed early in the 19th century, burst onto a broader stage in 1872 through Felix Klein's "Erlangen Program," which introduced a group-theoretical view of geometries. In this view, Euclidean geometry speaks of point sets in a plane or in a three- (or more) dimensional space, forming figures (triangles, circles, etc.) that are congruent, that is, can be made to coincide by "isometrics": rotations, reflections, and translations. Klein viewed this subject as the study of properties invariant under a distance-preserving group of transformations. In greater detail, he viewed the p l a n e E u c l i d e a n g e o m e t r y as the "Euclidean group of isometries E2," the p l a n e affine g e o m e t r y as the "Euclidean transformation generated by all parallel projections of the plane onto itself," and the p l a n e p r o j e c t i v e g e o m e t r y as the "group of projective transformations generated by all central and parallel projections of the plane onto itself." Klein further replaced the earlier distinction between Euclidean and non-Euclidean geometries (in terms of the number of lines through a given point parallel to a given straight line) by differences in their respective group structures. Considerations analogous to Klein's apply to the physics of systems in our three-dimensional space which is Euclidean over laboratory and microscopic length scales. Klein's approach combines at this point dynamical and geometrical elements of physics. Historically, even Euclid's geometry was itself influenced by and incorporated the physics of his day. Galilean and Newtonian mechanics indeed deal with the broader set of physical properties invariant under Galilean relativity transformations, including translations in time. Lagrangian and Hamiltonian mechanics extend this group by adding to it the canonical (or "contact") transformations that intermix position and momentum coordinates. The major stimulus to our subject resulted, however, from extending Klein's approach first to Einstein's relativity and then to quantum phenom-

Chapter 1. Introduction

ena. Electromagnetism had provided a critical spark by revealing how an observer in motion with respect to one---or more--electric charges perceives the presence of currents and of their ancillary magnetic field, in contrast to observers at rest with the charges. This breach of the invariance of physics as perceived by different observers was then overcome by introducing Lorentz transformations that view the electric and magnetic fields as components of a single tensor field. The magnitude of each component depends then on the observer's motion, just as the magnitude of any usual vector's components depends on the observer's orientation in space. These newly relevant transformations form the Lorentz and Poincar~ groups to be described in Chapter 11. Yet more striking revelations emerged early in this century upon extending experimentation to the atomic scale: Momentum transfers by crystal surfaces to incident atoms were found to proceed by discrete "quanta," inversely related to atomic spacings in crystal lattices; similarly, Einstein pointed out that energy transfers by a light beam to material surfaces are also discrete and proportional to the light's oscillation frequency. The "conjugation" relationships noted by Hamilton between position and momentum variables as well as between time and energy variables thus turned out to be actually a reciprocity~or "complementarity"~relation governed by Fourier transformations. (An analogous complementarity between linear and circular polarizations of light had emerged earlier.) Current experimentation is thus performed alternatively~depending on circumstances~ in either the "time" or the "frequency domain." These novel phenomena were initially described theoretically in two, seemingly incompatible, forms of quantum mechanics, namely, the Heisenberg matrix and the SchrSdinger wave mechanics. Dirac, however, soon showed not only how to "transform" these two forms mathematically into one another but further how their connection amounts only to a particular realization of a general t r a n s f o r m a t i o n theory" Any single form of atommolecular experimentation selects, in effect, a set of alternative, mutually exclusive ("orthogonal"), components of an initial "ensemble" of elements, each component being identified as an "eigenstate" of a specific physical variable. Sets of eigenstates of different variables are interconnected by a group of algebraic "unitary transformations." (The selection of alternative pairs of linearly polarized components of a light beam, by a suitably adjusted analyzer, affords a "two-level" prototype of this quantum process.) Although Lagrange and Hamilton had already cast classical mechanics as transformations between coordinate systems, quantum mechanics em-

1.1. S y m m e t r y

and the Selection of Variables

phasizes even more the central role of transformations between different "representations." This is in part because q u a n t u m mechanics precludes simultaneous specification of conjugate quantities such as position and mom e n t u m . Therefore, already at the outset, one is confronted with the choice of either the position or the m o m e n t u m representation in studying a system, giving an added dimension to other choices between alternative coordinate systems and representations. The first example of a unitary transformation is provided by the Fourier factor, exp(ipx/h), that carries from the position to the m o m e n t u m representation. 1 In Dirac's notation, this b r a c k e t , written as (xlp), is a basic prototype of other transformations between alternative representations. These themes, and even some further extensions to nonunitary transformations in the last chapters, underlie our study throughout this book.

1.1

S y m m e t r y and the S e l e c t i o n of Variables

This chapter introduces some of our principal objects of study, first through preliminary illustrative examples and then in more detailed analytical form. The discussion of this material then continues qualitatively in the final section, anticipating a succession of concepts, techniques, and applications to be treated in subsequent chapters.

1.1.1

Examples of

tensorial

Consider the vector e q u a t i o n / ~ Fx

-

ma:c,

Fy

equations

mff and its component equations, --

may,

Fz

-

maz.

(1.1)

The left- and right-hand sides of (1.1) depend on the orientation of coordinate axes. However, the equations hold irrespective of this orientation because the two sets of vector components, {Fx, Fy, Fz } and {ax, ay, az }, transform equally under coordinate rotations, while the scalar proportionality coefficient m remains invariant. (One refers to this property as the 1 With the hindsight of quantum physics, linear momentum may be identified with the gradient operation that causes translations in space. It is, therefore, properly measured in wave numbers, that is, in m -1 . In this view, Planck's constant arises only because of not having recognized this at the outset and introducing momentum as an independent entity with its own system of units (kgms -1). In this book, we will drop explicit reference to h, viewing linear momentum as an inverse length and angular momentum as a pure number.

C h a p t e r 1. I n t r o d u c t i o n

"covariance" of the two sides of the equations.) An analogous set of equations would be conceivable, in which m is replaced by a tensorial operator that changes the vector ~ into another vector. Nevertheless, the left- and right-hand sides of two or more equations would always transform equally under coordinate rotations. A second elementary example is afforded by the SchrSdinger equation for the degenerate eigenfunctions of an atomic level with angular momentum quantum number j, g u ~ ) - E.iu~). (1.2) Here the eigenfunctions depend on the orientation of an axis of quantization through the magnetic quantum number m - {j, j - 1 , . . . , - j } and experience a linear transformation under rotations of this axis, while the Hamiltonian operator H and its eigenvalue Ej remain invariant. Here again an equation with a tensorial, rather than invariant, operator is conceivable, but the expressions on the two sides of (1.2)--taken as a whole--nevertheless transform equally. We introduce now a less elementary example that demonstrates a concept fundamental for our purposes. Consider a medium subjected to an elastic distortion and indicate the displacement of a point ~' of the medium by a vector g'(r-). The distortion of the medium is described by the variation of g" from point to point in the vicinity of each f'. More specifically, it is represented by the components of the strain tensor, that is, by the set of nine derivatives of the three components {sx, sy, sz } of g" with respect to the three components {x, y, z} of ~'. These nine derivatives are conveniently laid out in a square array Ox

Oy

Oz

Os~

Os~

cOx

Oy

Oz

Ox

Oy

Oz

"

(1.3)

The strain is associated with a field of force within the medium, the stress, represented by a second tensor with nine components {Pxx, Pxy,..., Pzz}. The component Pxy represents, for example, the x component of the force transmitted within the medium across a unit surface perpendicular to the y axis. Other components are similarly defined. These nine components are not independent because their array similar to (1.3) is symmetric, with Pyx - Pxy, etc.

1.1. S y m m e t r y

and the Selection of Variables

7

The basic Hooke's law of elasticity states that a distortion is proportional to the applied stress, but it departs in general from a simple proportion as in (1.1) between force and acceleration. Elastic response does not quite imply that each component of the strain tensor is simply proportional to the corresponding component of the stress. The classical equations of elasticity relate the strain and the stress of the m e d i u m in a form that is indeed linear but each component Pij(fi, j} = 1, 2, 3 = {x, y, z}) is linearly dependent on all the components of the strain sij - 2( Oo~, 4, Oo_~). xj x,

Pij - Z

Cij~Iskl.

(1.3a)

kl

The 9 • 9 - 81 possible coefficients Cijkz that express the stress in terms of the strain form a "tensor of elasticity" which is of fourth rank. S y m m e t r y relations reduce drastically the number of independent coefficients from 81. Thus, the symmetric nature of the stress and strain tensors (Pij Pji, sij - sji) makes an immediate reduction to 6 • 6 - 36 coefficients. 2 Further reductions obtain as, for instance, in an isotropic medium, where only two constants suffice:

where p and A are the so-called Lam~ constants and ~ij is the Kronecker symbol, which equals unity for i - j and is zero otherwise. Alternatively, one may write

-- - (1/E)P~x - ( o / E ) ( P y y + P~ z ) , ~Oy -- - (1/E)Pyy - ( o / E ) ( P ~ + P~:~) - (1/E)Pz z - ( a / E ) ( P ~ 4, Pyy), 0z 0,~0~ 4- ~0y - (1 4 , a ) ( 2 / E ) P ~ y , + ~Oz - (1 + a)(2/E)Pyz Oy o,xoz 4, ~0~ - (1 4, t r ) ( 2 / E ) P ~ Ox

(1.4)

involving two constants of the medium, the Young's modulus E and the Poisson ratio a (see Problem 1.3). The appearance of two constants rather than one makes (1.4) more complicated in form than (1.1). It is not even 2In the language of group theory, this is a reduction of U(9), the unitary group that describes all possible transformations of nine elements, to its subgroup SO(9) of only orthogonal transformations.

Chapter 1. Introduction immediately obvious whether and how the left- and right-hand sides of (1.4) depend equally on the orientation of the coordinate axes. However, the answer to this question emerges clearly by an appropriate reorganization of this set of equations. The required simplification stems from the following key remark. The nine components of the strain tensor may be replaced, by linear substitution, with a new set consisting of subsets that transform independently of one another under coordinate rotations. The first two of these subsets have long been familiar in vector calculus, but the third one became familiar only with the advent of quantum mechanics. The three subsets consist, respectively, of a) the single element div ~ ' - o~, + ~Oy + oOz, . . ' Ox b) the three components of curl g', Oy

Oz

'

Oz

--

Ox

'

Oar

--

Oy

'

c) five residual elements linearly independent of those of a) and b), which are often chosen a s { ~ 0z - 0~), 0x - 0y , Oy

"Jr Oar '

Oz

Oy

'

Ox

Among these three subsets the div g" is, of course, a scalar; that is, it is invariant under coordinate rotations. Note that it equals the sum of the diagonal elements in the array shown in (1.3); that is, it equals the trace of the matrix represented by the array. The elements of subset b) transform, of course, as the components of a vector; they consist of the differences of the pairs of off-diagonal elements of the array (1.3) symmetrically placed with respect to its diagonal. The elements of subset c) transform under coordinate rotations according to a different law, to be discussed in later sections. 3 The first two elements of subset c) are linear combinations of the diagonal elements, orthogonal 4 to div g', while the remainder are the sums of the pairs of off-diagonal elements whose differences constitute subset b). Physically, div g' represents, of course, a d i l a t a t i o n of the medium in the vicinity of the point F, while curl g' represents a r o t a t i o n of the same region; such a rotation may occur in liquid or viscous media but is not a part of elastic distortions; that is, it vanishes for purely elastic deformations. 3This transformation relates to those subsets of a) and b) much as t h e transformation of the wave functions of atomic d-electrons relates to those of s- and p-electrons. 4These two combinations ( 2 , - 1 , - 1 ) and (0,1, - 1 ) , together with the (1,1,1) of div g, r e p r e s e n t three mutually orthogonal linear combinations of the diagonal derivative terms.

1.1. S y m m e t r y a n d t h e S e l e c t i o n of V a r i a b l e s

The components of subset c) represent a pure shear, a deformation at constant volume. Thus the three subsets corresond to physically distinct deformations. This replacement of the set of components of the strain tensor by separate subsets is but a first example of a general and fundamental procedure, the reduction of the set, which is described more formally below. In particular, the procedure applies equally to all sets of components ~ k of ordinary two-index tensors, which may be laid out in an array like (1.3), and specifically to the set of components Pik of the stress tensor described above. The symmetry of this tensor, indicated by Pik = Pki, causes subset b) to vanish identically in this case, as it does for the strain tensor in the case of elastic deformations. The invariant subset P=x + Pyy + P ~ coincides, when 1 with the p r e s s u r e prevailing at the point ~. Subset c) multiplied by - ~, represents a purely s h e a r i n g stress. We look upon the reduction as a change of the dependent variables in the equations of elasticity: The initial strain is replaced with a dilatation, a rotation, and a shear, each of which belongs to a different symmetry species. Similarly, the initial stress is replaced by the pressure and a shearing stress. Systematic introduction of variables belonging to distinct symmetry species represents a main feature of this book. The payoff of our change of variables occurs upon their application to the stress-strain relations (1.4): We find (1.4) to reduce to a pair of distinct and independent proportionality laws quite analogous to (1.1), between dilatation and pressure on the one hand and between shearing strain and shearing stress on the other. The first of these relations, a scalar equation, is obtained by summing the first three of Eqs. (1.4) and reversing the sign of the result; it is - d i v ~ ' - ~p, (1.5) where we have indicated by ~ = 3 ( 1 - 2cr)/E the compressibility of the medium and by p its pressure. On the left-hand side, - d i v ~" represents the negative dilatation, that is, the contraction of volume, induced by the pressure p. To establish the second relation, which is actually a set of five equations between the second-rank tensors of stress and strain, notice first that the last three of Eqs. (1.4) already represent a proportionality between corresponding components of the shear and of the shearing stress, since, for example, 2Pzy = P~y + Py~. The two remaining equations of the set of five result by combining the first three of Eqs. (1.4) so as to reproduce on the left-hand side the first two elements of the strain subset c); the resulting equations have the same form as the last three of (1.4).

Chapter 1. Introduction

10

Thus all five equations can be summarized by writing O~'= Q/p,

(1.6)

where O~" indicates the subset c) of strain components, that is, it represents the shear, Q indicates one-half of the corresponding shearing stress, and the proportionality constant p - E/2(1 + ~), introduced earlier as one of the Lam~ constants, is called the shear modulus. A striking conclusion emerges from our discussion of strain and stress. Had earlier physicists been familiar with the reduction of tensor components, they would have distinguished at the outset two types of elastic strain--dilatation and shear--and two corresponding types of stress-pressure and shearing stress. Hooke's law of elasticity would then have been translated directly into formulating (1.5) as a (negative) proportionality between the invariants--dilatation and pressure---and (1.6) as a proportionality between the shearing components that exclude any scalar or vector. It would have appeared natural to find the proportionality constants of these equations to reflect two different properties of the medium, its compressibility and shear modulus. Indeed, we used such an argument in writing (1.3b), which accommodates only two invariants. Our goal in this book is to refine and extend our understanding and practice of reduction procedures to the point of enabling us to analyze a great variety of phenomena by resolving them into separate pieces with independentparameters, much as we have resolved the equations of elasticity. In this manner, one focuses on the number and the significance of distinct parameters that characterize a physical system or phenomenon. Historically, the most familiar examples of reduction procedures occurred in the early development of wave mechanics. Consider two electrons, or other systems, which are regarded as noninteracting in an initial approximation and which have separate sets of degenerate eigenfunctions u~ ) and v(mJl), with fixed angular momentum quantum numbers j and j' and variable magnetic quantum numbers m and m', with degeneracies (2j + 1) and (2j'+ 1), respectively. The combined system has, in this approximation, degenerate eigenfunctions represented by the set of products {u~ )vm, (j')} with all different pairs (m, m'). This degeneracy is partly removed when the interaction between the two systems is taken into account. New eigenfunctions are then constructed by degenerate perturbation theory; they are represented by linear combinations of the set of products {u~ )"~m, (j') } and their energy eigenvalues depend on the total angular momentum quantum number J. This procedure--familiarly known as the "addition of angular

1.2. Algebraic Elements

11

momenta," ] + j~ - J, and discussed further in Chapter 5--constitutes a reduction of the set {u(m j) vm, (j') } of ( 2 / + 1)(2j' + 1) wave functions. This initial set is resolved into separate subsets {r each of which consists of 2J + 1 degenerate eigenfunctions with - J _ M - m + m' _< J, belonging to different symmetry species labeled by I J - J ' l _< J _< J + J ' . The stress-strain example considered earlier provides a simple illustration. The two vectors involved, Vr and ~', transform under rotations like angular momentum one, that is, j - jt _ 1. Linear combinations of the product are resolved into J - 0 , 1, and 2 as in a)-c) on p. 8. The above addition, or composition, of two angular momenta {j, j'} - 1 into sets with J - 0, 1, and 2 can be viewed in reverse. Starting with values of J - 2, they may be decomposed into i'+ i'. Proceeding next to the even simpler level, indeed the simplest nontrivial one, of J - 0 and 1, this suggests a decomposition in terms of 21-"+ }, that is, j _ j, _- !2. This hint of the existence of nonintegral j, while foreign to classical physics, takes on enormous importance in quantum applications. Indeed, the lowest 1 provides the fundamental representation and will nonzero value, j - ~, play a prominent role throughout the book.

1.2

Algebraic Elements

Our immediate task is to set forth in mathematical detail procedures that have only been implied in Section 1.1. Here, and throughout the book, we shall draw attention to many seemingly trivial points whose neglect has generated confusion. 1.2.1

Vectors, tensors, and related quantities

The components { Vx, Vu, Vz } of a vector 1~ are defined with reference to a specific system of Cartesian axes identified by unit vectors {&, ~), ~}. The same vector is represented with reference to a different system of Cartesian axes by another set of components, related to the first one by a linear transformation, V~ V,7 Vr

= = =

D~V~ + D~vVv + D~V~, D,TxVx + D,Tv Vv + D,7~ Vz , Dr V~ + Dr Vv + Dr Vz .

(1.7)

An array of vector components is often itself called a vector, especially by analysts. In physics, however, a vector is usually understood to be

Chapter 1. I n t r o d u c t i o n

12

a quantity characterized by a magnitude and by a direction of physical space. This quantity is then represented by combining the sets of vector components and unit vectors in the form v -

~v~ + yVy + ~v~.

(1.8)

A rotation of the coordinate axes brings about a transformation (1.7) of the components and a related transformation of the set of vectors {~, ~, ~}, but V itself, being associated with a fixed direction of space, is invariant under coordinate rotations. On the other hand, one may consider rotations of the vector itself, that is, changes of its direction. Such a rotation would bring about a transformation of the components alone, of the type (1.7), if the coordinate axes are kept fixed, or a transformation of the unit vectors {~, y, ~} alone if the axes rotate rigidly with the vector. This transformation of {~, ~, ~} to {~, ~,~} involves the inverse of the matrix D in (1.7) such that ~V, + ~Vy + ~Vz and ~Vr + T)V, + ~Vr represent the same physical object, the vector V. The transformation (1.7) of {V,, Vy, Vz } is therefore said to be "contragredient" to the transformation of {~, ~), ~} as we shall discuss further in Section 3.2, along with a notation borrowed from quantum mechanics (Section 3.2.3). In ordinary tensor algebra, a tensor T of degree n has components labeled with n indices, each index relating to one Cartesian axis {x, y, z}. For example, T=~y indicates a component of a tensor of third degree. A rotation of coordinates replaces T, yy by a transformation analogous to (1.7) with a new component T ~ , which is a linear combination of the 27 old components Tx=~, Txxy,..., Tzzz. We regard the tensor T itself as a geometrical entity represented, in terms of its components and of a set of unit tensors, by an equation analogous to (1.8). The unit tensors may be regarded as combinations of n unit vectors; for example, the nine unit tensors of second degree may be indicated as "dyadics" {~5:,5:9,&5,...,~}. A tensor T is itself invariant under coordinate rotations, whereas its components and the unit tensors depend on the orientation of the coordinate axes. Mathematical relations equivalent to (1.7) and (1.8) occur in quite different contexts. Thus, in quantum mechanics, an atomic system with angular momentum quantum number j has eigenstates U(m j) with magnetic quantum numbers m. A general state a of this system can be described as a superposition of eigenstates with different m by the formula ..o

C a - Z U(Jm)a(Jm)' m

(1.9)

1.2. A l g e b r a i c E l e m e n t s

13

which is analogous to (1.8). The quantum number m refers to an axis of quantization, that is, to a system of coordinates. A rotation of this system induces a linear transformation of the coefficients a~ ), which generates a new set of coefficients

a(J) - ~ ~,mD(J)a(mJ).

(1.10)

m

A rotation of the axis of quantization also induces a related or contragredient inverse transformation of the set of eigenstates u(m j) to u (j), and ~-~z u(J)a(J)equals ~m u(Jm)a~), both representing Ca. When the system itself rotates, as, for example, under the influence of a magnetic field, the change of Ca is described in the Schrbdinger representation by a transformation of the type (1.10) which leaves the states U(m j) fixed, or in the Heisenberg representation by a transformation of the base states u(m j) which leaves the coefficients ag) fixed. Because of the analogy between the transformations (1.10) and (1.7), induced by coordinate rotations, the name "tensor" has been applied occasionally to eigenstates u~ ) or to coefficients a~ ). In fact, among the states of a p-electron in an atom one can choose three basic ones which are called P~, Pu, and p~ and which have the same transformation laws as the unit vectors {~, ~), ~}. These states constitute a set of orthogonal unit vectors in the 3-dimensional Hilbert space of p states. The algebra to be developed in this book deals equally with sets of tensor components and with sets of unit tensors, with sets of coefficients a~ ) and with sets of eigenstates. To treat all these kinds of quantities equally, we shall call a "tensorial set" of order n any set of n quantities which are defined in connection with a system of space coordinates and which experience a linear transformation when this system rotates. To preserve the advantage of vector algebra, which deals with a vector as a unit, we shall regard as an element of our algebra a whole set rather than its individual elements. Although in tensor analysis the word "tensor" often indicates the set of components, the definition of tensorial sets is more general because it applies not only to a set of components but also to a set of unit vectors or tensors. In physics one often refers to rotations of a physical object of interest, rather than to rotations of coordinate axes. Strictly speaking, only the mutual orientation of a physical object and of coordinate axes is physically significant, and the axes themselves are usually identified by the orientation of a polarimeter or other instrument of observation. In this book, to avoid confusion, we shall refer normally to rotations of coordinate axes.

14

1.2.2

C h a p t e r 1. I n t r o d u c t i o n

Addition

and

direct

product

of tensorial

sets

Given two similar tensorial sets of quantities ai and bk, of the same physical dimensions and experiencing the same transformations induced by coordinate rotations, the sums of the pairs of corresponding elements of the set, ai + bi, constitute a new tensorial set, the sum of the initial sets. Given two similar tensorial sets, one of n quantities ai and one of m quantities bk, the n • m pairs of quantities aibk also experience a linear transformation whenever the ai and bk are so transformed. For this reason the set aibk constitutes a tensorial set, called the d i r e c t p r o d u c t of the initial sets. Both addition and product are elementary operations of tensor algebra but the product is more important for our purposes. When a rotation of coordinates replaces the set elements {ai} by new elements {aa} and the {bk} by {bz}, the product set {aibk} is changed into {ao,ba}. The linear transformation that relates the {a,~bz} to the {aibk} is obtained by forming the direct product of the two transformation matrices. The construction of products and the analysis of their transformations are discussed repeatedly in the following. Note, however, at the outset that the word "product" includes various manners of combining the elements of the pair, depending on the nature of the quantities ai and bk. Thus if ai and bk are the components of two tensors ') or the components a~ ), b(m j, , of two quantum-mechanical state functions, the symbol aibk indicates the ordinary algebraic product of ai and bk. On the other hand, if the two sets consist of eigenstates of two atomic systems with specified quantum numbers their product is symbolic; u(m j)"vlTbl 0') represents a joint state of the two systems in which one system is in the state u~ ) and the other in the state v(mJl). Still different is the combination of a tensorial operator T/ and a quantum-mechanical state u(m j), where the "product" T/u~ ) implies operation on the state by the operator. The product of two unit base vectors, ~y, is a symbol that represents a unit tensor of second degree, called a dyadic.

1.2.3

Linear

transformation

Given a tensorial set of n quantities ai and a matrix T of n rows and columns, the set of quantities

bk -- ~ Tkiai i

(1.11)

15

1.2. A l g e b r a i c E l e m e n t s

is clearly also a tensorial set. The transformations of the set ai induced by rotations of the coordinate system are special cases of (1.11). To prevent confusion we shall refer to the transformations induced by rotations as rtransformations. If the elements ai of a tensorial set are visualized as the components of a vector in a representative space with n dimensions, the transformation (1.11) may be regarded in two alternative ways: A) The transformation establishes a correspondence between this vector and another vector of the same space or of a different space of equal dimensionality. This interpretation is appropriate, for example, when Tki is the matrix of a quantum-mechanical operator that changes a set of eigenfunctions {ui} of a Hamiltonian into a new set of wave functions of the same system. B) The transformation represents the effect of a change of coordinates in the representative space. From this standpoint the representation vector remains unaffected by the transformation, while the form of its representation changes from {ai} to {bk}. This point of view implies that the transformation (1.11) is invertible and is accordingly appropriate only when the determinant of Tki is nonzero. It is particularly appropriate to the representation of r-transformations and of transformations that reduce a set into an aggregate of separate subsets. In the course of reduction and other procedures we shall often deal with the combined effect of two linear transformations to which point of view B) is applicable, typically of a reducing transformation represented by (1.11) and of an r-transformation represented by ao~ -- Z

(1.12)

Dalai. i

The reducing transformations, as well as many other transformations T, act equally on the corresponding elements of a set after or before it has been subjected to an r-transformation. (Thus, for example, the construction of the three subsets a), b), and c) on p. 8 proceeds equally regardless of coordinate orientation.) On the other hand, the transformation T in (1.11) will cause the r-transformation D~k of the set {bk} to differ from the rtransformation Dai of {ai} in (1.12). We write

bz - Z D;kbk - Z D;k Z Tkiai - Z Tzaaa , k

k

i

a

(1.13)

Chapter 1. Introduction

16

that is,

In matrix notation, Eq.(1.14) amounts to D'T = TD.

(1.14a)

An expression of D~k is then obtained from (1.14) by replacing ai with the reciprocal of the transformation (1.11), namely, ai = ~k(T-X)ikbk. Thus we have

or, in matrix notation, D' - T D T - 1 .

(1.15a)

Matrices D and D' connected by this well-known formula are said to be equivalent. Recall, however, that equivalence holds only for transformations

T that act equally, before or after an r-transformation. U n i t a r y m a t r i c e s . The r-transformation matrix D for the Cartesian components of a vector in (1.7) is unitary; that is, its reciprocal equals its Hermitian conjugate, D - 1 - D t - D*,

(1.16)

but the general r-transformation matrix D of (1.12) need not be unitary. 5 However, it may be shown by matrix analysis, using elementary group properties, that any r-transformation matrix can be reduced to unitary form (see Appendix A of Fano and Racah [1]). As we shall discuss in Chapter 3, unitarity of a transformation preserves the norm of a set to which it applies. (For this very reason, the transformation matrices of quantum mechanics are unitary to preserve normalization and probability.) Accordingly, we shall assume, unless otherwise stated, that all r-transformation matrices have been brought to unitary form. We shall also generally bring to unitary form the other transformation matrices T of interest. 5The dagger in (1.16) for Hermitian conjugation stands for the combined operations of transposition (,,,) and complex conjugation (*).

1.3. R e d u c t i o n P r o c e d u r e a n d I r r e d u c i b l e T e n s o r i a l S e t s

1.3

17

R e d u c t i o n P r o c e d u r e and Irreducible Tensorial Sets

When a tensorial set transforms under rotations according to (1.12), that is, as = ~ i D~iai, each as is, in general, a linear combination of the whole set {ai}. A reduction of this set is achieved by a transformation bk = ~ i T~iai when the r-transformation D' of the new set {bk}--given by (1.15)--is such that each of the transformed elements be depends only on a subset of {bk}. That is, the new r-transformation matrix D ~ has nonvanishing elements only within square blocks along its diagonal, as shown in Fig. 1.1; the new matrix with this property is called "block diagonal." Each subset of {bk} then constitutes a separate tensorial set whose r-transformation matrix is the corresponding submatrix along the diagonal of D ~. Recasting transformation matrices D into a block-diagonal form is regarded by group theory as the main aspect of reduction, because group theory focuses on transformations per se rather than on the sets to be transformed. The procedure of reducing a given tensorial set into separate subsets has an ultimate limit. The ultimate subsets are called i r r e d u c i b l e t e n s o r i a l sets and their r-transformations termed i r r e d u c i b l e r - t r a n s f o r m a t i o n s . The subsets obtained in the introductory examples of Section 1.1 are already irreducible; the three subsets a)-c) on p. 8 are the irreducible subsets of strain, the subsets { r with IJ - J'l _< J _< J + J' on p. 11 are the irreducible subsets of the product of two sets of angular momentum eigenstates. Indeed, it was the classification of the energy levels of atomic electrons which drew physicists' attention to irreducible sets. All irreducible sets with the same r-transformation belong to the same s y m m e t r y species. When one visualizes the elements ai of an initial set as the components of a vector in a representative space and their r-transformation Dai as representing the effect of a change of coordinates in that space---according to point of view B on p. 15--the reduction procedure takes the following aspect. A rotation of coordinate axes in physical space brings about a rotation of axes in the representative space. This rotation generally intermixes all axes corresponding to the vector components {ai}, but the axes corresponding to different subsets {b~} do not intermix. In other words, the axes corresonding to each irreducible subset of {bk} span a subspace that remains invariant under r-transformations. Each of these subsets may be regarded as the set of components of the projection of the initial vector onto one invariant subspace.

18

C h a p t e r 1. I n t r o d u c t i o n

F i g u r e 1.1" Diagram of a transformation matrix in reduced form. All matrix elements outside the shaded regions vanish.

1.3.1

An

analytical

example"

Reduction

of tensors

The sets of Cartesian components of ordinary tensors are reducible in general, except in the case of vectors (that is, of tensors of degree 1). Sets of base unit tensors are similarly reducible and in fact are reduced by the same transformation which reduces tensor components. Irreducible sets obtained by reducing sets of tensor components and of unit base tensors serve to construct new tensors in various ways. Consider first the simultaneous reduction of the set of components of an ordinary tensor T and of the corresponding set of base unit tensors. This operation splits the tensor T into a sum of tensors T (i) each of which consists of an irreducible set of components and of an irreducible set of base tensors. Each T (i) may accordingly be called an irreducible tensor. For example, a second degree tensor (such as the stress or strain tensors) represented by T -

&~:Tx~ + &~)T::y + . . .

+ 5ST~z

(1.17)

resolves into three parts T - T (~ + T (a) + T (2),

(1.18)

1.3. R e d u c t i o n P r o c e d u r e a n d I r r e d u c i b l e T e n s o r i a l Sets

19

where

T(~

-

1 -~(~ + f~f~+ ~ ) ( T . . + T~ + T~z) ,

T(1)

_

i -~[ (9~: - ~,9)(Tyz - T z y ) + (~:~c - ~c~,)(Tz:~ - T:~z)

(1.1Sa) -

+ ( ~ y - 9~)(T~ - T~) ], (1.18b) T (2)

-

1

89 ~ ( 2 ~ - i h - y~))(2T~ - T ~ - Tyy) + ( ~ - ~f~)(T.. - T~y)+ (f~ + ~ ) ( T ~ + Tzy) + ( ~ + ~5)(Tz~ + T~z) + (~y + y~)(T~y + Ty,) ].

(1.18c) This resolution into irreducible parts parallels exactly the treatment in a)-c) on p. 8 of the strain tensor. The form of the expressions (1.18) emphasizes the structure of the irreducible tensors T (~ T (1), and T (2) in terms of irreducible sets. However, each of these tensors may also be written out in the form (1.17)--that is, expanded into the initial reducible set { ~ , ~ y , . . . , ~ } - - a n d may thus be regarded as a tensor of second degree with nonindependent components. For example, T (~ has, when so written out, only three nonvanishing components corresponding to &&, 99, and 5i and all equal to (T:~ + Tyy + T z z ) / 3 . In (1.18a), T (~ is given as the product of two invariants ( ~ + yy + ~.~)/~/3 and (T:~:~ + Tyy + Tzz )/Vc3. 6 The tensor T (x) has, when written out in the form (1.17), six nonvanishing components that constitute an antisymmetric matrix. In the form (1.18b), T(1) consists of a set of three components, {(Ty~ - T~y)/V/2 }, etc., with the same r-transformations as a set of vector components, and of a set of base tensors {(~)~- 5/))/V~ }, etc., with the same r-transformations as the set {&,/~, ~}. The tensor T (2) has, when written out in the form (1.17), a set of nine components which constitute a symmetric traceless matrix when laid out in the array form (1.3). In the form (1.18c), T (2) consists of a set of five components and of a set of five unit base tensors. The base tensors, (2~ - ~~)~))/v/-6, etc., constitute the simplest set of irreducible unit base tensors of order five, analogous to the usual set of unit base vectors

{~,~,~}. In general, the simplest set of irreducible unit base tensors of order 2 n + 1 is the irreducible subset of highest order that results from the reduction of the n-th degree set { } n , & n - l y , . . . , ~ , } . Note how this construction yields only irreducible sets of odd order; sets of even order originate separately as we shall see later. From this perspective, whereas (1.18c) gives the simplest 6Tensors with this structure have been called "isotropic."

C h a p t e r 1. I n t r o d u c t i o n

20

irreducible unit base tensor of order five, Eqs.(1.18a) and (1.18b)are not the simplest sets of order one and three, respectively. Rather, the simplest irreducible units would be unity (&", etc., with n = 0) and the set of unit base vectors &, y, and ~. Irreducible tensors can be constructed by combining irreducible sets of the same order but consisting of unit tensors and of tensor components of different origin. In particular, one may take an irreducible set of components derived from the components of a reducible tensor of degree n and combine it with a contragredient set of base tensors of degree lower than n. For example, with the components of the second degree tensor T of (1.17) one can construct the ordinary scalar 1

x/~ (T** + Tyy + T~)

(1.19/

and the vector 1

(1.20) ) + y(T . + (T.y Vz which are irreducible tensors of order 1 and 3, respectively, but do not coincide with T (~ and T (11 as given by (1.181. If the set of components of T is the direct product of the sets of components of vectors g and b, i.e., if T** - a,b,, etc., Eqs. (1.19) and (1.20) coincide respectively with 6 . b and • b', to within normalization constants. (An alternative rendering in terms of quaternions is given in Section 1.6.) If ~ is the derivative vector operator x7 and b" a vector field #(x,y,z), that is, if T - x~/5, then (1.191 and (1.201 coincide respectively with div 15 and curl t7, to within normalization constants, as we have anticipated in Section 1.1. v

1.4

:

Further A s p e c t s of R e d u c t i o n

The concept of reduction, introduced in Section 1.3 for the r-transformations of tensorial sets, is equally applicable to any group of transformations of a set of quantities. (A g r o u p of transformations is a set of transformations that includes the identity, the inverse of any transformation, and the product of any two of these transformations.) Examples of such groups are coordinate translations, reflections on a plane, inversion at the origin, rotations by a fraction of 360 ~ permutation of particles, time reversal, and their various combinations. We shall continue to refer primarily to transformations induced by coordinate rotations but much of this section will be equally relevant to other groups.

1.4. F u r t h e r A s p e c t s of R e d u c t i o n

21

Recall that reduction serves to select variables of a phenomenon that remain independent of one another, such as the dilatation, rotation, and shear of a medium, which are separate components of its strain. The example of strain-stress equations rests on the isotropy of the medium, which requires the left- and right-hand sides of Hooke's equation to transform equally under coordinate rotations. Analogous circumstances underlie the search for the variables appropriate tcr the treatment of other phenomena. This section outlines broader aspects of reduction for purposes of general introduction, even though many of them will prove relevant only in later chapters of this book. 1.4.1

Reduction

procedures

In our initial examples the reduction of tensorial sets was presented as an "ad hoc" process, but we should, of course, seek procedures for reducing completely any given tensorial set. A leading concept for this purpose emerges from a remark in Section 1.3: In a vector space representation, the elements of each irreducible subset span a subspace that remains separate from the others under the transformation group of interest. We should then endeavor to identify these i n v a r i a n t subspaces. In the example of combined states of angular momentum j" and ~', the individual J values, IJ - J'l < J < J + J', provide such invariant subspaces with varying dimensions 2J + 1. To this end, we may view all the vectors g~ of an invariant subspace c~ as degenerate eigenvectors belonging to the same eigenvalue of some invariant operator Q [the squared angular momentum operator (j" + j',)2 in the above example]. The aggregate of the vectors {~',,, gZ,...} for all invariant subspaces {c~,fl,...}, together with arbitrarily chosen but unequal eigenvalues {q,~,q~,...}, identifies the operator Q. Reduction procedures apply this consideration in reverse direction, starting from the construction of suitable invariant operators, which are then diagonalized to find the eigenvectors corresponding to their several eigenvalues. Quite generally, full reduction of a tensorial set {ai} is achieved by constructing and then diagonalizing a m a x i m a l set of m a t r i c e s {Qr } operating on {a~} that commute with one another and with all r-transformations D of {ai}. ("Maximal" means that no further matrix, linearly independent of all the Qr, fulfills the same specifications as the Q~.) In the example of the products of quantum-mechanical eigenstates "~"(J)'(J')~" described in Section 1 1 the invariant operator to be diago) k

22

C h a p t e r 1. I n t r o d u c t i o n

nalized in the reduction process is the squared angular momentum If] 2 = ]j' + j"l 2. This operator was not mentioned explicitly in the other examples concerning the reduction of the strain tensor components or of general tensors, but it was indicated that the three subsets called a), b), and c) in Section 1.1 transform under coordinate rotations like the wave functions of s, p, and d electrons, which are in turn eigenfunctions of the squared angular momentum. We shall discuss explicitly in Chapter 2 how the operator Ill 2 represents not only the angular momentum in quantum mechanics but a more general geometrical entity relevant to any tensorial set. Indeed, diagonalization of the matrix of Jf]2 is a necessary step of the procedure for reducing any tensorial set. (This property of tensorial sets will also be relevant to the reduction of any set of variables with unitary transformations.) Circumstances under which additional invariant operators must also be identified and diagonalized will emerge below.

1.4.2

Labeling

of set elements

The elements of any irreducible set may be further transformed as convenient without impairing the set's irreducibility. In particular, the set elements may be selected initially, or transformed into, eigenvectors of suitable operators. For example, the products of quantum-mechanical states ~m'

are initially eigenvectors of the angular momentum component

along the axis of quantization, Jz - jz + J'z, with eigenvalues M - m + m', respectively; this property is preserved in the usual reduction procedure for diagonalizing Ill 2. Each element of the irreducible set thus obtained is then identified as a joint eigenvector of the pair of commuting operators Ill 2 and Jz with specified eigenvalues. The operator J~ is not invariant, of course, under all rotations of coordinate axes but only under their subgroup of rotations about z. This remark suggests that a second set of operators be introduced as a complement to the {Qr} which serve to select irreducible sets. The second maximal set {Ns } will consist of operators that commute with one another and with the {Qr } and remain invariant under a s u b g r o u p of the group of interest. J o i n t e i g e n v e c t o r s of all operators {Qr} and {Ns} constitute an aggregate of irreducible sets, each of whose elements is labeled by the corresponding eigenvalues of {Qr} and {N,}.

1.4. F u r t h e r A s p e c t s of R e d u c t i o n

1.4.3

23

Block diagonalization of the reduction

The labeling of individual set elements amounts to choosing a particular coordinate system in each invariant vector space g~. This choice is incidental in contrast to the separation of invariant subspaces but it has a major simplifying influence on reduction procedures. This is seen by reJ ) v m, (J')} are labeled as calling that the elements of the reducible set { (urn eigenvectors of Jz - j z + J~z because of being constructed as products of eigenvectors of j z and J~z. R e d u c t i o n of the product set by diagonalization of I l l 2 may then proceed s e p a r a t e l y for each subset with given M - m + m ~ . The diagonalizing matrix T becomes thus block d i a g o n a l , like the reduced r-transformation D ~ in Fig. 1.1, one block for each value of M. This simplification is applied routinely in the familiar example of the addition of angular momenta, ~ + ~' - J , as will be detailed in Chapter 5. We stress here its general relevance and hence the importance of labeling all set elements as eigenvectors of an operator set {N8 } invariant under a suitable subgroup of transformations.

1.4.4

Phase normalization

Labeling each element of a tensorial set, for example, a state u~ ), as an eigenvector of certain operators identifies the element only to within normalization. The magnitude of its normalization coefficient is generally prescribed by the initial definition of the set and by its successive transformations but phase normalization acquires a meaning only with reference to the phases of other set elements. Even though phase normalization of wave functions is in a sense arbitrary, performing a reduction or other algebraic transformation on tensorial sets with arbitrary phases would amount to doing analytical geometry in their vector space without a fully specified coordinate frame. Phase normalizations must indeed be specified. This operation has been done somewhat haphazardly in the past for lack of a clear system of normalization, with resulting confusion. Uniform systems of phase normalization can in fact be established with reference to symmetry elements that are preserved in the course of operations, be they r-transformations, construction of direct products, or reduction of such products. Normalization is imposed on the r-transformations of irreducible sets in Chapter 2 by requiring the transformation matrices to change into their complex conjugates under reflection on the x z plane, which serves as a zero azimuth for polar coordinates. Application of a fur-

24

Chapter 1. Introduction

ther symmetry akin to time reversal, in Chapter 3, will extend phase normalization to the construction of products and to their reduction. These developments, while somewhat laborious, will illustrate the necessity of exploiting a maximal amount of compatible symmetries when describing and treating analytically any system of interest.

1.4.5

Group

theory

Group theory underlies the treatment of tensorial sets since the set of all coordinate rotations in physical space forms a group. The set of rtransformation matrices of any tensorial set constitutes a "representation" of this group. The approach of group theory to the reduction procedures differs from our own in two main respects. First, group theory deals primarily with the group r e p r e s e n t a t i o n s , that is, with the r-transformation matrices in our case, and with their reduction, thus regarding the sets of quantities transformed by these matrices merely as the basis of reference of particular representations. We focus instead on the base sets themselves, tensorial sets in our case, which are generally of primary interest to physical theory; however, we shall often be confronted with connections between set elements and thus be drawn closer to group theory. Second, group theory deals with procedures of greater generality because its scope is not limited to a single group. Thus, for example, it does not center its attention on the particular operator Ill2; note, however, that we shall broaden our own scope progressively and show, among other items, how a generalization of Ill 2 is relevant to all continuous groups. Group theory strives, of course, to identify and exploit properties of the aggregate of all group operations, in contrast to those of single operations. It uses this approach to construct the invariants to be diagonalized in reduction procedures. Consider, for example, the product of two matrix elements of the same or of different group representations. This product is initially a function of the parameters that identify the particular rotation of coordinates to which the matrices correspond. However, a group invariant is obtained when this product is integrated, or averaged, over the whole range of variation of these parameters, that is, over all group operations. Important theorems are established by this approach, in Appendix A of [1] and in standard tests on physical applications of group theory. The vanishing of off-diagonal elements outside the diagonal blocks of the reduced matrix diagrammed in Fig. 1.1 is then established through the following argument. An invariant matrix Q commutes, by definition, with any r-

25

1.4. F u r t h e r A s p e c t s of R e d u c t i o n

transformation matrix; that is, we have QD - D Q = 0. After Q is brought to diagonal form, with eigenvalues q,, q z , . . . , this condition takes the form ( q a - qz)Daz = 0 and thus requires all Daz to vanish unless q~ = q~. This last step is quite familiar in other branches of mathematical physics.

1.4.6

Reduction

as an expansion

into eigenfunctions

The process of reduction by diagonalization of invariants has an additional implication which emerges from the dependence of each r-transformation matrix element on the parameters that identify a specific rotation of coordinate axes. (We shall see later how r-transformation matrices are most frequently represented as functions of three Euler angles.) The "Schur lemma" of group theory, presented in Appendix A.c of [1], shows that the set of all matrix elements of all inequivalent irreducible r-transformations forms a c o m p l e t e o r t h o n o r m a l set of f u n c t i o n s of the group parameters. [Two transformations D and D ~ are inequivalent if they cannot be related by (1.15a).] This result will also emerge in Chapter 4, where the matrix elements of the standard irreducible r-transformations will be shown explicitly to be eigenfunctions of systems of eigenvalue equations. The transformation that reduces an r-transformation matrix D determines the coefficients of the expansion of D into a complete set of eigenfunctions of the group parameters. In other words, the complete reduction of any tensorial set amounts to an eigenfunction expansion of the set's dependence on coordinate rotations. Here again the methods we are developing take the aspect of an extension of standard procedures of mathematical physics. More specifically, they follow the path of the Fourier analysis, which exploits the invariance in space or time under a simple translation of the origin of coordinate or the corresponding invariance under rotations about a fixed axis (see, for example, Appendices A and C of Fano and Fano [2]) to expand in terms of a complete set of trigonometric or other functions.The corresponding analysis of r-transformations involves, instead of simple sine (or exponential) functions of one variable, the more complicated irreducible representations which are functions of three Euler angles. As the Fourier expansion of a sharply varying function requires an admixture of high-frequency sine functions, the reduction of a tensorial set that depends sharply on coordinate orientation yields irreducible components corresponding to large eigenvalues of the invariant operator I l l 2. Quantummechanically, this reduction expresses the uncertainty relation between the conjugate quantities of angular position and angular momentum.

26

1.5

Chapter

1.

Introduction

S t r u c t u r e of t h e B o o k

The r-transformations form the orthogonal group SO(3) of three-dimensional rotations, whose real representations follow from the rotational transformations of Cartesian coordinates. With a view to quantum-mechanical applications, however, we shall deal from the outset with the complex rtransformations appropriate to handling probability amplitudes as in (1.10). Just as the complex probability amplitudes are viewed as square roots of probabilities, so also the "fundamental" (that is, lowest order) complex representation of rotations is two-dimensional and may be viewed as the square root of a rotation in real three-dimensional Cartesian coordinates. These two-dimensional objects known as spinors and their counterparts with higher even dimensions extend the list of scalars, vectors, and tensors that are familiar in classical physics. Indeed the matrices denoted by r)( 89 in (1 10) constitute the fundamental representation of the special unitary group in two dimensions SU(2). This group is identified as the c o v e r i n g g r o u p of SO(3), embracing both the rotations involved in SO(3) and their sign reversal. This sign reversal, represented by a factor -4-1, is familiar in quantum mechanics as the effect of a 2~r-rotation on a spin- 89variable. ~ It also arises upon combining rotations with reflections, whether through the origin or through a plane. 9- J f r l ~ 7 1 t

Part A of this book, comprising Chapters 2 through 5, deals with rtransformations viewed as representations of SU(2). It deals with the fundamental and higher-order representations of SU(2), with their symmetries under complex conjugation, with their products, and with their immediate relevance to quantum mechanics. This extended development reflects a sustained effort to describe how rather familiar mathematical formulations emerge from specific geometrical and physical elements. The origin of their symmetries, phases, and other normalization factors, as well as of angular dependences, which often prove obscure, should thus become apparent. Part B deals with triple or multiple tensorial products of quantummechanical states and especially their extension to operators, reflecting in part the multiparticle structure of matter. Alternative combinations of ZThe 4-1 factor of null- and 2r-rotation in SU(2) contrasts with SO(3), where b o t h these rotations reduce to the identity operation. Thereby, SU(2) contains double the number of elements in SO(3), equivalence between the two expressed more accurately as SO(3) _~ SU(2)/Z2, where Z2 is the group of integers modulo 2 or the group of the two

elements

(10)o (10) 0

1

0

-

1

"

1.5. S t r u c t u r e of t h e B o o k

27

multiple sets and alternative reductions play a major role because they emphasize different physical factors. Passage between them is the subject of Chapter 7, a major chapter of this book. The interplay of rotational symmetry with the permutational symmetry of identical particles, along with the treatment of incomplete atomic and nuclear shells, is the subject of Chapter 8. A main feature of this material, which forms the core of the "Racah-Wigner algebra" and of this book, lies in recasting all integrals over angular variables into algebraic expressions. This is an end product of the remarks in Section 1.4.6. Another important feature isolates dynamical parameters by factoring out all dependences on laboratory reference frames. In particular, magnetic quantum numbers are eliminated through applications of the Wigner-Eckart theorem. Further features deriving from the noncommutability of factors in multiple products are discussed below in Section 1.5.1. The final part, C, describes the combination of r-transformations with additional symmetry elements and variables, such as multiple reflections, higher dimensions, and translations in space, velocity, and time. The combination of these elements modifies group structures beyond the range afforded by the covering group. We shall stress, however, that the necessary extension of the r-transformation algebra can be largely achieved by modifying the fundamental SU(2). Part C also deals with extensions to operators that do not reflect symmetries of a system but have representations with continuous, rather than discrete (like Fourier coefficients) labels. This extension is essential especially for describing quantum systems with a continuous spectrum.

1.5.1

Alternative sets of c o m m u t i n g invariant operators

The simple examples of reduction described in this section have yielded irreducible subsets no two of which are of the same order or correspond to the same eigenvalue of jfj2. Put another way, since the index M in Section 1.4.3 is fixed at the value m + m', specifying the eigenvalue J labels every state of combined angular momentum uniquely. There was then no need for simultaneous diagonalization of additional invariants. This simplification occurs, in general, only when the initial set consists of the direct product of no more than two irreducible sets. (For the initial example of the components of the strain tensor ~ ' , the two factor set consists of the components of V and ~'.) On the other hand, the reduction of the direct product of three, or more, irreducible sets leads generally to an aggregate of irreducible sets, two or more

Chapter

28

1. I n t r o d u c t i o n

of which have equal order and thus correspond to equal eigenvalues of if[2. The characterization of these alternative sets of equal order requires considerations of a different kind than we have introduced thus far and will form the subject of Chapters 7 and 8. The main point of those chapters, which we anticipate here, is that the necessary set of several commuting invariant operators is generally not unique. A rather familiar example occurs in the study of two particles with nonzero spin moving in a central field; it involves four angular momenta--two orbital ([1, ~'2) and two spin (~'1, g2)--which may be coupled in the (LS) or (jj) modes to yield a constant Ill ~ = leq "4-8"1-~'e2-~'~'212. (Here Ill 2 - I/1 +[~12, etc.) One particular set of commuting invariants may be appropriate to the analysis of a particular problem, but more often different aspects of the same problem are best described in terms of the eigenstates of different, noncommuting operators. Thus, for example, the angular distribution of hght emitted by an atom depends on the orientation of the orbital currents of the atomic electrons, that is, in essence, on the expectation value of the vector orbital momentum (f_,); on the other hand, this expectation value is not a constant of the motion but varies in the course of time under the influence of fine or hyperfine structure interactions which depend on operators incompatible with f~. An effective treatment must then deal alternatively with different irreducible base sets of eigenfunctions and must accordingly shift--often repeatedly--from one base to another. The considerations that select the base appropriate to a particular stage of analysis are often dynamical and hence beyond the scope of this book. However, the transformations from one base set to another represent connections between the geometrical symmetry properties of alternative irreducible base sets of eigenfunctions. Accordingly, the study of these transformations constitutes a main part of our program, the part that is often called "Racah-Wigner algebra." The foundations of this algebra s have been clarified long ago, but more recent experience with the calculation of energy levels, of transition and collision rates, and especially of the angular distribution of reaction products, keeps opening up novel aspects thus expanding our subject to the frontier of current advance.

1.6

Quaternions

Together with real and complex numbers, quaternions and octonions are the only other sets of numbers whose algebra yields a well-defined product operation. The step from real and complex numbers to quaternions requires abandoning commutativity for this product operation and the further step to octonions abandons associativity as well. 8Although technically not an algebra as defined by mathematicians, this usage is now standard in atomic and nuclear physics.

1.6. Q u a t e r n i o n s

29

A quaternion a = (a0, ax, %, az) is a generalization of complex numbers, expressible as a = ao + a x i + a ~ j + a z k in terms of three independent square roots o f - 1 " i, j, and k. Scalars and vectors may be embedded into quaternions, a point of view once in vogue with Hamilton and Maxwell. Physics has instead been dominated by vector algebra and tensor analysis, but the quaternionic picture presents nevertheless attractive features. If a scalar S is embedded as the quaternion S = (S, 6") and a vector as the quaternion V = (0, tT), quaternionic multiplication, defined with the rules i 2 = j2 = k 2 = - 1 , i j = - j i = k (note the anticommutativity) and its cyclic permutations, embraces ordinary multiplication by scalars and scalar or vector products of two vectors. Thus, from two vectors and b, the product of the two quaternions a and b is another quaternion c: ab=c-(-if-b,

(1.21)

~ xb').

The scalar and vector products stand here together as the components of a general quaternion c. The quaternionic algebra is closed, a b also being a quaternion, whereas the dyadic product ~b of two vectors ~ and b includes not only a vector but also scalars and a tensor of rank 2 as, for instance, in (1.18). With quaternions, division too is well defined, a / b being another quaternion in general, whereas with vectors, the ratio ff/b has meaning only for parallel vectors. Further, square roots too can be extracted, again an operation not defined for vectors. Thus, the "vector" quaternion p = (0, ~g) has the square root ~(1,~), where 15 represents the unit vector y / p , and the "scalar" quaternion p2 = (p2,6) has the square root (0, ifi). The connection of these operations to the introduction of spinors into physics by Weyl, Pauli, and Dirac will be taken up in Section 2.4. Finally, note that the quaternionic derivative V = (0, V) combines with a "vector" quaternion F = (0, F) to yield both the div and curl operations V F = ( - d i v /~, curl F). These features of quaternions are also shared with the so-called "geometric (graded) algebra," originally due to Grassmann. Here again, a "geometric product" is defined which includes the scalar and vector product of two vectors ff and b as an "inner" and an "outer" product, respectively. There has been recent interest in reviving geometric algebra for use in classical and quantum mechanics [3].

Problems 1.1 W i t h reference to (1.3a), (a) Show t h a t the tensor of elasticity for a cubic crystal obeys the following relations: C ~ C x z z z , etc.

-

Cyvv u -

Czzzz,

C~vv

-

Cyv~

=

Chapter

30

1. I n t r o d u c t i o n

(b) C~= v and other such coefficients with an odd number of identical subscripts must vanish. Identify the symmetry operation on the cube which leads to each of these conclusions. Show, therefore, that only three nonzero constants C,x,x, Cx=vv, and C, wv suffice to specify the complete tensor of elasticity. 1.2 Isotropic noncrystalline materials have a further symmetry beyond those considered in Problem 1.1. Identify it and obtain the additional relation between the three nontrivial components of that problem. This leads to two surviving constants X and p, called the Lam~ constants:

1.3 Contrast (1.4) with the result in Problem 1.2 to obtain the following relations between the Lam~ constants and the Young's modulus and Poisson's ratio: #-

E E~r 2(1+~r) ' X - ( l + c r ) ( 1 - 2 c r )

"

1.4 Express the bulk compressibility to, defined in (1.5), in terms of the Lamd constants p and ~. 1.5 Construct tensorial sets by direct product of the components of V and of the electric field E or the magnetic field /3. Consider their reduction and show that Maxwell's equations are relations between irreducible sets. Discuss also the invariance of these equations under inversion of the coordinates (reflection at the origin). [Note, however, that inversion is separate from rotations.] Relate your work to the analysis of the reduction of tensors in Section 1.3.1. Extra teaser: Consider how to design test bodies that would measure div E, curl E, div B, and curl B at a point of space.

PART

A

STATE REPRESENTATIVES r-TRANSFORMATIONS: THEIR

CONSTRUCTION

AND PROPERTIES

AND

32

Part A

This initial Part A of the book, comprising Chapters 2 through 5, covers the basic algorithms of irreducible tensorial sets, typically of angular momentum eigenstates. The concepts and representations of infinitesimal rotations and the closely associated operators of quantum-mechanical angular momentum are introduced in Chapter 2, with primary reference to the r-transformations of the "fundamental" prototype, namely, of two-element (two-dimensional) sets which are isomorphic to eigenstates with angular momentum 71 (as in Section 1 5). As in the elementary examples of Sections 1.1.1 and 1.3.1, both an initial Cartesian base set and a more useful, irreducible, "standard base" set for describing r-transformations are defined. Chapter 3 deals at some length with the curious joint roles of "frame" reversal and complex conjugation in the standardization of r-transformations. These operations, closely allied to what is sometimes dubbed "time reversal," reverse the sign of angular momentum while leaving the coordinate vector unchanged. Their role is crucial to the proper definition of standard r-transformations, particularly to their proper phase normalization. The construction of standard r-transformations for N-dimensional irreducible sets is described in Chapter 4, with alternative explicit derivations of rtransformation matrices and their symmetries (including under coordinate inversion), together with several physical applications in macroscopic as well as quantum physics. Chapter 5 will deal finally with reducing products of irreducible tensorial sets by explicit orthogonal transformation. This chapter will be restricted to the products of two sets, called in quantum mechanics the addition of two angular momenta. The coefficients of reduction, variously termed Clebsch-Gordan or Wigner or 3-j coefficients, and their quantum-mechanical applications will be considered. More complicated reductions of products involving more than two sets will emerge in Part B of the book, which focuses on operators, complementing Part A's focus on single states.

Chapter 2

Infinitesimal R o t a t i o n s and A n g u l a r M o m e n t u m The study of the effects of coordinate rotations is intertwined with mechanical considerations owing to a discovery of quantum physics: The quantities previously known as momentum and angular momentum are in fact, when expressed in units of h, eigenvalues of infinitesimal translation and rotation operators, respectively. This remark has been extended by regarding the eigenvalues of reflections and other symmetry operations as dynamical quantities. Infinitesimal operations of continuous groups have particular relevance owing to Lie's basic remark: Their properties and representations determine all group transformations because any finite transformation can be generated as an integral over infinitesimal transformations. Accordingly, the study of continuous groups of transformations, and of the physical variables to which they apply, centers on the properties of infinitesimal operators. The construction of operator matrices depends of course on the identification and normalization of the sets of variables to be transformed. Accordingly, this construction bears on the reduction proceduresmand more generally on a full mastery of the use of symmetries--much as does the identification of set elements discussed in Section 1.4. Our main task in this chapter will be to show how purely geometrical considerations determine the commutation properties and the eigenvalues of infinitesimal rotation operators and hence those of the angular momentum components and of the squared angular momenta. The general 33

34 C h a p t e r 2. I n f i n i t e s i m a l R o t a t i o n s a n d A n g u l a r M o m e n t u m

procedure by which the commutation relations restrict the form of angular momentum matrices is familiar from quantum mechanics texts; it will be reproduced here for completeness and with a view to establishing any remaining indeterminacy of the operations. This indeterminacy will then be eliminated by requiring the angular momentum matrices, and indeed all r-transformations, to experience complex conjugation upon reflection of coordinates across the x z ( " z e r o azimuth") plane. The example of the f u n d a m e n t a l r e p r e s e n t a t i o n of these matrices i namely, the 2 • 2 "Pauli matrices" which are the smallest nontrivial o n e s 1 will finally be discussed. This representation serves far more than a purpose of illustration: Specification of the 2 x 2 angular momentum matrices is fundamental because it carries over to the matrices of higher order pertaining to larger irreducible tensorial sets. Indeed, the larger angular momentum matrices can be built from direct products of the fundamental Pauli matrices, much as all integers in arithmetic can be constructed by repeatedly adding unity. Because any unitary matrix can be specified in terms of its submatrices of order 2, this specification provides input for the unitary transformation matrices of any finite set of variables. A treatment of unitary transformations in terms of tensorial operator techniques will be given in Chapter 6. The form of Pauli matrices thus becomes relevant to all of quantum mechanics.

2.1

Basic Relations

Consider rotations of the coordinate system about a particular axis, say rotations Rz(99) of 99 radians about the z axis, and the corresponding rtransformation matrix of a tensorial set, Dz(~). When 99 is very small, 99 = c ~ 1, the matrix D z is approximately equal to the unit matrix being expressed as Dz(c) = 1 + e I z + O(e 2)

(2.1)

Iz - [ dDz(~)/d99]~o= o

(2.2)

where

and O(c 2) indicates, as usual, terms of order e2. A rotation by a finite 99 may be generated as a succession of a large number n of rotations by 99//n radians, so that

Dz(T ) - [D~(~/n)]~ - [1 + (~/n)L ]~ + O(1/n2),

(2.3)

2.1.

Basic

Relations

35

and, in the limit n ---+ oo, Dz(~)-

lim [ l + ( ~ / n ) I z ] n

n -exp(~Iz)-l+gIz+7~o

1

2 I z2 + . . .

. (9..4'~

-....* CX:~

A simpler differential analog of this transformation is seen in a formula t h a t represents the effect of translating an abscissa x's origin by a upon a function f ( x ) . This translation replaces the abscissa x of a point by z - a , thus replacing f ( x ) by 2

d

f ( x - a) -- f ( x ) - a f ' ( x ) + -~a f " ( x ) + . . . Here the operator - d / d x

-- e x p ( - a - ~ z ) f ( x

).

(2.5)

plays the same role as Iz does in (2.4).

Returning now to rotations, we consider an arbitrary axis fi instead of the ;~ axis. 1 To this end we introduce besides Iz analogous matrices Ix and I u pertaining to rotations about the axes x and y, yielding D~(e)

=

1 + e [uxlx + uyly +

=

l+efi-f+O(e

uzI, ]+ o(~2)

2)

(2.6)

and ..#

D a ( ~ ) - exp(~fi. I),

(2.7)

where the vector notation I indicates the set of three matrices Ix, Iy, and Iz. At this point we face a characteristic difference between rotations and translations. Whereas translations of the coordinate origin in different directions of space c o m m u t e with one another, rotations about different axes through a fixed origin do not. To determine the specific effect of nonc o m m u t a t i v i t y for infinitesimal rotations, consider the succession of four infinitesimal rotations indicated by

(2.8) which would yield no net rotation if the order of application were irrelevant. [The formula means that R y ( - c ) is applied first, then R x ( - 6 ) , etc.] The aThe description of rotations in three dimensions as three basic rotations with respect to the three coordinate axes hides a subtle feature, namely, that the proper description concerns planes. In two dimensions, the xy plane, there is only one independent rotation about the normal to the plane, conventionally described as the z axis, with coordinate qo as in (2.1) and (2.2). Similarly, in three dimensions, {ux,uy,uz ) should actually be thought of as normals to the yz, zx, and xy planes, respectively, as in (1.18b).

36 Chapter 2. Infinitesimal Rotations and Angular M o m e n t u m

corresponding r-transformation formula, obtained from (2.7) by expansion to second order in the angles, is e6I=eeI~e-6I=e-~I~ - 1 + 6c (I~Iy - IyI=).

(2.9)

The resultant of the four rotations is, in the same approximation, a single rotation by -6e radians about the z axis, R ~ ( - 6 e ) , as shown by the geometrical construction of Fig. 2.1, where the numbers mark the five successive positions of the tips of unit vectors along the coordinate axes.

1,5

2,3 4,5Z

X~I

5 N

1,2

J~ \Y 3,4

F i g u r e 2.1: Graphical demonstration of the commutation rule I= Iy - Iy I~: = - I z .

2.1. Basic R e l a t i o n s

37

Therefore the matrix (2.9) must equal D z ( - 6 ( ) = 1 - 6(Iz + 0(52(2),

(2.10)

I~Iy - IyI~ = - I ~ .

(2.11)

IyIz - IzIy = - I x ,

(2.11a)

IzI~ - I~Iz = - I y .

(2.11b)

whence follows Similarly one finds

This result will be verified analytically by a simple example in Section 2.2. The three equations (2.11) represent the essential information on rtransformations provided by the group structure of space rotations. An invariant infinitesimal operator is obtained, instead, by averaging the matrix (2.7) over the direction of the axis ft. In this case, expansion to second order in c is required to yield a nontrivial result, (Da(e))

-

( 1 + ((u~I~ + u y I y + UzIz) 1 2(u~I~ + uuly +uzlz) 2 + ... } + ~(

=

l+g

1 (2

(I x2 + I y 2+ I z ) +2 O ( ) .

(4

(2.12)

(The averaging causes any term to vanish that is odd in u~, uy, or Uz, 2 - (u~} - (u~) 2 - 1/3.) The matrix owing to symmetry; it also gives (u~} [II 2 - 12 + 12 + Iz2 is manifestly invariant under coordinate rotations. In particular this matrix is readily seen to commute with each of {Ix, Iu, I~ } on the basis of (2.11). The infinitesimal matrices {Ix, I~, I~ } are not Hermitian. Indeed their eigenvalues should be imaginary for the r-transformation matrix Da(~p) = exp(~p fi.I) to be unitary, that is, to have eigenvalues of unit modulus. (The differential translation operator d/dx is similarly anti-Hermitian.) Accordingly, one generally replaces these matrices by the set J,

= I~/i

,

Jy = Iy/i

,

Jz = I~/i .

(2.13)

Thereby (2.7) becomes Da(~) - exp(i~ ti..I)

(2.14)

and the commutation relations (2.11) take the form

JxJy - JyJx = iJz , JyJz - JzJy = iJ~,

JzJ~ - J~Jz = iJy

(2.15)

38 C h a p t e r 2. Infinitesimal Rotations and Angular M o m e n t u m

which is familiar for the quantum-mechanical angular momentum operators when expressed in units of h. In vector notation these relations take the form ,] x J - i,].

(2.15a)

-Ill

The operator IJI 2 has a broader--geometrical rather than dynamical --significance than is implied by its common name of squared angular momentum. We shall verify that scalars, vector components, and the components of the shearing strain or stress are all eigenvectors of IJI 2, corresponding to its eigenvalues 0, 2, and 6, respectively. The imaginary element has been introduced in (2.13) much as in wave mechanics, where it complements the non-Hermitian character of infinitesimal translations and where a sign reversal of the imaginary unit amounts to a permutation of p and x in their commutator. Here the corresponding permutation of the operators in (2.15) is associated with a symmetry operation: Throughout this section we have implicitly followed the usual convention of right-handed coordinate axes, accepted arbitrarily instead of the alternative equivalent assumption of a left-handed triad {x, y, z}. Complex conjugation is thus associated here with the transition from right- to left-handed axes. Note that sign reversal of the imaginary unit in (2.13) leads to a corresponding change in the commutator relation in (2.15), equivalent to the reversal J --+ - . ] . The broader context and implications of this reversal will be the subject of Chapter 3.

2.2

Analytical Example: Infinitesimal Transformation of Cartesian Coordinates

As an illustration of the formulas of Section 2.1, let us now construct explicitly the matrices {Ix, Iy, Iz } which transform the set of components x, y, z of a vector ~'. To this end we first show how a rotation of coordinate axes about the direction fi by a small angle c modifies the set {x, y, z} and then we cast the result as the transformation of the set {x, y, z} by the matrix (2.7). Recall that the change of the components of ~" resulting from a small rotation c fi of the coordinate axes is equivalent to the change due to a rotation of F by - c fi with the axes kept fixed. This rotation changes g into r-; - ~ ' - e ~ • ~'. If we indicate the components of ~" and r-; by one-column

2.2. A n a l y t i c a l E x a m p l e

39

matrices, the effect of this rotation is represented by

-

y

-~

u~x-u~z

Z

-

y

~xY-

+ eux

z

+O(c

2)

~yX

z

+ euy

0

-y

+ eUz

-x

x

0

+ O(e2).

(2.16)

The transformed set (2.16) is represented in the desired form by rewriting (2.16) as D~(e){x, y, z} - [ 1 + e(uxIx + uyIu + UzIz) + O(e2) ] {x, y, z}, with I,-

(o oo)(oo1) 0

0

1

0

-1

0

, Iy-

0

0

0

1 0

0

, I~-

(2.17)

(OLO) -1

0

0

0

0

0

.

(2.18) It is readily verified that this set of matrices satisfies the commutation relations (2.11); it also becomes Hermitian upon division by i because each matrix is antisymmetric. Similarly, we work out the explicit form of the averaged matrix (2.12) starting from the expansion of the rotated vector ~' to second order in e, namely, -" x 2/L x (fix r-")+ O(e 3) (2.19) Figure 2.2 illustrates the successive terms of this expansion, which builds r ~ starting from r Averaging over the orientation of fi yields

(#)

(2.20)

-

The set of components of this vector, { (x'), (y'), (z')}, can be expressed as (D~(e)) {x, y, z}, with the matrix (Da(e)) in the form (2.12), by setting

Ill ~ -

-2 0 0

0 -2 0

0 ) 0 . -2

(2.21)

It is readily verified that this matrix coincides with the sum of the squared matrices (2.18). That the matrix III 2 equals a multiple of the unit matrix

40 C h a p t e r 2. I n f i n i t e s i m a l R o t a t i o n s a n d A n g u l a r M o m e n t u m

in this example is consistent with the fact that the set of vector components {x, y, z} is irreducible and that their r-transformation matrix Da(~), with {I~, Iy, Iy } given by (2.18), is an irreducible representation of the rotation group. These properties may be contrasted with those of the corresponding matrices {Ix, Iy, Iz } and III 2 pertaining to the infinitesimal transformations of sets of tensor components with n _> 2 indices. Upon formation of the direct product of two vector spaces a and b, the infinitesimal rotation matrices of the product space are S Jaxb -- la x J b + J ~

(2.22)

x lb,

as is apparent from the definitions of direct product and of infinitesimal operation (Sections 1.2.2 and 2.1). The matrices {Ix, Iy,Iz} then have 3 n rows and columns with only 2n nonzero elements equal to =t=1 and are reducible. The matrix ]II 2 is nondiagonal; upon diagonalization, its eigenvalues are not equal, that is, III ~ is no longer a multiple of the unit matrix. As indicated in Section 1.5.1, diagonalization of 1~12 fails in general to reduce completely the initial set of tensor components except for tensors of second degree.

!c2 x(ux 2 6( 3

(/~X)3~

~

-c~xF

F i g u r e 2.2: Successive terms in the expansion of a rotated vector YI.

2(2.22) m a y b e viewed as a n a n a l o g of the differential f o r m u l a

~(ab) - a~b-I- (~a) b.

2.3. A n g u l a r M o m e n t u m

2.3

M a t r i c e s of Q u a n t u m

Mechanics

41

The Angular M o m e n t u m Matrices of Q u a n t u m Mechanics

The quantum mechanics of systems invariant under coordinate rotations usually deals with sets of eigenstates u~ ) which are irreducible--and hence eigenstates of the squared angular momentum IJI2--and which are also eigenstates of one component of this momentum, 3z, belonging to its eigenvalue m. The identification of individual elements of a tensorial set as eigenstates of a specific operator helps their further manipulation in the sense indicated in Section 1.4.1: Each element of a direct product set {u(jl,ml)v(j2,m2)}, constructed with sets of eigenvectors of 31z and of J2z, is itself an eigenstate of the total momentum component Jz - Jlz +J2z belonging to its eigenvalue ml + m2. A reduction of this product set may proceed separately for each subset of states with a given eigenvalue ml +m2, since this value remains invariant in the reduction process. The choice of set elements that are eigenvectors of J z helps in constructing the matrices of J , , Jy, Jz, and I.I[2 for these sets. This is done by a well-known procedure which we summarize here. Linear combinations of the last two equations in (2.15) yield the pair of relations J~(Jx -4- iJy) - (J~ -4-iJy)Jz = + ( J x -4- iJy).

(2.23)

The diagonality of J z is now exploited by writing explicitly the matrix elements of (2.23) between two eigenstates of J~ with eigenvalues m' and m. One finds the equation ( m ' - m :F 1)(J~ -4-iJu)m,m = 0,

(2.24)

which requires all matrix elements (J~ :i: iJu)m,m to vanish unless they are adjacent to the diagonal, with m' = m :[: 1. The two matrices J~ + iJ u thus play the roles of "raising" and "lowering" operators, respectively, because they change an eigenstate of J~ with eigenvalue m into one with m' - m + 1. They are also called "ladder" operators. The next step of the procedure considers the squared magnitudes of the nonzero elements of J~ :i: iJ u. Since the two matrices are Hermitian conjugates, we express each squared element as a diagonal element of a matrix product, I(J~ +

iay)m•

-- Z

(ax :F iJy)mm, (Jx -4- iJy)m, m

FF$ t

:

[(J~ :F iJy)(J~ :1: iJ u)].~m - (j2 + jy2 :]: Jz)s~m

42 C h a p t e r 2. I n f i n i t e s i m a l R o t a t i o n s a n d A n g u l a r M o m e n t u m

--

(1~1 ~ - J~ T J ~ ) ~

-

(1~1~)~

- m(m + 1).

(2.25)

This equation determines the spectrum of eigenvalues of J z and of ].~]2 through the requirement that its right-hand side match the nonnegative character of the squared modulus on its left. On the right of (2.25) the matrix elements ( I J l ~ ) ~ ~re themselves nonnegative; their values must also be identical for all elements of the irreducible set under consideration, that is, independent of m, because these elements belong to the same eigenvalue of I~12. The term - m ( m -[- 1), on the other hand, becomes increasingly negative as Iml incre~es. Consistency of (2.2g) then requires the sequence of eigenvalues of m to terminate at a sufficiently low value of Ira[ for m < 0 as well as m > 0. The values of m, in turn, must satisfy (2.24), which requires them to form a steady sequence, with unit intervals, which can terminate only at a zero value of (Jx • iJy)m'm. Consistency of (2.25) then requires its righthand side to take, at unit intervals of m, a sequence of nonnegative values which terminates with a zero value on either side of m = 0. The plot of these values, IJI ~ - m(m • 1), against m (Fig. 2.3) represents a pair of parabolas with vertices at m - :F 1 where they reach their maximum value iJi ~ + ,.1 Zero v~]ue~ of I~1~ - m ( m + 1) lie at the two intersections of each parabola with the m axis. The range of m compatible with (2.25) is limited by the intersection of each parabola nearest to m = 0, m+ and m_. It is the condition m + - m_ -- integer that specifies the spectrum of eigenvalues [~]2 as well as the range of values of m, that is, the spectrum of Jz. The well-known solution of this problem is represented by IJI 2 m

-

g(g + 1)1,

with

3

j - 0, 89 1, ~ , . . .

(2.26)

-j,-j+l,...,j-l,j.

The occurrence of alternative sets of solutions, with integer or half-integer values of j and m, results from the combined requirements of (a) integer intervals m ~ - m and (b) symmetry of the problem under sign reversal of m. Its far-reaching implications will be developed in the following. Equation (2.26) lists 2j + 1 alternative eigenvalues of m for each value of j. Accordingly, the list implies the existence of irreducible J-matrices, and therefore of irreducible r-transformations, of any positive order. For given values of j and m, Eq.(2.25) reduces now to ](Jx :k iJy)m+l,m[ 2 = j ( j + 1 ) - m ( m :k 1);

(2.27)

all other elements of J~ • iJy vanish. The matrices of Jz and of J~ -1- iJy

2.3. A n g u l a r M o m e n t u m M a t r i c e s of Q u a n t u m M e c h a n i c s

w

m_

0

F i g u r e 2.3: Dependence of

43

v

m+

I~12 - m(m -4- 1) on m.

of each order have thus been determined to within the complex phases of the matrix elements of 3~ + iJ u . In the language of group theory, r-transformations or rotations in threedimensional space constitute the orthogonal group 5'0(3). The three "generators," {J~, Ju, Jz}, defined through the commutation relations (2.15), completely prescribe this group.

2.3.1

Phase

normalization

The complex phase of (J~ + iJ u)m• is now determined by imposing certain symmetry requirements. This process implies a corresponding phase normalization of the elements of irreducible sets transformed by the .I matrices, in accordance with Section 1.4.4. As anticipated in Section 1.4.4, we choose to this end the z z plane of Cartesian coordinates as the zero azimuth of rotations about the z axis, that is, as the origin of the rotation angles ~o in (2.3). The operation R~z of reflection across this plane has the effect of changing the coordinate system from right- to left-handed and the r-transformation Dz(~O) into its complex conjugate (end of Section 2.1). This plane thus serves as a symmetry element with respect to complex conjugation, a role that is preserved upon constructing products of tensorial sets and upon reducing them. Rotations within this frame--that is, rotations about the y axis--are accordingly represented by real matrices Du(0 ). Equation (2.7) sets Dy(0) = exp(0Iy) = exp(i0Ju) , showing that a real Du(0 ) implies an imaginary matrix Ju" The matrix J~ must instead be real in order that D~(~) = exp(i~J~) be changed into its complex conjugate--as Dz(~,) is--by the reflection R~,. Altogether these specifications require (J~-4-iJy)m• to be real. The phase normalization of (J~ 4- iJu)m• will now be completed by

44 C h a p t e r 2. I n f i n i t e s i m a l R o t a t i o n s a n d A n g u l a r M o m e n t u m

following the "Condon-Shortley convention" which sets (J~ + iJy)m+l,m - [j(j -t- 1 ) - m ( m • 1)] 89.

