Quantum Theory at the Crossroads
arXiv:quant-ph/0609184v1 24 Sep 2006
Re onsidering the 1927 Solvay Conferen e
Guido Ba
iagaluppi Antony Valentini
Cambridge University Press
ISBN: 9780521814218
To the memory of James T. Cushing
Contents
Prefa e Abbreviations Typographi onventions Note on the bibliography Permissions and opyright noti es
1
2
page vii xiii xiv xiv xiv
Part IPerspe tives on the 1927 Solvay onferen e Histori al introdu tion
1.1 Ernest Solvay and the Institute of Physi s 1.2 War and international relations 1.3 S ienti planning and ba kground 1.4 Further details of planning 1.5 The Solvay meeting 1.6 The editing of the pro eedings 1.7 Con lusion Ar hival notes
De Broglie's pilot-wave theory
2.1 2.2
2.3
2.4 2.5
Ba kground A new approa h to parti le dynami s: 192324 2.2.1 First papers on pilot-wave theory (1923) 2.2.2 Thesis (1924) 2.2.3 Opti al interferen e fringes: November 1924 Towards a omplete pilot-wave dynami s: 192527 2.3.1 `Stru ture': Journal de Physique, May 1927 2.3.2 Signi an e of de Broglie's `Stru ture' paper 1927 Solvay report: the new dynami s of quanta Signi an e of de Broglie's work from 1923 to 1927 ii
1 3 3 6 10 16 20 22 24 26 30 30 37 38 43 54 57 61 72 75 84
Contents Ar hival notes
iii 88
3
89 3.1 Summary of Born and Heisenberg's report 90 3.2 Writing of the report 93 3.3 Formalism 94 3.3.1 Before matrix me hani s 94 3.3.2 Matrix me hani s 96 3.3.3 Formal extensions of matrix me hani s 100 3.4 Interpretation 102 3.4.1 Matrix me hani s, Born and Wiener 103 3.4.2 Born and Jordan on guiding elds, Bohr on ollisions105 3.4.3 Born's ollision papers 107 3.4.4 Heisenberg on energy u tuations 109 3.4.5 Transformation theory 111 3.4.6 Development of the `statisti al view' in the report115 3.4.7 Justi ation and overall on lusions 119 Ar hival notes 122
4
S hrödinger's wave me hani s
151
5
From matrix me hani s to quantum me hani s
4.1 4.2 4.3 4.4 4.5 4.6
Planning of S hrödinger's report Summary of the report Parti les as wave pa kets The problem of radiation S hrödinger and de Broglie The on i t with matrix me hani s 4.6.1 Early days 4.6.2 From Muni h to Copenhagen 4.6.3 Continuity and dis ontinuity Ar hival notes
123 124 126 128 133 137 139 140 142 146 150
Part IIQuantum foundations and the 1927 Solvay onferen e Quantum theory and the measurement problem 5.1 5.2
153 What is quantum theory? 153 The measurement problem today 155 5.2.1 A fundamental ambiguity 155 5.2.2 Measurement as a physi al pro ess: quantum theory `without observers'157 5.2.3 Quantum osmology 162 5.2.4 The measurement problem in `statisti al' interpretations of ψ 164
iv
6
Contents
Interferen e, superposition, and wave pa ket ollapse 168 6.1
6.2 6.3 6.4
7
8
9
10
11
12
Probability and interferen e 168 6.1.1 Interferen e in de Broglie's pilot-wave theory 169 6.1.2 Interferen e in the `quantum me hani s' of Born and Heisenberg172 Ma ros opi superposition: Born's dis ussion of the loud hamber177 6.2.1 Quantum me hani s without wave pa ket ollapse?178 Dira and Heisenberg: interferen e, state redu tion, and delayed hoi e182 Further remarks on Born and Heisenberg's quantum me hani s190
Lo ality and in ompleteness
194 194 198 202
Time, determinism, and the spa etime framework
204 204 209 213
7.1 7.2 7.3
8.1 8.2 8.3
Einstein's 1927 argument for in ompleteness A pre ursor: Einstein at Salzburg in 1909 More on nonlo ality and relativity Time in quantum theory Determinism and probability Visualisability and the spa etime framework
Guiding elds in 3-spa e
9.1 9.2
218 Einstein's early attempts to formulate a dynami al theory of light quanta218 The failure of energy-momentum onservation 221
S attering and measurement in de Broglie's pilot-wave theory 227
10.1 10.2 10.3 10.4
S attering in pilot-wave theory 228 Elasti and inelasti s attering: Born and Brillouin, Pauli and de Broglie232 Quantum measurement in pilot-wave theory 244 Re oil of a single photon: Kramers and de Broglie 246
Pilot-wave theory in retrospe t
248 250 258 259 265
Beyond the Bohr-Einstein debate
268 269 272
11.1 11.2 11.3 11.4
Histori al mis on eptions Why was de Broglie's theory reje ted? Einstein's alternative pilot-wave theory (May 1927) Obje tions: in 1927 and today
12.1 The standard histori al a
ount 12.2 Towards a histori al revision
Part IIIThe pro eedings of the 1927 Solvay onferen e
277 H. A. Lorentz † Fifth physi s onferen e
279 281
Contents
v
The intensity of X-ray re e tion (W. L. Bragg )
283 The lassi al treatment of X-ray dira tion phenomena 283 History of the use of quantitative methods 285 Results of quantitative analysis 289 292 Interpretation of measurements of F Examples of analysis 295 The me hanism of X-ray s attering 303 The analysis of atomi stru ture by X-ray intensity measurements308 The refra tion of X-rays 312 Referen es 316 Dis ussion of Mr Bragg's report 318 Notes to the translation 327
Disagreements between experiment and the ele tromagneti theory of radiation (A. H. Compton ) Introdu tion The problem of the ether The emission of radiation The photoele tri ee t Phenomena asso iated with the s attering of X-rays Intera tions between radiation and single ele trons Reliability of experimental eviden e Summary Dis ussion of Mr Compton's report Notes to the translation
329 332 333 335 342 348 352 354 356 372
The new dynami s of quanta (L. de Broglie )
374 I. Prin ipal points of view 374 382 II. Probable meaning of the ontinuous waves Ψ III. Experiments showing preliminary dire t eviden e for the new Dynami s of the ele tron390 Bibliography 397 Dis ussion of Mr de Broglie's report 399 Notes to the translation 407
Quantum me hani s (M. Born and W. Heisenberg )
408 Introdu tion 408 I. The mathemati al methods of quantum me hani s 410 II. Physi al interpretation 420 III. Formulation of the prin iples and delimitation of their s ope429 IV. Appli ations of quantum me hani s 434 Con lusion 437 Bibliography 438 Dis ussion of Messrs Born and Heisenberg's report 442
vi
Contents Notes to the translation
Wave me hani s (E. S hrödinger )
445
Introdu tion I. Multi-dimensional theory II. Four-dimensional theory III. The many-ele tron problem Dis ussion of Mr S hrödinger's report Notes to the translation
448 448 449 458 462 469 475
General dis ussion of the new ideas presented
477 477 498 501 521 525
Causality, determinism, probability Photons Photons and ele trons Notes to the translation Bibliography
Prefa e
And they said one to another: Go to, let us build us a tower, whose top may rea h unto heaven; and let us make us a name. And the Lord said: Go to, let us go down, and there onfound their language, that they may not understand one another's spee h. Genesis 11: 37
Anyone who has taken part in a debate on the interpretation of quantum theory will re ognise how tting is the above quotation from the book of Genesis, a
ording to whi h the builders of the Tower of Babel found that they ould no longer understand one another's spee h. For when it omes to the interpretation of quantum theory, even the most lear-thinking and apable physi ists are often unable to understand ea h other. This state of aairs dates ba k to the genesis of quantum theory itself. In O tober 1927, during the `general dis ussion' that took pla e in Brussels at the end of the fth Solvay onferen e, Paul Ehrenfest wrote the above lines on the bla kboard. As Langevin later remarked, the Solvay meeting in 1927 was the onferen e where `the onfusion of ideas rea hed its peak'. Ehrenfest's per eptive gesture aptured the essen e of a situation that has persisted for three-quarters of a entury. A
ording to widespread histori al folklore, the deep dieren es of opinion among the leading physi ists of the day led to intense debates, whi h were satisfa torily resolved by Bohr and Heisenberg around the time of the 1927 Solvay meeting. But in fa t, at the end of 1927, a signi ant number of the main parti ipants (in parti ular de Broglie, Einstein, and S hrödinger) remained un onvin ed, and the deep dieren es of opinion were never really resolved. The interpretation of quantum theory seems as highly ontroversial vii
viii
Prefa e
today as it was in 1927. There has also been riti ism on the part of historians as well as physi ists of the ta ti s used by Bohr and others to propagate their views in the late 1920s, and a realisation that alternative ideas may have been dismissed or unfairly disparaged. For many physi ists, a sense of unease lingers over the whole subje t. Might it be that things are not as lear- ut as Bohr and Heisenberg would have us believe? Might it be that their opponents had something important to say after all? Be ause today there is no longer an established interpretation of quantum me hani s, we feel it is important to go ba k to the sour es and re-evaluate them. In this spirit, we oer the reader a return to a time just before the Copenhagen interpretation was widely a
epted, when the best physi ists of the day gathered to dis uss a range of views, on erning many topi s of interest today (measurement, determinism, nonlo ality, subje tivity, interferen e, and so on), and when three distin t theories de Broglie's pilot-wave theory, Born and Heisenberg's quantum me hani s, and S hrödinger's wave me hani s were presented and dis ussed on an equal footing. * Sin e the 1930s, and espe ially sin e the Se ond World War, it has been ommon to dismiss questions about the interpretation of quantum theory as `metaphysi al' or `just philosophi al'. It will be lear from the lively and wide-ranging dis ussions of 1927 that at that time, for the most distinguished physi ists of the day, the issues were de idedly physi al : Is the ele tron a point parti le with a ontinuous traje tory (de Broglie), or a wave pa ket (S hrödinger), or neither (Born and Heisenberg)? Do quantum out omes o
ur when nature makes a hoi e (Dira ), or when an observer de ides to re ord them (Heisenberg)? Is the nonlo ality of quantum theory ompatible with relativity (Einstein)? Can a theory with traje tories a
ount for the re oil of a single photon on a mirror (Kramers, de Broglie)? Is indeterminism a fundamental limitation, or merely the out ome of oarse-graining over something deeper and deterministi (Lorentz)? After 1927, the Copenhagen interpretation be ame rmly established. Rival views were marginalised, in parti ular those represented by de Broglie, S hrödinger and Einstein, even though these s ientists were responsible for many of the major developments in quantum physi s itself. (This marginalisation is apparent in most histori al a
ounts written throughout the twentieth entury.) From the very beginning,
Prefa e
ix