(2.27a)

Together with (2.24) this implies 1 [j(j + 1 ) - m ( m + 1)] 896,,m+l,

(Ju)um - 7 : 8 9

(2.27b)

+ 1 ) - m(m + 1)] 89~#,m-t-1 9

(2.27C)

This phase convention determines the sign of the rotation in the Hilbert space of the coefficients a~ ) in (1.10), induced by a rotation of the coordinate axes in physical space. A rotation by the infinitesimal positive angle e about the coordinate axis y thus induces in Hilbert space the rotation [ D y ( c ) ] . m a(mj) m

=

a (j) + 89

{

~--~(1 + i~ Jy).ma(Jm ) m

1

7.10) 1)] ~.t,_1 - [ j ( j + l) - p ( . + ""7 l)j a .(J) +l

[J(J + 1 ) - , ( , -

}

(2.28)

In the particular case of a state uj(J) , that is , with a(m j) - 6mj , Eq . (2.28) (J) 1 1 1 yields aj_ 1 = - T e [ j ( j + 1) - (j - 1)j]7 - - e (j/2)7; the negative sign of this component coincides with that o f - e fi x § in the example of Section 2.2. The phase normalization introduced here persists when Dy(e) operates on the sum of two irreducible sets or, according to (2.22), when it operates on their direct product. This normalization is thus preserved by operations of tensorial algebra in accordance with the program outlined in Section 1.4.4. 2.3.2

Definition

of a standard

base

Irreducible tensorial sets of order 2j+ 1, whose infinitesimal r-transformation matrices J coincide with those of a set of the quantum-mechanical coefficients a~ ) in (1.10), will be said to constitute a s t a n d a r d base. The elements of their matrices J~ and Jy are given by (2.27b, c) and those of their matrix J~ by (J~).m = mS.m,

m = {-j,-j

according to the stipulations of Section 2.2.

+ 1 , . . . , j},

(2.29)

2.4. T h e F u n d a m e n t a l R e p r e s e n t a t i o n

45

Equations (2.29) and (2.14) imply that the r-transformation matrix induced by rotation ~ about the z axis, in the standard base, has the explicit form [Dz(~)]m'm

-

eim~~

(2.30)

Parametrization of an arbitrary rotation in terms of three Euler angles (r 0, ~) factors it into the product of three successive rotations, of which the first and last keep the z axis fixed, while the middle one has the y axis fixed. 3 The corresponding r-transformation matrix factors accordingly into three matrices, Dum(r 0, ~O) - [ D z ( r

e i"r [Dy(0)],,~ e imp.

-

(2.31)

Two of the three factors are thus given explicitly, while the third one is given in (2.28) only for infinitesimal values of 0. Construction and application of the matrices Dy(0) = e x p ( i O J y ) for finite 0 forms the subject of Chapter 4.

2.4

The Fundamental Representation

The infinitesimal rotation matrices are applied here to formulate the com1 This plete r-transformation for the smallest nonzero value of j, namely, 3" construction provides a key prototype. The infinitesimal rotation matrices for j - 71 themselves have important special properties The eigenvalues of Jz (and of J~ and Jy as well) are m - + 89 when j - 89 and the magnitudes of the nonzero elements of J~ + iJ~ equal 1 according to (2.27a). It is then convenient to represent all three matrices as the products of a factor ~1 and of certain standard "Pauli matrices" {a~, %, cry} with elements of unit magnitude. Setting j - ~ in (2.27b) (2.27c), and (2.29) gives

J-T

1 (~

,

O'x --

(01) 1

0

'

O'y --

(0 i

0

'

O'z --

(10) 0

--1

9

(2.32) Notice that ~r~ is diagonal, cr, real and symmetric, and cry imaginary and antisymmetric and that all three Pauli matrices have zero trace; the three matrices ~ri are in fact equivalent to one another in the sense of (1.15a); 3 T h e t h r e e r o t a t i o n s are m o r e p r o p e r l y d e s c r i b e d in sequence as first a r o t a t i o n t h r o u g h r a b o u t the z axis, t h e n t h r o u g h 0 a b o u t the resulting y axis (call it y') a n d finally t h r o u g h ~0 a b o u t the final z axis (called z'): D,,(cp)Dy,(O)D.(r Using ( 1 . 1 5 a ) a n d writing first Dz,(~o) = Dy,(O)Dz(~o)D],l(O), followed by Dy,(O) = Dz(r162 gives the m o r e convenient a l t e r n a t i v e ~endering as in (2.31) in t e r m s of fixed axes.

46 C h a p t e r 2. Infinitesimal R o t a t i o n s a n d A n g u l a r M o m e n t u m

that is, they transform into one another when the three coordinate axes are interchanged by a rotation. These matrices are easily seen to have the commutation and anticommutation properties ~

ay -

o'y O'x -

O'iO'k "~ O'itYk

--

2iO'z ,

(2.33)

26ik.

(2.34)

Note that (2.34) holds only for j - 7, while (2.33)is equivalent to (2 15). From these properties follows the useful identity ~7.ff 5 - b - f f . b + i Y . f f x b ,

(2.35)

where ff and b' are arbitrary vectors of physical space; when ff - b ' - /t, Eq.(2.35) reduces to (Y-ti) 2 - 1. (2.36) Note also that the above algebra of Pauli matrices is isomorphic to that of quaternions discussed in Section 1.6. This last equation enables us to represent the general r-transformation matrix (2.7) as a linear combination of Pauli matrices. Upon setting I i 71(7 in (2.7) and expanding into powers of 9, higher order terms condense into two terms only owing to (2.36), yielding D a ( p ) - exp(il~ ~. 5 ) - E ( i 719

it . ~ ) n / n l .

-

c o s 2~2

l+isinT~/t.Y.1

n

(2.37) This form of r-transformation in terms of an axis ti and of the rotation angle 9 proves less convenient than the parametrization in terms of three Euler angles (r 0, 9). [Equation (2.37) also depends on three parameters, that is, the value of 9 and two angular coordinates of/t.] The matrix thus factored, (2.31), is still expressed in terms of Pauli matrices or of their explicit form, D(r O, ~)

1 trz ) exp(il r ~rz) exp(i~1 00"y ) exp(i 79

[cos 89162162

o'~]

1 1 x [cos~101 + i sin ~10 cru][cos 791 + / s i n 79(r~] (2.38) (exp(il~) 0 x

0 1 ) ( cOslO1 exp(-iTr ) - sin 70

(exp(i 89 0

0 ) exp(-i-}9) "

l~

sin cos 10

(2.39)

2.4. The Fundamental Representation

47

From these equations one can construct the irreducible infinitesimal rotation matrices and the r-transformation matrices of all orders, as will be described in Chapter 4. The existence of the r-transformation matrices (2.37) or (2.38) of order 2 shows that the usual real transformation matrices of Cartesian coordinates, of order 3, are not the most compact ones, even though those of order 2 are complex rather than real. This fact is stated mathematically by saying that the r-transformations of order 2--which are actually the most general representation of the unitary unimodular group, SU(2)--constitute the "fundamental representation" of the group of rotations of physical space. The fundamental representation takes on broader significance in both classical and quantum physics because it models dichotomous v a r i a b l e s either/or, off/on, up/down, etc.Din general. Such applications to "twolevel systems" will be taken up in Section 6.3 but here we note that light polarization--linear polarization with respect to two orthogonal directions, or left/right circular polarization--affords a canonical example. Indeed, Stokes's analysis [4] in 1852 of a beam of light in terms of four parameters, the intensity I and three parameters (P1, P2, P3), can be seen as an early precursor of the relationship between SO(3) and SU(2) and, more generally, of the quantum physics of two-level systems. The parameters P stand for intensity fractions when the beam passes through alternative polarization settings, 1 and 2 for two linear settings at mutual 450 and 3 for selecting the right-circular component. These together provide a complete characterization of the light beam. Interpretation of these parameters as probabilities serves to make the step into quantum physics, the components acting as probability amplitudes.

2.4.1

Significance

of half-integer

j

The occurrence of irreducible r-transformations with half-integer values of j and m has a well-known consequence. For these values the matrix Dz(27r) = exp(27riJz), representing a rotation of coordinate axes by 360 ~ equals - 1 instead of 1 as one might have expected. The same holds for the matrices Da(27r) with an arbitrary axis ft. Since physical quantities generally remain invariant under such rotations--equivalent to a null rotation-one finds in physics no irreducible set of measurable quantities that corresponds to a half-integer j-value, that is, which has even order 2j + 1. However, sets of eigenstates or of probability amplitudes with half-integer j do occur in quantum mechanics. These sets do not represent any physically

48 Chapter 2. Infinitesimal Rotations and Angular M o m e n t u m

observable quantity directly but occur only as direct-product pairs in the expression of any observable quantity; such pairs do remain invariant under rotations by 3600 . They are analogous in this respect to complex numbers which afford simplifications of algorithms but do not represent experimentally accessible physical quantities directly; only real products of complex quantities with their complex conjugates enter into physical expressions. Linear combinations (that is, "superpositions") of sets with half-integer j are quite acceptable, because the relative phases of their several terms remain unaltered when transformed by D~(27r). What remains excluded, within ordinary quantum mechanics, is the superposition of mixed sets, with half-integer and integer j values; no such mixed set ever arises in the manipulation of tensorial sets. Irreducible sets with j - 3, called "spinors, also occur in the fundamental representation of the Lorentz group; here again physical applications always involve pairs of spinors. 4 Note that the irreducible infinitesimal transformation matrices of arbitrary order discussed in Section 2.3 differ from the matrices I" given by (2.18) for the irreducible Cartesian set {x, y, z} in several respects. The matrix J~ is diagonal in this section, in contrast to the matrices (2.18) and to their linear combinations with real coefficients, all of which are nondiagonal. Indeed the I matrices are real, and the corresponding J - I/i are Hermitian but imaginary and hence nondiagonal. The construction of the real set of matrices { Ix, Iy, I~ } of order 3 in Section 2.2 could be extended to analogous matrices of higher, but only odd, order. That is, real Cartesianlike, irreducible r-transformations exist only for integer values of j. The background of this property will emerge in Chapter 3. Thus the construction of J matrices for sets of eigenstates of J~ yields a wider range of tensorial representations than could be surmised from the example of Section 2.2. By dealing with tensorial sets whose elements are eigenstates of J z, we have introduced a characterization that remains invariant under formation of direct products and under their further reduction; we have also achieved a simplification by having all matrices Dz(~o) diagonal. These advantages are balanced in part by the fact that two of the matrices in (2.39) are not real; the resulting complications will be discussed in Chapter 3. (A corresponding alternative occurs in the formulation of Fourier series, which can also utilize complex exponential or real sine and cosine functions; see, for example, the discussion in Appendix A of Fano and Fano [2].) 4 M i x e d s e t s obey what is called a superalgebra and, along with the allied concept of supersymmetry, have recently been in vogue in particle field theories.

2.4. T h e F u n d a m e n t a l R e p r e s e n t a t i o n

49

Problems 2.1 In accordance with footnote 1 on p. 35, how many basic matrices I define an arbitrary rotation in (a) 4 dimensions, (b) n dimensions? Construct explicitly the antisymmetric matrices in (a) analogous to (2.18). 2.2 Verify, using the commutation relations for Jx, Jy, and Jz, that the quantum-mechanical equation of motion

d J / d t - i(HJ - JH)/h for the gyromagnetic precession, with H - -/7. B - - T J - B reduces to the classical form

d J / d t - f i x B. 2.3 Use matrices (2.18) to: (a) Verify (2.21) by explicit calculation of II~2 (b) Calculate the matrices of Iz and of Ix -4- ily in the base of the three unit vectors ~ ( 1 , i, 0), (0, 0, 1), ~ ( 1 , - i , 0) and compare your results with (2.23), (2.24), and (2.25). 2.4 From (2.27a-c), construct the 3 z 3 matrices J for j - 1. Verify explicitly the commutation relation in (2.15)" f x J - if. 2.5 Find the eigenvalues of the matrices I in (2.18). Note the imaginary elements and contrast with the eigenvalues of the matrices J in Problem 2.4. 2.6

(a) Transform the Pauli matrices in (2.32) into a representation in which ax is diagonal. (b) What form do cry and ~z take in this new representation? (c) Show that the unitary transformation U between the representations with az and a~ diagonal is given by exp(icry r/4)exp(ia~ r / 2 ) .

50 C h a p t e r 2. Infinitesimal R o t a t i o n s a n d A n g u l a r M o m e n t u m (d) Give a geometrical interpretation of U by considering the successive rotations of the {x, y, z} coordinate axes that it represents. 2.7 Prove that exp(i5, if) = cos a + (i~. if~a)sin a. 2.8 Consider the rotations De (Tr/2) and D~(Tr/2). Identify the net rotation Da(~p) if these two rotations are applied consecutively. Is there a difference depending on the order in which the two rotations are applied? 2.9 The group of Euclidean motions in two dimensions {x, y} consists of rotations through ~ and translations {ax, ay}. The general linear transformation ( cos~ sin~ a~ ) -sin~ cos~ ay 0 0 1

(0 10)(001)(000)

can be cast in terms of the infinitesimal generators

X0-

N~176

-1 0

0 0 0 0

, X1-

0 0 0 0 0 0

, X2-

0 0

0 1 0 0

( cry0 00 ) .

(a) Work out the commutation relations between these generators. Contrast with (2.11). (b) What is the counterpart of (2.21)?

.

Chapter 3

Frame Reversal and Complex Conjugation Symmetry under discrete operations--typically reflections of space or time coordinates~complements the symmetry under coordinate rotations and other continuous transformations. This chapter deals mainly with reversal of the frame of coordinate rotations from right- to left-handed or vice versa, that is, with the reversal of chirality. This operation has been associated with complex conjugation at the end of Section 2.1 for the case of rotations about a single axis; a more elaborate treatment is required in three dimensions. It is important for our development that symmetry under reversal from right- to left-handed frame is preserved in the construction of direct products and in their reduction. Imposing this symmetry will specify phase normalizations throughout our algebra, in accordance with the discussion in Section 1.4.4. In particular it will cause all reducing transformations to be real. Reversal of the infinitesimal rotation operators { J~, Jy, J~ } i - - t o be called f r a m e reversal~differs from the familiar inversion of a Cartesian frame of coordinate axes {&, ~), ~} in several respects. Each of the Cartesian unit vectors is an eigenvector of inversion at the origin, with odd parity because it is a polar vector; also, ~, y, and ~ commute freely and so do the components of infinitesimal displacements V. On the contrary J~, Ju, and Jz have even parity under inversion ( J is an axial vector) and do not c o m 1We discontinue here, for simplicity, the indication of matrices by boldface symbols, for example, J x, J y.

51

52

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

mute. The contrasting properties of these operations will bring into focus the roles and the interconnection of the Cartesian and standard forms of tensorial sets which have been used casually in Chapters 1 and 2. Frame reversal has been generally viewed as equivalent to time reversal, since its early analysis by Wigner [5], because of the quantum-mechanical association of f with angular momentum which is odd under time reversal. This association with time reversal may indeed be helpful but is essentially foreign to the introduction of f in Chapter 2 as a purely geometrical operator. The name frame reversal is used here to stress this distinction, but reversal of any infinitesimal operator, such as the translation ~1~ , actually belongs to the same class as frame reversal except for commutability. Broader relevance of frame reversal also stems from the mathematical isomorphism of r-transformations with the unitary representations of other groups, which has been mentioned in the introduction of this book and which will become fully apparent in Chapter 6 and beyond. Complex conjugation, on the other hand, serves to represent mathematically the sign reversal of whichever variable is represented by a coefficient of the imaginary unit, as illustrated by the following examples. In the wave mechanics of spinless particles, real wave functions--that is, standing waves--represent states with zero current. Representing states with nonzero current by superposition of standing waves with complex coefficients associates complex conjugation with the reversal of currents. Similarly, in the representation of a rotation by D(,(~) - exp ( i ~ J . zi), complex conjugation is equivalent to sign reversal of the rotation angle ~ provided J./L is real. This remark points to implications of using Euler angles and of standardizing fmatrices as in Sections 2.3 and 2.4: Arbitrary rotations are resolved into rotations about the z axis, inverted by complex conjugation because J~ is real and diagonal, and into rotations about the y axis which are real and unaffected by complex conjugation because Ju is imaginary. In our study, as in wave mechanics, we may then distinguish analytic transformations according to whether they are inverted or left invariant by complex conjugation. The standardization of real transformations thereby rests on a sign convention. This point of view stresses the usefulness of representations in which each component of f is either real or imaginary. The commutation relations (2.15) allow for this purpose only two alternatives: a) all three components are imaginary, as in the C a r t e s i a n base of Sections 2.1 and 2.2, or b) two components are real and one imaginary, as in the s t a n d a r d base of Sections 2.3 and 2.4. The Cartesian and standard bases thus belong to

Chapter

3. F r a m e R e v e r s a l a n d C o m p l e x

Conjugation

53

distinct classes. The connection of frame reversal to complex conjugation holds, indeed, for all Lie groups. Choosing their infinitesimal operators to be Hermitian implies imaginary values of the "structure factors," thus associating complex conjugation with a sign reversal of each operator in the basic formula A B - B A - ikCs C. The coefficients k are s t r u c t u r e c o n s t a n t s of the group, reducing to unity in (2.15) for the case of angular momentum operators. F r a m e r e v e r s a l is represented analytically by an operator K defined by the property I ~ f I ~ - 1 -- - J .

(3.1)

This operator changes a tensorial set {ai} into K { a i } . This new set is c o n j u g a t e to {ai} since a repeated operation of frame reversal coincides with unity to within a possible normalization factor. According to the preceding discussion, the operator K reduces to complex conjugation in the Cartesian base but must operate differently on the real and imaginary components of J in the standard base. Broader relevance of frame reversal stems from the circumstance that the operators of any finite vector space can be represented as polynomial functions of the J operators, as we shall see in Chapter 6. Even-degree terms of these polynomials are then even under the frame-reversal transformation (3.1); odd-degree terms are odd. Parity under frame reversal serves thus to classify all operators of a vector space. The analytical structure and properties of the frame-reversal operator K, particularly in the standard base, are described in Section 3.1. This operator will be seen to commute with all r-transformations D~(~), as one would expect because (3.1) has no reference to the orientation of coordinate axes. In non-Cartesian bases, K will be seen to factor into two parts, which operate separately on real and imaginary components of J . Each of the factors changes D~(~) into its inverse transformation. This property then leads to alternative procedures for constructing invariants such as the "norm" of a set and the scalar product of two sets (Section 3.2). These procedures underlie the construction of r-transformations and of reducing transformations presented in the following chapters. The contrasting roles of the standard and Cartesian bases are essential. The standard base is more general since it encompasses the occurrence of half-integer spin in quantum physics and allows the superposition of quantum states with complex coefficients. The Cartesian base, described in Section 3.3, is appropriate to the description of physical observables.

54

C h a p t e r 3. Frame Reversal and Complex Conjugation

A description of the transformations between the Cartesian and standard bases concludes the chapter.

3.1

Analytical

Representation

Implications

and

of Frame Reversal

We begin with transformations of the set of ~"vector components, {x, y, z}, in the Cartesian base. The components of f f o r this set are given by (2.18) divided by i, that is, are purely imaginary. Thus frame reversal reduces in this case to complex conjugation, generally indicated as an operator K0. This operation of complex conjugation reverses the sign of each component of J a n d the sign of the imaginary unit, thus leaving the commutators (2.15) unchanged. The squared operator Kg equals unity; nevertheless K0 is not unitary because it is antilinear rather than linear. That is, when K0 is applied to a superposition of elements {u, v,...} with coefficients {a, b,...} the result is Ko(au + bv + ...) = a*Kou + b*Kov + . "

(3.2)

instead of aKou + bKov + . . . . Our initial set {x, y, z} is s e l f - c o n j u g a t e because its elements are real. + ~v), ~ ( ~ Next we consider the set of eigenvectors of Jz , {_ ~(~ 1 iy), z}, obtained by applying to {x, y, z} the unitary transformation

T-

I -1 -i" 01 ~

0

~

0

0

.

(3.3/

1

Note that this set meets only some of the specifications of a standard base described in Section 2.3.2. The connection, it provides between the Cartesian and standard bases agrees with the choice in Condon and Shortley [6]. At the end of this chapter, following a discussion of phase normalizations, we will redefine the transformation matrix T and the resulting matrices J. To ensure that the operator K is a scalar and commutes with r-transformations D, it should be obtained from K0 by the equivalence transformation T as in (1.15a). Accordingly, we have K = T K o T - 1 = K o T * T -1 -- K o ( T T ) * -- KoU

(3.4)

3.1. I m p l i c a t i o n s of F r a m e R e v e r s a l

where

U - T'T-1

_

55

(!1 0) 0 1

0

0

.

(3.4a)

Two properties of the U matrix defined by (3.4a) are noteworthy: a) it is symmetric, being equal to (TT)*, and b) it differs from unity only insofar

asT#T*.

Even more important is the factoring of K into K0 and U in (3.4). The factors K0 and U have separate roles in satisfying (3.1) in the transformed base { - ~ ( x + i y ) , ~ ( x - i y ) , z}. In this base f i n obtained by transforming (2.15) and (2.18) by equivalence with the T of (3.3); the result is

/

J~-

o 0

o 0

-1

-1

~

-1/ / (_1o o)

:- -~1

, Jy-

o

Jz-

o 0

o 0

: ~s

i

-i

0

7~

0 0

1 0 0 0

,

(3.5)

.

Upon constructing the expression K J K - 1 _ K o U f U - 1 K o l we see that the K0 factor reverses the sign of the imaginary matrix Ju but leaves the real matrices J~ and Jz invariant; the U matrix (3.4a) instead reverses the sign of the real matrices Jx and Jz while leaving Ju invariant. On the other hand, K0 and U play equivalent roles when frame reversal is applied to the tensorial set { - - ~ 1 ( x + iy), ~ ( x - i y ) , z}" Either operation has the effect of interchanging the first two elements of the set (with a change of sign) while leaving the third one invariant. The combined operation K - KoU thus leaves the entire set invariant just as K0 left {x, y, z} invariant. In other words, the real set {x, y, z} remains selfconjugate upon transformation by T into the complex standard set. We regard self-conjugation as a generalization of the real character of a set much as the frame-reversal operator K - KoU constitutes a generalization of K0. However, in the standard base, the matrices (3.5) do not satisfy all of the specifications for J listed in Section 2.3.2, because the elements of J~ are negative. A minor change of T would remedy this discrepancy but a more significant point remains" The standard base was introduced in Section 2.3 without reference to an initial Cartesian set {x, y, z}, having in mind instead sets of eigenstates u(m j) represented by kets Ijm) rather than

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

56

by wave functions. These state symbols are n o t complex numbers; hence they are not modified by the operator K0 and cannot be self-conjugate. [Wave functions (0~ltm) are represented by complex numbers, in contrast to I j m ) , and may be self-conjugate as we shall see at this chapter's end.] Accordingly, the matrix U should not be constructed for the standard base by transforming an initial K0 into K = K o U , according to (3.4) through a matrix T defined as in (3.3) to construct the standard base from the Cartesian set {x, y, z}. We can, however, determine U by inspection for the standard base of the fundamental representation with j - 1 given -*

1

in (2.32). It is readily verified that (3.1) is satisfied by J - 7~ and

K -

KoU

,

U -

( 01 1 )-io'y-ei 89 _

(3.6)

0

We shall show later that any matrix fulfilling the specification of U for the standard representation of Section 2.4 differs from (3.6) at most by a phase normalization coefficient. Here we complement our earlier specification of the standard base by generalizing (3.6) to K-

KoU

,

U-

Dy(Tr)-

e i'~" ,

(3.6a)

regardless of the value of j. Together with Section 2.3.2, this specification serves to define the standard base for all j, independently of any other starting point such as the Cartesian base {x, y, z} for j = 1. The initial definitions for j = 1 at the beginning of this section, such as U in (3.4a) and Jy in (3.5), which satisfy U = -Dy(Tr), are then in conflict with this general standardization for all j. The conflict having arisen by the choice available for this specific value of j = 1, of constructing the standard base f r o m the Cartesian base through (3.3), a redefinition of the matrix T will be necessary at the end of this section and, thereby, of the matrices U and J in (3.4a) and (3.5). That Dy (Tr) meets the specifications of U is obvious since it commutes with Jy, which is imaginary, while reversing the sign of Jx and Jz through a rotation of axes by 180 o in the x z plane. The identification of U in (3.6a) is henceforth implied for all r-transformations in the standard base, r e d u c i b l e as w e l l as i r r e d u c i b l e w i t h a r b i t r a r y j - v a l u e . Note how this standardization is preserved by constructing direct products of j - 1 representations. The factorization of K into K0 and U is illustrated by considering that a) reversal of J~ by K0 amounts to the effect of reflection of the y axis through the x z plane; b) reversal of J~ and J~ is achieved by rotation of 1800 within the

3.1. I m p l i c a t i o n s of F r a m e R e v e r s a l

57

Z

y

/I Z

Z

F i g u r e 3.1: Inversion as product of rotation and reflection. plane; c) inversion of a three-dimensional frame at its origin is indeed equivalent to the product of reflection across any one plane and rotation by 180 ~ within that plane (Fig. 3.1). That the frame-reversal operator (3.6a) commutes with all r-transformations in their standard form, as functions of Euler angles zz

D(r O, ~o) -

e iCJ" e iaJ~ e i~'J" ,

(3.7)

is obvious because K i J , K - 1 - ( - i ) ( - J , ) - i d a , while i J y is real and commutes with K. On the other hand, e a c h of the two separate factors of K, namely, K0 and U, changes D(r 6, ~ ) i n t o D*(r 0, ~). Indeed we have KoD(r

~,)Ko I -

D*(r 0, ~)

(3.8)

58

C h a p t e r 3. F r a m e R e v e r s a l a n d

Complex Conjugation

by the definition of K0, while U = Dy(7r) gives

UD(r

qo)U-I= D* (r 0, qo),

(3.8a)

because U reverses the sign of J~ in the complex factors of (3.7) but commutes with the real factor exp(i0Jy) = Dy(O). The existence of a second transformation U which changes D into D* will prove important in Section 3.2, because the application of K0 to tensorial sets such as {u(m j)} is meaningless. Note that D* differs from the reciprocal of D, namely, D -1 = D*, only by transposition, a circumstance used in Section 3.2. Equation (3.8a) shows that complex conjugate r-transformations in the standard base are "equivalent" according to Section 1.2.3. The equivalence of the D and D* representations in (3.8a) is a characteristic of representations of the rotation group in three dimensions (not of all Lie groups), showing also that the conjugation relation of two sets, {hi} and K{ai }, remains invariant under coordinate rotations. 3.1.1

Explicit

form

of the

matrix

U

In the standard base, U is fully determined by its property of reversing the sign of J~ and Jz, that is, of anticommuting with them as indicated in Dirac notation 2 by (jm[UJ~ +J~:U[jm') = 0, (3.9)

(jmlUJz +JzUljm') -- 0.

(3.9a)

Since Jz is diagonal, Eq. (3.9a) reduces to (m'+m)(jmlU[jm') = 0, thus requiring the nonzero elements of U to have m ~ = - m whereby U is skewdiagonal. Equation (3.9) with m' = - ( r e + l ) then gives (mlUI-

m)(-mlJ~l- m - l ) 4 - ( m l J ~ l m + l ) ( m + l [ U I - m - l ) = O. (3.10)

Since (mlJ~lm+1 ) i s unchanged by sign reversal of its indices according to (2.27b), Eq. (3.10) requires successive elements of U along the skewdiagonal to be of alternate sign and equal magnitude. Note finally that 1. this value holds for all j as the element (jjlUIj -j) - 1 in (3.6)for j - 7, one can argue from the process of product formation and reduction. We conclude that (jmlUljm')- (--1)J-m6-m,m ' , (3.11) 2To distinguish between whole sets specified by j and individual members labeled by m, we use rounded brackets for the ket [jm') and the contragredient bra (jm[. Further details on the notation are given in Section 3.2.3.

59

3.1. I m p l i c a t i o n s o f F r a m e R e v e r s a l

that is,

U-

0 0 0

Dy(r)-

0 0 0

... -.. ..-

0 0 0-1 1 0

1 0 0

.

(3.12)

9

Equivalently, Uljm) = ( - 1 ) Y + ' ~ l j - m ) . Setting m = - j in (3.11)yields (J -JlUIjj) = ( - 1 ) 2j = ( - 1 ) 2j (jj]UIj -j). The matrix (3.12) has a nonzero diagonal element (jOlYljO)- (-1) j (3.12a) for integer j only. Further, U is Hermitian (skew-Hermitian) for integer (half-integer) j. For j = 1, this redefinition of 0 0 0-1 1 0

U-

1) 0 0

(3.12b)

from the earlier (3.4a) is accompanied by a change in sign of J , and Jz in (3.5),

/

0

1 /

~

o

/ ' ~/ 0

(1oo) i

o

~1

o

Jz -

o

0 0

0 0 0-1

-i

i

0

(3.13)

,

and a rearrangement of the standard base as (3.13a)

3.1.2

P r o p e r t i e s of t h e m a t r i x U

Some important properties of U hold regardless of the standardization which identifies U with Dy (Tr). The first of these is the uniqueness of U for irreducible r-transformations. Suppose that two matrices U and U' satisfy (3.8a) for all transformations D of an irreducible representation. We then have U - 1U ,D U'- 1U = U - 1D* U = D ; that is, U - 1U ~ commutes with all D of the representation. Any matrix A that commutes with all irreducible

60

Chapter

3. F r a m e R e v e r s a l a n d C o m p l e x

Conjugation

D coincides with unity to within a constant factor c, whose magnitude is unity if A is unitary; therefore U' = c U , with Icl = 1. Indeed, the two matrices in (3.4a) and (3.12b) are related through a change of sign, apart from a rearrangement of rows and columns. Transformations by equivalence, according to (1.1ha), preserve the identity of an operation while changing the form of its matrix representation. The operation U does not transform by equivalence, in general, while frame reversal does. Indeed, the equivalence relation of K, K'-

KoU'-

T K T -x - T K o U T -1 - K o T * U T -1,

implies that U'=T*UT

(3.14)

-1.

Failure to transform by equivalence means that U is, in general, not a physical observable. The type of transformation in (3.14) changes in general the identification of U, depending on T, much as (3.4) changes unity into U = T * T -1. However, Eq. (3.14) reduces to an equivalence transformation and thus preserves the identity of U whenever T is real. Conversely, preservation of the identity of U requires T to be real. In particular, all transformations that preserve the standard convention U = Dy(Tr) are real. This class includes all reductions and analogous transformations to be studied in Chapter 5 and beyond. Introduction and standardization of the frame-reversal operation thus manifests itself through a symmetry of the transformations of variables that are of central interest to the book. This role of frame reversal complements the standardization of the zeroazimuth plane x z which has caused the matrices of J~ and Jy to be real and imaginary, respectively, and the r-transformations exp(iOJy) to be real. The s y m m e t r y of U for an irreducible representation emerges from repeating the operation of frame reversal. This repeated operation amounts physically to an identity but its analytical representation K 2 - KoUKoU-

U*U

(3.15)

need not coincide with the unit matrix. Indeed, equivalence transformation of (3.8a) by U* shows that U*U commutes with all D; it may thus depart from unity by a factor c = +1 because U*U is both unitary and real. The result, U*U = c = -t-1, implies that this unitary matrix is either s y m m e t r i c or antisymmetric, U -

c (f -

:kU,

(3.16)

3.1. I m p l i c a t i o n s of F r a m e R e v e r s a l

61

depending on the value of c. For the matrix (3.11), c - ( - 1 ) 2j as noted, symmetry or antisymmetry being associated with whether j is an integer or a half-integer. Because T* - ~-1, this symmetry property is invariant under the transformations (3.14). The sign in (3.16) may accordingly be specified by considering only the standard base where U - Dv(Tr) and is real. Here we have U * U - U U - Ov(Tr)Ov(Tr)- Dv(27r). (3.17) We see in the first place that the product U*U in (3.15) amounts to a rotation of coordinates by 3600 , whose departure from unity was discussed in Section 2.4.1. More important is the substantive result, also given in that section, that the sign of Dv(27r ) and hence the sign in (3.16) is + or - depending on whether the indices j and m in (2.26) are integers or halfintegers. The symmetry of the U matrix thus coincides with the symmetry of boson or fermion wave functions under particle permutations. Indeed, because Dirac field operators change sign under double time reflection while scalar and vector fields do not, the time-reflection operator 7- defined in field theories has properties similar to U: a second time-reflection that follows T is given by T*, and T * T - +1. Thereby spin, and the corresponding statistics associated with the permutation of identical particles, relates to time reflection [7]. We conclude by stressing the internal consistency of properties of irreducible tensorial sets of the boson and fermion types, respectively: a) Sets of integer degree j are of odd order 2j+l and have a symmetric U matrix. Diagonalization of this matrix serves to reduce U to u n i t y - and hence to transform the set to the Cartesian base--because the set of eigenvectors of a symmetric unitary matrix can always be cast in the form of an orthogonal matrix R. That is, solution of the eigenvalue equation UR - R e x p ( - i A ) w w i t h A diagonal and real--enables one to cast U in the form

U - Re-iAR-

(ne-i 89189

(3.18)

The unitary matrix Rexp(i 89 plays the role of the matrix T of (3.4) which transforms a set of the Cartesian base with U - 1 to a different base with the U matrix (3.18). The T matrix that yields U - Dv(Tr) for the standard base constructed according to (3.4a) will be given in Section 3.3.1. Conjugation of a set of odd order by the frame-reversal operator with a symmetric U leaves at least one set element invariant

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x

62

Conjugation

to within phase normalization; this is usually the eigenvector of Jz with m - O, for example, the z element of the set {x, y, z}.

b)

3.2

Sets of half-integer degree j are of even order 2j + 1 and have an antisymmetric U. No Cartesian base exists for these sets because the U matrix of a Cartesian base is unity and hence symmetric. [Recall that the symmetry of U is invariant under the transformation (3.14).] Because each diagonal element of U vanishes, transformation of any set by the antisymmetric U changes each of its elements into a different element. The direct product of an even number of antisymmetric U matrices is symmetric; the corresponding product of irreducible sets with half-integer j must accordingly be reduced into sets with integer j.

C o n t r a g r e d i e n c e and the C o n s t r u c t i o n of Invariants

The relationship between complex conjugate r-transformations belongs to the broader class of contragredience relations. We introduce this subject by recalling that the r-transformations of tensorial sets {ai} consisting of orthogonal Cartesian components of vectors or tensors preserve the "norm" 2 Formally, this of the set. The norm is defined in this case as the ~i ai" property results from the fact that the r-transformations are represented by

a~ -- Z

Dkiai

(3.19)

i

with a matrix Dki which is o r t h o g o n a l , that is, such that

Z

DkiDkj -- 6ij.

(3.20)

k This gives (3.21) k

k

i

Equation (3.20) can be expressed in terms of a product of the matrix D and of its transpose D, bjk Dk, ~,, (3.20a) -

k

3.2. C o n t r a g r e d i e n c e a n d t h e C o n s t r u c t i o n of I n v a r i a n t s

63

or, in matrix notation, as DD-

1

(3.20b)

D -1.

(3.20c)

or equivalently b-

Equation (3.21) can accordingly be rearranged to read

Z a;2 -- E k

~-~ia i ( n - 1 ) i k

Ej nkjaj

k

-- E

ai"

"

i

Note that the inverse matrix transformation operates here on the right of the set elements ai. Equation (3.21) or (3.21a) amounts to a special case of the invariance (that is, conservation) of the scalar p r o d u c t of two different sets ai and bi, represented by

Ea~bk -- ~-~ [~-~jbj(D-1)Jk] [~iDkiai] - Eaib'" k

k

(3.22)

i

Upon generalization to tensorial sets pertaining to coordinates other than Cartesian, orthogonal, and real, the orthogonality of the r-transformations fails. One nevertheless extends the concepts of "norm" and scalar product as quantities invariant under r-transformations. This procedure starts from formulas with the structure of (3.21a) or (3.22) rather than of (3.21). One should consider two types of tensorial sets with different rtransformations. For one type, the r-transformation D operates from lhe left, for the other the reciprocal of the same matrix D operates from the rigM. The two types of sets are said to be c o n t r a g r e d i e n t to each other. (However, the word "contragredient" is used in more than one sense.) Invariance of the scalar product thus defined by combining two contragredient sets is obvious from (3.22). The r-transformation law

b~ - E

bj(D-')jk

(3.23)

J is equivalent to b~ - E ( D -1)kjbj, (3.23a) J with the matrix ~ - 1 applied from the left as D is applied to ai. These equations emphasize that contragredience is nontrivial whenever D-1 r D. We have been considering complex but u n i t a r y r-transformations, whose definition implies D - ' - D*. (3.24)

64

3.2.1

Chapter

3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

Contragredient

tensorial

sets

To each tensorial set {hi} whose elements possess complex conjugates is associated a contragredient set Ko{ai} = {a~}. Such pairs are the sets of eigenfunctions of angular momentum {(t~mi/?~)} and {(a~lt~m)}. Recall that quantum mechanics also considers tensorial sets of state representatives, bras (jm I = u~ )t and kets lyre)- u~ ) which are contragredient by definition, but which neither are complex conjugate to one another nor possess any complex conjugate. Sets with this property occur in other branches of physics. Regardless of complex conjugation, any tensorial set transforms into one contragredient to itself by the matrix U which turns D into D* by (3.8a). This transformation plays a major role in the following. Note that the sets Ko{ai) and U{ai} contragredient to {ai} coincide only when {ai} is self-conjugate. Contragredient sets also occur whose r-transformations are real rather than complex. Examples are the sets of vector or tensor components in a frame of Cartesian, real but oblique coordinate axes such as the axes of the direct and reciprocal lattices of many crystals. The norm of a lattice vector is then expressed by (3.21a) in terms of the contragredient sets of vector components along the two reciprocal sets of lattice axes. Similar circumstances prevail in the vector calculus for curved spaces. The matrix analogous to U that transforms a set into its contragredient conjugate is called a m e t r i c t e n s o r in these examples, the most famous one being Einstein's General Theory of Relativity. 3.2.2

Invariant

products

We have demonstrated the invariance of the norm of a set and of the scalar product of two sets when their r-transformations are orthogonal. We shall now extend the construction of invariant products of two sets, equal or different, to the case of tensorial sets whose r-transformations are unitary and complex rather than real. Equations (3.21) and (3.22) then turn into distinct inequivalent constructions. a) I n n e r p r o d u c t . The simplest of these constructions, most closely resembling the familiar scalar product, is appropriate to multiplication of two contragredient sets {hi} and {bi}. Equation (3.17) then applies directly. A typical example of this type is provided by (1.9), Ca - ~ m U(mJ)a(mJ),which represents the state Ca of a quantum system with angular momentum quantum number j as the superposition of

3.2. C o n t r a g r e d i e n c e a n d the C o n s t r u c t i o n of I n v a r i a n t s

65

a base set of states {u~ )} with the set of coefficients {a~)}; the two sets are contragredient, their inner product representing a state that remains invariant under rotation of the quantization axis. The inner product of two sets is invariant not only when the two sets experience r-transformations but also when they experience contragredient transformations by any unitary matrix T. Thus we have

k

A very i m p o r t a n t ~ a n d seldom recognized~example occurs when the transformation T reduces the initial sets. In this event the index k may be replaced by (c~jm) where (jm) pertains to a base and ~ classifies irreducible subsets of the same order, if any. The inner product (3.25) may then be expressed in the form

~aibi - ~-~ {~m [~iai(T-1)i'aJm] [~iTajm'ibi] } ' 9

(3.25a)

aj

separateinner productsof standard sets, {a(aj)} unit coefficients.

that is, as the sum of and {b(ai)}, with different ( ~ j ) a n d with

b)

H e r m i t i a n p r o d u c t . The construction of an invariant product becomes less trivial when the two sets and are not contragredient but have, instead, identical r-transformations. In this event, an invariant is constructed in two steps: The first one transforms the two sets into contragredient sets, and the second one constructs the inner product of the pair of sets thus transformed. If at least one of the two sets, say consists of elements having complex conjugates, the first step may replace {ai } by K0 {ai} - {a; }. The second step then generates the "Hermitian product" of the two sets, which is often indicated as (alb) - Z (3.26)

{a,}

{b,}

{ai},

a;bi.

i

not

Note that this product does have the commutation property in that (bla) = (alb)". (3.27) This definition of Hermitian product also applies to sets of complex elements whose r-transformations are real, as, for example, to sets of

66

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

components of complex vectors in a system of real orthogonal Cartesian axes. Familiar examples of Hermitian products occur in quantum mechanics. Thus when {ai} and {bi} are the sets of probability amplitudes that identify two states Ca = ~ i uiai and Cb in a common base set of states {u~}, their Hermitian product (3.26) represents the probability amplitude ("overlap integral") that systems prepared in the state Cb be found in the state Ca. c) I n v a r i a n t scalar p r o d u c t . Given any two tensorial sets {ai} and {bi} with the same (complex) r-transformations, one of them can be made contragredient to the other [step a), above] transformation with the U matrix introduced in Section 3.1, that is, for example, by replacing {bi} by U{bi} = {~k Uikbk}. The inner product of the set thus transformed and of the other set then yields their invariant "scalar product,"

E aiUikbk.

(3.28)

ik Here the matrix U plays the role of a metric tensor explicitly. In the 1 the matrix U is given by (3.6) elementary example of sets with j - 3, and (3.28) then takes the form

a 89

a_ 89

(3.28a)

The product thus defined reduces to the ordinary scalar product of (3.22) only when the U matrix reduces to unity. This product is symmetric or antisymmelric under permutation of its factor sets {ai} or {hi} depending on the symmetry of the matrix U, that is, depending on whether we deal with sets of the "boson" or "fermion" class. Antisymmetry of the scalar product of fermion-type sets is very important for us because it requires keeping accurate track of the order of numerous factors in practical applications. For example, we have already been led to apply the transformation U to the second of the two sets, {b~}, in defining (3.28), in contrast to the definition of Hermitian product (3.26). Thereby we have reversed a frequent practice (introduced on p. 28 of [1]) but extended the concept of scalar product to include the construction of a zero-angular-momentum state by vector addition of the momenta of two constituents; this construction is represented by (3.28a) for a pair of spin-71 particles.

3.2. C o n t r a g r e d i e n c e a n d t h e C o n s t r u c t i o n of I n v a r i a n t s

67

The scalar and Hermitian products of two sets coincide only for tensorial sets of the boson class that are self-conjugate; sets of the fermion class are not self-conjugate, whereby their scalar and Hermitian products differ.

3.2.3

Notation

Various notations are employed to distinguish sets contragredient to one another. Tensor calculus uses upper and lower indices, quantum mechanics uses the bra-ket notation for the purposes not only of r-transformations but of all transformations within its Hilbert space of state vectors. 3 In this book we shall instead borrow from quantum mechanics the notation of bra symbols, (jml, to indicate elements of irreducible standard sets, and the corresponding ket symbols, Ijm), for set elements contragredient to them. Additional labels that specify a particular set, as, for example, a letter a or b, may be inserted next to the index j, which specifies the degree of a set and its order 2 j + 1, by writing (ajm I . With this notation and with the standard form (3.12) of the U matrix, the invariant scalar product (3.28) of two standard sets takes the explicit form E (-

1)j-m (ajml(bj _ml.

(3.29)

m

The index j in the exponent of ( - 1 ) ensures that the coefficient of the term with m = j equals 1, whether j is integer of half-integer; omission of this index j in the construction of products with j integer reverses the sign of the product for odd values of j. The coefficient ( - 1 ) j - m in the scalar product (3.29) may be represented by a bracket symbol, with the letter S for "scalar product," (Sljm, j m ' ) --(--1)J-m~_m,m, ;

(3.30)

thereby one represents the scalar product (3.29) in the form

E (Sljm, jm')(ajml(bjm'l.

(3.31)

t-rim I

aReference [1] uses the notation a~ ) for the elements of irreducible tensorial sets of degree j whose r-transformations are standardized in Section 2.3.2 above and are applied from the left; it calls contrastandard the sets contragredient to them and indicates their elements by a~] (with their degree-index in square brackets). Boldface letters are used in [1] to indicate a whole set, standard or contrastandard, omitting then the index m that labels individual set elements. 9

68

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

For standard sets whose elements have complex conjugates, we use here an alternative notation, namely, a bracket symbol (jmla), in which the set label is entered on the opposite side of the indices j and m. By this we imply that (jmla)* = (aljm), as in quantum mechanics, and represents an element of a contragredient set. These notations do not imply, however, that we are actually dealing with quantum-mechanical quantities, except as may be apparent in the context of specific physical applications. To indicate a whole set, rather than a particular element, we replace the parenthesis of a bra or ket by an angle bracket, omitting the m index, thus writing (ajl or laY). The summation convention then applies to inner products, (alb) -(alj)(jlb ) - ~-~(aljm)(jmlb ). (3.32) 171

The description (1.9) of a quantum-mechanical state Ca is thus replaced by luj)(ujla ). This notation also enables us to represent the Hermitian product (3.26) in the same form as the inner product and the scalar product in (3.31) in the form

(Slaj, bj[ = (Slj, j)(ajl(bjl.

3.3

Cartesian Base for Integer j _

(3.33)

1

The Cartesian base is appropriate to tensorial sets that represent measurable quantities (for example, counting rates of radiation detectors arranged symmetrically about a source) and are naturally expressed in real form. Equations among these quantities are also conveniently expressed in the Cartesian base, as shown by examples in Chapter 1 and 2. This base has been used normally in tensor algebra but generally without separating out irreducible sets. Irreducible r-transformations in the real Cartesian base remain rather unfamiliar. All the J matrices are purely imaginary in the Cartesian base, while the corresponding I matrices are real as we have seen in Section 2.2 for j - 1. Prototypes of irreducible sets with real r-transformations are the sets of spherical harmonics {Ptm(O)cos m~, Pt,~(O)sin m~} with nonnegative m; these functions become harmonic polynomials in the coordinates {z,y,z} of a point upon multiplication by r t. The degree j of a set is often indicated by ~ or k when it is an integer. Standardizing conventions for real irreducible r-transformations in a Cartesian base are stated here, extending the conventions implied normally

3.3. C a r t e s i a n B a s e for I n t e g e r j > 1

69

for vectors: 1) The j2 operator is diagonal (though Jz is not) with eigenvalues m s where m = 0, 1 , . . . , g. Its eigenvalues with m > 0 are doubly degenerate but m s : 0 is nondegenerate. (For example, the prototype set elements proportional to cos mT and sin mT have the same m s value.) 2) The doubly degenerate eigenvectors of j2 are distinguished by a parity quantum number e = d=l. This index represents the parity under reflection through the plane zz, R,z, which changes ~ ---, -~o, for the prototype set {Ptr~ COSm ~ , Ptm sin m~} with inversion parity I = ( - 1 ) l. However, there are also sets of degree g with inversion parity I = ( - 1 ) TM. The parity e is made to apply uniformly to both types of sets by defining it as the eigenvalue of the product (-1)lDy(Tr) rather than as the eigenvalue of the reflection Rxz = IDy(Tr). [The 1800 rotation Du(7r ) is unrelated to U in the Cartesian base, where U - 1.] The nondegenerate eigenvectors of j2 with m - 0 have e - 1 according to our definition. Elements of irreducible sets are thus labeled in the Cartesian base by the eigenvalues of two commuting operators j 2 and e rather than a single one; both of these operators are invariant under frame reversal, in contrast to the single diagonal operator Jz of the standard base. 3) The matrix elements of Iy = iJy are real and are diagonal in e because Iy commutes with Dy(7r). Their values are (gmelly[grr~,

c') -

:t: 89189

[ j ( j + l ) - m ( r r ~ ) ] - ~ 6c~, , (3.34)

where the factor [1 + 6rnoC] 89 vanishes for (m adjusts the normalization for (m -- O, c = 1). 4) The matrix elements (gme[I~ [gm' e') -

(gml]LIgrn-1) are emh_~,~,hm,~,

-

O, e -

- 1 ) and

real and positive, _

0

5,.,~, .

(3.35)

Equations (3.34) and (3.35) extend (2.18) to e > 1 and may be derived from (2.18). The U matrix is, of course, unity in the Cartesian base. The conventions established above identify the Cartesian base for tensorial sets of any degree g. From these conventions irreducible r-transformation matrices and reduction matrices could be constructed. However, this construction proceeds far more simply in the standard base whose specifications have been designed appropriately. Accordingly, it has been found

70

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

convenient to perform analytical work in the standard base, whereas measurements are performed and analyzed more readily in the Cartesian base. The conversion formulas that connect the two bases are therefore of central importance and will be derived next.

3.3.1

Cartesian-to-standard transformation

The structure of this transformation is akin to that of (3.3) but its several details hinge, of course, on the specifications of each base described in Sections 2.3.2 and 3.3. As anticipated by (3.18), the transformation is represented analytically by

( s t a n d l T I C a r t ) - Re i 89

(3.36)

where R is an orthogonal matrix consisting of eigenvectors of the standard U given as (3.12), and exp(iA) consists of the eigenvalues of U arranged along its diagonal. However, these eigenvalues equal -4-1 and the specification of their square roots in the matrix exp(i 89A) is nontrivial. As a preliminary to the construction of the T matrix (3.36), notice that the U matrix (3.12) is cast into block-diagonal form by reordering its rows and columns, that is, by writing, for integer g, \ (gmlU,tandlgm,) _ (_ l )l_ m ( 01 1 } 6 m,m' O / ~

(gmlU*'="algm') = (-1)'6ram,

,

m :/:

0

(3.37)

~

m

=

O.

(3.37a)

The matrix T that diagonalizes (3.37) then consists of separate 2• blocks. To simplify notations, we indicate nonnegative quantum numbers m of the Cartesian base by fit. Note finally that the diagonal form of U stand belongs to the Cartesian base, by definition, and is indicated here by exp[iA(gfite)]. Accordingly, we diagonalize the matrix (3.37) by setting

1 (stand, emlTICart, grne) - N(m)---~ x/ "2

1

1/j/l

ei 89

(3.38) where N(m) is a sign normalization coefficient equal to 4-1. 4 To determine the diagonal factors N(m) and exp(i~1A) , a first operation transforms the 4The factor N(m) is missing in the nonstandard expression (3.3) of T.

3.3. C a r t e s i a n B a s e for I n t e g e r j > 1

71

diagonal matrix 18rand :rstand im6mm, to the Cartesian base, requiring the resulting matrix, T -lI*ztandT, to coincide with the expression (3.35) of I Car'. The coefficients N ( m ) cancel out in this process because g(m)-lIz'""dy(m) - I~ '~"d since I~ ' " ' d is diagonal. It is readily verified that indeed _

(gme'T-1l'tandT'gme')-

(

O

~0

(3.39)

,

provided we set

A(&he) - A'(&h) - 89

- e).

(3.40)

The residual dependence of the coefficient N on m and that of the eigenphase A' on rh are now determined by combining (a.as) with its transpose to form the matrix TT. This matrix represents U in the standard base according to (3.36) and (3.18) and must coincide therefore with (3.37). The two expressions do actually coincide if we set N ( m ) - (-1)'}('~+lml) ,

A'(e, rh) -- gr(mod27r).

(3.41)

The transformation (a.as) is thus specified adequately to verify that it transforms correctly the standard and Cartesian forms of J into one another. The sign of exp[i 89 still remains undetermined because it cancels out of the matrix transformations considered thus far. This residual sign must then be determined by performing a nontrivial transformation of tensorial sets rather than of matrices. To this end we transform the equation that defines the vector product of two vectors A and B in the Cartesian base,

(3.42)

C~ - A~ B v - A v B . ,

a transformation sensitive to the phase factor exp(i7I A) tor product must reverse its sign under frame reversal. { x , y , z } ofeach vector correspond to the values of (m, e) respectively, and are thus represented as superpositions ponents {A1, A_ 1, A0 } by _T-1

Ay As

(1 0 0

(A1) A-1

- e -i 89

because the vecThe components {1, 1; 1 , - 1 ; 0 , 1}, of standard com-

•

A0

0 0 ~ 0 0 1

)/1 1 0/( i 0 0)(A1) ~ ~1 0

~- 1 0

0

1

1 0

0 1

A-1 Ao

,(3.43)

72

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

using the preceding expressions of T [which differs from (3.3), being in fact the complex conjugate of that matrix]. Substitution of these expressions for A, B, and C transforms (3.42) into

e-i 89

- ie-ia'(t=l)(A1B-1 - A-1B1).

(3.44)

The expression A IB-1 - A_IB1 represents the m = 0 component of the irreducible set of degree j = 1 resulting from the reduction of the direct product set AiBk ; it is antisymmetric in Ai and Bk with a sign matching that of the scalar product (3.28). (This last consideration underlies the sign determination in the reduction process as we shall see in Chapter 5.) Comparison of the phase factors on both sides of (3.44) then yields e i 89

(3.45)

= i.

Qualitative arguments on the construction and reduction of multiple products of vectors indicate then that (3.45) extends for g > 1 to e i~A'(lrn) =

it.

(3.45a)

The value of A'(g, rh) in (3.41) is thus equal to er exactly rather than mod2r. The imaginary factors thus introduced in (3.44) and in analogous formulas ensure that frame reversal of the equation amounts to the permutation of factors due to transition to a left-handed frame. The explicit form of the transformation matrix (3.38) is obtained finally by combining it with (3.40), (3.41), and (3.45a), to yield

(stand's

((-1)m-i(-1)ml i ) -'~1 it (3.46)

for rh > O, and

(stand, ~OITICart, s

e = 1) = i l,

(3.46a)

for rh = 0. 3.3.2

Phase

normalization

of spherical

harmonics

Implications of the complex phase factor of the transformation (stand[T[Cart) are illustrated by an analysis of the wavemechanical probability amplitude OO

e

-

"~

.

t=0

=

i

(2e+

j

+ 89

§

9

l--0

(3.47)

3.3. C a r t e s i a n B a s e for I n t e g e r j > 1

73

This familiar equation may be viewed as the combination of two transformations in Dirac notation,

( ke0)(ke01g)

(3.47a)

t The first factor on the right of (3.47a) is viewed as a free particle eigenfunction with orbital and magnetic quantum numbers (g, m - 0); the set of these eigenfunctions with - g < m < g constitutes a standard base according to the ordinary wave-mechanical conventions of Section 2.3.2. The second factor, (kgOlk), represents instead the probability amplitude of the state IkgO)in the wave vector eigenstate Ifc). Notice now that the fe is an ordinary vector, defined in the Cartesian base, while IkgO) belongs to a standard base as noted above. Accordingly, the factor (kgOIk) includes the standard-to-Cartesian transformation matrix element for m - 0, which equals i l according to (3.46a) and indeed constitutes the imaginary element on the right of (3.47). 5 The eigenfunctions (Ylkgm) include as factors the spherical harmonics Yem(O, ~). Both sets, (~'lkgm) and ]Qm(0, ~), transform as kets of a standard base (Section 3.2.3) but do not fit the definition of self-conjugate sets in Section 3.1, because they are not changed into their complex conjugates by the transformation U "rand. Self-conjugate sets that transform as kets of a standard base are instead obtained from real sets of the Cartesian base by the transformation (CartlT-11stand) reciprocal to (3.46). A self-conjugate set of spherical harmonics that transform as kets of the standard base will be indicated in this book by the Dirac symbol (O~lgm). This set is obtained from the Cartesian base set of real harmonics {Ylm~} -- [(2g+1)/4~r] 89{Ptm(O)cosrh~,Ptm(O)sinrh~}

(3.48)

by applying to its right the transformation T-1 - ~ . , which yields

(O~lgm)

=

~r177 Yemr Y~(O,~o)i-e-(emlO~) * .

gm)

(3.49)

Clarification of circumstances leading to (3.49) for self-conjugate sets of spherical harmonics has been achieved slowly, this definition underlying recent works where symmetry under frame/time reversal is essential. 6 5The wave function (r']kgm) should include a factor i - t reciprocal to the i t factor of the s y m b o l ~* were i n t e r p r e t e d literally as a vector of the Cartesian base as is i n t e r p r e t e d above. However, the f u r t h e r analysis of wave functions in Section 4.3.2 implies t h a t ~" is not so r e g a r d e d in fact. 6 T h e phase n o r m a l i z a t i o n a d o p t e d in (5.17) of reference [1] was incorrect for its failure to a p p r e c i a t e all relevant circumstances.

(kg0lf~) if

74

C h a p t e r 3. F r a m e R e v e r s a l a n d C o m p l e x C o n j u g a t i o n

Problems 3.1 On the basis of (2.15) and assuming that each of {J:~, Jy, Jz} is an eigenvector of the complex conjugation operator K0, show that the product of their three eigenvalues of K0 equals - 1 . 3.2 Verify that the matrix T -~ defined by (3.43) and (3.45) correctly transforms the standard base matrices f , Eqs. (2.18), into the Cartesian base forms (3.34-35). 3.3 With U and Ju defined as in (3.4a) and (3.5), verify that u = - D u ( 7 r ) , an equation that differs in sign from the standardization in (3.6a). Verify on the other hand that the definitions (3.12b) and (3.13) conform to the present standardization. 3.4

(a) Transform the f matrices in (3.5) under K as given in (3.4) to verify that K J x K -1 = - J x , K J u K - I - Ju and K J ~ K - I =

--Jz. (b) Repeat with the standard form of f in (3.13) and U as in (3.12b). 3.5 The frame transformation matrix K for j - 1 is given in (3.6). Transform the Pauli matrices (2.32) under this operation to verify (3.1). 3.6 Find the eigenvalues and eigenvectors of U for j = 1 in (3.12b) and show explicitly that U can be cast into the form (3.18).

Chapter 4

Standard r-Transformation M a t r i c e s and Their A p p l i c a t i o n s

Having specified in Chapters 2 and 3 a standard form for the infinitesimal rotation matrices, we can now derive the matrices for arbitrary finite r-transformations with the same standardization. The present chapter describes the structure of these matrices, many of their properties, and some of their wide-ranging applications to macroscopic and quantum physics. First, various alternative derivations of these matrices are presented in Section 4.1. The spinor method utilizes in Section 4.1.1, as the basic building block, the fundamental representation of order 2 in (2.38) of Section 2.4. Symmetries under transposition of the matrix indices and under frame reversal are considered in Section 4.1.5 before turning to applications in the next two sections. Particularly important is the treatment of particle transmission through the inhomogeneous field of a Stern-Gerlach magnet. Closely paralleling the passage of light through polarizing elements, this system is central to quantum physics. The last section of this chapter deals with the nontrivial combination of general r-transformations with the operation of coordinate inversion. 75

Chapter 4. Standard r-Transformation Matrices

76

4.1

Explicit Form and Properties

We have seen how rotations of coordinate axes are conveniently parameterized in terms of Euler angles rather than by identifying a fixed axis fi and the angle of rotation about it. That is, one factors as in (2.31) each rotation of axes, R, and the corresponding transformation matrix, D ( R ) , into a sequence of three rotations, by an angle ~o about the z axis, by an angle 0 about the y axis, and by an angle r about the z axis, R

=

R~(r

(4.1)

D(R)

=

e i~a,ei~176

(4.2)

The diagonality of the standard Jz makes the corresponding Dz matrices diagonal and very simple. The fact that Ju is imaginary according to (2.27c) makes the D u matrix real but, in general, not simple. The elements of this matrix are important functions of mathematical physics, to be discussed below; they are given here explicitly but can also be expressed in terms of hypergeometric polynomials, "(j) (0) ~'m,~

-

[(j + m ' ) ! ( j - m ' ) ! ( j + m ) ! ( j - m)!] 89 ~ r ( - 1 ) r (j - m' - r)'(j. + m - r)'r!(r. + m ' - m)'. • (cos 10)2J-m'+m-2r (sin 89 2r+m'-m

=

1 1 C(sin ~0)P(cos ~0)q 1 1 • F(-~(p + q) + j + 1, ~(p + q) - j; q + 1; sin 2 89

(4.3) The range of values of the index r is limited by the requirement that the arguments of the factorials in the denominator be nonnegative; a negative value would cause the factorial to diverge and its contribution to ~ to vanish. The parameters of the last expression in (4.3) are p = I r a ' - ml, q = ]m ~ + ml, while C depends on the signs of m' and m. Upon setting 0 - 7r in (4.3), one verifies that d~! m coincides with the matrix (3.12). The elements of the complete matrix (4.2), in alternative notations, are then D(mJ!m(r 0, 9) - ( j m ' [ R ( r

,0 ~o)[jm) - e im'r dm.,~( 0

e imcp .

(4.4)

When either m or m ~ vanishes, the function (4.3) reduces to an associated Legendre p o l y n o m i a l and (4.4) to a s p h e r i c a l h a r m o n i c , to within normalization; when both m and m' vanish, Eqs.(4.3) and (4.4) reduce to the ordinary L e g e n d r e p o l y n o m i a l s of degree j. This connection will be

77

4.1. Explicit Form and Properties

made apparent by the following discussion of the equations obeyed by the d-functions. Explicit matrices of the d-functions in (4.3) for j _< 33 are displayed in Table 4.1. The functions (4.3) and (4.4) have manifold, seemingly unrelated aspects. These will be illustrated by describing first four alternative methods for constructing them. We regard the first one as the most basic, therefore setting the subsequent three subsections as complementary to the main flow of the chapter. Each of them contains, however, specific elements of interest for higher order r-transformations. The main treatment resumes in Section 4.1.5 with an outline of the symmetry and integral properties of the d-functions.

Table 4.1: Table of d(mJ!m(0) for j _< 3/2 1 t~ = cos 30

m'\m 1 J--2

i

~ 1

(

31

-31

C~

m'\m

1 j-1

0

-1 ml\m 3_ 2

1 j-~

0

-i

~2

v~c~D

~2

~2 _ ~2

~/~

--x~c~D

o~2

-v~ ~2 3_ 2

1 2

1 2

_~,~ _~3

m

~

3 2

~3

O~3

2 2

)

1

3

_3_

1 /3 -- sin 30

a ( 1 - 3/32)

(3~ 2 - 1)~

v~~ ~

- ( 3 . ~ - 1)Z

. ( 1 - 3d 2)

v%2~ ~3

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

78

4.1.1

Spinor

method

Many properties of irreducible standard sets of arbitrary degree j can be derived in straightforward fashion by constructing such sets from the direct product of 2j identical standard sets of degree 89 The initial sets of order 2 are called "spinors." The direct product of 2j sets of order 2, with elements { ( 1~1l a ) , ( ~1- ~ [1a ) } , consists in general of 2 2j different elements, but only 2j + 1 of these are different when the initial sets are identical. These 2j + 1 distinct elements form a standard irreducible set of degree j, properly normalized. This spinor method for constructing sets of higher order from the spinors of order 2 exemplifies the basic role of the latter representation according to the remarks in Section 1.5. The corresponding group of transformations SU(2) is, therefore, central to the study of all symmetries under rotations and reflections. To obtain the proper normalization, one starts from the invariant Hermitian product of one set with itself, represented in the form (3.32),

(ala) -

1 1 (al~)(~la) - ( a l 1~1) ( ~1l1a )

1 1 1 + (al~1 -~)(~ -~la).

(4.5)

This product is then raised to its 2j-th power, by the explicit binomial expansion, (ala) 2j

=

2j ~--~(2J)(algg

ggLa

(alg

) -p(g-gla)

-p

p=0

+m) (alg7

(a17 -g

m=-j

•

=

11

)j+m (1

]

(4.6)

lla)J-m(j

E ( a 2jljm)(jmla2j), m

1

where (jmla 2j) _ ( 89 la)i+m(7, _ 1 [a ) j - m (j +m) . Note in the first step the recasting of the summation index p into the index m which is symmetrically distributed on either side of zero, with the attendant splitting of the binomial coefficient into two symmetric square-root factors. It will now be shown that these set elements belong in fact to an irreducible standard set. The r-transformations of the set (jla 2~) are unitary, since the Hermitian product of this set with itself is invariant. The 2j + 1

4.1. E x p l i c i t F o r m a n d P r o p e r t i e s

79

elements (jmla2~) are eigenvectors of Jz, with eigenvalues m ranging from j to - j , since a coordinate rotation by ~ radians about z multiplies each of them by ei 89189 - eim~. Their J~, Jy, and U matrices also fit the specifications of standard sets, thus completing the required proof. The construction of the transformation matrices (4.4) and (4.3) by the spinor method requires only a few steps. We label for simplicity the spinor 1 1 symbols (7 :t:71a) as a+ and recall from (2.38) that their transformation under coordinate rotations takes the form a;+

-

ei 2x_r (cos 70 1 el'5s0 a + + s i n 7 01 e -i 89 a _ ) ,

a'

--

1 e - i 5~a 1 e -i-}r ( - sin 89 e i 89176a+ + cos 70 )

_

~

(4.7) .

Following (4.6) and (4.7), we then represent the elements of a transformed set (j[a '2j) by

(jm'la '2~)

_

_

_

..Ij+m' t~ + a 0 "-m' (j +2jm ' ) 71 2j ~

[

"~(.7+m')r eiT(j+m -28)qo

~ $

s=l

c o s J + m ' - s 1 0 sin s 10 -J+m'-SaS

•

2j • ~

7

( j - r e ' ) e - ' 89162

r=l

•

7 u+

_

(_ sin 89 ~ cosj - m ' - ~ !92

ei 89

r aJ-m'-r -

+m')

"

(4.8) S e t t i n g s - m ' - m + r a n d collecting factors, this f o r m u l a reduces to

J (jm'[a'2J)

-

eim'r

y~

{ Zr2J__.l

m---j

"

x c o s 2~ + m - m ' - 2~

• --

eim'r Z

[(j +m')~(j -m')~(j +.~)~(j -m)~] 89 (--1)rr,(j_m,_r),(m,_m+r),(j+m_r),

1 sin "~i-rn+2r gO

10 }

eimqoaJ++maJ_-m (j+m)2J 89

d(Jm!m(o)eimq~ ( j m l a 2 J )

,

m

(4.8a1

where ~'m'm'S(J)(0) coincides with its expression (4.3). This construction casts 1 and sin 70, 1 "m'm's(J) as a polynomial of degree 2j in cos 70 as seen in the illustrative examples provided in Table 4.1. The usefulness of the spinor method may be illustrated further by deriving the explicit form (3.29) of the scalar product for an arbitrary value of 1 namely (771a)(7 11 1 - T1 l b ) - (71 -71 [a)( 7117 [b) 9 j from its special form for j - 7,

80

Chapter

4. S t a n d a r d

r-Transformation

Matrices

Raising this expression to the 2 j - t h power and e x p a n d i n g the binomial as in (4.6), we have

[( 89189189 - 89

-

2j

:

E ( ~ l 1a )1P ( : - : l1b ) P1 ( - 1 ) 2 J - P ( ~ - ~ l1 a ) 1

1 2j - p ( 1_~Ib)2J-P(2/)

p=0

-

~

(-11 #-m

11

-Fro i

(::la)#

1

2j

(:-:la)J-m(j+m)

•

(4.9)

-:I

~--~(-1)J-m(jmla2J)(j _mlb2~). m

This result shows how the a l t e r n a t i n g sign of the scalar product ( 3 . 2 9 ) and of the U m a t r i x elements (3.12)--derives from the - sign in the scalar p r o d u c t of two spinors (3.28a). This - sign derives in turn from the antis y m m e t r y of the m a t r i x t~u and of the r - t r a n s f o r m a t i o n (2.38) for r = ~ = 0 and 0 = T r .

4.1.2

Algebraic approach

This approach focuses directly on the transformation D, viewed as a passage from one axis to a rotated axis. Consider first the combination of an arbitrary r-transformation D with an infinitesimal r-transformation matrix, J-~i, with arbitrary axis ~. The rotation R corresponding to D changes ~ into a new axis 6'. The fact that the standard matrices J have the same form in any frame of reference is thus represented by J-~'= or equivalentlyby

D J . ~i D -1 ,

J . a' D = D J - ~ i .

(4.10)

(4.10a)

Equation (4.10a) is now regarded as a system of algebraic equations for the matrix D corresponding to the R that changes ~i into ~i'. The coefficients f of the system are given by (2.27) and (2.29). Because the portion of D to be studied is the matrix d (J) m i r a (9) for rotations R~(8) about the y axis, it will be sufficient to write (4.10a) explicitly for three alternative independent pairs of directions (~i~, ~), such that ~ is transformed into ~ = R(8)d by a rotation of arbitrary magnitude 8 about the y axis. We lay 6~ alternatively along the three coordinate axes {~; ~); ~}; the corresponding vectors

81

4.1. E x p l i c i t F o r m a n d P r o p e r t i e s

then have the three sets of components [{cos 8, 0 , - sin 8}; {0, 1, 0}; {sin 0, 0, cos 0}]. The three equations thus obtained form the system J=d(O) J:
= = =

d(O)(,Ix cosOd(O)a:, d(O)(,lx sin O +

J:sinO), (4.11)

J: cosO).

Linear combinations of these equations, which involve the simpler matrices of the ladder operators ,Ix -4-iJu as given in (2.27a), yield recurrence relations among (~) m i r a (0) with values of m' and m differing by unity:

[j(j + 1)- rn'(rn'-t- 1)] 89(1 -4- cos O) d~!:i:a,m --

T(m'-4-m)sinSd~!,~ + [j(j + 1 ) - r e ( m - 1)] 89(1 -4- cosO) "i(~)%~,m_',

(4.12) in obvious analogy to the infinitesimal rotation considered in (2.28). Appendix D of [1] shows how to unravel this algebraic system, for arbitrary j, to obtain the matrix ,t9-,+(~) (a)in the form (4.3) ~ , / # ~,, t

4.1.3

First order differential system

A differential approach starts by recognizing that either side of (4.10a) may result from the expansion of the product of D and of an infinitesimal r-transformation matrix, ei~a"YD, or De'~aY Consider now that the effect of an infinitesimal transformation upon the matrix D may also be represented by appropriate shifts of the Euler angles that replace D(I~, 0, qa) by D(tb + 5tb, 0 + 60, ~ + 6~p) = [1 + 51bO/Otb + 500/00 + 6~O/Oqa]D. One thus constructs the alternative equations, i~'. fD

- r

,

iD it. f = it. B D ,

(4.13)

in terms of vector operators/7, a n d / 7 linear in {O/01b, 0108, 0 1 0 ~} whose explicit form has been constructed by differential geometry in Appendix E of [1]. Each of the two equations in (4.13) constitutes a system of first order differential equations for the matrix elements r)(3) Setting ~' or ~ equal to 5 determines simply the -L/rnt rn 9 I/,- and v - d e p e n d e n t factors of (4.4). Setting ~' and ~ equal to ~ :E ~) generates chains of recurrence relations between the functions d (j) (8) with values of m' or m differing by unity, as in the algebraic method. Here, however, one equation for rn I = m = j remains uncoupled and can be integrated, the other matrix element following by recursion, as shown in [1]. Normalization results from the condition d(:!m(

-- 6m' m 9

(4.14)

82

Chapter

4.1.4

4. S t a n d a r d

r-Transformation

Matrices

S e c o n d o r d e r differential equation

Iterating the first equations in (4.13) and averaging the result over the direction of fi', in analogy to the procedure of (2.12), yields the scalar equation - I f l 2 D IAI2D, which no longer interlinks different elements of the matrix D. Substituting here the exphcit form of/~, from Appendix E of [1], and separating out the simple dependence of D on r and ~oyields the second order wave equation for each matrix element a(J) t * r n t m (0) ,

+ cotg0 ~-~ I.

sin 2/9

,,~,m J

~'m'm

(4.a5)

This construction has imitated the transformation of Maxwell's first-order coupled equations into separate wave equations for the components of/~ and/~. Equation (4.15) is analogous to that of Legendre polynomials and indeed reduces to that for the associated polynomials when either m or m' vanishes. Notice that ]fl 2 plays the role of eigenvalue in this equation; in fact its eigenvalues j ( j + 1) could be determined by solving (4.a5). Notice also that setting m and/or m' to zero amounts to eliminating one or both of the variables r and ~o, thus restricting the scope of the equation. This elimination of variables applies only to the solutions with integer j for which m and/or m' may be set to zero; here again the functions with half-integer j display a different behavior. That the irreducible r-transformation matrix elements D m,m~b,t? (J) / , ~o), for all integer j, m, and m', form a complete orthonormal system of functions of {r 9, ~o}--as anticipated in Section 1.4.6--follows from the self-adjointness of the operators in the eigenvalue equations for the three factors of D. The D functions with half-integer j constitute a separate complete system for which D~ (2r) = - 1 . They do not form representations of the rotation group because of this change of sign--to each rotation will correspond two D matrices with different signs. Avoiding this two-valuedness and getting proper representations requires either a double rotation group in which rotation through 27r differs from the unit element or the equivalent, isomorphic group SU(2) of unimodular matrices (pp. 157-160 of Wigner [8]). Notice finally that the second order equation in (4.15) fails to establish the relative normalization of the different matrix elements, in magnitude and phase, for lack of any interlinkage. Similarly, the d'Alembert equation provides less information about the electromagnetic field than does the system of Maxwell equations.

4.1.5

Symmetries of t h e s t a n d a r d r - t r a n s f o r m a t i o n s

S y m m e t r y properties of the r - t r a n s f o r m a t i o n matrices will be derived here from qualitative arguments, even though they could also be established by

83

4.1. Explicit F o r m a n d P r o p e r t i e s

inspection of the analytical expressions in (4.3) and (4.4). An important preliminary property emerges from the factorization (4.1) of a rotation of axes in terms of Euler angles. The angle 0 of rotation about the y axis ranges only from 0 to rr; hence the inverse of the operation R~ (0) is not represented as R~(-0) but by the sequence of operations R,(-r)R~(O)R,(Tr). The inverse of the general operation (4.1) is then given by

[R,(r

(O)Rz (~o)]- ~ = =

R,(-~)R,(-,~)R~(O)R,(,~)R,(-r

R , ( - T r - ~o)R~(O)R,(Tr-

r

(4.16)

On this basis we can now state the formal consequences of three symmetry properties of the matrices d(O) that represent Rv(O)" a) The transpose of the unitary and real matrix d equals its inverse. We have then

d-(~!m

-

d (~) (O)- e -'m'*z(~) trt~

!

tbm1771

(O)e''" -(_i)m-m' d~!,,,(e)

"

(4.16a) b) The commutability of the real matrix d with the frame-reversal operator K in (3.6) and (3.8), implies that d - K d K -1 - UdU -~, and hence

d(Jm?m(o) -[Ud(J)(O)U-1]m'rn - ( - 1 ) m' -m.(~) '*-m',-m (O).

(4.16b)

c) A rotation about the y axis by ~r-0 may be resolved as R~l(O)R,y(~r) R.(-~r)P~(O)Rz(Tr)Ry(Tr) represented by the matrix _

( _ l ) - m ' Eam,m,,Aj)(O)(_l)m"Um,,m f r t II

= 4.1.6

(4.16c)

(--:)J-m'~?_m(0).

Integrals

As indicated in the Introduction to Chapter 1, our program will effectively bypass the integrations over multiple products of angular functions. A main tool to this end replaces products of r-transformation matrix elements by a single such element, through the reduction procedure to be developed in Chapter 5. Here, however: a) we stress that the integral over the variables of a single D(Jm!m function with j ~ 0 vanishes because of orthogonality to D~~ ) - 1, and b) we point out how the squared modulus of a single element,

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

84

which appears in normalization integrals, drops out altogether by reducing in effect to a unit factor. Consider the integral of a squared eigenfunction of (4.15) over the three Euler angles

I(~),m -

dO

/0

sin OdO

/0

dg0[D~!m(r 0, 9o)

9

(4.17)

A rotation of coordinate axes would replace this integral by a linear combination of analogous I (j) integrals with different values of m' and m. Yet this transformation, when applied to the factors n(J) and D m(j)* of the igml m tm integrand, would simply replace each of them by a single equivalent matrix element, owing to the group property of r-transformation matrices, thus in effect leaving r(J) a r m s m unchanged. (This argument is developed formally in Appendix A of [1] as a main procedure of group theory.) Accordingly t(J) ~m'm is in fact invariant under coordinate rotations and hence independent of m' and m for any fixed value of j. We may then write

I(~)m - (2j + l ) - ' E

Im, m

(4.18)

mt

and replace, accordingly, in the integrand of (4.17) [n~!m[ 2

~

(2j + 1)-1 E [ n ~ ! m [ 2 m I

---- (2j + 1)-1 E

D(mJ!m*n(j)~lm,m

(4.19)

m !

=

(2j + 1) -1 E ( n ( J ) - l ) m m , n~!m - ( 2 j + 1) -1 . m p

The integration then reduces trivially to the volume 87r2 spanned by the combined range of the Euler angles, whereby the normalization integral over the squared matrix element equals the ratio of this volume to the order of the representation, Imj)'m -- 87r2/ ( 2 j + 1) . (4.20) This result constitutes an elementary but fundamental contribution of group theory to our subject. Familiar results of the explicit evaluation of normalization integrals in mathematical physics may be regarded as special cases or analogs of (4.20). In particular, when m ~ - 0 and the contribution 27r of the f de drops out, (4.20) reduces to the normalization integral for spherical harmonics. When m - 0 as well, a second factor 2~r from f d~ drops

4.2. M a c r o s c o p i c A p p l i c a t i o n s

85

out and (4.20) reduces to the integral 2/(2j + 1) over the square of the Legendre polynomial Pj (cos 0). 4.1.7

r-Transformations

in the

Cartesian

frame

For integer j - t~, the standard from (4.2) of the r-transformation, given explicitly by (4.3) and (4.4), is recast in the Cartesian frame by the transformation (stand ITI Cart) of (3.36),

DCart(R) - (em' e'lT-:e~J~Wd(J)(O)e~J~Tleme)

,

m, m' > O .