however, there were some notes of aution: for example, when Bohr's landmark paper of 1928 (the English version of his famous Como le ture) was published in Nature, an editorial prefa e expressed dissatisfa tion with the `somewhat vague statisti al des ription' and ended with the hope that this would not be the `last word on the subje t'. And there were a few outstanding alarm bells, in parti ular the famous paper by Einstein, Podolsky and Rosen in 1935, and the important papers by S hrödinger (in the same year) on the at paradox and on entanglement. But on the whole, the questioning eased in all but a few orners. A general opinion arose that the questions had been essentially settled, and that a satisfa tory point of view had been arrived at, prin ipally through the work of Bohr and Heisenberg. For subsequent generations of physi ists, `shut up and al ulate' emerged as the working rule among the vast majority. Despite this atmosphere, the questioning never ompletely died out, and some very signi ant work was published, for example by Bohm in 1952, Everett in 1957, and Bell in 1964 and 1966. But attitudes hanged very slowly. Younger physi ists were strongly dis ouraged from pursuing su h questions. Those who persisted generally had di ult areers, and mu h of the areful thinking about quantum foundations was relegated to departments of philosophy. Nevertheless, the losing de ade of the twentieth entury saw a resurgen e of interest in the foundations of quantum theory. At the time of writing, a range of alternatives (su h as hidden variables, many worlds,
ollapse models, among others) are being a tively pursued, and the Copenhagen interpretation an no longer laim to be the dominant or `orthodox' interpretation. The modern reader familiar with urrent debates and positions in quantum foundations will re ognise many of the standard points of view in the dis ussions reprodu ed here, though expressed with a remarkable
on ision and larity. This provides a wel ome ontrast with the generally poor level of debate today: as the distinguished osmologist Dennis S iama was fond of pointing out, when it omes to the interpretation of quantum theory `the standard of argument suddenly drops to zero'. We hope that the publi ation of this book will ontribute to a revival of sharp and informed debate about the meaning of quantum theory. * Remarkably, the pro eedings of the fth Solvay onferen e have not re eived the attention they deserve, neither from physi ists nor from
x
Prefa e
historians, and the literature ontains numerous major misunderstandings about what took pla e there. The fth Solvay onferen e is usually remembered for the lash that took pla e between Einstein and Bohr over the un ertainty relations. It is remarkable, then, to nd that not a word of these dis ussions appears in the published pro eedings. It is known that Einstein and Bohr engaged in vigorous informal dis ussions, but in the formal debates re orded in the pro eedings they were relatively silent. Bohr did
ontribute to the general dis ussion, but this material was not published. Instead, at Bohr's request, it was repla ed by a translation of the German version of his Como le ture, whi h appeared in Naturwissens haften in 1928. (We do not reprodu e this well-known paper here.) The appending of this translation to the published pro eedings may be the ause of the
ommon misunderstanding that Bohr gave a report at the onferen e: in fa t, he did not. Born and Heisenberg present a number of unfamiliar viewpoints on erning, among other things, the nature of the wave fun tion and the role of time and of probability in quantum theory. Parti ularly surprising is the seeming absen e of a ollapse postulate in their formulation, and the apparently phenomenologi al status of the time-dependent S hrödinger equation. Born and Heisenberg's `quantum me hani s' seems remarkably dierent from quantum me hani s (in the Dira -von Neumann formulation) as we know it today. De Broglie's pilot-wave theory was the subje t of extensive and varied dis ussions. This is rather startling in view of the laim in Max Jammer's lassi histori al study The Philosophy of Quantum Me hani s that de Broglie's theory `was hardly dis ussed at all' and that `the only serious rea tion ame from Pauli' (Jammer 1974, pp. 11011). Jammer's view is typi al even today. But in the published pro eedings, at the end of de Broglie's report there are 9 pages of dis ussion devoted to de Broglie's theory, and of the 42 pages of general dis ussion, 15 ontain dis ussion of de Broglie's theory, with serious rea tions and omments
oming not only from Pauli but also from Born, Brillouin, Einstein, Kramers, Lorentz, S hrödinger and others. Even the well-known ex hange between Pauli and de Broglie has been widely misunderstood. Finally, another surprise is that in his report de Broglie proposed the many-body pilot-wave dynami s for a system of parti les, with the total
on guration guided by a wave in on guration spa e, and not just (as is generally believed) the one-body theory in 3-spa e. De Broglie's theory is essentially the same as that developed by Bohm in 1952, the only
Prefa e
xi
dieren e being that de Broglie's dynami s (like the form of pilot-wave theory popularised by Bell) is formulated in terms of velo ity rather than a
eleration. * This work is a translation of and ommentary on the pro eedings of the fth Solvay onferen e of 1927, whi h were published in Fren h in 1928 under the title Éle trons et Photons. We have not attempted to give an exhaustive histori al analysis of the fth Solvay onferen e. Rather, our main aims have been to present the material in a manner a
essible to the general physi ist, and to situate the pro eedings in the ontext of urrent resear h in quantum foundations. We hope that the book will ontribute to stimulating and reviving serious debate about quantum foundations in the wider physi s
ommunity, and that making the pro eedings available in English will en ourage historians and philosophers to re onsider their signi an e. Part I begins with a histori al introdu tion and provides essays on the three main theories presented at the onferen e (pilot-wave theory, quantum me hani s, wave me hani s). The le tures and dis ussions that took pla e at the fth Solvay onferen e ontain an extensive range of material that is relevant to urrent resear h in the foundations of quantum theory. In Part II, after a brief review of the status of quantum foundations today, we summarise what seem to us to be the highlights of the onferen e, from the point of view of urrent debates about the meaning of quantum theory. Part III of the book onsists of translations of the reports, of the dis ussions following them, and of the general dis ussion. Wherever possible, the original (in parti ular English or German) texts have been used. We have ta itly orre ted minor mistakes in pun tuation and spelling, and we have uniformised the style of equations, referen es and footnotes. (Unless otherwise spe i ed, all translations of quotations are ours.) Part I (ex ept for hapter 2) and the reports by Compton, by Born and Heisenberg and by S hrödinger, are prin ipally the work of Guido Ba
iagaluppi. Chapter 2, Part II, and the reports by Bragg and by de Broglie and the general dis ussion in Part III, are prin ipally the work of Antony Valentini. Chapters 2, 10 and 11 are based on a seminar, `The early history of Louis de Broglie's pilot-wave dynami s', given by Antony Valentini at the University of Notre Dame in September 1997, at a onferen e in honour of the sixtieth birthday of the late James T. Cushing.
xii
Prefa e *
To James T. Cushing, physi ist, philosopher, historian and gentleman, we both owe a spe ial and heartfelt thanks. It was he who brought us together on this proje t, and to him we are indebted for his en ouragement and, above all, his example. This book is dedi ated to his memory. Guido Ba
iagaluppi wishes to express his thanks to the Humboldt Foundation, whi h supported the bulk of his work in the form of an Alexander von Humboldt Fors hungsstipendium, and to his hosts in Germany, Carsten Held and the Philosophis hes Seminar I, University of Freiburg, and Harald Atmanspa her and the Institut für Grenzgebiete der Psy hologie und Psy hohygiene, Freiburg, as well as to Ja ques Dubu s and the Institut d'Histoire et de Philosophie des S ien es et des Te hniques (CNRS, Paris 1, ENS) for support during the nal phase. He also wishes to thank Didier Devriese of the Université Libre de Bruxelles, who is in harge of the ar hives of the Instituts Internationaux de Physique et de Chimie Solvay, Université Libre de Bruxelles, for his kindness and availability, and Brigitte Parakenings (formerly Uhlemann) and her sta at the Philosophis hes Ar hiv of the University of Konstanz, for the ontinuous assistan e with the Ar hive for the History of Quantum Physi s. Finally, he should wish to thank Je Barrett for suggesting this proje t to him in Utre ht one day ba k in 1996, as well as Mark van Atten, Jennifer Bailey, Olivier Darrigol, Feli ity Pors, Gregor S hiemann and many others for dis ussions, suggestions,
orresponden e, referen es and other help. Antony Valentini began studying these fas inating pro eedings while holding a postdo toral position at the University of Rome `La Sapienza' (199496), and is grateful to Mar ello Cini, Bruno Bertotti and Dennis S iama for their support and en ouragement during that period. For support in re ent years, he is grateful to Perimeter Institute, and wishes to express a spe ial thanks to Howard Burton, Lu ien Hardy and Lee Smolin. We are both grateful to Tamsin van Essen at Cambridge University Press for her support and en ouragement during most of the gestation of this book, and to Augustus College for support during the nal stages of this work. Guido Ba
iagaluppi Antony Valentini
Lake Maggiore, August 2006
Abbreviations
xiii
Abbreviations AEA: Albert Einstein Ar hives, Jewish National and University Library, Hebrew University of Jerusalem. AHQP: Ar hive for the History of Quantum Physi s. AHQP-BSC: Bohr S ienti Corresponden e, mi ro lmed from the Niels Bohr Arkiv, Copenhagen. AHQP-BMSS: Bohr S ienti Manus ripts, mi ro lmed from the Niels Bohr Arkiv, Copenhagen. AHQP-EHR: Ehrenfest olle tion, mi ro lmed from the Rijksmuseum voor de Ges hiedenis van de Natuurwetens happen en van de Geneeskunde `Museum Boerhaave', Leiden. AHQP-LTZ: Lorentz olle tion, mi ro lmed from the Algemeen Rijksar hief, Den Haag. AHQP-RDN: Ri hardson Colle tion, mi ro lmed from the Harry Ransom Humanities Resear h Center, University of Texas at Austin. AHQP-OHI: Oral history interview trans ripts. IIPCS: Ar hives of the Instituts Internationaux de Physique et de Chimie Solvay, Université Libre de Bruxelles. Ann. d. Phys. or Ann. der Phys.: Annalen der Physik. Bayr. Akad. d. Wiss. Math. phys. Kl.: Sitzungsberi hte der Mathematis h-Physikalis hen Klasse der Königli h-Bayeris hen Akademie der Wissens haften (Mün hen). Berl. Ber.: Sitzungsberi hte der Preussis hen Akademie der Wissens haften (Berlin). A ad. Roy. Belg. or Bull. A . R. Belg. or Bull. A . roy. de Belgique or Bull. A . roy. Belgique or Bull. A . roy. Belg. or Bull. A . R. Belg., Cl. des S ien es: Bulletin de l'A adémie Royale des S ien es, des Lettres et des Beaux-arts de Belgique. Classe des S ien es. Bull. Natl. Res. Coun.: Bulletin of the National Resear h Coun il (U.S.). Comm. Fenn.: Commentationes Physi o-mathemati ae, So ietas S ientiarum Fenni a. C. R. or C. R. A ad. S . or Comptes Rendus A ad. S i. Paris: Comptes Rendus Hebdomadaires des Séan es de l'A adémie des S ien es (Paris). Gött. Na hr.: Na hri hten der Akademie der Wissens haften in Göttingen. II, Mathematis h-Physikalis he Klasse. J. de Phys. or Jour. de Phys. or Journ. Physique or Journ. d. Phys.: Journal de Physique (until 1919), then Journal de Physique et le Radium. Jour. Frank. Inst.: Journal of the Franklin Institute.