(4.21)

The three factors of the standard D are conveniently transformed separately, because the rotations about z are diagonal in m and that about y diagonal in e in the Cartesian frame [see (3.34-5)]. The explicit form (3.46) of T yields

(gin' c'--t-llT-:eiJ'r

s i n m ' r )Sin, cosm'r m (4.21a) for m ~ > 0 and unity for m ~ = 0, e -- 1. For e iJ'~', one simply replaces m ' r in the above by rap. The matrix ,~(l) (0) in the standard frame has indices (re'm) of either sign; negative values of these indices are eliminated by (4.16b) when both of them are negative and by (4.16c) when a single one is. There results '

(em' e'lT-leiJ~'~

x

4.2

c-q-l)-

'

( cosm'r sin m ' r

- 5,,,

(-1)m'md(me!,~(O) + ( - 1 ) ' e d mSlT1 (') (Tr - 0)

5,1(- l)m v/2dom (0)

eeld~/o)(O)~elPl(cos

0)

,

m I, m > O,

,

m ~ = 0 , m > 0,

/TiP -- m - -

0.

(4.21b)

Macroscopic Applications

The matrices of r-transformations, and their dependence on the Euler angles that relate two sets of coordinate axes, acquire a more concrete significance in the context of examples where different coordinate frames pertain to different parts of a physical system. A familiar association of a coordinate frame with a material object occurs for the moment of inertia of a body, whose components reduce to three nonvanishing ones, {I**, Iyy, Izz }, when the coordinate axes coincide with the principal axes of inertia of the

Chapter 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

86

body; the rotational motion of the body is then studied in terms of the Euler angles relating this body frame to a laboratory frame. Less familiar examples are tensorial sets representing properties of laboratory apparatus (magnets, radiation detectors, etc.) which may interact with the sets of microscopic systems. As a trivially simple example of relations between frames appropriate to different systems, consider the energy of a magnetic needle with a dipole moment fi in a magnetic field/~, which is represented by the scalar product of two vectors, E - - f t . / ~ . This expression of the energy is an invariant, defined without explicit reference to any set of coordinate axes, but two distinct axes are in fact relevant, one parallel to the field direction and the other to the direction of the needle. In terms of the first axis the field is identified by its strength B alone, while the needle's magnetic moment is identified by its magnitude # alone when related to the second axis. The explicit expression of the energy in terms of the angle 0 between the two axes

E

=

-fi- B--t,

cos0B,

is a prototype of application of the r-transformation matrix between frames with axes parallel to fi and /~, since cos 0 is the explicit expression of the matrix element Here we have j = 1 because we deal with vectors, whose components are sets of degree 1, and we have m ~ - m - 0 because the standard sets of components of fi a n d / ~ reduce to their single elements With m - 0 in their respective special frames. This reduction to a single element reflects the axial symmetry of vectors and would already fail in examples involving moments of inertia. As a more substantial example, consider the electric potential energy of a number of charges lying in the proximity of a point P which we take as the origin of coordinates. We regard the potential V(r-') as a sufficiently smooth function of f" to permit its expansion into harmonic polynomials,

d~l)(o).

-

(4.22)

-

lrn

lrn

where the expansion coefficients Vim - (emlV) consist of derivatives of Y(r') at P. Ifn charges {e~, e 2 , . . . , en} lie at {f'~, f'2,..., #'n}, their combined potential energy is represented in terms of their combined electric multipole moments

(Mitre)

n

- ~ i=l

(4.23)

87

4.2. M a c r o s c o p i c A p p l i c a t i o n s

by the sum of inner products

E - ~-~(MItm)(gmlV)- ~--~(MIt)(glV). lm

(4.24)

l

Assume now that the charges are rigidly connected to one another but free to rotate collectively about the point P. We deal thus with two coordinate frames of reference, one attached to the charges and one to the field; their mutual orientation is identified by Euler angles {r 0, ~}. The energy E is then represented as a function of these angles by a modification of (4.24), namely,

E(r O, ~)

-

~

(Mlgm')(tm'lR(O, O, ~)ltm)(t'~lV)

lm'm

-

(4.24a)

~(MIt)(fIR(O, O, ~)lt)(tIV). t

Here the standard r-transformation matrices play their typical role, as a complete set of harmonics of the Euler angles into which we have expanded the energy E ( r O, ~p). If the system of charges (or the potential field V ( ~ , or both) have an axis of symmetry which may serve as the z ~ axis of their coordinate frames, the sum over m ~ (or over m, or both) reduces to a single term and each of the transformation matrices reduces to a spherical harmonic. The directional distribution of radiation scattered by a small object, or diffusing within a medium through a sequence of scattering processes, involves typically two different frames of reference: one pertaining to a radiation source and the other to any detector--actual or ideal--that probes the distribution, even when the scattering object or medium is itself isotropic. Here again the full r-transformation matrices are appropriate to the analysis only when neither the source nor the detector has axial symmetry, otherwise their simpler elements with m and/or m ~ = 0, namely, spherical harmonics, are adequate. Applications to elementary scattering processes and to the representation of wave fields will be indicated in the next section and in following chapters. When the distribution of radiation throughout an isotropic medium is analyzed into irreducible components, that is, into eigenvectors of IJ~ 2, the distribution of each component can be calculated separately, and the several components have to be superposed only to fit the specifications of the source and of a detector. (The appropriateness of this and related forms of analysis to take advantage of invariances has been stressed by Wigner [8].) This form of analysis can be dispensed with and

Chapter 4. S t a n d a r d r - T r a n s f o r m a t i o n Matrices

88

replaced by analysis of simpler trigonometric functions of the Euler angles, when neither the source distribution nor the elementary scattering processes introduce any sharp dependence on angular variables. For example, Chandrasekhar's classic treatment of Radiative Transfer ([9], especially Sections 17 and 26) handles thus the diffusion of light in stellar atmospheres. Analysis of irreducible components has instead proved convenient for treating the diffusion of X rays, whose scattering is peaked forward at high photon energies (see [10], especially pp. 697-8).

4.3 4.3.1

Applications to Quantum Physics Particle

transmission

through

a Stern-Gerlach

magnet Consider a beam of particles with angular momentum j, prepared by filtration through a Stern-Gerlach polarizer magnet that lets through only particles with magnetic quantum number m. The state of these particles is represented by the ket symbol Izjm) in a coordinate system with its z axis parallel to the field of the polarizer magnet. Suppose now that the beam is analyzed by a second magnet, with field direction parallel to a different axis z ~. We seek the probability that a particle emerging from the first magnet will be transmitted by the second in its channel corresponding to the magnetic quantum number m ~. The probability amplitude of the event of interest is represented by the projection (z'jm'lzjm) of the initial state Izjm) onto the bra eigenstate (z~jm'l in the field of the second magnet. This projection equals a single element of the irreducible r-transformation matrix that transforms bras of degree j from the coordinate system with axis z to the system with axis z~; that is, we have

(z'jm'lzjm) - (jrn'lR(r O, ~ ) l j m )

9

(4.25)

The probability we seek is then

Pm,m = I(jm'lR(r O, ~)ljm)l 2 - [d~!m (0)] 2,

(4.26)

where 0 is the angle between the field directions in the two magnets. The correct normalization of these probabilities, ~-~m'=-jJ P m ' m - 1, is implicit in the argument embodied in (4.19). One of the earliest derivations of the expression (4.3) of d~! m was actually motivated by this application. In the

4.3. Applications to Quantum P h y s i c s

89

1 example of particles with spin j - ~, Eq. (4.26) takes the explicit forms (see Table 4.1)

(89 2 [d!l(O)]

-

1 + cos 0), cos 2 ~10 - ~(1

(4.26a)

22

[d(_89 89 (0)]2 _ sin 2 ~10 -

4.3.2

Angular

distribution

89 - cos 0).

of a particle

(4.26b)

in orbital

motion This problem is normally treated by solving the rotational part of the Schr5dinger equation for a particle in a central field. Its eigenfunctions are the spherical harmonics Ylm(O, ~p), which are special cases of r-transformation matrix elements as noted below (4.4). Here we rederive the result from the point of view of r-transformations; thereby we obtain a new insight, especially on normalization, and a lead to further generalizations. (This approach originates from Chapter 19 of [8].) Consider a spinless particle moving in a central field and the probability amplitude that it be found in a specific direction § = (0, ~). In an intrinsic coordinate frame with its z' axis parallel to § the particle's state must be an eigenvector of the orbital angular momentum component gz' with zero eigenvalue, because the centrifugal force associated with nonzero angular momentum would otherwise keep the particle away from this axis. Now suppose that the particle's state is known to have quantum numbers (g, m) in a laboratory frame. According to the treatment in Section 4.3.1, the probability amplitude (O~lgm) amounts to

(o,,,:,lem)

=

(o' = oleo)(~Oln(r o, ~)lem),

(4.27)

where (0' = 0[g0) indicates the still unknown probability amplitude of finding the particle along the z' axis (0' = 0) when m' = 0. The r coordinate in (4.27) can be dropped or set to 0 since (4.27) is independent of it for m I --0. To determine the unknown (tO' = 0it0), we write the condition that (4.27) be normalized to unity, namely,

=

I(r

2 f / d cos0 d~ I(t01R(0, 0, ~,)ltm)l 2 .

(4.28)

90

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

The integral in this equation equals agJ) of (4.20) divided by 2r to allow for 0m the lack of integration over r in (4.28). Alternatively, we can proceed by observing that the integral is independent of m whereby nothing is changed by summing over m and dividing by (2g + 1). Since ~ , ~ I(gOIRIgm)l 2 = ~m(gOIR-1lgm)(gm[R-a[gO)~ , . , . , ( g O I R R - a l g O ) - 1, the integral equals 4 r / ( 2 g + 1). Hence we have ( 0 ' - 0 l g 0 ) - e i`~ [(2g + 1)/47r] 89

(0 lem) -

[(2e + 1)/47r] 89(gOIR(O , O,

(4.29)

)lem).

(4.30)

where e ia is a still arbitrary phase factor. Equation (4.30) may be regarded as a definition of the spherical harmonic Ylm; ordinarily one sets a = 0 for simplicity. However, the transformation matrix element (0' = 01g0) should actually have as a factor the inverse of the T matrix for m = 0 in (3.46a), namely, i - l , if one wants the bra (0~oI to transform like the components of a unit vector F in a Cartesian frame. Including this factor amounts to setting c~ - - 7 171"g. Setting a - 0 as one usually does, implies that the tensorial set of spherical harmonics (O~olgm) is not self-conjugate but, instead, antiself-conjugate for odd values of g. Indeed the complex conjugates Y~'m - (gmlO~~ differ by a factor of ( - 1 ) l from ~ m ' Umm'(~.mtlO~) . Accordingly the ordinary spherical harmonics, with the Condon-Shortley normalization, should be viewed as the elements with m ~ = 0 of a standard r-transformation matrix, renormalized by a real factor, rather than as arising from real harmonic polynomials in the Cartesian frame by transformation to the standard frame through the matrix T of (3.36).

4.3.3

Rotational eigenfunctions and eigenvalues for symmetric-top polyatomic molecules and heteronuclear diatomics

Molecules are usually described with reference to intrinsic coordinate systems determined by the relative positions of their nuclei. We consider here molecules with one axis (z ~) of cylindrical symmetry. An axis of n-fold symmetry, with n >_ 3, is equivalent to full cylindrical symmetry for inertial purposes since the moments of inertia about all axes perpendicular to it will be equal (see, for example, NH3 or CH3C1). We can then consider separately the representation of eigenstates in the intrinsic coordinate system and its transformation to laboratory coordinates.

4.3. A p p l i c a t i o n s to Q u a n t u m P h y s i c s

91

In the intrinsic system the energy eigenstates are eigenstates of J~,; therefore, they can be characterized by an angular momentum quantum number A with respect to the symmetry axis and will be indicated by [A). Coordinate rotation about this axis by an angle r multiplies this state representative by a factor e - i A r (Effects of degeneracy between ]A) and I - A) involve considerations of symmetry under inversion of all coordinate axes to be discussed in the next section.) If the angular momentum arises only from rotation of atoms within the molecule about the symmetry axis, the corresponding energy is A2/2C, (4.31) where C is the moment of inertia about this axis. Otherwise the energy dependence on A involves a study of the electron motions within the molecule. Consider now the further rotation of the molecule about any axis orthogonal to ~', disregarding effects of this rotation upon the eigenstates IA). The combined rotation, about ~' and the orthogonal axis, yields a total angular momentum J, whose component about a laboratory axis, for example, J~, is a constant of the motion with eigenvalue m. The eigenvalue of IJ~ 2 is similarly indicated by j ( j + 1). The full rotational state of the molecule in the laboratory frame is then represented~in analogy to (4.27)~by IA) (jAIR(r , 0, ~ ) l j m ) [ ( 2 j + 1)/4r] 89

(4.32)

where the normalization factor corresponds to integration over all possible orientations (0, ~) of the intrinsic coordinate axis of the molecule, the influence of variations of r being included in the normalization of IA). The eigenvalue of the squared angular momentum is conveniently represented here in terms of the components of J in the intrinsic system

[j[2 _ j ( j + l )

- j2,+j2,+j2

_ j 2 + j ~ , + A 2.

(4.33)

The rotational motion about axes perpendicular to the symmetry axis contributes therefore to the squared angular momentum the amount

j~, + j2, _ [j(j + 1 ) - A2],

(4.34)

and to the rotational energy the amount

[j(j + 1) - A2]/2A,

(4.35)

where A indicates the moment of inertia about any axis orthogonal to z'.

92

4.3.4

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

Spinor and vector harmonics

The wave functions of an electron (or other particle) with spin may be considered from the point of view of Sections 4.3.2 and 4.3.3 above. If the particle state is represented by Ijm) in a laboratory coordinate frame, the probability amplitude of observing it in a direction ~ _= (0, 9) and 1 with spin orientation +§ that is, with angular momentum component ~7 in the direction § is proportional to the r-transformation matrix element 1 (j +~[R(r 0, 9)lyre). On this basis one can construct wave functions of a Dirac electron in a central field, equivalent to those given in textbooks and derived originally by Darwin, but differing from them by relating the spin orientation to the radial vector § rather than to the laboratory frame. The Dirac equation can thus be separated by a procedure more transparent than Darwin's. Similarly, one can represent 2l-pole spherical electromagnetic waves, with angular momentum quantum numbers Is by expressing the field components at a point ~' =- {r,~9, 9} in a frame depending on ~', as, for example, with a Cartesian axis z' along § The transverse part of the electric field is then represented by a superposition of terms with alternative circular polarizations,

(E~, 5= iEy,) (e ,I[R(r 0, ~)[/m).

(4.36)

Thus one can construct vector field harmonics equivalent to those which are usually found in the treatment of a multipole field (see, for instance, p. 1801 of Morse and Feshbach [11]) but which differ from them by representing the fields in a frame attached to § and by the explicit appearance of (J) (0) functions instead of combinations of the usual spherical harmonics ITlt ITI This approach permits a separation of the Maxwell equations in spherical coordinates. For these spinor and vector harmonics, however, as for the molecular rotation wave functions, a complete treatment should consider parity eigenfunctions, which are linear combinations of functions with quantum numbers =t:m' in the direction ~', much as the real, trigonometric functions, cos m9 and sin rag, are linear combinations of e 4"imp, shown in the next section.

4.4. Coordinate Inversion and Parity Eigenfunctions

4.4

93

Coordinate Inversion and Parity Eigenfunctions

The reflection of Cartesian coordinate axes at the origin, which yields the transformation { x , y , z} ---. { - x , - y , - z } , is an operation that commutes with all rotations of coordinate axes, since f remains unchanged under this inversion. This operation, to be indicated by I (not to be confused with the infinitesimal operator f of Chapter 2[), has the eigenvalues +1 since 12 - 1 and is called the inversion of coordinate axes at the origin. 1 It is also often called p a r i t y for brevity (though there are many parity transformations) and often indicated by a script i or 7r. The inversion I differs from the frame-reversal operator K in (3.1) as already discussed in Chapter 3. In the familiar application to the orbital motion of a spinless particle, it is well known that orbital momentum eigenfunctions (rOcfl[ngm) are also parity eigenfunctions with the eigenvalue ( - 1 ) l. This is verified by substituting in the wave function the polar coordinates of F according to F - {r, 0 , 9 } -

{x,y,z}

---. - F -

{r, 7 r - 0 , ~ + T r } - { - x , - y , - z } .

(4.37)

All sets of tensor components of degree t have the parity ( - 1 ) l , if we take "tensor" to mean tensor proper rather than pseudotensor; sets of pseudotensor components of degree g have the opposite parity - ( - 1 ) l . Notice that the operation I combines with a rotation about y, Dy(Tr), to yield a reflection through the plane x z containing the z axis, as indicated by 2 (see also Fig. 3.1) IDy(7r) - R~z - ~rv(xz).

(4.38)

Note also that for tensorial sets of degree t~, with parity ( - 1 ) t , the eigenvalue of ~rv(xz) - (-1)lD~(Tr) coincides with the quantum number e introduced in Section 3.3 for the Cartesian base. We proceed now to parity eigenstates for spinning particles and for the molecules and vector fields considered in Section 4.3. In this event, the inversion of coordinate axes affects not only the polar coordinates of F as in (4.37) but also a set of "body frame" coordinates, attached to F, 1 This a r g u m e n t a c t u a l l y holds only for bosons and m u s t be revised a n d elaborated for fermions, as will be seen below. 2 S t a n d a r d g r o u p t h e o r y n o t a t i o n uses av(xz) instead of Rzz and C2y instead of

Dy(r).

Chapter

94

4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

as well as the third Euler angle r that identifies the orientation of this frame about the axis ~'. The body frame serves, for example, to identify the position of various nuclei of a polyatomic molecule with respect to its symmetry axis, the electron positions in a diatomic molecule with respect to the internuclear axis, or the spin or vector field components with respect to the position vector Y. To identify the effect of I on the body frame, we consider the relation between the Cartesian coordinates {x, y, z} of an arbitrary point of space in the basic "lab frame" and the coordinates {x', y', z'} of the same point in a body frame with the same origin. This relationship is represented, in terms of the Cartesian frame r-transformation of vector components, that is, by (4.21) with g = 1,

yt zI

__

cos r cos 0 cos ~

cos q, cos 0 cos ~

- s i n ~b c o s 0 c o s ~o

- s i n r c o s 0 s i n ~0

- c o s r s i n ~0

+ c o s r c o s ~0

sin 0 cos ~

sin 0 sin ~o

- cos ~ sin 0 sin

r

sin 0

Y

"

z

cos 0 (4.39)

The inversion operation has, in the first place, the effect represented by (4.37). This operation reverses the sign of {x, y, z} and of each element of the bottom row of the matrix in (4.39), leaving z I unaffected, meaning that the positive z I axis of the body frame maintains its direction. In particular, this direction remains parallel to the radius vector Y itself--drawn from the origin of the lab frame to that of the body frame when this frame is so displaced--or parallel to the molecular axis in a heteronuclear configuration. In addition, the body frame coordinates must change "handedness" as a result of the inversion; since z ~ remains unchanged, this is achieved, for example, by reversing the y' coordinate. The complete result is thus attained by adding to (4.37) the substitution r

--+ 7 r -

r

(4.40)

T h a t is, applying (4.37) and (4.40) to the right-hand side of (4.39) affects its left-hand side as indicated by

v', z'} -+

z'} =

y', z ' }

(4.41)

An equivalent description of inversion is given in Section 86 of LandauLifshitz [12].

4.4. Coordinate Inversion and Parity Eigenfunctions

95

Now we meet another essential point. Whereas the orbital wave functions (rO~oln~m)~and the r-transformation matrix elements L-,'0trt r~(t) (r 0, ~) within themmare eigenfunctions of I, as defined by (4.37), the general matrix elements DO! m (r 0, ~o) are no longer eigenfunctions of I when (4.37)is complemented by (4.40). We have instead, using (4.4), (4.16b)and (4.16c),

ID~!,.n(r

0, ~) - D~!m(~r - r r - 0, 7r + ~) - (-1)J-m'D(__/),,~ (r 0, ~). (4.42) This result goes hand in hand with still another important fact, namely, that the change of handedness of the body frame implies a sign reversal of the angular momentum component of the molecular angular momentum, spin, or vector field along the body-frame axis. This reversal is represented formally by the effect of the operator av(x'z') upon the molecular wave function, spin, or vector field representation in the body frame, which appears besides the D function in (4.32) or (4.36). We indicate the carrier of body-frame angular momentum by I~), a symbol that may represent any of the following: a) the state of rotation of a polyatomic molecule about its axis of inertial symmetry, or b) the electronic state of a diatomic molecule, or c) the spin state of an electron at F, or d) electric field components at F. In each of these cases, a rotation X of the body frame coordinates about z' multiplies the ket I~) by e -m• changing r into r + X in the n(J) (r 0 ~o). Accordingly, the expression Ifl) r)0) ~'nm(r 0, ~) - If~)(jftlR(r 0, ~')[jm)

(4.43)

remains invariant under rotation of the body frame. To determine the effect of I on the complete expression (4.42) we must still construct the matrix (121cr,~(z'z')ll2') which operates on Ift). To this end, consider that o'v(z'z') commutes with J], but anticommutes with Jz, since it changes Jz, into - J z , , as represented by I n ' ) n ' + n(nl

ln') = (n +

In') = 0.

(4.44)

That is, a~ is represented by a 2 x 2 skew-diagonal matrix, which connects only [ft) with If~') - I - g t), keeping 12 + ft' = 0. (For 12 = 0, cry amounts to unity.) In addition, we may represent the inverse of a, by applying first a rotation by 7r about z', then cry itself, and finally the inverse rotation by - l r about z', (g2Jtr~-i ]a ') - (~2]ei-J,, cr~e-i~J, , [g2') - ei~(~-n')(~lcr~ [g2,).

(4.45)

96

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

Since ~' - - ~ and e 2i'~a = 4-1, depending on whether f~ is integer or half-integer, we find the matrix of cr~ to be symmetric for bosons and antisymmetric for fermions. 3 Finally, the matrix of or,2 must coincide with the unit matrix, to within a factor - 1 in the case of fermions. It follows that we can set (al(r~(x'z')lft') - ( - 1 ) ~ + a b _ a , , a

,

(4.46)

where a is a still undetermined integer for bosons and half-integer for fermions. Combining (4.45) with (4.41), application of I to (4.42) yields Ila) (jalR(r

0, ~)ljm)

--

(4.47)

( - 1) J - a( J - ~ l R (O, O, ~ )[J m )

~

(-1)J+~l -f~)(j -f~lR(r 0, ~)ljm) ;

it thus reverses the sign of f~ and adds a real phase factor independent of the sign of f~. This result permits us to conslrucl eigenstates of inversion, as symmetric or antisymmetric superpositions of two expressions (4.42) with equal and opposite values of f~. We thus obtain, for any ]f~) ~ 0, two eigenstates that display opposite parity when transformed by I, I

( =

la)(Jfl[R(O'O'~~ : i : l - ~ ) (j

-nln(O,o,v)ljm)

:t:(_l)j+c,

{

}[

] }[ 1

2j + 1

87r(1 + 6ao)

la)(jalR(C,o,~o)ljm) +l-a)(j -aln(r

2

2j + 1 87r(1 + 6a0)

1_

(4.48)

The normalization factor has been so written that the formula applies also for f~ - 0, in which case the matrix of ~r~ reduces to a single element ( - 1 ) 4 . The parity + ( - 1 ) j+~ of the superpositions in (4.47) depends on three parameters: a) the angular momentum quantum number j, b) the symmetry + of the superpositions, and c) the parameter c~, which depends on the structure of the system represented by f~. In the example of the orbital motion of a single electron in a diatomic molecule~where f~ is usually called A--one finds c~ - 0. [In this case the pairs of eigenstates with the same A and opposite parity 4-(-1)J differ slightly in rotational energies, a 3We arrive at the same conclusion by considering t h a t av t r a n s f o r m s eiXJz ' into its c o m p l e x c o n j u g a t e as Dy(r) does for any D m a t r i x of space r o t a t i o n . However, Dy(r) is itself one of the D while, for the g r o u p of r o t a t i o n s a b o u t a fixed axis, av is a reflection operator.

4.4. C o o r d i n a t e I n v e r s i o n a n d P a r i t y E i g e n f u n c t i o n s

97

difference called "A-doubling."] By contrast one finds a = 1 for the twoelectron ground state orbital of the oxygen molecule, which has A = 0 and is designated as E - , with the - sign indicating the parity factor ( - 1 ) ~. When If~) represents the electric field of a 2j-pole spherical wave, we have a = 0; eigenstates with f~ = 1 or 0 and the + superposition combine to form electric multipole states, while those with f~ = 1 and the - superposition pertain to magnetic multipole states. When If~) represents the spin projection of an electron at ~, we have a - - 3 1 and the + or - superposi1 respectively. In any specific tions pertain to states with ~ - j - g1 or j + 5, example, the value of c~ depends on aspects of the system under consideration that are beyond our scope; thus for a spinning electron it depends on the relativistic aspects of the Dirac equation, that is, on properties of the Lorentz group. The example of the parity eigenstates of a diatomic molecule is treated in much greater detail in Chapter 6 of Judd [13].

Problems 4.1

(a) Evaluating the equations (4.11) between states obtain the recurrence relations (4.12).

(jm' I and Ijm),

(b) Establish a special case of the recurrence relation 1 ( j + m ) ~ s i n 3 01

djm ( j ) - ( j - m + 1 ) S c o s1 3 0 d1j m

(J) 1"

(4.49)

(c) From the above and the unitarity relation for any row or column of the d matrices, ~,~L[d(J) jm(0)] 2 -- 1, show that

()r

!

Note the illustration of (4.50) in Table 4.1. The top rows of these matrices correspond to terms in a binomial expansion except for replacing their numerical coefficients by square roots. 4.2 From the explicit expression (4.3), verify the various symmetries (4.16ac) of the r-transformation matrices. 4.3

Y'~m=-.iJ mld(Jm)'(O)l2 ~--.,m=-jJ m2]mm' d(j) (0)12 --

(a)Evaluate (b) P r o v e

1).

1 _12 -}j(j+l)sin 2 0 + g,u (3cos 2 0 -

98

C h a p t e r 4. S t a n d a r d r - T r a n s f o r m a t i o n M a t r i c e s

Hint" Use the identity -~(j) (/9) - d W(j) (-/9) to recast the expressions tt rtllTI t ire t so that finally the sum on m can be carried out by closure and the evaluations reduced to diagonal matrix elements of operator combinations of J~ and Jy. 4.4 An atomic beam consisting of 52Cr in its ground state passes through a Stern-Gerlach magnet with vertical field, isolating the component with highest upward deflection. This beam then enters "suddenly" a second Stern-Gerlach magnet with field 60 o from the vertical. Determine the relative intensities of the component beams that emerge from this second magnet. 4.5 A beam of nitrogen atoms in their 4S3/2 ground state is filtered through a Stern-Gerlach magnet whose field points in the positive z direction. The component with highest deflection in this direction then enters "suddenly" a region with a magnetic field H in the z direction and again exits "suddenly" from it after 10 -4 s. It is then analyzed by a second Stern-Gerlach magnet identical to the first one. All four intensity components are monitored. Plot their relative intensities as oscillating functions of H. Disregard the effects of nuclear spin.

Procedure: Analyze the initial beam into components with respect to a vertical (z) axis of quantization, take into account the time dependence of each in the field H, and recombine the components in the frame of the analyzer magnet. 4.6 Consider a pair of quadrupoles, each consisting of two charges q, placed at opposite corners of a square of size d • d, and of two charges - q placed at the other pair of opposite corners. The quadrupoles lie at a distance r of the order of 10d. Calculate the potential energy of interaction between the two quadrupoles and its dependence on the quadrupoles' orientation. Disregard terms of relative order 1/100. [The derivation of (4.24a) provides a model.]

Chapter 5

Reduction of Direct Products (Addition of Angular Momenta) This chapter describes and discusses the transformation matrix that reduces the direct product of two standard sets. Its main item constructs irreducible products of standard sets. Recall from Section 1.1 that the product of the sets ~ and the strain ~"can be reduced to sets of order 0, 1, and 2. The end of that section viewed this result as a special instance of a more general reduction that has become familiar with the advent of quantum physics. Indeed the construction of angular momentum eigenfunctions for two or more interacting particles became familiar in quantum mechanics as the "addition of angular momenta." The construction process extends to the products of indefinite numbers of tensorial sets, as well as of sets contragredient to one another. This extension permits, among other things, the harmonic analysis of atomic charge densities and the construction of tensor operators. We shall give special attention to the determination and symmetries of the reduction matrix, which is called a Clebsch-Gordan matrix or a Wigner 3-j symbol. Its derivation in Section 5.1 follows the spinorial approach as in Section 4.1.1. An alternative recurrence method is also pointed out. After a dis99

100

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

cussion of normalization and symmetry properties, Section 5.2 extends the reduction of products of two sets to products of their r-transformation matrices. The irreducible product sets, their symmetries, and a convenient diagrammatic representation for them, together with extension to contragredient sets and thereby to operators in quantum physics, are the subjects of Sections 5.3 and 5.4. These sections provide the basis for the further discussion of operators and multiple products of sets in Part B of the book.

5.1

Structure and Properties of the Reducing Matrix

The direct product of two standard sets of degrees jl and j2 consists of (2j~ + 1)(2j2 + 1) elements. The matrix that reduces it is unitary, of order (2jl + 1)(2j2+ 1). We shall indicate it in abbreviated notation by (jlP[jlj2), where j ~ t h e degree of each reduced subset~takes in general several alternative values. As already noted at the end of Section 1.1.1, addition of angular momenta in quantum mechanics identifies these values as ranging from j = j~ +j2 to j = IJ~-J21 in unit steps, where j ( j + 1) is the eigenvalue of the operator (J~ + j~)2. To each value of j corresponds a rectangular submatrix with (2j + 1) rows and (2jl + 1)(2j2 + 1) columns. Elements of the matrix P are often indicated in quantum-mechanical notation by (jmlPIj~ ml , j2m2 ) - ( j ~ j 2 j m l j l m l , j2m2) '

(5.1)

where the pair of indices (ml, m2)identifies a column and the pair ( j m ) a row of the "Clebsch-Gordan matrix." We will develop and discuss several alternative notations in current use. The specifications of standard sets have been designed to ensure that: a) The matrix P commutes with the diagonal matrix Jz = Jlz + J2z. Therefore it can be made block diagonal, with one diagonal block for each eigenvalue of Jz, m = ml + m2, by arranging its rows and columns according to the value of m rather than according to those of j, jl, or m2. b) The matrix P preserves the standardization in (3.12) of the operator U as equal to the rotation Dy(Tr); accordingly P is real. Each diagonal block of (5.1) is square, that is, has equal numbers of rows and columns. This number is limited by the conditions Ira1[ <_ j~,lm2[ <_ j2,

5.1. S t r u c t u r e a n d P r o p e r t i e s of the R e d u c i n g M a t r i x

ml +m2 ml m2 m

4 2 2

1 3

2 1

3 1 2

101

-5 -2 -3

0 3

,i

5

*

*

*

*

*

*

*

*

*

+,

-5

F i g u r e 5.1: Diagram of the matrix of Wigner coefficients which reduces the direct product of two irreducible sets of degrees jl - 2 and j~ - 3. Matrix elements not indicated by an asterisk vanish. and by the condition ml + m 2 : m (Fig. 5.1). It equals for each value of m, the smallest of {jl + j2 - m, 2jl § 1, 2j2 -I- 1}; this is the number of alternative values of ml - r n 2 , for a given m, and also the number of alternative values of j within the range jl + j2 >_ j ___Max(m, [jl - j2[).

(5.2)

In Fig. 5.1, the maximum dimension of the diagonal blocks attains its maximum value of five for m = 0,-t-1. Each diagonal block thus transforms degenerate eigenvectors of Jz = Jlz + J2z, replacing eigenvectors of Jlz - J2z by eigenvectors of IJ1 + J212. This transformation is referred to as the passage from the uncoupled to the coupled representation of two angular momenta. The limitations on the values of j can be cast in the form of "triangular relations" stating that each of the numbers a=j2+j-jl,

b=j+jx-j2,

c=jl+j2-j,

(5.3)

102

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

is a nonnegative integer; the numbers jl, j2, and j can thus stand for the lengths of the sides of a triangle. Note the reciprocal relations 2jl=b+c,

2j2=c+a,

2j=a+b.

(5.3a)

These relations have the same function in determining the eigenvalue parameters of irreducible products as the rules for beat frequencies have in Fourier analysis. It is important that no two sets with the same j occur in the reduction of products of two irreducible sets. Note also the duality of the two triads of nonegative numbers {jl, j2, ja} and {a, b, c}: each of them determines the other but two of the j's may be half-integers whereas each of {a, b, c} is an integer. Because the transformation matrix P is real and unitary, its contragredient transformation is represented by its transpose (JxJ21PlJ), whose elements (jlml,j2m21jlj2jm) are identical to the elements (5.1) with the same indices. This coincidence is consistent with the quantum-mechanical notation, where transposition of the entries in the bra and ket amounts to complex conjugation of the number given by their product (... ]...). The calculation of the matrix elements (5.1) may proceed by alternative methods. We describe here particularly a spinor method which keeps in sight the symmetry properties of the matrix.

5.1.1

Spinor

approach

Standard sets of any desired degree j can be constructed from products of the elements of sets of order 2, as indicated in Section 4.1.1. Here we indicate the elements of such spinor sets by (g1 • 1 specifying that their values are given by (4.7) with {a+ = 1,a_ = 0} and with angles (r that represent polar coordinates of the unit vector ft. (The angle ~o which corresponds to a reference azimuth about ti may be disregarded here.) Spinor methods combine invariant products of spinors to generate invariant products of higher degree that display desired properties. Our application requires two standard sets of degrees ja and j~ constructed according to (4.6), {jllti 2j') and (j21f~2J~), whose direct product is to be reduced by the transformation P as per (5.1). An invariant that contains the reduced product should be formed using a third set contragredient to the others, ( tb2j IJ}. This invariant shall thus have the structure I-

(~J Jj){jIPIjlj2} (jx [~i2y,) (j21~3212).

(5.4)

5.1. S t r u c t u r e a n d P r o p e r t i e s of t h e R e d u c i n g M a t r i x

103

Building blocks for constructing the invariant I are invariant products 1 1 1 of three spinors (~lfi), (~[~)), and (wl~/There are three such invariant products, ,

,

1

,19 ) ,

(SII~

1

1

(5.5)

where (SI 89189is the scalar product matrix of (3.33). From Section 4.1.1, particularly (4.6), we gather that the invariant (5.4) must be of degree 2j in 1 (@1 89 of degree 2jl in {~[fi), and of degree 2j2 in (~~]~)). The combination of the invariants (5.5) that fits these specifications is I

-

N

[ ( ~ 1 ~ 1> 1< ~ 1 ' ~ ) ] "

[(wl 89

,

a

1 1 c , [(Sl ~11~><~1,~)<~1,~)]

with the exponents b, a, and c given by (5.3); the normalization coefficient N is to be determined to fit specifications of (5.4). The binomials in (5.6), 1 1 1 such as (d21 89189 - ( d 2 1l 1~ ) ( ~1 1] f i ) + ( w l1~-~)(~-~)lfi), should now be expanded and combined so as to reproduce the structure of (5.4). This program anticipates that the elements of the matrix P will be expressed in terms of factorials. The symmetry of this matrix under permutation of jl and j2 may also be anticipated from that of the spinor invariants (5.5), among which the scalar product is odd under permutation of/L and b. Equation (5.6) has then the parity (-1) c under this permutation, as well as under the frame-reversal operation which also permutes the factors of the scalar product. Notice also the role of the last factor of (5.6), which drops out when j attains its largest value, jl + j2, whereby c vanishes according to (5.3). For j < jl + j2, lower values of the exponents a and b suffice to yield j - 89 + b), but the additional factors ( 89 and 1 (~1~)) that contribute to (5.6) when c :/: 0 provide the sets (jllfi 2jl) and (j21~2j2) that appear in (5.4). Such additional factors are incorporated in the scalar product without raising the value of j. R e s t r u c t u r i n g t h e invarlant: Expansion of the three binomials in the manner of (4.6) yields the explicit form of (5.6) 11

• E

( 89a

1 _ 89 89

~b-f-~ 1

1 }) la-l-~ (t~311 _~) 1 89189 I (W[ 2"

1

89

_~.l~) 89

-y •

11

1

1

89

9

(5.7)

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

104

While seemingly complicated algebraically, this expression is a straightforward rewriting of (5.6), each line representing the binomial expansion of each of the three factors in (5.6). Recall now from (4.6) the definition of the standard sets that should appear in the expression (5.4), 11)j+m (tbl~-g 1 1)j-m , (j_l j ) 89 (tblgg

(~2~ljm)-

(jlmll it2j ) -

(j2m2l~j~)

_

(7glti) j

(g-girl

(y,+m,2J' i ,

(llj~)j2+m2(l_ll?))j2-m2(

(5.8

2j2 )3 " 1

~,j~+m2

The spinor product in each of these expressions indeed coincides with the corresponding product in (5.7), as verified by recalling (5.38) and identifying the m-quantum numbers as re=a+/3

,

ml = f l + 7

,

m2=-7+c~.

(5.8a)

Note that one among the parameters {c~, fl, 7} remains free when {m, ml, m2 } are fixed, since rn = ml + m2. Accordingly the invariant I can now be expressed in terms of the tensorial sets (5.8) divided by the corresponding binomial coefficient -

N

j + m

rnzrn2m

(2jl )-- 89 x

jl+ml

j2+m2

)- 89(jl ml Jti2j')(/em2

[~2J')

9

(5.9)

This expression has indeed the same structure as (5.4), in that its expression in the braces represents the desired matrix element (5.1) to within the normalization coefficient N which remains independent of {m, ml, m2}. 5.1.2

Normalization

To determine the coefficient N we notice that the value of the invariant I in (5.4) is set by the orthogonality of the reduction matrix (jlPIjlj2},

E (JlJ2jmljlml'j2m2)(jlml'j2m21Jlj2j'm')-5JJ'5'nm''

(5.10)

rnlm2

and

jm(jlmz,j2m21jlj2jm)(jzj2jmljlml,j2m2)-

E

I

9

I

$m,ml 5t,71,2 t.rl,/

.

(5.10a)

5.1. S t r u c t u r e a n d P r o p e r t i e s of t h e R e d u c i n g M a t r i x

105

These orthogonality conditions for the real reduction matrix P of (5.4) are examples of unitarity, that is, conservation of the sets' norm. The condition (5.10) might be enforced by setting the P matrix equal to the expression in the braces of (5.9) multiplied by N and adjusting the value of N. However, the required identities of factorial algebra are best derived in a context of combinatorial analysis. We proceed then by studying the expressions (5.4) and (5.6) of I further and comparing the averages of their squared moduli over the orientations of the vectors u and ~). The expression of Ill 2 derived from (5.4) includes products of the sets (jllu 2jl) and (j21v 2j~) and of their Hermitian conjugates; these products must reduce to scalars upon the averaging. This remark, combined with the normalization ~-~ml I(jlml 1~2J')l 2 = ~ , ~ I(j2m21~2J~)l2 - 1, implies that

dfi(jlrnllu2J~)(iz2jlljlml)

=

1 L 2j1+1

d~ (j2m2]i~2J~)(i~2j~lj2ml)

=

2j~+1 9

!

'

(5.11)

Substituting these expressions and of (5.10) then reduces Ill 2 to

A_]4~ dfi 4@f di: Ill 2 --

(2jl-l-1)(2j2+l{J|d:2') )(tD2z]J)

:

[(2jl

-~- 1)(2j2 + 1)] -1

(5.11a)

To evaluate the corresponding expression derived from (5.6), we consider first the explicit dependence of the squared moduli of the invariant factors of (5.6) upon the vectors {~i, ~, tb}. The squared moduli of the inner products are seen to be represented by (4.26a) and that of the scalar product by (4.26b). Thus we have III 2 - i 2 [ 89 + tb. ti)] b [ 89 + ~b. ~)]~ [1(1 - ti. ~))]~

(5.12)

In the averaging over the direction of ti, the variable 1(1 + tb-u) is simply 1 to be integrated from 0 to 1; similarly 7(1 + t~. ~)) ranges from 0 to 1 when averaged over ~). These operations must, however, be preceded by expressing the last factor of (5.12) according to spherical trigonometry as [g(1-fi.~?)] {1-tb-titb.7? -[(1tb. fi)2(1 - tb. ~)2]~1 71 (ei~o

(5.12a) 4- e -i~~

)

}c

,

and averaging it over the angle ~ between the planes (tb, ~i) and (t~, ~?). Only terms independent of ~ contribute to the average of the quadrinomial

106

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

on the right of (5.12a); one thus finds ( [ ~1( 1 -

ft. tb)] r ) v

-

[ 89 + d~. a)] q [1-(1 - w. ~)]~-q 2 q=0

x [1( 1 2 + ~h. 7))]~-q [ 89 - if-~))]q . (5.13) Substitution of the average (5.13) into (5.12) can now be followed by 1 + ~h./L) and 89 + 7h. 7)) from 0 to 1, as separately integrating over 7(1 anticipated above. These integrals are expressed in terms of factorials (Eq. 6.2.1 of [14]) leading to ~-~

dfi ~

d~)]I] 2 -

N2

(~)2(b+q)!(C(b+r q--0

(a+~-q(a+r g ,

_

g 2

=

a!b!c!(aTbTc+ 1)! N2(b+c+l)!(a+c+l)!(a+b+i)! ,

-

c':a'b'

(b+~+x)~(~;+~+~)~

~

(b+q)(aTc-q) , q ,

(5.14)

~-q

q=0

whose last equality follows from the identity (11.15) of Feller [15]. Comparison of (5.14) with (5.11a) and reference to (5.3) then yields N2 _ ( 2 j 1 ) [ ( 2 j 2 ) ! ( 2 j + 1)[ - a[b!c!(ji + j2 + j + 1)["

(5.14a)

Note that the last expression (5.14) exhibits full symmetry under permutations of the indices {a, b, c}, a symmetry vitiated in (5.14a) by the special role of jl and j2 in (5.11a) reflecting the asymmetry of (5.4). Multiplication of the expression in the braces of (5.9) by the value (5.14a) of N, replacing residual indices {a, b, c} and {c~,fl, 7} according to (5.3) and (5.8a), gives finally the explicit form of the reduction matrix element (jlj2jmljlml,j2m2)

• ~-'~(_ l)r

- [(2j + 1)a!b[c[/(a + b + c + 1)[1~

[(j+m)!(j-m)!(~,+m,)!(j,-m,)!(j2+m2)!(j2-m2)!] 89 (j-j2Wma +r)!(jx-ml-r)!(j2Tm2-r)r(j-jl-m2Tr)r(c-r)!r!

7"

(5.15) 1 where we have also set r - 7 c - 7. This parameter r is an integer whose range is finite and confined to nonnegative values. [Negative values of r, or of the argument of any factorial in (5.3), cause such factorials to diverge, thus yielding a vanishing contribution to (5.15); for this reason, the limit to the sum over r need not be stated explicitly.]

5.1. S t r u c t u r e a n d P r o p e r t i e s of t h e R e d u c i n g M a t r i x

107

The expressions (5.15) are often called Wigner coefficients since Wigner derived them first in 1931 in equivalent form. They are also called ClebschGordan coefficients after the men who introduced their concept much earlier, in the context of an expansion of group representations. Various alternative notations are used to indicate them. Table 5.1 gives a sample of these coefficients for the low values of jl and j2 that occur most commonly in applications to atomic and nuclear physics. Alternatively, their values are tabulated extensively in the equivalent symmetrized form of "3-j coefficients" to be described in Section 5.4. Practical applications in modern numerical calculations for atomic, molecular, and nuclear systems generate these coefficients most conveniently by subroutines that derive them through recurrence relations. 5.1.3

Recurrence

relations

An alternative approach to the calculation of the Wigner coefficients proceeds along the line indicated in Section 4.1.2 for the calculation of the d matrices. The reducing matrix P must transform the infinitesimal matrix of direct products (glz+iJly)+(J2,+ig2y) into the matrix Jz+iJy = P[(glx+ iJly)+(J2~ + iJ2y)]P -1 or rather (J~ + iJy)P = P(Jx~ + iJly + g2x + iJ2~). The explicit form of the latter equation

llJ~ + iJyljm)(jlj2jmljl ml, j2m2) = (jlj2jmA-1ljl rnlA-l,j2m2)(j~ m~+llJ~

(j m +

+ iJlyljlml)

+ (jlj2j m+lljlm~,j2 rn2+l)(j2 m2+l]J2. + iJ2ylj2m2) (5.16) and its Hermitian conjugate establish a system of recurrence relations among the matrix elements (5.1)with given (j,j~,j2) and different m values. This algebraic system leads again to (5.15) but requires a seemingly arbitrary convention to establish the relative sign of the matrix rows with different j. The spinor procedure avoids this requirement by providing the expression (5.15) with a definite sign for all values of the j indices. [The normalization coefficient is understood to be inserted with the positive sign of the square root from (5.14a).] 5.1.4

Symmetries

Two symmetry properties of the reducing matrix (j[PlJlj2) are important, namely its parities under permutation of jl and j2 and under frame reversal,

108

Chapter

Table

5.

Reduction

of Direct

Products

1 1 , and 73 7,

5 . 1 : C l e b s c h - G o r d a n or W i g n e r coefficients for j2 1 (jim1, ~m2lj~ 89

m2 1 j~ + -~

jl-

1

•

~/

jl :t:m+ } 2j1+1

1 -~

2jl+l

(jlml,lm2ljlljm) m~ - + 1

m2 -

r

(jl:t:m)(jl-t-m-t-1)

jl+l

+re+l) (2j1+ 1)(jl-'t- 1)

(2j1+ 1)(2j i-t-2) ::Fr (J l -l-m )(j l ~rn + 1) 2j1(jl-t-1)

jl

jl--1

0

y~l(jl+l)

/(j 1:t:m)(j 1::Fro+ 1) 2j1(2j1+l)

--

r

, - m ) ( j , +m3 jl(2jl+l)

( j i m 1 , 3m21jl 3 j m ) m2--~ jlWg

j1-1 jl--g

3

m2 -

(jl d:,,m-- 89 )(jl d=rn+ 89)(jl :t=m+ 23-) (2ix + 1)(2jl +2)(2j 1+3)

3

jl-t- 1

3

(2j, + 1 )(2j 1+2)(2j I +3) (jl=l=m+ 1)

:~:./3(j 1-t-rn - 89)(jl =Era+ ~ )(jl :Fro+ 23-) Y 2j 1(2j i + 1)(2jl +3)

~/ q:

3(jl.4.m_ 89)(jl :Fm. b x

V

=F(Jl =F3m+ ~ )

9

(2jl-- 1)(2jl + 1)(2jl-}-2)

2jl (2jl-- 1)(2jl-}- 1)

1

:i:~

--(jl+3rn-- 89 )

+

C

2jl(2j1..}-l)(2jl+3)

2j1--1)(2jl+l)(2j1+2)

a(j, ~ m + 89 + ~ - ~ ) ( j , - m 2j,(2j1-1)(2/~ + 1)

89

5.1. S t r u c t u r e a n d P r o p e r t i e s of t h e R e d u c i n g M a t r i x

109

that is, under sign reversal of all m indices. Both parities equal

(-1)J,+J~-J = ( - 1 ) ~

(5.17)

as we had anticipated in Section 5.1.1 from the antisymmetry of the scalar product (5.7) and from its role in the generating function (5.6). These parities can also, of course, be verified from the explicit form (5.15) of the reducing matrix. The commutation of frame reversal with the transformation (5.15) is represented analytically by

(jlj2jmJjlml,j2m2)

-

~ t~r.Ll

(jmlUJjm') I

i

I~D, 11r/'I, 2

X (jlj2jm'Jjlmll, j2m12)(Jlmll =

IUljlml)(j2m~lUlj2m2)

(5.18)

(--1)J-m(jlj2j-mJjl-ml,j2-m2)(--1)J~-rn~(--1) j2-rn2 ,

since (5.15) is real and U -1 - /). Considering further that m - ml + m2 and that j - m is an integer, reversal of the sign of all m indices is seen to have the parity (5.17). Note that (5.18) requires the Wigner coefficients to vanish for ml = m2 - m - 0 unless jl + j2 - j is even. 5.1.5

Reduction

in the

Cartesian

frame

The transformation matrix that reduces a direct product of two sets in the Cartesian frame is obtained by transforming the standard-frame matrix (5.15) to the Cartesian frame by the matrix T given in (3.46) and (3.46a)"

(ei~2~'JmJEJ eljmlj~l'e2jm2j~2) -

Z

(~JmJ~[T-lJ~'m/) (5.10)

The matrix (5.19) is no longer block diagonal in m = m l + m 2 , of course; restrictions to its nonzero elements are more complicated, analogous to those in the reduction of products of Fourier series terms when cast in cosine rather than exponential form. On the other hand, the transformation (5.19) preserves the parity of the direct product elements under the operation Dy(zr), as represented by the block-diagonality condition (-1)te -

( - 1 ) l~+12 ele2.

(5.20)

110

Chapter 5. R e d u c t i o n of Direct P r o d u c t s

This condition ensures that the matrix elements (5.19) are real even though the matrices T are complex. Additional details on reduction in the Cartesian frame are given in [16]; however, the phase normalization of the T matrix differs in that reference.

5.2

Reduction of r-Transformation Products

Having learned how to reduce direct products of two standard sets, for example, (jlml[(j2m~[, by the matrix (5.1), we should also be able to reduce the direct product of their r-transformation matrices, (jlm~ [R(r 0, ~o) 9 I [jlmi) x (32m2[R(r , 0, ~o)[j2m2). In matrix notation, this reduction may be indicated by casting it as pei~"(fl+f=)P -1 - e ia$, or conversely, e ia(fl+$=) = p - l e i ~ " $ P . One may also say that an r-transformation of the reducible product set (jlmll(j2m2[ could proceed alternatively in two ways, with the same end result: a) by multiplying the product set by the product of the two matrices D1 and D2, or b) in successive steps, by reducing the initial product set, followed by r-transformation of each irreducible subset of degree j and finally--if need be---by inverse transformation of the result to the original direct product representation. Alternatives a) and b) correspond to the two sides of the equation (jl m~ JR(t, O, ~o)ljlml )(j2m~2nR(r 0, ~o)[j2m2) =

"m'2]JlJ2J ml E ( j l m l ', 32 J x (jm~+m~lR(r (5.21)

x ( j l j 2 j ml + m 2 1 j l m l , j 2 m 2 ) .

This key equation embodies the very concept of reduction; in fact, it occupies the central position in the group theoretical description of reduction, because group theory deals mainly with transformations rather than with the sets to be transformed. Replacement of the explicit form (4.4) of the r-transformation matrices into (5.21) shows the reduction to be nontrivial only for the -~(j) factors, because the exponentials e~'~ etc. factor out d(J 1) {O~d(J~) m,lml , ] m,2m2(0) x d(J! rnl +rn~

ml+m

-

, E ( j l m ~ ,~ j2m2lj~j2j m~~ +m~) J

-

2

(O)(jlj2j ml + m 2 1 j l m l , j2m2)

9

(5.22)

For our purposes, mainly those of quantum-mechanical applications, Eq. (5.21) provides the tool for evaluating integrals over the multiple products

5.2. R e d u c t i o n of r - T r a n s f o r m a t i o n P r o d u c t s

111

of r-transformation matrices which occur as the angular factors of wave functions. Multiplication of (5.22) by d~! m followed by integration over angles yields, owing to (4.20), 0* sin 0 dO a(J) (O)d(J 1 =

(j2

. m ~2 1 j l j 2 j m ' ) 2-]T-f ( j i m 1 , 32 2 ( j l j 2 j m ] j l m l, j2 m2)

X 6mr ,m 1'+m~ 6m,rnl +m2 9

(5.22a)

(We take here the point of view of Section 4.3, according to which even the usual spherical harmonics are special cases of r-transformation matrices; the same point of view can be extended to transformations under operations of additional, or altogether different, groups.) As (5.21) replaces the product of two matrix elements by a linear combination of single matrix elements, it can be applied in succession to the product of an indefinite number of factors, until this product reduces to a linear combination of single matrix elements. At this point only the elements, if any, with j = 0 contribute to an integral over (r 0, T), as stressed in Section 4.1.6, in which case the integral reduces to 87r2 because D~~ - 1. Alternatively, and equivalently, the reduction of the product could terminate with a linear combination of products of two matrix elements, whose integral is given by the orthonormality of the D functions. The procedure outlined here does not eliminate angular integrations altogether. Rather, it replaces the integrations by the algebraic procedure of evaluating sums of products of matrix elements (5.15). The summation indices, which replace the angular variables, are discrete with a range limited by triangular conditions. This remark illustrates the familiar quantum-mechanical procedure of working within alternative--in particular, conjugate--representations. Evaluation of integrals within a coordinate representation proves equivalent to summations over quantum numbers in the conjugate representation, in obvious analogy with Fourier analysis, where discrete Fourier coefficients provide information contained in a continuous periodic function. The summations which replace integrations might nevertheless be themselves quite laborious but will be further reduced in Chapter 7 to a sequence of simpler standard steps. We also defer the elementary evaluation of quantum-mechanical matrix elements of spherical harmonics for atomic wave functions until Chapter 6, where it will be framed within the general treatment of tensor operator matrices.

112

5.3

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

Irreducible Product Sets

Having seen how to reduce the direct product of two standard sets, (j~mll(j2m21, in Section 5.1, we may now introduce a notation that labels individual irreducible product subsets obtained by reduction. To this end we adopt once more the notation of quantum-mechanical transformation theory, as in Section 3.2.3, extending it as appropriate. We shall thus indicate by

(jlj2jm I - ~

(JlJ2jmljlml,j2m2)(J~m~l(J2m21

(5.23)

mlm2

the elements of the "irreducible product of degree j" obtained by reducing the direct product of two sets <Jll and <J21 by the transformation matrix (5.15). The entire set of elements (5.23) will be indicated by {jlj2jl, or by <(JlJ2)jl, thus grouping the indices of the two factors in parentheses when their meaning might prove unclear. Products of sets contragredient to the standard sets will be indicated by the corresponding ket symbols, as in

Ijlj2jm)-

~

Ijlml)lj2m2)(jlml,j2m21jlj2jm).

(5.23a)

mlm2

Additional labels may again be introduced next to the indices j as needed. 5.3.1

Special

cases

The product of degree j = 0 of two standard sets, which exists only when jl = j2, coincides with their scalar product (3.28) to within a normalization faclor. Indeed, setting j = 0 in the triangular conditions (5.3) implies j2 = jl and a = b = 0. Ratios of factorials in the summation of (5.15) then reduce to unity while its normalization factor before the ~ sign reduces to (2jl + 1)- 89 yielding

(jljlOOIjlml,jlm2)- (2j~ + 1)- 89

(5.24)

The j-dependent normalization factor in this and analogous expressions ensures orthogonality of the reducing matrix. The reduction is thereby inverted simply by transposing the reducing matrix,

(jlrnll(j2m21 -- ~-~(jl ml, j2m2ljlj2jm)(jlj2jml. jm

(5.25)

5.3. I r r e d u c i b l e P r o d u c t Sets

113

This inversion requirement is foreign to the definition of scalar product in Chapter 3, which was patterned after the product of two vectors, that is, without regarding the scalar product as part of a complete set of irreducible products. The normalization factor in (5.24) was, therefore, absent in (3.30). [The normalization factor y/ 89appropriate to jl - 1 appeared instead in (1.19), which derives from a reduction procedure.] The product of degree 1 of two irreducible sets of the same degree 1 corresponds to a vector product--as anticipated in (3.42-44)wagain to within a normalization factor arising from the orthogonality of the reduction matrix. The general analytical form of the Wigner coefficients (5.15) simplifies considerably for small values of the j indices, up to 1, a, or 2, as listed in Table 5.1 and in reference books. [The reference to sign differences of products, in a note on p. 37 of [1], is not relevant here, because such differences have been avoided by adjusting the sign conventions in (3.28) and in (3.4243).] 5.3.2

Symmetry

Equation (5.17) implies that irreducible product sets are even or odd under permutation of their factors depending on the parity of j l + j 2 - j . (Effects of noncommutability of the factors in a product of operators will be considered separately in Chapter 7.) This result extends the familiar property that the scalar product of two vectors is even while their vector product is odd under permutation of its factors. As the vector product of a vector by itself vanishes, so do all products of an irreducible set of degree jl by itself vanish for odd values of 2jl - j. Note that this rule excludes odd values of j for integer jl but even values for half-integer jl. Self-conjugation of irreducible sets under frame reversal is preserved by construction of their irreducible products, owing to the combined effect of (5.17) and (5.18)--with the understanding that frame reversal implies commutation of factors, as it does for ordinary vector products. 5.3.3

Products

of contragredient

sets

The direct product of two contragredient sets, as, for example, of a set of bra and one of ket symbols, also constitutes a reducible tensorial set. Its reduction is achieved by the matrix (5.15), or by its inverse, a f t e r the two sets have been made cogredient by transforming one of them by the matrix U in (3.11) or by its inverse. A complication is met, however, in spelling

114

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

out this procedure. When the two initial sets are of equal degree, their irreducible product of degree zero should be defined so as to coincide with the ordinary inner product, to within a positive normalization coefficient. A curious effect occurs in the construction of the irreducible product of degree zero of initially contragredient sets that involves lwo successive applications of the U matrix, one to remove the contragredience and the other to build the product according to (5.24); this combination reverses the sign of sets with half-integer j. The sign reversal will be avoided by removing the contragredience by means of the operation U -1, rather than by the usual U, in the construction of irreducible products. This stipulation should be applied, however, only when transforming the second factor of a direct product to which the irreducible product of degree zero applies the U matrix. Transformation of the first factor by the U matrix, followed by construction of the irreducible product of degree zero, yields instead the inner product of the initial sets as desired, whereby product definitions depend on the factor to be transformed. Accordingly we shall denote, in the product symbol, the index of the factor transformed by the U or U -1 matrix by a cross, writing,

Ijlj~jm)- ~

Ijlml)(-1)J~+'n2(j2-m21(jlm~,j2m21jlj2jm),

(5.26)

1711171 2

(jlj2jml- ~

(jlj2jmljlml,j2m2)ljl-m1)(-1)Jl-"l(j2m21.

(5.26a)

mlm2

Notice the use of the exponent +m2 in (5.26), whereas - m l appears in (5.26a). (This stipulation departs from the definition 7.19 of [1], but [1] was led to modify that definition in its applications.) 5.3.4

Wave-mechanical

examples

Early examples of construction of irreducible products dealt with the wave functions of a pair of particles with orbital momentum quantum numbers gl and g2, fl(rl)Y~l,nl(Olqal) and f2(r2)Y12m2(02~2). A set ofwave functions of this pair, with total angular momentum quantum number L, is represented in our notation according to (5.23a) as (r1~l~l,r2t92~2[i-1g2LM)

-

E (01~pll~lml)fl(rl) Erl llTI 2

• (02~21s

(5.27)

The flexibility afforded by the concepts of direct and inverse transformation is utilized when the wave functions of the two separate electrons are

5.3. I r r e d u c i b l e P r o d u c t Sets

115

given initially as superpositions Ca(f'l) and Cb(F'2) with specified coefficients (~flm~la) and (e m lb). One represents then the r162 as a superposition of wave functions (5.27) by the combination of two reciprocal transformations,

~-]~(r~O~,~,r202~2lele2LM)(ele2LMlab)

r162162

(5.28)

LM

where (5.27) is to be substituted and

(f.I~2LM]ab)-

Z

(~ll2LMl~lml'f'2m2)(~lmlla)(~2m2lb)"

(5.28a)

llml12m2 Direct products of contragredient sets occur in the angular distribution of the probability density of a single particle, for example, Ir

(alem')f(r)(em']O~,).

m

m

(5.29)

I

We resolve this density distribution into a superposition of its multipole components by combining (5.26) and (5.28), Ir

-

E kq

=

[~-'mm'(O~lem)(-1)l+m' (g -m'lO~,)(em, em'leekq)] X [Emm, (eekqlem, ern')(ernla)(-1) l+m' (~le-m')] (0

,leetkq)(eetkqlaat).

kq

(bao) In this example, the dipole term (with k - 1) and other odd multipoles vanish owing to the combined symmetries of spherical harmonics, Yt:-m, (-1)m'Ylm, and of permutation symmetry (5.17)in the reduction matrix elements. This effect reflects the inversion symmetry of the density distribution of all eigenstates of I~ ~. 5.3.5

Multiple

products

The direct product of an arbitrary number n of standard sets can be reduced completely by a sequence of n - 1 transformations of the type (5.1), applied to two sets at a time. Thus, for n - 3, one may construct the irreducible product with elements

((jlj2)jl2j3jml

=

~

((jlj2)jl2jajljlml,j2m2,j3m3) x (jlmll(j2m21(j3m31,

(5.31)

116

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

whose coefficients are products of matrix elements (5.1), ((jl j2)j12j3j IJl ml , 22 m2 , j3m3 ) "- (j12j3jmljt2m12, j3m3)(jlj2j12m121jl m1, j2m2) .

(5.31a)

This construction is not unique, as shown by comparison with the alternative product (J1(J2J3)J23jml-

E

(Jlj23jmljlml'j23m23)

rrtlm2m3

• (j2j3j23m231j2m2, j3m3)(jlml ](j2m21(j3m31.

(5.32)

The irreducible set elements (5.31) are eigenvectors of the whole set of six commuting matrices

{ I~12, I~12, IAI 2, I~ + ~12, I~ + f2 + ~12, Jl~ + J2z + J3z },

(5.33)

while the elements (5.32) are eigenvectors of the alternative set

{ I~12, I~12, IAI 2, I~ + AI ~, I~ + i2 +/312, J~ + J=~ + J3~ }. (5.34) Although associativity holds for this operation of combining more than two sets insofar as the resulting irreducible products have the same values of ]J-1 + J'2 + ~]~ and Jl~ + J ~ + J3~, the pairs of matrices differ in including alternative intermediate operators [J] + j~j2 and ],~ + ~]2 that do not commute. In general, two matrices ] ~ i ~12 and j ~ k X j2 commute if, and only if, the two sets of indices {i} and {k} are either mutually exclusive or one of them is entirely included in the other. Many alternative sets of 2n commuting matrices can thus be found for n > 3. All involve the n individual squared angular momenta j~[2 together with the total angular momentum and its z component, leaving open choices for the ( n - 2) remaining "intermediate" matrices. Irreducible products consisting of eigenvectors of such alternative sets of matrices are said to differ in their "coupling schemes," an expression derived from the "coupling" (or addition) of angular momenta in quantum mechanics. Irreducible products of equal sets constructed according to different coupling schemes are related by orthogonal "recoupling" transformations, to be described in Chapter 7. 5.3.6

Coupling

diagrams

Diagrammatic methods serve to represent reduction operations~originally the equivalent addition of angular momenta--especially for multiple cou-

5.3. I r r e d u c i b l e P r o d u c t Sets

117

plings and for recouplings. Basically we shall use the symbol

(

j2

jl (5.35)

to indicate the rectangular submatrix of (jlj2jmljlml, j2m2) with a fixed value of j. In essence a line open to the right (left) represents a ket (bra) symbol. Additional details or modifications of (5.35) will be introduced when appropriate. The submatrix of the reciprocal transformation, (jlml,j2m21jlj2jm), whose elements are equal to those represented by (5.35), is diagrammed by reflecting (5.35) through a vertical line

j2

9

(5.36)

A main diagramming procedure joins a bra and a ket line with the same j index to indicate summation over the corresponding quantum number m. Thus, for example, the combination of matrix elements (5.31a), representing the reduction of a triple product, is diagrammed by

j3

j12

(

j2

jx (5.37)

even though only a single term, with m 1 2 - - m l -t- m2, gives a nonzero contribution to the reduction matrix. We apply here the diagram technique to illustrate an alternative structure of the Wigner coefficient formula (5.15). This structure hinges on the remark that the Wigner coefficients, as well as their construction, simplify greatly when the three infinitesimal rotation operators {~1, ]2, j} are "parallel," in the sense that the index j equals jx + j2 whereby c = jx + j2 - j

118

C h a p t e r 5. R e d u c t i o n of D i r e c t P r o d u c t s

vanishes, 1

(Jlj2jmljlml,j2m2) -- (a+b)!(j,+rnl)!(j2Tm2)!(j, =

(a--I-b)- 89 a

) 89

jl + rnl

)!(j -rn2

) 89

, for j - jl + j2.

jl -- ml

(5.38)

On this basis, one may rearrange the factorials in the braces of (5.9) or (5.15) to represent (5.15) as a sum of products of three Wigner coefficients of the special type (5.38),

1ac~)(Si ~c7, 1 1 (jlj2jmljl ml, j2m2) - N E(lb-21ajmllbfl, 7 -~ 7c -7) • (152 ~ , lc,.fllblcj 1/7/1)(1~ c

1 9 -7 , l a ~ 1 1 -~c-~a.12m2),

(5.38a)

where N is given by (5.14a) and j takes any value in its usual range from [jl - - j2[ to jl "~" j2. The right-hand side of (5.38a) is represented by the diagram 1 ~a

1 > ~C

J2

89 ~

jl

f

J

!b 2

(5.39)

in which we have introduced the additional symbol

(

1 ~c

1 ~c

(5.40) to indicate a scalar product. Note the arrow's essential role in the diagrams (5.39) and (5.40)in representing the order of factors. Its reversal would introduce a factor ( - 1)c in the evaluation of a matrix element corresponding to permutation of jl and j2 in the diagram (5.35).

5.4. S y m m e t r i z a t i o n of W i g n e r Coefficients

119

Together with the diagrammatic rendering in Chapter 7 of recouplings of more complicated multiple products, combinations of sets and operators can be conveniently handled through such graphs. :lust as the famous Feynman diagrams have proved powerful in quantum field theory, these angular momentum diagrams have proved increasingly useful in atomic and nuclear physics [17]. The geometry of diagrams often helps to recognize symmetries and relationships that are not as immediately evident from algebraic expressions. Diagrams serve directly for calculations as well, just as Feynman diagrams do. Once basic diagrams have been defined for Clebsch-Gordan coefficients as in (5.35), for other recoupling coefficients, and for operators, their combinations can be immediately constructed. Subsequent manipulations then yield simplifications for the final results. Indeed, the topological equivalence of (5.35) and the basic Feynman vertex with three interaction legs afford integrating Feynman and angular momentum diagrams. General rules have been given for such graphical calculations of the interaction between many electron (or other fermion) systems through multipole or other operators and for handling the angular momentum couplings described by such diagrams [17, 18, 19, 20].

5.4

Symmetrization of Wigner Coefficients: Invariant Triple Product and 3-j Coefficients

The definition of the reduction matrix elements (5.1) ("Wigner coefficients") implies some form of symmetry under permutation of their indices jl and j2, which was indeed discussed earlier, but no such symmetry under permutation of either of them with j. Yet a degree of symmetry among all three indices (jl,j2, j) of the Wigner coefficients is suggested by the triangular relations (5.3) and by the structure of their explicit forms, (5.15) and especially (5.38a). We shall now explore this more extended symmetry from both functional and structural points of view. The functional symmetry emerges by considering the triple product of degree zero, that is, the invariant product of three standard sets with the seemingly different representations (5.31) and (5.32). Upon setting j = 0 in the expression (5.31a) of the transformation coefficient of (5.31), we see that its index j12 must equal j3, that m12 -" -m3, and that the right-hand

Chapter

120

5. R e d u c t i o n

of Direct Products

side of (5.31) reduces to (5.41)

(--1)J a +m3 ( 2j3 -4- 1)- 89(jl j2j3 - m a ljl rn l , j2 rn2 ) .

There is in fact only one irreducible invariant product in this case; hence it must coincide, to within a :t: sign, with the product of degree j - 0 constructed by the alternative coupling scheme (5.32). To verify this conclusion in (5.32), we must set j = 0,j23 -- j l , rn:~3 "- - r n l , whereby the coefficient of (5.32) reduces to ( - 1)j ' - ' n ' (2jl + 1)- 89(j2j3jl - m l

Ij2m2, j 3 m 3 )

(5.41a)

9

The two sets of coefficients (5.41) and (5.41a) should now coincide to within a sign; they actually coincide exactly, as one verifies by replacing in (5.15): j by j3, m by - m 3 , and r by r - j 3 + jl - m l - m 3 . To bring out the identity of the two formulas, we first notice that the different factors, (2j3 + 1)- 89 and (2jl + 1)- 89 just cancel the obvious asymmetry of the normalization factors of the Wigner coefficients in (5.15) or (5.38a). The remainder of the normalization factor is wholly symmetric in the j indices as well as in the three indices {a, b, c}. With regard to the main portion of the Wigner coefficients, one can see that the magnitude of individual terms of the sum within (5.38a) also depends quite symmetrically on the three j indices and on the {a,b, c}. The one asymmetric element in (5.38a) is the scalar product matrix element. We should then modify the structure of (5.38a) a little further, to manifest its higher symmetry. To this end, the first Wigner coefficient on the right-hand side of (5.38a) may be transformed in four steps, none of which changes its value: a) replace it by 1 1 its transpose , b) permute sa and sb, c) apply (5.18), and d) replace the phase factors from (5.18) by scalar product matrix elements according to (3.30). Thus we write

(Tb-fa~~ Jm[ 89 89 ol l ~ l ll,,.n.I

I 'bjm')(jm, x ( 89 ' , ~'b~'l 5a

jm'lS).

(5.42)

Upon substitution in (5.38a), we remove an asymmetry by permuting fl and ~/' in the scalar product matrix (S] 89 ~1 b~') , inserting the equivalent factor ( - 1)b; an apparent asymmetry between the triads {a, ~, 7} and {c~', ~', 7'} still remains, but it is also removed by interchanging all labels fl and /~', with no effect since both indices are summed over. We also separate the asymmetric factor (2j + 1) 89 and the factor ( j m , j m ' l S ) = (--1)J--mt~_m,,~,. Equation (5.38a) thus becomes

121

5.4. S y m m e t r i z a t i o n of W i g n e r Coefficients

(jlj2jmljlml,j2m2) = (2j + 1) 89 X

J-m

(Sl ~c;, ~ ~' ; ) ('S l ~ . ~ , •

~'~'~'

~' . ~ ' ) ( S l ~' b ~ , ' ~b~)'

--m)

(5.43) The expression in the summation now shows full circular symmetry in the equivalent sets of indices {jl, j2, j} and {a, b, c}. The factorization of (2j + 1) 89in the symmetrizing process (5.43), which was previously implied in (5.22a), will recur in (5.47) and again extensively in Chapter 7 and beyond. The invariant considered in Section 5.1, constructed with a Wigner coefficient and with the three sets (c2J[j), (jI[A2J'), (j21B2J2), may now be viewed as the triple product of degree zero, constructed according to (5.31), (5.26), and (5.41),

E

(C2ilJm)(jij2jmljlml'j2m2)(jlmllA2j~)(j2m21B2j~)

t'tl m l m 2

=

( 2 j + 1) 89

E mlm2rn

(JlJ2jtOO[jlml'j2m2'jm~) I

(jlmllA2J')(j2m21B2J~)(-1)J+m' (c2Jlj -m'),

(5.44)

renormalized by the factor (2j + 1) 89 Comparison with (5.43) shows the triple product coefficient to coincide with the symmetric expression in the braces of (5.43) to within the factor ( - 1 ) b. The occurrence of this factor shows the triple factor product to be still not quite symmetric in its three factors (see also Chapter 10 of [1]). The fully symmetric summation in (5.43) is called a "3-j coefficient," forming a main ingredient of the "Racah-Wigner algebra" developed in later chapters. Its explicit form does not preserve its manifest symmetry once one starts carrying out the sums in (5.43) taking into account the Kronecker symbols in its (Sljj) matrices. Accordingly we define here the 3-j symbols in terms of the Wigner coefficients, as given by (5.15),

jl ml

j2 j3 ) m2 -- m3

(- 1)J~+j~-j2 (jlj2jaOOljl ml , j2m2, j3 -m3) (2j3

+ 1)- 89(-1)J'+J~-J~(-1) j~-m~ x (jlj2jamaljlml,j2m2).

(5.45)

The first symbol here represents the 3-j symbol in its usual form, whereas the equation's middle part represents the coefficient of the triple product in (5.44) modified by a phase factor. On the right-hand side of (5.45), the

122

C h a p t e r 5. Reduction of Direct P r o d u c t s

first factor (2j3 + 1)- 89 represents the change of normalization discussed above, while the next two appear in the previous equations as the values of ( - 1)b and of (jm, j'm'lS). The change of normalization allows for including the summation over three rather than two indices in the orthogonality statement of the 3-j coefficients, E

(jl

j2

ml

mlm2m3

E

j3 ) 2

m2 --m3

(jlj2jaOOIjlml,j2m2,j3-m3) 2 -

1.

(5.46)

mlm2m3

The 3-j symbols have been discussed and tabulated extensively, particularly by M. Rotenberg et al. [21]. The complete symmetry properties of the 3-j symbols stem from those of the Wigner coefficients by (5.45) and the summation in (5.43): a) invariance under circular permutation of its columns of indices (j m), b) parity (--1) jx+j2+j3 under permutation of any two columns, c) parity (-1)Jl+J2+J3 under sign reversal of all "m" indices. The symmetrization procedure of this section is illustrated by deforming the diagram (5.39), from its representation of the construction of a Wigner coefficient, to that of a 3-j coefficient, as shown and discussed below,

ga

jl

1(

7c 89

~

j2

J

la( (2j+l) 89

b

>

)

jl

89 (5.47)

5.4. S y m m e t r i z a t i o n of W i g n e r Coefficients

123

The major features introduced by deforming the left-hand side of this diagram result from twisting the j line on its left to parallel the jl and j2 lines 1 on its right. This process stretches the 5b line around the back of the rightz line. The twisted hand diagram, folding and compressing instead the ~a section of the j line appears on the right as a semi-circle detached from the rest and represents the factor (jm, ire'IS ) of (5.42) that is not included in the summation in (5.43) but is allowed for by the factor ( - 1 ) j3-m3 in (5.45). Arrows on the ~al and ~lb lines on the right complement the initial arrow on the 89 line to represent a uniform flow. But the symmetry of the {a, b} pair is thereby spoiled, an effect compensated by the coefficient ( - 1 ) b. The additional coefficient (2j + 1) 89stems finally from (5.43).

Problems 5.1 Consider a system of three spinless particles whose independent states are classified, respectively, as p, d, and f. Using the triangular conditions, determine the number of eigenstates of the total angular momentum/~2 of this system with various values of L. Verify your result by using alternative coupling schemes. 5.2 Consider a system of two spinless particles in a central field, whose independent states are classified as p and d. For each possible value of the total orbital quantum number L write the wave function with M = L. (An adequate table of Wigner coefficients is found in Table 5.1 or on p. 76 of Condon-Shortley [6].) 5.3 The ground state of the NO molecule has a single unpaired electron in a 7r state, that is, with unit orbital momentum component about the molecular axis; call its orbital wave function f(p, z')e +i• The spin of this electron can take either orientation with respect to this axis, yielding the two levels of a doublet II3/2, IIz/2. Write all the rotation wave functions of NO molecule in this doublet state with j - 51 and give their parity. (Disregard the other electrons and the vibrational motion.) (This problem utilizes material from Chapter 4.) 5.4 Majorana pointed out that the coefficients a(m j) of the state of a particle of spin j, ~-~m u~)a~), may be identified by the orientations of 2j unit vectors /Li. Which are these orientations for the state identified by the coefficients (jmla 2j) in (4.6) or for the states (jllti2J'), (j21~?2jr) in (5.4)?

124

Chapter 5. R e d u c t i o n of Direct Products

5.5 The similarity of the explicit expressions, (5.15) for the ClebschGordan coefficient and (4.3) for the rotation functions, reflects their relationship, particularly in the "classical limit" of large angular momenta: (jJJ'm + Mljm, JM) ~ (-)J-J+J'd (j) (0), with cos0 w

-- J, rrt

M[J(J + 1)]- 89 Here uppercase symbols denote numerically large values, although the relation remains valid down to surprisingly low values. Verify the above for the particular case j = m = 1, J' = J + 1 when the right-hand side of the expression reduces from Table 4.1 to d~11) - 89 + cos 0).