xiv
Permissions and opyright noti es
Lin ei Rend.: Rendi onti Lin ei. Man hester Memoirs: Man hester Literary and Philosophi al So iety, Memoirs and Pro eedings. Math. Ann. or Mathem. Ann.: Mathematis he Annalen. Naturw. or Naturwiss. or Naturwissens h. or Naturwissens haften: Die Naturwissens haften. Nat. A ad. S i. Pro . or Pro . Nat. A ad. S i. or Pro . Nat. A ad.: Pro eedings of the National A ademy of S ien es (U.S.). Phil. Mag.: Philosophi al Magazine. Phil. Trans. or Phil. Trans. Roy. So .: Philosophi al Transa tions of the Royal So iety of London. Phys. Rev.: Physi al Review. Phys. Zeits. or Phys. Zeits h. or Physik. Zts.: Physikalis he Zeits hrift. Pro . Camb. Phil. So . or Pro . Cambr. Phil. So . or Pro . Cambridge Phil. So .: Pro eedings of the Cambridge Philosophi al So iety. Pro . Phys. So .: Pro eedings of the Physi al So iety of London. Pro . Roy. So . or Roy. So . Pro .: Pro eedings of the Royal So iety of London. Upsala Univ. Årsskr.: Uppsala Universitets Årsskrift. Z. f. Phys. or Zts. f. Phys. or Zeit. f. Phys. or Zeits. f. Phys. or Zeits h. f. Phys. or Zeits hr. f. Phys.: Zeits hrift für Physik.
Typographi onventions The following onventions have been used. Square bra kets [ ℄ denote editorial amendments or (in the translations) original wordings. Curly bra kets { } denote additions (in original types ripts or manus ripts). Angle bra kets < > denote an ellations (in original types ripts or manus ripts).
Note on the bibliography The referen es ited in Parts I and II, and in the endnotes and editorial footnotes to Part III, are listed in our bibliography. The referen es
ited in the original Solvay volume are found in the translation of the pro eedings in Part III.
Permissions and opyright noti es
Part I Perspe tives on the 1927 Solvay onferen e
1 Histori al introdu tion
Quantum re on iliation very [added, deleted℄ unpleasant [deleted℄ tenden y [deleted℄ retrograde [deleted℄ questionable [added, deleted℄ idea [deleted℄ ippant [deleted℄ title leads to misunderstanding. Ehrenfest, on the onferen e plans
1
The onferen e was surely the most interesting s ienti
onferen e I have taken part in so far. 2 Heisenberg, upon re eipt of the onferen e photograph
The early Solvay onferen es were remarkable o
asions, made possible by the generosity of Belgian industrialist Ernest Solvay and, with the ex eption of the rst onferen e in 1912, planned and organised by the indefatigable Hendrik Antoon Lorentz. In this hapter, we shall rst sket h the beginnings of the Solvay onferen es, Lorentz's involvement and the situation in the years leading up to 1927 (se tions 1.1 and 1.2). Then we shall des ribe spe i ally the planning of the fth Solvay onferen e, both in its s ienti aspe ts (se tion 1.3) and in its more pra ti al aspe ts (se tion 1.4). Se tion 1.5 presents the day-by-day progress of the onferen e as far as it an be re onstru ted from the sour es, while se tion 1.6 follows the making of the volume of pro eedings, whi h is the main sour e of original material from the fth Solvay onferen e and forms Part III of this book.
1.1 Ernest Solvay and the Institute of Physi s Ernest Solvay had an extensive re ord of supporting s ienti , edu ational and so ial initiatives, as Lorentz emphasises in a two-page 3
4
Histori al introdu tion
do ument written in September 1914, during the rst months of the rst world war:3 I feel bound to say some words in these days about one of Belgium's noblest
itizens, one of the men whom I admire and honour most highly. Mr Ernest Solvay .... is the founder of one of the most ourishing industries of the world, the soda manufa ture based on the pro ess invented by him and now spread over Belgium, Fran e, England, Germany, Russia and the United States. ....
The fortune won by an a tivity of half a entury has been largely used
by Mr Solvay for the publi bene t.
In the rm onvi tion that a better
understanding of the laws of nature and of human so iety will prove one of the most powerful means for promoting the happiness of mankind, he has in many ways and on a large s ale en ouraged and supported s ienti resear h and tea hing.
Part of this a tivity was entred around the proje t of the Cité S ienti que, a series of institutes in Brussels founded and endowed by Ernest Solvay and by his brother Alfred Solvay, whi h ulminated in the founding of the Institutes of Physi s and of Chemistry in 1912 and 1913.a This proje t had originally developed through the han e en ounter between Ernest Solvay and Paul Héger, physi ian and professor of physiology at the Université Libre de Bruxelles (ULB), and involved a ollaboration between Solvay, the ULB and the ity of Brussels. In June 1892, it was agreed that Solvay would onstru t and equip two Institutes of Physiology on land owned by the ity in the Par Léopold in Brussels.b There soon followed in 189394 an Institute for Hygiene, Ba teriology and Therapy, funded mainly by Alfred Solvay, and a S hool of Politi al and So ial S ien es, founded by Ernest Solvay in 1894, whi h moved to the Cité S ienti que in 1901, and to whi h a S hool of Commer e was added in 1904. The idea for what be ame known as the rst Solvay onferen e in physi s goes ba k to Wilhelm Nernst and Max Plan k,c who around 1910
onsidered that the urrent problems in the theory of radiation and in the theory of spe i heats had be ome so serious that an international meeting (indeed a ` oun il') should be onvened in order to attempt to resolve the situation. The further en ounter between Nernst and Solvay provided the material opportunity for the meeting, and by July 1910, a The following material on the Cité S ienti que is drawn mainly from Despy-Meyer and Devriese (1997). b One was to be ome property of the ity and given in use to the ULB, while the other was to be leased for thirty years to and run by Solvay himself.
In the rest of this and in part of the following se tions, we draw on an unpublished
ompilation of the ontents of the Solvay ar hives by J. Pelseneer.4
1.1 Ernest Solvay and the Institute of Physi s
5
Nernst was sending Solvay the detailed proposals. He had also se ured the ollaboration of Lorentz (who was eventually asked to preside), of Knudsen and naturally of Plan k, who wrote: .... anything that may happen in this dire tion will ex ite my greatest interest and .... I promise already my parti ipation in any su h endeavour. For I an say without exaggeration that in fa t for the past 10 years nothing in physi s has so ontinuously stimulated, ex ited and irritated me as mu h as these a quanta of a tion.
Lorentz set up a ommittee to onsider questions relating to the new experimental resear h that had been deemed ne essary during the
onferen e (whi h took pla e between 30 O tober and 3 November 1911). This ommittee in luded Marie Curie, Brillouin, Warburg, Kamerlingh Onnes, Nernst, Rutherford and Knudsen. Lorentz in turn was asked to be the president. Further, at the end of the onferen e, Solvay proposed to Lorentz the idea of a s ienti foundation. Lorentz's reply to Solvay's proposals, of 4 January 1912, in ludes extremely detailed suggestions on the fun tions and stru ture of the foundation, all of whi h were put into pra ti e and whi h an be summarised as follows.6 The foundation would be devoted prin ipally to physi s and physi al
hemistry, as well as to questions relating to physi s from other s ien es. It would provide international support to resear hers (`a Rutherford, a Lenard, a Weiss') in the form of money or loan of s ienti instruments, and it would provide s holarships for young Belgian s ientists (both men and women) to work in the best laboratories or universities, mostly abroad. The question of a link between the foundation and the `Conseil de physique' was left open, but Lorentz suggested to provide meeting fa ilities if Solvay wished to link the two. Lorentz suggested instituting an administrative board ( onsisting of a Solvay family member or appointee, an appointee of the King, and a member of the Belgian s ienti establishment) and a s ienti ommittee (whi h ould initially be the one he had formed during the rst Solvay onferen e). Finally, Lorentz suggested housing the foundation in an annex of one of the existing institutes in the Cité Universitaire. During January, Solvay sent Paul Héger to Leiden to work with Lorentz on the statutes of the foundation, whi h Lorentz sent to Solvay on 2 February. Solvay approved them with hardly any modi ations (only su h as were required by the Belgian legislation of the time). The a Exquisite ending in the original: `.... dass mi h seit 10 Jahren im Grunde ni hts in der Physik so ununterbro hen an-, er-, und aufregt wie diese Wirkungsquanten'.5
6
Histori al introdu tion
foundation, or rather the `Solvay International Institute of Physi s', was o ially established on 1 May 1912, whi h predates by several years the establishment of the omparable Belgian state institutions (Fondation Universitaire: 1920; Fonds National de la Re her he S ienti que: 1928). In this onne tion, Lorentz hoped `that governments would understand more and more the importan e of s ienti resear h and that in the long run one will arrive at a satisfa tory organisation, independent of the individual eorts of private persons',7 a sentiment e hoed by Solvay himself.a The institute, whi h Solvay had endowed for thirty years, ould soon boast of remarkable a tivity in supporting s ienti resear h. The numerous re ipients of subsidies granted during the rst two years until the rst world war in luded Lebedew's laboratory, von Laue, Sommerfeld, Fran k and Hertz, W. L. Bragg (who was later to be ome president of the s ienti ommittee), Stark, and Wien. In 1913, an Institute of Chemistry followed suit, organised along similar lines to the Institute of Physi s.
1.2 War and international relations The rst meeting of the s ienti ommittee, for the planning of the se ond Solvay onferen e, took pla e on 30 September and 1 O tober 1912. The onferen e was held the following year, but the a tivities of the institute were soon disrupted by the start of the rst world war, in parti ular the German invasion of Belgium. Immediate pra ti al disruption in luded the fear of requisitions, the di ulty of ommuni ation between the international membership of the s ienti ommittee and, with regard to the publi ation of the pro eedings of the se ond Solvay onferen e, the impossibility of sending Lorentz the proofs for orre tion and the eventual prospe t of German
ensorship.a The war, however, had longer-term negative impli ations for international intelle tual ooperation. In O tober 1914, a group of 93 representatives of German s ien e and ulture signed the manifesto `An die a `Mr Solvay also thinks that it is the role of the state to subsidise and organise s ienti institutions, and he hopes that in thirty years the state will ful ll this duty better than it does today.'8 a The pro eedings of the rst Solvay onferen e had had both a Fren h and a German edition. Those of the se ond Solvay onferen e were printed in three languages in 1915, but never published in this form and later mostly destroyed. Only under the hanged onditions after the war, in 1921, were the pro eedings published in a Fren h translation ( arried out, as on later o
asions, by J.-É. Vers haelt).