PART

B

T E N S O R I A L A S P E C T S OF QUANTUM PHYSICS

126

Part B

Part A of this book has dealt with the classification, transformation, and combination of irreducible tensorial sets. Its bearing on quantum physics has been limited to identifying and treating state representatives, namely, "ket" I) and "bra" (I symbols, and probability amplitudes. Directly observable elements of quantum physics are, however, represented by the richer manifold of operators, which can also be cast in terms of tensorial sets. Quantum-mechanical operators have the form of bilinear products of a ket and a bra: I)(1. Section 5.3.3 has introduced the handling of such products but Part B of the book centers on a broader task that has stimulated major developments in the algebra of tensorial sets. The relevance of tensorial algebra to quantum physics rests on the role of the two-dimensional unitary algebra SU(2) as a building block common to tensorial algebras and to the Hilbert space of quantum mechanics. This connection is detailed and developed in Chapter 6, reaching a point where emphasis on the geometrical representations of operators affords casting quantum mechanics into geometrical and real, rather than complex, terms. The reduction of multiple products of tensorial sets may follow different paths corresponding to alternative dominance of various physical factors. A prototype example occurs in the reduction of two-electron states, which may stress either the spin-orbit coupling of each electron or the interaction between the paired electron orbits (j j- vs LS-coupling). The irreducible sets constructed by such alternative paths are interconnected by an algebra stimulated by physical contexts, which is invariant under r-transformations and described in Chapter 7. That chapter considers varied recoupling transformations, the L S ~ j j being one example, among multiple products of sets and operators, their factorizations, and their applications to matrix elements of quantum physics that involve such multiple products. The final chapter, 8, of Part B deals with the specific problem that motivated the developments of Chapters 6 and 7, namely, the identification and classification of energy eigenstates of incomplete atomic or nuclear shells. This problem will also introduce the interplay of rotational and permutational transformations of identical-particle states. The unique classification of states of an incomplete shell, particularly for higher j, involves sets of operators that do not commute with the Hamiltonian of the multiparticle system. The resulting "noninvariance" groups and symmetries are instances of a broader class of entities, including the so-called "noncompact" groups to be treated in Part C.

Chapter 6

Tensorial Sets of Quantum Operators The relevance of irreducible tensorial algebra to quantum systems is obvious for isolated ground state atoms and for other systems whose energy eigenstates are also eigenstates of angular momentum. More generally, countable sets of base states of any quantum system can be arranged in one-to-one correspondence w i t h - - t h a t is, "mapped" onto--one or more irreducible sets of angular momentum eigenstates, as will be illustrated by various examples. This chapter deals with operators of any such system. Casting of any operator as a linear combination of reducible tensorial set elements is the first step of this task. Reduction of these base sets will then recast the operator into one or more irreducible tensor operators. Consider a quantum system with base states represented by an irreducible set of kets {]jm)}, directly or by mapping. The product of a ket Ijm') and of a bra (jml, namely, ]jm')(jm], may be viewed as an operator that replaces the ket ]jm) with the ket Ijm'), when applied to Ijm). This operator is not Hermitian, but the symmetrized Hermitian combination

Ijml)(jml + Ijm)(jm'l

(6.1)

amounts to an operator that permutes the elements (m, m I) of the set {Ljm)}. An operator represented by the matrix T,nm' is similarly represented by

T - ~

Ijm)Tmm, (jm'

mSm

127

I,

(6.2)

128

C h a p t e r 6. Tensorial Sets of Q u a n t u m O p e r a t o r s

that is, as a linear combination of the set of elementary operators

{ Ijm)(jm'l }.

(6.3)

The operator set (6.2) is reducible, its reduction being accomplished, in accordance with (5.26), by

Ijm')(jml(- 1)j-m (jm', j -mlkq) - Ijj t kq),

(6.4)

rl-gtm

whose right-hand side introduces the symbol for elements of an irreducible standard tensorial set of unit operators. A corresponding reduction can be applied to the matrix elements Tram', viewed as tensor components,

y ~ (kqjjtljm',j -m)(-1)J-mTm,m - (kqjjtlT), m'm

(6.5)

of the matrix Tram'. An alternative representation of the operator T, equivalent to (6.2), is thus afforded as the inner product of irreducible sets on the right of (6.4) and (6.5),

Ijj t kq)( kqjjt lT).

(6.6)

kq Illustrations of this formal development by specific examples will be provided in the following sections. Just as products of two setswof kets and b r a s n a r e involved in defining operators in quantum physics, they also are central to the so-called "Liouville representation" of quantum mechanics wherein the density matrix, not the wave function, is the primary object of study. Section 6.1 introduces this subject, and Sections 6.2 and 6.3 will deal extensively with prototype examples involving a spin-~1 particle and quantum systems equivalent to them, dubbed "two-level systems." Sections 6.4 and 6.5 will extend this treatment to the orientation of particles with arbitrary spin j and to their isomorphic "(2j + 1)-level systems." Section 6.6 will introduce a further extension to systems whose base sets consist of, or can be mapped onto, aggregates of irreducible tensorial sets. The final Section 6.7 focuses on the calculation of matrix elements for all such quantum systems.

6.1

The Liouville Representation of Quantum Mechanics

As noted at the outset in Section 2.4, and explicitly in Section 4.3.1, any observable property of a particle with spin 71 can be regarded as a function of

6.1. T h e Liouville R e p r e s e n t a t i o n of Q u a n t u m M e c h a n i c s

129

its spin polarization/~. Cartesian components of this vector are measured by the particle's mean deflection within Stern-Gerlach magnets oriented along the axes and are represented theoretically as the mean values of the Pauli operators {cry, ~ru, a~} defined in (2.32). Accordingly, the magnitude IPI may range from zero to unity as in Eqs. (4.26). The tensorial algebra developed in previous chapters serves to extend the concept and treatment of spin polarization in two distinct but complementary and compatible directions: A~ Since the states of all so-called "two-level systems" can be "mapped" onto the states of orientation of a particle with spin ~1 (that is since all these systems are isomorphic), all properties of two-level systems can be similarly represented as functions of a polarization vector P. This remark enables one to transfer the experience gained in the study of spins to the analysis of any new two-level system. Its importance for acquiring familiarity with quantum mechanics has been stressed by Feynman et al. [22]. Current use of the expression "photon spin echoes" rests on the isomorphism of photon-induced transitions between two atomic levels and of reversals in the orientations of a spin -1 particle. no The state of polarization of particles with spin j > 1 is represented by a set of parameters analogous to/5. This set consists of the mean angular momentum (J)-- which is analogous to that of spin 1 and represents the particle's o r i e n t a t i o n , of a mean quadrupole tensor (Q) which represents the particle's a l i g n m e n t , and of higher multipole parameters for increasing values of j extending up to a maximum multipole order of 2j. The time-dependent SchrSdinger equation for the particle polarization can be cast as a system of linear first order differential equations in these parameters. The Bloch equation (Problem 2.2) for the precession of a magnetic moment about an external magnetic field is the prototype of such first order equations for the precession of higher multipole moments in arbitrary inhomogeneous fields.

Extensions A and B are combined by translating any (2j + 1)-level quantum problem into a polarization problem for a particle with spin j. More accurately, the quantum system is mapped onto an operator set isomorphic to polarization and to higher multipole moments. Systems of any size can be considered from this point of view. The very process of mapping

130

Chapter 6. Tensorial Sets of Quantum Operators

one representation onto another also introduces symmetry elements such as parities under reflection of reference axes or complex conjugation, as well as recurrence relations. Once the mapping is established, the concepts and techniques developed in previous chapters for the treatment of tensorial sets apply to the analysis of any physical system. The parameterization of quantum mechanics in terms of the mean values of operators isomorphic to the polarization parameters of a spin j is called the "Liouville representation" owing to the role it plays in statistical mechanics. This chapter deals specifically with constructing complete sets of polarization parameters, but its importance rests largely on the isomorphism of all quantum systems with equal numbers, 2j + 1, of mutually orthogonal states. Extension A will be considered first, beginning with a summary treatment of spin polarization.

6.2

Q u a n t u m Mechanics of Particles with Spin 1

We review here briefly the quantum mechanics of particles with spin 89in order to adapt it to our purposes. The states of orientation of spin- 71 particles are represented in terms of a pair of base states, for example, "spin up" and "spin down." A "pure state" Ia) of spin orientation directed along t 3 - (8, ~) is represented, according to (4.26a, b) and (2.38) (with r = 0), by the standard set of probability amplitudes 1 1 1 (77]a) - cos 70

ei

89

,

1 (71 -71 a) - sin 7I 0 e - i 1~, "

(6.7)

Consider an observable property of such a particle represented by a 2 • 2 Hermitian matrix G. Its mean value in the state Ia) is represented by (aIGla). Our point of departure represents G instead as a linear combination of the unit matrix and of the three Pauli matrices {~r,,~ry,Crz} introduced in (2.32), G -- gol + t~" ~ . (6.8) 1 It is readily verified that the coefficients satisfy go __ 7TrG, gx _ _ 71 W r ( G O . x ) etc., and that the eigenvalues of G equal go :t= [][. With this notation, the mean value of G is expressed as

(a[G[a) - go + g" P ,

(6.9)

6.2. Q u a n t u m M e c h a n i c s o f P a r t i c l e s w i t h Spin

1

131

that is, as an explicit linear function of the orientation vector P. The casting of a general 2 • 2 matrix G in terms of unity and ~ constitutes of course a reduction, much as the reduction of (1.3) to the three subsets a)-c) on p. 8. The orientation/7 can be determined experimentally by measuring the mean values of a sufficient number of different quantities G. For our immediate purposes it is sufficient to set G equal to each of the three matrices { J,, Jy, Jz } whose mean values equal one-half of the three Cartesian components of/~, respectively. Setting G equal to any three linearly independent combinations of {Jx, Jy, Jz } would yield, of course the same final result. This operational point of view for characterizing a state is essential when dealing with states other than pure ("mixed"), which are not represented by probability amplitudes (6.7). In this event we may still cast the mean value of G in the form (6.8) provided we replace the unit vector 15 by a vector P of unspecified magnitude. This magnitude remains, however, restricted by the conditions that" a) the mean value of any operator G may not exceed its largest eigenvalue go + Ig[, b) the mean value of any positive G be positive, and c) the mean value of a unit G be unity. These requirements amount to 0 < P < 1. The value of P represents the d e g r e e of p o l a r i z a t i o n of the particle and vanishes for a state of fully random orientation. A general--pure or nonpure--state of the particle is represented by 1 in the form its d e n s i t y m a t r i x p, which may be represented, for spin :,

p - : [ l 1+

/:~. 9 ]

(6.10)

Density matrices serve to represent the mean value of any observable G through the trace expression Tr(pG), as in

(G)p - Tr(Gp) - go + ]" f t .

(6.11)

[Equation (6.11)is verified by substituting (6.8) and (6.10) and utilizing as well as the fact that Tr(Y-u) vanishes for arbitrary ft.] In particular, the particle's angular momentum Y-"- 1~ has the mean value

(2.35)

(f} - :P;: "

(6.12)

hence the mean value of G is a linear function of (J). Since the particle's Hamiltonian is itself represented in the form (6.8), H - h0 -I- h-(:, its time-dependent SchrSdinger equation takes the form

dp/dt - - i ( H p -

pU) - 5 . h x / ~ ,

(6.13)

C h a p t e r 6. Tensorial Sets of Quantum Operators

132

according to (2.35), yielding the equation of motion of the polarization P in the form of a ("Bloch") precession equation dP/dt-

d(~)/dt-

h • P.

(6.14)

This form of the equation of motion also holds under the conditions of statistical mechanics, where the action of fluctuating interactions between the particle and its environment is represented by fluctuations of the vector h. [More accurately, the fluctuations are represented by the autocorrelation function (h(O)h(t)).] Equation (6.14) will be extended to general quantum systems in the following chapter. A particularly noteworthy feature will be the use of a small set of base operators, analogs of {1, ~}, in terms of which all the operators of the system such as G, H, and p (including their products) may be expressed, thereby reducing subsequent consideration to coefficients such as h and P and their relationships as in (6.14). The formal solution of (6.13) is p(t) - e -iHt p(O)e ill'

(6.13a)

Substitution into (6.11) gives the mean value expression (G)

-

Tr[Gp(t)] - Tr[Ge-iHtp(O)eiHt]

=

Tr[eiH'Ge-iH'p(O)]-

(6.13b)

Tr[G(t)p(O)],

which displays the equivalence of the Heisenberg and SchrSdinger representations through regrouping of the four factors in the trace. Commutability of H and p, which occurs when f~ and fi are parallel, implies d p / d t - 0 and constancy in time for all mean values (G). 6.2.1

Base

sets of matrices

and

operators

The matrices Y in the preceding equations are expressed as {ax,cru,crz}, that is, with reference to a Cartesian frame. In view of later generalization to higher dimensionality we shall now introduce an alternative set of matrices, defined with reference to the standard frame. More precisely we shall use standard matrices that transform like kets under coordinate rotations of physical space, that is, in a manner contragredient to standard sets. The four matrices of the standardized base set belong to two irreducible subsets, one invariant and one of degree 1. They are represented in terms of Pauli matrices as in (3.3) 1

1

(6.x5)

6.2. Q u a n t u m Mechanics of Particles with Spin 89

133

that is, { (~ 0

0 ) ~2'

(0-i) 0 0

'

( ~2 0 ) (0 0)). 0--~2 ' 1 0

(615a) "

The set of three matrices of degree 1 transforms like the set of three spherical harmonics of the same degree in (3.13a), {Yll, Y10, YI-1}. The transformation of the Cartesian frame matrices {crx, Cry, Crz} into the set of (6.15) is substantially the same as that represented by (3.46) and (3.13a) for a set of vector components, except for the introduction of the normalization factor :~1 and for omission of the phase factor i l in (3.46). The factor i t has been omitted in (6.15) for simplicity and in accordance with common practice, as it is omitted in (3.13a) in the harmonics Ytm; the implications of this omission will be discussed later. The real normalization coefficient has been so adjusted that the matrices (6.15) constitute an orlhonormal base in the following sense: a) the product of each matrix and of its Hermititan conjugate has trace unity;

b) the product of each matrix and of the Hermitian conjugate of every other matrix (6.15)has trace zero. The four matrices (6.15) are represented analytically in terms of Wigner coefficients by (_l) 89 ~rn, ~1 __mrI 11 ~kq), (6.16) where 1 m' - : i : ~1, m--=l=~,

k-0or

1, q - k , k - 1 , . . . , - k .

(6.16a)

We have replaced here the usual indices j m by a new pair, kq, following the common practice of labeling the eigenvalues of IJI 2 and Jz for sets of tensorial matrices or operators by a pair kq, reserving j m for pure states' labels. The degree k of the matrix sets (6.16) equals only 0 or 1 for particles of spin 1 but attains higher values for higher spin as we shall see in Section 6.4. The matrices we have been considering represent operators in the particular base of eigenvectors lYre) of Ill 2 and Jz. However, they can be converted to the abstract representation of quantum-mechanical operators utilizing a set of elementary unit operators, constructed with ket and bra eigenvectors of IJI 2 and Jz with the general form Ijm)(j'm'l .

(6.17)

134

C h a p t e r 6. Tensorial Sets of Q u a n t u m O p e r a t o r s

The matrix of the operator (6.17) has a single nonzero element equal to 1 and lying in the row (jm) and in the column (j'm'); this operator, when applied to a complete set of kets, projects out the single element ]j'm') replacing it by [jm). A set of four elementary operators (6.17), namely, 1 1 189+:)(1 +:[, converts our base matrices (6.16)into a base set of lensorial operators for spin-:1 particles. The conversion yields irreducible sets constructed in accordance with (5.26), to be indicated here by ]:11' / c q ) ) - ~--~]:177./)(177/! : ] ( - - 1 )1: - (rn'l r n , :

1 __TT/']::/cq). 11

(6.18)

~-12t m

A double parenthesis has been introduced on the left-hand side for symbols representing operators, here and in (6.17), whereas the corresponding pair in (5.26) had no such implied meaning. The orthonormality of the matrix sets (6.15) or (6.16) is reflected in the orthonormality of operator sets (6.18), represented in quantum-mechanical notation by (( 89189 11$ 7 k q ) ) - 6kk,6qq,. (6.19) The set of four operators (6.18) plays the role of base vectors spanning a Hilbert space of operators, quite distinct from the usual Hilbert space of spinor states I:177/). The present chapter deals with the representation of quantum mechanics in this alternative Hilbert space, used extensively in statistical quantum mechanics and called the Liouville r e p r e s e n t a t i o n . 1 The density matrix of a spin-:1 particle as well as any operator G of this system may also be regarded as ket vectors of the Liouville representation; their expansions (6.8) and (6.10) are then indicated by

IG))-

iI~[ ii? I:: kq))((:: kqlG)),

(6.20)

:it I::_ kq))((:11t kqlp)).

(6.20a)

kq

Ip))kq

These representations of G and p have been designed to facilitate their extension to particles with higher spin. The notation is elaborate for this i T h e n a m e stems from the n i n e t e e n t h c e n t u r y physicist Liouville who studied how an ensemble of particles, d i s t r i b u t e d with density p over points { q i , p i } of a multidimensional classical phase space, p r o p a g a t e s in the course of time. Its H a m i l t o n i a n e q u a t i o n d_s dt --~ - { H , p } holds in q u a n t u m mechanics as well, with the Poisson bracket replaced by a c o m m u t a t o r as in (6.13): H. Goldstein: Classical M e c h a n i c s , second ed. (Addison-Wesley, New York, 1980), p. 426.

6.3. T w o - L e v e l S y s t e m s

135

reason, but one should recall, by comparing with the initial formulas, that, for example, the invariant set element ( ( ~1 1t O O I p ) ) equals ~ for any state of orientation and that the set { ( ( l~i t lqlp))} consists of the standard components of __~/3. The latter part of this book will deal with set elements

n logou to

6.3

llt

kqlG))

11]"

kqlp))

more

eeli

tions.

Two-Level S y s t e m s

The name "two-level system" applies colloquially to physical systems for which one may consider one pair of orthogonal pure states--and all their superpositions--as effectively isolated from the other states. The mechanics of these systems can be represented by the same formalism as applies to the orientation of spin- 89 particles; that is, their respective mathematical representations are "isomorphic." An early application of this isomorphism describes the proton and neutron as alternative charge states of a single particle, a "nucleon." These 11 1 two states are regarded as the eigenvectors, 177) and 171 -7), of an "isotopic spin" operator Iz, while operators I, +iIy change protons and neutrons into one another. Similar or more extended treatments apply throughout particle physics. The mapping of the states of any two-level system onto the states of 1 a spin-7 particle proceeds by two intials steps" 1) Identifying one specific I state of the two-level system as corresponding to "spin up, ,, that is, to I~I 7), 1 1 while the single state orthogonal to it is identified as 17 -7)" 2) Identification of a specific superposition of these two states as corresponding to the spin orientation P - ~; the coefficients of this superposition must have equal magnitude, because ~ is orthogonal to ~, but the phase normalization of 1 1 17 -7) is a matter of convention. An interesting example of a double two-level system, reflecting degeneracies unique to the nonrelativistic hydrogen atom, is provided by its set of states with n = 2. The n 2 = 4 elements are subdivided in the spherical coordinate representation as one s(g = 0) and three p(g = 1) states: {Igm)} = {100), Ill), I10), I1,-1)}. Alternatively, the parabolic coordinate representation casts them as {Ijlj2mlm2)} with jl - j2 - l ( n - 1) - ~1 and (ml, m2) - + 7,1"that is, as 2 • 2 - 4 states of two "spin-71" systems jx and j2. The existence of alternative, equivalent classifications (separation of variables) and the additional degeneracy of the s state with the p states, rotational or SO(3) symmetry of the Coulomb Hamiltonian only requiring

136

C h a p t e r 6. Tensorial Sets of Q u a n t u m O p e r a t o r s

the (2~ + 1) degeneracy of each f, points to a higher symmetry. This higher symmetry possessed by the 1/r potential is well known to rest on invariance of the Hamiltonian under SO(4) corresponding to rotations in four dimensions. The conserved Laplace-Runge-Lenz vector ff of the CoulombKepler problem (Section 36 of ref. [12]) provides three generators besides the conserved angular m o m e n t u m / , yielding in all the six generators of four-dimensional rotations. The linear combinations, ~1,2 - ~1([-1- ~) behave like two independent angular momenta obeying the usual commutation relations (2.15). Whereas ff lies in the plane of the orbit, [ i s perpendicular to it, yielding ~'-ff - 0. The two combinations ]1,2 are therefore of equal magnitude and further, upon recasting the Hamiltonian in terms of them, it follows (Sections 36 and 37 of ref. [12]) that they are simply related to the principal quantum number n "jl - j2 - 8 9 1). We will return to these matters in greater detail in Section 10.1. The above analysis of hydrogen states illustrates what is termed "dynamical symmetry." The unitary group U(4) appropriate to the four n = 2 states has 16 unit operators (Problem 6.7), and the two alternative "decompositions" in terms of subgroup chains represented by U(4) D SO(4) D SO(3) D SO(2)

u(4) su(2)•

so(2),

associated respectively with the {[~m)} and {[jlj2mlm2)} descriptions. Note that the quantum number m of SO(2) is common to both chains, m = ml + m2, because both spherical and parabolic descriptions share the coordinate ~ of azimuthal symmetry. The relation [ - ~l + j2 connects the two descriptions or the two subgroup chains by the simple process of adding two angular momenta in Chapter 5. The matrix of coefficients (jlj2fmljlml, j2m2) in (5.1) provides the explicit transformation between spherical and parabolic states. Note that f, m, and a take only integer values, analogous to the set {a, b, c} in (5.3), whereas jl, j2, ml, and m2 may also be half-integers. Other examples of dynamical symmetry, wherein alternative subgroup chains provide quantum numbers to label states and their energies, will be considered in Chapters 8 and 10. 6.3.1

Atom

in a radiation

field

The name "two-level system" relates particularly to the example of a pair of atomic states [a) and Ib) with different energies, Ea and Eb, and opposite parities, strongly coupled by a radiation field oscillating near resonance.

6.3. Two-Level Systems

137

Other states of the same atomic system need not be considered insofar as they are not admixed with la) and Ib) by the action of the radiation or other agents. (Isolation of the pair {la), ]b)} on this basis constitutes of course an approximation; separation of the orientation of a particle from its translational position also constitutes an approximation although possibly 11 1 1 a more accurate one.) The mapping { la) ~ 13~), [ b ) ~ 13-3) } affords writing the Hamiltonian of the two-level atom in the absence of radiation as So - ~l(Ea q- Eb)-k 89 Eb)O'z', its interaction with radiation may then be represented initially by - F ( t ) d where d is the component of the atomic dipole moment parallel to the electric vector F(t) of the (classical) radiation field. This mapping onto spin states also affords adjusting the phases of ]a) and Ib) to make the dipole matrix element (aldlb) real. The complete Hamiltonian is then represented by

H -

89

+ Eb)l + 89

- E b ) ~ -(aldlb)F(t)cr~.

(6.21)

As an exactly solvable model, this "Jaynes-Cummings" model [23, 24] underlines current quantum optics literature. Note that the indices z and x in (6.21) pertain to directions in the model space (of spin orientation) wholly unrelated to the direction of the field ~5 in physical space. The radiation's field strength is unrestricted here except for disregarding its effect on states other than la) and Ib). The model represents the atom's state under the influence of the Hamiltonian (6.21) by a vector/5, with P _ 1, whose variation in the course of time obeys the precession equation analogous to (6.14)

dP/dt - [ ( E ~ - E b ) ~ - 2(aldlb)F(t)&] • P .

(6.22)

Resonance occurs when F(t) oscillates with frequency very near to ( E a Eb)/li. In this event, the z component of/3 varies considerably, as it would for a spin perturbed at resonance, regardless of the magnitude I(aldlb)FI of the radiation coupling remaining much smaller than l E a - Ebl. Off resonance, the vector j6 merely precesses about the ~ axis with a nearly constant value of P~.

6.3.2

Light polarization and Stokes parameters

The polarization of a plane light wave provides a classical example of a twolevel system. Any single pair of orthogonal polarizations such as right- and left-circular polarization maps onto the "spin-up" and "spin-down" states of

138

Chapter 6. T e n s o r i a l Sets of Quantum Operators

a particle. A specific linear polarization is then represented by the vector P = ~. Remaining states of linear polarization are then represented by vectors P in the (~, ~)) plane, while vectors oblique to this plane represent elliptical polarizations. The four parameters introduced by Stokes in 1852 [4] to represent the intensity and polarization of an unspecified light beam coincide with the intensity I and the three components of the vector I/5. Here again the value of IPI represents the degree of polarization. The vector P is thus confined generally within a sphere of unit radius in the model space, called the "Poincard sphere" (Figure 6.1). The effect of an optically active medium upon the polarization of a light beam is represented by a precession of P, with type and strength characterized by the axis and rate of the precession. Several further aspects of this mapping may be noted. Light, as a vector field, is said to have spin 1, meaning that its spherical waves resolve into eigenvectors of IJ~ 2 whose eigenvalues have integer j > 1. Light may nevertheless be mapped on spin- 89states because its plane waves, eigenvectors of J~, belong to only two of its eigenvalues, namely, to circular polarizations with Jz = -4-1; an eigenvalue Jz = 0 would correspond to nonexistent longitudinal polarization. The mapping is classical of course, having been developed by Stokes and Poincar@ before 1900, but the occurrence of partial polarizations, with P < 1, is essentially a quantum phenomenon, identified and defined only in terms of mean values of operators or by equivalent considerations; classical treatments of partial polarization remained indeed awkward prior to quantum mechanics. 6.3.3

Further

applications

Occupation,

creation,

of two-level

systems-

and annihilation

operators

States of systems consisting of N fermions (for example, the electron systems of atoms and molecules) are often represented in terms of independentparticle base states, with the N particles assigned to as many single-particle states ("orbitals") u~, uo, etc. Since the number of available orbitals usually exceeds N (it is often infinite), each of them can be "occupied" or "empty" in a particular N-particle base state. Since spectral phenomena deal with shifts--actual or virtual--of particles from one orbital to another, it has proved convenient to generalize the concept of orbital state to include an indication of "filled" or "empty," that is, to replace the symbol ur by a pair This single pair can then be treated formally as a "two-level" system

6.3. T w o - L e v e l S y s t e m s

139

right

linear X /

left

F i g u r e 6.1: The Poincard sphere and axes of polarization. with operators (1, Y) or combinations thereof. A set of operators commonly used in this connection is indicated by

-a~a(

O)

1 0

__1

3(l+~rz)-

() 1

0

0

0

, a~ - 7l(~rx + i~ry) '

a(a~

(0 1) (oo)

3(1 - crz) --

0

0

0

'

1

.

(6.23) Pairs of states (u(f, u(~) are eigenstates of a~a( (with eigenvalues 1 and 0, respectively) and of a(a~ (eigenvalues 0 and 1). A single-particle state ( corresponding to an energy eigenvalue E( contributes to its Hamiltonian an amount represented by E(a~a(. Operations that add or subtract particles from a particular orbital are represented in terms of a~ and a( ("creation" and "annihilation" operators). This analytical representation was first applied to atomic problems by Heisenberg in the early 1930s. Its application to many-electron atoms has been systematized by Judd [18] with considerable success, as we shall see in Chapter 8. (States of many-boson systems are isomorphic to those of harmonic oscillators, rather than to those of spin-71 systems. See Problem 6.5.)

140

C h a p t e r 6. T e n s o r i a l Sets of Q u a n t u m O p e r a t o r s

6.4

P a r t i c l e s w i t h Spin j > 1. W i g n e r - E c k a r t Theorem

The treatment of the orientation of particles with spin 1 has been formulated in Section 6.2.1 in a manner that permits ready extension to any value of the index j. Formally, the extension consists merely of replacing the value ~1 of the index j, in (6.16) and in (6.18), by any desired value of j. This substitution also extends the range of values of the integer k, the degree of the sets of standard matrices and operators in (6.16), to the larger interval 0 _< k _< 2j. (6.24) The interpretation and evaluation of the sets of tensorial parameters ((jjt kqlG)) and ((jjt kqlp)) to be entered in (6.6) and (6.20a) require, however, detailed considerations. (Recall that the double parentheses indicate bras or ket of the Liouville space.) The representation of the density matrix p requires only some discussion but the calculation of the tensorial parameters of operators G involves a more extensive development. 6.4.1

Density

matrix

With regard to the density operator p, we recall that its matrix form, in the ]jm) base, is obtained by replacing in (6.20a) the unit operators Ijjtkq)) by the corresponding matrices (6.16), yielding 2j

(jm[pljm') -- ~

k

~

(-1)J-m'(jm, j -m']jjkq)((jj t kqlp)) .

(6.25)

k=Oq=-k

This equation represents the density matrix as a sum over tensorial components. Notice first that each component with k ~- 0 is traceless, owing to (5.24) and to the orthogonality of Wigner coefficients, Tr[(-1) j - m ' ( j m , j =

-m']jjkq)] - ~-'~(-1)J-m(jm, j -mJjjkq)

(2j + 1) 89~_,(jjOOIjm,

j -m')(jm, j-m'ljjkq) - (2j + 1) 89

mm t

(6.26) Several properties of density matrices restrict the values of tensorial components ((jjtkqlp)) and even specify one of them. The requirement

1 Wigner-Eckart Theorem 6.4. P a r t i c l e s w i t h Spin j > 7:

141

that the unit operator have unit mean value in any state implies T r p - 1 and hence ((jjtOOJp))- (2j + 1)- 89 (6.27) The requirement that any operator with nonnegative eigenvalues have a nonnegative mean value implies 0 < Tr(p") < 1.

(6.27a)

Unit value of Tr(p n) characterizes pure states. For n - 2, Eq. (6.27a) takes the form Tr(p2) _ ~kq j((jjt kqjp))j2 _- 2j 1+ 1 + k=~~(PJjjt 2~ k))((jjt kip) < 1. (6.27b) For spin j, Tr(p 2) varies therefore from a minimum value of (2j + 1) -1 to 1 the variation a maximum of 1 (pure state). For the special case of j - ~, is between 71 and 1; see ( 6 . 1 0 ) . Notice finally that each term of the matrix expansion (6.25) has its nonzero elements confined to a line parallel to the main diagonal, owing to the factor ~q,m-,~, in the Wigner coefficient; that is, the [jjtkq)) are "ladder operators" that raise the index m' by q units. With regard to k ~: 0, recall that for j - ~1 the set ofelements ( ( ~111" lqJp)) represents components of the polarization vector P in the standard base, to within a normalization coefficient ~'~2" For j > 1, the parameters ...#

((jjtkqJp)) are to be similarly interpreted as the components of tensors of different degrees k, again in the standard base. These tensors have been variously called "statistical tensors" or "state multipoles" (state 2k-poles, to be specific); their ensemble, for all values of k from 1 to 2j, characterizes the state of polarization of a particle of spin j, just as a single polarization vector/5 does for j - 89 More specifically, it is apparent from (6.25) that the parameter ((jflkqJp)) is defined as the trace of the product of the matrix (jmlpljm') and of the matrix (jjkqJjm, j - m ' ) ( - 1 ) j-re'. According to (6.16) and (6.18)--generalized to j > 89 last matrix represents the unit operator ((jjt kqJ, the Hermitian conjugate of Jjjtkq)). In other words, the state multipoles are tensors whose components consist of the values of the unit operators ((jjt kq I for the state with the density matrix p. They thus provide an operational identification of the state just as the mean value of (J-~ alone does for j - 7" 1 More specifically, the identification of the state of orientation of a particle with spin j requires in general measurements to be performed with (2j + 1) 2 - 1 - 4j(j + 1) different arrangements to determine all the state multipoles with k ~- 0. This

142

C h a p t e r 6. Tensorial Sets of Q u a n t u m O p e r a t o r s

number is reduced only when symmetries or other circumstances in the preparation of the particle restrict the independence of these multipoles. It may be added that the state multipoles of degree 1, ((jjt lqlp)) ' are proportional to the mean values of the set of matrices ~1( j , _

{ 7~ -1 (J,

+ iJy )t , Jz,

ijy)t } , Hermitian conjugates of the standard components of J,

1 Similarly, the state multipoles with k > 1 are proas they are for j - 3" portional to the mean values of irreducible k-fold products of the matrix set {J~,, Jy, J~ }. The difference of normalization arises from our having imposed the orthonormality requirement (6.19) upon the unit tensorial operators. As an illustration, we give here the explicit form of the nine matrices (-1)J-m'(jm, j-rn'ljjkq) for j - 1"

k-l,q-1

k-O,q-O

/

0

o ~1

o 0

0

0

~

k-l, 0

/

0 k-2,

0-v o

o~ 0 0

//-ill 0

q--1 0 0

0

k-l,

o ~-1

~o 0

0

0

0

0

k-2, q-2 001

o

o

ooo

~2

0

0

0

q-O

o 0 -1

~77

/

k-2, q- 1 0 7~

0

o

o

0

0

0

0 0 l0 0 0/ (000) q-0

k-2,

0

o

~6

q--1

0

o

-1 ~

k-2,

q--2

0

0

0

0

o

1

0

0

(6.28) These nine matrices are the generators of the unitary group U(3). All except the first of them are traceless, forming the eight generators of SU(3). Similar sets of 4 j ( j + 1) matrices with 1 _< k <_ 2j describe the larger unitary groups SU(2j + 1). The time-dependent Schr6dinger equation for the density matrix may be cast as the precession equation for a 4j(j + 1)-dimensional generalized polarization vector P whose components are the aggregate of the state multipole components [25]. This equation presents interesting features but its formulation requires us to extend to j > 1 the Eqs. (2.33-36) on the products of Pauli matrices, as will be done in Chapter 7.

1 6.4. Particles w i t h Spin j > 3" Wigner-Eckart Theorem

6.4.2

Multipole

expansion

of operators

143

G

A formal extension of the operator representation for j - 1 in (6.8) is readily achieved since the matrix of any Hermitian operator G consists of (2j + 1) 2 independent elements, thus being represented as a linear combination of any set of (2j + 1) 2 linearly independent standard matrices. More importantly, however, we are aiming at representing G as a linear combination of multipole operators [jj~kq)) characterized by their transformation properties under rotations of coordinate axes. This feature suggests resolving any operator G acting on a particle with spin j into a sum of multipole operators before rather than after calculating its matrix (jmlGIjm'). This procedure involves a systematic expansion into multipoles, whose prototype lay in the expansion (4.22) of a potential field, constituting the core of the Racah-Wigner operator algebra. Accordingly, we regard each operator G as expanded in the form

G- E

[gkq)(kqlG)'

(6.29)

kq

whose coefficients (kqlG) do not operate on our particle (that is, are " cnumbers"). In the next step, we consider the tensorial set elements Igkq) as operating specifically on a particle of spin j, that is, as components of a vector of the Liouville representation, to be indicated by Ijjtgkq)). However, this representation includes one--and only one--vector with the r-transformations of a ket of degree k, namely, the one with components Ijj ~kq)). Vectors (... I with equal k therefore coincide to within a constant factor. This remark, equivalent to the famous Wigner-Eckart theorem, reduces the study of the action of our operator G upon the particle to the evaluation of a single constant for each value of k _< 2j. We represent this constant by writing

Ijjtgkq)) - Ijjtkq))(2k + 1)- 89

(6.30)

in a notation established before the introduction of orthonormal bases of unit tensorial operators. The symbol (jllg(k)llj), with the double bars, is called the diagonal r e d u c e d m a t r i x e l e m e n t of the tensorial operators g~k) _ igkq), with angular momentum j < k/2. The essential point of (6.30) is that the elements of the reduced matrix are independent both of the index q of a tensorial operator and of the indices (m, m') of the rows and columns of an ordinary matrix. The symbol (jllg(k)llj) includes nevertheless row and column indices, both equal to j,

144

Chapter 6. Tensorial Sets of Quantum Operators

allowing for its further extension to off-diagonal elements (jJlg(k)llj ') when the operator g~k) changes the particle's angular momentum (Section 6.6). The Wigner-Eckart theorem is commonly formulated as a property of the ordinary matrix of a tensorial operator, which is itself indicated as g~k) rather than in the ket notation Igkq),

(jm[g~k)[jm,) _ (jllg(k)l[j)(_l).i_m (

j -m

k j) q m~

"

(6.31)

This equation states that the matrix elements of g(k) depend on q, on m, and on m ~ in a standard manner represented by the 3-j symbol, defined in (5.45), and by the factor ( - 1 ) j-re. Equation (6.31) means that the dependence of a tensor operator's matrix upon the orientation of coordinate axes is purely geometrical, accordingly being represented by the standard 3-j coefficient; the only intrinsic element of the operator is its invariant reduced element. Equations (6.31) and (6.30) are equivalent and can be transformed into one another. The reduced matrix element to be entered in (6.30) may be calculated by evaluating just one ordinary matrix element on the left-hand side of (6.31), for a particular set of (q, m, m') and the 3-j symbol on the right-hand side. Choosing, for example, q = 0 and m -- m ~ = j, one finds

(jl]g(k)llj) - (jjlg(k)ljj)[(2j + k +

1)!(2j - k)!] 89

(6.32)

from the particular values of (5.45) or (5.15), with two equal j indices and one vanishing m. The nuclear parameters commonly known as "dipole moment" and "quadrupole moment" of a particle or other system are in fact matrix elements of tensor operators represented in our notation by (jjlp~l)ljj) and (jjIQ~2)[jj), respectively. Calculations for particular operators g(k) will be described in Section 6.7. Once the reduced matrix elements have been obtained, further study of an operator's action involves only the handling of the unit operators [jj*kq)) and of the c-number tensors (kqlG) without further reference to G itself. It is for this reason that a single number like the magnetic moment of a nucleon or a nucleus suffices to characterize what is actually a vector quantity fi with various components. All dependences on these components, that is, all the rn, m', and q dependences of matrix elements of/7, are given by the standard factors in (6.31) which multiply

(jjlp(1)ljj).

6.4. Particles with S p i n j > 1. W i g n e r - E c k a r t T h e o r e m

6.4.3

145

Physical implications of the triangular r e l a t i o n

k<_2j

Atomic systems with low excitation have generally stationary states [jm) with low values of j. The same holds for the joint state of particles colliding with low kinetic energy. The triangular relation k _< 2j limits the degree of any tensorial operator involved in the dynamics of such a system, the multipolarity of the fields generated by a particle or the degree of any rtransformation D (k) that relates the orientation of the particle to that of observation instruments. For an atomic or nuclear system to have a nonzero multipole moment of order k, the angular momentum of its state cannot be less than k/2. No dipole moment occurs with j - 0, no quadrupole 1 moment with j < ~, etc. In particular, elementary particles like electrons, protons, neutrons, etc., that is, elementary spin-~1 objects, only possess a monopole and dipole moment, electric charge and magnetic moment, respectively. Quadrupole or higher moments have no meaning for these objects. (That the electric dipole moment also vanishes is an aspect of time reversal symmetry to be discussed in Section 7.3.4.) Physically, the action of any anisotropic operator upon a particle involves an exchange of angular momentum, indicated by a triangular diagram of vectors ~, fc and ~. The largest transfer that may be experienced by a particle with spin j results from a flip-over of its spin amounting to 2j units. This fundamental limitation is analogous to the limitation set by the uncertainty principle upon the time rate of elementary processes in which a particle can absorb only a limited amount of energy (see, for example, p. 163 of [2]). No time-dependent observation can vary by a substantial fraction within a time v unless its Fourier analysis contains a substantial amount of frequencies ~, --, l / r , that is, unless the observation process involves the exchange of energy quanta of order hv - h/7". Most systems cannot receive an excessively large quantum hv without blowing apart. Likewise with angular momentum, where a spin-j object just cannot exchange more than 2j units of angular momentum and thereby observation can reveal no structure dependence on its orientation finer than ~ -,- 1/(2j). Detailed angular dependences (shapes) displayed by macroscopic drawings or macroscopic objects necessarily imply the exchange of a huge number of angular momentum units with the observing apparatus.

146

6.5

C h a p t e r 6. T e n s o r i a l S e t s o f Q u a n t u m

Operators

Systems with 2 j + l Levels

We consider here how to map the states of a system with 2j + 1 levels onto those of a particle with spin j, thus extending the mapping described in Section 6.3 for two-level systems. One just deals with a mapping of the vectors of any (2j + 1)-dimensional Hilbert space and of their transformations which form the group called SU(2j + 1). (The letter S here means that simple renormalization of all phases by a factor e i~ is excluded from the group.) This mapping has not yet been used extensively in the sense, for example, in which it has served for j - 71 to illustrate the behavior of a two-level atom coupled to radiation, but is now under consideration [26]. It will serve here for two applications: a) to rederive and illustrate the classification of the infinitesimal operations of unitary groups introduced long ago by Cartan, b) to illustrate the time dependence of a complete set of operators in the Heisenberg representation. For j - 71 we have utilized the base set of three infinitesimal operators {~,, ~ru, ~rz } in one-to-one correspondence with infinitesimal rotations of coordinate axes in physical space. For any j, the number of these operators is (2j + 1) 2 - 1 = 4j(j + 1); we choose them in the form introduced in (6.18) 1 namely for j - 7,

]jjtkq))

- ~ [Jm)(jm'J(-1)J-m'(jm, j-m'ljjkq) , mm

(6.33)

I

where Ijm), with m = j, j - 1 , . . . , - j , stands for one base vector of the representation. Among these operators, the one with k = q = 0 equals (2j + 1)- 89times the trivial unit operator irrelevant to S U ( 2 j + 1). The 2j operators with q = 0 and k > 0 commute with one another, because their matrices are diagonal in the Ijm) basis; none of the other operators (6.33) commutes with any of them. Thus the 2j operators with q = 0 and k > 0 form a maximal set of nontrivial commuting infinitesimal operators, with the common eigenvectors Ijm). They perform together the same role as Jz alone does for j - 71 and for the rotations of coordinates in physical space. They may also be expressed as polynomials of degree k in Jz. The operators with q :/: 0 act instead as l a d d e r o p e r a t o r s , as w e have seen in Section 6.4, stepping up the index m by q units, or stepping it down for negative values of q. The analog of f2 for this group is the

6.5. S y s t e m s w i t h 2 j + 1 Levels

147

invariant product constructed with all the operators (6.33) 2j

k

~

Ijjtkq))ljjtk--q))(-1) q ,

(6.34)

k = l q=-k

called the C a s i m i r o p e r a t o r to within normalization. The analytical procedure of Cartan's classification is described, for example, in Racah's lectures [27]. The eigenvalues of the Casimir operator and of the commuting operators IjjtkO)) serve to classify a representation of SU(2j + 1). The terms with k - 1 alone yield of course just lj-'[2. Incidentally, the operators (6.33) are not Hermitian for q ~- 0 and have complex r-transformations. The Hermitian conjugate of Ijjtkq)) can be represented as ((jjtkq[ but also coincides with Ijjtk --q))(-1) q, as one can verify from (6.33) and from the explicit examples in (6.28). The connection

[Ijjtkq))] t - ( j j t k q l -

Ijjtk - q ) ) ( - 1 ) q

(6.35)

resembles a frame-reversal transformation but differs from it characteristically by omission of a factor (-1)k. Indeed the base sets of unit operators (6.33) can be replaced by Hermitian base sets through a unitary transformation TH 1 that is" a) related to the inverse of the transformation (3.36), which changes kets of the Cartesian to the standard base, but b) differs from T -1 by omission of the factor i l of T. Recall that the factor i l had also been removed when constructing the matrices (6.15) from the Hermitian ~; failure to remove it would have made the q = 0 operators anti-Hermitian rather than Hermitian in (6.15) and hence also in (6.33). By the same token, transformation of the operators (6.33) by the full matrix T -1, instead of TH, would make them anti-Hermitian; specifically the operators with k = 1 would become analogs of the {Ix, I v, Iz } of Chapter 2 instead of coinciding with { J~, Jy, Jz }. We shall then transform the operators (6.33) by the matrix TH -- ikT -1 applied on the right, obtaining the Hermitian sets

[jjtklql'e-1)) [jjtklq], e - - 1 ) )

=

1 ~7~ [ Ijjtklq[))(-1)q + ljjtk -[ql)) ] ( l + ~ q ~ 1 8 9 :~i [ Ijjtklql))(-1 ) q + l + [JJ tk -[ql)) ] -

(6.36) Hermitian conjugation of the operator with e -- - 1 reverses the sign of the coefficient i, but this sign reversal just compensates that of the expression in the brackets. The coefficient (1 +6q0)- 89of the operator with e = 1 allows for the identity of the two terms when q - 0, in which case the ( = - 1 operator vanishes.

C h a p t e r 6. T e n s o r i a l S e t s of Q u a n t u m

148

Operators

Notice that using TH instead of T changes the standard U matrix to ( - 1 ) k l rather than to the unit matrix itself, meaning that the Hermitian base operators have parity ( - 1 ) k under frame reversal, as they should, at least in the case of k - 1 which includes J itself. -r

To map the states of an atomic system with energy levels {El < E2 < 9 9 9E n . . . E2j + 1}--decoupled from other levels--onto the polarization states

of a particle, one may set IE1) ~ [ j , - j ) , IE2) ~ I J , - J + 1), etc., with a suitable phase normalization, thus generalizing the two-level application in Section 6.3.1. The Hamiltonian is then represented as a linear combination of operators I j j t k O ) ) . A general operator G can be expanded in tensorial components gq(k) , all of which are traceless except g~0). The components g(k) 9k ) remain constant in time in the Heisenberg representation, while the g with q r 0 oscillate in the course of time with the spectrum of frequencies ( E ~ + q - E . ) / h . Note the likely behavior of (G) as a function of time when the number of levels increases toward a macroscopic limit" The fraction of constant components with q - 0 decreases as 1 / j , while the spectral density of frequencies En+q - E n ~ g e n e r a l l y i n c o m m e n s u r a t e - - i n c r e a s e s with j. Accordingly the components with q ~ 0 tend to average out by interference and (G) tends to vanish within a time interval of the order of the reciprocal spectral density. One may see here a model for relaxation phenomena.

6.6

Transfer of A n g u l a r M o m e n t u m

In this chapter we have considered thus far tensorial operators that serve to identify the state of orientation of a particle, or other atomic system, or that transmit to a particle some external action without changing its internal structure. We extend our considerations now to include operators that increase or reduce the squared angular momentum of an atomic s y s t e m - atom, molecule, nucleus, or other particle--instead of merely changing its state of orientation. Formally this extension proceeds as readily as those introduced earlier. Instead of considering only base sets of states [ j m ) with a single value of j, we shall also consider aggregates of tensorial sets with two or more different values of j. In fact, we may even want to add further labels to distinguish sets of states with the same j but differing by some other quantum numbers; when these quantum numbers are specified, they will be added next to the j index, when they are not, we shall use a generic index, a , / ~ , . . . , labeling a

6.6. T r a n s f e r of A n g u l a r M o m e n t u m

149

state, for example, as lajm). Accordingly, we shall now no longer consider only base sets of unit operators Ijjtkq)) with two identical indices j but, more generally, operators with different indices as, for example,

laja'j 't kq)) - Z Z IaJm)(a'J'm'l(-1)J'-m'(jm'j' m

m

-m'ljj'kq) "

(6.37)

!

The reduced matrices of tensorial operators will then also include offdiagonal elements (jllg(k)llj ') or also (o~jllg(k)llo~'j'). As a prototype tensorial operator that induces changes of j, we mention the electric dipole moment -eY of an atomic electron, whose interaction moment with longwave radiation yields transitions between atomic states with values of j and j~ which generally differ by one unit. In (6.37) the range of k is set by the triangular relations I J - J~l <_ k _< j + j', yielding a total of )-~k(2k + 1 ) - (2j + 1)(2j' + 1) operators. Hermitian conjugation of the unit operators (6.37) changes now, in the first place, the direct product unit operator ]jm)(j~m~l into ]j~m~)(jml, thus involving also the permutation of (jm) with (j~m ~) in the rest of the formula. Taking into account both of the symmetries (5.17) of the Wigner coefficients as well as the conventions in (5.26), we find

[Ijj 't kq))] t

-

Ij'jtk -q))(-1) j'-j+q .

(6.38)

Construction of a Hermitian base of unit operators now becomes a two-step procedure, involving a linear combination of operators (6.37) with q values of opposite sign, and of operators with pairs of indices (jj, t) and (j,jt). The first of these steps is performed by (6.36) with (jj?) replaced by (jj'?). The second step involves the introduction of a new index 7?, analogous to e, as indicated by

IJJ'tklqle'r]- l)) - ~2 [ Ijj'?klqle) + (-1)J-J'lJ'Jtklqle)) ] (1 + Ijj'?klqle, r ] - _ l ) ) - :~ [IJJ'tklqlc)-(-1) j-j Ij'jtklqle))] i

t

~JJ')- 89 '

.

(6.39) Both of these sets are Hermitian, as one can verify considering that, when

j' =/=j, [Ijj'tklql~))] t -(_l)J-J'lj'jtklqlc)). The two Hermitian operators (6.39) are distinguished by having opposite parity under frame reversal. Here again one can verify that frame reversal yields Kljj'?klqle))If -1 - (-1)J-J'+kljj 'tk]qJe)), a parity that reduces to ( - 1 ) k in the case of j' - j, as noted after (6.36). Substitution of this result in (6.39) yields then

KIjj 't klqle•))I.( -1 -

(-1)kr]ljj 't klqier])).

(6.40)

150

Chapter 6. Tensorial Sets of Quantum Operators

This characterization of unit Hermitian operators of opposite parity under frame reversal originates mainly from the 1969 thesis of M. Lombardi [28]. The reduced matrix elements with j' r j may be obtained from the off-diagonal form of (6.31), namely,

(jmlg~k)lJ'm') - (Jllg(k)llJ')(-1)J-m ( -rn j kq

J,)

m'

'

(6.41)

yielding the analog of (6.32)

(j][g(k)l[j,) - (jj]g~j,]j,j,) [(k

+ j + J'+(2j),(2j'),l)'(J+

j' - k)!] ~

(6.42)

These tensorial sets of operators g~k) are self-conjugate under Hermitian conjugation, meaning that g~k)t _ g(k)(_ 1) q and hence

(jmlg~k)lj'm ') -(j'm'lg~k)tljm)* -(j'm']g(k)]jm)*(_l)q.

(6.43)

It follows from this formula and from (6.41) that

(j'l[g(k)t[Ij)- (_l)J'-J(j]]g(k)]lj')*. 6.7

(6.44)

C a l c u l a t i o n of M a t r i x E l e m e n t s

The explicit expression of tensorial operator matrices depends, of course, on their normalization as well as on the phase normalization of the states on which they operate. The properties discussed in previous sections, concerning unit operators, their matrices, and the general treatment of other operators, depend instead only on the orthonormality of the base sets of angular momentum eigenstates Ijm). Additional elements emerge, however, when evaluating reduced matrix elements, as outlined below in the context of specific examples. Phase normalizations depend, for example, generally on the specification of transformation matrices between alternative base sets; yet the reduced matrix of the set of vectorial operators {J~, Jy, Jz} remains independent of phase normalizations for two reasons: a) the magnitude and phase of the eigenvalues of Jz is fixed in the standard base, and b) this operator is diagonal in the quantum number j, whereby the phase normalization of lyre) cancels out. We enter then in (6.32), j~l) _ Jz, (jjlJzljj) - j and k = 1, yielding (jllJ(~)l]j) -[j(j + 1)(2j + 1)] 89 (6.45)

6.7. C a l c u l a t i o n of M a t r i x E l e m e n t s

151

The next most important set of tensorial operators consists of the multipole moments M (k) pertaining to particle charges within an atomic system; the definition of multipole moment was introduced in (4.23). We restrict ourselves initially to the multipole moments, M~ k) - erkY(k)(O, ~o), of a single particle with orbital angular momentum eigenstates [agm). Accordingly, we have to evaluate the matrix element with m = g and m' = g, to be entered in (6.42). This matrix element splits into radial and angular factors

(eelY , le'e')

so)l 'e'e')

(6.46)

the first one depending on the radial functions of the system under consideration and of no immediate concern. The second factor consists of an integral over the product of three spherical harmonics, I

l-l

-

/0

sin OdO

~o) /0 d~oY(l)*(O, ~o)Yl(k_~,(O,~o)Y(t')( O,(6.47)

whose evaluation constitutes a prototype application of our methods, proceeding through four steps" a) Identification of the two harmonics y(k)l_t and y(t') with the r-transformation matrix elements D 0,l-t, (k) and ~0t, r)(t') , respectively, to within normalization factors. b) Application of (5.22) to expand this product of D functions into a superposition of functions D~Jt)(0, 0, ~p)cx Y(J)(O, ~), with Wigner coefficients. c) Application of the orthonormality of the harmonics to eliminate all terms with j ~: g, thus reducing to unity the integral over the term with j - ~, to within the appropriate coefficient. d) Setting the appropriate normalization. The result's essential feature lies thus in the coefficient of the j = g term in the expansion (5.22) of the product y(k),y(,t') This term's coefficient l-l consists of the two Wigner coefficients (k0, gOIkggO) and (kggglkg-g', gg'). The second of these cancels the 3-j coefficient in (6.41), to within a factor, thus reducing the left-hand side of that equation essentially to the desired reduced matrix element, (t?llY(k)l[g'). The first one of the Wigner coefficients, with all of its m indices equal to zero, establishes an important parity

152

C h a p t e r 6. Tensorial Sets of Q u a n t u m O p e r a t o r s

selection rule, because it vanishes unless k +g' +g is an even number. [This coefficient is clearly invariant under frame reversal and yet it must equal ( - 1 ) k+~'-~ times its own value according to the parity factor (5.17).] The entire coefficient is then expressed in terms of binomial coefficients, without any summation. The normalization coefficients for reducing the spherical harmonics to D functions are given by (4.29). Setting c~ in the phase factors e ia equal to -!a'g, with appropriate g, yields harmonics that form self-conjugate sets, 2 whereby all their irreducible products are also self-conjugate. This practice is, however, seldom followed, that is, one sets simply a = 0 in (4.30); the equations given here will also assume c~ --- 0, resulting in a slight loss of symmetry for the three indices {g,k, gl}. With this stipulation, one finds the reduced matrix element

0

,+l'-k

, 89 k+e-,'

89 ~'+~-~ ~89 l+,'+k ~- 89

(6.48) whose last four factors are binomial coefficients. Phase normalization of all spherical harmonics by setting a - =t=717rg would replace the phase factor ( - 1)l in the middle expression above by the more symmetric ( - 1) 89 thus eliminating the phase factor on the right-hand side of (6.48). An alternative normalization of the harmonics y(k) is often encountered that dispenses with the factor [(2k + 1)/47r] 89here as in (6.48), with the symbol C (k) of Eq.(5.19) of [1] replacing y(k) in the reduced matrix element. A variant of the matrix element (6.46)serves to evaluate the multipole field components generated near the center of an atom by an orbiting electron, components that are indicated by (kqlV) in the notation of (4.22). This variant merely replaces the factor r k in the radial integral in (6.46) by r -(k+l). However, most applications of the reduced matrix of multipole moments occur in more complicated problems, where the orbital motion of the relevant particle is coupled to a spin variable and/or to the orbital motion of other particles. A typical application, of central importance to the calculation of energy levels and more generally throughout atomic and molecular physics, evaluates the reduced matrix elements for the electrostatic interaction energy

6.7. C a l c u l a t i o n of M a t r i x E l e m e n t s

153

between two particles at positions r'l and ~'~,

e2

( rk / rk+l ) c~k)(o1,991) c(_k:(02, ~2)(-1) q ;

= e2 ~

(6.49)

kq here r< indicates the smaller one of the distances rl and r2 from the origin, r> indicates the larger one, and the notation C (k) indicates spherical harmonics shorn of the factor [(2k + 1)/47r] 89 Each of these applications requires extending the methods developed thus far, to deal with products of tensorial operatorsmfor example, of C~k)(01 ~1) and (:',(k)(02 ~2)--and with unit operators acting on multiparticle states or on combined orbital and spin motions. The necessary extensions form the subject of the following chapters. A further type of application concerns the reduced matrix elements of operators that select particles or radiation collimated in a beam or particles or radiation emerging in a specified direction from a collision or decay process. The state of particles collimated in the z direction and with momentum P - P~ is represented by the density matrix

(#lppl#')

f) 6(#' - f ) .

-

(6.5o)

This matrix is transformed to a base of orbital momentum eigenstates Ip2, m - 0) by a matrix (p2OIp-') consisting mainly of normalization factors, (peOlp-3

-

[4 (2e + 1)] 89

(6.51)

[The phase normalization factor i l of (3.45a) has been removed here in accordance with the discussion in Section 6.5.] Thus we have

(p20lpplp'2'o) - 4r[(22 + 1)(22' + 1)] 89p-2 6 ( p - P ) 6 ( p ' - P).

(6.52)

The reciprocal of the transformation (6.25) yields finally the state multipoles ((g# tkqlpp)) - 6qO(-1) t' (g2'k0le0 , 2'0) • 47r[(22 + 1)(22' + 1)] 89p-2 5 ( p - P)8(p'- P).

(6.53)

An operator selecting particles radiated in a direction P amounts to the Hermitian conjugate of (6.53).

154

C h a p t e r 6. T e n s o r i a l Sets of Q u a n t u m

Operators

Problems 6.1 Show that the general 2 • 2 matrix in (6.8), with ff real, has eigenvalues go 4-Ig-]. Find the corresponding eigenvectors, expressing them in terms of the spherical polar coordinates (~, ~) of ~'. 6.2 One of the famous problems of modern physics, with applications to nuclear magnetic and electron spin resonance, concerns the precession of a spin-~1 particle under the combined influence of a steady strong longitudinal magnetic field h0 and a weak transverse one hi that rotates at radio frequency: h - (h 1 cos wt, h 1 sin wt, h0) , with h0 -~ 103hl. Solve the Bloch equation (6.14) to show the variations of the polarization P(t). Interesting effects occur at "resonance" when w ~_ -Th0. The original treatment of this phenomenon by F. Bloch in Phys. Rev. 70,460 (1946) included phenomenological effects of field fluctuations that cause P to relax toward its thermal equilibrium value. The first fuller description of relaxation in terms of field fluctuations was given by Wangsness and Bloch, Phys. Rev. 8 9 , 7 2 8 (1953). 6.3 Write down explicitly the base set of tensorial operators for spin-~1 particles 189189 tJ:q)) from the expression given in (6.18). Verify that these matrices satisfy the orthonormality relations (6.19). 6.4 Paralleling the discussion of the n - 2 stages of the hydrogen atom on p. 135, consider the stages with n - 3. Write down the nine states in both spherical and parabolic coordinate representations, and construct the explicit unitary transformation connecting them. Table 5.1 provides the Clebsch-Gordan coefficients required for this construction. 6.5 The fermion creation and annihilation operators a t and a in (6.23) obey the anticommutation relation {a,a t} - aa I + ata - 1. Contrast this relation with the similar operators for a (bosonic) harmonic oscillator whose commutator equals the unit operator" [a, a t] - 1.

6.7. C a l c u l a t i o n o f M a t r i x E l e m e n t s

155

Denoting the eigenstates of the operator N - at a by In), N I n ) - nln ), find the eigenvalues of g for the states aln I and a t In). Consider both fermionic and bosonic cases. Since the eigenvalues of N are nonnegative, what restrictions arise on the values of n? In particular, show that the fermionic case has only n - 1 and 0, the "occupied" and "empty" states, as in the explicit matrix representation in (6.23). Note that {a, a} - {a t, a t } - 0 and N2-N. 6.6

(a) Cast the Hamiltonian (6.21) for a two-level atom in a radiation field in the form (6.8), that is, H - h0 + h . ~. Show that 1

ho - = ( E . + Eb) Z

h" - { - ( a l d [ b ) F ( t ), O, ~ (Eo

-

(b) Write the density matrix p - Ir162 for the pure state Ir Cala) + r as a 2• matrix in the basis {la), Ib)} and identify the polarization vector P from (6.10). 6.7 Consider, as described on p. 135, the manifold of states of the hydrogen atom with n = 2, disregarding spin and relativistic effects. Identify its alternative sets of 16 unit operators (6.33) or (6.36) through the physical properties proportional to each of their mean values. (This problem is studied in ref. [29].)

This Page Intentionally Left Blank

Chapter 7

Recoupling Transformations: 6-j and 9-j Coefficients We have seen in Chapter 5 how to construct alternative irreducible products of three or more tensorial sets according to different "coupling schemes" and thus characterized as alternative eigenvectors of different, incompatible operators. The present chapter treats transformations that relate products with different coupling schemes, outlining their very extensive applications. We shall find these transformations to be invariant under rotations of coordinate axes; in other words, commuting with all rotations of physical space. Their technology, called "Racah-Wigner algebra," accordingly makes no reference to coordinate axes or to magnetic quantum numbers. Many phenomena studied in physics concern closed systems in the absence of external vector fields that would single out special directions in space. Irrelevance of the orientation of coordinate axes and invariance under their rotations are central to many a study. The development and applications in this chapter form, in a way, the core of this book. The need for this development has been anticipated repeatedly in earlier chapters. We recall here the sources of this requirement: a) Wave functions of two or more particles with spin can be constructed by adding first their orbital angular momenta and, separately, their spins (that is, by LS-coupling) or instead by combining first the spin and orbital momentum of each particle (j j-coupling). Either because 157

158

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

of the physical separation of the particles, or because of some other factor that may apply, one or the other construction may be more appropriate. We seek here the transformation between the two types of wave functions. b) Wave functions of many particles in the same shell must be symmetrized under particle permutations. Any single coupling scheme is inherently asymmetric because of the order in which the particles are coupled, and hence inadequate. The construction of symmetrized functions, to be discussed in Chapter 8, requires a superposition of different coupling schemes and hence a mastery of their relations. c) The Hamiltonian of most atomic systems (whether invariant under coordinate rotations or including interactions with anisotropic external fields) depends on the coupling of such different tensorial variables as orbital and spin momenta of different particles and as multipole moments of their charge and current distributions. The Schr5dinger equation's structure requires us to calculate the mean value of tensorim operator products for eigenstates of the total angular momentum of all particles, an operation that generally involves extensive recoupiing. d) The SchrSdinger equation's expression in terms of state multipoles, that is, the generalization of the precession equation (6.14), requires expanding products of unit multipole operators, thus reducing to a recoupling transformation, as we shall see. This chapter's task may be viewed in a broader frame: Chapter 6 has shown how to construct sets of operators that fill the Liouville representation space of physical systems with 2j + 1 orthogonal states (j arbitrarily large) completely. Each of these complete sets provides a coordinate frame for the physics of a system. We now seek procedures for passing from one such frame to another. A small subgroup of these transformations is isomorphic to rotations of the coordinates of physical space, or even coincides with this rotation group; we are concerned here with extending the transformation to the broader class that fills the entire Liouville representation. As an illustration, Fig. 7.1 indicates two alternative combinations of the space coordinates ~,- of four particles. These combinations belong to the class of "Jacobi coordinates" that represent the relative positions of two particles or two groups of particles and are indicated diagrammatically by "trees." Figure 7.1a shows the "standard" tree for four particles

159

wherein each particle's position relates to the center of mass of the previous assembly; the corresponding analytical structure of the Jacobi coordinates Xi is shown alongside. Figure 7.1b represents instead a nonstandard tree with two pairs of particles combined separately rather than sequentially. It has been pointed out that the connections between alternative J acobi trees and the transformations that connect them are isomorphic to the recoupling transformations among products of standard sets reduced according to alternative coupling schemes [30]. This remark enlarges the scope of recoupling transformations still further but remains to be implemented widely. Formulating recoupling transformations presents a rather straightforward task, to be undertaken first. The great variety of the resulting transformation matrices can then be reduced substantially by expressing them in terms of a few standard types of highly symmetrical coefficients. This further development will finally be followed by applications to various classes of operator products.

1

2

~)

x= -

1

b)

2

3

4

[~(~ + ~2)-

_. fyx ~ - V~ 89

ea]

- ~'2)

X2 - ~ ( ~ ' 3 ~]/~(r'l -~" r"2 -- ~3 -- r'4) V

"

F i g u r e 7.1: Alternative Jacobi trees for four particles.

160

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

7.1

Transformation Matrices and Their Analysis

As an initial example, consider the two triple products indicated by ((jlj2)jl2j3jm] and by (j1(j2j3)j23jm I in (5.31) and (5.32). As each of these products is defined through the reducing transformation that relates it to the direct product set (jl ml(j2ml(j3m], their connection is established by multiplying one of these transformations by the reciprocal of the other. Specifically, we first express the direct product in terms of ((jlj2)jl2j3jm I by inverting the reducing transformation (5.31), that is, we write

(jlmll(j2m21(j3m31

=

~

(jlml,j2m2,j3m3](jlj2)jl2j3j'm')

j12j'm'

x ((jlj2)j12j3j'm' I .

(7.1)

Next we substitute this equation on the right-hand side of (5.32), finding (jl(j2j3)j23jml --

~

(jl(j~j3)j23jml(jlj2)jl~j3j'm') ((jlj2)jl2j3j' m'l ,

j12j'm'

(7.2) where the coefficients on its right-hand side

(jl (j2j3)j23jml(jlj2 )j12j3j' m') = ~ (jl (j2j3)j23jmljlml, j2m2, j3m3) mlm2m3

x (jim1, j2m~, j3m31(jlj2)j12j3j'm')

(7.3)

constitute the desired matrix that transforms the set of products with indices j12 into the other set with indices j23. Observe now the behavior of the matrix (7.3) under rotation of coordinate axes in physical space. Formally this matrix experiences an rtransformation by application of D(J)(r 9, ~) on its left and of the contragredient D (j')-I (r 9~) on its right. Yet its new elements should be identical to the initial ones, being constructed in every detail according to the same analytical prescription. Hence the matrix should in fact represent an invariant. Indeed according to the "Schur lemma" (for example, Appendix A of [1]), the irreducibility of D (j) and D (j')-' implies that: a) the matrix (7.3) vanishes for j r j', and b) it equals a constant times the unit matrix for each j - j~.

7.1. T r a n s f o r m a t i o n M a t r i c e s a n d T h e i r A n a l y s i s

161

These implications are formulated by the equation (j1(j2ja)j23jml(jlj~)j12j'm') -- (j1(j2j3)j231(jlj2)J12j3) (j) ~jj'~mm', (7.4) stating that the matrix (7.3) is block diagonal in j and m, with identical blocks for equal j and different m. Each diagonal block has its rows labeled by j23, that is, by an eigenvalue of IJ~ + J-'al2, and its columns by j12, that is, by an eigenvalue of IJ~ + J~l 2. The row and column indices take all the values consistent with the constraints of triangular relations. The considerations presented above for a particular case of triple products are equally applicable to all n-fold products of standard sets (jxml[(j2m21... (jnmn 1. Each recoupling transformation may be visualized as a rotation of axes within the vector space of these products, which has 1-Ii=l(2ji + 1) dimensions. This rotation affects separately each subspace labeled by (jm), that is, consisting of eigenvectors o f [ Z / j~12 and of ~ i Ji~. It affects identically all the subspaces with equal j and different m. Each recoupling matrix is a function of the same constant parameters j l , j 2 , . . . , j n , and of j. Its rows are labeled, in accordance with the classification of products in Section 5.3.5, by the eigenvalues of a maximal set of commuting operators [ ~ s p J-~ 12 where Sp indicates a subset of the integers {1, 2 , . . . , n}; its columns are labeled by the eigenvalues of a second maximal set of operators corresponding to a different selection of subsets n

s;. The recoupling transformations have no analog among the identities of vector algebra, because the concept of expanding one product into all elements of a different class of products of the same factors, implied by the ~j12 of (7.2), is foreign to vector algebra. 7.1.1

Diagrams

The diagramming procedure introduced in Section 5.3.6 also serves to illustrate recoupling transformations. For example, the matrix (7.3) is represented by

ja

/

'"

I III

\

j2

\_

/ (7.5)

Chapter 7. Recoupling Transformations

162

A heavy bar has been added here to the line corresponding to the resultant of each "vector addition," to represent a normalization factor (2j + 1) 89 with the value of j corresponding to that line. This factor is included in each Wigner coefficient, as in (5.15), but it is often removed in the course of symmetrization as, for example, in (5.45). Removal of the heavy bar in the diagram will denote henceforth the removal of the corresponding factor. Note that the central lines in the diagram (7.5) are stacked in the order of the m i indices on the right-hand side of (7.3). The block-diagonality of the matrix (7.3) is represented in its diagram by the occurrence of a single open-ended bra-line on the left and a single openended ket-line on the right. Any diagram with only two such open terminals must similarly represent an invariant, diagonal in j and independent of m. As the matrix (7.3) is independent of m, the value of each of its elements remains unchanged by summation over m and division by (2j + 1). The sum is represented by looping together the terminals of the diagram (7.5), while the division by (2j + 1) is represented by removal of two heavy bars, as indicated by the equation

j

j2 \

j3

\

jl

J J

j3

(7.5a) The diagram on the right-hand side reflects the equivalence of the four indices (jl, j2, j3, j), which are constant parameters of one diagonal block of the transformation, whereas j23 and j12 label the rows and columns of that block. The closed diagram on the right reflects the rotational invariance of the recoupling matrix even better than the occurrence of single terminals on its left-hand side. 7.1.2

Group

properties

Within any given vector space of direct products ( j l m l l . " ( j n m n l and for given values of j and m, the recoupling transformations form a class of operations that shares many--but not all--the properties of a group. This class is accordingly called a g r o u p o i d . A group of transformations

7.1. T r a n s f o r m a t i o n M a t r i c e s a n d T h e i r

Analysis

163

includes the product of any two elements of the group. The product of two recoupling transformations U and V is instead defined only in a restricted manner" A transformation (BIUIA) that changes a coupling scheme A into a scheme B combines into a product with all transformations (CIVIB) that change B into any other scheme C; this product can be extended to include further factors (DIW[C), etc. On the other hand, no product of (B[U[A) and (D]WIC) alone is defined because the two transformations cannot be combined sequentially by themselves. Subject to this limitation, the transformations of a groupoid are treated much as those of a group. In particular, they are represented by real orthogonal matrices. In the example of n - 3, the group includes also, for example, the transformation ((j~j2)j~2j31(jlj3)jl3j2) (j). Because of the different ordering of ji symbols in this transformation, we include in the groupoid also simple permutation operations, such as ((jxj~)jl2jal(j2jl)j~2ja) (j) = (-1) jl+j~-j, illustrated by a diagram with only a twist of two lines. Orthogonality of transformations places, of course, a constraint on their matrix elements which is expressed by s u m rules. Such sum rules represent their groupoid property as, for example, ((jl j2 )jl 2j3 ijl (j2j3)j23)(J )

= ~ ((jlj2)j12jal(jlj3)j13j2)(J)((jlj3)j13j21jl(j2j3)j23)(J). (7.6) j13

The summation over j13 of the central ket-bra which involve this index amounts to unity. This closure operation reflects the completeness of the subset of states with alternative j13 for describing the three-particle system. 7.1.3

Factorization

of transformations

The groupoid properties afford resolving complicated transformations into products of simpler ones, belonging to different subgroupoids that leave one or more of the operators [ ~ r fr[ 2 invariant. In fact any recoupling transformation of n-fold products can thus be reduced to a sum of products of triple product transformations. Diagrams prove particularly useful in finding the simplification most appropriate to any particular recoupling. We consider here examples drawn from each of the three main classes of transformations of quadruple products: a) The transformation matrix ((jlj~)j12(j3j4)j341[(jlj2)j123j4) (j) is block diagonal in the idex j12; that is, it commutes with the operator

164

Chapter 7. Recoupling Transformations IJ-1 + J212. Accordingly it is equivalent to the triple product matrix (j12(j3j4)j341(j12j3)j123j4) (j). This simplification amounts to dropping from the diagram

j4 j3

\

j2 j12

\

/

/

jl

(7.7) the trivial portion

j2 j12

/

~

j12 (7.7a)

The portion thus eliminated represents a unit matrix, namely, the product of (jlj2j12m121jl ml, j2m2) and of its reciprocal; this example suggests a first general rule: Any portion of a diagram connected to the remainder by only two lines represents the unit matrix, or a multiple thereof; such a portion can be factored out by rejoining the two connecting lines as indicated by

J j4

\

j34

ja

\

....\ jl

j12

j2

\

~

/

j12

j12 /

J j12a

jl (7.7b)

165

7.1. Transformation Matrices and Their Analysis

After factoring out the trivial subdiagram at the bottom, the nontrivial diagram that is left represents the coupling of three quantities j12, j3, and j4 to a total j. b) A second recoupling matrix may be resolved by inspection into the direct product of two triple product matrices, ((jl j2 )jl 2(j3j4)j34 [[jl (j2j3)j23]j123j4 )(J )

=

(j12(j3j4)j341(j12j3)j123j4)(J)((jlj2)j12j31jl(j2j3)j23) (j'2~) (7.8)

Of the two matrices on the right-hand side of (7.8), the first one commutes with IJ-~ + J-212, whose eigenvalues label the rows on the left, while the second commutes with IS1 + f2 + LI 2, whose eigenvalues label the columns of the left-hand-side matrix. The diagram of this transformation may be drawn with successive modifications that display the detailed analytical procedure leading to (7.8),

j4 .j34

/

" "

~ j12 ...

j3 / \

~

j23 /

jl

j123

(7.9) The breaking up of this diagram starts by considering the orthonormality--or, rather, completenesswrelation

(j~m~,, jbmbljajbj~rnr162162

jbm'b) -- 6,~.m, 6,~m~,

j c lTt c

which is represented by the diagram jb

X" /__.,

j~

~ ~

~ / j~

jb

jc

_ ~

jb

~

\_____ ja

ja (7.9a)

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

166

The right-hand side of this diagrammatic equation may be identified with any pair of lines in a diagram and replaced by the diagram on the left. In the case of the diagram (7.9) we set ja = jl~ and jb - j3; insertion of (7.9a) brings then (7.9) to the form

J j4

j12

(7.9b) We shall call this operation the p i n c h i n g off of the pair of lines j12 and j3 into the single line jc. (Pinching off may be viewed as a reduction operation since it replaces a pair of lines, pertaining to separate sets, with a single one that represents their product.) The portion of the diagram (7.9b) between the lines jc and j12a represents by itself a triple product transformation matrix, which must be invariant, as in (7.4), with a factor ~jtj,2a which reduces the ~ j o to a single term. This two-terminal invariant matrix constitutes in fact the second factor on the right-hand side of (7.8); it may be separated out from the rest of the diagram, as was the subdiagram in (7.7b). The remainder of the diagram (7.9) now represents the first factor on the right-hand side of (7.8). From this example we can draw a second general rule for the analysis of recoupling diagrams: Any nontrivial portion of a diagram connected to the rest by three lines only--in our example the portion connected by j12 and ja on its left and by j123 on its right--can be factored out as a separate transformation. c) The remaining type of recoupling matrix for fourfold products does not resolve into a simple product of factors because of its different topological structure. This matrix is of particular importance in both concept and practice and is represented by

((jlj2)j12(j3j4)j341(jlj3)j13(j2j4)j24) (j),

(7.10)

7.1. Transformation Matrices and Their Analysis

167

and by the diagram

J

J j4 .,

j3

"'j2

' ~ ..... j 2 4 ~

(7.11) This matrix represents the transformation of a two-electron wave function from LS- to j j- coupling when (jl, j~) represent the orbital momenta of two electrons and (ja,j4) represent their spins. Notice how the lines j2 and J3 interchange in the center of the diagram (7.11). To resolve this transformation into simpler ones, we proceed to separate out the portion involving jl, j2, and j3 only, by "pinching together" the lines j12 and j3 as we did in (7.9); we also pinch together the lines j13 and j2 on the right of the diagram, an operation that was unnecessary in (7.9). The diagram (7.11) is thus shown to be equivalent to

j4

j4 ,,

j2

, j123

j123 j2

J

3

~

.

\

. jl,

3 .... j13 ] o

(7.11a) The subdiagram in the lower central part is now separated out, but the index j123 remains to be summed over, because its value is not identified by either the row or the column indices of the matrix (7.10). Topologically, a summation over j123, or analogous parameter, is made necessary by the fact that no nontrivial portion of the diagram

168

Chapter 7. Recoupling Transformations (7.11) is connected to the rest by three lines only. This point of view guides us, however, to observe that the left- and right-hand portions of the diagram (7.11a) are indeed connected by only three lines, ja23, j4, and j. The remaining step in resolving the diagram is thus readily performed. One pinches together the pair of lines j123 and j4, corresponding to the operators Ja + J2 + Ja and J4 whose addition gives the total f. Thereby the diagram resolves into the sum of products of triple product diagrams _#

j4

j

...

..#

...

j4 / ~jl2a

~

j24

j2

j123

ja j12

j~

j:

j~

(7.11b)

The corresponding analytical formula ((j lj2 )j12(jaj4 )j34 I(Jl j3 )jl 3(j2 j4 )j24 )(J)

=

Z(j12(j3j4)j341(j12j3)j123j4)(J)((jlj2)j12j31(jlja)j13j2)

(j1~3)

j123

• ((j13j2)j123j4[j13(j2j4)j24) (j)

(7.12)

serves actually to evaluate the matrix elements because the matrices for triple recoupling are evaluated more readily than the left-hand side of (7.12), as we shall see.

7.2

Symmetrized Recoupling" Coefficients

6-j and 9-j

The matrices of recoupling transformations can be represented in terms of highly symmetric functions of their indices, much as was done for the reducing matrices in Section 5.4. In the initial form of a recoupling matrix

7.2.