1.2 War and international relations
7
Kulturwelt!', denying German responsibilities in the war.a Among the signatories were both Nernst and Plan k. This manifesto was partly responsible for the very strong hostility of Fren h and Belgian s ientists and institutions towards renewal of s ienti relations with Germany after the war. No Germans or Austrians were invited to the third Solvay onferen e of 1921. The only ex eption (whi h remained problemati until the last minute) was Ehrenfest, who was Austrian, but who had remained in Leiden throughout the war as Lorentz's su
essor. Similarly, no Germans parti ipated in the fourth Solvay onferen e of 1924. Fren h and Belgian armies had o
upied the Ruhr in January 1923, and the international situation was parti ularly tense. Einstein had (temporarily) resigned from the League of Nations' Committee on Intelle tual Cooperation, and wrote to Lorentz that he would not parti ipate in the Solvay onferen e be ause of the ex lusion of the German s ientists, and that he should please make sure that no invitation was sent.9 Bohr also de lined to parti ipate in the onferen e apparently be ause of the ontinued ex lusion of German s ientists (Moore 1989, p. 157). S hrödinger, however, who was Austrian and working in Switzerland, was invited.a Einstein had distinguished himself by assuming a pa i st position during the war.b Lorentz was pointing out Einstein's ex eptional ase to Solvay already in January 1919: However, in talking about the Germans, we must not lose sight of the fa t that they ome in all kinds of nuan es. A man like Einstein, the great and profound physi ist, is not `German' in the sense one often atta hes to the word today; his judgement on the events of the past years will not dier at all from 11 yours or mine.
a The main laims of the manifesto were: `.... It is not true that Germany is the
ause of this war. .... It is not true that we have wantonly [freventli h℄ infringed the neutrality of Belgium. .... It is not true that the life and property of a single Belgian itizen has been tou hed by our soldiers, ex ept when utter self-defen e required it. .... It is not true that our troops have raged brutally against Leuven. .... It is not true that our ondu t of war disregards the laws of international right. .... It is not true that the struggle against our so- alled militarism is not a struggle against our ulture ....' (translated from Böhme 1975, pp. 479). a Van Aubel (a member of the s ienti ommittee) obje ted strongly in 1923 to the possibility of Einstein being invited to the fourth Solvay onferen e, and resigned when it was de ided to invite him. It appears he was onvin ed to remain on the
ommittee.10 b For instan e, Einstein was one of only four signatories of the ounter-manifesto `Aufruf an die Europäer' (Ni olai 1917). Note also that Einstein had renoun ed his German itizenship and had be ome a Swiss itizen in 1901, although there was some un ertainty about his itizenship when he was awarded the Nobel prize (Pais 1982, pp. 45 and 5034).
8
Histori al introdu tion
In the meantime, after the treaty of Lo arno of 1925, Germany was going to join the League of Nations, but the details of the negotiations were problemati .a As early as February 1926, one nds mention of the prospe t of renewed in lusion of German s ientists at the Solvay
onferen es.13 In the same month, Kamerlingh Onnes died, and at the next meeting of the s ienti ommittee, in early April (at whi h the fth Solvay onferen e was planned), it was de ided to propose both to invite Einstein to repla e Onnes and to in lude again the German s ientists. On 1 April, Charles Lefébure, then se retary of the administrative
ommission, wrote to ommission members Armand Solvay and Jules Bordet,a enquiring about the admissibility of `moderate gures like Einstein, Plan kb and others'16 (Bordet telegraphed ba k: `Germany will soon be League of Nations therefore no obje tion'17 ). On 2 April, Lorentz himself had a long interview with the King, who gave his approval. Thus, nally, Lorentz wrote to Einstein on 6 April, informing him of the unanimous de ision by the members of the ommittee present at the meeting,c as well as of the whole administrative ommission, to invite him to su
eed Kamerlingh Onnes. The Solvay onferen es were to readmit Germans, and if Einstein were a member of the ommittee, Lorentz hoped this would en ourage the German s ientists to a
ept the invitation.18 Einstein was favourably impressed by the positive Belgian attitude and glad to a
ept under the altered onditions.19 Lorentz pro eeded to invite the German s ientists, `not be ause there should be su h a great haste in the thing, rather to show the Germans as soon a Lorentz to Einstein on 14 Mar h 1926: `Things are bad with the League of Nations; if only one ould yet nd a way out until the day after tomorrow'.12 Negotiations provisionally broke down on 17 Mar h, but Germany eventually joined the League in September 1926. a Lefébure was the appointee of the Solvay family to the administrative ommission, and as su h su
eeded Eugène Tassel, who had died in O tober 1922 and had been a long-standing ollaborator of Ernest Solvay sin e 1886. Armand Solvay was the son of Ernest Solvay, who had died on 26 May 1922. Bordet was the royal appointee to the ommission, and had just been appointed in February 1926, following the death of Paul Héger.14 b A
ording to Lorentz, Plan k had always been helpful to him when he had tried to intervene with the German authorities during the war. Further, Plan k had somewhat quali ed his position with regard to the Kulturwelt manifesto in an open letter, whi h he asked Lorentz to publish in the Dut h newspapers in 1916. On the other hand, he expli itly ruled out a publi disavowal of the manifesto in De ember 1923.15
Listed as Marie Curie, Langevin, Ri hardson, Guye and Knudsen (with two members absent, W. H. Bragg and Van Aubel).
1.2 War and international relations
9
as possible our good will',20 and sent the informal invitations to Born, Heisenberg and Plan k (as well as to Bohr) in or around June 1926.21 As late as O tober 1927, however, the issue was still a sensitive one. Van Aubel (who had not been present at the April 1926 meeting of the s ienti ommittee) replied in the negative to the o ial invitation to the onferen e.a Furthermore, it was proposed to release the list of parti ipants to the press only after the onferen e to avoid publi demonstrations. Lorentz travelled in person to Brussels on 17 O tober to dis uss the matter.23 Lorentz's own position during and immediately after the war, as a physi ist from one of the neutral ountries, had possibly been rather deli ate. In the text on Ernest Solvay from whi h we have quoted at the beginning of this hapter, for instan e, he appears to be defending the impartiality of the poli ies of the Institute of Physi s in the years leading up to the war. Lorentz started working for some form of re on iliation as soon as the war was over, writing as follows to Solvay in January 1919: All things onsidered, I think I must propose to you not to ex lude formally the Germans, that is, not to lose the door on them forever. I hope that it may be open for a new generation, and even that maybe, in the ourse of the years, one may admit those of today's s holars who one an believe regret sin erely and honestly the events that have taken pla e. Thus German s ien e will be 24 able to regain the pla e that, despite everything, it deserves for its past.
It should be noted that Lorentz was not only the s ienti organiser of the Solvay institute and the Solvay onferen es, but also a prime mover behind eorts towards international intelle tual ooperation, through his heavy involvement with the Conseil International de Re her hes, as well as with the League of Nations' Committee on Intelle tual Cooperation, of whi h he was a member from 1923 and president from 1925.b Lorentz's gure and ontributions to the Solvay onferen es are movingly re alled by Marie Curie in her obituary of Lorentz in the pro eedings of the fth Solvay onferen e (whi h opens Part III of this volume). a Lefébure's omment was: `be ause there are Germans! Then why does he stay in the Institute of Physi s?'22 b The Conseil International de Re her hes (founded in 1919) has today be ome the International Coun il for S ien e (ICSU). The Committee on Intelle tual Cooperation (founded in 1922) and the related International Institute of Intelle tual Cooperation (inaugurated in Paris in 1926) were the forerunners of UNESCO.25
10
Histori al introdu tion
1.3 S ienti planning and ba kground What was at issue in the remark that heads this hapter,a s ribbled by Ehrenfest in the margin of a letter from Lorentz, was the proposed topi for the fth Solvay onferen e, namely `the on i t and the possible re on iliation between the lassi al theories and the theory of quanta'.26 Ehrenfest found the phrasing obje tionable in that it en ouraged one to `swindle away the fruitful and suggestive harshness of the on i t by most slimy un lear thinking, quite in analogy with what happened also even after 1900 with the me hani al ether theories of the Maxwell equations', pointing out that `Bohr feels even more strongly than me against this slogan [S hlagwort℄, pre isely be ause he takes it so parti ularly to heart to nd the foundations of the future theory'.27 Lorentz took Ehrenfest's suggestion into a
ount, and dropped the referen e to re on iliation both from the title and from later des riptions of the fo us of the meeting.a The meeting of the s ienti ommittee for the planning of the fth Solvay onferen e took pla e in Brussels on 1 and 2 April 1926. Lorentz reported a few days later to Einstein: As the topi for 1927 we have hosen `The quantum theory and the lassi al theories of radiation', and we hope to have the following reports or le tures:
1 2 3
W. L. Bragg. New tests of the lassi al theory. A. H. Compton. Compton ee t and its onsequen es. C. T. R. Wilson. Observations on photoele trons and ollision ele trons by
the ondensation method.
4 5
L. de Broglie. Interferen e and light quanta. (short note):
Kramers.
Theory of Slater-Bohr-Kramers and analogous
theories.
6
Einstein. New derivations of Plan k's law and appli ations of statisti s to
quanta.
7
Heisenberg. 29 theory.