Symmetrized Recoupling: 6-j

a n d 9-j Coefficients

169

some of its indices represent the degree of the sets that are being recoupled, another one, j, labels a diagonal block of the matrix, and the remaining ones label the rows or columns of the diagonal block. In the symmetrized form the indices play fully--or at least nearly--equivalent roles. A part of the initial asymmetry derives from the presence of factors of the type (2j12 + 1) 89 which derive from the normalization of reducing matrices and which have been represented by thick bars in our diagrams. Generally one such coefficient appears for each index that serves to label a row or column. One step of the symmetrization procedure consists of factoring these coefficients out of each recoupling matrix element. Thus, for example, the matrix element represented by (7.4) and (7.6) includes the pair of factors (2j1~ + 1) 89 + 1)~. Separation of these factors requires them to be reintroduced explicitly in any sum rule that reflects orthonormality or other matrix product relations. Following the separation of these normalization factors and removal of the heavy bars that represent them in a closed diagram, each diagram consists of a network of lines--one for each of the j indices--each of which joins two nodes. Each node is joined to three others by as many lines. The number of lines in a closed diagram then equals a multiple of three, 3n, and the number of nodes equals 2n. The various nodes and lines have nevertheless different roles in the construction of the matrix elements, which may result in sign reversal under permutations of the j indices that twist-but do not tear--the diagram. In a straightforward approach, the study of such residual asymmetries resolves a matrix into a product of matrices, typically into 3-j coefficients; one may then examine the symmetry of the product under various permutations of indices. The study may be conducted analytically or in terms of diagrams, keeping track of the signs of m quantum numbers, reflected by arrows in the diagrams of Chapter 5, and of the order of lines about each node. Elaborate techniques have been developed for this purpose (see particularly [17] and [20]). These rules remain rather laborious and somewhat obscure. Here we shall only outline the principal symmetrized forms and their main properties; a compact summary of formulas is given in the introduction to the tables of 3-j and 6-j coefficients by Rotenberg et al. [21].

170

7.2.1

Chapter 7. Recoupling Transformations

6-j coefficients

The symmetrized form of the matrix for recoupling of triple products provides our basic building block. Removal of the two heavy bars from the diagram on the right of (7.5a) reduces it to a network consisting of 6 lines and 4 nodes, topologically equivalent to a tetrahedron. Closer examination shows full tetrahedral symmetry to result by separating out a +1 factor defined as (-1) j~+j2+j~+j. The remainder of the diagram is a symmetric function of the four triads of indices {(jlj~j12), (j3j2j23), (jljj23), (j3jj12)}, which is called a 6-j coefficient, with each of the six j indices belonging to two triads. [The original less symmetric form of this function, including the factor (-1) j~+j2+j3+j, is called a R a c a h coefficient.] The standard symbol for a 6-j coefficient includes a list of the 6 separate indices grouped into triads according to a code that fails to reflect the full tetrahedral symmetry owing to typographical constraints. The general symbol is { jl •1

j2 ~2

j3 } t3 '

(7.13)

with the implied convention that one of the four triads consists of the three upper tier indices, while each of the others consists of two indices in the lower and one in the upper tier, all of them in different columns. Each triad fulfills, of course, the triangular conditions (5.3). The diagrammatic representation of the 6-j coefficient

(7.14) shows each of its four vertices joining one triad of edges labeled by j indices. An alternative to the three-dimensional figure of a tetrahedron for exhibiting all four triads equivalently, unlike (7.13), will follow below in (7.22), which views the 6-j symbol as a special case of a 9-j symbol. According to our discussion, the matrix (7.4) is expressed in terms of a 6-j coefficient by

7.2. S y m m e t r i z e d Recoupling: 6-j and 9-j Coefficients

171

(jl (j2j3)j23](jlj2 )j 12j3 )(J) =

1) 89

(2j12 + 1) 89

jl+j2+ja+j { jl j3 j2 j j12 j23 } .(7.15)

The following properties of 6-j coefficients should be noted: a) The value of the symbol (7.13) is invariant under all permutations of its three columns of indices. b) The same value is also invariant under permutations of upper and lower tier indices within any two columns. The invariances a) and b) correspond to the symmetries of the tetrahedron in (7.14) under rotations with respect to its various edges. c) Since the triangular conditions restrict the number of half-integer indices in a triad to 0 or 2, the half-integer indices of a 6-j coefficient, if any, can either occupy all four positions of two columns or three positions in different columns, one of them in the lower tier. d) Any j-index that vanishes can be shifted to the lower tier of the last column using properties a) and b); the value of the 6-j-coefficient is then

jl

j2 j3 "~ _

/71 ~72 0 J

(_l)j~+j:+ja(2jl + 1)- 89

1)- 89

x 6llj~hl2j,~(jlj2j3), (7.16) where $(jlj2j3)is a Kronecker symbol (Eq.(10.19) of [l]) that vanishes if its three indices fail to obey the triangular condition. e) The orthogonality of the matrix (7.15) implies the sum rule { jl

~-~'(2j3+11

j3

j2

j3 } { jl

el e2 e3

=

j2

j3 }

el e2 e~ +

1) -1

.

(7.17)

f) The associative (that is, groupoid) property of recoupling matrices [see, for example, (7.6)] implies a second sum rule, ~--~(_1)j~+~+~;(2j3+1){ j~ jl

t?2 /?3

j2 /71 /73 }"

j2

j3 } { J~ j2

j3 }

(7.1s)

Chapter 7. Recoupling Transformations

172

The most familiar definition of the 6-j coefficients represents them as sums over products of four 3-j coefficients, one of them for each triad of j indices; the sum is taken over the values of all m indices (Eq. (11.6) of [1]). Since a single 3-j coefficient serves to construct one invariant product from the direct product of three standard sets, the product of four 3-j coefficients can be regarded as an element of the matrix that constructs four invariant triple products from the direct product of twelve standard sets (jllA), (jllA'), (j21B), (j21B'), . . . , (~31F'). A different invariant product of the same 12 sets consists of the six scalar products (Sljljx) (jl IA)(j~ IA'), etc. The 6-j coefficient may then be regarded as the transformation matrix element that connects these two different invariant products of 12 sets. A more direct analytical expression of the 6-j coefficients has been given by Racah (Eq. (11.9) of [1]). The numerical values of the 6-j coefficients are given in [21] for extensive ranges of their indices. They are also obtained by standard computer codes as well as from a convenient family of recurrence relations such as the one in (7.18) and an identity to be cast as Problem 7.10. The 6-j coefficients are frequently applied to transform products of two or three 3-j coefficients into other 3-j coefficients. Relevant formulas are given in [21]. The principle of these transformations may be illustrated by a diagram showing how to transform a product of Wigner coefficients by multiplication with an appropriate unit matrix j3 9

j2

\

j3

j3 x

__

j12

jl

J

j3

j3 J

J

(

j12

(7.19) The elements of any recoupling transformation matrix can be expressed as sums of products of 6-j coefficients, and thus evaluated, by reducing them first to sums of products of triple product recouplings by the methods of Section 7.1. Entire classes of operations of this type have been programmed

7.2. S y m m e t r i z e d R e c o u p l i n g : 6-j a n d 9-j Coefficients

173

for automatic computing. 7.2.2

9-j coefficients

Various types of recoupling transformations have been symmetrized and thus reduced to the form of 3n-j coefficients. Of these only the 9-j coefficients, resulting from the symmetrization of the matrices (7.10) or (7.11), have drawn much attention; they pertain to the transformation between LS- and j j-coupled systems and to other important applications, as we shall see. The 9-j coefficients are also related to transformation matrices even more directly than 6-j coefficients and have been regarded as more fundamental--in certain respectswthan the 6-j's themselves (see [31] and Section 7.2.3 below). A 9-j coefficient is obtained from the transformation matrix element (7.10), or from its diagram (7.11), by separating out only the normalization coefficient [(2j12+ 1)(2j34+ 1)(2j13+ 1)(2j24+ 1)] 89 The remaining diagram, consisting of 9 lines and 6 nodes, may be viewed as a recoupling matrix element connecting two alternative sets of invariant triple products of nine standard sets. We write

((jl j2 )j~2(j3j4 )j34 I(J~J3)jl 3(j2 j4 )j24 )(J) =

, [' jl [(2j~2 + 1)(2j34 + 1)(2j13 + 1)(2j24 + 1)]5 / j3 j13

j2 j4 j24

j12 j34 j

} (7.20)

and

jl j2 j3 } kl k2 ka tl

~'2 i3

= ((jlj2j3)O(klk2k3)O(t.lt.2g.a)Ol(jlkltl)O(j2k2t2)O(j3k3t.3)O), (7.21) whose symbol in braces is called a 9-j coefficient. This coefficient is a function of 9 indices, grouped into 3 triads in two different manners (rows and columns). Each triad obeys the triangular conditions. The explicit rendering of (7.21) in which the 9-j coefficient is written as a sum of products of six 3-j's is given in Eq. (12.10) of [1]. The 9-j coefficient is invariant under circular permutation of three rows of

174

Chapter 7. Recoupling Transformations

indices or of its three columns. Permutation of two rows or of two columns multiplies a 9-j coefficient by +1 depending on the parity of the sum of all its 9 indices, as one can see from permuting the triple products on the right-hand side of (7.21). Any index that vanishes within a 9-j coefficient reduces a transformation of quadruple products to one of triple products, thus reducing the 9-j coefficient to a 6-j coefficient. Since that index can be permuted to the lowest row and to the last column, it suffices to give the formula for this special case, namely, j, kl ~1

j2 k2 ~2

j3 } k3 0

~t112~jak3(--1) j2+ja+k'+l' [(2j3 + 1)(2el + 1)]- 89 •

jl k2

j2 kl

j3 } tl "

(7.22)

Orthogonality and associative properties of the recoupling transformation (7.10) can be cast in the form of sum rules for the 9-j coefficients analogous to (7.17)and (7.18):

(26 + 1)(26 + 1) lll2 =

{jikl j2k2 j3}{ji j2 k3 kl k 2 ]r ~1

if2

~3

~1

~j3j;~k3k,a~(jlj2j3),~(klk2k3)~(j3k3g3)(2j3

~2

~3

+ 1)-1(2k3 + 1) -1 ,

(7.23)

(2el + 1)(26 + 1)

{jlkl j2k2

k3

k2 jl

--

]:2

j~

kl g2

j2

j2 j3 } kl t:3 k~ ~3

9

(7.24)

Applications to expand products of 3-j coefficients, analogous to that outlined in (7.19) for the 6-j, also occur for 9-j coefficients. Relevant formulas are given in [17] and in other standard references. For the purposes of numerical calculations, 9-j coefficients are nowadays usually expressed as sums of products of 6-j coefficients (see Eq. (12.12) of [1]); therefore direct tabulation of the 9-j's has become unimportant although sometimes they

7.2. S y m m e t r i z e d R e c o u p l i n g : 6-j a n d 9-j Coefficients

175

can be useful as listed in [32]. The interest of 9-j coefficients centers on their frequent role in analytic manipulations to be described in the following sections.

7.2.3

Alternative

perspectives

6-j, 9-j, and higher 3n-j coefficients may be viewed, as in the above sections, as recoupling transformations that arise upon considering multiple products of (n + 1) tensorial sets with n > 2. In this view, the 6-j coefficient is the basic object and all higher coefficients may be expressed as sums of products of 6-j's. A rather different starting point, one emphasized by some authors (as, for instance, ref. [31]), regards the 9-j coefficient as basic, with the 6-j as a special case with one index vanishing as in (7.22). This perspective stems from associating 9-j coefficients with rotations in four-dimensional space in analogy with associating 3-j coefficients with rotations in three-dimensional space. That is, the 9-j coefficients have the role of Clebsch-Gordan coefficients for adding two four-dimensional angular momenta, much as the 3-j coefficients do for adding two three-dimensional angular momenta--see (5.1) and (5.45). Whereas rotations in three dimensions involve the three operators {/?,, s s }, the three generators of SO(3), those in four dimensions involve six basic operators whereby SO(4) has six generators. Besides ~, the other three transform like a vector in SO(3) and may be denoted by ff with

(7.25) As in the specific example of the hydrogen atom's SO(4) considered in Section 6.3, where ~ is the Laplace-Runge-Lenz vector, the combinations jl,2-" - ~1(~'-I-if) behave like two independent three-dimensional angular momenta, each obeying (2.15). This feature, that a four-dimensional rotation reduces to a pair of three-dimensional angular momenta, makes the addition of two four-dimensional angular momenta equivalent to adding four threedimensional ones, thereby introducing 9-j coefficients in the corresponding Clebsch-Gordan algebra [13]. We will return in Section 10.1 to a more detailed discussion of rotational eigenfunctions in four dimensions, the socalled four-dimensional spherical harmonics which generalize the Yl,,-,(O,~) in (4.30).

176

7.3

Chapter 7. R e c o u p l i n g Transformations

P r o d u c t s of O p e r a t o r s

The construction of the orbital momentum operator of a particle ~ ' - / ' x t7, as the vector product of two vector operators, affords a simple example of procedures that combine tensorial aspects with operator algebra. Equations (6.10) to (6.14) afford a less obvious but equally basic example, where the Schrbdinger equation,

dp/dt = - i ( H p - p H ) ,

(7.26)

involves products of the scalar operators H and p, both of which include the vector operators Y. In this example the eventual form of the Schrbdinger equation in (6.14) no longer involves operators: Operator products have first been reduced to single operators, after which the expansion of both sides of the equation into the base set (1, ~) recasts the problem into an equation among the expansion coefficients. This section will extend the example of Section 6.2 by developing a general procedure that attains the same result for whichever quantummechanical system is mapped onto the states of a spin-j particle. Recoupling transformations complement the expansion into a set of unit operators by resolving the entire structure of quantum-mechanical operator equations into sequences of standard procedures. Their net result embodies the quantum aspects of equations into elements of standard recoupling transformations that appear as coefficients of equations among expansion parameters, that is, among c-numbers, as the example of (6.14) embodies recoupling in the symbol of vector product.

7.3.1

Unit operators

We begin by considering the products of standard unit operators. The standard operators themselves are constructed by reducing elementary operators of the general type Iajm)(a'j'm'l, which are direct products of ket and bra symbols. A primitive ingredient of our calculation lies in the orthonormality relation of bra and ket symbols of the same complete set (7.27) This orthonormality implies that the product of two elementary operators contracts into a single operator, as indicated by

I~j~m~)(~j~m~ll 2J2 2)(c~232rn21- 6=',==6jIj26m~m=lc~ljlml)(c~232m21.

(7.28)

7.3. P r o d u c t s of O p e r a t o r s

177

This expression and, in particular, the Kronecker symbol ~j~xj 2 arising from orthonormality hold the key to the reduction of products of operators into a single operator. We now return to the standard unit operators, as defined by (6.37), to construct their product in accordance with the identity (7.28). We thus find in a first step 9 "'t (3232)k2kq)) ~-~jj~rnl).,, (32rn2J(-1 )j;-m;

I(J131t)k1 "'

=

mlm 2

~

I

( _ ) j1' ~ - r n '

1

"' ' (jlm1,31-mlJjlj~klql)

I

mlqlq2

x (j2m2,32-m213232/c2q2)(k~ q~ , Ic2q2JlcllC2kq)~j'j~m'~m~,. (7.29) 19

I

9

"1

The coefficient ~ m , lqlq . . - o n the right-hand side of this equation consists of four Wigner coefficients, three of which appear explicitly while the fourth one emerges by writing from (5.24) and (5.17), /

!

/-

/

( - 1 ) ~'-m~ 6 , , ~ m ' , m ~

-- (--1) ~j`

(2j'~+1)~ (j~j~OOIj~-mx,j~m2) 1

I

(7 29a)

A transformation of this coefficient analogous to that displayed in (7.19) turns the right-hand side of (7.29) into the product of a unit operator and of a recoupling matrix element 9

"I

!

J(jlj~t)kl(j232t)k2kq)) (_l)2J,(2j~ + 1)~i~j~ x jj~j~t kq))((j~j~)k(j~j2)OJ(j~j~)kl(j2j~)k2)(k). 1

-

(7.30)

Notice that the coefficients (-1)2J'1 and (2j~ + 1) 89 of this formula derive from (7.29a); the former occurs because the bra factor (j'm' I, with its coefficient ( - 1 ) j'l-m'x , is the first rather than the second factor of an invariant product [Eqs. (5.26)], the second coefficient occurs because of a normalization mismatch to be compensated below. Note also the role of two trivial (or "saturated") triads in combining with four nontrivial coupled triads in achieving this central result (7.30) for a coupled product of two standard unit operators as a single unit operator. The two factors of the product involve jl and j~ coupled to kl, and j2 and j~ coupled to k2. Subsequently, these kl and k2 are coupled to the product's k. The recasting as a single operator with jl and j~ coupled to k proceeds through the aid of jl +j2 - 0 arising from the Kronecker symbol in (7.28) and (7.29a) and an attendant trivial triad k + 0 - k. The main coefficient of (7.30), that is, the recoupling matrix element, appears to belong to the general class of fourfold transformations (7.10).

178

C h a p t e r 7. Recoupling Transformations

In fact it differs from (7.10) by a permutation of the indices j2 and j~ on its right-hand side and especially by the vanishing of one index. A closer matching with (7.10) is attained by permuting j2 and j~ in the coupling 9 "I k symbol (1232) 2," this permutation constitutes a further recoupling whose matrix element equals (-1)J~+J; -k2. After this preliminary, Eq. (7.20) expresses the complete matrix element in terms of the 9-j coefficient,

( (jlj~ )k(j~ j2 )Oljlj~ )k l (j2j'2 )k2 )( k ) (-1) j~+j;-k~ [(2k 4- 1)(2kl + 1)(2k2 + 1)] 89

=

jl

j~

j~ kl

j2 0 k2 k

. (7.31)

The occurrence of a null index implies that the 9-j coefficient reduces to a 6-j coefficient through application of (7.22) preceded by a circular permutation of the index rows that brings the null index to the lower right corner of the 9-j pattern. This procedure reduces (7.30) to the explicit form

I(JlJ~ t)kl (j2j'2t)k2kq)) =

[jxj'2tkq))(-1) j'+j~+k (2kl 4- 1) 89{ j~kl jxk2 J~k

}

6j1~2.,.(7.32)

This final form contains a 6-j coefficient as could have been anticipated, because (7.29) is a sum of products of four Wigner coefficients. Note how graphical analysis leads to this result more directly. The recoupling matrix element in (7.30) is represented by

j2

/

jl

/

k2

\

(7.32a) where the curve with an arrow represents the null index as in the diagram (5.39). This diagram's structure, with 4 nodes and 6 lines, coincides with that of the 6-j symbol in (7.32). Careful analysis of the ordering of lines converging to each node by the method of [27] is, however, required to identify the sign factor in (7.32).

7.3. P r o d u c t s of O p e r a t o r s

7.3.2

General

179

operator

The reduction of the product of two unit operators, through (7.29-32) opens the way for constructing the product of any two operators G1 and G2 through the following sequence of steps, or an equivalent one. Each of the operators can be expanded in tensorial form according to (6.29) and (6.30),

GIG2

gql

1 k2q2

--

[Zklqljlj~l

;~q2

'JlJ~tklql))(2kl+1)-89

1

[ ~-~ Ij2j'2tk2q2))(2k2-4-1)- 89 k2q2j2j;

9

(7.33) The direct product of the two expansions is then reduced by applying the matrix (klql,k2q21klk2kq) to the unit operators and its reciprocal to the c-number expansion coefficients, yielding

GiG2

I(jlj~t)kl(j2j~t)k2kq))(2kl + 1)- 89

=

+ 1)- 89

klk2kqjlj~j2j~

• (j~llg~k')JJjl)(j2Jlg~k~)Jlj~)(kik2kqJa~G2). (7.33a) Equation (7.32) now transforms the unit operator, yielding

G1G2 -

E

IJlj2tkq))(2k + 1)- 89

klk2kqjlj~ (7.33b) where we have introduced the reduced matrix

(jlllG(k,k~k)llj~) - ~ (Jlllg~k*)llJ2) j2 {kl k2 k} x (-1)/~+J;+k(2k + 1) 89 j; jl j2

(J211g~k2)IIJS).

(7.33c)

Note how (7.33b) represents an expansion of the product GIG2 into unit operators of the system, conducted in accordance with (6.29) and (6.30) and constructing the reduced matrix of a single tensor operator G (k~k2k) from those of operators g(k~) and g(k2) pertaining to the separate expansions

Chapter 7. Recoupling Transformations

180

of G1 and G2. The essential recoupling coefficient appears explicitly only in the construction of the reduced matrix of G (k~k:k) The passage from a product of unit operators in (7.33) to an expansion in a set of single operators with index k in (7.33b) generalizes the identity (2.35) for Pauli 1 It , and a commutator version that follows below, will spinors to j > 3" play a crucial role in Section 7.3.4 in reducing the Schr6dinger equation to a Bloch-like precession equation for general j, paralleling Section 6.2's 1 treatment for spin ~.

7.3.3

Commutators

Equations (7.32) and (7.33) afford constructing the commutator of any pair of operators in tensorial form. The commutator G1G2 - G2G1 will nevertheless involve in general the commutator of their reduced matrices, which remains to be calculated. A major simplification and an interesting result emerge, however, in the case of operators acting on a particle with fixed spin quantum number j, or in the equivalent case of a (2j + 1)-level system (Sections 6.4 and 6.5). In this event we set jl - j~ - j2 - j in (7.33), whereby all factors of (7.33b) and (7.33c) commute under permutation of G1 and G2 except (klk2kqlGiG2), which has the parity ( - 1 ) kl+k2-k according to (5.171. Accordingly the result G I G 2 --

G2Ga

-

Z

Ijjtkq))(2k + 1)- 89

kq

•

[i -

( - 1 ) k'+k2-k]

(kak2kqlG~G2), (7.34/

receives nonvanishing contributions only from terms with odd values of kl + k 2 - k. This important selection rule ensures the preservation of frame-reversal symmetry, as shown by the following considerations. As we have seen following (6.361, Hermitian unit operators of degree k have parity (-1)k under frame reversal. Accordingly the combined parity of each term on the righthand side of (7.34) is ( - 1 ) kl+k2. The requirement that kl + k2 -- k be odd seems to reverse this parity but in fact preserves it, because the commutator GIG2 - G2G1 is anti-Hermitian, being normally made Hermitian by multiplication with the imaginary unit. This conservation of parity is reflected in the Schrhdinger equation, as shown below. A corresponding result holds when commuting the unit operators of Section 6.4. Application of (7.32) to such a commutator requires reducing

7.3. P r o d u c t s of O p e r a t o r s

181

each product at the outset. We have then

Ijjtklql))ljjtk2q2))- Ijjtk2q2))ljjtklql)) = ~

{l(JJt)kl(Jjt)k2kq))- I(jjt)k2(jjt)klkq)) } (klk2kqlklql,k2q2)

kq

=

Zl(jjt)kq))(_l)2j+k(2kl+l)89189

]gl ]g2 ]g } J J J x [1 - ( - 1 ) k~+k~-k] (klk2kq[klql, k2q2).

kq

(7.35) Here again only terms with odd values of kl % k 2 - k yield a nonvanishing contribution. In contrast to j - ~, 1 where the commutators of any two Pauli matrices ~7 are reexpressed in terms of a single ~ as in (2.33), the commutators (7.34) and (7.35) for j > 1 involve a linear combination of many {kq} matrices of the set. The result that only odd values of kl + k2 - k occur in the expansions in (7.34) and (7.35) has an important implication for the structure of the U(2j + 1) group: Its infinitesimal operators with odd values of k form a subgroup, since for odd values of both kl and k2 the commutator (7.35) yields an expansion into unit operators that contains only odd values of k. This is the subgroup of infinitesimal operators that are odd under frame reversal (see p. 149) and become anti-Hermitian when transformed to the Cartesian base. It constitutes the orthogonal subgroup S0(2j+1) of U ( 2 j + 1) for integer values of j and the "symplectic" subgroup Sp(2j + 1) for halfinteger j, as we shall see in Chapter 8. The same argument shows that the operators with k = 1 form the subgroup corresponding to rotations of physical space, since (7.35) has nonzero terms only with k = 1 when k l = k 2 = 1. 7.3.4

SchrSdinger

equation

f o r a (2j + 1 ) - l e v e l s y s t e m

The formulation of this equation was postponed in Chapter 6 pending the evaluation of operator products. The equation is now obtained by straightforward application of (7.34), with the operator form given in (7.26). For the density matrix p we use the analog of the expansion (6.20a) for arbitrary j, setting the Hamiltonian as in (7.33),

H - ~

Ijjtklql))(2kl + 1)- 89(Jllh(kl)[[j)(klql]H).

(7.36)

klql Using these expansions, and the expression (7.34) of the commutator H p pH, both sides of the SchrSdinger equation are represented as expansions

182

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

in the unit operators Ijjtkq)). The equation thus resolves into separate equations for the coefficients of each operator, namely,

d((jjtkqlp))/dt-

- i ( - 1 ) 2~+k ~-~(2k2 + 1) 89{ kl

x [ 1 - ( - 1 ) kl+~2-k]

klk2

J

k2

k }

J

J

(Jllh(k,)llj)

Z(klk2kqlklql,k2q2)(klqllH)((jj?k~q2lp)). qlq2

(7.37) This equation represents the time derivative of each state multipole ((jjtk~q21p)) as a linear combination of all state multipoles ((jjtk2q21p)), thus extending the precession equation (6.14) for spin 89where only one multipole (k = 1) occurs. Note also that, consistently with Section 6.4.3, the highest multipole k is restricted to 2j as reflected in one of the triangular relations of the 6-j coefficient. Equation (7.37) possesses characteristic symmetries because its righthand side is restricted to odd values of kl + k2 - k, and because of the odd symmetry of the Wigner coefficient. These symmetries display conservation rules that are implicit in the Schrbdinger equation, namely: a) The total squared magnitude of all state multipoles, Ekq is conserved in the course of time, as one would expect.

[((jjt kqlp))12,

b) The Hamiltonian terms of degree kl = 1, which represent the action of a uniform magnetic field upon the system, induce only a simple precession of the particle. Indeed for kl -- 1, odd values of kl + k 2 - k occur only for k2 = k with a Wigner coefficient proportional to the matrix element (kqlJql [kq2) of infinitesimal rotations of coordinate axes. This result expresses the familiar proportionality between magnetic moment and angular momentum. c) All terms of the Hamiltonian with odd values of kl, which represent magnetic field actions upon the particle, induce variations of state multipoles of odd (magnetic) or even (electric) parity which are linear functions of state multipoles of the same parity, odd or even respectively. That is, magnetic actions of any order do not intermix magnetic with electric properties of a system. This conservation law reflects the symmetry of the Schrbdinger equation under time reversal if one recalls the equivalence of the frame reversal and time reversal operations for the dynamics of a particle with given spin. The time

7.4. Combining Operators of Different Systems

183

derivative of a state multipole of even (odd) parity results from a magnetic action--odd under time reversal--upon state multipoles of even (odd) parity.

d)

The complementary law also holds: All terms of the Hamiltonian with even values of kl, which represent electric field actions, induce time derivatives of state multipoles of either parity by acting on multipoles of opposite parity. That is, electric actions of any order intermix electric and magnetic properties of a particle. Thus, for example, the action of a nonuniform electric field upon the electric quadrupole moment of a particle alters the magnitude of the particle's magnetic dipole moment and of its mean angular momentum {J). This phenomenon has been demonstrated experimentally by Lombardi [28].

These considerations of parity and time reversal symmetry also suffice to explain why elementary objects, such as electrons, protons, or neutrons, possess magnetic but no electric dipole moments according to present evidence. Such a moment must be proportional to the spin (ar) of the particle, which is the only vector object available for an elementary system. However, an electric dipole moment (er-] behaves oppositely to J under parity or time reversal, thus being ruled out so long as these symmetries hold. The quest for increasingly stringent experimental upper limits on the electric dipole moment of a neutron is aimed at testing time reversal invariance of weak interactions (invariance under space parity alone is already known to be violated in these interactions that couple it to charge symmetry). Notice finally that these symmetries are not restricted to the equation for a spin-j model even though its Hamiltonian may not involve electric or magnetic fields. It is the parities of the operators under space and time reversal that are relevant to the symmetries. These parities must of course be considered when specifying the mapping.

7.4

Combining Operators of Different Systems

We have dealt in Section 7.3 with products of unit operators constructed from ket and bra symbols of a single system, such that a bra-ket combination (~jm I I~'j'm') implies projection of two states onto one another. We consider now products of operators pertaining to different systems, for instance, two different particles or orbital and spin variables of the same

Chapter

184

7. R e c o u p l i n g T r a n s f o r m a t i o n s

particle. In this case a bra symbol of one operator does not project onto the following ket of the other one as it did in (7.27) and (7.28). This type of product occurs whenever one combines different systems. The "product" of two operators pertaining to separate systems amounts to a single operator of the combined system and its construction involves no "contraction" analogous to (7.28). To construct standard unit operators of the combined system one reduces products of states and products of operators separately. These operations take place in the Hilbert as well as in the Liouville space of the combined system, respectively. Let us indicate ket symbols of the two subsystems by Ijim1) and Ij2m2), respectively, with the understanding that a bra (j~m~ll does not project on a following ket Ij2m2). The direct product of one standard unit operator of system number 1 with one of system number 2 is defined as

9 ",1 k~q~)) 132~2 , )j'-m', ljim1 )(j[m I [(-- 1 1

Ijlj~tklql)) ---tnlrn

,

,

,

(jl m l ,31 - m l IJ131 ]r ql )

! 1

x

Ij2m2)(j;m2l' ( -

1)J;-m'

, 9 ~ (j2m2,j2., -m2lJ2J2k2q2).(7.38)

m2m; This expression includes direct products of pairs of tensorial sets in three different spaces: the product Ijlml)lj2m2) in the Hilbert space of kets, the product (j~m~ll(j~m~l in the corresponding bra space, and the product of operators Iklql))lk2q2))in the Liouville space. Its reduction need not be performed simultaneously in the Liouville and/or in the Hilbert space, but the kets and bras are usually dealt with together. The reduction of the direct product in Liouville space on the left-hand side of (7.38) yields sets of standard unit operators of the combined system,

I(JlJ~t)kl(j2J'2?)k2kq)) - ~

IJlJ~ tklql))lJ2J'2 t k2q2))(klql, k2q2lklk2kq) .

qxq2

(7.39) Reduction in the Hilbert spaces yields instead eigenstates of the total squared angular momentum lJ-1 + J'2l 2 represented by kets Ijlj2jm) and "1 "1 "1 bras (31323 m I I. From these combined eigenstates one constructs the standard unit operators

[(jlj2)j(j~j~)j'tkq)) - ~

IJlj2jm)(j~J~J'm'l(-1)J'-m'(J m, j' -m'ljj'kq) .

mm I

(7.40)

of Different Systems

7.4. C o m b i n i n g O p e r a t o r s

185

We thus have two sets of unit operators, with the same r-transformations and spanning the same portion of the Liouville space. The operators (7.39) are classified by the pair of indices (klk2), the operators (7.40) by the pair (jj'). The unit operators of either set can be expanded into linear combinations of the operators of the other set. The combined coefficients are obtained by projecting the two sets onto one another--"projection" means here "trace of the product" taken in both spaces number 1 and number 2. They are readily seen to be elements of the recoupling transformation ((jlj2)j(j~

j'~)j' I(J~J~)kl (j2j~ )k2 )(k),

(7.41)

which belongs to the class (7.10). The recoupling transformation performs here a change of base in the Liouville space of operators, whereas the transformation (7.10) from LS- to jj-coupling performs a change of base in the Hilbert space of wave functions; however, the two transformations perform equivalent mathematical roles. Applications of this recoupling occur mainly in the following context. Operators G of the combined system which represent, for example, the interaction of its two parts are generally expanded into operators of the (k~) and g2q~ The characteristics of these separate subsystems, such as glq~ separate operators are then embodied in their respective reduced matrices (Jxllgl(k,)llJ[) and (J2llg2(k;)llj~). On the other hand, the study of the combined systems is appropriately conducted in terms of its angular momentum eigenstates Ijlj2jm) and of the corresponding standard operators (7.40). This study requires reduced matrix elements (j~j2jllG(k)llj'lj~j') of the operator G~k) defined by

(kl)"(k2)(klql G~k) -- E glql ~2q2 qlq2

k2q2lklk2kq),

(7.42)

and

G - 6162 - Z e~ k) Y~ (klk2kqlklql' kq qlq2

k~q2)(klqllG1)(k2q2lG2).

(7.42a)

The reduced matrix elements of G (k) result from comparing the expansions of G into the two operator sets (7.39) and (7.40) as represented by

,.,., 9

(JxJ2JllG(k)ll3~J~3 ) - ~

klk2

[

(2k-I-1)

(2kl + 1)(2k2 + 1)

x ((JxJ2)J(J~J'2)J'l(JxJ~)kl(j2j;)k2)

(k)

] 89

(jx IIg~')llJ~)(J211g~k~)llJ'2). (7.43)

186

Chapter 7. Recoupling Transformations

An analogous formula holds for combining the state multipoles of two component systems merging in a collision or absorbing radiation, (((jlj2)j(j~j~2)j 't kq]p,p2)) -

E

((J'J2)J(J~J~2)J'l(JlJ~)kl(J2J~2)k2)(k)

klk2

• E

(klk2kqlklql'k2q2)((Jlj[tklqllPl))((j2j~tklqllp2))

.(7.44)

qxq2

7.5

Illustrations

The transformations of operator products described in Sections 7.3 and 7.4 find very extensive applications. Products of operators of the same system (Section 7.3) are straightforward needing no further discussion, as (7.33) provides essentially the reduced matrix of a product in terms of the reduced matrices of its factors. On the other hand, the combination of two systems into a single one (Section 7.4) and its inverse process--namely, the break-up of one system into two, as in the emission of radiation--present a great variety of aspects. Extrinsic differences among such processes may then obscure their common element, namely, the relevance of a change of base in the Liouville space. We aim here at illustrating this relevance and identifying its specific function through several examples. Actually it is not uncommon for a single treatment to utilize a sequence of transformations (7.43) that play quite different roles. The main roles of recoupling can be outlined with reference to the familiar semiclassical model of adding spin and orbital angular momenta (Fig. 7.2): a) The interaction between two parts of a system characterized, for example, by the orientation of their angular momenta L and S that depends on the angle 0 between these vectors. In the simplest case of dipole interaction, this dependence is represented by 2 L . S - J ( J + I) - L ( L + I) - S ( S + I) ,

(7.45)

but multipole interactions also occur, especially interactions that do not commute with L or S. Calculation of such an interaction centers then on a recoupling transformation such as appears in (7.43). b) The joint precession of/~ and S about f averages out to some extent the interaction of either of the two systems with an external field, even though their angle 0 remains invariant. The effective average

7.5. I l l u s t r a t i o n s

187

/ '\ J \ \

J

F i g u r e 7.2: Coupled spin and orbital angular momentum of L or S--or of any property represented schematically by these vectors--amounts to its projection on J, shown in Fig. 7.2. Here again the projection is calculated by a simple vector formula in the familiar example of the Zeeman effect but a recoupling transformation is generally appropriate. Repeated projections may be required, for example, when f~ and S themselves result from the vector addition of the momenta of several particles. c) Correlations between variables of the two systems represented by /~ and S are observed when both systems emit radiations whose correlation depends on the angle 0 in Fig. 7.2, with other relevant circumstances mostly taken into account by recoupling transformations. The following paragraphs describe particular examples of these various classes in some detail. 7.5.1

Interaction

matrix

elements

As prototype examples, we consider the fine or hyperfine interaction operators, L - S or 1. J. Their transformation to the frame of total squared angular momenta, IJ-/2 or is performed in elementary manner by vector identities of the type (7.45) without any reference to (7.43). However, the right-hand side of (7.45) is in fact proportional to the 6-j coefficient { 1 1 1 } S S J to which the transformation matrix of (7.43) reduces for

188

Chapter 7. Recoupling Transformations

jl = j~ - L, j2 - j~ - S, j - j' - J, kl - k2 - 1, and k - 0. The elementary analog of (7.45) for quadrupole coupling (that is, for kl = k2 = 2), developed by Casimir, is rather more complicated. The full formula (7.43) becomes essential for still higher values of kl and ks, but especially when the relevant operators are not diagonal in the pairs of quantum number (jlj~) and/or (j2j~). One may recall that the normalized product f~.S~/[L(L+ 1)S(S+ 1)] ~ is familiarly interpreted as representing the cosine of the angle between the "vectors" f~ and S with resultant f ; the 6-j coefficient{ L S S

Jk } is similarly interpreted as a Legendre p~176

Pk(cos 0) to which it indeed becomes proportional for large values of L, S, and J : L

L

k }

S S J

~- (-

)L

+s+J[(2L+ I)(2S+ l)]- 89

(7.46)

An important application of (7.43) evaluates matrix elements of the electrostatic interaction (6.49) for a pair of charged particles with orbital quantum numbers (gl,t~) and (t~2,g~), respectively. The interaction operator is a scalar (k = 0, ky = k2) with a tensorial expansion represented by (6.49). Equation (7.43) then takes the explicit form

E(L)-

(lflg2L]J

1

(--1)ti+t'~+L(2L + 1) 89 • E i(uk

+

(7.47)

kl} t~2

L

'

whose coefficients X are called "Slater integrals":

xk(ele2,elt;) - (elllC( )lle1)(e211c(k)lle;) x

r12dr1

r22dr2r< k r>(k+l)Rtl(rl)Rt2(r2)Rt,(rl)Rt,2(r2).

(7.47a) Note how the expression (7.47a) fails to factor out completely into reduced matrix elements of operators g~k,) and g~k2), through nonseparability of the radial integrals. The matrix element (7.47) is generally complemented by an exchange term differing from it by permutation of the quantum numbers gl and ~ and of the radial functions bearing those labels. In contrast to the simple dependence (7.45) of the spin-orbit coupling upon J, the dependence (7.47) of the interaction energy on L is rarely simple owing to prevalence of quadrupole terms with kl = 2 and of even

7.5. I l l u s t r a t i o n s

189

higher multipole terms. On the other hand, the contributions of different multipole orders kl to the whole expression (7.47) can be separated out from experimental values E ( L ) of its left-hand side for different values of L. The procedure utilizes the orthonormality of 6-j coefficients as in (7.17), applied to the 6-j coefficient in (7.47) by constructing the expansion

L

~2

g2

L _

:

=

1

(--1)1', +12(2k + 1)- 89

(7.48) Note also that the matrix element (7.47) seldom occurs by itself in a calculation, being usually accompanied by other transformation matrix elements that separate the orbital motion of the two particles from their spins and/or from their coupling to other particles of the same system. The presence of many particles in the same shell will be treated in Chapter 8.

7.5.2

Projection

of operators

As a prototype of the processes to be considered here, we take again one treated familiarly in the anomalous Zeeman effect by elementary considerations. The difference between the gyromagnetic ratios of spin and orbital currents requires us to project their angular momenta separately on the net total momentum f~ + S - J. The elementary procedure utilizes vector identities averaged over the precessional motion of S (or L) about the constant J, namely, -

(S.JJ-(SxJ)

-

!2 [ y ( y +

1)-

x J)

S(S +

1)-

L(L +

1)]{Y) ~

(7.49)

yielding the Land~ factor. The expression on the right of (7.49) is again proportional to a 6-j coefficient, in this case { SJ

SJ

L1 } ' thatsymmetrizes

the recoupling matrix element on the right-hand side of (7.43) upon setting jl-j~-S, j2-j~L,j-j'J, k - k 1 1, a n d k 2 - 0 . Note the implication of setting k2 - 0: The orbital motion affects the magnitude of (S) only by inducing the precession of ff about f but remains otherwise a spectator.

190

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

In the Zeeman effect and in analogous phenomena, the projection of a relevant operator measures the effective interaction of a multipole m o m e n t ~ the spin in our case--with a magnetic field external to the atomic system. In other phenomena, the corresponding projection represents the interaction of an entire precessing system with another portion of the same atomic system. Thus, for example, the hyperfine interaction between atomic electrons and a nucleus results from the electric and magnetic field generated by the electrons at the nucleus; each electron's contribution to each multipole component of the field must be averaged over the electron's precession about the total momentum f of all atomic electrons. For example, Eq. (7.43) is applied here setting jl - j~ equal to the relevant orbital or spin momentum of the electron under consideration and j2 - j~ equal to the net momentum of all other electron motions, then once again setting j - j' - J, kl - k equal to the multipolarity of interest, and finally ks - 0 . Spin-orbit interactions act primarily between the spin and orbit of each atomic electron taking a definite value for electrons with quantum numbers f2 _ 1~'+ s-j2, being evaluated appropriately for electron groups treated in jj-coupling. The electrostatic interaction (7.47) among several electrons, on the other hand, generally predominates over their spin-orbit couplings and is thus treated appropriately in LS-coupling. Here again it becomes necessary to project the ['and g"of each electron onto the combined/~ and S of the electrons. Projection of dipole, or other multipole, operators is also prominent in the calculation of the relative line intensities of the spectral lines in "transition arrays" between the manifold levels of two configurations. The configurations differ generally only by the orbital motion of a single electron which emits or absorbs light. It is the change of orbital motion which gives rise to the total line s t r e n g t h . However, an electron's change of orbital motion is generally accompanied by various changes of its interaction with other electrons or with the nucleus with resultant splitting of the transition's initial and final levels. The transition strength is thereby distributed among numerous lines. Parceling out of the total strength among these lines results from projecting the radiating 2~-pole moment onto the moment associated with the transition from one configuration 's level, with angular momentum j, to a particular level j' of the other; the strength of each line is thus specifically proportional to the square of the projected matrix element (7.43). This class of projections finds a more varied application in beam-foil spectroscopy, where observations are conducted with high time resolution at

7.5. I l l u s t r a t i o n s

191

the expense of spectral resolution. In this event the observed light intensity, or analogous quantity, depends on the coherent superposition of energy eigenstates with different angular momenta j and/or j~, being proportional to a sum of terms derived from (7.43)

[((JlJ2)JlJ~J;)J'{(JlJ'l)k(J2J'2)O)(k)(Jlllg (k)~IlJ~)]2

cos(wjj,t) jj'

(7.50)

Here wjj, - (Ej - E j , ) / h indicates an oscillation frequency that may be observable electronically if jl and j~ belong to the same configuration, but lies instead in the optical range for different configurations. In the case of a single configuration, Eq. (7.50) represents the Fourier analysis of a quasistatic squared 2k-pole parameter of the configuration; the time dependence of 2k-poles of an excited atom has been thus analyzed by Fano and Macek [33]. In the case of a single configuration, the beats of different terms of (7.50) are observable in the time domain. 7.5.3

Correlations

Two quantum-mechanical quantities are "correlated" when the mean value of their product, as determined by a single experiment, differs from the product of their mean values as determined by separate experiments. (Correlations are particularly striking when the separate mean values vanish but the mean value of the product does not.) Correlations between the states of two components of a single system, number 1 and number 2, are displayed by representing their joint density matrix as mean values of products of unit multipole operators of the two component systems. These mean values are obtained by inverting (7.44) into the form 9!

.

.!

((JlJ1?klql,3232

k2q21p)) - Z ( k l q l , k 2 q 2 ] k l k 2 k q ) kq

• ((jljl)kl(j2j'2)k2[(j~j2)j(jlj~)j')(k)(((jlj2)j(jlj'2)j'tkq[p)).

(7.51) Whereas this equation and (7.44) represent formally reciprocal transformations, they pertain to characteristically different circumstances. Equation (7.44) pertains to the formation of a combined system, the direct product of px and p~ representing thus an absence of correlations. Equation (7.51) pertains instead to the analysis of a system whose density matrix need not necessarily factor out into px and p2.

192

Chapter

7. R e c o u p l i n g T r a n s f o r m a t i o n s

Performing a separate measurement on the component system number 1 alone amounts to performing also on system number 2 a trivial measurement of its unit operator, thus determining the value of a state multipole (7.51) with k2 - 0. Similarly, measurements on number 2 alone determine the values of state multipoles with kl - 0. Correlations are displayed in this representation by nonzero departures of the expression (7.51), with nonzero kl and k2, from the products of values obtained with either k2 - 0 or kl - 0,

C(klql,k2q2) - ((j131"tk 1qz , j2j~tk2q21p)) - ((jzj~tklqi , j~j2t00[p)) x (2j~ + 1) 89 ((j~j~OO, j2j'2k2q2lp))(2jl + 1) 89 . (7.52) Extensive studies have been performed, experimentally and theoretically, on the angular correlation of particles or photons emitted in atomic, nuclear, or particle processes. The probability of detecting one particle with momentum P1 and another with momentum P2 ejected from a system in the state (7.51) is given by the inner product of operators representing the two detectors and of the density matrix (7.51). Detector operators Dp~ and Dp2 are Hermitian conjugates of the density matrix (6.53) of particles incident from a collimated beam, as noted at the end of Chapter 6. Each of these operators is suitably constructed in a frame with its polar axis attached to the detector, just as the matrix (6.53) was constructed with reference to a coordinate axis parallel to t3. Accordingly, the construction of inner products of operator and density matrix sets must be accompanied by appropriate r-transformations such as appear in (4.24a). The resulting structure of the probability of joint detection of the two particles is then , -(k~) [((Dp, [jljltklql))Dq, q, (~)1,~1 , ~1 )]

J

jkq

9 .i .,,,,D(k~) • [((D 132j2k2 2, q;q

] ((j j .i k q ,j2j2k2q21p)), 9 .1

(7.53) where the Euler angles {r 0i, ~i } identify the orientation of the coordinate frames attached to /9/ with respect to the frame of the state multipoles. Nonzero indices q~ of the detector operators occur when the detectors analyze the particle spins. [Spin had been disregarded for simplicity in (6.53).] The construction of angular correlation functions such as (7.53) has been an important motivation for developing the techniques presented in this book. Variants of the general (7.53) apply to different experimental circumstances. Notice first that the problem was formulated here in terms of

7.5. I l l u s t r a t i o n s

193

emission of two particles from a collision complex, for simplicity, but (7.53) actually pertains to detection of one particle and of the whole residue of the complex. If two particles were actually emitted leaving behind a residue, the density matrix (7.51) of the initial complex should be represented in terms of three components rather than two, namely, the two emitted particles plus the residue; the residue's variables should be averaged out by setting k,.e8 = 0, if unobserved. Alternatively, the state representation (7.51) might be regarded as concerning the state of the emitted particles preaveraged over the unobserved variables of the residue. Another aspect of the phenomenon concerns the initial state of the collision complex, which generally results from the impact of an incident particle (or radiation) on a target. The target is often, though not necessarily, unpolarized; in this event, each of the state multipoles on the right-hand side of (7.51) is proportional to the corresponding multipole of the incident beam given by (6.53) or by its variant appropriate to the specific problem. (The proportionality factors consist of rotationally invariant elements of the complex's transition matrix.) When this circumstance is reflected explicitly in the density matrix of (7.53), this equation also represents the angular distribution of the ejected particles with respect to the direction of incidence. As a last example, we consider a correlation problem for which numerical data are readily obtained by direct evaluation of a recoupling coefficient. It concerns the correlations between the orientation of nuclear and electron spins in one pure state of the hyperfine doublet ground state of Na, 3s!. 2 The nucleus has spin I - 3 and we consider here the hyperfine component state with F = 1 and M = 1. Accordingly we enter on the right-hand side of (7.51) jl - j l - I - 3, j 2 - j ~ 89 j - j ' F - 1. The state multipoles of the whole atom on the right-hand side of (7.51) are nonzero for k = 0, 1, 2, since F = 1, and for q = 0 only, owing to the axial symmetry of the state M = 1 due to its being an eigenstate of Jz. Their values are given by (6.28) as the elements in the upper left-hand corner ( i = m' = 1) of the three matrices with q = 0,

k

( ( ( 3~1 ) 1 ( ~3)1 1

kOIp))

0

1

V~ (7.54)

194

Chapter

7. R e c o u p l i n g

Transformations

These multipole components are multiplied in (7.51) by three separate recoupling matrices, one for each value of k, but of each transformation we need only the single column with j = j' = 1, whose elements are

kl

k2

0 0 1 2 1 2 3

0 1 0 0 1 1 1

k =

0

1

2

(3/8)~ -1/4 5 89 (5/8) 89

(3/16) 89 -(1/40) 89 0 (63/80) 89

0

(5/s) 89

(7.5s)

Zero entries occur in this table for odd values of kl + k2 + k owing to symmetry of the matrix under permutation of the primed and unprimed variables which coincide in our case. Blank spaces occur for values of ( k l k 2 k ) that do not fulfill the triangular conditions. Inserting on the right-hand side of (7.51) the appropriate entries from (7.54) and (7.55) together with the values of the Wigner coefficients, we obtain on the left-hand side the mean values of unit-multipole products, 33~ 11t ((5 klql, 25 k2q2[P))

lgl

ql

k2

q2

o

o

o

o

o

o

1

o

0 1 1

0 0 :kl

1 0 0

+1 0 0

2

o

o

o

2 1

+1 +1

0 1

0 :F1

1

0

1

0

2

:t:1

1

:F1

2

0

1

0

_(1) 89

1

:F1

1

0

3 (~)(~) 89 3 3 89 -(~)(g)

3 3

• 0

(~)~ -(~) 0 ( ~1) ( ~5) 89 0

(~) 0 (1~)(~)~ 5

1 - (~)(~)~ __(1~)(~) 5 89- (~)(~) 1 89 1 (~)(3)~

(7.56)

7.5. I l l u s t r a t i o n s

195

The correlations (7.52) are accordingly found to be

k:

q:

k2

q2

i

-I-i

1

:FI

2

+:

:

::

2

0

I

0

3

+1

1

:i

3

0

i

0

C(klql,k2q2) 1 5 (~)(~) 1

1 89 -- (-i~)(1A-d) 89

5 i 3 : (:1(:):

-(:1

-I-(:)

- -(:)

3 3 -(:)(:)~

(7.57)

Problems 7.1 Calculate the angular (0, ~) and spin wave functions of a pair of electrons in a p and a d state with total angular momentum J = 1 a) in LS-coupling b) in j j-coupling c) Write down explicitly the recoupling transformation matrix that relates the wave functions in the two alternative coupling schemes. 7.2 The spin-orbit interaction for a pd state is given by Gpgl "sl -}-Gdg2"[~, where Gp and Gd involve radial integrals. Calculate the matrices of spin-orbit energy between pd J = 1 states in both cases (a) and (b) of Problem 7.1. Show how these matrices are related by the transformation matrix (c). 7.3 The spin-orbit interaction for an atomic electron involves the matrix element (gsjm]i. ~gsjm), which may be evaluated by either of two procedures. One relies on (7.45) to cast the interaction in terms of diagonal operators, whereas the second utilizes (7.43) and (6.45). Show that both methods give the same final result. 7.4 During a process of (single) photodetachment or photoionization, the released electron is best described initially in LS-coupling while remaining close to the residual atom but finally in j j-coupling after

196

C h a p t e r 7. R e c o u p l i n g T r a n s f o r m a t i o n s

moving far away toward a detector. Consider a closed shell atom or negative ion with noble gas configuration p6 1S0 and the J = 1 states formed upon photoabsorption, with the outgoing photoelectron in either an s or a d wave. Enumerate the J = 1 states of photoelectron plus residue in LS- and jj-coupling and work out the LS --~ j j transformation matrix. 7.5 The magnetic moment of a deuteron includes contributions from both orbital and spin currents f i - (e/2Mc)(Lp + gpSp + gnSn). Viewing the deuteron as a triplet spin symmetric combination of the neutron and proton, fi can be cast as --'

- ( ~ p + ~ . ) s + ~ ""

1~

,

where L and S are the deuteron's orbital and spin angular momenta, respectively, and /zv = 2.7925, # , = -1.9128 in units of the nuclear magneton. a) Using expressions such as (7.49), work out the magnetic moment for different values of L and S. b) Show that the simplest choice compatible with J = 1, namely L = 0 and S = 1, gives Pd - - P p + Pn, thus failing to account for the observed Pd - - 0.8574. c) The deuteron has even parity, restricting a possible admixture to L = 2, S = 1. Show that a 4% admixture of d wave accounts for the observed Pd. 7.6 Enumerate the Slater integrals X in (7.47) that occur in evaluating the electrostatic interaction between two electrons in the following alternative configurations: a) p2 b) pd c) d 2 d) f2 e) df 7.7

a) Draw a diagram for the recoupling matrix element in (7.20) (which is essentially a 9-j symbol) ((jl j2 )jl 2 (j3j4)j34 {(jl j3 )j13(j2j4 )j24 )(J). b) Set j2 - 0 and reduce a) diagrammatically to its form

6j lj 126j,j2, (jl (j3j4)j34 {(jx j3 )jl 3j4)U), which is a diagrammatic rendering of (7.22).

7.5. Illustrations 7.8

197

a) According to property (a) on p. 171,

{ jl j2 j3 }_ { j2 j3 jl }_ { j2 jl j3 } Identify the symmetry operations on the diagram (7.14) that correspond to these permutations. b) What symmetry operation on the tetrahedron expresses property (b) on p. 171: {Jill

j2t2 J3t l }~- - {3

jl

~2 ~3 j2 j3 } ?

7.9 Draw the diagram of the 5-set recoupling (( (jl j2 )j12 (j3j4)j34 )j1234j5 I((jlj3)jl3(j2j5

)j25 )jl 3:~5j4)(J)

This 12-j coefficient diagram can be cast as an octagon with four internal "diagonals," each of them running from a vertex to another vertex three steps away from it. 7.10 Show, by analytical and graphical manipulation, how to resolve the recoupling matrix (7.8) into a sum of products analogous to that shown in (7.11b)and (7.12). Compare your result with (7.8) and derive from this comparison a "Biedenharn-Elliott" identity satisfied by 6-j coefficients (Appendix I of ref. [1]).

This Page Intentionally Left Blank

Chapter 8

P a r t i a l l y F i l l e d Shells o f A t o m s or N u c l e i Atomic or nuclear many-particle wave functions that are eigenfunctions of the squared angular momentum have been constructed in Chapter 5 through simple addition of angular momenta, starting from independent particle functions. This general procedure requires a major adaptation to the constraints of the exclusion principle when the radial wave functions of the various particles are equal rather than orthogonal. The full manyparticle wave function must change sign upon interchange of the angular momentum and spin labels for any pair of identical particles. Exclusion thus restricts the kind and number of states of identical particles in a system (see, for example, Chapter 15 of [2]). Its application to electrons or nucleons in a partially filled shell has presented a major challenge to theory. This problem was approached in the 1930s by elementary procedures, with progress confined to systems with only a very few particles (or vacancies) in an unfilled shell. Essential progress was achieved in the 1940s by Racah's development of new methods for atoms, extended to nuclei in the 1950s, and slowly but increasingly refined since that time. Judd introduced a new fruitful approach in the 1960s but much may remain to be done to attain greater clarity and elegance, thus completing the solution of the problem. The developments of previous chapters were largely motivated by the problem to be considered now, finding here varied applications and extensions. A qualitative introduction in Section 8.1 includes the appearance of 199

200

C h a p t e r 8. P a r t i a l l y Filled Shells of A t o m s or N u c l e i

new "fractional parentage coefficients" for three identical particles. It is followed in Section 8.2 by an outline of the newer "second quantization" techniques due to Judd [18] and finally by their application in Section 8.3. The discussion will refer mainly to atoms, with connections to the analogous nuclear problem. When three or more identical particles of increasingly large j are coupled together, the increasing multiplicity of states with the same total orbital angular momentum, spin, and parity requires additional labels of "seniority" and "quasi-spin" when g = 2. The f-electrons (g = 3) require even further extensions. In particular, operators that are not contained in the Hamiltonian or commute with it, constituting higher dimensional groups, become necessary for a unique classification. They are called "noninvariance" groups because they do not represent symmetries of the Hamiltonian, each of their representations embracing states with different energy or even different numbers of particles, thus anticipating features to be described in Part C.

8.1

Qualitative Discussion

Theoretical spectroscopy has aimed traditionally at determining the energy eigenvalues of the Hamiltonian for a system of particles, together with the main characteristics of the corresponding eigenfunctions. To this end, one generally constructs the Hamiltonian matrix in an appropriate base and then diagonalizes it. Modern computing can handle the diagonalization rather well. Remarkably, new theoretical concepts are required in the preliminary stages of the problem, just to specify an appropriate base. The construction itself of the Hamiltonian matrix proceeds by rather straightforward application of the methods of Section 7.5, provided one knows the combined angular momentum of the relevant orbital and spin motion for each pair of particles. For identical particles within a shell, all pairs are equivalent but any one pair is embedded in the shell with other particles. The problem thus arises of identifying (or "labeling") the rows and columns of the Hamiltonian matrix, that is, a complete set of base states of a partially filled shell, in a manner suited to govern the admixture of alternative angular momenta for each pair. This construction of base states must, of course, be carried out within the constraint of the exclusion principle. In other words, the problem lies in designing and laying out a suitable set of vectors that span the Hilbert space of a partially filled shell. Since the n particles of a configuration ~ can be distributed among 2(2/?+ 1)

8.1. Qualitative Discussion

201

different spin-orbitals, the number of distinct states, that is, the dimension of the relevant Hilbert space, is (4t-1-2~. There are 15 states of p2 or p4 configurations and 20 states of p3. The desired base states shall be vectors of this dimensionality identified according to Section 1.4.1 as eigenvectors of a maximal set of commuting operators. The operators ISI 2, S~, I/~l2, and Lz belong in such a maximal set but do not complete it when t > 1. Thus, while no duplication of a specific (S, L) occurs in any pn configuration, the same no longer holds for d n states; for example, the d 3 configuration gives rise to two 2D states, calling for additional distinguishing labels. k

n

/

Rotational invariance of the Hamiltonian causes its matrix to be block diagonal in a base of eigenvectors of [J-'l2 and Jz. For atomic applications, one is usually satisfied with assuming initially block-diagonality in the separate orbital and spin variables, that is, in a base of eigenvectors of [/~2[, L~, ISIs, and S~, with appropriate corrections to be introduced later. It is the labeling of rows and columns for each diagonal block which presents a substantial problem for a system of n electrons in a shell with a given orbital quantum number g, even for values of g as low as 2. The value of n is limited by the capacity of the shell, 2(2g + 1). The problem is still trivial for n - 2, but complications set in thereafter for g > 2 as in the example of d 3 2D states cited in the previous paragraph. Much progress stems from considering the problem simultaneously for all values of n < 2(2g + 1); one observes a particle-hole symmetry, that is, a correspondence between base states with n and with 2(2g + 1 ) - n particles. Thus, two distinct 2D states also occur in the d z configuration, just as they do for d 3. This symmetry, noted by Racah, has been utilized more extensively by Judd [34]. Considering simultaneously states of different n introduces a new philosophical element of enormous importance. We are no longer constraining ourselves to look at symmetries of a Hamiltonian or a system. Freedom to consider several systems together, with varying n, introduces novel symmetry elements among which particle-hole symmetry is but one noteworthy example. Relativistic quantum mechanics and field theories invoke naturally these particle-hole aspects, while more recent explorations of supersymmetry go further in considering mixed systems of fermions and bosons. Such extensions enlarge the role of symmetries in physics, much as statistical mechanics stepped from canonical (with a fixed number N of particles) to grand canonical (with N fluctuating about some average) ensembles enlarging the domain of classical thermodynamics. Part C will elaborate further on this enlarged view of symmetries arising here.

202

8.1.1

Chapter 8. P a r t i a l l y Filled Shells of A t o m s or Nuclei Two-particle states

In the case of two identical fermion particles in a central field, with the same radial and orbital quantum numbers (ng), one proceeds simply by reducing the products of their orbital and spin states, that is, by adding their orbital and spin momenta, either in LS- or in jj-coupling. The symmetry of irreducible products under permutation of their factors implies that permutation of particles in the orbital and spin states of a fermion pair multiplies its reduced state by ( - 1 ) L+S+I in LS-coupling or by ( - 1 ) J+l in j j-coupling. Antisymmetry is thus secured by restricting attention to states with L + S = even,

(LS-coupling),

(8.1)

J = even,

(jj-coupling) .

(8.1a)

That is, exclusion simply selects a subset of the states of two equivalent particles constructed without regard to their being identical. The 36 states of two p electrons, described as 1S, 3S, 1p, 3p, 1D, and 3D, reduce thus to the 15 states 1S, 3p, and 1D that survive when the two electrons share identical (nl) quantum numbers. In nuclear problems, the isotopic spin parameter of each nucleon, isomorphic to its spin, must also be included besides the orbital and spin variables; that is, a quantum number T appears with S and L. For this reason, no triplet-spin analog of the deuteron (np with T = 0) with S = 1, L = 0 occurs in the dineutron (nn) or diproton (pp) systems with T = 1.

8.1.2

S t a t e s of t h r e e or m o r e e q u i v a l e n t particles

The construction of states of three equivalent particles may begin by adding the spin and orbital angular momenta of a third particle to those of the first two particles previously antisymmetrized by selection according to (8.1) or (8.1a). We indicate an initial set of LS-coupled states of two equivalent electrons, with spin and orbital quantum numbers s and g, by ](82)S)[(g2)L), with an even value of L + S. The initial set of states obtained by adding to this set the angular momenta of a third equivalent electron will then be indicated by I(s2)SsS) I(g~)LgL). (8.2) These states are not antisymmetric under permutation of the third particle with either of the first two. One should then construct fully antisymmetric states superposing initial sets (8.2) with different (S, L) and equal (S, L).

8.1. Q u a l i t a t i v e Discussion

203

We indicate here some aspects of this problem, deferring its solution to Section 8.3. The antisymmetric superpositions of the initial sets (8.2) are usually indicated by

I}g3o~SL) - ~ _

I(s2)SsS)I(g2)LgL)(g2SL~.I}g3~SL) ,

(8.3)

_

SL

with the sum restricted to S + L - even. The coefficients of this superposition are called coefficients of f r a c t i o n a l p a r e n t a g e , or "cfp," because they represent the fraction of each set of "parent" states ](s2)S)I(/2)L) included in the antisymmetric states ]}g3~SL). The cfp pertain to superpositions of irreducible sets being accordingly independent of M quantum numbers. Equation (8.3) introduces the characteristic notation I} that signifies an antisymmetrized state. The index c~ represents any of the additional quantum numbers generally required to distinguish the different antisymmetric states of a configuration tm in addition to (SMsLML). Note that theoretical spectroscopy normally indicates the spin quantum number S on the left of L, the spin being the first factor in the construction of irreducible spin-orbit product states. For a configuration of n equivalent electrons, g~, Eq. (8.3) takes the general form

]}g'~SL) - ~

]g~-16~SLgSL)(g~-l~SL~]}gn~SL). _

(8.4)

_

~SL

The notation for the cfp in (8.3) and (8.4) is standard, while the state symbols vary; the indications of spin s and spin configurations s n are generally omitted. In the case of jj-coupling, Eq. (8.4) takes the form

I}j'~J) - ~ ](jn-l(~]jJ)(j'~-x6~jjJ]}c~J) ; ~J

(8.4a)

j j-coupling is generally appropriate to nuclei but is then also accompanied by a notation referring to isotopic spins. A main feature of the cfp lies in their forming rectangular, rather than square, matrices because the aggregate of antisymmetric states ]}gnaSL) spans a smaller space than the aggregate of nonsymmetric states 1(I}~-I)6~SLgSL) with the same values of (SL). The cfp have dimension (4l+2~ smaller than (4g + 2)(4t+2] Thus the superpositions (8.4) x n jr \n-l/" with all different c~ project the aggregate of nonsymmetric states onto the corresponding smaller subspace of antisymmetric states. In other words,

204

Chapter 8. P a r t i a l l y Filled Shells of A t o m s or N u c l e i

each column of a rectangular matrix of cfp represents an eigenvector, corresponding to the degenerate eigenvalue unity, of a projection operator ,4(SL) that represents antisymmetrization. The problems facing us are then first to construct this operator for all configurations of more than two equivalent particles and then to construct and characterize an orthonormal set of its degenerate eigenvectors. These tasks will be undertaken at the end of the chapter.

8.1.3

Quantum

numbers

for many-particle

states

We outline here the general, group theoretical, point of view for the classification of states of many equivalent particles introduced by Racah. Recall how we classified the elements of standard tensorial sets as eigenvectors of a maximal set of commuting operators, IJ~2 and J~, when dealing with their r-transformations. We deal now with the unitary transformations of all the antisymmetric states of n equivalent electrons resulting from transformations of the single-electron base set Ism,)ltm), or Ijm)in the case of j j-coupling. Infinitesimal operators of these single-electron transformations take the form of the standard unit operators introduced in Chapter 6, namely Isstk~ql))l~tk2q2)) for LS- and Ijjtkq)) for jj-coupling. The goal is now to classify n-electron states as eigenvectors of a maximal set of commuting operators of the relevant group, namely, U(2s 4- 1) • U (2t + 1) for LS- and U ( 2 j + 1) for jj-coupling. Since eigenvectors of Jz are invariant under the subgroup of space coordinate rotations that leave the z axis fixed, we shall classify states as eigenvectors of operators which remain invariant under certain subgroups of U(2s + 1) • U(2t + 1), or of U(2j + 1). Labeling our states by (SL) (or by J) represents already a first step of this program, because the group SO(3) of rotations of space coordinates constitutes a subgroup of the unitary transformations U (3) of single particle states. It is in fact the subgroup whose infinitesimal operators have kindices equal to unity, that is, whose quadratic Casimir operator, as given in (6.34), equals [~q[2, [/,[2, or ill2 for U(2s + 1), U(2t + 1), or U(2j + 1), respectively. [For a shell of nucleons, rather than of electrons, the group of rotations of the isotopic spin t - ~1 combines with the spin and orbital transformations to form a group U(4(2s + 1)) = U(2t + 1) x U(2s + 1) x U(2e + 1). The eigenvalues T(T + 1) of its quadratic Casimir operator 17~12 label states of the shell together with those of I~12 and I/~12.] The next step of the program utilizes a larger subgroup of U(2s + 1) [or of U(2j + 1)]. This subgroup consists of all the transformations C that

8.1. Q u a l i t a t i v e D i s c u s s i o n

205

leave invariant the scalar product of two tensorial sets, defined by (3.33) as (Sljj) (ajl(bj I, with (Sljj) = U =_ Dy(Tr). [When the sets (ajl and (bjl consist of wave functions (jmll), (jm[2) of two particles, their scalar product represents the two-particle state with J = 0, whether in LS- or jj-coupling, to within normalization.] Invariance of the scalar product requires (ajl and U(bjl to transform contragrediently, according to (3.22-24). Contragredience means that U(bj] transforms into C*V(bjl when (ajl ---. C(ajl and U(bj]---+ UC(bjl , that is, it requires that C*U = UC. This amounts to requiring that the finite operations of the subgroup satisfy the same (3.8a) as the r-transformation and hence that the corresponding infinitesimal operators satisfy the same (3.1) as the infinitesimal rotations J, that is, that they are odd under frame reversal. The subgroup of V(2j + 1) with these properties has been identified in Section 7.3.3 as the one with infinitesimal operators of odd degree k. This subgroup is the orthogonal subgroup SO(2j § 1) of U(2j + 1) for integer values of j, specifically for the U(2g + 1) group pertaining to orbital states Igm). For half-integer values of j, which occur in the case of j j-coupling, there exists no Cartesian base with real transformations and the relevant subgroup of U(2j + 1) is called symplectic and indicated by Sp(2j + 1). The distinction between orthogonal and symplectic subgroups relates to the even or odd parity of a scalar product under permutation of its factors. Physically, the invariance of two-particle states with J = 0 under the subgroup's operations serves to sift out of a state I}gnSL) any rotationally invariant component consisting of particles paired off with J = 0. The residue, with quantum numbers (SL), will instead experience nontrivial transformations under the subgroup SO(2j + 1) or Sp(2j + 1). This subdivision of the n particles into those that are or are not paired off with J = 0 will be seen in following sections to depend on the eigenvalue of the subgroup's Casimir operator. This eigenvalue is a quadratic function of an integer quantum number v, much as the If] 2 of space rotation depends quadratically on j. Specifically an eigenfunction of this quadratic Casimir operator labeled I}g'~vSL) consists of 89 - v) pairs with J - 0 and of a residue [}g"SL). In the example of the two 2D states in d 3, one of them can be associated with the one d 2 pair state with S = L = J = 0; upon addition of the residual third d particle, this requirement implies a 2D state. Accordingly, for this state, we have n = 3 and v = 1. The second 2D state without any J = 0 character in its parentage is then assigned v = 3. The number of particles v is thus the lowest one for which there occurs a state

206

Chapter 8. Partially F i l l e d Shells of A t o m s or N u c l e i

that is an eigenvector of I:~12 and ]/~12 with quantum numbers (SL) and of the quadratic Casimir operator with quantum number v. For this reason v is called the s e n i o r i t y q u a n t u m n u m b e r . Classification by the quantum numbers (vSLMs ML) suffices to identify all states of a complete set ]}gnvSL) with ~ - 2 (see Table 8.1). The highest multiplicity, threefold, occurs for the 2D states of the d ~ configuration which can be assigned v - 1, 3, and 5 to distinguish them uniquely. Thus, this single additional label--or, equivalently, the quadratic Casimir operator of the group SO(5)--supplementing (SLMsML) of the two SO(3)'s and SO(2)'s of orbital and spin angular momentum, completes the unique characterization of all d n states (0 < n < 10). The Casimir operators involved are those of the group chain U(5) D SO(5) D SO(3) D SO(2). An alternative rendering of the first two labels, n associated with U(5) and v with SO(5), in terms of more familiar angular momentum language ("quasi-spin" Q) will be considered in Section 8.3.2. For g - 3, that is, for the configurations fn of lanthanides and actinides, further quantum numbers are required. The chain analogous to the one above, namely, U(7) D SO(7) D SO(3) D SO(2), does not suffice here, given the higher multiplicity now involved. Thus, for example, there occur ten 2F and 2D states in the f7 configuration and the analog of seniority from SO(7) can only account for a maximum of four. Indeed, even for n - 3, the twofold multiplicity of 2F can be handled through the SO(7) label by considering one of these states as built on the 1S, J - 0 state of f2 much as we did for states of the d 3 configuration. The twofold multiplicities of 2D, 2G, and 2H of f3 do not yield to such treatment, requiring consideration of an additional subgroup of U(2g + 1) - U(7) and of its orthogonal subgroup SO(7), intermediate between SO(7) and SO(3). This is the subgroup, called G2, whose infinitesimal operators are the standard unit operators with k - 1 and k - 5, thus differing from SO(7) by excluding the octupole operators with k - 3. That the operators with k - 1 and 5 form a group follows from a special property of the commutator (7.35) for j - 3 and for kl - k2 - 5, namely, that its term with k - 3 cancels out. The cancellation occurs because the recoupling matrix element with k - 3 in (7.35) contains the factor { 5- 035 "3 }3

(8.5)

The vanishing of this 6-j coefficient is regarded as "accidental," meaning that it has not been related to other effects thus far. The

Table 8.1: d" configuration terms arranged by seniority index and quasi-spin quantum numbers.

MQ

n

5

10 9

$

207

i s

c

$

2

1

0

1

1s

'

7

f 0

6 5

's

4

's

-1

3

-2

21

_ -i

_ _t

o

2

1

2

0

2

3

4

5

2D

'S

1

_ _1

3

I J 2D

J J

I

2D

J

J -2D

's

I

J

' S ' P ' D 3 D ' D ' F ' F 'G 3G ' H ' I

J

II

J

I '5' ' P ' D ' D ' D ' F ' F 'G 'G ' H ' I J

1J

I J

J

I

I

I p G - G T - 1J

I J 2D

1s

I

[ [

' S ' S ' D * D ' F ' G *G ' I 1

208

Chapter 8. P a r t i a l l y Filled Shells of A t o m s or N u c l e i

orthogonal transformations of the subgroup G2 leave invariant the 4S states, 3 I}f 3 S - ~, L - 0), much as all those of SO(7) leave the g2 1S invariant. Therefore, the classification of invariants of the subgroup G2 relates to the number of particles of a configuration fn that are coupled into subsets of f3 4S. The SO(7) and G2 labels suffice for the complete classification when n < 4 but even further labels are necessary for 5 < n _< 7. Racah's introduction of SO(5), $O(7), and G2, and of the corresponding quadratic Casimir operators of these groups, has been enormously influential in the spectroscopic classification of g - 2 and g - 3 multiparticle systems. These are "noninvariance" groups because the Hamiltonian of the corresponding atom or nucleus only has the rotational SO(3) symmetry and is itself not invariant with respect to the operators of these higher groups. Nevertheless, their introduction and use may be seen, at least in hindsight, as the precursor of the later and very extensive use of noninvariance groups and of "dynamical symmetries" in particle physics and nuclear physics for unique classification of states/particles [35, 36]. The term d y n a m i c a l s y m m e t r y (see the last paragraph of Section 6.3) concerns Hamiltonians expressible in terms of the Casimir operators alone of groups in a chain such as U(N) D . - D SO(3). The Hamiltonian's eigenvalues turn thereby into simple algebraic functions of each subgroup's Casimir operator eigenvalues. Unlike the example of the hydrogen atom at the end of Section 6.3 where a full physical understanding emerges from showing explicitly how each subgroup's generators commute with the Hamiltonian, no similar understanding is available in the present context. Nevertheless, the success of the seniority and other labels associated with subgroups such as SO(7) and G2 in providing a classification of states, and the economy thereby afforded in dealing with a (small) set of their Casimir operators rather than with all N 2 generators of U(N), support the usefulness of dynamical symmetry. We will return to this topic in Part C.

8.2

Shell-wide Treatment

Qualitative and quantitative relationships among the spectra of various groups of elements have long suggested a unified analysis of each group in contrast to the separate study of each spectrum. A systematics of partially filled shells is particularly conspicuous for shells of rather small radius, the d shells of transition elements and the f shells of lanthanides and actinides. A conspicuous feature was mentioned above, namely, the symmetry be-

8.2. S h e l l - w i d e T r e a t m e n t

tween the spectra of an element with n electrons another with n vacancies in the same shell. Even the interaction of any two electrons in an open shell respects, of the presence of other electrons in that

209

in a given shell and of more important is that is independent, in many shell.

Mapping of all the states [gnavSL) of a given open shell as vectors of a single Hilbert space facilitates a unified analysis even though such states belong to different elements. A further step maps all these states and their operators onto vectors of a Liouville representation according to the procedures of Chapter 6. One of these procedures, namely, expansion into unitary operators, sorts out intrinsic parameters, represented by reduced matrix elements, from extrinsic variables--including the number of electrons in the shelluwhich will appear only as indices of 3-j or 6-j symbols. The joint mapping of states [~avSL) with different values of n on a single Hilbert space does not suffice by itself to interconnect them. Their connection is provided by introducing operators, akin to the pair {a~, a~} of (6.23), which add an electron into an empty shell orbital or remove an electron from a filled orbital. The breadth of these application to a whole shell envisaged here contrasts with their application to a single orbital in Chapter 6, where a~ and a~ joined into the quadratic expression a~a~and

a~a~ to form a set isomorphic to the spin operators {1, Y}. Here we follow instead a procedure introduced by B. R. Judd in the 1960s [34], which replaces the operator pair {a~, a~ } with the notation aqq, an operator that raises, when small mq 7, or lowers (mq - 89 the index M O of a shell's fractional occupation. This i n d e x u t o be defined in Section 8.2.1--serves as the main variable parameter of a shell bookkeeping patterned after that of magnetic quantum numbers, being accordingly dubbed the q u a s i - s p i n method. The process of shell filling (or emptying) takes thereby the aspect of an r-transformation, much as charge transfers among nucleons are treated as rotations of an abstract "isotopic spin" in nuclear dynamics. These analogies favor a formal blending of the shell-filling process into the rtransformations of orbital and spin state representatives. This blend indeed led Judd to formalize a powerful t r i p l e - t e n s o r procedure that treats shell filling on a par with the r-transformations of spin-orbital states, such as the product of two elements in (8.2) viewed as "double tensors." The following paragraphs introduce the notation and properties of operations in quasi-spin space and their relations to seniority and fractional parentage.

Chapter 8. P a r t i a l l y Filled Shells of A t o m s or Nuclei

210

8.2.1

Triple tensors and their matrices

Since an electron can be added to or removed from different orbitals of a shell, represented by kets Ism~,~mt), the operator pair a(mq~ acting on a specific orbital must be labeled by this orbital's quantum numbers. We ,,(q,t) , by analogy with the notation used for tensors indicate it then as ~,~q,~.mt gqk of Chapter 6. As the kets Ism8 , ~mt) experience r-transformations upon rotation of the space coordinates, the same r-transformations apply to the indices (ms mr) of operators .,(qsl) t a f l ~ q ~ s 7T~t 9 These r-transformations are direct products of separate transformations acting on the spin and orbital indices m8 and mt as long as spin and orbit remain uncoupled" the symbol .(qst) behaves in this respect as a "double tensor." Proceeding further to treat quasi-spin rotation in Hilbert space on the same footing as rotations of physical space leads us finally to regard the operators ~fl,~q ,~(q,t) triple rr~ s rr~l a s t e n s o r s which experience separate transformations of their three indices The detailed correspondence between triple tensor components with indices mq - :]: ~1and the operator pairs {a~, a~ } involves the sign conventions of (5.26) and of Chapter 6 regarding the conversion of bra to ket in the construction of irreducible products and of operators. The resulting definition is

a (q't)

-

-

a t('t) : a~

:

a(q[ t)

(-1)'-m~

(8.6)

(The second equation of (8.6) differs in sign from the corresponding definitions on pp. 32 and 42 of [18].) Equation (8.6) ensures that the correspondence between particle occupancies and vacancies in a shell is represented by frame reversal in the quasi-spin space. Its notation affords ready adaptation to various circumstances. Application to a shell of j j-coupled states [jnavJ) replaces the triple tensors (8.6) by double tensors

a (qj) 89 -- at(J) mj ,

a (qj) (j) - 89-~r,j -- ( - 1 ) J - m J a mj"

(8.6a)

Extension to a nucleon shell with j j-coupling would yield triple tensors a(qtj) mqTr~tmj which include their dependence on the isotopic spin labels (tm~) The triple tensor operators (8.6) have been identified thus far through their action on the occupation of a single orbital by a single electron. Let us consider now how this action modifies the quantum numbers of a multielectron state I } g n a v S M s L M L ) . By adding (removing) an electron in an o bit l

(S.6) with

-

ch .g s the q u a n t u m num-

ber ML into M~ = ML :l: ml by algebraic addition, and L into L' by vector

8.2. S h e l l - w i d e T r e a t m e n t

211

a d d i t i o n / } - / ~ 4- t-'. Similarly it changes Ms into M's - Ms 4- m, and S into S' by S~ - $4- g'. These combinations, by algebraic or vector addition, affect quantum numbers pertaining to eigenvalues of operators of physical space, that is,/~, Lz, S, Sz. The quantum numbers (qmq) in (8.6) pertain instead to rotations in the quasi-spin model space, where multielectron operators (~ and Qz act isomorphically to the pairs (/~L~)and (SS~). i new pair of quantum numbers (QMQ) should then represent eigenvalues of [(~]2 and Qz, combining with (qmq) as (LML) combine with (grnl). This new pair of quantum numbers (QMQ) should prove equivalent to the pair of shell occupation parameters (n, v) of the states ]}~avSL) because (qmq) pertain to changes of shell occupation. A treatment of the multielectron operators (~ and Qz involves products of the single-orbital operators (8.6) developed in the next section. We anticipate here the expressions of the number n of shell electrons and of the seniority v in terms of (QMQ),

n-2~A- I+2MQ ,

v-2~A-1-2Q,

(8.7)

illustrated in Fig. 8.1. Note particularly how the diagram centers at the half-filled level of shell occupation, n = 2g + 1. This feature accords with representing the symmetry between occupation and vacancy by frame reversal. 1 Note also how the seniority v appears here as a complement to the quantum number Q, in that v + 2Q = 2g + 1. The factor 2 that multiplies MQ and Q in (8.7) permits Q and MQ to take half-integer values while (n, v) remain integers. Indeed Q represents the number of 1S particle pairs required to half-fill the shell as a complement to the initial set of v electrons that form the base state [}~V~vSL). Thus, for the example considered earlier of d 3 2D states, the v = 1 and 3 correspond to Q = 2 and 1, namely, to the number of d 2 1S pairs needed to complete d and d 3 to the d 5 which marks a half-filled d shell (see Table 8.1). Antisymmetrized states of a configuration tm will now be indicated as

[iaQMQ,SMs, LML), with the values of n and of the seniority v defined by (8.7) and with the antisymmetry sign ]} implied by the presence of Q quantum numbers. With this notation and applying the Wigner-Eckart theorem in its form (6.41) to each index of the triple tensor operator (8.6), we factor out its matrix in the form 1This particle-hole symmetry under frame reversal relates to the time reversal hnk between particle and antiparticle in field theories noted at the outset of Chapter 3.

212

C h a p t e r 8. P a r t i a l l y F i l l e d S h e l l s o f A t o m s or N u c l e i

I (ea I QI MQ S'M~s L'M~Lla(q't)

--

le~QMQSMs LML)

(_I)Q'-M;+S'-M;+L'-M~

•

/

-Mq

L

S

mq Mq

-M~s m,

)(

Ms

x (~'Q'S'L'i[a(q'~)II~QSL).

L'

g

--M'L mt

L)

ML

(8.8)

This equation has separated out the matrix element's dependence on the numbers of particles in the initial and final states by representing it in the explicit form of a 3-j coefficient, as anticipated above.

n 2(2g+1)

MQ .

.

.

.

.

.

.

'1

I

Q

2~+1

0

v

-Q I .

.

.

.

F i g u r e 8.1" Relationship between seniority (v) and quasi-spin (Q,MQ) quantum numbers and number n of electrons in a shell.

8.2. Shell-wide T r e a t m e n t

8.2.2

Coefficients

213

of fractional

parentage

Since the matrix element (8.8) relates two antisymmetrized states with the difference of electron number In~ - n] = 1, its reduced form must contain essentially the same information as the cfp defined by (8.4). To establish this connection we obtain the cfp from (8.4) by projecting that equation onto the bra state

(~-:aSLtSMs,LML] -

Z

(SsSMsnSMs,sm,)

K't s lVl L m , m t

• (LtLMLILML,tmt)(~-:~SMs, LML](sm,,tmtl.

(8.9)

Notice now that the projection of the n-electron ket I~aSMs, LML) on the left-hand side of (8.4) onto the single electron orbital (sms,C.mtl is ,(st) represented in terms of an electron removal operator by ~.m,m~]}~aSMs, LML). Switching to the notation of (8.8) and (8.6), the projection of (8.4) yields then the cfp expression

(taQSLtI}taQSL) -

Z

(SsSMsNSMs'sm')(LgLMLILML'Imt)

IVI s lV1L rrt s m t

•

•

(t&QMQ,SMs, LML l"_89

--~t llaQMQ' SMs, LML).(8.10)

The matrix element of the triple tensor, as given in (8.8), may now be entered in the expression (8.10), but the effect of this substitution is made more transparent by utilizing an alternative form of (8.8). The suitable form is obtained by: a) reversing the sign of all m indices of the 3-j coefficient in accordance with item c) in Section 5.4, b) expressing each of these coefficients in terms of a Wigner coefficient by (5.45), and c) transposing each Wigner coefficient. These steps yield

(-1)s-m'+e-mt(gStQMq, SMs , LML[a(qst),

[s

, SMs , LML)

= (-1) (2+Q+~+s+L+L-t [(2Q + 1)(2S + 1)(25 + 1)]- 89 • (QMQ,q 89 • (LML, ~mtlLiiUL)(&QSL]ia(qst)ilaQSL ) 9 (8.8a) Upon entering this expression in (8.10), its summation reduces to the inner product of reciprocal Wigner coefficients. The cfp is thus represented

214

C h a p t e r 8. Partially Filled Shells of A t o m s or N u c l e i

as the reduced matrix element of a triple tensor multiplied by a Wigner coefficient which displays its explicit dependence upon the number of electrons, through the quantum number MQ, and by an explicit normalization coefficient,

(~(~QSL~[ } ~ Q S L ) = (--1)(2+Q+S+S+L+L-I[(2Q + 1)(2S + 1)(2L + 1)]- 89 1 • (5~QSL]Ia(qstllI~QSL)(QMQ, qTIQqQMQ).

(8.11)

The calculation of the reduced matrix element of a (qsl) rests on further considerations to be developed in the next section.

8.3

Algebra of Triple Tensors and Its Applications

The concept and the technique of triple tensors find extensive application by constructing products of creation-annihilation operator sets a (qst). Thus far products of two such factors have been considered, being exploited only to a limited extent; accordingly we shall only make passing remarks about higher order products. Conceivable extended applications of triple-tensor techniques also emerge in the following sections. The basic double product of triple tensors may be defined as kqk,kt) X(qqq,qt

_

~

a(qSt) a(qSt) mqmsmt rrtlqmlsmlt I

• (qmq, qmtq[qqkqqq)(Sms, Sm~sIssk, q,)(tm,,

tm'~lttktqt).

(8.12)

The values of kq and ks are limited to 0 and 1 since q - s - 3, x while kt ranges up to 2g. Expressing various quantities in terms of these operators yields the explicit dependence of their matrix elements on the number of particles in a shell through the Wigner-Eckart theorem as in (8.11). It also brings out the effect of symmetry selection rules by considering triangular conditions and of parity under frame reversal; this parity represents here the symmetry of atomic shell states under interchange of electrons and vacancies. The more substantive applications to be outlined here concern the correspondence of quasi-spin and seniority and the determination of fractional parentage coefficients.

8.3. A l g e b r a of T r i p l e T e n s o r s a n d Its A p p l i c a t i o n s

8.3.1

Interpretation

215

o f X (kqk~k~)

The concept and notation of triple tensors are comprehensive to the point of including elements of rather heterogeneous physical significance, thus requiring some analysis of their structure and interpretation. Recall from (6.23) that the operator a~a~ represents the probability of occupation of a spin-orbital I~) while a~a~ represents the complementary probability of the orbital being empty. Accordingly, their sum amounts to the unit matrix

a~a~ + a~a~ -

1.

(8.13)

Their difference, which represents the departure from an average level of half-occupation, is represented by the matrix a~, that is, by the qq - 0 component of the vector operator ~. Equation (8.12) also involves operator products amounting to a~a,, a,Ta~, a~at, and a,a, 7 with 7/:/= ~. These operators represent particle creation and annihilation processes in different spin orbitals, processes that are independent except for symmetry considerations. Recall also that, in the construction of independent-particle wave functions by Slater determinants, the spin-orbitals have to be arranged in a standard order; addition or removal of any single orbital is understood to occur in the last (or first) position of a determinant and to be accompanied by permutation into its standard place. These considerations imply that all creation or annihilation operators with ~ ~: 7/ anticommute. Their combination with (8.13) is summarized by the anticommutation relation a(q,l)

a(q ,t)

rn a m ~ m t

= =

m Iq r n ~, r n tl

[(2q + 1)(2

• (r

+ a (q'l)

a(q,t)

m ~qr n ; m t~

+ 1)(2

rn q m s m t

(8.14)

+

m',)(ttoolt

, tm ).

[The corresponding Eq. (39) of [18] has opposite sign on the right-hand side owing to a difference in the definition corresponding to (8.6).] Equation (8.14) serves to examine the structure of the operators X in (8.12) on the basis of its symmetry under permutation of the primed and unprimed indices. It also permits us to eliminate any consideration of vacancy-counting operators a~at,, concentrating instead on the particlecounting, atrial. Let us then reduce (8.14) with respect to its dependence on the m indices, by multiplying that equation by the Wigner coefficients

C h a p t e r 8. P a r t i a l l y Filled Shells of A t o m s or Nuclei

216

of (8.12) and summing over all m. Notice that, under permutation of the primed and unprimed indices, the product of the three Wigner coefficients has the parity (--1)2q-kq

+2s-k,+2l-kt

(-1)k~ +k~

=

,

(8.15)

owing to (5.17) and to the even value of 2q + 2s + 2~. Accordingly, the two terms on the left-hand side of (8.14) yield equal or opposite contributions, when reduced, while the right-hand side of (8.14) simplifies by orthonormality of the Wigner coefficients. Thus one finds [l+(--1)ka+k'+kt]X -- - "(k~k'k~) qqqsqt ~ 2(2g+1)~

(8.16)

6kqO6ksO6ktO~qqO6qsO6qtO

since [(2q + 1)(2s + 1)] 89 - 2. That is, X (kq~.k') vanishes unless either kq + ks-4-kt is odd or kq - ks - kt = 0. In the latter case, Eq. (8.16) represents only the trivial fact that (8.13) yields 2(2g+ 1) when summed over all ~ = (msmt); accordingly we need consider only odd values of kq+ks +kt. For kq = 0 and odd values of k, + kt, it suffices to consider the terms of (8.12) with mq - 1, adding an equal amount for mq -_ -3"1 Applying (8.6) and (q 89 q- 89 ~7~2,Eq. (8.12)reduces to

O_s_tq q

~

msmt

9

t

t i r r t s r r t 9r r t l r r t l

• ( s m s , s m ' l s s k , qs)(gmt,gm'llggktqt) ,

k, + k t -

odd. (8.17)

Comparison with (5.26) identifies the summation in this equation as the reduced product of double tensor sets at(St)a (sl), called W (k'kt) by Judd [34]. Accordingly we rewrite (8.17) as _ X/2 W (k'k') A O(0k,k,) qsqt -"" q s q t

for

ks-1 ks-O

kt-even, kt - odd.

(817a)

This operator represents, when ks = 1, a density of spin magnetic moment weighted by an even multipole moment of the entire particle density in the shell, and when k, = 0 a magnetic multipole moment of the orbital current contributed by all shell particles, being anyhow odd under frame reversal. For kq = 1, the terms of (8.12) with qq = 0 have equal coefficients (q+ 89 q ~ 8 9 ~ being again combined and expressed in terms of W (k'k*), for even values of ks + kt, including a nonzero contribution from

8.3. A l g e b r a of Triple T e n s o r s a n d Its A p p l i c a t i o n s

217

the right-hand side of (8.14). Thus one finds x ( lOq,qt k.kt)

(k.kt) _ (2g + 1) 896k,06qs 06k~O6q~ 0 , "- V/'2Wq.qt

for ks + kl -- even (8.18)

representing a density of multipole moments even under frame reversal. Finally, for kq -- 1 and qq -- -l-l, the operators in (8.12) are both a t or both a, respectively, whereby their anticommutator (8.14) vanishes owing to its factor 6-mq mq, The analog of (8 917a) with kq -- 1 to be indicated by A t or A, represents the creation or annihilation of a pair of particles paired to a spin S -- ks and orbit L = kt, with an even value of S + L = ks + kl in accordance with the exclusion (8.1). Thus we write msmt

"~qsqt

-(lk.kt)

'

~'-lqsqt

_

--

v/-~A(k.k~ ) qsqt '

for

ks + kt - even .

(8.10) 8.3.2

Quasi-spin

and

seniority

0 ) play a special role being totally invariant under roThe operators y~.( 1qq000 tations of space coordinates. For qq = 1, 0, or-1 these operators represent the creation of a pair of particles in a 1S state, the number of such pairs in the shell, and the annihilation of a pair, respectively. They represent infinitesimal operations in quasi-spin space only, specifically simple rotations in this space of degree kq = 1. The commutator of any two components of X ~]00) 00 yields just another one of these components according to Section 7.3, thus forming a subgroup of all infinitesimal triple tensor transformations. In other words, the set of operators X ~]00) 00 is isomorphic to the set

of infinitesimal rotation operators J, components of the quasi-spin vector operators Q. To be exact, X (1~176 differs from (~ by a normalization factor, to be determined by comparing the commutators of their three components explicitly. The commutators of {X~ 00 1~176 v(l~176v-'-100 ( l ~ 1 7 6 are obtained by procedures ,--000 analogous to the derivation of (7.35), using the basic property (8.14), while the commutators of {Q~, Qu, Q z } coincide with those of {J~, Jy, Jz }. We omit this calculation, lifting the result from p. 43 of [18], with a change of sign reflecting the different conventions in (8.6): X (100) -- 2 ( 2 e @

1)- 89

(8.2o)

The squared magnitude 1012 has accordingly the eigenvalues Q ( Q + 1), as anticipated in Section 8.2. Both Q and It~[2 can be expressed directly in

218

Chapter 8. P a r t i a l l y Filled Shells of A t o m s or N u c l e i

terms of the double scalar operators W (~176A t(~176and A (~176by means of (8.17) and (8.19). As we know, the action of any component of f on the elements of an irreducible tensorial set only changes the set elements into one another, leaving invariant the degree j of the set which corresponds to the eigenvalue j(j + 1) of {j-~2. This isomorphism of (~ and f implies that any component of (~ can add, for example, a pair of particles coupled to 15, to a state of the shell, thus raising its quantum number MQ by one unit, but leaving the quantum number Q invariant. Recall now from Section 8.1 that the seniority quantum number v was defined as characterizing a state that remains invariant by addition (or removal) of a 1S pair; it was also indicated that each value of v corresponds to an eigenvalue of the Casimir operator for the orthogonal subgroup of U(2t~ + 1). The operational equivalence of Q and v for the classification of states, which we have verified here but was anticipated in Section 8.2 by writing the explicit relationship (8.7), implies also that the Casimir operator of S0(2~ + 1) is a function of I(~]2. This function is obtained from (8.7) and from the explicit form of the quadratic Casimir operator eigenvalues f(v) given by Racah (Eq. (19)of [37]),

f(v)

-

:

~v[4(g + 1 ) - v ] - S(S + 1)

( ~ + ~1 ) ( i + - ~ ) - Q ( Q + I ) - S ( S + I ) .

(8.21)

No more direct relationship between I(~]2 and this quadratic Casimir operator seems to have been established. As indicated in Section 8.1, the combination of the quantum numbers Q and MQ with the usual (S, Ms, L, ML) suffices to label a complete set of states of all particle configurations dn. The classification of all terms of these configurations is illustrated in Table 8.1, which displays the entire family of dn states with its six-label classification. One may verify that the total number of states, (2S + 1)(2L + 1), for each term 2S+lL in a row of the table, equals the expected value (4l+2]. Viewing the table in \ n ] terms of columns, states with equal 2S+lL but varying n form multiplets with the same quasi-spin value Q and MQ running over the (2Q + 1) values - Q , - Q + 1 , . . . , Q. Parentage follows the arrows so that the only nonzero cfp's pertain to states in boxes connected by an arrow.

8.3.3

Quasiparticles for the f shell

Completing the classification of states for higher values of/? requires additional operators, as indicated on p. 208, because of the even larger mul-

8.3. A l g e b r a of Triple Tensors a n d Its Applications

219

tiplicities of 2S+iL that occur for t~ > 3. Note here that the addition to an f shell of three electrons in a state [}f3 4S) is presumably achieved by

a triple product of operators a(qst) which might be indicated by y ~- q ,0O ~)

"

This extension of the triple tensor formalism does not seem to have been developed, nor have possibilities been explored to identify still larger subgroups of particles or their relations to the Racah analysis of the subgroups

of U(2/+ i). A recent exploration by Judd and co-workers [38] introduces quasiparticles dubbed "quarks" because of their analogy to the usage in elementary particle physics. Its initial motivation emerged from tabulated matrix elements and coefficients of fractional parentage for f electrons in rare earth and actinide spectroscopy, showing unexpected proportionalities between entries and seeming selection rules foreign to SO(7), G2, or SO(3) symmetries. Observe that the total number of fn states, 0 < n < 14, is ~ n ( 4 t + 2 ~ __ 214 = 16384, and can be recast as (8) 4 • 2 x 2. An eightfold symmetry basis arises by complementing the seven f wave functions (each of which would vanish upon spherical averaging) with an unspecified isotropic dynamical element. (This element is likely to consist of a breathing mode--swelling or contraction--of closed shells; similar circumstances will emerge in later chapters.) Having thus expanded the SO(7) symmetry of f states to SO(8), one considers a set of four eight-dimensional elements with two parity labels. These elements are dubbed "quarks"mseven fQ and one sQ--in [38], presumably by assonance with the eight infinitesimal elements of the SU(3) symmetry group of subnucleon quarks. Figure 8.2 gathers on its left the states of maximum multiplicity of configurations fn, separately for even and odd n values (see Table 8.2), contrasting them with corresponding states of configurations (sQ + fQ)2. This arrangement reinterprets all states of actual f electrons (with alternative spin alignments) as coupled states of two fictitious "quarks." \

fl

~'

These observations, and the many simplifications of f-electron spectroscopic matrix elements generated in [38], hold the promise that analogous models and higher symmetry groups such as SO(8) may prove useful in analyzing many-fermion systems with g > 3. Recently, quark-like models have also been applied to d electrons to provide an alternative classification to the seniority and quasi-spin schemes [38].

220

C h a p t e r 8. P a r t i a l l y F i l l e d Shells of A t o m s or N u c l e i

D

D P

P

G S,F

G S,F F

F F

I f

OR f6

S

H

H,I

f3

f5

f7

f4

f2

fo

S

s8

2 qsq

Figure 8.2: Alternative rendering of states in the f shell" Maximumspin states of fn configurations on the left and of pairs of fictitious "quarks" on the right [38]. Bracketed pairs such as S, F denote that linear combinations in one scheme are necessary in reexpressing in terms of the other scheme. 8.3.4

Determination

of fractional

parentage

The coefficients of fractional parentage have been introduced in Section 8.1 as the eigenvectors of an antisymmetrization operator ,4. Subsequently, Eq. (8.11) factored out their dependence on the number of electrons in a shell, whereby the residual cfp and the operator ,4 depend only on the seniority, spin, and orbital quantum numbers. Two qualitative circumstances suggest the following procedure for constructing the operator .,4 and the cfp: a) Antisymmetrize the combination of another electron with an electron detached from a previously antisymmetrized state, selecting their joint state according to (8.1). The construction of the cfp for a configuration ~ rests thus on prior knowledge of the cfp of gn-1 pertaining to its separation into ~ - 2 + g. b) In the special case of n = 3, the detachment of an electron from g2 poses no problem because antisymmetric g2 states are identified by (8.1) without any recourse to fractional parentage. One can thus construct the cfp for g3 and thereafter proceed by recurrence to solve the problem for all tm.

8.3. Algebra of Triple Tensors and Its Applications

221

Table 8.2: States of f'~ with multiplicities shown in parentheses (adapted from ref. [6]) Configuration

Number of states

States (multiplicity)

f o , f14

1

1S

f, f13

14

2F

p , f12

91

1S 3p 1D 3F 1G 3 H 11

f3, fl 1

364

4S 2p 2D(2) 4D 2F(2) 4F 2G(2) 4G 2H(2) 21 41 2K 2L

f 4 , flO

1001

1S(2) 5S 3p(3)1D(4) aD(2) 5D 1F 3F(4) 5F 1G(4) 3G(4) 5G 1H(2) 3H(4)1I(3)31(2) 5I 1K ~;C(2) 1L(2) ~L ~M ~N

f5, f9

2002

4S 2p(4) 4p(2) 6p 2D(5) 4D(3) 2F(7) 4F(4) 6F 2G(6) 4G(4) 2H(7) 4H(3) 6H 21(5) 41(3) 2K(5)4K(2) 2L(3) 4L 2M(2) 4M 2N 2O

f6, f8

3003

1S(4) ~S 1p 3 p ( 6 ) s p 1D(6) 3D(5) 5D(3) 1F(4)3F(9) 5F(2) 7F 1G(8) 3G(7) 5G(3) 1H(4) 3H(9) 5H(2) 11(7) 31(6) ~I(2) 1K(3) 3K(6) 5K 1L(4)3L(3) 5L 1M(2) 2M(3) 1N(2) 3g 30 1Q

f7

3432

2S(2) 4S(2) 8S 2p(5) 4p(2) 6p 2D(7) 4D(6)6D 2F(10) 4 F ( 5 ) 6 F 2G(10) 4G(7)6G 2H(9) 4H(5) 6H

~;(v) ~1(5) ~1 ~/;(7) 4/,;(3) 25(5) 4L(3) 2M(4) 4M 2N(2) 4N 20

2Q 16384

222

C h a p t e r 8. P a r t i a l l y Filled Shells of A t o m s or Nuclei

This procedure does not actually progress from each value of n to the next higher one but only from each seniority v to v + 1, beginning with v = 2. More accurately, the procedure works recursively from the configuration gn to ~m-2, according to a) above, that is, from each value of Q = Q to the next lower value Q - Q - 89beginning with Q - g - 89down to Q - 0. The recurrence relation that will determine the matrix .4 and the cfp is formulated in terms of the reduced matrix element of a triple tensor operator, analogous to that in the general expression of a single cfp in (8.11). Specifically, we start from the reduced matrix element of (8.16), recalling from (6.32) that the reduced element of a unit operator (j]Jl(~ equals (2j 4- 1) 89 [1 + ( - 1 ) kq+k'+k~] (~'Q~'s

- 2(2g + 1)6kq06k,06kt0 9

(s.22)

Viewing now X(ka k,~t) as a sum over operator products according to (8.12), we apply the recoupling transformation (7.33c) to the left-hand side of (8.22)--in triplicate since we deal with a triple tensor--thus reducing (8.22) to [1 +

(--1)kq+ks+kt][(2]Cq+ 1)(2k, + 1))(2kt + 1)] 89 x

y~

(--1)2(2+~'q( q

x (-1)~'+~+k' { S'

q kq }

0 0 Q'

k,} )L'+L+k,{e 5'

S'

(-1

L'

L

kl} L'

• (~'Q~'L'IIa(q'~)II~'Q'S'L')(~'Q'S'L'IIa(q't)II~QSL) =

2(2g+ 1)6k,o6k,o6kto.

(8.23)

This equation will serve as a recurrence relation because the quasi-spin vector formula Q' -" - Q~ + ~" with q - 1 allows just two values for the quantum number Q', namely Q-4- 89 The reduced matrix elements between and Q~ - Q + ~1 will be regarded as known from a previous cycle of recurrence, since we proceed from large to low values of Q. The reduced matrix elements between Q and Q - Q - ~1 are instead determined from (8.23). Equation (8.23)is bilinear in the matrix elements

( 'Q'S'L'Ila(q' )II

aQSL) which we seek to determine. However, these matrix elements are eigenvectors of the projection operator A, as we know; indeed subsets of the bilinear terms of (8.23) actually constitute a matrix element of A, namely,

8.3. A l g e b r a of Triple Tensors and Its Applications

223

( &'QS' L' I.A(QSL ) I~QSL ) E(&'QS'L'[[a(qSt)[[aQSL)(aQSL[la(aSl)[[aQSL).

(8.24)

=

Ot

Our procedure amounts thus to regarding (8.23) as an inhomogeneous system of linear equations that determines the matrix elements (8.24). Note that the determinant of the matrix (8.24) has the structure of a Gram determinant which vanishes provided the number of terms in the ~ a is smaller than the number of columns ((~SL) as stated in Section 8.1. On this basis, the determination of the matrix (8.24) with unit eigenvalue will be straightforward once the matrix A has been calculated. Equation (8.23) may now be cast in more convenient form by shifting to the right-hand side its terms with Q' - Q - Q + 1, obtained from a previous cycle, and by constructing linear combinations of the equations with different value of kq, ks, kt with the coefficients

(--1)2(2+$'+S+L'+L+kq+k'+kt[(2kq + 1)(2ks + 1)(2kt + 1)] 89 x

Q, Q

Q

S'

S

S

L'

L

n

" (8.25)

The sums over products of 6-j coefficients simplify through (7.17) and (7.18), yielding the system of equations

Z [ ( 2 Q + 1)(2S + 1)(2L + 1)]-16s,S6L,L + (--1) 2Q-s+S+s'+L+L' S,L ~

x

{

q

Q

Q

}{

s

S"

S'

qQQ sS S x (&'QS'L'IA(QS'L')I6~QSL)

}{

g

L'

gL

L'

L

}

(--1)2(2+S+S+L+L4(2g + 1)6~,SQ.,L + ~ ( - 1 ) 2(2+~+s+~+L+L 8SL

• (~'QS'L'Ila(qSl)I$5OSL)(&O, SLIla(qSl)II&QSL ).

(8.26)

Solution of this system provides the matrix .A(QS' L') for each pair of values of (S'L') and hence the reduced matrix elements of a (qst) to be entered on the right-hand side of (8.11).

224

8.3.5

C h a p t e r 8. P a r t i a l l y F i l l e d Shells of A t o m s or N u c l e i

Operator

matrices

Interactions among electrons of a partially filled shell, or between such electrons and the nucleus or external fields, are represented by operators that conserve the number of shell electrons even though they may shift electrons from one orbital to another. (As examples, the linear Stark coupling to an external field can shift an atomic electron between 2s and 2p orbitals; similarly the Coulomb interaction of an electron pair connects 2s 2 aS and 2p 2 1S y(k,k.k,) with states.) These operators may be expanded into triple tensors "oq.q~ their index qq - O, which remain diagonal in the quantum number MQ, that is, in the number n of electrons in the shell. Two positive results emerge from this approach: a) Each operator's matrix depends on M Q - - t h a t is, on the electron number n~explicitly through the standard 3-j coefficients through an analogue of the Wigner-Eckart Eq. (8.8) for triple tensors. b) Selection rules on the quasi-spin--that is, seniority--quantum number Q result from the triangular relation among {Q~, kq, Q}. The triple-tensor formula leads naturally to previously known selection rules on seniority. In particular, the triple tensor feature kq _< 1 reflects the fact that single-electron operators shift only a single orbital, leading to the selection rule AQ - 0, 1, that is, Av - 0, 2. (The seniority of a state is even or odd according to its number of particles, as apparent in Table 8.1; correspondingly the values of Q and MQ are half-integer or integer.) Two-electron operators, such as the electron-electron interaction e2/]r*l -- r*21 in (7.47), may instead shift two electron orbitals yielding triple tensor expansions with kq ~_ 2. Their selection rule is then AQ - 0, 1,2, that is, Av - 0, 2, 4. Early examples of this analysis are presented in Chapter 8 of ref. [18], but a complete approach to spectroscopy--atomic and n u c l e a r ~ b y triple tensor algebra remains largely undeveloped.

PART SYMMETRIES

C OF HIGHER

DIMENSIONS

226

Part C

Many aspects of the r-transformation symmetries described and developed in Parts A and B apply also to symmetry elements of more extended (or restricted) dimensionality, belonging to various groups. Their treatment in the context of diverse physical phenomena forms the subject of this book's remaining chapters. Molecules and crystals are often invariant under reflection through several planes or other elements. This feature restricts their rotational invariance to rotations about specific axes and by specific angles only. There result diverse sets of compatible symmetries, called "point groups," which underlie much of crystallography as well as much of the physics of molecules in their respective "body frames." (Each molecule also rotates freely about its center of mass, of course, in gas phase, an operation to be combined with the relevant point group.) Chapter 9 deals with general aspects of these symmetries, omitting details covered by standard textbooks [39]. Direct extension of r-transformations to phenomena invariant under rotations of mathematical "spaces" in four or more dimensions appears in a way more obvious but is examplified by less familiar phenomena than crystal symmetries. An important prototype is afforded by the isoenergetic transformations of Kepler orbits with alternative orientations and eccentricities and by their quantum counterpart, namely, the transformations among degenerate states of H atom levels (to within fine structure). The relevant group, SO(4), has six infinitesimal operations, that is, rotations about six orthogonal axes in momentum space; its elements are accordingly represented by pairs of S U ( 2 ) matrices. "Configurations" of N particles at positions defined by N three-dimensional vectors {Y/} are transformed into one another by rotations in a 3(N - 1)-dimensional space, about the center of mass, combined with a uniform stretch of their radius of inertia R -- ( ~ i N _ 1 m i r - 7 / M ) 8 9 Basic aspects of these examples are described in Chapter 10. A novel element emerges in the treatment of the four-dimensional Lorentz transformations, which contrast with the SO(4) group by conserving the expression x 2 § y2 § z 2 _ (ct)2 with one negative sign. Whereas SO(4) has six infinitesimal transformations--three of which are rotations (pseudovectors) and three are stretches (vectors)--the stretches are replaced in the Lorentz group by relativistic "boosts" equivalent to imaginary stretches and to translation in velocity space. These boosts are dealt with formally by an SU(2) representation with imaginary angle parameters. The Lorentz group is accordingly indicated by the symbol SO(3, 1) implying 3-dimensional rotations combined with an imaginary boost, to be described in Chapter 11.

Part C

227

Chapter 12 will finally introduce transformations that represent a stretch of energy, much as the Lorentz "boost" represents a stretch of an observer's velocity. The stretch of energy reaches generally to infinity, in analogy to a boost's representation as the t a n h - l ( v / c ) , a feature reflected in the labeling of SO(3, 1) as a "noncompact group." Energy stretches are also noncompact within discrete spectra in the case of Rydberg systems that include an infinity of levels, though not for vibrational spectra that consist generally of a finite number of levels below a dissociation threshold. Several instances of such noncompact groups will be described in Chapter 12. These illustrations in Part C exemplify the richness in the number and types of symmetries brought in by more dimensions. An enlargement in the number of dimensions or degrees of freedom means a larger group of transformations with a larger variety of subgroup structure. Alternative sets of generalized coordinates, or of alternative clusters of them, analytical and other continuations, etc., serve to simplify or connect descriptions of physical systems or phenomena.

This Page Intentionally Left Blank

Chapter 9

Discrete Transformations of C o o r d i n a t e s Crystals and many molecules are invariant under various groups of coordinate rotations and reflections about a specified point, often the center of one atom. These groups of operations, called "point symmetry groups," are subgroups of the coordinate rotations and reflections treated in earlier chapters. Restriction to a subgroup does not generally bring about simplifications, it being often more complicated to deal with discrete than with continuous sets of operations. The description and classification of point groups constitutes by itself a sizable task. This subject involves the following three activities: a) identifying the various discrete symmetry elements, primarily axes of finite rotation, reflection planes, and their combinations into various groups; b) developing relevant elements of the formal theory of groups, primarily the theory of the "character tables" utilized in many books on molecules and crystals [39]; c) learning how to apply a) and b) to specific examples. We shall work on all these activities simultaneously, that is, with reference to examples of complication barely sufficient to be nontrivial, such as the molecules H20, NH3, CH4, CH3D as well as rings of potential wells that occur in benzene and in multitudes of other organic molecules. 229

230

Chapter 9. D i s c r e t e T r a n s f o r m a t i o n s of Coordinates

We deal with discrete transformations of sets of quantities, quite analogous to the r-transformations of tensorial sets. Irreducible sets consist once again of eigenvectors of a maximal set of operators that commute with all operators of a discrete group. A principal task identifies such a maximal set for each of the groups that will be listed. The search for a maximal set of commuting operators is central to quantum-mechanical applications, such a set giving a complete and unique labeling of stationary states of the system.

9.1

Point Symmetry Operations and Their Groups

The study of point groups deals with the basic symmetry operations of an object's rotations about an axis and of reflections through a plane. Denoting by Cn the operation of rotation by an angle 27r/n about an axis, n successive rotations represented by Cn" leave the body unchanged in its orientation. The standard notation for this identity operation is E so that Cn n - (C,~) ~ = E. Reflection through a plane is usually denoted by cr whereby er2 = E. Since one often considers combined rotations and reflections, a useful distinction is made between reflections through a plane containing a certain rotational axis and those in a plane perpendicular to that axis; they are denoted err and erh, respectively, the axis being considered vertical, and v and h denoting vertical and horizontal planes. The operations C,~ and ~rh commute (that is, Cnah = a h C n ) , the combined operation being called S~. Rotation by 7r accompanied by a reflection in the horizontal plane, that is, $2 = C2o'h = ahC2, is called an i n v e r s i o n I (Fig. 9.1). Clearly, 12 = E. Objects that remain invariant under I are said to have a center of symmetry. Among other combinations of rotations and reflections, consider ~rv~r,,, that is, successive reflections through two planes intersecting on a vertical axis (Fig. 9.2). This combination is equivalent to a rotation of 2r about that axis, where r is the angle between the planes. Together with E, a set of rotation and reflection operators constitutes a s y m m e t r y g r o u p if the product of any two elements of the set is also contained in the set and if the inverse operation for every element is also an element of the set. The number of elements of a discrete group is called its o r d e r . Two elements A and B are c o n j u g a t e s of each other if an element C of the group satisfies A C = C B , that is, A = C B C -1. The set of all conjugate elements forms a class; every group can be subdivided

9.1. P o i n t S y m m e t r y O p e r a t i o n s a n d T h e i r G r o u p s

231

F i g u r e 9.1: The operation of inversion. into classes, each element falling into one and only one class. The set {E; C2}, one of the simplest examples, forms a group of order two, each of the elements constituting its own class. On the other hand, the set {-I-1, +i, +j,-I-k} of the quaternions defined in Section 1.6 is a group of order eight, split into the five classes 1 ; - 1 ; i,-i; j,-j; k,-k. The first set {E, C2} is an example of an A b e l i a n g r o u p each of whose elements forms a class by itself. The quaternionic group in contrast is n o n - A b e l i a n . A necessary and sufficient condition for a group to be Abelian is that all its elements commute among themselves upon multiplication. The point groups include the following. The rotation Cn about an axis and its various powers Cn r, r = 1,2,..., up to Cn n = E, constitute the cyclic g r o u p , Cn. Its elements commute among themselves, and each of the n elements forms its own class: this is an Abelian group. Also Abelian is the group of 2n elements formed by complementing {Cn~} with the n elements {Cn"~rh} which combine rotation through 27rr/n about the vertical axis and reflection through the horizontal plane. This group is denoted Cnh. On the other hand, consider adjoining to C2 the reflection av in a vertical plane. According to the above discussion of Fig. 9.2, this introduction immediately calls for another reflection ~rv, through a second, vertical plane orthogonal to cr,, the combined operation Crva~,, amounting to rotation through r, that is, to C2. The four elements {E, C2, a,, a,,} form a group C2v, the symmetry group of the H20 molecule (Fig. 9.3). Likewise, adjoining cry to a C~ yields ( n - 1) further, evenly spaced vertical planes of reflection; the total of 2n elements constitutes the non-

232

Chapter 9. Discrete Transformations of Coordinates

p.' O p,,

A

F i g u r e 9.2: Successive reflections through two vertical planes intersecting along axis AB. Point P at r 0 carried into P ' at r + 0 and then P " at - ( r + 0), equivalent to net rotation OP ---+OP" through 2r around

AB. Abelian group C , , , whose elements are arranged into ( n + 3 ) / 2 or ( n / 2 ) + 3 classes depending on whether n is odd or even, respectively. The pyramidal NH3 molecule has C3~ and the benzene ring C6~ symmetry. For a rotation C2n with even-valued index, the 2n elements {C2, r ~rh }, r = 1, 2 , . . . , 2n, themselves form an Abelian group called $2n. Note that $2 has the two elements {E, I} where $2 - I represents the inversion operation of Fig. 9.1. Contrast with other second order groups such as C2 - {E, C~} and Clh -- {E, ~rn}, also called C,. The group S2 is also labeled Ci and the group $4n+2 as C~,.,+~,i because it can be formed through the "direct product" of the elements of C2n+1 and Ci. The rotations C,~ about a vertical axis combine with rotations by r about a horizontal axis; specifically the n elements Cn r join with rotations by r about n such horizontal axes (at mutual angles 7r/n) to form the group D~. Supplementing these 2n elements with n reflections {cr,} and n operations {Cn"O'h} yields the group D,h of 4n elements. The planar BF3

9.1. P o i n t S y m m e t r y O p e r a t i o n s a n d T h e i r G r o u p s

3 0

233

z

y

1 H

F i g u r e 9.3:

2

/X

/-/

C~v symmetry of the water molecule.

molecule provides an example of D3h symmetry. On the other hand, supplementing Dn with n reflections {cry} and n combined reflections {(rv~rh} yields a second group of 4n elements denoted Dnd. All the above groups consist of rotations and reflections with respect to horizontal and vertical directions. Other point groups center on symmetries of geometrical figures (Platonic solids) such as the tetrahedron and icosahedron, useful in discussing symmetries of relevant molecules with these geometries. The tetrahedral group T results by combining the four elements of D2 with four oblique axes C3 that join the center to the vertices. The three rotations through 7r constituting (with E) D~ have mutually perpendicular axes that connect midpoints of opposite edges of the tetrahedron. The group T contains 12 elements divided into four classes: E, the three r rotations, the four 63 rotations, and four C32( -- C31). TWO related groups, Td and Th, with double this number of elements, are formed by adjoining additional symmetry operations: Adding to T a center of symmetry yields the direct product T • Ci = Th with 24 elements, split into eight classes. The center of symmetry adds to the three axes of 7rrotations reflections through three mutually perpendicular planes through them, whereas the four C3 axes become the transformation ,96. The group Td contains instead all the symmetry transformations of the tetrahedron, including in addition to T, six reflections through planes and six transformations $4 and $43. The molecule CH4, with C at the center and the H atoms at the vertices of the tetrahedron, has Td symmetry (Fig. 9.4); replacement of a single H atom by its deuterium isotope or a halogen such as CI reduces the symmetry to C3v. The octahedral group O has 24 elements coinciding with the symmetry operations of a cube (Fig. 9.5): three C4 axes that join midpoints of oppo-

234

Chapter 9. D i s c r e t e T r a n s f o r m a t i o n s of Coordinates

site faces of the cube, four C3 axes through opposite vertices, and six C2 axes that join midpoints of opposite edges. There are five classes in all: E, eight rotations C3 and C32, six rotations C4 and C43, three rotations C42, and six rotations C2. The group of all symmetry transformations of the cube is called Oh obtained by adding a center of symmetry: O • Ci. An example of Oh symmetry is seen in the SF6 molecule. The cube and the octahedron are "dual" objects, the midpoints of the faces of one defining the vertices of the other. Thus the same groups O and Oh are the symmetry groups relevant to these two Platonic solids. The tetrahedron is a self-dual Platonic solid. There remain two Platonic solids, the icosahedron (with 20 triangular faces) and its dual the dodecahedron (with 12 pentagonal faces). The "icosahedral groups" Y and Yh, of more limited interest than the tetrahedral and octahedral groups, describe their symmetries, with 60 and 120 elements, respectively.

H

H

H

F i g u r e 9.4: The tetrahedral Td symmetry of methane.

9.2. C h a r a c t e r s of G r o u p R e p r e s e n t a t i o n s a n d A p p l i c a t i o n s 235

F i g u r e 9.5: One each of the three types of rotations constituting octahedral symmetry.

9.2

9.2.1

Characters of Group Representations and Their Applications Abelian

groups

The search for a maximal set of commuting operations has a trivially simple answer when all operations of the group commute with one another (that is, for "Abelian" groups). All group operators can then be diagonalized simultaneously. Each of their simultaneous eigenvectors will never be changed into another one by a group operation, thus constituting by itself an irreducible set. Each set of quantities irreducible under an Abelian group of operators thus consists of a single element. The transformations of different irreducible sets can differ only by the operator eigenvalues in the form exp(i~) since we deal with unitary transformations. Moreover, if n-fold repetition of this operation yields unity, ~p must equal r2r/n. The so-called "table of characters" for an Abelian group consists of the alternative combinations of eigenvalues for all group operations. See, for example, Table 9.1 for the group C2, C 3, or C2h. Each (a-th) row of these tables lists the eigenvalues G~a)(i - 1 , 2 , . . . , g) of the various group operators Oi for one of the possible "species" of (single-element) irreducible sets. The simplest one,

236

C h a p t e r 9. D i s c r e t e T r a n s f o r m a t i o n s of C o o r d i n a t e s

C2, has two such sets, the z axis and the x (or y) axis. The former is invariant under either of the operations E or C~, whereas the x and y axes are rotated through ~r(ei'~ - - 1 ) by the operation C2. The characters of C3, on the other hand, divide into three irreducible sets. Once again, one set, corresponding to the z axis, is unchanged by any of the three elements {E, C3, C32} of this group. This trivial one-dimensional representation is usually denoted by A. But next, under the operation of C 3 - e - e 2i~/3, there are two eigenvectors of this rotation through 120 ~ namely, x + iy. These complex conjugate representations, often not distinguishable, are regarded as a single two-dimensional representation called E. As we know, the irreducible transformation matrix elements of a group form an orthonormal system. For Abelian groups, each matrix consists of a single eigenvalue. The sets of eigenvalues in the various rows of a table may then be regarded as components G (~) of orthonormal vectors f(~) satisfying the conditions E

(9.1)

G("~)G~Z)* - g S ~ z ' F

(9.1a)

-

C~

where g equals the number of operations ("order") of the group. Thus, in Table 9.1, the sum of the squared moduli of each row adds to 2 for (72, 4 for C4, etc. Also, products of corresponding entries in two rows add to zero. Any quantity f that transforms into O~f under the r-th group operator can be expanded into irreducible set elements, f - ~ f(~), by setting O,.f - E Ol

- Z 7"

r

Z

O"f(~) - E

- Z

OC

(9.2)

G~)I(~) '

Oc

- 9I

(931

O~

In the simple example of Ci with its two elements, the identity operation E and the inversion I about a point (the origin x - 0), we have E f ( x ) - f(x), I f ( x ) - f ( - x ) for any function of x. The set of relations (9.2) now takes the form E f ( x ) - g(x) + u ( x ) ,

(9.2a)

If(x)

(9.2b)

- g(z) - u(x) ,

according to the entries in Table 9.1, the irreducible elements in this example being the even and odd parts of f ( x ) ' g ( x ) - [f(x) + f ( - x ) ] / 2 and

9.2. C h a r a c t e r s of G r o u p R e p r e s e n t a t i o n s a n d A p p l i c a t i o n s 237

u(x) : [ f ( x ) - f(-x)]/2.

To illustrate (9.3), apply Ag and Au to (9.2a) and (9.2b) to yield the obvious identities EI(

)+ x

=

El(x)- If(x) = 2u(x).

(9.3c) (9.3d)

The order g of the group equals 2 in this example.

9.2.2

Non-Abelian

groups

When not all operations commute, they constitute a number of "classes." Each operator of a class with more than one element is represented by matrices G (a) that do not commute, at least for some of the "species" (that is, for some of the irreducible sets c~ transformed by the operator). This is because the very definition that two different operations t and r belong to the same class requires G~a)G(a)(G~a)) -1 - GI a). However, the matrices G (a), G~a), etc., of each class have a common ingredient that commutes with all other operations; this is a certain amount of "unit matrix" character, represented by their trace. In the spirit of the expansions into a base set of operators (Section 6.2.1), we may think of each G (a) having dimensions ha x ha as expanded into some orthonormal set of matrices including one proportional to the unit matrix I (a) with its ha nonzero elements along the diagonal equal to 1. The relevant term of this expansion is

1 Tr { G~a) Ia}Ia h--~

Tr [G(a)lh~lIa.

(9.4)

This term of the expansion commutes with all matrices G~a) of the group which change G (a) into GI a), being accordingly identical for all G (a) of each class. The value of Tr (G! a)) - X~a) , (9.5) labeled by the class subscript c, is called the c h a r a c t e r of G (a). Since the terms in (9.4) for all operations r of a group commute with one another, and with all G! a), they constitute a set of commuting operators. This set is also maximal. The eigenvalues of this set coincide with the set of characters X!a) for all classes of the group, constituting the set of "quantum numbers" that identify the "species" (transformation properties) of any one irreducible set of quantitites to be transformed. The most symmetric set, with unit values for the characters of each class, is designated A as in Table 9.1.

238

C h a p t e r 9. D i s c r e t e Transformations of Coordinates

The sets of characters of each non-Abelian group are analogous to the sets of all operator eigenvalues for the Abelian groups. They may be regarded as sets of components of orthonormal vectors, the orthonormality pertaining to both row and column indices, with the properties analogous

to (9.1), - Z r

-

9.6)

c

x~,

-

g~r

(9.6a)

Ot

r

Ot

As an illustration, consider the characters of C3v in Table 9.1. The six elements of this group, E, C3, C32, and three vertical planes of reflection a~, divide into three classes E, C3, and av, with g~ =1, 2, and 3, respectively. There are also three irreducible representations A1, A2, and E. Products of corresponding entries in two different rows, when weighted by go, sum to zero, whereas similarly weighted self-products of a row with itself sum to 6, which is the value of g. This example illustrates (9.6). On the other hand, for any two columns, the sum of the products of corresponding entries vanishes if the columns are different, but equals g/g~ if the same, so that this sum of squares of the three columns equals 6, 3, and 2, respectively, in conformity with (9.6a). Conversely, turning the remarks in the above paragraph around, tables of characters of irreducible representations can be constructed through the equations (9.6) supplemented sometimes by relations such as x ( C 3 2) Ix(C3)] 2. Thus, with reference to the group C3~ or D 3, one starts with the first row, which is the trivial representation A with unit entries for each element of the group. Next, the requirement from (9.6a) that the sum of the squares in each column be g/g~ yields entries 1 and 2 to within signs for the first column, 1 and 1 (again to within a sign) for the second, and 1 and 0 for the third. Orthogonality of different irreducible representations (between different rows) finally fixes the signs. The sets of characters serve to determine the number and type of species contained in any reducible set but do not identify any labeling of the elements of irreducible sets or the individual matrices G (~) whose rows and columns correspond to these labels. This labeling is obtained, as for the space rotations, by reference to eigenvectors of a sufficient number of operators that do not commute with all operations of the group. This labeling

9.2. C h a r a c t e r s of G r o u p R e p r e s e n t a t i o n s a n d A p p l i c a t i o n s 239

function has been performed for space rotations by the eigenvectors of J z in the spinor base, and by those of the pair { gz2, ( - 1)lDy (r) } in the Cartesian base. No corresponding standard base appears to have been established for the classes of irreducible representations of point groups. As is also clear from Table 9.1, wherein different groups share the same entries, sets of characters alone cannot specify relevant symmetries or symmetry groups. 9.2.3

Characters

of the rotation

group

SO(3)

The role of group characters, central to application of point groups, is illustrated further by viewing them in the broader context of continuous group characters, even though these characters played no role in Parts A and B. The group of rotations in three dimensions is a continuous group in terms of the angle T of rotation about a relevant axis. Characters for its representations can be defined in analogy to those for discrete groups. Thus, consider first orbital angular momentum t = 1. Rotations with respect to any axis are specified by matrices analogous to (4.39); we choose the canonical form for the z axis cos~

-sin~ 0

sin~ cos~ 0

0 ) 0 1

which coincides with (4.39), setting there r - O namely, the trace of this matrix, is therefore X(1) - 1 + 2 c o s ~ .

(9.7)

0. The character,

(9.8)

The same result follows from the matrices in Table 4.1, where again, for j-l, 3a 2 - ~ 2 l+2cos0. Rewriting (9.8) as X (1) - e i~ + 1 + e - i ~ ' , (9.9) provides a general and useful expression, which also embraces the halfinteger values j of the group SU(2)"

~(J)- E eim~"

(9.10)

rn---j

The irreducible representations with any value of j with multiplicity (dimension) 2j + 1 are labeled further by the eigenvalues m of the operator J~,

240

C h a p t e r 9. D i s c r e t e T r a n s f o r m a t i o n s of C o o r d i n a t e s

each of these one-dimensional representations contributing e i~~ 1 we have Thus, from Table 4.1, for j - 7,

to X (j)

;~( 89 - 2 cos(9/2).

(9.11)

Performing the geometric series sum in (9.10) yields x(j) = sin(j + 89 sin 71 9

(9.12)

t an equation equivalent to 1 + 2 }-~m=l cos m9 for integer values of j - g and to 2 }-~m=0 J- 89cos(m + 7)9, 1 for half-integer values of j. The function in (9.12) also coincides with the Chebyshev polynomial of the second kind, g 2 j + l ( C O S 79,, 1 ) which will be encountered again in Section 10 1.2. The orthogonality of different irreducible representations, cast algebraically as (9.6) for discrete groups, has its continuum counterpart for pairs of different representations ;~(J) and x(J'):

( x(j ) ~(j') ) _ ~jj '

f02~ d9 !

~

1 sin(j + 89 sin(j' + 89 2 sin 2 79 sin 89 sin 79 1

(9 13) *

*

The arrangement of the right-hand side, casting an obvious orthonormality relation obeyed by sine functions in the interval (0,27r) in terms of the functions (9.12), shows that the requisite integration measure for these X is (1 - cos 9)(dg/2~r). 9.2.4

Reduction

of representations

A major contribution of character tables of irreducible representations lies in resolving a general representation D of a symmetry group in terms of irreducible representations D (") that label rows in Table 9.1: D - Z

a~'D(~')"

(9.14)

The coefficients a~ of such a reduction follow upon writing the corresponding expression for the characters of D, X- Z

a~'x(v) '

(9.14a)

multiplying it by X(u), and applying the orthonormality relations (9.6) to yield au - (X (u), X). The coefficients av often emerge by inspection of the small set of numbers in any of the tables in 9.1.

9.2. C h a r a c t e r s of G r o u p R e p r e s e n t a t i o n s

and Applications 241

Consider thus the vector representation V : ( x , y , z ) under the point group symmetry C 3. The element E transforms the vector into itself, its matrix operation being the unit 3 • 3 matrix with trace •(V)(E) = 3. The other two elements, C3 and C32, transform (z, y, z) according to rotations about the z axis through 27r/3 and 47r/3 whereby the trace (9.8)of the corresponding matrices in (9.7) vanishes in either case. Thereby, ~:(y) for the vector representation is (3, 0, 0) and the table of characters for C 3 (recall 1 + e + e2 = 0) identifies the vector representation as A + 2E. On the other hand, the same vector yields under the symmetry group C3v , x ( E ) = 3 and x(C3) = 0, but a reflection in a vertical plane shows one of the three coordinates to change sign yielding X(a~) = 1. The vector is thus represented by X(y) - (3, 0, 1), reducing upon inspection to A1 + E. Likewise in the symmetry group D3, where again x ( E ) = 3 and x(C3) = 0, two coordinates change sign under rotations by 7r about a horizontal axis to give X = - 1 , representing a vector by x(v) = (3, 0 , - 1 ) which decomposes to A2 § E. Particularly noteworthy now is that in D 3, unlike in C 3 and C3,, the vector representation does not include the identity representation A1. Thus no system with D 3 symmetry can support a nonvanishing vector physical quantity such as a dipole moment. On the other hand, C3~ symmetry represents an electric dipole moment (a polar vector) by (3, 0, 1), the NH3 molecule providing an example. Note that an axial vector which behaves like a polar vector under rotations but oppositely to it under reflections is represented by (3, 0 , - 1) under C3, which reduces, upon inspection of Table 9.1, to A2 + E. Once again, the absence of A means that the NH3 molecule possesses no axial vector character such as a magnetic dipole moment. In general, groups which include reflections operate either on electric or magnetic dipole moments but not both simultaneously. Angular m o m e n t u m representations can also be discussed in terms of irreducible representations of point groups. This analysis becomes relevant for atomic orbitals in molecular or crystalline environments with discrete symmetries, lower than the isotropy of a free atom. Indeed, g - 1 orbitals, which transform like vectors, are described just as in the last paragraph. For a C 3 symmetry, the characters of g = 1 are given by the expression (9.8) with T = 0, 2 r / 3 , and 4 r / 3 for the elements E, C3, and C32, respectively. Thus, we have X(1) = (3, 0, 0)exactly as before for X(y), which again decomposes to A + 2E, lifting the threefold degeneracy of the free atom partially, separating the energy levels with m = 0 from Iml = 1. Consider on the other hand, the g = 1 orbital in the tetrahedral symmetry T. From Table 9.1, the relevant group elements are E, C~, C3, and

242

C h a p t e r 9. Discrete Transformations of Coordinates

C3 2, X(1) being thus evaluated in (9.8) for ~ - 0, r, 2 r / 3 , and 4 r / 3 , that is, X(1) _ ( 3 , - 1 , 0 , 0). This pattern coincides with that of the irreducible tetrahedral representation F (sometimes called T) where the threefold degeneracy of g : 1 persists. Likewise for Td, the vector representation with character (3, 0 , - 1 , 1 , - 1 ) coincides with F2. Note also, as in the last paragraph, that upon reduction these vector representations no longer contain the identity A whereby molecules such as CH4 possess no dipole moment, whereas CH3C1, with C3~ symmetry, does have an electric dipole moment.

9.2.5

Reduction

of set products

Products of irreducible representations D (~) and D (v) with point symmetry can also be reduced to sums of irreducible representations D(~) • D ( ~ ) -

Z D(~

(alPv> 9

(9.15)

(7

This expansion provides the Clebsch-Gordan series for the point symmetry group, analogous to the corresponding series for SO(3) studied in Part A. The reduction is effected as in Section 9.2.4 because the characters of the product on the left-hand side in (9.15) are simply products of the characters of X(~) and X(~). Thus, the product E x E in Ca~ has the character (4, 1, 0) from Table 9.1, which resolves into A1 + A2 + E by inspection. Therefore, all three coefficients of (9.15) for a - 1, 2, 3 equal unity. On the other hand, multiplication of any representation by the identity representation leaves it unchanged; for example, A1 x E, with character ( 2 , - 1 , 0 ) , yields E itself. Products of the vector representation V with the various irreducible representations play a particular role in providing selection rules for electromagnetic transitions. The electric dipole operator vector resolves into irreducible representations as discussed in Section 9.2.4, yielding, for example, V - A1 § E in Cav symmetry. As indicated in Table 9.1, A1 may be identified with linear polarization along the z axis while E describes linear or circular polarizations in the x y plane. Polarization along z amounts to the identity yielding A1 • A1 - A1, A1 x As - As, and A1 x E - E; the selection rule for light absorption or emission with z polarization reduces thus to AI --+ A1, As --+ As, and E -+ E. On the other hand, the selection rules, for the x y polarization E of V, yield A 1 , A 2 --+ E and E --+ A 1 , A 2 or E, because E • As - E and E x E - A1 + As + E as discussed in the previous paragraph. The product of two vector representations, V x V, also reduces in this manner, finding many applications. Thus, under C3. , where X (v) =

9.3. S y m m e t r i e s of M o l e c u l e s a n d Crystals

243

(3,0, 1), the characters of V x V are (9,0,1), usefully resolved into an antisymmetric part X(v• - = (3, 0 , - 1 ) just as in the axial vector and a symmetric second-rank tensor with X (y• = (6, 0, 2). Stress and strain tensors, the Pij and sij of Section 1.1.1, or the electrical conductivity (rij, are examples of such symmetric tensors. Once again they reduce under C3v by inspection to 2(A1 + E). The occurrence of two copies of the identity representation signifies that the most general such system with C3, symmetry has two independent parameters serving to characterize one of these symmetric second-rank tensors. On the other hand, the tetrahedral group T has •(y) _ ( 3 , - 1 , 0 , 0 ) , yielding ;~(v• _ (6, 2, 0, 0), whence it follows from Table 9.1 that (V x V)+ - A + E1 + E~ + F. The occurrence of a single A, that is, of a single identity representation, means that these tensors reduce to a single scalar quantity, such as the conductivity (r, for tetrahedral symmetry. The perspective thus generated through group characters also applies to combining angular momenta. Consider two angular momenta j - 1 with the characters X(~) 1 - 2 cos !2~ according to (9.10)or (9.12) or from Table 4.1. Their product 4 cos 2 !2~ can be rewritten as 2 + 2 cos ~ which equals ~(0) + X(1) according to (9.8), a familiar result for adding two j - 71 angular momenta. Similarly, from the expression (9.12), a product x(Jl)x (j2) resolves into the sum of X(j) with ]jl - J21 _< J _< jl + j2.

9.3

Symmetries of Molecules and Crystals

The study of discrete coordinate transformations aims at classifying stationary states of molecules and crystals, just as the study of infinitesimal r-transformations has served for single atoms or single nuclei. Treating molecules or crystals faces, however, the coexistence of electronic and nuclear motions that remain independent only in zero-order approximations. Similarly, nuclear physics views a single nucleon's motions as independent only in the "shell model," which needs complementing by the study of collective motions. Crystal structures are invariant not only under point groups but also under lattice translations that are discrete but repeatable to quasi-infinite extent, whereas the transformations studied in this chapter return to the identity, E, after finite cycles. Lattice translations of crystals and collective motions of nucleons exceed the scope of this book. The standard procedure to classify stationary states of molecules and

244

Chapter 9. Discrete Transformations of Coordinates

crystals views atomic nuclei at the outset as fixed at specified positions with appropriate symmetry. This symmetry extends, of course, to the Coulomb potential field generated by the nuclei and thus to the electronic energy eigenfunctions governed by that field. Degenerate eigenfunctions form accordingly irreducible tensorial sets, properly labeled by quantum numbers of the relevant group and of one of its subgroup chains. The initial assumption of fixed, symmetric positions of the nuclei is, of course, not quite realistic. The preceding paragraph's procedure, called the "Born-Oppenheimer approximation" in molecular physics, includes a chain of analogous steps to determine, for each electronic stationary state, the dependence of its energy on shifts in nuclear positions that preserve the initial symmetry. The resulting eigenenergy's dependence on nuclear positions, E(R1, R~,...), serves then as the potential term for a second SchrSdinger equation governing nuclear oscillations about their lowest-energy positions. Eigenvalues of this second equation represent energy levels of "vibrons" for molecules and of "phonons" for crystals, once again labeled by symmetry quantum numbers analogous to those of electronic spectra. Molecular treatments involve a third sequence of analogous steps concerning the energy levels of molecular rotations in free space. This sequence utilizes combined electronic and vibrational eigenfunctions to calculate the molecule's inertia tensor and therefrom its rotational spectrum. (This third step is not relevant to crystals, whose orientation is generally fixed in a laboratory. A crystal's unit cell amounts to an analog of a molecule described in its "body-frame" coordinates.) 9.3.1

Symmetry

combinations

The symmetries of electronic, vibrational, and (where relevant) rotational states combine, in essence, much as the symmetries of different orbitals did in Part B for atomic electrons of open shells. That is, products of electronic, vibrational, and rotational eigenfunctions constitute reducible representations of molecular or crystalline groups. Reduction of these representations singles out eigenfunctions of whole molecules or crystals. The interactions of electronic and nuclear motions are evaluated at this stage, separately for each eigenfunction, thus removing initial degeneracies. Simple examples of this procedure are presented in Section 9.3.4; a more systematic treatment of the whole process occurs in Chapters 11-13 of the Landau-Lifshitz book on quantum mechanics [12], more detailed ones in treatises of molecular or

9.3. S y m m e t r i e s of Molecules a n d C r y s t a l s

245

crystal physics [39]. 9.3.2

Vibrational

motions

The treatment of molecular vibrations is simplest for diatomic molecules, whose electronic energy dependence on nuclear positions reduces to a single variable function U(R) of the internuclear distance R. This function has a single minimum for vibronic states at their "equilibrium distance" of the nuclei R = Re. Small departures of R from Re are represented by the expansion U(R)- U(Re)+ 89 R~) 2 + - . - , (9.16) whose subsequent "anharmonic" terms are often disregarded. The resulting eigenvalues and eigenfunctions are provided, in this approximation, by the usual treatment of a harmonic oscillator with frequency ,, - (k/M) 89 M indicating the reduced mass of the nuclear pair. For polyatomic molecules, the analogue of the expansion (9.16) replaces the squared departure from /~e by a quadratic polynomial in the nuclear positions {/~1,/~2,...,/~N}. Diagonalization of this polynomial--which determines the vibrational "normal modes"--reduces it to the form _.~i "~kiQi,1 2 where Qi - (Ri - Re) represents the normal mode coordinate R~'s departure from the equilibrium position/~. Independent energy eigenvalues and eigenfunctions then describe the various normal mode vibrations. The complete potential functions, including their anharmonic terms, namely, U(R) for diatomics and U(Qi) for the polyatomic normal modes, generally flatten out at large values of R or Qi--where chemical bonds begin to fail--attaining a limiting value U(oo), called the "dissociation threshold." Vibrational excitation energies in excess of U(cx)) belong to the continuous spectra of dissociating states. These spectra also include "repulsive"eigenstates of nuclear motions whose potential functions attain their equilibrium values at R - oo or Qi - oc, rather than at a finite R or at Qi - 0 . 9.3.3

Molecular

rotations

Standard variables of rotational eigenfunctions are sets of Euler angles {~o, 0, r connecting the molecule's inertial axes--that is, its "body f r a m e " to a laboratory coordinate frame. The eigenfunctions themselves are generally the D functions described in Chapter 4, or linear superpositions thereof. The corresponding energy eigenvalues are trivially simple---proportional

246

C h a p t e r 9. D i s c r e t e Transformations of C o o r d i n a t e s

to J(J + 1)--for inertially isotropic molecules with equal inertial axes. They are represented analytically for axially symmetric ("symmetric top") molecules having two equal axes by eigenfunctions with three labels (J, K, M) that represent the total angular momentum J and its components along the molecular symmetry axis and a laboratory axis, respectively. (States with different M are usually degenerate as they are for atoms.) The rotation of "asymmetric top" molecules, with three different axes of inertia, requires instead a numerical evaluation of both eigenvalues and eigenfunctions, labeled by two standard quantum numbers (J, M) in addition to an index of the eigenvalues' sequence.

9.3.4

Stability analysis of nuclear positions

The Born-Oppenheimer approach to molecular structure hinges on electronic energy levels having a lowest value for a specific set of nuclear positions. The determination of normal modes for polyatomic molecules implies the occurrence of a stationary point of the electronic energy at which all Qi vanish but remains incomplete in the presence of degeneracy, that is, when two, or more, of the ki parameters coincide. A further analysis of the potential's expansion beyond its quadratic terms is then required. An important example of this phenomenon, called the "Jahn-Teller effect" ([40], and Sec. 102 of [12]), was detected in the early days of molecular physics, in the behavior of linear molecules. Some of the Qi coordinates represent in this example "stretches" of the bond lengths along a chain, others representing "bendings" of bond angles away from 180 ~ The latter, indicated by a pair of degenerate coordinates (Qj, Qk), are orthogonal to one another and to the molecular axis. Stability analysis of the linear structure of excited states was performed by expanding the electronic energy into powers of two coordinates Q/ and Qk and each of these coordinates, in turn, into powers of the departures of the original nuclear positions from axial symmetry. This analysis revealed in a number of instances an instability along certain directions in the (Qi, Qk) plane. It thus alerted physicochemists to the phenomenon of molecules having a ground state stable in a linear configuration, yet bent out of linearity in a class of stationary excited states. The water molecule provides a standard example of a bent ground state configuration (Fig. 9.3). Symmetry groups reduce the determination of "normal modes," that is, of eigenvalues w and of corresponding eigenvectors to a smaller set than the nine coordinates Qi of the three nuclei [39]. As already noted, H20 has C2~ symmetry, the molecule being invariant under

247

9.3. S y m m e t r i e s of M o l e c u l e s a n d C r y s t a l s

a rotation through r about a z axis (C~) and reflection through the x z plane (or,). It is also unchanged under reflection through the yz plane of the paper (t/v). The only contributions to the characters for these operations stem from those of the nine coordinates Qi that remain invariant since only these stand on the diagonal of the 9 • 9 matrix representing the operation and thereby contribute to the trace in (9.5). Thus, under C2, it is only the O atom we need consider and the symmetries z3 ---* z3, x3 ~ - x 3 , and Y3 ---* - y 3 yield X(C2) = - 1 . Similarly, under cry, once again only the O contributes, the two H atoms interchanging their coordinates, yielding I Y3 --~ -Y3 with x3 and z3 unchanged, giving X(O'v) - 1. The reflection a v leaves all three atoms fixed; that is, the z and y coordinates are unchanged I for each atom and only x ---* - z so that X(O'v) - 3. With the trivial x(E) = 9, the set of characters for {E, C2, a,,,Crv,} is, therefore, ( 9 , - 1 , 1,3). The most general transformations considered above embrace far more than the normal modes of vibration because they include also overall translations of the center of mass and rotations with respect to it. There are six of these operations which account for ( 6 , - 2 , 0, 0) of the characters noted above so that the three vibrations alone make up the difference (3, 1, 1, 3). Turning to Table 9.1, this vibrational contribution resolves for C3~ into 2A1 + B2. These three normal modes of vibration are shown in Fig. 9.6. Actual computation of the three eigenfrequencies requires of course specifics about the interatomic potentials.

*O H

~O

~'-O

H

H,, (r

T +z

""

A1

H

A1

""

B2

F i g u r e 9.6: The three normal modes of vibration of the water molecule.

248

C h a p t e r 9. D i s c r e t e T r a n s f o r m a t i o n s of C o o r d i n a t e s

Table 9.1: Characters of irreducible representations of point groups (from ref. [12], with kind permission of Butterworth-Heinemann Ltd.). Boldface symbols indicate groups. Columns are labeled by classes of symmetry operations, E being the identity, Cn rotations, and a reflections; numerical prefactors denote multiplicity. Rows are labeled by irreducible representations, A or A1 denoting the trivial one-dimensional representation and E a two-dimensional one. Labels of coordinate axes shown alongside provide examples; thus C 3 z : z , C a ( x .4- iy) = -4-e(x -4- iy). e =_ exp(2iTr/3) and w - exp(2iTr/6). Ci

E

I

C3

E

o"

[ E

2 5

Ag Au;x,y,z

A;z B;x,y

C2h C2

D2 Ag

Bg A,~; z Bt,; x, y

A1;z B2;y A2 B1;x

A B3; x

B1;z B2;y

A'", x , y A'"~Z

1 1 1 -1

E E

oh av

C2 C2

1 1 1 -i 1 1 1 -i

1 -i -I 1

A

B;z

E; x -4- iy

E2; x -4- iy

E

-i -I

--03 03 2

--03

1

1 1

1

-i

1 1 --03 0,32 03

2

--03

A2;z E;x,y

C2 C2

-1

C3

A1

C4 5'4

1 1 1

1 032

E E

D3 A1;z A2 E;x,y

1 1

C6 -1

e2

C3,~

E E

$4 A;z B

E1

1

E C~ C~ C~

C4

C6 A;z B

E; x :t: iy

I a~i

i

C4 3 $4 3 1 -1 -i

-1 -1

C2

1 -1 1 1 -1 -1

C3 2

{Ill1 e1 c12

'

. . . .

Ca

i

C3 2 1

C6 5 1

1

-1

2 O3

--03

--03

032

--03

--03 2

O32

O3

r

2(?3 3c% 2C3 3U2 1 1 1 -1 -1 0

249

9.3. S y m m e t r i e s of Molecules a n d C r y s t a l s

Table 9.1: Continued

E

C 4 t)

D:~d A1 A2 B1 B2; z E;x,y

A1 A2;z B1 B2 E;x,y

D6

C6,~ A1 A2;z B1 B2 E~ E1; x, y

D3h A~ A~ A~I' A~2~;z E~ ; x, y E"

A1;z A2 B2 B1 E2 E1; x, y T A E

~

F;x,y,

z

E 1 1 1 3

O

T~ A1 A2 E F2 F1; x, y, z

A1 A2 E F2 ; x, y, z F1

2~'~

E 62 264 2U2 2U~ E C2 2S4 2U2 2od 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 2 -2 0 0 0

D4

A1;z A2 B1 B2 E;x,y

C2 2C4 2~,

E E E

C2 C2 ah

1 1 1 1 1 -1 1 -1

2 2 2 -2 3C2 1 1 1 -1

2C3 2C3 2C3

2C6 2C6 2S3

1 1 1 1

1 1 -1 -1

-1 -1

1 -1 -1

-1

1

0 0

0 0

4C32 1 c2 c 0

E

8C3 3C2 6C2 6C4

E

8C3

3C2

1 1

1 1

1 1

2 3 3

-1 0 0

2 -1 -1

6ad

3U~ 3q~ ! 3a~

1 -1 1

-1 1

4C3 1 ~ e2 0

3U2 3~ 3U2

6S4

1

1

-1

-1

0 1 -1

0 -1 1

250

Chapter 9. Discrete Transformations of Coordinates

Problems 9.1 Following the procedure in the two paragraphs after Eqs. (9.6), construct from these equations the table of characters for the groups T and C4~. 9.2 How is the degeneracy of an g = 2 atomic orbital affected for an atom lying at the center of a system with (a) C3v, (b) T symmetry? 9.3 An atom is subjected to a field with hexagonal symmetry D6. a) Analyze the splitting of orbitals with g = 1 and 2. b) Compute any further splitting generated by reducing the hexagonal system to D 3 symmetry. 9.4 Apply the second paragraph of Section 9.2.4 to selection rules for electric dipole transitions in the tetrahedral molecule CH4. 9.5 Through a consideration of characters, show that the combination of two angular momenta jl and j2 gives the Clebsch-Gordan series with one each of the values IJl-J21, [jl-j2] + 1 , . . . , jx +j~, as we know it. 9.6 Classify the normal modes of NH3 vibration according to irreducible representations of its C3v group. Apply the same procedure to the planar molecule BF3 with D3h symmetry.

Chapter 10

Rotation Groups in Higher Dimensions: Multiparticle Problems Symmetries play two different roles in the study of physical systems. The more obvious one proceeds by identifying the invariance of a system's mechanics under specific transformations, typically under rotations or reflections of coordinates. Such transformations form the invariance group of the system. Less direct procedures disregard initially certain aspects of the system's mechanics, thus utilizing broader symmetries of a preliminary formulation in further analysis. Transformations preserving such symmetries form noninvariance groups. Familiar examples of these procedures include the shell models of atoms and nuclei, which treat electrons or nucleons initially as independent particles moving in symmetric average potentials. The identification of a relevant invariance group often begins by viewing a system in its center of mass frame: A two-body system reduces thereby to a single effective particle with a "reduced mass," whose position is represented by a single vector Y with origin at the pair's center of mass. Similarly, one views an N-body system in its center-of-mass frame representing its constituents' positions by 3 ( N - 1) coordinates. Rotations of the system about its center of mass form generally its invariance group, for example, SO(6) for the three-body problem. Extensions of our treatment to such broader groups will be considered in Sections 10.2 and 10.3, together with their noninvariance subgroups. 251

252

Chapter

10. R o t a t i o n

G r o u p s in H i g h e r D i m e n s i o n s

Extending the r-transformations of physical space to higher dimensions may proceed by adding one dimension at a time. The first step of this procedure, described in Section 10.1, has particular interest because it deals with the Coulomb-Kepler system of two-body motion, with interparticle potential proportional to l / r ; its invariance complements space coordinates with an eccentricity parameter, forming thus a four-dimensional space with the invariance group SO(4). This invariance proves exceptional by reducing to a product SO(3) x SO(3). Its extension to still higher dimensions will be the subject of Section 10.2, including the introduction of hyperspherical harmonics that replace the familiar Ylm(O, ~v). Section 10.3 will deal instead with further developments, concerning primarily the interplay of noninvariance and invariance groups and their physical implications. Further extensions of rotational symmetry will occur in the following Chapters 11 and 12, through the device of introducing imaginary angles that, for example, extend the symmetry of physical space coordinates to include the time variable of special relativity. Another extension includes couplings to an external field, most notably the electromagnetic field, yielding transitions between different energy states. The resulting groups are called noncompact owing to the occurrence of infinite-dimensional representations which accommodate all the states that can be transformed into each other through the action of the operators (for example, the dipole operator ~ constituting the group.

10.1

F o u r - D i m e n s i o n a l Rotations: The Coulomb-Kepler Problem

The attractive central force problem with potential proportional to ( - l / r ) is well known to have additional symmetries besides its obvious isotropy in space. The classical Kepler orbits are closed ellipses whose eccentricity is embodied in a constant vector 5, with magnitude proportional to the eccentricity and direction along the major axis: d5/dt = 0. This "LaplaceRunge-Lenz" vector i provides three further conserved parameters besides 1 Laplace recognized this conserved vector quantity for Kepler orbits long ago in his Traite' de Mecanique Celeste (1799). W.R. Hamilton discovered it independently, calling it the eccentricity vector. A clear description in the language of vectors was given by J.W. Gibbs in 1900 and repeated by C. Runge in a German text Vektoranalysis (Hirzel, Leipzig, 1919). It has become customary to attach the names of Laplace and Runge along with that of W. Lenz who first discussed this vector for the quantum Coulomb problem [41]. For further details on this vector and its history, see Goldstein [42].

10.1. F o u r - D i m e n s i o n a l R o t a t i o n s

253

the three components of the angular momentum g. Six is precisely the number of independent rotations in a four-dimensional space (see Problem 2.1 and footnote 1 on p. 35); indeed, these six quantities imply invariance of the Coulomb-Kepler problem (at fixed energy) under SO(4) rotations. This invariance persists in quantum mechanics, being reflected in the additional degeneracy of energy eigenvalues of the hydrogen atom with respect to the quantum number g besides the degeneracy in m expected from the SO(3) spherical symmetry in physical space. Defining ~ as

f-f

x

6+

§

(10.1)

where p is the reduced mass of the electron in a hydrogen-like atom and E , is its energy, each component of this vector commutes with H =

(f2/2~)-

(10.2)

There also emerge the commutation and orthonormality relations [gi, aj] - ieijkak , [ai, aj] -- ieijkgk , [" ~ - - O,

(10.3)

where_fi/k is the completely antisymmetric symbol in three indices. Therefore, g is axial, whereas ~ is a polar vector; the latter lies in the plane of the orbital ellipse while g'is perpendicular to it. Together, [ a n d ff serve as the generators of the four-dimensional rotation group SO(4). The Hamiltonian in (10.2) is expressed in terms of the generators as H-

- ( ~ e 4 / 2 h 2 ) / ( P + ~2 + 1),

(10.4)

depending only on the combination ~ + if2 _ (~'+ ~)2 _ ( [ _ if)2, which is a "Casimir invariant" of SO(4), that is, an operator that commutes with all the generators and is thereby the counterpart of ~ for three-dimensional rotations. The action of a component of ~"or ~ on any energy eigenstate changes it into other states of the same energy. While ~, and ty change the azimuthal quantum number m by +1 without changing the g values, the operators ~ also change the ~ value by 4-1 as implied by (10.3), accounting for the degeneracy of energy eigenstates in both g and m. 10.1.1

Spherical

and parabolic

representations

Alternative sets of three mutually commuting operators, including the Hamiltonian, afford thus alternative types of hydrogen-atom eigenstates. The set,

254

C h a p t e r 10. R o t a t i o n G r o u p s in H i g h e r D i m e n s i o n s

{H, ~ , gz }, with associated eigenkets Ingm), corresponds to the usual spherical representation, while a second set, {H, az, gz }, with kets In, n l - n2, m), belongs to a "parabolic representation," nl and n2 being parabolic quantum numbers with values 0, 1, 2 , . . . , ( n - Im[ + 1). The necessity for two further labels, whether (g, m) or (nl, m), reflects the rank 2 of the 5'0(4) group, the rank of a group being the number of labels in addition to the invariance label n required for a unique specification of the states. Passage between the two sets proceeds most conveniently in terms of the further combinations 2 (10.5) which, as a consequence of (10.3), momenta: [jli, jlj] [j2i, j2j] [jli, j2j]

behave like two independent angular

----

iQjkjlk , iQjkj2k, O.