Adaptation of the foundations of dynami s to the quantum
Another report, by the ommittee's se retary Vers haelt,30 adds, on erning point 5 : `(at least, if Mr Kramers judges that it is still useful)'; a In the original: `Quantenverzoening <{zeer} antipathik<e>>
<{bedenkelijk<e>}> [?℄ titel wekt misverstand'. Many thanks to Mark van Atten for help with this passage. a To Bohr in June 1926: `.... the on i t between the lassi al theories and the quantum theory .... '; to S hrödinger in January 1927: ` .... the ontrast between the urrent and the earlier on eptions [Auassungen℄ and the attempts at development of a new me hani s'.28
1.3 S ienti planning and ba kground
11
it further lists a few alternative speakers: Compton or Debye for 2 , Einstein or Ehrenfest for 6 , and Heisenberg or S hrödinger for 7.a Thus, the fth Solvay onferen e, as originally planned, was to fo us mainly on the theory of radiation and on light quanta, in luding only one report on the new quantum theory of matter. The shift in fo us between 1926 and 1927 was learly due to major theoreti al advan es (for example by S hrödinger and Dira ) and new experimental results (su h as the Davisson-Germer experiments), and it an be partly followed as the planning of the onferen e progressed. S hrödinger's wave me hani s was one of the major theoreti al developments of the year 1926. Einstein, who had been alerted to S hrödinger's rst paper by Plan k ( f. Przibram 1967, p. 23), suggested to Lorentz that S hrödinger should talk at the onferen e instead of himself, on the basis of his new `theory of quantum states', whi h he des ribed as a development of genius of de Broglie's ideas.31 While it is un lear whether Lorentz knew of S hrödinger's papers by the time of the April meeting,a S hrödinger was listed a week later as a possible substitute for Heisenberg, and Lorentz himself was assuring Einstein at the end of April that S hrödinger was already being onsidered, spe i ally as a substitute for the report on the new foundations of dynami s rather than for the report on quantum statisti s.b Lorentz losely followed the development of wave me hani s, indeed
ontributing some essential ritique in his orresponden e with S hrödinger from this period, for the most part translated in Przibram (1967) (see hapter 4, espe ially se tions 4.3 and 4.4, for some more details on this orresponden e). Lorentz also gave a number of olloquia and le tures on wave me hani s (and on matrix me hani s) in the period leading up to the Solvay onferen e, in Leiden, Itha a and Pasadena.32 In Pasadena he also had the opportunity of dis ussing with S hrödinger the possibility that S hrödinger may also give a report at the onferen e, as in fa t he did.c S hrödinger's wave me hani s had also made a great impression on Einstein, although he repeatedly expressed his unease to Lorentz at the use of wave fun tions on on guration spa e (`obs ure',34 `harsh',35 a `Mysterium'36 ), and again during the general dis ussion (p. 488). a For details of the other parti ipants, see the next se tion. a Cf. se tion 4.1. b Note that S hrödinger (1926a) had written on `Einstein's gas theory' in a paper that is an immediate pre ursor to his series of papers on quantisation.
Lorentz was at Cornell from September to De ember 1926, then in Pasadena until Mar h 1927.33 On S hrödinger's Ameri an voyage, see Moore (1989, pp. 23033).
12
Histori al introdu tion
One sees Lorentz's involvement with the re ent developments also in his orresponden e with Ehrenfest. In parti ular, Lorentz appears to have been stru k by Dira 's ontributions to quantum me hani s.a In June 1927, Lorentz invited Dira to spend the following a ademi year in Leiden ,38 and asked Born and Heisenberg to in lude a dis ussion of Dira 's work in their report.39 Finally, in late August, Lorentz de ided that Dira , and also Pauli, ought to be invited to the onferen e, for indeed: Sin e last year, quantum me hani s, whi h will be our topi , has developed with an unexpe ted rapidity, and some physi ists who were formerly in the se ond tier have made extremely notable ontributions.
For this reason I
would be very keen to invite also Mr Dira of Cambridge and Mr Pauli of Copenhagen. .... Their ollaboration would be very useful to us .... I need not
onsult the s ienti ommittee be ause Mr Dira and Mr Pauli were both on 40
a list that we had drawn up last year .... .
Lorentz invited Pauli on 5 September 1927 (Pauli 1979, pp. 4089) and Dira sometime before 13 September 1927.41 On the experimental side, some of the main a hievements of 1927 were the experiments on matter waves. While originally de Broglie was listed to give a report on light quanta, the work he presented was about both light quanta and material parti les (indeed, ele trons and photons!), and Lorentz asked him expli itly to in lude some dis ussion of the re ent experiments speaking in favour of the notion of matter waves, spe i ally dis ussing Elsasser's (1925) proposals, and the experimental work of Dymond (1927) and of Davisson and Germer (1927).42 Thus, in the nal programme of the onferen e, we nd three reports on the foundations of a new me hani s, by de Broglie, Heisenberg (together with Born) and S hrödinger. The talks given by Bragg and Compton, instead, re e t at least in part the initial orientation of the onferen e. Here is how Compton presents the division of labour (p. 329): Professor W. L. Bragg has just dis ussed a whole series of radiation phenomena in whi h the ele tromagneti theory is on rmed.
....
I have been left the
task of pleading the opposing ause to that of the ele tromagneti theory of radiation, seen from the experimental viewpoint.
Bragg fo usses in parti ular on the te hnique of X-ray analysis, as the `most dire t way of analysing atomi and mole ular stru ture' (p. 284), a This orresponden e in ludes for instan e a 15-page ommentary by Lorentz on Dira (1927a).37
1.3 S ienti planning and ba kground
13
the development of whi h, as he had mentioned to Lorentz, was the `line in whi h [he had℄ been espe ially interested'.43 This in ludes in parti ular the investigation of the ele troni harge distribution. At Lorentz's request, he had also in luded a dis ussion of the refra tion of X-rays (se tion 8 of his report), whi h is dire tly relevant to the dis ussion after Compton's report.44 As des ribed by Lorentz in June 1927, Bragg was to report `on phenomena that still somehow allow a
lassi al des ription'.45 A few more aspe ts of Bragg's report are of immediate relevan e for the rest of the onferen e, espe ially to the dis ussion of S hrödinger's interpretation of the wave fun tion in terms of an ele tri harge density (pp. 307, 312, se tion 4.4), and so are some of the issues taken up further in the dis ussion (Hartree approximation, problems with waves in three dimensions), but it is fair to say that the report provides a rather distant ba kground for what followed it. Compton's talk overs the topi s of points 2 and 3 listed above. The expli it fo us of his report is the three-way omparison between the photon hypothesis, the Bohr-Kramers-Slater (BKS) theory of radiation, and the lassi al theory of radiation. Note, however, that Compton introdu es many of the topi s of later dis ussions. For instan e, he dis usses the problem of how to explain atomi radiation (se tion on `The emission of radiation', p. 333), whi h is inexpli able from the point of view of the lassi al theory, given that the `orbital frequen ies' in the atom do not orrespond to the emission frequen ies. This problem was one of S hrödinger's main on erns and one of the main points of on i t between S hrödinger and, for instan e, Heisenberg (see in parti ular the dis ussion after S hrödinger's report and, below, se tions 4.4 and 4.6). Compton's dis ussion of the photon hypothesis relates to the question of `guiding elds' (pp. 331 and 354) and of the lo alisation of parti les or energy quanta within a wave (pp. 339 and 348). These in turn are losely
onne ted with some of de Broglie's and Einstein's ideas (see below
hapter 7, espe ially se tion 7.2, and hapter 9 ), and with de Broglie's report on pilot-wave theory and Einstein's remark about lo ality in the general dis ussion (p. 488). Bohr had been a noted s epti of the photon hypothesis, and in 1924 Bohr, Kramers and Slater had developed a theory that was able to maintain a wave pi ture of radiation, by introdu ing a des ription of the atom based on `virtual os illators' with frequen ies equal to the frequen ies of emission (Bohr, Kramers and Slater 1924a,b).a A stationary state of a As Darrigol (1992, p. 257) emphasises, while the free virtual elds obey the
14
Histori al introdu tion
an atom, say the nth, is asso iated with a state of ex itation of the os illators with frequen ies orresponding to transitions from the energy En . Su h os illators produ e a lassi al radiation eld (a `virtual' one), whi h in turn determines the probabilities for spontaneous emission in the atom, that is, for the emission of energy from the atom and the jump to a stationary state of lower energy. The virtual eld of one atom also intera ts with the virtual os illators in other atoms (whi h in turn produ e se ondary virtual radiation) and in uen es the probabilities for indu ed emission and absorption in the other atoms. While the theory provides a me hanism for radiation onsistent with the pi ture of stationary states ( f. Compton's remarks, p. 335), it violates energy and momentum onservation for single events, in that an emission in one atom is not onne ted dire tly to an absorption in another atom, but only indire tly through the virtual radiation eld. Energy and momentum onservation hold only at a statisti al level. The BKS proposal was short-lived, be ause the Bothe-Geiger and Compton-Simon experiments established the onservation laws for individual pro esses (as explained in detail by Compton in his report, pp. 350 .). Thus, at the time of the planning of the fth Solvay onferen e, the experimental eviden e had ruled out the BKS theory (hen e the above remark: `if Mr Kramers judges that it is still useful').a The short note 5, indeed, dropped out of the programme altogether.b The des ription of the intera tion between matter and radiation, in parti ular the Compton ee t, ontinued to be a problem for Bohr, and
ontributed to the development of his views on wave-parti le dualism and
omplementarity. In his ontribution to the dis ussion after Compton's report (p. 360, the longest of his published ontributions in the Solvay volumea ), Bohr sket hes the motivations behind the BKS theory, the Maxwell equations, i.e. an be onsidered to be lassi al, the virtual os illators and the intera tion between the elds and the os illators are non- lassi al in several respe ts. a Bothe and Geiger had been working on their experiments sin e June 1924 (Bothe and Geiger 1924), and provisional results were being debated by the turn of the year. For two diering views on the signi an e of these results for instan e see Einstein to Lorentz, 16 De ember 1924 (the same letter in whi h he wrote to Lorentz about de Broglie's results)46 and the ex hange of letters between Born and Bohr in January 1925 (Bohr 1984, pp. 3026). By April 1925, Bothe and Geiger had lear- ut results against the BKS theory (Bothe and Geiger 1925a,b; see also the letters between Geiger and Bohr in Bohr 1984, pp. 3524). b On the BKS theory and related matters, see also hapter 3 (espe ially se tions 3.3.1 and 3.4.2), hapter 9, Darrigol (1992, hapter 9), the ex ellent introdu tion by Stolzenburg to Part I of Bohr (1984) and Mehra and Re henberg (1982, se tion V.2). a See below for the fate of his ontribution to the general dis ussion.
1.3 S ienti planning and ba kground
15
on lusions to be drawn from the Bothe-Geiger and Compton-Simon experiments and the further development of his views. Lorentz, in his report of the meeting to Einstein had mentioned `SlaterBohr-Kramers and analogous theories'. This may refer to the further developments (independent of the validity of the BKS theory) that led in parti ular to Kramer's (1924) dispersion theory (and from there towards matrix me hani s), or to Slater's original ideas, whi h were roughly along the lines of guiding elds for the photons (even though the photons were dropped from the nal BKS proposal).a Note that Einstein at this time was also thinking about guiding elds (in three dimensions). Pais (1982, pp. 44041) writes that, a
ording to Wigner, Einstein did not publish these ideas be ause they also led to problems with the onservation laws.b Einstein was asked by Lorentz to ontribute a report on `New derivations of Plan k's law and appli ations of statisti s to quanta' (point 6 ),
learly referring to the work by Bose (1924) on Plan k's law, hampioned by Einstein and applied by him to the theory of the ideal gas (Einstein 1924, 1925a,b). The se ond of these papers is also where Einstein famously endorses de Broglie's idea of matter waves. Einstein thought that his work on the subje t was already too well-known, but he a
epted after Lorentz repeated his invitation.47 On 17 June 1927, however, at about the time when Lorentz was sending detailed requests to the speakers, Einstein informed him in the following terms that he would not, after all, present a report: I re all having ommitted myself to you to give a report on quantum statisti s at the Solvay onferen e. After mu h re e tion ba k and forth, I ome to the
onvi tion that I am not ompetent [to give℄ su h a report in a way that really
orresponds to the state of things. The reason is that I have not been able to parti ipate as intensively in the modern development of quantum theory as would be ne essary for this purpose. This is in part be ause I have on the whole too little re eptive talent for fully following the stormy developments, in part also be ause I do not approve of the purely statisti al way of thinking on whi h the new theories are founded .... Up until now, I kept hoping to be able to ontribute something of value in Brussels; I have now given up that hope. I beg you not to be angry with me be ause of that; I did not take this lightly but tried with all my strength .... (Quoted in Pais 1982, pp. 4312)
Einstein's withdrawal may be related to the following ir umstan es. On a Cf. Slater (1924) and Mehra and Re henberg (1982, pp. 5436). See also Pauli's remark during the dis ussion of de Broglie's report (p. 401). b Cf. Einstein's ontribution to the general dis ussion (p. 486) and the dis ussion below in hapter 9.
16
Histori al introdu tion
5 May 1927, during a meeting of the Prussian A ademy of S ien es in Berlin, Einstein had read a paper on the question: `Does S hrödinger's wave me hani s determine the motion of a system ompletely or only in the sense of statisti s?'48 As dis ussed in detail by Belousek (1996), the paper attempts to de ne deterministi parti le motions from S hrödinger's wave fun tions, but was also suddenly withdrawn on 21 May.a The plans for the talks were nalised by Lorentz around June 1927. An extra t from his letter to S hrödinger on the subje t reads as follows: [W℄e hope to have the following reports [Referate℄ (I give them in the order in whi h we might dis uss them): 1. From Mr W. L. Bragg on phenomena that still somehow allow a lassi al des ription (re exion of X-rays by rystals, dira tion and total re e tion of X-rays). 2. From Mr Compton on the ee t dis overed by him and what relates to it. 3. From Mr de Broglie on his theory. I am asking him also to take into a
ount the appli ation of his ideas to free ele trons (Elsasser, quantum me hani s of free ele trons; Dymond, Davisson and Germer, s attering of ele trons). 4. From Dr Heisenberg
or
Prof. Born (the hoi e is left to them) on matrix
me hani s, in luding Dira 's theory. 5. Your report [on wave me hani s℄. Maybe another one or two short ommuni ations [Beri hte℄ on spe ial topi s 50 will be added.
This was, indeed, the nal programme of the onferen e, with Born and Heisenberg de iding to ontribute a joint report.a
1.4 Further details of planning In 192627 the s ienti ommittee and the administrative ommission of the Solvay institute were omposed as follows. a The news of Einstein's ommuni ation prompted an ex hange of letters between Heisenberg and Einstein, of whi h Heisenberg's letters, of 19 May and 10 June, survive.49 The se ond of these is parti ularly interesting, be ause Heisenberg presents in some detail his view of theories that in lude parti le traje tories. Both Einstein's hidden-variables proposal and Heisenberg's rea tion will be des ribed in se tion 11.3. a See se tion 3.2. Note that, as we shall see below, while Bohr ontributed signi antly to the general dis ussion and reported the views he had developed in Como (Bohr 1949, p. 216, 1985, pp. 357), he was unable to prepare an edited version of his omments in time and therefore suggested that a translation of his Como le ture, in the version for Naturwissens haften (Bohr 1928), be in luded in the volume instead. This has given rise to a ommon belief that Bohr gave a report on a par with the other reports, and that the general dis ussion at the onferen e was the dis ussion following it. See for instan e Mehra (1975, p. 152), and Mehra and Re henberg (2000, pp. 246 and 249), who appear further to believe that Bohr did not parti ipate in the o ial dis ussion.
1.4 Further details of planning
17
S ienti ommittee: Lorentz (Leiden) as president, Knudsen (Copenhagen) as se retary, W. H. Bragg (until May 1927, London),a Marie Curie (Paris), Einstein (sin e April 1926, Berlin), Charles-Eugène Guye (Geneva),b Langevin (Paris), Ri hardson (London), Edm. van Aubel (Gent). Administrative ommission: Armand Solvay, Jules Bordet (ULB), Mauri e Bourquin (ULB), Émile Henriot (ULB); the administrative se retary sin e 1922, and thus main orrespondent of Lorentz and others, was Charles Lefébure. The se retary of the meeting was Jules-Émile. Vers haelt (Gent), who had a ted as se retary sin e the third Solvay onferen e.52 The rst provisional list of possible parti ipants (in addition to Ehrenfest) appears in Lorentz's letter to Ehrenfest of 29 Mar h 1926: Einstein, Bohr, Kramers, Born, Heisenberg (Jordan surely more mathemati ian), Pauli, Ladenburg (?), Slater, the young Bragg (be ause of the ` orresponden e' to the lassi al theory that his work has often resulted in), J. J. Thomson, another one or two Englishmen (Darwin?
Fowler?), Léon
Brillouin (do not know whether he has worked on this, he has also already been there a number of times),
Louis
de Broglie (light quanta), one or two
who have on erned themselves with dira tion of X-rays (Bergen Davis?, 53 Compton, Debye, Dira (?)).
Lorentz asked for further suggestions and omments, whi h Ehrenfest sent in a letter dated `Leiden 30 Mar h 1926. Late at night': Langevin, Fowler, Dira , J. Fran[ ℄k (already for the experiments he devised by Hanle on the destru tion of resonan e polarisation through Larmor rotation a and for the work he proposed by Hund on the Ramsauer ee t undisturbed passage of slow ele trons through atoms and so on), Fermi (for interesting
ontinuation of the experiments by Hanle), Oseen (possibly a wrong attempt at explanation of needle radiation and as sharpwitted riti ), S hrödinger (was perhaps the rst to give quantum interpretation of the Doppler ee t, thus
lose to Compton ee t), Bothe (for Bothe-Geiger experiment on orrelation of Compton quantum and ele tron, whi h destroys Bohr-Slater theory, altogether a ne brain!) (Bothe should be onsidered perhaps
before S hrödinger),
Darwin, Smekal (is indeed a very deserving onnoisseur of quantum nesses, only he writes so
frightfully
mu h).
Léon Brillouin has published something re ently on matrix physi s, but I 54 have not read it yet.
a W. H. Bragg resigned due to over ommitment and was later repla ed by Cabrera (Madrid).51 b In 1909 the university of Geneva onferred on Einstein his rst honorary degree. A
ording to Pais (1982, p. 152), this was probably due to Guye. Coin identally, Ernest Solvay was honoured at the same time. a For more on the spe ial interest of the Ramsauer ee t, see se tion 3.4.2 below.
18
Histori al introdu tion
At the April meeting (as listed in the report by Vers haelt55 ) it was then de ided to invite: Bohr, Kramers, Ehrenfest, two among Born, Heisenberg and Pauli, Plan k, Fowler, W. L. Bragg, C. T. R. Wilson, L. de Broglie, L. Brillouin, Deslandres, Compton, S hrödinger and Debye. Possible substitutes were listed as: M. de Broglie or Thibaud for Bragg, Dira for Brillouin, Fabry for Deslandres, Kapitza for Wilson, Darwin or Dira for Fowler, Bergen Davis for Compton, and Thirring for S hrödinger.a The members of the s ienti ommitee would all take part ex o io, and invitations would be sent to the professors of physi s at ULB, that is, to Pi
ard, Henriot and De Donder57 (the latter apparently somewhat to Lefébure's hagrin, who, just before the
onferen e started, felt obliged to remind Lorentz that De Donder was `a paradoxi al mind, loud [en ombrant℄ and always ready to seize the word, often with great maladroitness'58 ). Both the number of a tual parti ipants and of observers was to be kept limited,59 partly explaining why it was thought that one should invite only two among Born, Heisenberg and Pauli. The hoi e initially fell on Born and Heisenberg (although Fran k was also onsidered as an alternative).60 Eventually, as noted above, Pauli was also in luded, as was Dira .61 Lorentz was also keen to invite Millikan and possibly Hall , when he heard that Millikan would be in Europe anyway for the Como meeting (Einstein and Ri hardson agreed).62 However, nothing
ame of this plan. When Einstein eventually withdrew as a speaker, he suggested Fermi or Langevin as possible substitutes (Pais 1982, p. 432). For a while it was not lear whether Langevin (who was anyway a member of the s ienti
ommittee) would be able to ome, sin e he was in Argentina over the summer and due to go on to Pasadena from there. Ehrenfest suggested F. Perrin instead, in rather admiring tones. Langevin was needed in Paris in O tober, however, and was able to ome to the onferen e.63 Finally, the week before the onferen e started, Lorentz extended the invitation to Irving Langmuir,64 who would happen to be in Brussels at the time of the onferen e.a Lorentz sent most of the informal invitations around January 1927.65 In May 1927, he sent to Lefébure the list of all the people he had `provisionally invited',66 in luding all the members of the s ienti ommittee and the prospe tive invitees as listed above by Vers haelt (that is, a A few days later, Guye suggested also Auger as a possible substitute for Wilson.56 a To Langmuir we owe a fas inating `home movie' of the onferen e; see the report in the AIP Bulletin of Physi s News, number 724 (2005).