(10.6)

In terms of these jl and j~, the six generators of SO(4) resolve into two independent sets of SO(3) generators thus factoring into the direct product" SO(4) D S01 (3) x SO2(3), or, more precisely, SU1 (2) x SU2(2), since the quantum numbers ( j l , j 2 ) also take half-integer values. Representations of four-dimensional rotations factor, therefore, into the products of two rtransformations, D(Jl)D (j2) The SO(3) algebra of earlier chapters adapts immediately to provide the irreducible representations of SO(4). The invariant ~ + ~2 turns into 4jl~ - 4~'~, the equality owing to the vanishing of the second Casimir invariant g'. 5. The energy eigenstates of (10.4) with quantum number n correspond to jl - j2 - - } ( n - 1), these angular momenta running over the integers and half-integers, with degeneracy (2jl + 1)(2j2 + 1 ) - n 2. The parabolic coordinates corresponding to these angular momenta form the space of the u n c o u p l e d representation with j21 , j 2 , j l z , and j2z as mutually commuting operators, while the spherical coordinates are coupled with momenta j 2 , j 2 , j 2 , and j~, where j - jl + j-'2 - g-'. The unitary matrix that transforms one description into the other is thereby the Clebsch-Gordan matrix of angular momentum addition:

1~2m) - ~

Inem)(j~j2emlj~jlz,j2j2z),

(10.7)

g 2With g'an axial, and ~ a polar, vector, the objects 3'1 and j2 in (10.5) form a chiral pair, each transforming into the other under parity. See also Section 11.2.2.

10.1. F o u r - D i m e n s i o n a l R o t a t i o n s

255

with

jlz,2zInln2m ) --

~1 ( m -4- r/1 :]: n 2 ) l r / l n 2 m >.

(10.8)

The results of Section 5.1 thus carry over to the treatment of the hydrogen atom and, in particular, to interconnect the alternative descriptions in terms of spherical and parabolic symmetry which arise from the "higher" SO(4) symmetry of this problem. Historically, Pauli's treatment by matrix mechanics of the hydrogen atom followed this route wherein the quantization of H in (10.4) reduces to that of angular momenta jl and j2. We can also record the action of the other operators in this system in terms of angular momentum algebra:

jl:i:[nln2m) -

89

l :F(m+nl-n2)][n- l •

89 zhl, n2, m), (10.9)

with a similar expression with 1 and 2 interchanged for the action of j:+ on the parabolic eigenkets. SO(4) is a higher symmetry or "invariance" group of the nonrelativistic Hamiltonian of the hydrogen atom. Every representation of SO(4) labeled by the quantum number n representing the invariant ~ + 52 _ 412 _ 4 ~ , contains in it one each of the states with g - 0, 1 , 2 , . . . , ( n - 1). Each of these belongs to a base set of SO(3) and contains (2g + 1) states characterized by the invariant m of the next subgroup SO(2). This hierarchical buildup of representations is characteristic of the orthogonal groups. For comparison, an isotropic rigid rotor in three dimensions with Hamiltonian H - ~ / 2 I , with I the moment of inertia, has only SO(3) as its highest invariance group. Each level with a given g has a distinct energy. Once again, all the states with g less than some integer n can be grouped into a single representation of SO(4) which is no longer an invariance group but a "noninvariance" group of the rotor since 5 does not commute with ~ / 2 I . Likewise, we will see in Section 10.3 that states of hydrogen with n _< u can be accommodated into a single irreducible set of the next orthogonal group, SO(5), the noninvariance group of their set.

10.1.2

Rotations

in four

dimensions

An alternative to viewing SO(4) as a product of two SO(3)'s looks at it directly as dealing with rotations in a four-dimensional space indexed by one radial and three angular coordinates, 01, 0, and ~, collectively denoted by

256

Chapter

10. R o t a t i o n

G r o u p s in H i g h e r D i m e n s i o n s

f~. The unit sphere in this four-dimensional space utilizes four coordinates ~0

--

COS 01 ,

~z (~ (y

-

sinOlcosO, sinO1 sinOcos~o, sinOx sinOsin~o,

-

-

(10.10)

with the three independent parameters: O
O
O<~o<2~r,

(10.11)

[ dFt - 27r2.

(10.12)

and surface element df~ - sin 2 O1 sin OdOx dOd~o,

J

The great-circle arc length between two points ( a n d ('on the sphere equals 2 ( 1 - cosw)

-

( ( _ (,)2 +

_ ~)2,

(r

(10.13)

with

~'OC~)- -

01 sin 0~ cos 7

(10.13a)

cos 7 - ~" ~' - cos 0 cos 0' + sin 0 sin 0' cos(~o - ~ o ' ) .

(10.13b)

CO8r

- - ~ - ~ - t "4-

COS 01 COS 0~ -~- sin

and

Four-dimensional spherical harmonics Y(01,19, 9) are eigenfunctions of the angular part of the Laplacian in four dimensions, that is, u=0 ~ u 2

Y ~ l m ( a ) - ( n - 1)(n + 1)Y~tm(a).

(10.14)

In analogy with the squared angular momentum in three dimensions, ~ , the operator on the left-hand side of (10.14) may be dubbed the squared four-dimensional angular momentum A42 - ~ u > , g2u,' with antisymmetric rotation matrices, gu,,/~, u = 0, 1,2, 3, obeying the commutation relations [gu,, g0~] = i(6,,ogup + 6uog,,o - 6,oguo - 6uog~0) ,

(10.15)

whose equivalence to (10.3) emerges by setting gij = eijk/~k and gi0 = ai for i, j, k = 1, 2, 3. This correspondence is displayed by the array

g u ,,

-

0

--ax

--ay

a ~,

0

g ,,

ay az

-gz gy

0 -g~,

-az -

gy

g,: 0

'

(10.16)

10.1. F o u r - D i m e n s i o n a l R o t a t i o n s

257

with the six 4 x 4 rotation matrices displayed explicitly as 0 0 0 0 (10.17a) 0 i 0 0 0 0 i 0 0 0 0 i -i 0 0 0 0 0 0 0 0 0 0 0 ax " 0 0 0 0 ay 9 - i 0 0 0 az " 0 0 0 0 0 0 0 0 0 0 0 0 -i 0 0 0 (lO.17b) These, purely imaginary, Hermitian matrices provide the Cartesian base generalization to four dimensions of the similar sets in (2.13) and (2.18) for three-dimensional rotations. Defining a "dual" object ~.

0

0

0

0

0 0 0

0 0 0

0 0 i

0 -i 0

iv.

0

0

0

0

0 0 0

0 0 -i

0 0 0

i 0 0

i .D - 71Z

~.

e..oot;o ,

0

0

0 0 0

0 i 0

0 -i 0 0

(10.17c)

pa where e,,pa is the completely antisymmetric (under interchange of any two indices) Levi-Civita symbol in four dimensions, note that g and - ~ are mutually dual. The vectors j~ and j~ in (10.15) are instead, anti self-dual and self-dual, respectively" jD _ --jl, jD _ 32. The four-dimensional spherical harmonics in (10.14) are eigenvectors of the operator A~ - ~ > ~ g~, in the form A42 Y n l m ( a ) -- A()~ -4- 2)Ynlm(a),

(10.18)

with A - n - 1 coinciding with 2jl - 2j2 of (10.5). Explicitly, the harmonics are products of the three-dimensional spherical harmonics and of Gegenbauer polynomials (Chapter 22 of [14]) in 01" gns

-t-1

(~-~) -- (7t+1 ~ n - l - 1 ( c o s O 1 ) g t m ( O , ~)

(n + e)[

sin 101 d(cosO1)

(10.19)

Un_l(COS01),

(10.19a) where Un(x) - C ~ ( x ) are Chebyshev polynomials of the second kind, defined through the generating formula (Chapter 22 of [14]) OO

(1 - 2 x z + z2) -1 - Z n--0

z"U,.,(x).

(10.19b)

258

Chapter 10. Rotation Groups in Higher Dimensions

A related and useful expression is U~(cos01) -- [sin(n + 1)01J/sin01.

(10.19c)

The harmonics Y~tm are orthonormal

f Y,~*,em,(~)Y,~em(~)d~ - 6,.,,-,,6tt'6mm'.

(10.20)

The arc length (10.13) between f~ and f~ expands into [2(1 - cosw)] -1 -

Z(2~2/n)Y~*t,,(ft)Y,.,lm(~').

(10.20a)

ntm

A single invariant, the operator A42 , with eigenvalue A(A + 2), characterizes this set of spherical harmonics, with (2jl + 1)(2j2 + 1) - (A + 1) 2 - n ~ degenerate elements spanning the space of a "symmetric tensor" representation of SO(4), denoted by [A0]. The second index, zero, is the eigenvalue of the "second" Casimir invariant of SO(4), namely of the operator g. 5 - ~ijk eiikgijgko, which also commutes with all six generators {g, 5}, but vanishes identically for the representation (10.17) appropriate to the hyd~'ogen atom. More general representations of S0(4), denoted by [Aw], with A - jl +j2, w - j l - j 2 , yield the two Casimir invariants 1 Zu~ g~v and "21E i j k eiJ k~'iJ~.kO with eigenvalues A(A + 2) + w 2 and w(A + 1), respectively. Their analogs prove of interest in Chapter 11 for the Lorentz group. 10.1.3

Hydrogen

atom

in momentum

space

The Schr6dinger equation for the hydrogen atom in momentum, rather than coordinate, space displays its full invariance under four-dimensional rotations. Fourier transformation of the operator 1/r casts this momentumspace equation as = Zr

(10.21)

Introducing the energy parameter p0 - (-2ttE)-~, which is real for bound states and equal to li/nao, turns the coordinates of the unit sphere in (10.10) into ( _ 2poff/(p2o+ if2), ~o - ( P ~ - ff2)/(pg + ~1, (10.22 / with the volume element

dft- 8p3dff/(pg + itT~)3.

(10.23)

10.1. F o u r - D i m e n s i o n a l R o t a t i o n s

259

The coordinate w, defined by (10.13a), serves to express the momentum difference lift-i~l 2 as

IP- ff]-2 - p~[(p~ + ~2)(p2o -4-p-,2)sin 2 89 1,

(10.24)

Entering (10.24) and the rescaled wave function

[4p~/(p2o + ~72)2]Y(12),

r

(10.24a)

in (10.21)reduces it to

y(~)_

n /

Y(12')dl2' 2(1 - cosw)"

(10.25)

Finally, the expansion (10.20a) for the denominator ( 1 - c o s w ) identifies the scaled momentum wave function Y(f2) as indeed the four-dimensional harmonic Yntm(f2), as demonstrated first by Fock in 1936 [43]. 10.1.4

Alternative atom

subgroups

in external

of SO(4)"

The

hydrogen

fields

Having thus shown the Hamiltonian of the hydrogen atom to have SO(4) symmetry, we view the set of its degenerate states at each Bohr energy, -e2/(2aon2), as forming a base set of SO(4). In the absence of any external fields, the ordinary spherical symmetry of the problem leads to the grouping of these states according to g = 0, 1, 2 , . . . , (n - 1), each of the g states itself (2~ + 1)-fold degenerate in the quantum number m. In the language of group theory, this scheme corresponds to the occurrence of the subgroups SO(3) and S O ( 2 ) i n the chain SO(4) D SO(3) D SO(2) whose two links reflect the rank 2 of SO(4). Applying an external field modifies these symmetries; specifically, a static electric ("Stark") field ~'in the z direction preserves the axial symmetry and thereby the quantum number m but intermixes degenerate states of different g. The relevant symmetry is now the parabolic symmetry of Section 10.1.1, with the Hamiltonian of combined Coulomb and electric field, H = ( i g 2 / 2 p ) - (e2/r) + (eez), (10.26) separating in parabolic coordinates. The set of mutally commuting operators for this symmetry, {H, gz,az}, identifies the parabolic states Inln2m) as the eigenstates of the linear, first order in e, Stark effect. Since jlz and

Chapter 10. Rotation Groups in Higher Dimensions

260

j2z are conserved, the relevant group chain is SO(4) D SO1

(3) • SO2(3)

S O 1 ( 2 ) • ,S'O2(2).

Less familiar and more interesting is the case of a hydrogen atom in a weak static magnetic field /~ along the z axis. The paramagnetic, or linear Zeeman, interaction - ( e h / 2 t ~ c ) g . B - - ( e h B / 2 p C ) g z does not mix degenerate states whence, again with the standard labeling [ngm), each separate state receives an additive energy contribution proportional to m. The diamagnetic interaction, on the other hand, (e2/81~c2)(B X r-~2 -( e 2 / 8 p r 2 ) B 2 r 2 sin 2 0, has the more interesting effect of mixing degenerate states with different g (but with the same m and parity). The matrix 1 2sin 2 0 in a degenerate manifold coincide with those of a elements of 7r combination of jl and j2 [44]" 1 2 sin 2 0 ~r

~1 n 2 a02[3n2 -k- 1 -- 4m 2 + ( 2 0 j l z j 2 z - 8 L " ]2)] , 2a02In 2 + 3 + ( g z +24 ~ -_ ~n

2 - 5 a 2 )] .

(10 27a) ( 10. 27b)

Therefore, eigenstates of the Coulomb plus diamagnetic interaction are obtained by simultaneous diagonalization of the mutually commuting set { H , g z , 4 6 2 - 5az2}. Note from (10.3) that g~ commutes with az and with 2 2 a x -k-ay.

The set of eigenvalues of (10.27) can be grouped into two classes, each with a simple symmetry classification. Those with 5 lying close to the z axis, that is, with ~2 .,~ az2 ' are identified by {H , g~ , az} , coinciding thus with the same parabolic set that describes the linear Stark effect. For the states with ~ lying instead close to the x y plane, that is, with ~2 "~ a~2+au2 _-

(jlx-

j2x) 2 -k-(jly

--jzy) 2,

Eq. ( 1 0 . 2 7 b ) c a n be recast as

-~1n2a2[n2 + 3 + 4(a~2 + au2 + g 2 ) _ 3/2] .

(10.27c)

The variables of (10.27c) amount to the squared length of a vector fi: fi - ( a~, a u, gz ) - ( j l ~ - j 2 ~

, Jlu - J 2 u , jl~ +j2~ ),

(10.28)

and of its z component. Rotations of this vector /7, forming an SOu(3 ) subgroup of SO(4), different from the usual SO(3) of orbital angular momentum, g ' - j~ + j-'2, provide the symmetry relevant to the class of eigenstates of the diamagnetic interaction with ~ transverse to B. The energies of these states emerge now by replacing the factor a~2 + a u2 + g2 in (10.27c) by its value Ifil2 - p(p + 1) to yield 1 2 a2o[n 2 + 3 + 4p(p + 1) - 3m2]. -~n

(10.29)

10.1. F o u r - D i m e n s i o n a l R o t a t i o n s

261

The corresponding eigenstates are represented as superpositions of parabolic states by Int tin) -- E ( - 1 ) J ~ "

+(n-1)/~lnl n 2 m ) ( j l j l z , j2j2z ljlj21tm).

(10.29a)

rl 1

This SOu(3) symmetry turns out to be associated with a partial (to order B ~) separability of the Schrbdinger equation in elliptic cylindrical coordinates, type I, in momentum space [45]. In terms of J acobian elliptic functions called sine amplitude sn, cosine amplitude cn, and delta amplitude dn (Chapter 16 of [14]), the coordinates ff in (10.10) are parametrized as

~o

=

cn~ d n ~ ,

~z ~: ~u

= =

dna sn~ , snadnflcos~f, s n a d n ~ sin~,

(10.30)

with (~2 + ~ + ~02 invariant. Similarly, one might envisage other interactions that mix degenerate states of the hydrogen atom, introducing other subgroups with yet more labels analogous to nl, t~, or p, that run over the integers from 0 to ( n - l ) . The corresponding subgroups of SO(4), analogous to S 0 1 ( 2 ) • SO2(2), SO(3), or SOu(3), represent symmetries appropriate to such interactions.

10.1.5

Clebsch-Gordan coefficients for products of SO(4)

The dominance of electron-nucleus over electron-electron interactions in atoms allows hydrogenic wave functions to provide a first, and often fairly good, qualitative picture of many-electron phenomena. Just as the coupling of multielectron angular momenta extends the relevance of the SO(3) Clebsch-Gordan algebra of Chapters 5-8, the analogous algebra of SO(4) might play a role in dealing with phenomena that mix all the degenerate states of each electron's energy. We consider here the first step in this algebra for a two-electron atom, introduced by Herrick and Sinanogolu [46], namely, the Clebsch-Gordan reduction of the product of two SO(4)'s, which is now recognized as useful to the study of doubly excited states. In considering the product SO(1)(4)• SO(~)(4), with superscripts labeling the two electrons, we rely on the parabolic description of Section 10.1.1 that views each SO(4) itself as the product SO1 (3)• SO2(3) with the angular momenta jl and ]2. In this language, a combined two-electron state of total orbital angular momentum [ - ~~1) .~ g'~2) may be viewed as resulting

262

C h a p t e r 10. R o t a t i o n G r o u p s in H i g h e r D i m e n s i o n s

from the addition of four angular momenta j'(11), ~21), ~(12), and ~2). The Clebsch-Gordan algebra of SO(4) reduces thus to the subject of Section 7.2.2, namely, the recoupling of four r-transformations by 9-j coefficients. One chain of couplings proceeds by ~11) + ~1) _ g71) and ~2) + ~2) _ fie), followed by g-(l) + g-~2)_/~, and an alternative chain by ~(11) -i- ; ~ 2 ) _ J1,

J2

q" 32

-- J2,

J1 -+- J2

L,

(10.31)

providing two alternative base sets. The first chain leaves the two electrons separate, SO(1)(4) x SO(2)(4), whereas the second introduces operators of a single "coupled" SO(4) of the combined system, with six generators 1 embodied by the vector pair/~ and 7(J~ - J ~ ) - A - a-41)+ a-'(2) Equation (7.20) casts the Clebsch-Gordan coefficients of SO(4) relating these two base sets, - Z

]//(1)n(2)J1J2)(n(1)n(2)JiJ2]n(1)g(1)' n(2)g(2)) ' (10.32)

as

(n(1)n(2)J 1J2]n(1)t(1), n(2)g(2)) = [(2e

+ 1)(2e

x

+ 1)(2J1 + 1)(2J2 + -}(n(1) 1) 7(n (1) 1) g(1) } . 89 (2) 1) 7(n (2) 1) g(2) L J1 J2

(10.32a)

The additional coupling of the azimuthal or magnetic quantum numbers, gl) + g72) _ f, and m (1) + m (2) - M, introduces a further SO(3) ClebschGordan coefficient (t(1)m(1), g(2)m(2)lt(1)t(2)LM ). The chain (10.31) is not unique in coupling the four angular momentum vectors, thus mixing elements of both particles at an intermediate step. An alternative considers )*(11)+ ~22) -- J~,

~(12)-~- ~(21) -- J~,

J~ --[--J~ -- i .

(10.33)

Once again, the unitary matrix connecting S0(~)(4) x S0(~)(4) states of the uncoupled representation to the single S0(4) representation provided by J[ and J~ rests on the expression (10.32) with the primed quantities replacing J1 and J-2. Whereas ~ and J~ in (10.31) involve the sum of the Laplace-Runge-Lenz vectors ~1) and ~2), the J~ and J~ in (10.33)introduce their difference ~ 1 ) _ ~2). In recent years [47], the mixing of different angular m o m e n t a t-~l) and ~-~2)in doubly excited states of atoms of a given

10.2. O r t h o g o n a l G r o u p s in H i g h e r D i m e n s i o n s

263

L has been well approximated by the coupling scheme (10.33). States with largest values of [~(1) _ ~(2)[ confine the two electrons to opposite sides of the nucleus; they thus minimize the electron repulsion and dominate the two-electron spectrum, being labeled by two quantum numbers associated with J~ and J~ and commonly called (K, T).

10.2

Orthogonal Groups in Higher Dimensions

Rotations in dimensions larger than four involve, of course, increasing numbers of generators and invariants, with applications to physics. As anticipated in Chapter 2 (footnote 1 and Problem 2.1), just as SO(4) involves (4 x 3/2) - 6 generators, rotations in D dimensions have D(D- 1)/2 generators, the number of independent two-dimensional planes in D dimensions. Correspondingly, rotations are described by antisymmetric orthogonal D • D matrices. As in SO(4), these matrices may combine into subgroup chains of SO(D), one of which relates to the D-dimensional hypersphere and to a hierarchy of ( D - 1) hyperangles that define a point on the sphere as in (10.10). This parametrization corresponds to the hierarchy of subgroups in the group chain SO(D) D SO(D- 1) D SO(D-2) D " - D

so(3)

so(2).

In this view, each step from SO(d) to SO(d + 1) requires adding d generators which transform as a vector in the d-dimensional space, just as Section 10.1 added the three-dimensional vector ~ to the generators t of SO(3) to yield the six generators of SO(4). Adding next four more generators M1, M2, M3, and/144, which transform like a four-dimensional vector under SO(4), yields the ten generators of SO(5). A further step subdivides this set of four M's into a three-dimensional vector A/I - {M1,M2,M3} and a scalar/1//4 under SO(3), a pattern continuing for higher orthogonal groups. Section 10.2.1 details this description of SO(D), whereas Section 10.2.2 will turn to an alternative scheme adapted to multiparticle systems. In either case, the total number of generators equals D(D- 1)/2 and the number of Casimir invariants--that is, the rank of the group--equals D/2 or ( D - 1)/2 depending on whether D is even or odd, respectively. As in the case of SO(4), a major role belongs to the so-called symmetric tensor representation [A00--.] in which all but the first invariant vanish.

264

C h a p t e r 10. R o t a t i o n G r o u p s in H i g h e r D i m e n s i o n s

10.2.1

Hypersphere

in D d i m e n s i o n s

As with spherical coordinates {r,O, ~} in three dimensions, and with the similar set (10.10) in four dimensions, "hyperspherical" coordinates in D dimensions can be defined in terms of D Cartesian coordinates xi through the "canonical scheme" R sin 0x sin 02 sin 0 3 " ' ' sin OD-3sin 0 sin ~o, Rsin01 sin 02 - . - s i n 0 D - 3 s i n 0 c o s T , Rsin01 sin 02 -'-sin0D-3 cos0, Rsin01 sin 02--.sin0D_4 cOS0D-3,

X1 - x2 x3 -x4 -XD_I

--

R sin 01cos 02 ,

XD

--

Rcos01 .

(10.34)

Among the ( D - 1) hyperangles, we have set OD-2 -- 0 and OD-1 -- ~o conforming to the standard usage in three dimensions. Except for ~p's range (0, 2~r), all the O's range from 0 to 7r. The infinitesimal element of solid angle takes the form d~ - (sin 01)D-2(sin 02) D - 3 " " -sin 20D-3 sin OdOld02...dOD_3dOd~o, (10.34a) with the total angular integral over the unit sphere given by

~D

--

27rD/2 __ { dFt - F(D/2) -

(27r)D/~ (D-2)!! 2(27r) (D-~)/2

(n-2)~!

D

even,

D

odd,

(10.34b)

with the double factorial (p. 258 of ref. [14]) standing for ( D - 2)![ ( D - 2 ) ( D - 4 ) ( D - 6 ) . - . 2 or 1, ( - 1 ) ! ! - 0 ! [ - 1. The D-dimensional Laplacian has the alternative Cartesian and polar forms

(O0__~i)

V2 - Z

1 0 RD_IO - R D-1 OR OR

A2D R 2'

(10.35)

g

with the grand angular momentum operator defined A~)-Zg2

,

p,u-

as 3

1,2,...,D.

(10.35a)

/~>v 3Unlike in Section 10.1.2 for SO(4), where the indices ~t and v took the values 0,1,2,3 with an eye to a later specific extension in Chapter 11, we choose here indices running from 1 through D for the general D-dimensional problem.

265

10.2. Orthogonal Groups in Higher D i m e n s i o n s

The last term in (10.35) involving the grand angular momentum represents a centrifugal potential in the radial (R) SchrSdinger equation. The generators of SO(D) operators i s , may be cast as D ( D - 1)/2 antisymmetric rotation matrices analogous to (10.16) and (10.17), with commutation relations (10.15). Their explicit D • D matrix form with nonzero entries :ki only in the corresponding (pv) positions is easily expressed as in (10.17). The sum-of-squares Laplacian operator (10.35a) commutes with each of the generators, being a Casimir invariant of SO(D). Its eigenvalues, with eigenfunctions called hyperspherical harmonics [48, 49], solve the equation (10.36)

A 2 y ~ { z } ( a ) - A(A + D - 2)Y~{~,}(a),

where {p} stands for a set of (D - 2) other quantum numbers identifying degenerate harmonics for each ~. Just as/? in g(t? + 1) of three-dimensional angular momentum may be viewed as a quantum-mechanical zero-point contribution, so also in D dimensions, each of the ( D - 2) angles beyond 01 providing an additive )~ to the eigenvalue in (10.36). Of special importance, as before with SO(4), is the symmetric tensor basis provided by the hyperspherical harmonics in (10.36) wherein all other invariants of SO(D) vanish. The degeneracy, or "dimension," of these [)~00...] sets equals

dim(D) - (D + 2 ~~-[ ( 2)(D + ~ - 3)! D - 2)[ '

(10.37)

reducing to the familiar values of (2A + 1) for D - 3 and (A + 1) 2 for D = 4. Other representative values are shown in Table 10.1. Each entry in Table 10.1 is the sum of entries in the previous column down to that row. Thus, the [30] space of SO(4) is spanned by S, P, D, and F atomic states whereas the [30] of SO(5) includes the [30] + [20] + [10] + [00] of SO(4) with ( S + P + D + F) + ( S + P + D) + ( S + P) + S states. The explicit expression of the grand angular momentum operator in terms of the hyperspherical angles, -

1 0 (sin 01)D-2 0 (sin 01)D-2 001 001 1 0 (sin02)~ + (sin 2 01)(sin 02) D-3 00----2 1 0~ + sin 2 01 sin 2 02...sin 2 0 0~o2'

0

002 (10.38)

266

Chapter

10. Rotation

Groups

in Higher

Dimensions

T a b l e 10.1" Dimensions of the [A00...] representation of orthogonal groups.

S0(3) S0(4)S0(5)

5'0(6) S0(7)

1

1

1

1

1

3 5 7 9

4 9 16 25

5 14 30 55

6 20 50 105

7 27 77 182

displays a hierarchical structure in terms of the successive operators A~_ 1, A~_2, etc. Indeed, the Y.x{u} harmonics considered here are simultaneous eigenfunctions of the invariant operators of SO(D- 1), SO(D- 2), etc., whereby the labels {p} - {Pl,P2,...,PD-2} denote invariants of these SO(D) subgroups" A~_IY~{u ) A~)_2Y,~{u}

-

124 YX{p} A~Y~{u}

--

--

]/1(]~1%-D - 3)Y~{u}, ]22(P2 + D - 4)Y~{u ) , (10.39)

( - V 0) 2

Y~{u}

_

+ 2)Y~{,}, L(L %-1)Y~{u}. M 2 Y~{u},

~D_4(]2D_

4

with A _> pl _> #2 _> "'" _> # D - 4 >__ L >_ IM[ > 0. Once again, we set PD-3 -- L and #D-2 -- [M[ in terms of the familiar angular momentum quantum numbers, illustrating explicitly the use of invariants #i of the chain of subgroups SO(D) D SO(D- 1) D . - - D SO(3) D SO(2) in labeling the canonical hyperspherical harmonic eigenfunctions to distinguish among the degenerate states in [A00...]. These hyperspherical harmonics consist, as in (10.19), of Gegenbauer polynomials (Chapter 22 of ref. [14]) in the hyperspherical angles: D-3

Ya{u}(a) - Aa{.} H

(sinOi+l)*"+'e•176

_u.+.

+x).

i=0

(10.40) with the polynomials C defined in (10.19a), a normalization constant A~{u}, and p0 - A. Note in the above sequence of products that the factor with i r lMI -(cos 0). coincides with D - 3 , and PD-3 -- L , PD-2 _ [MI, namely ~L-I

10.2. Orthogonal Groups in Higher Dimensions

267

the Associated Legendre polynomials that occur in the spherical harmonics YLM, and the factor i = D - 4 with /to-4 = n - 1 coincides with the C nL _L_l(COS01) +I of the four-dimensional harmonics in (10.19a). Typically, the constant A in (10.40) ensures the normalization

/

Y~,{u}Y~,,{u,}dgt - 6~,6{u}{u, }.

(10.40a)

These harmonics also satisfy an addition formula

Zv.(a)v(a,)

_ (D + 2A - 2)(D - 4)!!c(D_2)/2(it. fi,),

(#} " A { U } ' A { # } --

(10.41)

SD

where fi and fi~ are two D-dimensional unit vectors specified by the hyperangles ~ and ~', respectively [48, 49].

10.2.2

Hyperspherical coordinates for multiparticle systems

Upon separating the motion of the center of mass, the dynamics of an Nparticle system reduces to that of a configuration point in D = 3(N - 1)dimensional space. Thus, the three-body problem deals with six Cartesian coordinates embodied in two radial vectors (~1, ~2) from the center of mass to particles 1 and 2. The third particle's position is automatically specified as - ( m l r'l + m2~2)/(ml + m 2 + m3). The recasting of the 3 ( g - 1) coordinates 71, r'2,..., r'u-1 in terms of (3N - 4) hyperangles is, however, far from unique. Increasing N leads to an explosive growth in the number of alternative recastings of coordinates represented by "Jacobi trees" as in Fig. 7.1. The canonical choice of 3 ( N - 1) coordinates arranged in sequence as the Cartesian components of a single D-dimensional vector on the left-hand side of (10.34) corresponds to the hyperspherical description of Section 10.2.1. The harmonics are then characterized by A and {#} without further reference to the individual particles i = 1, 2 , . . . , N - 1. The individual angular momenta gi of particles play no role in this classification of harmonics. For many purposes, however, it is desirable to keep the single-particle labels gi, thus calling for the following casting. Figure 10.1 illustrates these two alternatives for the N = 3 example, the lower half showing the scheme of Section 10.2.1 and (10.34), the upper half the scheme of the present

268

Chapter 10. Rotation Groups in Higher Dimensions

section. 4 The five hyperangles of the latter scheme consist of the usual spherical coordinates of § i - 1,2, and of a fifth "angle" c~ defined through the radial distances" R-(r~

+ r~) 89

tana

-

(10.42)

r2/rx.

More generally, for the D - 3 ( N - 1) problem of an N-particle system, the angles § - (0~, ~0~) provide 2 ( N - 1) of the ( 3 N - 4) hyperangles, the remainder being "angles" cq expressing ratios of radial distances: 5 rN- 1

--

R cos ~ N - 2 ,

rN-2 rN-3

---

Rsin Rsin

r2

--

RsinaN_2sinag_3.--sina2cosal

rl

--

R sin C~N_2 sin aN_ 3" " "sin c~2 sin a 1 9

~ N - 2 COS ~ N - 3 , aN-2

sin

O~N_ 3 COS ~ N - 4 ,

(10.43) ,

Alternatively one might set R~ - Z ~ = l r~ leading to N-1

R 2-

R~v-1 = Z r~, i=1

sin 2 a i

-

R2_I/R~

(10.43a)

(10.43b)

9

Note that 0i and ~i have the usual range, (0, 7r) and (0, 2r), respectively, but the ai range from 0 to r / 2 . This definition of hyperangles affords an indication of directions and radial distances relevant to each particle. In particular, the angles c~i pertain to radial correlations. The volume element is here N-2

R3N-4dRdQ-

R3N-4 H i=1

N-1 cOs2

~176

H

sin OidOidcpi.

i=1

(10.44) 4 Many othe r alternative grouping ( "clusterings" ) of the coordinates are possible, particularly with increasing n u m b e r of particles. T h e y m a y prove relevant to situations where such clusterings are a p p r o p r i a t e to specific p h e n o m e n a u n d e r consideration [50]. 5For N = 3 , the single angle c~a is essentially the one defined in (10.42), except t h a t historical usage makes c~ the r / 2 - c o m p l e m e n t of C~l, t h a t is, the role of 1 a n d 2 is interchanged. For an assembly of particles with differing masses, the equations are readily a d a p t e d to "mass-weighted" coordinates. Thus, for instance, R represents the 2 andM=Emi radius of inertia, with (10.43a) replaced by MR 2 = E miri'

269

10.2. O r t h o g o n a l G r o u p s in H i g h e r D i m e n s i o n s

c~ R

Xl

Yl

Zl

x2

Y2

z2

R

F i g u r e 10.1: Alternative hyperspherical rendering of six Cartesian coordinates of a three-particle system. Angles involved are shown alongside each node. The D-dimensional Laplacian reads V2 _ R_(3N_4)/2 0 R(3N_4)/2 0 OR OR

A5 R 2'

(10.45)

with A ~ ) ( f l ) - A~(N_I)(flN_I) , 2

A i(~i)

=

02 oa~_,

( 3 i - 1) cos 2 a , - 1 - 2 sin ~ a i - 1 sin a , _ , c o s a , _ ,

-[-

Ag(,_,) ( f l , _ l )

sin~ ~,-,

+

,

COS2 ~ t-- 1 "

(10.45a) 0

Oai_,

(10.45b)

The grand angular momentum A~ includes, therefore, the angular kinetic energy associated with the ai coordinates together with the individual orbital angular kinetic energy corresponding to ~i, scaled by trigonometric factors depending on the ai coordinates [48]. The eigenvalues of t ~ are again as in (10.36) A~r

- A(A + D - 2)r

(10.46)

with corresponding degenerate eigenvectors whose further unique specification requires ( 3 N - 4) other labels {nigimi}. Those labels and the corresponding hierarchy of other operators with simultaneous eigenfunctions r

270

C h a p t e r 10. R o t a t i o n G r o u p s in H i g h e r D i m e n s i o n s

differ, of course, now from (10.39) in Section 10.2.1. These operators are denoted in (10.45b) by A/2 with eigenvalues and eigenfunctions indicated by

A23i(ai)r

- Ai(Ai + 3 i - 2)r

(10.47)

where

Ai = Ai_l + gi + 2ni ,

ni = 0, 1 , 2 , . . . ,

(10.47a)

and

r

(9~i_ : )(cos eq_:)e' (sin e~i_:)~'-'

A,_a + ( 3 i - 5 ) / 2 , l , + 89( c o s 2o~ i_ X p ni

1)Yt,m, (Oi, ~Pi),

(10.47b)

with Jacobi polynomials P (Chapter 22 of [14]). Setting )~1 = el and AN-1 = A, the recursive relation in (10.47b) represents the eigenfunctions in (10.46) as products of Jacobi polynomials in the angles ai and of spherical harmonics of (Oi, pi). The eigenvalue A equals N-2 A

-

2Z i=1

N-1

ni + Z

g.i,

(10.48)

i=1

each eigenvalue A :/: 0 being degenerate with respect to the alternative partitions involved in the sum over g's and 2n's. The labels gi and mi represent, of course, the orbital angular momenta of the i-th particle, while n i may be termed a "radial correlation" quantum number, indexing the number of nodes of P,~. Each quantum number ni is multipled in (10.48) by a factor of two reflecting the smaller range of the anges ai " (0, 2) contrasting with the range (0, 7r) of the angles 0i. Besides A, the scheme in (10.46) and (10.48) provides then 3 N - 4 labels, to be contrasted with the D - 2 = 3 N - 5 values {pi} in the alternative scheme of (10.36). The difference of unity between these two numbers is accounted for by the parity label which is a separate independent quantity in (10.36), wherein only the total L of the N-particle system is specified, whereas parity is automatically included in (10.46) as the parity of ~ gi. The eigenfunctions CX{n,e,m,} are products of the Jacobi polynomials in c~i introduced in (10.47b) and of spherical harmonics. Since the magnetic quantum numbers enter only in the latter, the product IIYl.m, can be coupled to total L and M in the usual manner of Part B of this book. Applications to many-electron or many-nucleon systems afford coupling spin wave functions to total spin and carrying out the antisymmetrization required by the Pauli principle according to the standard procedures of Chapters 7 and

271

10.2. O r t h o g o n a l G r o u p s in H i g h e r D i m e n s i o n s

8. For this reason, the hyperspherical scheme of this Section 10.2.2 appears more convenient than that of Section 10.2.1, which does not readily adapt to symmetry under permutations of particle coordinates.

10.2.3

Transformation

between

alternative

schemes

The two previous subsections have provided examples of the many alternative hyperspherical schemes for a set of D - 3 ( N - 1) coordinates. Such alternatives include the alternative "Jacobi trees" in Fig. 7.1. Any two sets of eigenfunctions are related by unitary transformation coefficients, appropriately labeled "timber coefficients," which have been developed and studied particularly in nuclear physics [30]. When the interaction potential between particles is patterned after isotropic harmonic oscillators, these transforms involve so-called "Raynal-Revai" or "Talmi-Moshinsky" coefficients, distant analogs of Clebsch-Gordan coefficients [49]. The familiar SO(3) rotation group is replaced here by the unitary group SU(1, 1) and the symplectic group Sp(2, R). We do not consider them any further here but turn to the simple example of the three-body problem with N - 3 and D - 6, where the SO(4) description of Section 10.2.1 and that of SO(6) in Section 10.2.2 for L - 0 states both involve only one hyperangle, being essentially coincident. The first extension beyond the usual spherical coordinates 0 and ~ to hyperangles occurs in the SO(4) description of Section 10.2.1 introducing the hyperangle 01. Likewise, in the many-particle analysis of Section 10.2.2 there arises a hyperangle a l for N - 3, besides the angular coordinates rl and § For L - 0 states of the system, where [ g l l - [/2[, the six variables r-'t and r'2 reduce to three: R, a and 012 - arccos(§ § This set maps one-to-one to the set R, 01, and 0 of the SO(4) scheme of Section 10.2.1 with similar correspondences between eigenvalues and eigenfunctions. Note first from (10.48) that A - 2nl +gl +g2 is always even for such L - 0 states, whereby the SO(6) eigenvalue A(A + 4) with A even in (10.46) recasts into 4(A/2)[(A/2) + 2], with (A/2)running over all positive integer values. Apart from the factor of 4, this expression coincides with (10.36) in the SO(4) scheme. That same factor of 4, which reflects the difference in the ranges of 01 "(0, 7r) and a "(0, 7r/2), occurs also in the corresponding Laplacians of (10.45) and (10.38)" 1 O O sin 2 01 sin 201 001 00---]- =

1 1 O 0 sin2(O1/2) 4sin2(Oa/2) 0(0~/2) O(Ox/2)

272

Chapter 10. Rotation Groups in Higher Dimensions 1 1 0 0 4 sin2 ~ Oc~ sins ~ 0--~"

(10.49)

The Jacobi polynomial in (10.47b), p~li+ 89189 with ~1 - ~?~, reduces likewise to the Gegenbauer polynomial in (10.40), making the two schemes essentially identical, and accounting in part for the success of the fourdimensional SO(4) classification (particularly for L = 0) for doubly excited states of atoms, a three-body problem which would otherwise involve Euler angles in a six-dimensional configuration space.

10.3

Further D e v e l o p m e n t s

Previous sections have dealt with physical systems leading to orthogonal groups larger than SO(3). On the one hand, N-particle systems generally display invariance under the group of rotations in the full D-dimensional configuration space of SO(D), with D - 3 ( N - 1). On the other hand, even the two-body Coulomb problem involves symmetries higher than under SO(3), through the combined action of the Laplace-Runge-Lenz vector and of the angular momentum vector ~. Larger groups arise precisely from such extended transformations, whether more directly by introducing additional particles or because of special relevant symmetries or circumstances. Since behavior under r-transformations, and under the corresponding classification according to angular momentum, is always relevant to physical systems, states are labeled both by SO(3) quantum numbers and by those of SO(D). Particularly with increasing D, both the building up from SO(3) and the reduction from a starting SO(D) toward SO(3) involve a larger number and variety of intermediate subgroups, whose quantum numbers should also prove useful for classifying or interconnecting states of a system. The example of particle-hole symmetry relating states of atomic configurations dn and d (5-n) has already occurred in Section 8.1.3, providing the seniority label of d" states and similar considerations for fn states. 10.3.1

Invariance

and noninvariance

groups

When all the generators of a group commute with the Hamiltonian of a system, one speaks of an i n v a r i a n c e group, the system being invariant under all the group's transformations. These transformations reflect, therefore, symmetries of the Hamiltonian, affording energy eigenstates to be labeled

10.3. F u r t h e r D e v e l o p m e n t s

273

by the group's Casimir invariants, and displaying the immediate utility of the invariance group for classification. N o n i n v a r i a n c e g r o u p s , some of whose generators fail to commute with the Hamiltonian, prove nevertheless useful by isolating specific elements of the Hamiltonian, whose influence can thereby be assessed. Thus the proven relevance of the seniority label, mentioned above, for the classification of spectral levels implies that the addition of electron pairs with 1S coupling hardly disturbs other electrons in an open shell. Consider now the following contrast. On the one hand, SO(4) is an invariance group for the hydrogen atom, all six generators {~,5} commuting with the Coulomb Hamiltonian. Every set [A0], with A - n - 1 - 2jl - 2j2, has a high degeneracy, encompassing all the n 2 states with the common energy - 1 3 . 6 e V / n 2. Distinguishing between them requires, for instance, the further labels g - 0, 1 , . . . , n of SO(3), and m - - g , . . . , g 1,g of SO(2), successive subgroups of SO(4) (or labels of alternative subgroups in parabolic coordinates). On the other hand, the isotropic rotor with Hamiltonian where I represents the moment of inertia, affords a contrasting example" It does not display SO(4) symmetry because the additional three generators ff of four-dimensional rotations do not commute with the rotor Hamiltonian. SO(4) serves nevertheless as a noninvariance group, for example in the presence of an external field that couples states with different g-'. Thus, for a charged rotor in an electric field, with coupling proportional to cos 0, a state of given g is coupled to the neighboring states g+l.

~/2I,

Similar considerations apply to treating the hydrogen atom. The representation [00] of SO(4) pertains only to its state ls, and [10] to the four n - 2 states. Electric dipole transitions connect ls dominantly to the 2p states, through the so-called "resonance transition." The states with n - 1 and 2, belonging to different SO(4) representations, can be fitted into a single basis set of the next larger orthogonal group SO(5), whose representation [10] includes both the [00] and [10] of SO(4) according to Table 10.1. The electric dipole operator ~', which does not commute with the hydrogenic Hamiltonian, may thus be included in further considerations, by progressing from the invariance group SO(4) to the noninvariance group SO(5). The previously defined Laplace-Runge-Lenz vector 5 in (10.1) arises as a linear combination of the two vectors ff x g and ~' that lie in the plane of the Kepler orbit. The dipole operator ~"may, therefore, be resolved into the sum of 8 and of a second linear combination, 1~ - / 7 x g'- § Together with a scalar operator ~'./7, the ten generators {g, 5, M, ~'-~ form the noninvari-

274

Chapter 10. Rotation Groups in Higher Dimensions

ance group SO(5). In Chapter 12, the electric dipole operator will similarly connect the ls state explicitly not only to 2p but to all the higher p states of the hydrogen atom (namely, to the whole Lyman series and attached continuum).

10.3.2

"Dynamical

symmetries"

for atoms

and

nuclei

The reduction of SO(5) down to SO(3) and the buildup from SO(3) to SO(5) need not be viewed in stages involving invariants of the intermediate SO(4). An alternative looks at representations of SO(5) directly in terms of their "angular-momentum content," that is, directly as SO(5) D SO(3). The five-dimensional [10] base set in Table 10.1 is then viewed as a single set in its transformation under S0(3) rotations, and thereby identified with the d states (~ = 2) rather than with is; 2s, 2p as in the previous paragraph. Similarly, [20] in this direct scheme includes the 14 states transforming under SO(3) rotations as dg rather than as Is; 2s, 2p; 3s, 3p, 3d in the intermediate SO(4) scheme. Such a direct reduction of any higher SO(2g+ 1) to SO(3) views the [100.-.] base set of the former as including the states of a particle of angular momentum g; for this reason, SO(5) and S0(7) proved useful in Section 8.1.3 for classifying states in the d and f shells of atoms. The seniority quantum number v, providing in (8.21) the Casimir operators' eigenvalues, serves to label the states of d '~ and fn configurations. The use of these labels and of the corresponding SO(2g + 1) groups in a subgroup chain was identified in Section 8.1.3 as expressing a dynamical symmetry of the states of these shells. Turning to other representations of SO(2g + 1), the [1100..-] base set is next in importance. It contains the antisymmetric (in space) states of g2 and therefore has the dimension of the number of generators of the group; thus, g(2g+l) for SO(2g+l). The 10 generators of the [11] representations of SO(5) may be classified as pf according to their transformation properties under SO(3) rotations and the 21 generators in [110] of SO(7) as pfh; that is, these are the antisymmetric or odd-parity objects indexed by k = 1 , 3 , . . . , ( 2 g - 1)in Section (8.1.3)for S O ( 2 g + 1). When a subset of these objects generates an intermediate subgroup, as do the 14 ph states (k - 1 and 5) which generate the group G2, the chain: SO(7) D G2 ~ SO(3) is useful in classifying fn states, providing another example of dynamical symmetry as discussed in Section 8.1.3. The orthogonal groups of even dimension, SO(2g + 2), introduce an additional element of great generality and usefulness. Their base set [100.--]

10.3. F u r t h e r D e v e l o p m e n t s

275

with (2/? + 2) elements has one more element than the [100...] of the previous S0(2~. + 1) group, its r-transformations forming a set of angular momentum e. This additional element transforms as a scalar under SO(3) rotations, that is, as an t = 0 object. Thus, for example, the states of SO(6) transform as a pair sd, those of SO(8) as s f, etc. Indeed, SO(8) and a classification scheme based on states of (s + f)~, analogous to those of a pair of particles, have proved useful in analyzing the f shell in Section 8.3.3. Similarly successful has been an (s + d) configuration scheme for nuclear spectra. The so-called "interacting boson model" views nucleons as paired into bosons of angular momentum 0 and 2, providing the effective degrees of freedom for low-lying excitations of nuclei [36]. Quadrupole excitations are well known to be important for nuclei so that the /? = 2 or d symmetry is an obvious candidate. The addition of an overall volume change through the ~ = 0 degree of freedom permits distortions more flexible than quadrupolar. Together, these six degrees of freedom and the corresponding symmetry group U(6) prove capable of encompassing a large number of nuclear spectra. Three subgroup chains and, correspondingly, three types of dynamical symmetries, occur in the reduction of U(6) to SO(3), with evidence of their respective influence in the spectra of different nuclei. Two of these chains involve the higher orthogonal groups SO(5) and SO(6): (I):

(II): (m):

U(6) D U(5) D SO(5) D SO(3), U(6) D SU(3) 2) SO(3), U(6) D SO(6) D SO(5) D SO(3).

(10.50)

The occurrence of only these particular chains and, in particular, of I and III involving the intermediate orthogonal groups, follows once again upon considering the generators and their symmetries under rotation. Analyzing subgroup structures in terms of single-particle angular momenta, labeled s, p, d, etc., has proved convenient pending further identification of the underlying physics. This labeling, that is, classification under rtransformations, will serve here, but its preliminary role needs bearing in mind. The 36 generators of U(6) corresponding to (s + d) 2 are restricted in (I) first to the 25 arising from d-symmetry alone, providing the generators of U(5). Of these, the ten antisymmetric ones, transforming under rotations as p f (k = 1 and 3 of Section 8.1.3), form the generators of SO(5). In the final step of the reduction, the three p alone remain to serve as the generators of the final SO(3) in this group chain. On the other hand, (III) represents an alternative scheme whose first step restricts the 36 (s + d) 2

276

C h a p t e r 10. R o t a t i o n G r o u p s in H i g h e r D i m e n s i o n s

states to the 15 generators of S0(6), transforming under rotations as pdf. Among these, the ten odd-parity states p f (k = 1,3) correspond to the ten antisymmetric states of d 2 from (s -+-d)2, whereas the d arises from the cross-product sd. The d is discarded next to leave behind the p f as the generators of SO(5) and, once again, the final step retains only the p to give SO(3). The third scheme (II) in (10.50) represents yet another alternative where first only the 8 generators of SU(3), transforming as pd, are retained and then the d discarded to reach the final SO(3). Starting from the states of (s A-d) 2, these three schemes account for all the possible subgroup chains in the reduction of U(6) to the final SO(3). In particular, SO(4) with its two vector generators (that is, with rtransformation property pp) does not occur as a subgroup whereby in both (I) and (III) the final step goes directly from SO(5) to SO(3) without an intervening SO(4). When the spectrum of a nucleus--energy levels, transition strengths, etc.--can be expressed in terms of the Casimir invariants of the groups in one of the chains, the nucleus is said to exhibit the corresponding dynamical symmetry [36]. Thereby, the Hamiltonian may be expressed as a superposition of common operator sets' Casimir operators multiplied by corresponding coefficients, without requiring 36 coefficients for each of the 36 generators of U(6). As in the examples of d n and fn classifications in Section 8.1.3, the very economy afforded by dealing with a small subgroup set and its Casimir operators, encompassing large amounts of spectral data, makes applications of dynamical symmetry attractive, even though short of a complete understanding of the underlying physics, comparable to Section 6.3's prototype example of the hydrogen atom's SO(4) symmetry.

10.3.3

Adjoining

an extra

degree

of freedom

The above example of the interacting boson model illustrates how more extended symmetries and structures may be described by adjoining a degree of freedom. Thus, the chain (I) is already present for quadrupolar deformations considered by themselves, with d 2 accounting for the generators of all the groups involved in this chain. However, it is only with the adjoining of the s generator that the other two chains become possible. The cross-term sd in (s-4-d) ~ is responsible for an antisymmetric d (k = 2) required to provide five of the generators of SO(6). Such an interplay between s and d, or an s and an f, is also familiar already in the building up of the elements of the Periodic Table. The transition elements and the lanthanides and actinides involve competitions between filling an outer s subshell vs occu-

10.3. F u r t h e r D e v e l o p m e n t s

277

pying an inner d or f subshell. Dramatic effects of configuration mixing for ground and low-lying states have long been recognized for these elements, particularly when the inner subshells are nearly filled or half-filled. There are, of course, no real s or d bosons in a nucleus, which is actually a collection of many fermions. The s and d describe rather quasiparticles accounting for low-lying nuclear excitations. The use of quasiparticles or effective degrees of freedom is common throughout physics; indeed, Section 8.3.3 provided a "reverse" example, bosonic states of t~ = 3 being regarded instead as fermionic "quarks" after adjoining another bosonic ~ = 0 degree of freedom. The adjoining of one degree of freedom and the subsequent simplification in describing dynamics and symmetries that is illustrated by the above examples also seem to have broader and more general applications in physics. One instance is of simple harmonic motion in one dimension x. While the frequency ~ is fixed, the amplitude of the motion varies continuously between - L and L. However, if this motion is embedded into two dimensions by adding an identical motion in the y direction, then the two-dimensional motion becomes even simpler when viewed in circular coordinates p and ~. It consists of uniform circular motion with both p - L and d ~ / d t = w fixed. The more complicated behavior of the amplitude for the original motion in x is thus seen to arise entirely in the projection of the uniform motion of the point on the circle down to the x axis. Such an embedding becomes even more significant when the frequency itself varies with time: w(t). Although both the amplitude and the frequency for the one-dimensional motion may now be complicated functions of time, the associated two-dimensional motion is still one of fixed angular momentum p2(d~/dt), with p also often having a simpler time dependence. 10.3.4

Alternative multiparticle

reduction

schemes

for

systems

Increasing dimensions imply, of course, a rapid increase in the number of subgroups and of alternative subgroup chains involved in the reduction of an original large starting group. Multiparticle systems, which start with ( N - 1) degrees of freedom and, therefore, an initial set of [ 3 ( N - 1)] 2 generators of U ( 3 N - 3), pose these questions in particularly stark terms. Focusing on behavior under rotations, Section 10.2 has considered two dominant reduction schemes for S O ( 3 N - 3), one hierarchical through all the intervening S O ( D ) down to the final SO(3) of the total orbital angular too-

278

Chapter 10. Rotation Groups in Higher Dimensions

mentum of all the particles, and a second that decomposes SO(3N- 3) into a product SO(1)(3) • SO(2)(3) x . . . of all the ( N - 1) coordinates ~/. These are the two major (extreme) chains of physical interest, the first abandoning completely the individual identities of ~/with all 3 ( N - 1) coordinates seen on equal terms as Cartesian axes of the full 3(N - 1)-dimensional space, and the second retaining a role for the s of each particle. Clearly, many other possible decompositions may be identified with partial retention of the generators of single particles or of clusters in describing the physics of the system. Each reduction scheme represents, therefore, a particular statement of relevant dynamical symmetries just as in (10.50) and other examples above. Exploring alternative dynamical symmetries and the transformations interlinking them as generalized recouplings of generators remains largely undeveloped. Applications in the quantum physics of atoms and nuclei have been confined along the lines of Section 10.2 to threeand four-body systems. For the kinetic energy operator in (10.45) and for Coulomb or oscillator potentials between two particles, matrix elements between the hyperspherical harmonics in (10.40)or (10.47) are available [48, 49], and the Schrhdinger equation describing doubly and triply excited states in atoms has, for instance, been analyzed along these lines [51, 52]. The crux of the method, viewing each degree of freedom as an axis of a multidimensional space and then considering rotational (or other) transformations of the full space or of its subspaces can be carried over to other problems. For instance, the motion of three particles on a line may be viewed in SO(3) just as the example of harmonic oscillations in x in Section 10.3.3 was viewed in SO(2). [In that example, the introduced y coordinate may be described either as a second dimension of motion of one particle as we did or, alternatively, as the (identical) motion of a second particle on the same line]. Once embedded thus in an SO(3), the Racah-Wigner algebra can be brought to bear with interactions and dynamics recast in the language of r-transformations. States of the three particles on the line may be described as Ijm), a delta-function or "hard-core" interaction between the particles viewed as an interaction diagonal in j, etc. Thereby, r-transformations and the other generalizations to higher orthogonal groups considered in this chapter become applicable to a variety of problems having little to do with actual rotations.

Chapter 11

Lorentz T r a n s f o r m a t i o n s and t h e Lorentz and Poincar Groups The operations of rotations and reflections discussed in earlier chapters are bounded in that their relevant parameters, typically angles of rotation, have finite ranges of variation. Further, these operations preserve the squared distance between any two points in space. Their irreducible representations are finite-dimensional, the corresponding groups being called "compact". These features are immediately evident for the discrete groups of Chapter 9 with their finite (indeed, small in number) tables of characters as in Table 9.1. The same holds for the group SO(3) of three-dimensional rotations of earlier chapters, where a representation with angular momentum j has (2j § 1) dimensions. Larger SO(D) groups of higher dimensions in Chapter 10 also have finite-dimensional (even if large--see Table 10.1) representations. The transformations of rotations and reflections are classed as "isometries" that preserve geometrical properties of objects such as their shapes and sizes. Their relevance to physics stems from the expectation that transformations between coordinate systems do not influence physical laws. The equations of physics are form-invariant under these symmetry transformations. A broader context for symmetries and for the invariance of physics is provided by other transformations that involve stretches, possibly infinite, 279

280

Chapter 11. Lorentz Transformations

in other variables. The most important of these, the subject of this chapter, concerns the invariance of physical laws when viewed from frames of reference at different velocities. The Special Theory of Relativity places on an equivalent footing all frames that differ only in the uniform velocity of their relative motions. The "Lorentz transformations" between these frames relate the respective space and time coordinates as did rotations and reflections (for the spatial variables alone) but with important differences: These transformations preserve the space-time interval, rather than the spatial distance between events. As a result, the noncompact Lorentz group, SO(3, 1) consists of transformations involving the four dimensions of space and time, with irreducible representations that may be of finite or infinite dimensions. The Lorentz transformations, and the ensuing theory of Special Relativity, stemmed from a basic feature of electromagnetism: an observer at rest with respect to an electric charge detects its electric action and no magnetism; an observer in motion with respect to that charge perceives an electric current generated by the charge's motion and its magnetic effects. Observers moving in the opposite direction perceive a current of opposite sign. Lorentz formulated his transformations with the objective of ensuring that Maxwell's equations held equally for all observers regardless of their own constant velocity. Atomic phenomena, particularly the interactions of particles moving at speeds comparable with the light velocity, also prove independent of their observer's constant velocity when formulated in accordance with Lorentz transformations. Novel phenomena emerged thus from Dirac's adaptation of quantum mechanics to invariance under those transformations. This adaptation involved both representing states of physical systems by probability amplitude symbols (wave functions, kets and bras) and the operators that combine with them to yield observable parameters. Charge and current densities experience thus Lorentz transformations, revealing the occurrence of particle pairs with opposite charges. The formulation of "elementary" high energy phenomena hinges thus not only on the Lorentz transformations but also on the properties of irreducible base sets on which they operate and which represent states of relevant particles and fields. The following Sections 11.1, 11.2, and 11.3 describe the Lorentz transformations and their relevant groups with increasing scope. Section 11.4 will then outline their physical applications. The following Chapter 12 will deal with other analogous noncompact groups with variables of infinite range such as energy or radial coordinates.

11.1. Lorentz Transformations

11.1

281

Lorentz Transformations

The geometrical considerations of the previous chapters apply to spatial variables alone, based as they are on a static description. The equations of physics, however, describe the evolution of systems in time. This dynamic evolution lies outside the domain of the purely geometric description of space. Adjoining time as an additional coordinate enlarges geometry to embrace also evolutions in time thereby giving a geometrical interpretation to kinematical constructs such as velocity and energy. The Special Theory of Relativity pointed to the necessity for this enlargement by removing any sharp distinction between statics and dynamics once different inertial frames of reference were considered together. The fundamental geometrical constructs are thereby no longer points in coordinate space, but "events," that is, coincidences in both space and time. The coordinates s of a particle, along with a clock reading the time t, are together ascribed to an event in any one inertial frame. Different events may be connected by sending a signal from one place to another. Experiments, plus Einstein's interpretation of Maxwell's laws of electromagnetism, show that the signal has a maximum velocity c. This velocity is known empirically to equal the velocity of light c, but such identification is irrelevant to the geometry of space and time. Mathematically, it suffices to regard c as a scale factor with dimensions of velocity, serving to define a fourth coordinate zo - ct with the dimension of length. Indeed, modern metrology sets the speed of light as a fixed, fundamental number (equal to 299,792,458 ms -1 exactly), thus fixing the unit of length in terms of the unit of frequency or of time. Were we to identify the four vector (11.1)

z~, - ( c t , g)

with a four-dimensional vector such as the ( in (10.10) and consider the four-dimensional rotations of SO(4) described by (10.17), we would have a Euclidean geometry that keeps the "distance" between two points, c2(tl t2) 2 + (~'1 - 272)2, invariant. But the equations of mechanics are not even approximately invariant under such a geometry. Instead, the equations of relativistic mechanics and electromagnetism are invariant under space-time transformations that preserve the interval distance defined as

C2(ti t2)2 --

--

(~'1

--

s 2,

with a crucial difference in sign between space and time.

(ii.2)

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

282

The occurrence of this crucial minus sign also emerges in the propagation of waves, whether in electromagnetism or in quantum or classical mechanics. A base set of plane waves propagating in a homogeneous medium is invariant under translations, with surfaces of constant phase

k . F'- (w/c)ct - constant,

(11.3)

where c is the phase velocity of the wave. Clearly, transformations under rotations in the three spatial dimensions leave the scalar product k. Y unchanged but (11.3) is also invariant under transformations that mix space and time. Thus, with k chosen parallel to the z axis, should the sign in (11.3) be replaced by a plus sign, then r-transformations of the pairs (w/c,k) and (ct, z) would leave kz + (w/c)ct invariant. The minus sign, however, causes the appropriate transformation to employ an imaginary angle, being represented by the matrix B-(

coshx -sinh X

-sinhx) cosh X

(11.4) '

with a real variable X ranging from - o o to cx). Unlike its r-transformation counterpart ( r

\

9o -sm~

sin~ )

cos~o ]

B is nonunitary while remaining unimodular,

that is, with det B = 1. The minus sign in (11.2) and (11.3) is thereby associated with the similar sign in the identity for hyperbolic functions, cosh2 X - sinh2 X - 1, as against cos 2 ~ + sin2 ~ _ 1 for trigonometric functions. Setting -- (1 -

v2/c 2) =

tanh X =

cosh ;~,

v/c,

(11.5) (11.5a)

the transformation (11.4):

(c,,z , ) - B (c:)

(11.6)

is precisely the Lorentz transformation between two inertial frames with relative velocity v along the z direction. It is called a L o r e n t z b o o s t rather than a rotation (through an imaginary angle) because of its physical effect of boosting the velocity of a frame by v. Together with its two counterparts for similar boosts along the x and y directions, the full four-dimensional

11.2. G e n e r a t o r s and Representations of the L o r e n t z G r o u p 283

form of this L - t r a n s f o r m a t i o n is x~ x~ z~ x~

_

coshx 0 0 -sinhx

0 0 1 0 0 1 0 0

-sinhx 0 0 coshx

x0 X1

(11.7)

z2 x3

to be contrasted with a pure rotation or r-transformation of earlier chapters, such as the rotation through ~ about the z axis x~) x~ x~ x~

_

1 0 0 0 cos~p - s i n ~ 0 sin~ cos~p 0 0 0

0 0 0 1

x0 xl x2 x3

"

(11.8)

In the limit when v / c (( 1, the transformation in (11.4) and (11.5) reduces to the Galilean transformation, z --~ z - vt, t ---. t, of nonrelativistic Newtonian mechanics. This chapter aims at extending the results of Parts A and B to include the L-transformations (11.7). To this end, we complement the Ltransformations by their infinitesimal partners. The L-transformations turn out to factor into products of two (complex) r-transformations affording the mathematical apparatus of previous chapters to be readily adapted to the algebra of L-transformations.

11.2

Generators and Representations of the Lorentz Group

The task of blending the boost operator B into the r-transformation system will occupy much of this chapter. Historically, however, its development in the 1920s proved exceedingly seminal for physics: it led van der Waerden to introduce the term "spinor" (the concept itself having arisen earlier in the work of the mathematician Elie Cartan) that was to prove crucial in describing electron spins. Factoring of the quadratic invariants of relativity, x2 + y2 + z 2 _ c2t 2 and E 2 - p2c2 - ( E - p c ) ( E + p c ) , introduced sign reversal symmetries of E and t, which--together with spinors--led Dirac to his relativistic equation, thus laying the foundation for the concept and discovery of "antimatter."

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

284

11.2.1

Four-vectors

and

the

Lorentz

metric

The minus sign in the invariant interval (11.2) or in the analogous invariant squared "length," x02 - ~ 2 , of a four-vector in (11.1), suggests the introduction of a related four-vector with upper indices (11.9)

x ~ = (ct,-~)

so that the squared length can be factorized as

X2 -- E X~X~ #

c2t 2 _

_

s

(11.10)

As in Section 10.1, the index p takes the value 0,1,2, and 3 and the summation may be implied when the same index is repeated as an upper and a lower one. Alternatively, Eq.(11.10) or other similar scalar products such as ~ k ~ x ~ where k ~ - ( w / c , - k ) can be formulated by introducing a four-dimensional m e t r i c t e n s o r ri'~ that raises or lowers indices"

q~,v _ rit,~ -

1

0

0

0

0

-i

0

0

0

0

- 1

0

0

0

0

-1

(11.11) '

so that we have x ~' - ~"~,, ri~'~' x v and x 2 - ~ x~,q ~" x , , . A general Lorentz transformation A is represented by a 4 x 4 matrix, x I = Ax, including all the operations of pure rotations such as (11.8), pure boosts or L-transformations such as (11.7), reflections (x0 ~ x0, s ~ - s and time reversals (x0 ~ - x 0 , s ---* ~), which preserve the interval distance between two events. Together they form the Lorentz group 0(3, 1). A subgroup, called the p r o p e r L o r e n t z g r o u p SO(3, 1), includes all the operations continuously connected to the identity, such as the boosts and rotations, excluding inversions through the origin. Whereas real orthogonal matrices O of SO(4) satisfy the condition o T o = I , Lorentz transformations satisfy ATriA =7/. (11.12) The operator A is sometimes referred to as "pseudo-orthogonal" and the 3 and 1 in the symbol 0(3, 1) reflect the occurrence of three minus and one plus sign in the metric (11.11). Taking the determinant of both sides in (11.12) yields (detA) ~ - 1 whereby detA - 4-1. The operators with d e t A - 1 constitute the proper Lorentz group. The Lorentz transformations A resolve into four classes depending on the sign of det A and

11.2. G e n e r a t o r s a n d R e p r e s e n t a t i o n s of t h e L o r e n t z G r o u p 285

ct

Z

F i g u r e 11.1- The light cone: time-like (xux u > 0) regions are shown shaded and space-like (xUx u < 0) unshaded in the z-ct plane. The 450 lines define the null interval of signal propagation.

on whether A ~ is >_ 1 or < - 1 . The latter condition distinguishes so-called orthochronous transformations with A~ >_ 1 that preserve the sign of the time component of a time-like interval (those with x 2 > 0), meaning that events lying within the forward and backward light cone (Fig. 11.1) remain within their region. A time-reversal operation, on the other hand, with A0~ < - 1 interchanges these two regions.

11.2.2

Generators

of the proper

Lorentz

group

The number of generators is not affected by the plus or minus signs in the metric, remaining at six just as in SO(4). Indeed, three of them are identical to the g in (10.17a), obtained in the usual way as infinitesimal versions of the matrix (11.8); idA/dr As expected, these generators of pure rotations coincide with (2.13) and (2.18), with an added first row and column of zero entries pertaining to the untouched time coordinate. On the other hand, the three generators for Lorentz boosts that follow from expressions (11.7) through ida/dXlx= o differ from the SO(4) generators in

C h a p t e r 11. L o r e n t z Transformations

286

(10.17b), being represented by

NX

m

0 -i

-i 0

0 0

0

0

O0

0

0

0

0 0

0 0 '

Nv-

0

N~-

0

0

0

-i

0 0 -i

0 0 0

0 0 0

0 0 0

0 0

-i 0

0 0

-i

O

0

0

0

0

0

0

"

(11.13) Unlike the ~ of SO(4) in (10.17b), these matrices are anti-Hermitian rather than Hermitian, with an imaginary factor reflecting again the step from real rotations to pseudorotations. The six generators g and N of the proper Lorentz group form a closed algebra through the commutation relations [gi, gj] [gi, Nj]

-

i~ijkgk , ieijkNk ,

[N~, N~]

-

--ie~kek

(11.14) .

The first two commutators are, as expected, identical to the relations (10.3) for SO(4), identifying {as the operator of angular momentum or rtransformations, with N transforming under rotations as a three-dimensional vector. The third relation in (11.14) differs, however, from (10.3) in its crucial minus sign that reflects the passage from the compact SO(4) to the noncompact 0(3, 1). The generators N of Lorentz boosts are odd under reflection of coordinates, behaving like a polar vector (whereas {is of course axial). It is also useful, again as in Section 10.1.2, to define the six operators as the six antisymmetric second-rank tensors guy in space-time with gij cijkgk and gio - Ni. The commutation relations (11.14) take then the form

[e..,

-

- i ( , 1 . o e . p + ,7.peso - ,7.pe, o - ,7,~e.p),

(11.15)

similar to the corresponding (10.15) for SO(4) with -77 from (11.11) replacing the Kronecker 6. Just as the linear combinations of g and 5 in (10.5) obey the simpler commutation relations in (10.6), we define similarly combinations of g and N, but now with the minus sign in (11.14) leading to an imaginary unit: J-] J2

- 89 ilV)' - ~ ( 1e - i'"~ ) .

(11 16)

11.2. G e n e r a t o r s and R e p r e s e n t a t i o n s of the Lorentz Group 287

In terms of this pair, the commutation relations (11.14) reduce to a form identical to (10.6): [Jli, Jlj] [J2i, J2j]

= =

iQjkJlk , ieiikJ2k,

[Jli, J2j]

=

0.

(11.17)

The imaginary element in (11.16) prevents (11.17) from representing two independent SU(2) algebras as in (10.6), being instead isomorphic to the group of linear transformations in a complex two-dimensional space as we will see. The operators [ a n d i/V are Hermitian and mutually dual just as ~'and are in Section 10.1.2. Their combinations (11.16) are, therefore, self-dual and anti self-dual. As with earlier combinations in (10.5), the opposite parities of ~'and/V under reflection (P" s - s prevents J-1 and J-2 from having well-defined parity, being instead related by P J 1 P -1 - J2.

(11.18)

These two operators are also connected by the frame-reversal transformation K of Chapter 3 which reverses [while leaving i/V unchanged: KJ1K

-1 - - J 2 .

(11.19)

In the present context of Lorentz transformations, the connection of frame reversal to time reversal leads to the same conclusion as in (11.19). The reversal of time reverses both ~" and the Lorentz boosts JV while, at the same time, the complex conjugation inherent in this operation changes the sign of the imaginary unit in (11.16). 11.2.3

Lorentz

transformations

to r-transformations

Viewing J-1 and J2 as infinitesimal operators affords factoring the Lorentz transformations into products of r-transformations operating on separate spaces. The basis vectors spanning these spaces are complex, as the (J1, J2) matrices satisfying (11.17) also are; their components are labeled by quantum number pairs analogous to those of angular momentum eigenstates, specifically by integer or half-integer values of J1 or J2 and by corresponding magnetic quantum numbers. This specification holds for finite-dimensional representations of the proper Lorentz group. (Infinite-dimensional representations also occur for complex values of J1 - J2 or when Jx + J2 and [J1 - J2[ are not simultaneously integers or half-integers.) The finite-dimensional

288

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

representations become then products of two matrices of the class described in Chapter 4, namely, i(JiJ2) --D('/~) ( ) n ( J 2 ) ( mlm2,#,#2 -- ml#x m2P2

)

'

(11.20)

with - J i <_ {mi, lli} (_ Ji, and variables described below. Both D (J') and D (J2) depend on three indices playing the role of Euler angles but in general complex. This formulation might seem at first sight to involve six complex or twelve real parameters but the spaces of J1 and J2 are not independent, being linked by the condition analogous to (11.19), J1 - - J-;.

( 11.21)

As a result, six real parameters specify a Lorentz transformation and an irreducible representation (J~, J2) of S0(3, 1). Writing these parameters as a complex vector ~ + i;g involving six real components {~, ~, X, 2} casts the Lorentz transformation (11.20) as exp[i(~ + i)~)-J~] exp[i(~-i)~)-J-2],

(11.22)

that is, as the direct product of two complex r-transformations. Alternatively, Eq. (11.22)can be recast as exp[i~, g] exp[-i;~. N]

,

(11.23)

since J~ and J-2 commute, allowing these operators to be combined in the exponent with the aid of (11.16). The first factor is a usual unitary rtransformation but the second is antiunitary because N is anti-Hermitian. Indeed, this second factor represents a Lorentz boost with )~ the direction of the velocity 73 of the coordinate frame and fl = tanh X = v/c. The group's two Casimir invariants, ~2 _ j~2 a n d / ' . AT, can be expressed alternatively in terms of J~l and J~. Note again the minus sign, contrasting with ~ + if2 in Section 10.1. Thus, there occurs a denumerable infinity of finite-dimensional irreducible representations of the proper Lorentz group SO(3, 1), labeled (J1, J2), with the same (2J1 + 1)(2J2 + 1) dimensions as in SO(4). The magnitudes 1 1 , . . . with A (J1 ' J2) being single-valued of J1 and J2 take the values 0, 7, if J1 + J2 is an integer, double-valued otherwise. This behavior is illus1 and (3, t 0) , both trated by the first pair of nontrivial representations (0, 3) of which have two degrees of freedom but operate in different spaces. These so-called Weyl representations operate on the states of massless neutrinos

11.2. G e n e r a t o r s a n d R e p r e s e n t a t i o n s of t h e L o r e n t z G r o u p 289

as we shall see in Section 11.2.5, whereas their superposition, (0, 7 ) + ( 7~, 0) , with four components, operates on the states of Dirac electrons. The repre1 1 sentation (3, 3) with four degrees of freedom operates on four-vectors while (1,0) + (0, 1) with six degrees of freedom operates on antisymmetric tensors describing the electromagnetic field as formulated in Section 11.2.6.

11.2.4

Spinor r e p r e s e n t a t i o n s

Here, as in Section 2.4, r-transformation matrices of order 2 which provide the "fundamental" representation for the lowest nonzero value, 89 of J1 and J2 prove convenient in constructing general Lorentz transformations. 11 Denoting the base spinors on which J1 operates by ul - (uls~) and u2 = 1 1 (ul~ -~), its transformation by the first factor of (11.20) takes the form Ull

--

aul

u~

-

c u l + du2 ,

Jr- b z t 2 '

(11.24)

where a ' b, c, and d stand for the transformation matrix elements 1-)( a J f l , l m89! with 1 m, m' - ~7. These elements coincide with those considered in Chapter 4 when the Lorentz transformation amounts to a real rotation but the four elements of L-transformations (boosts) no longer constitute a unitary matrix, even though satisfying the unimodular condition ad-

bc

= 1.

(11.25)

A second pair of base spinors, fi,, distinguished by a dot, is required [53] to illustrate the transformations by the operator ~ , these transformations being complex conjugate to (11.24) and expressed by u~~

-

a ' u 1 9 + b * U2 9 ,

U~~

--

C* U 1 ~

+ d* u2..

( 11.26)

Under the unimodular condition (11.25), unaffected by complex conjugation, the set of complex numbers {a, b, c, d} corresponds to the six-(real) parameter vector-pair set {~, ~} in Section 11.2.3. 1 The spinors {ul, u2} form the base representation with J1 - 3, J2 - 0, 1 while {ulo, u2~ provide the corresponding base for J1 - 0, J2 - 3" The "spinor" method of Section 4.1.1 can thus provide directly the bases of higher order representations of (J1, J2) by raising the elementary ones to requisite integer powers: 2Jl-kl

k2

(11.27)

Chapter 11. Lorentz Transformations

290

with 0 <_ kl <_ 2J1, 0 <_ ks <_ 2J2. The process of forming direct products of operators, followed by their subsequent reduction, applies to the dotted and undotted spinors separately, with the understanding that transformations in the two separate spinor spaces are no longer unitary. All of the mathematical apparatus of Chapters 4 and 5 applies thus to Lorentz transformations, providing the finite-dimensional irreducible representations of the proper Lorentz group S 0 ( 3 , 1). The occurrence of separate transformations in the spaces pertaining to J1 and J2 causes each of these representations, and of their associated state representatives, to be labeled by the pair of indices (J1, J2). The reduction process also applies directly to (11.27) in the sense of adding the operators (J1, J2) themselves as in Chapter 5. Thus, reduction produces a new basis in which A (J',J2) is no longer block diagonal because dotted and undotted spinors transform differently. Even so, the reduction is useful because transformations corresponding to pure rotations, with J1 - J2 - ~g, become indeed block-diagonal. Under this condition of pure rotation, the coupling of two spin- 89bases forms a vector and a scalar which together amount to a four-vector analogous to the coordinate vector x u or the vector potential A u whose zeroth component is a scalar while the other three form a vector. Explicitly, the linear transformations (11.24) and (11.26) transform a bilinear form, "~

"~

1

"~

Cll~

~ -~- C 1 2 ~

~ -~- C 2 1 . U 2 U 1 ~

-~- C 2 2 ~

~ ,

(11.28)

into itself, the determinant C11oC22~

-- C12~

o

(11.28a)

remaining invariant. If the expression (11.28) is Hermitian, c11. and c22, are real while c12. and C21o are complex conjugates, thus constituting a real four-dimensional space with coordinates {ct, x , y , z } according to the prescription c21. = x + iy c12- -x - iy (11.29) Cll*

--

Z -~- Ct

C22"

----

- - Z -~- C t .

The invariant determinant (11.28a) is thereby identified as c2t ~ - (x 2 + y2 + z2), placing the Lorentz invariant interval in one-to-one correspondence with the constancy of the determinant in (11.25) under the linear

11.2. G e n e r a t o r s a n d R e p r e s e n t a t i o n s of t h e L o r e n t z G r o u p 291

transformations of a complex two-dimensional space. The product of a dotted and undotted spinor with mutually complex conjugate coefficients, (ClUl +c2u2)(C*lUlO +c~u2~ has precisely the form (11.28) with c11, = ClC~ and c22~ - c2c~ real and c12~ - c~1~ - ClC~, thus giving rise to fourdimensional space-time through the identifications (11.29). Pure rotations correspond to the transformations (11.24)and (11.26)with d = a* and b = -c* because c11~ + c22, = 2ct remains fixed and only x, y, and z transform among themselves in a three-dimensional rotation. On the other hand, a pure Lorentz boost along the z axis corresponds to a scaleU2 transformation of the spinor components u~ (1 +/~) 89 u 1, u~ - (1 +/?)- 89 with fl = v/c. The proper Lorentz group S0(3, 1) and the group SL(2, C) of linear transformations of a complex two-dimensional space are thus equiv-

-

alent. Since the identity of the Lorentz group maps onto both and ( - 1 0

(10) 0

1

0 ) o f S L ( 2 , C) ' the latter groupisviewed asthe "covering"

-1

group of S0(3, 1), bearing to it the same relationship as SU(2) does to S0(3) in Section 2.4 or SU(2) x SU(2) to S0(4) in Section 10.1.1.