1.4 Further details of planning
19
as yet without Pauli and Dira ). All had already replied and a
epted, ex ept Deslandres (who eventually replied mu h later de lining the invitation67 ). Around early July, Lorentz invited the physi ists from the university,68 and presumably sent a new invitation to W. H. Bragg, who thanked him but de lined.69 Formal letters of on rmation were sent out by Lefébure shortly before the onferen e.70 Around June 1927, Lorentz wrote to the planned speakers inviting them in the name of the s ienti ommittee to ontribute written reports, to rea h him preferably by 1 September. The general guidelines were: to fo us on one's own work, without mathemati al details, but rather so that `the prin iples are highlighted as learly as possible, and the open questions as well as the onne tions [Zusammenhänge℄ and
ontrasts are lari ed'. The material in the reports did not have to be unpublished, and a bibliography would be wel ome.71 Compton wrote that he would aim to deliver his manus ipt by 20 August, de Broglie easily before the end of August, Bragg, as well as Born and Heisenberg, by 1 September, and S hrödinger presumably only in the se ond half of September.72 (For further details of the orresponden e between some of the authors and Lorentz, see the relevant hapters below.) The written reports were to be sent to all parti ipants in advan e of the onferen e.a De Broglie's, whi h had been written dire tly in Fren h, was sent by Lorentz to the publishers, Gauthier-Villars in Paris, before he left for the Como meeting. They hoped to send 35 proofs to Lorentz by the end of September. In the meantime, Vers haelt and Lorentz's son had the remaining reports mimeographed by the `Holland Typing O e' in Amsterdam, and Vers haelt with the help of a student added in the formulas by hand, managing to mail on time to the parti ipants at least Compton's and Born and Heisenberg's reports, if not all of them.74 Lorentz had further written to all speakers (ex ept Compton) to ask them to bring reprints of their papers.75 Late during planning, a slight problem emerged, namely an unfortunate overlap of the Brussels onferen e with the festivities for the
entenary of Fresnel in Paris, to be o ially opened Thursday 27 O tober. Lorentz informed Lefébure of the lash writing from Naples after the Como meeting: neither the date of the onferen e ould be hanged nor that of the Fresnel elebrations, whi h had been xed by the Fren h President. The problem was ompounded by the fa t that de Broglie had a Mimeographed opies of Bragg's, Born and Heisenberg's and S hrödinger's reports are to be found in the Ri hardson Colle tion, Harry Ransom Humanities Resear h Center, University of Texas at Austin.73
20
Histori al introdu tion
a
epted to give a le ture to the So iété de Physique on the o
asion.a Lorentz suggested the ompromise solution of a general invitation to attend the elebrations. Those who wished to parti ipate ould travel to Paris on 27 O tober, returning to Brussels the next day, when sessions would be resumed in the afternoon. This was the solution that was indeed adopted.77
1.5 The Solvay meeting The fth Solvay onferen e took pla e from 24 to 29 O tober 1927 in Brussels. As on previous o
asions, the parti ipants stayed at the H tel Britannique, where a dinner invitation from Armand Solvay awaited them.78 Other meals were going to be taken at the institute, whi h was housed in the building of the Institute of Physiology in the Par Léopold;
atering for 5055 people had been arranged.79 The parti ipants were guests of the administrative ommission and all travel expenses within Europe were met.80 From the evening of 23 O tober onwards, three seats were reserved in a box at the Théatre de la Monnaie.81 The rst session of the onferen e started at 10:00 on Monday 24 O tober. A tentative re onstru tion of the s hedule of the onferen e is as follows.82 We assume that the talks were given in the order they were des ribed in the plans and printed in the volume, and that the re eption by the university on the Tuesday ontinued throughout the morning. It is lear that the general dis ussion extended over at least two days, from the fa t that Dira in his main ontribution (p. 491) refers expli itly to Bohr's omments of the day before.a • Monday 24 O tober, morning: W. L. Bragg's report, followed by dis ussion. • Monday 24 O tober, afternoon: A. H. Compton's report, followed by dis ussion. a In Lorentz's letter, the date of de Broglie's le ture is mentioned as 28 O tober, but the o ial invitations state that it was Zeeman who le tured then, and de Broglie the next evening, after the end of the Solvay onferen e. A report on de Broglie's le ture, whi h was entitled `Fresnel's ÷uvre and the urrent development of physi s', was published by Guye in the Journal de Genève of 16 and 18 April 1928.76 a In a letter to Vers haelt, Kramers refers to `the general dis ussion of Thursday', but that in fa t was the day of the Fresnel elebrations in Paris. The photograph of Lorentz in luded in the volume, a
ording to the aption, was also taken on that day. Sin e the elebrations opened only at 8:30pm, it is on eivable that there was a rst dis ussion session on Thursday morning. Pelseneer states, however, that sessions were suspended for the whole day.83
1.5 The Solvay meeting
21
• Tuesday 25 O tober, starting 9:00 a.m.: re eption oered by the ULB.
• Tuesday 25 O tober, afternoon: L. de Broglie's report, followed by dis ussion. • Wednesday 26 O tober, morning: M. Born and W. Heisenberg's report, followed by dis ussion. • Wednesday 26 O tober, afternoon: E. S hrödinger's report, followed by dis ussion. • Thursday 27 O tober, all day: travel to Paris and entenary of Fresnel.a • Friday 28 O tober, morning: return to Brussels.
• Friday 28 O tober, afternoon: general dis ussion.
• Saturday 29 O tober, morning: general dis ussion,b followed by lun h with the King and Queen of the Belgians. • Saturday 29 O tober, evening: dinner oered by Armand Solvay.
The languages used were presumably English, German and Fren h. S hrödinger had volunteered to give his talk in English,c while Born had suggested that he and Heisenberg ould provide additional explanations in English (while he thought that neither of them knew Fren h).86 The phrasing used by Born referred to who should `explain orally the ontents of the report', suggesting that the speakers did not present the exa t or full text of the reports as printed. Multipli ity of languages had long been a hara teristi of the Solvay
onferen es. A well-known letter by Ehrenfest87 tells us of `[p℄oor Lorentz as interpreter between the British and the Fren h who were absolutely unable to understand ea h other. Summarising Bohr. And Bohr responding with polite despair' (as quoted in Bohr 1985, p. 38).d On the last day of the onferen e, Ehrenfest went to the bla kboard and evoked the image of the tower of Babel (presumably in a more a Most of the parti ipants at the Solvay onferen e, with the ex eption of Knudsen, Dira , Ehrenfest, Plan k, S hrödinger, Henriot, Pi
ard and Herzen, travelled to Paris to attend the inauguration of the elebrations, in the grand amphithéatre of the Sorbonne.84 b The nal session of the onferen e also in luded a homage to Ernest Solvay's widow.85
Note that S hrödinger was uent in English from hildhood, his mother and aunts being half-English (Moore 1989, hapter 1). d Both W. H. Bragg and Plan k deplored in letters to Lorentz that they were very poor linguists. Indeed, in a letter explaining in more detail why he would not parti ipate in the onferen e, W. H. Bragg wrote: `I nd it impossible to follow the dis ussions even though you so often try to make it easy for us', and Plan k was in doubt about oming, parti ularly be ause of the language di ulties.88
22
Histori al introdu tion
metaphori al sense than the mere multipli ity of spoken languages), writing: And they said one to another: .... Go to, let us build us .... a tower, whose top may rea h unto heaven; and
let us make us a name ....
And the Lord said: ....
Go to, let us go down, and there onfound their language, that they may not understand one another's spee h. (Genesis 11: 37, reported by Pelseneer, his a emphasis )
Informal dis ussions at the onferen e must have been plentiful, but information about them has to be gathered from other sour es. Famously, Einstein and Bohr engaged in dis ussions that were des ribed in detail in later re olle tions by Bohr (1949), and vividly re alled by Ehrenfest within days of the onferen e in the well-known letter quoted above (see also hapter 12). Little known, if at all, is another referen e by Ehrenfest to the dis ussions between Bohr and Einstein, whi h appears to relate more dire tly to the issues raised by Einstein in the general dis ussion: Bohr had given a very pretty argument in a onversation with Einstein, that one ould not hope ever to master many-parti le problems with threedimensional S hrödinger ma hinery. (more or less!!!!!!):
He said something like the following
a wave pa ket an never simultaneously determine EX-
ACTLY the position and the velo ity of a parti le. Thus if one has for instan e TWO parti les, then they annot possibly be represented in
three-dimensional
spa e su h that one an simultaneously represent exa tly their kineti energy and the potential energy OF THEIR INTERACTION. Therefore......... (What omes after this therefore I already annot reprodu e properly.) In the multidimensional representation instead the potential energy of the intera tion appears totally sharp in the relevant oe ients of the wave equation and one 90 does [?℄ not get to see the kineti energy at all.
1.6 The editing of the pro eedings The editing of the pro eedings of the fth Solvay onferen e was largely
arried out by Vers haelt, who reported regularly to Lorentz and to Lefébure. During the last months of 1927 Lorentz was busy writing up the le ture he had given at the Como meeting in September.91 He then died suddenly on 4 February, before the editing work was omplete. The translation of the reports into Fren h was arried out after the
onferen e, ex ept for de Broglie's report whi h, as mentioned, was a This may have been Ehrenfest's own emphasis. Note that, if not ne essarily present at the sessions, Pelseneer had some onne tion with the onferen e, having taken Lorentz's photograph reprodu ed in the pro eedings.89
1.6 The editing of the pro eedings
23
written dire tly in Fren h. From Vers haelt's letters we gather that by 6 January 1928 all the reports had been translated, Bragg's and Compton's had been sent to the publishers, and Born and Heisenberg's and S hrödinger's were to be sent on that day or the next. Several proofs were ba k by the beginning of Mar h.92 Lorentz had envisaged preparing with Vers haelt an edited version of the dis ussions from notes taken during the onferen e, and sending the edited version to the speakers at proof stage.a In fa t, stenographed notes appear to have been taken, typed up and sent to the speakers, who for the most part used them to prepare drafts of their ontributions. From these, Vers haelt then edited the nal version, with some help from Kramers (who spe i ally ompleted two of Lorentz's
ontributions).93 A opy of the galley proofs of the general dis ussion, dated 1 June 1928, survives in the Bohr ar hives in Copenhagen,94 and in ludes some ontributions that appear to have been still largely unedited at that time.b By January, the editing of the dis ussions was pro eeding well, and at the beginning of Mar h it was almost ompleted. Some ontributions, however, were still missing, most notably Bohr's. The notes sent by Vers haelt had many gaps; Bohr wanted Kramers's advi e and help with the dis ussion ontributions, and travelled to Utre ht for this purpose at the beginning of Mar h.95 At the end of Mar h, Kramers sent Vers haelt the edited version of Bohr's ontributions to the dis ussion after Compton's report (pp. 360 and 370) and after Born and Heisenberg's report (p. 444), remarking that these were all of Bohr's ontributions to the dis ussions during the rst three days of the
onferen e.c In ontrast, material on Bohr's ontributions to the general dis ussion survives only in the form of notes in the Bohr ar hives.d (Some notes by Ri hardson also relate to Bohr's ontributions.97 ) A translation of version of the Como le ture for Naturwissens haften (Bohr 1928) was in luded instead, reprinted on a par with the other reports, and a
ompanied by the following footnote (p. 215 of the published pro eedings): a A
ording to D. Devriese, urator of the IIPCS ar hives, the original notes have not survived. b We have reprodu ed some of this material in the endnotes to the general dis ussion.