11.2.5

Neutrino

and

electron

spinor

states

Representations of the Lorentz group have a key role in the analytic treatment of elementary particles. Under the twin requirements of quantum mechanics and special relativity, states of an elementary system should be represented by the smallest possible set of probability amplitudes consistent with the superposition principle and invariant under Lorentz transformations. Otherwise, any decomposition of states into smaller subsets, each of them invariant itself under Lorentz transformations, would imply a relativistically invariant way of subdividing the state representatives so that they would no longer qualify as elementary. Irreducible representations of the Lorentz group serve, therefore, to represent elementary particles in physics. Treating the mass and linear momentum of a particle requires actually an extended version of S0(3, 1) representations including space and time reflections as well as translations. The relevant "Poincar~ group" will be introduced in Section 11.3. At this stage, we can however already identify several irreducible representations of S0(3, 1) as appropriate to specific elementary particles according to their spin because the base pair (J1, J2) suffices to specify the behavior under pure rotations.

292

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

The (0, 3)a and (3,1 0) representations are most important, since they transform state representatives under rotations as appropriate to particles with angular momentum 3" 1 One may thus represent the two-component spinors by u - (::) and u. - (:1o) and write Lorentz invariant equations 2* of motion i(-a0 + E1 0, i(-Oo - E

(1/c)O/Ot

-

o,

(11.3o)

O/axk.

where 00 = and Ok These are the "Weyl equations" for neutrinos with the differential operator 0 u - ( 0 0 , - V ) consistent with the four-vector in (11.1), and the 2 x 2 Pauli spin matrices ak defined as in Section 2.4. Note that 5 and - 5 appear in the two equations which may be identified with two essentially different types of neutrinos. As seen from (11.18) and (11.19), the two types are parity and frame-reversal conjugates of each other. In terms of a so-called "helicity" operator to be defined in Section 11.3, they are referred to as left- and right-handed. The particles emitted with electrons in beta-decay are right-handed, while those emitted with positrons are left-handed. They are called antineutrinos and neutrinos, respectively, much as positrons are regarded as the antiparticles of electrons. 1 and (3, 1 O) , on the other Combination of the two representations (0, 3) hand, can yield parity eigenstates and serve to represent the spin -1 electron according to Dirac. Defining the pair of two-component superpositions Ca-

1 ~(U+Uo)-

(r r

'

(11.31)

1

(rr

'

(11.31a)

~bb -- ~ ( u - - u . ) - -

they may be arranged as a four-component Dirac spinor,

r ~-

~b

~2

"

(11.32)

~3 Likewise, the Y and - ~ are arranged into the 4 x 4 Dirac spin matrix

%-

--(rk

0

'

(11 33)

allowing (11.30) to be rewritten as the Dirac equation for the electron:

(iETuOU - mc) ga- O. tt

(11.34)

11.2. G e n e r a t o r s a n d R e p r e s e n t a t i o n s of t h e L o r e n t z G r o u p 293

To the three 7k, we have added here a fourth Dirac matrix 70 -

(I

0

0 ) -I '

(11.35)

and a unit 4 • 4 matrix for the mass term, the electrons possessing a rest mass unlike the massless neutrinos. That is, unlike massless entities which are always constrained to propagate with velocity c, systems with m # 0 afford a frame of reference in which they are at rest and which can be reached through suitable Lorentz transformations. Note that the Dirac 7 matrices and the four-component bi-spinor r are formed as combinations of pairs of two-dimensional Pauli matrices and spinors. Alternative forms of 4 • 4 matrices 7u are available but all must satisfy the requirement that the generators gu~ of the Lorentz group S0(3, 1) in (11.15) be expressed in terms of them as the antisymmetric product of two 7u's:

gt,~' _ _~auv1 _ -~i[Tu, 7~].

(11.36)

The basic commutation relationship (11.15)of the Lorentz group translates into the corresponding defining equation for 7u given by Dirac: {7u,7~} - (7u7~ + 7~%) - 2r/u~-

(11.37)

These anticommutation relations define a closed algebra for the Dirac matrices, called a "Clifford algebra." Independently of any specific form for the matrices as in (11.33) and (11.35), the relationships (11.37) alone suffice to factorize the relativistic expression linking energy, mass, and momentum: (11.38)

( E / c ) 2 _ fi2 _ rn2c 2 = 0 , p2__ m2c2 _

~

Z pv

7uTvPuPU-

m2c2

(11.38a)

v

where Pu - (E/c,p-'),pU - (E/c,-p-'). Indeed Dirac was led to (11.34) through this factorization. In arriving at (11.38b), the symmetry of pUp~' 1 was used to rewrite ~-~u~, ~Tu~'pup~' - -2 ~~uu{Tu, 7,'}PUP v as ~u~, 7uTuPuP"" The Dirac equation for the electron, as given in (11.34) with (11.32), can be written

( (E/c)-mC~.p

(E/c)-~'P+mc ) (r162

(11.39)

294

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

These are coupled equations for the two-component Ca and Cb. In the nonrelativistic approximation with E = m c 2, Ca is large compared with Cb, their relative ratio being ]p-~/2mc. In the extreme nonrelativistic limit, the electron may be ascribed, therefore, the two-component r alone as its wave function, returning us to the description of earlier chapters. But there are also negative energy states where the situation is reversed, Cb alone surviving in the nonrelativistic limit and identified with states of the antiparticle, the positron. In all other situations apart from this limit, both Ca and Cb, that is, all four components of the Dirac bi-spinor, are required to describe either the electron or the positron.

Electromagnetism and its quantum

11.2.6 1

1

The (~, ~) representation of SO(3, 1) has four degrees of freedom and transforms a four-vector Cu. Under rotations, the four-vector reduces to a vector and a scalar. Therefore, a spin-1 element, with three degrees of freedom and intrinsic angular momentum one, is acted upon by (1, 89 only upon supplementing with an auxiliary condition, typically ~ u 0uCu - 0. The vector and scalar potential of electromagnetism, At, - ( r afford the most familiar example where the "Lorentz gauge condition" ~ u cOUAu - 0 serves this purpose in defining the vector field for the quanta of electromagnetic energy (photons). In this particular case, however, the masslessness of the states leads to a further reduction by one of the degrees of freedom, the photon having only two such degrees of freedom which are identified with two orthogonal polarizations as in Section 11.4.2. Next, the (1,0) representation has again to be considered in combination with (0,1) in order to have well-defined parity for the objects transformed. This combination with its six degrees of freedom has the same dimension as the generators of the group. 1 In electromagnetism, it represents the electric (/~) and magnetic (B) fields as components of the antisymmetric tensor Fg~" Foi - - E l ,

Fij - - Z

e.ijkBk,

(11.40)

k

1The use of (dl, d2) to label representations of S0(3,1) differs from that of SO(4) in Section 10.1.2. There, and in Table 10.1 for other orthogonal groups, we used instead 1 1 [dl+ d2, J1 - ,/2]. Thus the ( ~-, 7) of S0(3,1) is the [10] representation of Chapter 10 and the (1,0) ~ (0,1) is the [11] of SO(4). The dimensions of corresponding representations of the two groups are identical, the comma playing no role in this counting.

11.3. The Inhomogeneous L o r e n t z ( P o i n c a r 6 ) G r o u p

295

that is, F,,, -

0

-Ex

-Ey

-Ez

E~: Ey

0 Bz

- Bz 0

By -Bx

Ez

-By

B~

0

"

(ll.40a)

The fields E and/~ are analogous to 57 and e, respectively, in terms of their behavior under space and frame (or time) reversal. The linear combinations B -4- i E that parallel (11.16) are transformed into each other under these reversals just as in (11.18) and (11.19), whereas the combinations g2 _ ~2 and B . E remain invariant under Lorentz transformations. Once again/3 and iE are mutual duals in the sense of (10.17c), and g ~ iE are self-dual and anti self-dual, respectively. Similarly, higher-spin elementary particles correspond to other irreducible representations of the Lorentz group. Thus, the (1,1) representation with 9 degrees of freedom, together with four supplementary conditions, might fit a particle with spin 2. The four conditions are required to subtract out a scalar and vector part contained in the D(1)D(1) of (11.20) which, as per the Clebsch-Gordan series for r-transformations in Chapter 5, equals D (~) + D (1) A- D (~ In general, one requires 4JiJ2 of such supplementary constraints. These higher spin representations are, however, not of as much interest in physics as the spinor and vector ones.

11.3

The Inhomogeneous Lorentz (Poincar Group

)

The space-time interval (11.2) between two events is invariant not only under the pure rotations and Lorentz boosts of the proper Lorentz group S 0 ( 3 , 1) and under the space and time inversions included in the full Lorentz group 0(3, 1), but also under translations of an observer in space and time. That is, the extension of x' = Ax to x.

A~,x. + a~,,

(11.41)

V

with four-vectors a u independent of ~ and t, represents the full set of operations that preserve the interval distance (11.10). The general, inhomogeneous Lorentz transformation {a, A} and its associated group, called the Poincar~ group, do not factor, however, into a direct product of Lorentz

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

296

transformations and of translations because these operations do not commute. Indeed, the product of two inhomogeneous transformations yields (11.42)

{a, A}{a', h'} - {a + Aa', AA'}.

A transformation {a, A} is characterized by ten parameters, with four components a, supplementing the six of A. The expression (11.41) may be cast in terms of 5 • 5 matrices by adjoining a fifth row and column to Lorentz transformations" x~ x~ -

x~ x~

11.3.1

0

Generators

0

and

0

0

a0 al

x0

a2

z2

a3 1

x3 x4

Xl

commutation

(ll.41a)

relationships

The Poincar~ group has ten generators corresponding to its ten parameters. Besides the /?~, of Section 11.2.2, the Hermitian operators p, - i0, = (iOo,-i~) provide the generators of translations in space and time. These operators commute among themselves (as in the two-dimensional example of Problem 2.9), [p~,p~]- 0. (11.43) The commutators of g,~ components are listed in (11.15) and those between s and p~, easily follow from their definitions, ~,

epa] -- i(rl~,ppa - ~7~,aPp).

(11.44)

These ten generators thus form a closed algebra. In terms of the/? and N generators in (11.14), the above relations are equivalent to [p0,t]

-

0,

[gi,Pj] [p0, Ni]

--

-

ieijkpk , ipi,

[pi, Nj]

-

ipo6ij .

(11.45)

The first two of these relations express that p0 is a scalar and 17 a vector under r-transformations, while the second pair shows that a Lorentz boost along any direction i interchanges p0 and Pi, leaving the other two components of t7 unchanged.

11.4. F i e l d R e p r e s e n t a t i o n s

297

Casimir invariants, that is, combinations of Pu and gu,, that remain invariant under all inhomogeneous Lorentz transformations, because they commute with each of the generators, are p2 _ ~~u PuPU - m 2 c 2 and W 2, the square magnitude of the "Pauli-Lubanski" four-vector [54]

1 W u = --~Eeu,,poUPp~

(11.46)

vpr

=- (g'./7, p0g'- 17 x N),

(11.46a)

with euvpa defined as in (10.17c). The W u components obey the relations [gu,,, Wp] - i(~7,,pWu - ~TupW~,),

[W u , p.] - 0, [Wu, W,,] - i E

~u,,ooWPp ~ ,

(11.47)

(11.47a) (11.47b)

pa

E

Wop ~ - 0 .

(11.47c)

~

The invariants p2 and W 2 provide, therefore, two labels for irreducible representations of the Poincard group. Only nonnegative values of p2 _ m 2 c 2 are of physical interest. The first of these invariants is the squared length of the energy-momentum four vector, identified with the particle mass. The interpretation of the second invariant will emerge in Section 11.4, being associated with a fundamental distinction between systems with m :/: 0 and m - 0. The former admits a frame of reference in which the particle is at rest whereas the latter does not, leading to different meanings of W 2, as essentially the spin angular momentum ~.2 for m 7(: 0 but the polarization or "helicity" for massless entities.

11.4

Field Representations

By incorporating all the generators of space-time transformations, namely, the energy-momentum and angular momentum operators, the inhomogeneous Lorentz group transforms the fundamental entities of physics, particles and fields. These entities are characterized first by the values of the group invariants p2 and W 2 The identification of further invariants for unique specification proceeds differently for the two cases of vanishing or nonvanishing values of p2, that is, for massless or massive objects, respectively.

Chapter 11. Lorentz Transformations

298

11.4.1

Massive s y s t e m s

When p2 = m2c 2 > 0, the sign of p0(= E / c ) commutes with all the generators thus providing another invariant. Representations (p, W) subdivide further according to this sign, pertaining to particle/antiparticle for positive/negative energy states. With rn ~ 0, shifting to the rest frame through appropriate Lorentz transformations reduces p, to (P0, 0). In this frame, Eq. (11.46a) also simplifies to

W , - (0, mcs-'),

(11.48)

where g has reduced to the spin vector g'. Irreducible kets are thus eigenvectors of p2 and W 2 with labels rn and s"

p2[m,s,...> W21m,

s, . . .)

-

m2c2{m,s,...},

-

-m=c2s(s

+

(11.49)

1)Jm, s,...).

The dots stand for any further labels necessary to identify a state uniquely. Typically, the invariant description of a free particle or field proves relevant in any consideration of scattering or interactions in physics, the initial and final states corresponding to noninteracting particles "at infinity." These asymptotic states correspond to particles of definite linear momentum/Y prepared at a source or registered by a detector. Therefore, ig provides a third label for such eigenkets. A final, fourth operator that commutes with p2, W 2, and/Y is the component W0 because, as seen from (11.47a), W0 - g - f commutes with/Y whereas t~ by itself does not. This projection of the spin angular momentum (using again g" in place of t~) on the direction i6 of propagation is called helicity: W0/lp~ - g'" ifi,

(11.50)

the eigenvalues of this fourth Lorentz invariant being labeled ~. In all, therefore, eigenkets Ira, s, f, ~) represent uniquely the states of any massive free particle. In the particle's rest frame, the helicity operator becomes indeterminate but ~ as an invariant retains its meaning, being identified with any component of l ~ / m c - g, the usual choice being Sz. The general state of a massive free particle may, therefore, be described by starting with the standard state Ira, s, O, Sz = ~) at rest with spin pointing in the z direction and then subjecting it to appropriate translations, r- and L-transformations [a rotation through an angle 0 of the z axis to bring it into the direction p, followed by a Lorentz boost along i5 with ;~ = tanh-~(IpT/mc)] to bring it to the state Im, s,/7, )t}.

11.4. F i e l d R e p r e s e n t a t i o n s

11.4.2

299

Representations of massless entities

The case p2 _ m ~c2 _ 0 has an important role in physics because carriers of long-range interactions, most notably the electromagnetic field, are massless. However, W 2 retains meaning as an invariant of the Poincar~ group, and massless representations divide into two classes according to the value of this invariant, whether negative or zero. When both W 2 and p2 vanish, these two invariants do not suffice to specify a representation. As a result of (11.47c), we now have W~ proportional to p~. Since p0 - IP-~for these massless representations, the constant of proportionality W o / p o reduces precisely to (11.50), that is, to the helicity. Thus, the label A of helicity may be attached to the representations. For a given m o m e n t u m p~, we will see that there is one state for A - 0 and two different states for all A ~ 0. Even in the absence of a rest frame, it is always possible to select a frame in which two components of 17 vanish so that p~ - (p0,0, 0,p0). Correspondingly, the parallel four-vector W~ reduces to A(p0, 0, 0, p0). From (11.50), A may be identified as s~. The "spin" of such massless entities with W 2 - 0 has, therefore, only one operator, that of helicity or Sz, differing fundamentally from the massive case wherein three SU(2) operators of spin lead to a value of s with - s _< s~ _< s. This helicity is invariant under rotations and also under Lorentz boosts. However, Eq. (11.50) changes sign under the parity transformation, helicity being a pseudoscalar. Except for A - 0, we always have two values +A for any value of Sz, identified with two independent polarizations. Any long-range interaction is characterized, therefore, by two polarizations regardless of the value of Sz. Photons, with spin 1, display helicities ("circular polarization") + l , 1 gravitons of the gravitational field with spin 2 helicities +2. For the spin-~ neutrinos treated in Section 11.2.5, A - - ~1 is associated with left-handed neutrinos and A - + 89with the right-handed antineutrinos. We can also consider massless fields with W 2 r 0. Choosing again the frame with p , - (p0, 0, 0, p0), we have now from (11.46a) W~, - p o ( s z , w l , w2, s~),

(11.51)

W 2 - -p2o(W21 + wg).

(11.51a1

with

In this frame with p~ - py - 0, it follows from (11.47b) that [Wl, w2] - 0. Defining, therefore, the combinations w • - wl + iw2, we find the commu-

300

C h a p t e r 11. L o r e n t z T r a n s f o r m a t i o n s

tation relationships [s~, w•

= +w• ,

[w+, w_] = 0.

(11.52) (11.52a)

The group formed by this set of three operators {sz, w+, w_ } is isomorphic to the Euclidean group E2 in two dimensions, with sz generating rotations and w• translations in the plane (see Problem 2.9). The w+ act to raise or lower the eigenvalues of s~. Since the two operators now commute, Eq. (11.52a) differs from the analogous commutator of s z 5: i s u of threedimensional rotations which is nonvanishing. Also, the quadratic Casimir operator of E2 which commutes with all the three generators {sz, w+, w_ } is w~ +w~ and, from (ll.51a), essentially coincides with the Lorentz invariant W 2. There is no link between this invariant and sz similar to the SU(2) relation s z2 + s u2 + 2 s_z> 2 s z so that sz now is not constrained to lie in any range ( - s , s) nor constrained to integer or half-integer values. Indeed, for any given value of s~, the action of w+ generates an infinite tower of other S z , spaced by integer values. The case p2 = 0 but W 2 :/: 0 is associated, therefore, with infinite-dimensional representations and entities which have an infinite number of polarizations. Such entities do not seem to be realized in nature so that the only massless entities of interest to physics are those having W 2 = 0 with two orthogonal polarizations, whether circular, linear, or elliptical.

Chapter 12

Symmetries of the Scattering Continuum This final chapter deals with even more extended symmetry transformations, in situations where they interrelate an infinite number of physical elements. The operations considered so far of rotations and reflections, whether in three dimensions in Parts A and B, or generalized to higher dimensions in Chapters 9 and 10, transform a finite number of elements, typically the (2j + 1) components of a tensorial set with index j. Correspondingly, the generators of these r-transformations constitute a compact group whose unitary representations have finite dimensions. Even when considering extensions as in Section 10.3 to operations that are not invariances of the Hamiltonian, such as particle-hole interchange which links physical systems with different numbers of fermions, the groups involved were still compact, with finite-dimensional representations (albeit large, such as the 2 TM elements of the f shell). Chapter 11 has introduced a genuinely noncompact group, the Lorentz group 0(3, 1) of L-transformations, but only its finite-dimensional representations are of physical interest. Only upon turning to phenomena where transition operators couple an infinity of set elements do we encounter infinite-dimensional representations of noncompact groups. This situation pertains naturally and generally to atoms and molecules, where the basic Coulomb attraction between oppositely charged particles has "long range," being thus able to support an infinity of states in radial extension. Transitions between them, most importantly under the dipole operator of the electromagnetic field, inter301

302

C h a p t e r 12. S y m m e t r i e s of the Scattering Continuum

connect the entire infinite manifold whose full treatment and whose coupling to the radiation field lead to noncompact groups with their infinitedimensional representations. We turn, once again, in this chapter to the hydrogen atom as a prototype, its infinite extent arising from the radial problem. Section 12.1 focuses first on just this part involving the noncompact group SO(2, 1), a lower-dimensional counterpart of the proper Lorentz group SO(3, 1) considered in Chapter 11. Although the features of noncompactness and infinite-dimensional representations are exemplified already by this treatment of the purely radial problem, a full treatment of the hydrogen atom and of couplings by the dipole operator ~' requires, of course, the angular aspects as well. Section 12.2 considers the resulting full "noninvariance" group SO(4,2) of the hydrogenic Hamiltonian. Section 12.3 indicates a few broader current applications to other problems in scattering, many further ones being likely in the future as we develop greater familiarity with noncompact groups. Transitions to noncompact behavior will be cast as analytical continuations of parameters to their imaginary domains, much as continuation of rotation angles in Section 11.1 turned trigonometric into hyperbolic functions.

12.1

Symmetries of Radial Eigenfunctions

Since infinite families of hydrogenic states arise from the radial rather than angular aspects of the system, it is natural to look first at the purely radial rather than the complete system for the appearance of noncompact group representations. For each value of spin and orbital angular momentum, the hydrogenic spectrum has an infinite number of discrete energy levels labeled by the principal quantum number n and a continuum. Scalar operators such as r (or its higher powers) have nonzero matrix elements between the corresponding eigenfunctions, which are solutions of the radial SchrSdinger equation. The operators involved in this equation are p~, l / r , and the unit operator. Here pr indicates the radial momentum operator Pr = ( h / i ) ( O / O r ) + h / r

(12.1)

[r, p~] = iti.

(12.1a)

conjugate to r: The operators relevant to defining radial wave functions as well as couplings between them may alternatively be viewed as r and rp~. Together

12.1. S y m m e t r i e s of R a d i a l E i g e n f u n c t i o n s

303

with rpr, these three operators form a closed set of commutation relationships that follow from (12.1a): [rp., rp7 ] [rpr, r]

=

ihrpT , -ihr,

(12.2)

Further, also from (12.1a), we have (rp2r)r + r(rp~) - 2(rp~) 2 - 0.

(12.2a)

These relationships resemble--but differ in crucial minus signs from--those of r-transformation operators in (2.15) and (2.23), namely, the commutators [J~, J+] and [J+, J_], and the invariance of J + J _ + J _ J + + 2J~ - 2(J~ + Jff + Jz2). The three operators in (12.2)may, therefore, be regarded as a vector set f analogous to J. Indeed, just as the passage in four dimensions from the commutation relationships (10.3) of SO(4) to those in (11.14) for S 0 ( 3 , 1) replaces the invariant x~ + x~ + x~ + x~ by the invariant x ~ - x~ - x~ - x~, a noncompact group SO(2, 1) in three dimensions arises from SO(3) through the replacements: J1,2 ---* iT1,2, J3 ---" T3. As a result, the commutators (2.15) become [T2,T3] - i T s , IT3, T1] - iT2, (12.3) [T, , T2 ] -iT3, with a minus sign appearing on the right-hand side of the last commutator. In place of J21+J2 + J 23 as the invariant of SO(3) with values j ( j + l ) , we now have ( T 2 3 - T ~ - T ~ ) , which commutes with each T/, as the Casimir invariant of S 0 ( 2 , 1) with eigenvalues k ( k - 1). It is this "three-dimensional Lorentz group" that expresses the symmetry of the radial hydrogen Hamiltonian by identifying the dimensionless combinations [55]

lao]

- T1

(12.4)

![rff 2/(pe2) - r/ao] -- T3 2

(12.4a)

+

2

(F. ~ / h )

-

i

- T2 .

(12.4b)

Representations of SO(2, 1) are labeled by the eigenvalues of (T~ T 2 - T ~ ) and of one component of T, typically T3. Linear combinations T+ - T1 4-iT2 of the other two components raise or lower the eigenvalue of 7"3 by unity: (T~ - T~ - T~)lkn) - k(k - X)lkn), (12.5a)

304

C h a p t e r 12. S y m m e t r i e s of the Scattering C o n t i n u u m

T31kn) - nlk ) ,

(12.5b)

(T1 + iT2)lkn) - [n(n ~ 1) + k(1 - k)]{lkn 4- 1}.

(12.5c)

Once again, a crucial sign difference in (12.5a) and (12.5c) from the corresponding relations for Ijm) states of SO(3) leads to a profoundly different spectrum. With Ta2 _> (T32 - T 2 - T~), the first two of the above relations give n >_ k, and combine with the third one to yield n - k, k + 1, k + 2 , . . . , the values increasing without limit. This behavior contrasts with that for angular momentum where a given value j of the Casimir operator bounds the allowed values of m both above and below. As a result, the representations of S 0 ( 2 , 1) are infinite-dimensional unlike those of S 0 ( 3 ) which are finite. The identification of n with the principal quantum number and of k with (g + 1) follows upon recasting the Schr6dinger Hamiltonian in (10.2) in terms of the operators in (12.4), 1 [(Eao/e 2) 4- -~]T1 -[(Eao/e

2)_1

~] T 3

_

1.

(12.6)

The third operator T2, which does not appear explicitly in (12.6), may be used to affect a rotation by the angle

O - l e n ( - 2 E a o / e 2)

(12.7)

in the plane (TI, T3), namely,

exp(-iOT2)T3 exp(iOT2) exp(-iOT2)7'l exp(iOT2)

-

T3 cosh 0 + 7'1 sinh 0, 7'3 sinh 0 + T~ cosh 0,

(12.8)

reducing (12.6) to the simpler form ( - 2 E a 0 / e 2) 89 T3 - 1.

(12.9)

The eigenvalues of T3 given in (12.5b) imply then the Bohr formula E - e 2 / 2 a o n 2 for the Ikn} states [55]. Both the Schrhdinger Hamiltonian in (10.2) and the operators in (12.4) exhibit an azimuthal symmetry without reference to the quantum number m, the eigenvalue of 6z. The [kn) kets may thus also be denoted as [knm), forming a complete set of states equivalent to the familiar set of physical eigenstates [n6m}, the indices being related according to k - 6+ 1, n > 6+ 1, [m[ < 6. The exact connection of the two sets involves the same "tilting" rotation in (12.8), this time for states instead of operators, in the form

1 In6m) - - exp(iOT2 )lknm). n

(12.10)

12.2. T h e Full N o n i n v a r i a n c e G r o u p of H y d r o g e n

305

Whereas the physical eigenstates Ingm} are orthonormalized by integrating over d~, the corresponding normalization of Ikgm} involves d~'/r, reflecting the initial multiplication of (10.2) by r to yield (12.6). Alternatively, whereas the radial squared momentum operator p2 proves Hermitian under integration over r2dr, the operator rpr2 requires the differential element rdr. The denumerably infinite complete set }kn) embraces both the discrete and continuum solutions of the radial Schr6dinger equation for hydrogen, coinciding, to within multiplicative factors, with the so-called "Sturmian functions" [56], also obtained by the process of multiplying (10.2) by r. Aside from its noncompactness, the group S0(2, 1) bears several resemblances to its compact counterpart SO(3). In particular, the similarity of the relationships (12.3) and (2.15) of the two groups leads to correspondences in the form of certain radial and angular matrix elements. Some striking correspondences had long been observed before being explained by recognizing the S0(2, 1) symmetry. Thus, the expectation value of r, (12.11)

(~> - ~o ~~[3n ~ - g(g + 1)] ,

is obviously reminiscent of P2(cosO) - 113 cos 2 0 - 1]. Likewise, (r 2) is proportional to [5n2-3g(e+ 1)] which corresponds to the factor (5cos ~ 0 - 3 ) of P3(cosO). Indeed, the semiclassical connection,

(,.,)- (ao.~),

1)' (n) + n

Ps+l g.~_~i

,

s > -1

(12.11a)

stems directly from the similarity of S0(2, 1) and SO(3). The above relations and (12.5) show n in the role of m while g is common to both. Selection rules, such as the one stemming from the angular momentum addition g'] + g2 - L in a matrix element (gl [o[L][g2), hold for the radial part of the problem as well: (nglr-Slng'l = 0 for s = 2 , 3 , . . . , I~- g'l + 1 [55].

12.2

The

Full

Noninvariance

Group

of Hydrogen A complete description of the hydrogen atom's symmetries, including both their angular and radial aspects, requires combining the ~ of the previous section with the {gz,gy,gz} of orbital motion. These two sets operate on different three-dimensional spaces, the former as S0(2, 1) pseudorotations of the infinite set of radial functions while the g pertain to real rotations

306

C h a p t e r 12. S y m m e t r i e s of t h e S c a t t e r i n g C o n t i n u u m

of the physical three-dimensional space. Their direct product S O ( 3 ) | SO(2, 1) describes operations in a six-dimensional space which is thereby the domain for the full hydrogen symmetry. Such an embedding of a real physical system in a much larger number of dimensions yields, of course, richer structures and interconnections. The six-dimensional noninvariance group considered here extends far beyond classifying the symmetries of the Hamiltonian and of its individual stationary states. It encompasses all the atomic states together with their transformations [57], including the transformations between states of different energy. The group generators include operators such as ~'in its combined radial and angular aspects, that is, in its full role as the dipole operator of electromagnetic transitions. Just as 3 and 6 are the numbers of antisymmetric entries serving as generators of rotations (or pseudorotations) in three and four dimensions, respectively, the corresponding number in six dimensions is 15. From the direct product of the previous paragraph, ~" and T provide six of these. The indices 1 , 2 , . . . , 6 of the six dimensions are apportioned to the two independent three-dimensional spaces of angular and radial aspects, with 1, 2, and 3 for the former and 4, 5, and 6 for the latter. The previous section's T i , i - 1,2 and 3, are now relabeled 5, 6, and 4 respectively. In the form of a 6 • 6 antisymmetric matrix g~,v,p,u- 1 , 2 , . . . , 6 , we arrange therefore [ a n d T in two nonoverlapping diagonal blocks as in Fig. 12.1. The identification of the remaining nine generators and therefore of the remaining entries in Fig. 12.1 builds on previous chapters. A complete description of hydrogen clearly includes its symmetry under SO(4) which is associated with states degenerate in energy that are transformed into one another by the Laplace-Runge-Lenz vector A. Therefore, this set of three quantities is also included in the relevant set of generators. Indeed, expecting SO(4) to be a relevant subgroup in six dimensions means that A is adjoined to tT"as the p or ~, - 4 entries in Fig. 12.1. With SO(4) the largest invariance group for hydrogen, it is clear that the remaining generators will not commute with the Hamiltonian. Once again, three of these six have already been identified in Section 10.3.1, as another vector M that arises from the electric dipole or radial coordinate operator ~'. With A identified as one linear combination of the vectors f • t and ~' that lie in the plane of the Kepler orbit,/1~ may be identified with the other combination, ~' itself then viewed as/VI- A. There remain three further operators of a vector set to complete the fifteen t ~ , forming a closed algebra under commutators as in (11.15). In terms of dimensionless combinations of coordinate and

12.2. T h e Full N o n i n v a r i a n c e G r o u p of H y d r o g e n

307

momentum operators, the complete set of generators is given by I ~-ij -- g

-

~.i4 -- A

--

(tte2)-l[ 89

gi~ - M

--

(~t62)-1 [ l r ' p 2 -- p~r'-p~] -+- ( r ' / 2 a o ) ,

-

rfi/h,

gi6 -- F

~46

-

T1

2 --/7(~'. p-')] - (~'/2ao), (12.12)

--

t45 -- T2

-

(r

g56 - Ta

_

1

) +

9

0 -t~ gy -A~ - M~

gz 0 -g~ -A u -My

-gu g~ 0 -A~ - M~

A~ Au Az 0 -7"2

-Fx

-Fy

-Fz

-T1

Mx Mu M~ 7"2 0 -Ta

Fx Fu F~ T1 T3 0

F i g u r e 12.1" The generators of S 0 ( 4 , 2) and its subgroups. The above set of fifteen operators satisfies the commutation relations (11.15) as can be verified using only the basic commutator [xi,pj] = ih6ij. The six-dimensional metric 7/in (11.15) is ( - 1, - 1, - 1, - 1, 1, 1). The resulting group of these transformations is, therefore, S 0 ( 4 , 2), the noncompact group of rotations and pseudorotations in six dimensions which incorporates SO(3), SO(4), SO(3)| S 0 ( 2 , 1), and S 0 ( 4 , 1) as subgroups. It is the full non-invariance group of the hydrogen atom [57]. It has the same rank 3 as SO(6) with three Casimir operators, C2 C3 C4

-

~ . ~ tu~t "~ , ~uupaa~ Cuupa6~t"v gpa g6~ , ~-~uupoguugUPgpagau ,

(12.13)

where, as in Section 11.2.2, gu,, - ~'~po qupq,otpa, and r is the completely antisymmetric symbol of six dimensions. Note that these invariants, which commute with all the generators in (12.12), are products of 2, 3, and 4 generators, respectively. ~The definition of A in (12.12) differs from the earlier g in (10.1) by proportionality factors.

308

Chapter 12. S y m m e t r i e s of the Scattering C o n t i n u u m

The operator set (12.12) yields a nonzero value only for the quadratic invariant in (12.13), namely, -

P

+

-

-

+

-

-

-

(12.14)

-3,

whereas C3 - 6'4 - 0. Also, since the vectors /~ and F lie in the plane of the Kepler orbit as does A, we have g'. A~ - g'. F - 0 just as g'. A vanishes in Chapter 10. 12.2.1

Alternative

decompositions

noninvariance

of the

group

The previous section generated S0(4, 2) as a direct product of the angular SO(3) and the radial SO(2, 1) thereby providing one particular decomposition, namely, SO(4,2) D SO(3)| S0(2, 1), appropriate to hydrogen in spherical polar coordinates. Noncompactness is then manifest entirely through the radial coordinate which has infinite range and a corresponding infinite spectrum of radial excitations. As in previous chapters, a large group such as S0(4,2) has alternative decompositions. Indeed, the increased symmetry of higher dimensions can be expected to exhibit richer structures and substructures. One such alternative views the hydrogen atom in parabolic coordinates as in Section 10.1.1, with two quantum numbers nl and n2 ranging over an infinite set of values corresponding to the two parabolic coordinates r 4- z. Two different S0(2, 1)'s, operating in two different subspaces, treat these states through sets of three generators that are linear combinations of T, Az, Mz, and Fz:

'(Tx +

,

!(T1 - Mz), 2

89

-

r,)

89 + F~)

, ~

89 + A,), 89

A~)

(12.15)

"

In terms of the commutators among the operators in (12.12), both sets in (12.15) obey the commutation relationships (12.3) of S0(2, 1). These two sets are themselves independent, any member of one commuting with all three of the other. The parabolic description maintains the azimuthal symmetry and its quantum number m associated with the third coordinate ~ conjugate to g~. The full description of relevant subgroups for the parabolic description is, therefore, SO(4, 2) D SO(2) | SO(2, 2),

(12.16a)

SO(2, 2) D S0,(2, 1)@ S0~(2, 1),

(12.16b)

12.2. The Full Noninvariance Group of Hydrogen

309

the first step separating out the ~obehavior, and the resulting four-dimensional noncompact group SO(2, 2) further decomposing into the two S 0 ( 2 , 1)'s of the two remaining coordinates with infinite range. This latter step is analogous to Chapter 10's decomposition SO(4) D SO~ (3) | SO2(3) for states of fixed n extended now to all n. With reference to Fig. 12.1, the decomposition in (12.16) corresponds to a subdivision of the 6 x 6 matrix into two diagonal blocks, the top one a 2 x 2 matrix involving lz alone, and the bottom one a 4 x 4 involving the operators (12.15). Other submatrices of Fig. 12.1 may similarly be interpreted in terms of other structures. Thus, its first four rows and columns amount to the maximal invariance group SO(4) of hydrogen, base sets of representations at each fixed discrete energy E~ = - e 2/2a0n 2 containing the n 2 degenerate states with jl - j2 - 8 9 1) as in Chapter 10. For the positive-energy continuum states, n extends to imaginary values in' = i(e2/2aoE) 89 Correspondingly, we set jl = j2 = - 89+ in', preserving the relation ]1 § ~ - / . This "analytic continuation" of SO(4) leads to the noncompact S 0 ( 3 , 1) whose infinite-dimensional representations pertain to the hydrogenic continuum states with fixed energy E and an infinity of values of ~ : 0, 1, 2 , . . . , c~. (This approach yields another view of SO(3, 1) as a subgroup of the noninvariance SO(4, 2) group of hydrogen, supplementing its analogue in Chapter 11 as the proper Lorentz group of space-time transformations.) The relation j~ § - [led to the transformation from parabolic to spherical descriptions of discrete states in Chapter 10. It remains true now for continuum states but takes the following meaning: Since the parabolic description pertains to a Coulomb wave function propagating asymptotically along the z axis, its recasting in terms of spherical partial waves involves the Coulomb phase shift, a r g r ( t q- 1 - in'); thereby the Clebsch-Gordan coefficient in (10.7), (jlj2iO[jlj~,j~j2~), with complex angular momenta jl - j2 - - 89+ in', corresponds to this phase shift [58]. The first four rows and columns of Fig. 12.1 constitute then the invariance subgroups of S0(4, 2), with SO(4) for states of fixed negative energy En, and SO(3, 1) for states with a definite positive energy E. The next higher subgroup, SO(4, 1), includes the generators in the first five rows and columns of Fig. 12.1, which themselves form a closed set under commutations. Among these, as already discussed in Section 10.3, the operators /Q in particular introduce the dipole operator ~" which couples states of different energy. In that section, the states n - 1, 2 , . . . , v, so coupled for any v were seen to provide the basis for a single representation of the fivedimensional SO(5). A similar extension to all the continuum states with

310

C h a p t e r 12. S y m m e t r i e s of the Scattering Continuum

positive energies less than a value E provides the basis for a representation of SO(4, 1), with the various SO(3, 1) of fixed energy as its subgroup representations. This group SO(4, 1) may, therefore, be regarded as an analytic continuation of SO(5) just as S0(3, 1) derived from SO(4) in the previous paragraph. Yet another slice through the full spectrum of the hydrogen atom, namely, the set of all discrete and continuum states with n > u + 1, also provides a basis for a representation of S0(4, 1).

12.3

Dynamics and Symmetry Transformations

Turning from states to operators that transform them, the electric dipole operator Yhas already been discussed and included among the group generators of SO(4, 1) or S0(4, 2) in (12.12) as M - A. Higher tensor operators of these groups can be formed as combinations of these generators. Scattering form factors, for instance, involve the combination exp[iK-r-]. Just as in Parts A and B of this book but now for SO(4, 2) instead of SO(3), algebraic operations involving recoupling coefficients of this group provide matrix elements of operators between states of the hydrogen atom [59]. Thus, the formulation of states and operators for these noncompact, noninvariance groups reduces the dynamics of radial behavior to symmetry transformations (but in an extended sense) just as angular and spin aspects were so rendered in earlier chapters. We take here another step toward viewing all physics as geometry, also viewing radial dynamics as transformations from one symmetry to another in accordance with the preliminary outline in the Introduction. Indeed, this extended view of symmetries is natural in quantum mechanics, where discrete and continuum states combine naturally into a single spectrum of the Hamiltonian [51]. This view casts naturally this chapter's considerations that extend symmetries and transformations from discrete, finite manifolds to their continuous and infinite domain. Reversing the energy sign shifts the corresponding wave number from its real to its imaginary domain, thus requiring the analytic continuation from compact to noncompact groups as in the hydrogen example. The only other applications to radial dynamics so far have dealt with the P6schl-Teller and Morse potentials [60] which are closely related to the Coulomb problem, but such characteristics of the extension discussed in this chapter will apply more generally. Applications to other physical problems and other noncompact

12.3. D y n a m i c s a n d S y m m e t r y Transformations

311

groups remain to be developed but hold promise. Each step expanding the domain of symmetry considerations, whether by embedding in higher dimensions, appending new degrees of freedom, or analytic continuation of relevant variables, brings in greater articulation and richer insights into the nature of physical structures and of their transformations.

This Page Intentionally Left Blank

Bibliography [1] U. Fano and G. Racah, Irreducible Tensorial Sets (Academic Press, New York, 1959). [2] U. Fano and L. Fano, Physics of Atoms and Molecules (University of Chicago Press, Chicago, 1972). [3] D. Hestenes, New Foundations of Classical Mechanics (Reidel, Dordrecht, 1986). [4] G.G. Stokes, Trans. Camb. Phil. Soc. 9,399 (1852). [5] E.P. Wigner, Nachr. Ges. Wiss. G5ttingen, p. 546 (1932). [6] E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1963). [7] J. Schwinger, Phys. Rev. 82, 914 (1951); P. Roman, Theory of Elementary Particles (North-Holland, Amsterdam, 1964), chapter IV, section 3. [8] E.P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959). [9] S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960). [10] U. Fano. L.V. Spencer and M.J. Berger, Penetration and Diffusion of X-rays, in Encyclopaedia of Physics, S. Fliigge, Ed. (Springer, Berlin, 1959), Vol. 38/2, p. 661.

[11] P.M. Morse and H. Feshbach, (McGraw-Hill, New York, 1953).

Methods of Theoretical Physics

[12] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, third edition (Pergamon Press, Oxford, 1977). 313

314

Bibliography

[13] B.R. Judd, Angular Momentum Theory of Diatomic Molecules (Academic Press, New York, 1975). [14] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

[15] W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley, New York, 1950). [16] U. Fano, J. Math. Phys. 1,417 (1960). [17] A.P. Yutsis, I.B. Levinson and V.V. Vanagas, Theory of Angular Momentum (Israel Program for Translations, National Science Foundation, Washington, D.C., 1962). [18] B.R. Judd, Second Quantization and Atomic Spectroscopy (Johns Hopkins University Press, Baltimore, 1967). [19] P.G.H. Sandars, Adv. Chem. Phys. 14, 365 (1969). [20] J.S. Briggs, Rev. Mod. Phys. 43, 189 (1971). [21] M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., The 3-j and 6-j Symbols (Technology Press, Cambridge, Ma., 1955). [22] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Ma., 1965), Vol. III. [23] E. Jaynes and F.W. Cummings, Proc. IEEE 51, 89 (1963). [24] S. Haroche, in New Trends in Atomic Physics, G. Grynberg and R. Stora, Eds. (North-Holland, Amsterdam, 1984), Vol. II, Part I. [25] U. Fano, Rend. Fis. Acc. Lincei, Serie 9, 6, 123 (1995). [26] R.T. Hioe and J.H. Eberly, Phys. Rev. Lett. 47, 838 (1981). [27] G. Racah, Group Theory and Spectroscopy, in Springer Tracts in Modern Physics, Vol. 37, G. H6hler, Ed. (Ergebnisse der Exakten Naturwissenschaften) (Springer, Berlin, 1965). [28] M. Lombardi, Thesis, Grenoble (unpublished, 1969); J. Phys. (Paris) 30,631 (1969). [29] G. Gabrielse, Phys. Rev. A 22, 138 (1980).

Bibliography

315

[30] Yu.F. Smirnov and K.V. Shitikova, Sov. J. Part. Nucl. 8, 344 (1971); N.J. Vilenkin, G.I. Kuznetsov and Ya A. Smorodinskii, Sov. J. Nucl. Phys. 2, 645 (1966).

[31] M. Danos and V. Gillet, Angular Momentum Calculus in Quantum Physics (World Scientific, Singapore, 1990).

[32] H. Matsunobu and H. Takebe, Progr. of Th. Phys. 14, 589 (1955); K.M. Howell, Tables of 9-j Symbols, Res. Rep. 59-2, University of Southampton, England, 1959; A.P. Yutsis, Tables of 9-j Coefficients, Comput. Center AS USSR, Moscow, 1968. [33] U. Fano and J. Macek, Rev. Mod. Phys. 45, 553 (1973). [34] B.R. Judd, Operator Techniques in Atomic Spectroscopy (McGrawHill, New York, 1963). [35] M. Gell-Mann, Phys. Rev. 125, 1067 (1962); M. Gell-Mann and Y. Ne'eman, The Eightfold Way (Benjamin, New York, 1964). [36] F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987); F. Iachello, Rev. Mod. Phys. 65, 569 (1993). [37] G. Racah, Phys. Rev. 76, 1354 (1949). [38] B.R. Judd and G.M.S. Lister, Phys. Rev. Lett. 67, 1720 (1991); M.R. Godefroid, B.R. Judd and N. Vaeck, J. Phys. B 27, 3635 (1994). [39] M. Lax, Symmetry Principles in Solid State and Molecular Physics (John Wiley, New York, 1974); D.A. Robson, N.D. Mermin, D.S. Rokhsar and D.C. Wright, Rev. Mod. Phys. 63, 699 (1991). [40] H.A. Jahn and E. Teller, Proc. Roy. Soc. A 161,220 (1937) [41] W. Lenz, Z. Phys. 24, 197 (1924). [42] H. Goldstein, Am. J. Phys. 43, 735 (1975) and 44, 1123 (1976). [43] V.A. Fock, Z. Phys. 98, 145 (1936). [44] J.C. Gay, D. Delande, F. Biraben and F. Penent, J. Phys. B 16, L693 (1983). [45] D.R. Herrick, Phys. Rev. A 26,323 (1982).

316

Bibliography

[46] D.R. Herrick and O. Sinanogolu, Phys. Rev. A 11, 97 (1975); D.R. Herrick, Adv. Chem. Phys. 52, 1 (1983). [47] C.D. Lin, Adv. At. Mol. Phys. 22, 77 (1986). [48] N.J. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol. 22 (Am. Math. Soc., Providence, R.I., 1968). [49] J. Avery, Hyperspherical Harmonics: Applications in Quantum Theory (Kluwer, Dordrecht, 1988); A. Novoselsky and J. Katriel, Phys. Rev. A 49,833 (1994)and Ann. Phys. (N.Y.) 196, 135 (1989); A. Novoselsky and N. Barnea, Phys. Rev. A 51, 2777 (1995). [50] M. Cavagnero, Phys. Rev. A 30, 1169 (1984); 33, 2877 (1986). [51] U. Fano and A.R.P. Rau, Atomic Collisions and Spectra (Academic Press, Orlando, Fla., 1986). [52] C.H. Greene and C.W. Clark, Phys. Rev. A 29, 177 (1984). [53] B.L. van der Waerden, Group Theory and Quantum Mechanics (Springer-Verlag, Berlin, 1974). [54] H.F. Jones, Groups, Representations and Physics (Adam Hilger, Bristol, 1990), Section 9.4. [55] L. Armstrong Jr., Phys. Rev. A 3, 1546 (1971). [56] M. Rotenberg, Adv. At. Mol. Phys. 6, 233 (1970). [57] A.O. Barut and R. Raczka, Theory of Group Representations and Applications (World Scientific, Singapore, 1986), chapter 12; B.G. Adams, J. Cizek and J. Paldus, Int. J. Q. Chem. 21,153 (1982); B.G. Wybourne, Classical Groups for Physicists (John Wiley, New York, 1974). [58] A.M. Perelomov and V.S. Popov, Sov. Phys. JETP 27, 967 (1968). [59] A.O. Barut and Raj Wilson, Phys. Rev. A 40, 1340 (1990). [60] Y. Alhassid, F. Giirsey and F. Iachello, Phys. Rev. Lett. 50,873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); Chem. Phys. Lett. 99, 27 (1983).

Index A Abelian group, 231,232,235,236 Actinides, 208,276 Alignment, 129 Angular momentum, s e e a l s o r transformation addition, 11, 99, 254,305 classical limit, 124 commutation relations, 36, 37, 49 complex, 309 coupling, 116-122 diagrams, 116-122, 161-168, 178 eigenfunctions, 64 four-dimensional, 175,255-258 grand, s e e Grand angular momentum graphical analysis, 116-122,161168, 178 matrices, 41-44, 55, 59 operators, 37 quantum numbers, 6, 42, 91, 96, 114 Annihilation operator, 138, 139, 154, 209,210,214-217 Antineutrino, s e e Neutrino Antisymmetrization, 66,203,204, 211, 220, 222, 223, 270, s e e a l s o Fractional parent317

age; Seniority Associated Legendre polynomial, 68, 76, 82,267 Autocorrelation, 132 Axes of inertia, 85, 246 Axial/Azimuthalsymmetry, 87, 136, 259,304

B Base set Cartesian, s e e Cartesian base reduction, s e e Reduction standard, s e e Standard base tensorial, s e e Tensorial Base spinor, 130,239,289 Biedenharn-Elliott identity, 197 Bi-spinor, 292-294 Bloch equation, 129,132,137,142, 154, 180, 182 Body frame, s e e Frame Boost, s e e Lorentz Born-Oppenheimer approximation, 244 Boson 66, 96, 201,275-277 Bracket, s e e Dirac

C Cartesian axes, 11

318 Cartesian base, 32, 52, 55, 61, 62, 68-73, 85, 132, 205,239 four-dimension, 257 Casimir operator, 147, 208, 254, 258, 263, 265, 273, 274, 276, 307,308 Lorentz group, 288 Poincar~ group, 297 quadratic, 204,206,218,300, 3O8 SO(2, 1), 303, 304 s o ( n , 1), 288 S 0 ( 4 ) , 253,265 SO(4, 2), 307, 308 Center of mass, 226,251,267 cfp, see Fractional parentage coefficient Character, 229, 235, 237-243 of rotation group, 239,240 of r-transformations, 239,240 table for point groups, 235, 248-250 Chebyshev polynomial, 240,257 Chirality, 51,254 Class, 230, 231,233,237 Clebsch-Gordan coefficient, 32, 104-115, 119, 124, 154,271,309 SO(4), 175,261,262 matrix, 99, 100, 142, 254 series, 242,250,295 table, 108 Clifford algebra, 293 Coefficient Clebsch-Gordan, see ClebschGordan fractional parentage, see Fractional parentage coefficient Racah, 170

Index recoupling, see Recoupling timber, 271 Wigner, see Wigner Cogredience, 113 Completeness, 165 Compressibility, 10 Conjugate, 230,292 Hermitian, see Hermitian of tensorial sets, 61 Continuum hydrogenic, 309 symmetries, 301-311 Contragredience, 12, 20, 58, 6264, 67, 68, 99, 102, 112, 132,205 Contrastandard, 67 Correlations, 187, 191-195, 268, 270 Coulomb-Kepler problem, 136,252 Laplace-Runge-Lenz vector, 136, 252,262, 272,273,306 Coulomb phase shift, 309 Coupling angular momentum, 116-122 j j - , see j j-coupling L S - , see LS-coupling spin-orbit, 186-190, 195 Covariance, 6 Covering group, 26 of SO(3), 26, 47 of S 0 ( 3 , 1), 291 Creation operator, 138, 139, 154, 209,210, 214-217

D D ~ )~, , d ~ L , , see also r-transformation

matrix differential equation, 81, 82

Index explicit expression, 45, 76, 77, 89 table, 77 normalization, 81 recurrence relation, 81, 97 sum rule, 97 symmetries, 83, 95 ~ij, see Kronecker symbol ~ ( j l j 2 j 3 ) , 171 d shell, 201, 205-208, 211, 218, 219, 272 table of states, 207 Degeneracy, 10, 11 hydrogen atom, see Hydrogen atom orthogonal group, 265,266 symmetric tensor representation, 265,266 Density matrix/operator, 128,131, 132, 134, 140-142, 191, 192 Diagram angular momentum, 116-122, 161-168, 178 factoring, 163, 164 Feynman, 119 "pinching off", 166-168 recoupling, see Recoupling 6-j coefficient, 170 tetrahedron, 170 Diamagnetic interaction, 260 group structure, 260,261 Dilatation, 8-10, 21 Dimension, 265,266 Dipole operator, 241,242,252,273, 302, 307,309, 310 Dirac bi-spinor, 292-294 bracket, 5, 68

319 equation, 92, 97, 292,293 notation, see Quantum physics spinor, 292 transformation theory, 3 Direct product, see Tensorial set Dissociation, 245 Double tensor, 209, 210, 216 Doubly excited state, 261,262,272, 278 Dual, 234, 257,287, 295 Dyadic, 12, 14 Dynamical symmetry, 136,208,274278

E Euclidean group cijk, 253,254, 286, 287 e~po, 257,297 e~,upo6e, 307 rluu , see Metric tensor Eigenfunction, 6, 10, 11, 64, 93, see also Harmonics expansion, 2, 11-13, 25, 64 of a pair, 114, 115 Elastic distortion dilatation, 8-10, 21 shear, 9, 10, 21 modulus, 2 Young's, 7, 30 tensor, 29, 30 Electric dipole moment, 241,242, 302,309 Electromagnetic field, 289,299,301 Electromagnetism, 280,294,295 Electron Dirac equation, 292,293 E 2 , see

Index

320 representation, 289, 291-294, 298 Electrostatic interaction, 152,153, 188, 190, 197, 224 Elliptic function, 261 Equivalence of transformations, 16, 25, 58, 60 Erlangen program, 3 Euclidean geometry, 3,281 group, 50, 300 transformation, 3 Euler angles, 45, 46, 52, 57, 76, 81, 83-88, 94, 192, 245, 272 Event, 281,295 Exclusion principle, 199,202 Expansion into functions, 2, 25

F

9

f shell, 206, 208, 218-221, 275, 301 multiplicity, 219,221 quasiparticle, 218 table of states, 221 Factorization of diagrams, 163-165 of transformations, 163 Fermion, 66, 96,138,201,277, see also Spinor Feynman diagrams, 119 Form factor, 310 Four dimension, see Rotation; SO(4) Fourier analysis, 2, 25, 48, 102, 111, 145, 191 Four vector, 281, 284, 290, 294297 electromagnetism, 294

Pauli-Lubanski, 297 Fractional parentage coefficient (cfp), 200, 203, 209, 213, 214, 220,222 expression, 213 Frame body, 86, 93-95,226,244,245 Cartesian, 85, 90, 109 center of mass, 251 laboratory, 86, 91, 94, 245 reversal, 32, 38, 51-57, 60, 71, 83, 93,103,107,147,149, 180-182, 205, 211, 214, 217,287, 295 Fundamental representation, 1, 11, 26, 34, 45, 47, 56, 289

G G2,206, 219,274 Galilean transformation, 3,283 Gegenbauer polynomial, 257,266, 272 Generators Euclidean group, 50,300 Lorentz group, 283,286,293 Poincar~ group, 296 rotations, 35-37,263 S 0 ( 2 , 1), 303,308 SO(3), 37, 43, 55, 59

S0(3, i), 286 S0(4), 253-257 S0(4, 1), 309 SO(4, 2), 306-308 SO(5), 263,273-275 SO(D), 263, 265, 275

u(3), 142 Geometric algebra, 29

321

Index product, 29 . Geometry, 3,310 Erlangen Program, 3 Euclidean, 3,281 isometries, 3,279 non-Euclidean, 3 plane affine, 3 Euclidean, 3 projective, 3 Graded algebra, 29 Gram determinant, 223 Grand angular momentum eigenfunctions, 265-267 eigenvalues, 265,266,269,270 Laplacian, 264-266,271 Grassmann algebra, 29 Graviton, 299 Group, 3, 20 Abelian, 231,232,235,236 chain, 136,206,208,259,260, 263,266,274-276 character, see Character compact, 279,301 continuous, 24, 33 E2, see Euclidean group G2, 206, 219, 274 O(3, 1), 284,286,295, 301, see also Lorentz group SL(2, C), 291

S0(2), 136,206

S 0 ( 2 , 1), see S 0 ( 2 , 1),

S0(2, 2), 308, 309 S0(3), see S0(3) S 0 ( 3 , 1), see S 0 ( 3 , 1); Lorentz group S0(4), see S0(4) S 0 ( 4 , 1), 307, 309,310 S0(4, 2), see S0(4, 2)

S0(5), 206, 208, 255, 263, 265,266, 273-276,310 S0(6), 251,266,271,275, 276 S0(7), 206,208, 219, 266, 274

SO(8), 219,275 SO(9), 7 SO(D), 263,265,266,279 SO(2j + i), 18i, 205,274, 275 Sp(2, R), 271 Sp(2j -4- 1), 181 SU(1, 1), 271

su(2),

su(2)

SU(3), 142, 219, 275,276 S U ( 2 j 4- 1), 142, 146 U(3), 142,204 U(4), 136

u(5), 206, U(6), 275,276 U(7), 206 U(9), 7 U(2j + 1), 181, 204, 205, 218,219 covering, see Covering group cyclic, 231 discrete, see Point group Euclidean, see Euclidean group generators, see Generators icosahedral, 234 invariance, see Invariance Klein's Erlangen Program, 3 Lie, 33, 53, 58 Lorentz, see Lorentz group multiparticle, 251,261-278 non-Abelian, 231,237, 238 noncompact, see Noncompact group

322 noninvariance, see Noninvariance group octahedral, 233-235 of transformations, 20 order, 230-232,236 orthogonal, see Orthogonal group Poincar~, see Poincar~ group point, see Point group rank, 254, 259, 263,307 representation, see Representation symplectic, 181,205, 271 tetrahedral, 233,234,241-243 theory, 2, 24, 84, 93 unitary, 47, see also SU(2); Group U() Groupoid, 162, 163, 171 Gyromagnetic precession, 49, 186, 189 ratio, 189

H Harmonics hyperspherical, see Hyperspherical harmonics spherical, see Spherical harmonics spinor, 92 vector, 92 Heisenberg representation, 13,132, 146, 148 Helicity, 298 graviton, 299 massive particle, 298 massless particle, 297, 299 neutrino, 292,299 photon, 299 Hermitian

Index conjugation, 16, 41,105,133, 141, 147-150, 153 product, 65, 66, 68 Hilbert space, 2, 13, 44, 126, 134, 184, 185, 200, 201, 209, 210 Hooke's law, 7, 10 Hydrogen atom degeneracy, 135,136,208,253 in external field, 259-261 invariance group SO(4), 136, 175, 252-258, 273,276 Laplace-Runge-Lenz vector, see Laplace-Runge-Lenz vector momentum space, 258,261 noninvariance group S 0 ( 2 , 1), 302-309 S 0 ( 4 , 1), 307, 309, 310 S 0 ( 4 , 2), 302,306-310 255

parabolic, 135, 154,253-255, 259-261,308 spherical, 135, 154,253,255, 308 Sturmian function, 305 Hyperfine interaction, 187, 190 Hypergeometric function, 76 Hyperspherical coordinates, 264,267-269 harmonics, 252,265-267,278 addition formula, 267 normalization, 266,267 quantum numbers, 265,266, 269,270

Index

Inner product, 64 Interacting boson model, 275,276 Invariance, 64 group, 251,272, s e e a l s o Group; Hydrogen atom Lorentz, s e e Lorentz parity, s e e Parity r-transformation, 18 rotation, 12, 25, 157,201 time reversal, 1,145,182, 183, 284, 295 translation, 1,295 Invariant operator, 21, 25, 37 product, 64, 66, 102-104, 119, 172, 173 subspace, 21 Inversion of coordinate axes, 1, 30, 51, 57, 69, 91, 93-96,284 operator, 69, 93-96,230,231 Irreducible r-transformation, s e e r-transformation representation, s e e Representation set, s e e Set tensor, 18-20 Isometry, 3, 279 Isotopic spin, 135, 202, 204, 209, 210 Isotropic rotor, 255,273 Isotropy, 1, 19

J J-matrices, 41-44, 55, 59

323 jj-coupling, 28, 126,157,195,196, 202-205,210 Jacobi coordinates, 158, 159 polynomial, 270,272 trees, 158, 159, 267, 271 J acobian elliptic function, 261 Jahn-Teller effect, 246 Jaynes-Cummings model, 137

K Klein's Erlangen Program, 3 Kronecker symbol, 7, 121,177,286

L A-doubling, 96, 97 L-transformation, 283,288, s e e a l s o Lorentz transformation LS-coupling, 28, 126, 157, 190, 195, 196,202, 204,205 Ladder operator, 41, 141, 146 Lam@ constant, 7, 10, 30 Land@ factor, 189 Lanthanides, 208,276 Laplace-Runge-Lenz vector hydrogen atom, 136,175,252255,272, 273, 306,307 in diamagnetic interaction, 260 in doubly excited states, 262, 263 Laplacian D dimensions, 264-266,269 four dimensions, 256,257,271 Legendre polynomial, 76, 82, 85, 188,305 Levi-Civita symbol four dimensions, 257,297

324

Index

six dimensions, 307 Lie group, 33, 53, 58, see also Group Light cone, 285 Line strength, 190 Liouville representation, 128,130,134, 143, 158,209 space, 140, 184-186 Lorentz boost, 226,227,282,285-289, 295,298 generators, 285, 286 gauge, 294 group, 4, 97, 226, 227, 258, 279, 280,284 Casimir invariant, 288 commutation relations, 286 fundamental representation, 289 generators, 283, 286,293 inhomogeneous, see Poincar~ group proper, 284, see also S 0 ( 3 , 1) representation, 48,287-295 three-dimensional, 303,309 transformation, 4, 226, 279290,295,296

M Magnetic moment, 86, 129, 144, 183, 196,241 Matrix (J ) ) Dram,, see D(J ram' element, see Quantum

physics; Reduced matrix element orthogonal, 62, 70 r-transformation, see r-transformation matrix

unitary, 16, 65 Metric tensor, 64, 66, 284, 307 Mixed state, 131, see also density matrix Modulus elasticity, 2, 7 shear, 10 Young's 7, 30 Molecule, 226,229,243 diatomic, 93-97,245 NO, 123 heteronuclear, 90, 94 polyatomic, 94,245-247 BF3,232 benzene, 229,232 CH3C1, 90, 242 CH3D, 229 CH4, 229, 233, 234, 242, 250 H~O, 229, 231,233,247 NH3, 90,229,232,241,250 SF6,234 symmetric top, 90 Morse potential, 310 Multiparticle, 199,251,263,277 hyperspherical coordinates, 264, 267-269 Multipole moment, 92, 115, 129, 145, 151, 152, 189, 194, 195 electric, 86, 97, 182 magnetic, 97, 182 Multipole operator, 143, 182, 190

N Neutrino, 291 helicity, 292,299 representation, 292

Index Weyl, 292 9-j symbol, 157,168,173-175,178 for quadruple products, 173, 178, 196 orthogonality, 174 reduction to 6-j, 174 sum rules, 174 symmetries, 173, 174 Noncompact group, 126,227,252, 280,286,301 Noninvariance group, 126,200,208, 251,272,273 hydrogen atom, 255,302-310 isotropic rotor, 255,273 Normalization, 81, 104,258, 267, s e e a l s o Phase normalization Normal mode, 247-249

O Octahedral, 233-235 Operator angular momentum, s e e Angular momentum annihilation, s e e Annihilation operator antisymmetrization, 220,222, 223 Cn, 230-236 Casimir, s e e Casimir operator creation, s e e Creation operator density, s e e Density matrix dipole, s e e Dipole operator E, identity, 230-234,236,238 electrostatic interaction, 153, 188, 224,278

325 fine structure, 187, 188, 195 frame reversal, 53, 83 grand angular momentum, 264266, 271 helicity, 298 hyperfine structure, 187 I, inversion, 69, 93-96, 230, 231 ladder, s e e Ladder operator lowering, 41,303 multipole, s e e Multipole operator product, 176, 177, 179, 183 radial momentum, 302 raising, 41,303 reflection, 30, 95, 230-232 or, 230 ~rh, 230-233 ev, 93-96,230-233,247-249 Sn, 230 tensorial, s e e Tensorial unit, s e e Unit operator Orbital, 138, 201, 209, 210, 213, 224 Order, 13, 19, 20, 43, 44,230-232, 236 Orientation, 129-131,135, 141 Orthochronous, 285 Orthogonal group, 26, 43,205,255, 263-266,273-276,294 Orthogonality, 104,122,140, 163, 171,174

P Parentage, s e e Fractional parentage coefficient Parity, 51, 53, 69, 93, 103, 122, 180, 183, 216, 254, 286,

326

287 eigenfunction, 93, 96, 97 selection rule, 152,214,241 state multipoles, 182, 183 Partial wave, 2 Particle-hole symmetry, 201, 211, 272, 301 Pauli -Lubanski vector, 297-299 matrices, 34, 45-47, 49, 50, 74, 130-133, 181,292 principle, 199,202,270 Permutation, 1, 20, 27, 61, 66, 158,202,271 Phase normalization, 23, 43, 62, 72, 73,148,150,152,153 Condon-Shortley convention, 44, 54, 90 of angular momentum matrices, 23, 43, 44, 54, 72, 73 Photon helicity, 299 polarization, 294, 299 spin echo, 129 Plane wave, 1, 138 Platonic solid, 233,234 Poincar~ group, 4, 279,291,295 Casimir invariants, 297 commutation relations, 296 generators, 296 representations, 297-300 sphere, 138, 139 Point group, 226, 229-239, 241243,247-250 Poisson's ratio, 7, 30 Polarization light, 47, 92, 137, 138,242 photon, 294,299,300

Index precession, 129,135,140, 152 spin, 129, 132, 141 Polynomial Chebyshev, 240, 257 Gegenbauer, 257,266,272 Jacobi, 270,272 PSschl-Teller potential, 310 Precession equation, s e e Bloch equation Probability amplitude, s e e Quantum physics Product of tensorial sets, s e e Tensorial Pseudorotation, 286,305 Pseudotensor, 93 Pure state, 130, 133, 141, 193

Q Quadruple product, 163,165-168, 173, 178, 196 Quadrupole, 98,144,145,183,188 Quantum number angular momentum, 42, 91, 96, 114 hyperspherical, 265,266,269, 270 multiparticle, 204, 211, 265, 266,269, 270 parabolic, 254,255 parity, 69, 93, 95, 96 quasi-spin, 211-218 radial correlation, 270 seniority, 211,212, 218,274 spherical, 42, 96, 114,253 two-electron, 263 Quantum physics/theory, 2-5, 33 angular momentum, s e e Angular momentum

Index antisymmetrization, s e e Antisymmetrization base set, 127, s e e a l s o Base set; Tensorial set classification of levels/states, 18, 126, 148 complementarity, 4 continuum states, 301, 302, 309, 310 degeneracy, s e e Degeneracy density matrix, s e e Density matrix Dirac equation, s e e Dirac Dirac notation, 5, 55, 58, 64, 67, 68, 73, 126 eigenfunction, s e e Eigenstate eigenstate, 4, s e e a l s o Eigenfunction Hilbert space, s e e Hilbert space hydrogen atom, s e e Hydrogen atom joint eigenstate, 22, 114, 115, 157, 158,202-205,209 Liouville representation, s e e Liouville many particle, s e e Multiparticle matrix element, 41-44, 128, 144, 150-152, 187, 188 mixed state, s e e State operator, 6, 14, 15, 126-128, 133, 134, 146, 147, s e e a l s o Operator partial wave, 2 probability amplitude, 66, 72, 92, 130, 131,280 pure state, s e e State representation, s e e Representation

327 transformation, s e e Transformation two-level system, s e e Two-level system "Quark" d shell, 219 f shell 219,220,277 Quasiparticle, 218,277 Quasi-spin, 200,206,207,209, 211218,222-224 Quaternion, 28, 29,231

R r-transformation, 15, 23, 94,278, 283, 288, 301, s e e a l s o Angular momentum; Rotation character of, 239,240 in Cartesian frame, 85 infinitesimal, 34-40, 80, 81 in standard frame, 56, 57 invariance under, 17, 18 irreducible, 17, 18, 25, 42, 47, 56, 59, 68, 69, 82, 88, 160 matrix, 15, 17, 26, 34, 37, 39, 40, 43-46, 62, 63, 75-77, 79-97,110,151,192,254 products of, 110, 111, 254, 288 recurrence relation, 81, 97 standardization, 32, 52, 67, 68, 74 symmetries of, 82, 83, 93-97 Racah coefficient, 170, s e e a l s o 6 - j coefficient

Index

328 -Wigner algebra, 27, 28,121, 143, 157,278 Radial correlation, 268,270 Rank, s e e Group Recoupling coefficient, see 6 - j coefficient; 9-j coefficient diagram, 161-168 transformation, 116,126,185 diagram, 161-168 L S to j j , 28,167, 173, 196 of 5-sets, 197 of quadruple products, 165168, 173, 174, 262 of triple products, 160-162 Recurrence/Recursion relation, 81, 97, 107 Reduced matrix element, 143,150, 152, 179,185,222,223 Reducing transformation, 100,160 Reduction, 9, 10, 20-25, 60 matrix, 104, 107-109 of a set, 15, 18-21,127, 274 of direct product, 41, 99-102, 109, 110, 115, 242 of representations, 240-243,275 of tensors, 18-20, 128 of triple product, 116, 117 procedure, 17, 21-23 Reflection operator, 30, 95, 230232 Relativity Einsteinian, 3,252,280,281, see a l s o Lorentz Galilean, 3,283 Relaxation, 148 Representation, 5 coupled, 101,254 fundamental, see Fundamen-

tal representation group, 24, 39, 107,235-238 Heisenberg, s e e Heisenberg representation infinite-dimensional, 300-302, 304 irreducible, see r-transformation; Lorentz group Liouville, see Liouville Lorentz, see Lorentz massive particle, 298 massless particle, 288,294,299, 3OO momentum, 5 parabolic, 135, 154,253-255, 259-261,308 Poincar@, 297-300 position, 5 reduction of, s e e Reduction rotation, s e e Rotation

so(),

so()

SchrSdinger, see SchrSdinger spherical, 135, 154, 253-255, 309 spinor, 289-292 symmetric tensor, 258, 263, 265,266 tetrahedral, 242 uncoupled, 101,254 vector, 241,242 Weyl, 288 Resonance, 154,273 Rotation, 8, 35, see a l s o r-transformation four dimensions, 175,252-258 group structure, 20, 37, s e e also

SO(3)

irreducible representation, 4O

Index infinitesimal, 33, 37, 40, 47, 51, 81 operator, 36, 37, 39 matrices, 55, 59 commutation relations, 37, 257 four dimensions, 49, 256, 257 multiparticle problems, 251

S S0(2, 1), 302-309 Casimir invariant, 303,304 commutation relations, 303 generators, 303,308 representations, 303-305 S0(3), 26,135,136,175,206,260, 266,271-278,303 Casimir invariant, 265 characters, 239, 240 commutation relations, 37, 49 covering group, 47 generators, 37, 43, 55, 59 representations, 39, 82, 265, see also r-transformation SO(3, 1), 226, 227, 280,284,288, 290, 291,293, 294, 302, 309, see also Lorentz group S0(4), 136 Casimir invariants, 253,254, 258, 265 commutation relations, 253, 254, 260 generators, 175,226,253-257, 273 invariance group, 175, 252255, 273,276 noninvariance group, 255,273

329 rank, 259 representations general, 258, 294 symmetric tensor, 258,265, 266 S 0 ( 4 , 1), 307, 309, 310 S0(4, 2), 306-310 Casimir invariants, 307, 308 commutation relations, 307 generators, 306, 307 S O ( D ) , see Group SU(2), 26, 47, 78, 82, 136, 226, 3OO SU(3), 142, 219,275,276 Scalar product, 63 Schr5dinger equation, 6, 129, 131,. 142, 158, 176, 181, 182, 244, 261,265 hydrogen atom, 259, 302, 304 momentum space, 258,259 representation, 13, 132 Schur lemma, 25, 160 Second quantization, 200 Selection rule, 242 parity, 152, 214, 241 quasi-spin, 224 seniority, 224 Self-conjugate, 54, 55, 64, 67, 73, 113, 150, 152 Self-dual, 287,295 Seniority, 200,206,207,209,211214, 217, 218, 222, 272, 274 Set, see also Tensorial base, see Base set Cartesian, 52, 55 contragredient, 12, 20, 58, 63,

330

64, 67, 112, 132 contrastandard, 67 irreducible, 19, 22, 27, 67, 78, 102, 112, 132 labeling, 22 maximal, 21, 146, 161 of products, 41, 63-66, 102, 112 reduction, see Reduction self-conjugate, see Self-conjugate standard, 52, 67, 102, 128 tensorial, see Tensorial Shear, 9, 10, 21 6-j coefficient, 157, 168-172, 187189, 197,206,222,223 diagram, 170 orthogonality, 171 sum rule, 171 symmetries, 171, 197 Slater determinant, 215 integral, 188, 196 Space-time interval, 280,281,290 Special Theory of Relativity, 252, 280, 281 Spherical harmonics, 68, 73, 76, 86, 89, 90,111,114, 115, 151-153,266 four dimensions, 175,256-259, 266 phase normalization, 72, 73, 133, 152, 153 Spin isotopic, see Isotopic spin massive particle, 299 massless particle, 299 Spinor, 26, 48, 78, 91, 103, 180, 283 base, 130,239,289

Index electron, 291-293 four-component, 292,293 method, 78-80, 102-104,289 neutrino, 292 representation, 289-292 two-component, 292,294 Spin-orbit coupling, 186-190, 195 Standard base, 32, 44, 52-59, 7072, 132, 141,239 Standardization of r-transformations, 52, 68, 74 Stark effect, 224,259, 260 State mixed, 131, s e e a l s o Density matrix multipoles, 141-149,158,182, 183, 186, 192 pure, 130, 133, 141, 193 Statistical tensor, 141, see a l s o State multipoles Stern-Gerlach apparatus, 75, 88, 98, 129 Stokes parameters, 47, 137, 138 Strain, 6, 7, 17, 38,243 Stress, 6, 7, 9, 38,243 Sturmian function, 305 Subgroup, 22,181, see a l s o Group chain Sum rule, 97, 163, 171, 174 Superalgebra, 48 Supersymmetry, 48, 201 Symmetric tensor representation, 258 dimension, 265,266 orthogonal groups, 263, 265, 266 Symmetry, 2, 18, 113, 310, s e e a l s o Invariance accidental, 206

Index axial/azimuthal, 87,136,259, 304 continuous, s e e Group discrete, s e e Group doubly excited state, 261,262 dynamical, s e e Dynamical symmetry inversion, s e e Inversion hydrogen atom, s e e Hydrogen atom of 9-j coefficient, 173, 174 of r-transformation, 82, 83, 93-97 of 6-j coefficient, 171, 197 of tetrahedron, 197 of 3-j symbol, 1 2 2 particle-hole, 201, 211, 272, 301 permutation, 1, 20, 27, 61, 66, 158, 202 point, s e e Group SO(4), s e e Hydrogen atom; S0(4) time reversal, 1, 73,145,182, 183, 284,295 translation, 1,295 Symplectic group, 181,205,271

T Tensor, 11, 13, s e e a l s o Tensorial double, 209, 210, 216 elasticity, 7, 29, 30 irreducible, 18-20 isotropic, 19 metric, s e e Metric tensor rank, 9 reduction of, 18-20, 128

331 statistical, 141, s e e a l s o State multipoles strain, 6, 7, 18, 19, 243 stress, 6, 7, 18, 243 triple, 209-217,222-224 unit, 12, 19 Tensorial equation, 5 operator, 6, 14,127,134,144151,153, 179 commutator, 180, 181 set, 13, 14, 90, 126, 127 contragredient, 20, 64-67 degree, 12, 18-20 direct product, 14, 21, 40, 63-66, 102, 112,153, 184 irreducible, 17, 18, 41, 67, 78, 126, 128, 134 multiple product, 115,142, 161,175, 184 order, 13, 19, 20, 42-44,230232 quadruple product, 163,165168, 173, 178 rank, 9 reduction of, 18-21, 25,113, 127 triple product, 115,116,119121, 160-162 Tetrahedron, 170, 197, 233, 234, 241,250 3-j symbol, 32, 99, 107, 121,122, 144, 169, 172, s e e a l s o Clebsch-Gordan; Wigner coefficient normalization, 122 symmetries, 122 Tilting, 304 Timber coefficient, 271

332 Time reversal, 1, 24, 32, 52, 61, 73, 145, 183, 211, 284, 287,295 Trace, 19,131,132,140,141,185, 237, s e e a l s o Character Transformation Cartesian to standard, 54, 56, 70-74, 147 contragredient, s e e Contragredience Dirac bracket, s e e Dirac discrete, 229, s e e a l s o Inversion; Parity Einsteinian relativity, see Lorentz equivalence of, 16, 54, 58, 60 Euclidean, 3 Fourier, 4 frame reversal, s e e Frame Galilean, 3,283 inequivalent, 25 inhomogeneous, 295-297 L - , s e e L-transformation L S to j j , 126, 167, 173, 185, 196 linear, 13-15,290 Lorentz, s e e Lorentz matrix, 17, 72, 160 nonunitary, 282,290 of quadruple products, 163168, 173, 177, 178 of triple products, 160-162, 170 orthochronous, 285 parity, s e e Parity projective, 3 r - , s e e r-transformation recoupling, s e e Recoupling spherical to parabolic, 136,254 theory

Index Dirac, 3, 4 Klein, 3 quantum-mechanical, 112 tilting, 304 time reversal, s e e Time reversal unitary, 4, 5, 16, 49, 54, 63, 78, 154, 204, 271 Translation, 1, 5, 35, 37, 50, 295, 296,298 Transposition, 16 Trees, s e e J acobi Triangular relation, 101,119,145, 149, 161, 170, 171, 177, 182, 194, 214 Triple product, 115, 116,119-121, 160-162,219 Triple tensor, 209-217,222-224 12-j coefficient, 197 Two-level system, 4, 47, 128,129, 135-138, 155 (2j + 1)-level system, 128, 129, 180-182

U Unimodular, 1, 47, 82, 282,289 Unitary group, 47, 142, s e e a l s o Group;

su(2); su(3) matrix, 16, 65 transformation, s e e Transformation Unit operator, 128, 144, 147, 149, 158, 176-185,222

V Vector, 5, 8, 11

333 field, 93, 138, 294, 295 four, see Four vector harmonics, 92 representation, 241,242 Vibration, 246-249

213, see also Clebsch-Gordan; 3-j symbol -Eckart theorem, 27,140,143, 144, 211,224

Y W Weyl equation, 292 Wigner coefficient, 32, 99, 101, 107115, 117-124, 133, 140,

Young's modulus, 7, 30

Z Zeeman effect, 187, 189, 190,260

This Page Intentionally Left Blank