Note that Bohr also asked some brief questions after S hrödinger's report (p. 469). Kramers further writes that Bohr suggested to `omit the whole nal Born-Heisenberg dis ussion (Nr. 1823) and equally Fowler's remark 9'. Again, thanks to Mark van Atten for help with this letter.96 d This material is not mi ro lmed in AHQP. See also Bohr (1985, pp. 357, 100 and 4789).
24
Histori al introdu tion
This arti le, whi h is the translation of a note published very re ently in
Naturwissens haften,
vol. 16, 1928, p. 245, has been added at the author's
request to repla e the exposition of his ideas that he gave in the ourse of the following general dis ussion. It is essentially the reprodu tion of a talk on the
urrent state of quantum theory that was given in Como on 16 September 1927, on the o
asion of the jubilee festivities in honour of Volta.
The last remaining material was sent to Gauthier-Villars sometime in September 1928, and the volume was nally published in early De ember of that year.98
1.7 Con lusion The fth Solvay onferen e was by any standards an important and memorable event. On this point all parti ipants presumably agreed, as shown by numerous letters, su h as Ehrenfest's letter quoted above (reprodu ed in Bohr 1985), Heisenberg's letter to Lefébure at the head of this hapter, or various other letters of thanks addressed to the organisers after the onferen e:99 I would like to take this opportunity of thanking you for your kind hospitality, and telling you how mu h I enjoyed this parti ular Conferen e. I think it has been the most memorable one whi h I have attended for the subje t whi h was dis ussed was of su h vital interest and I learned so mu h. (W. L. Bragg to Armand Solvay, 3 November 1927) It was the most stimulating s ienti meeting I have ever taken part in. (Max Born to Charles Lefébure, 8 November 1927)
Per eptions of the signi an e of the onferen e diered from ea h other, however. In the o ial history, the fth Solvay onferen e went down (perhaps together with the Como meeting) as the o
asion on whi h the interpretational issues were nally lari ed. This was presumably a genuine sentiment on the part of Bohr, Heisenberg and the other physi ists of the Copenhagen-Göttingen s hool. We nd it expli itly as early as 1929: In relating the development of the quantum theory, one must in parti ular not forget the dis ussions at the Solvay onferen e in Brussels in 1927, haired by Lorentz.
Through the possibility of ex hange [Ausspra he℄ between the
representatives of dierent lines of resear h, this onferen e has ontributed extraordinarily to the lari ation of the physi al foundations of the quantum theory; it forms so to speak the outward ompletion of the quantum theory .... . (Heisenberg 1929, p. 495)
1.7 Con lusion
25
On the other hand, the onferen e was also des ribed (by Langevin) as the one where `the onfusion of ideas rea hed its peak'.100 From a distan e of almost 80 years, the beginnings of a more dispassionate evaluation should be possible. In the following hapters, we shall revisit the fth Solvay onferen e, fo ussing in parti ular on the ba kground and ontributions relating to the three main `lines of resear h' into quantum theory represented there: de Broglie's pilot-wave theory, Born and Heisenberg's quantum me hani s and S hrödinger's wave me hani s.
26
Notes to pp. 39
Ar hival notes 1 Handwritten remark (by Ehrenfest) in the margin of Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h). 2 Heisenberg to Lefébure, 19 De ember [1927℄, IIPCS 2685 (in German). 3 AHQP-LTZ-12, talk X 23, `Ernest Solvay', dated 28 September 1914 (in English). Cf. also the Fren h version of the same, X 10. (The two pages of the latter are in separate pla es on the mi ro lm.)
Historique des Instituts Internationaux de Physique et de Chimie Solvay depuis leur fondation jusqu'à la deuxième guerre mondiale,
4 Pelseneer, J.,
[1962℄, 103 pp., AHQP-58, se tion 1 (hereafter referred to simply as `Pelseneer'). 5 Plan k to Nernst, 11 June 1910, Pelseneer, p. 7 (in German). 6 Lorentz to Solvay, 4 January 1912, Pelseneer, pp. 2026 (in Fren h). See also the reply, Solvay to Lorentz, 10 January 1912, AHQP-LTZ-12 (in Fren h). 7 Lorentz to Solvay, 6 Mar h 1912, Pelseneer, p. 27 (in Fren h). 8 Héger to Lorentz, 16 February 1912, AHQP-LTZ-11 (in Fren h). 9 Einstein to Lorentz, 16 August 1923, AHQP-LTZ-7 (in German). 10 Van Aubel to Lorentz, 16 April, 16 May and 19 July 1923, AHQP-LTZ-11 (in Fren h). 11 Lorentz to Solvay, 10 January 1919, Pelseneer, p. 37 (in Fren h). 12 Lorentz to Einstein, 14 Mar h 1926, AHQP-86 (in German). 13 Lefébure to Lorentz, 12 February 1926, AHQP-LTZ-12 (in Fren h). 14 Cf. Lefébure to Lorentz, 12 February 1926, AHQP-LTZ-12 (in Fren h). 15 Letter by Lorentz, 7 January 1919, Pelseneer, pp. 356 (in Fren h), two letters from Plan k to Lorentz, 1915, AHQP-LTZ-12 (in German), Plan k to Lorentz, Mar h 1916, Pelseneer, pp. 345 (in German), and Plan k to Lorentz, 5 De ember 1923, AHQP-LTZ-9 (in German). 16 From Lefébure [possibly a opy for Lorentz℄, 1 April 1926, AHQP-LTZ-12 (in Fren h). Obituary of Tassel,
L'Éventail,
15 O tober 1922,
AHQP-LTZ-13 (in Fren h). 17 Cf. Lefébure to Lorentz, 6 April 1926, AHQP-LTZ-12 (in Fren h). 18 Lorentz to Einstein, 6 April 1926, AHQP-86 (in German). 19 Einstein to Lorentz, 14 April 1926 and 1 May 1927, AHQP-LTZ-11 (in German). 20 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German). 21 Compare the invitation of Lorentz to Bohr, 7 June 1926, AHQP-BSC-13 (in English), and the replies of Bohr to Lorentz, 24 June 1926, AHQP-LTZ-11 (in English), Plan k to Lorentz, 13 June 1926, AHQP-LTZ-8 (in German), Born to Lorentz, 19 June 1926, AHQP-LTZ-11 (in German), and Heisenberg to Lorentz, 4 July 1926, AHQP-LTZ-12 (in German). 22 Van Aubel to Lefébure, 6 O tober 1927, IIPCS 2545 (in Fren h), with Lefébure's handwritten omment. 23 Lefébure to Lorentz, 14 O tober 1927, IIPCS 2534, and 15 O tober 1927, IIPCS 2536, telegramme Lorentz to Lefébure, 15 O tober 1927, IIPCS 2535, and Lefébure to the King, 19 O tober 1927, IIPCS 2622 (all in Fren h). 24 Lorentz to Solvay, 10 January 1919, Pelseneer, p. 37 (in Fren h). 25 There is a large amount of relevant orresponden e in AHQP-LTZ.
Notes to pp. 918
27
26 Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h), with Ehrenfest's handwritten omments. 27 Ehrenfest to Lorentz, 30 Mar h 1926, AHQP-LTZ-11 (in German). 28 Lorentz to Bohr, 7 June 1926, AHQP-BSC-13, se tion 3 (in English), Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 29 Lorentz to Einstein, 6 April 1926, AHQP-86 (in German and Fren h). 30 Vers haelt to Lefébure, 8 April 1926, IIPCS 2573 (in Fren h). 31 Einstein to Lorentz, 12 April 1926, AHQP-LTZ-11 (in German). 32 See for instan e the atalogue of Lorentz's manus ripts in AHQP-LTZ-11. 33 See also Lorentz to S hrödinger, 21 January and 17 June 1927, AHQP-41, se tion 9 (in German). 34 Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (in German). 35 Einstein to Lorentz, 22 June 1926, AHQP-LTZ-8 (in German). 36 Einstein to Lorentz, 16 February 1927, AHQP-LTZ-11 (in German). 37 En losed with Lorentz to Ehrenfest, 3 June [1927, erroneously amended to 1925℄, AHQP-EHR-23 (in Dut h). 38 Lorentz to Dira , 9 June 1927, AHQP-LTZ-8 (in English). 39 Cf. Born to Lorentz, 23 June 1927, AHQP-LTZ-11 (in German). 40 Lorentz to Lefébure, 27 August 1927, IIPCS 2532A/B (in Fren h). [There appears to be a further item also numbered 2532.℄ 41 Dira to Lorentz, 13 September 1927, AHQP-LTZ-11 (in English). 42 Cf. de Broglie to Lorentz, 22 June 1927, AHQP-LTZ-11 (in Fren h), and Ehrenfest to Lorentz, 14 June 1927, AHQP-EHR-23 (in Dut h and German). 43 W. L. Bragg to Lorentz, 7 February 1927, AHQP-LTZ-11 (in English). 44 Cf. also W. L. Bragg to Lorentz, 27 June 1927, AHQP-LTZ-11 (in English). 45 Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 46 Einstein to Lorentz, 16 De ember 1924, AHQP-LTZ-7 (in German). 47 Einstein to Lorentz, 12 April 1926, AHQP-LTZ-11, Lorentz to Einstein, 28 April 1926, AHQP-86, and Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (all in German). 48 `Bestimmt S hrödingers Wellenme hanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?', AEA 2-100.00 (in German), available on-line at http://www.alberteinstein.info/db/ViewDetails.do?Do umentID=34338 . 49 Heisenberg to Einstein, 19 May 1927, AEA 12-173.00, and 10 June 1927, AEA 12-174.00 (both in German). 50 Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 51 W. H. Bragg to Lorentz, 11 May 1927, AHQP-LTZ-11 (in English). 52 Tassel to Vers haelt, 24 February 1921, AHQP-LTZ-13 (in Fren h). 53 Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h). 54 Ehrenfest to Lorentz, 30 Mar h 1926, AHQP-LTZ-11 (in German). 55 Vers haelt to Lefébure, 8 April 1926, IIPCS 2573 (in Fren h). 56 Guye to Lorentz, 14 April 1926, AHQP-LTZ-8 (in Fren h). 57 Cf. for instan e Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 58 Lefébure to Lorentz, 22 O tober 1927, AHQP-LTZ-12 (in Fren h). 59 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German), Lorentz to Lefébure, 9 O tober 1927, IIPCS, 2530A/B (in Fren h).
28
Notes to pp. 1819
60 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German), Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (in German). 61 For the latter, f. also Brillouin to Lorentz, 20 August 1927, AHQP-LTZ-11 (in Fren h). 62 Lorentz to Einstein, 30 January 1927, AHQP-86 (in German), Einstein to Lorentz, 16 February 1927, AHQP-LTZ-11 (in German), Ri hardson to Lorentz, 19 February 1927, AHQP-LTZ-12 (in English). 63 Brillouin to Lorentz, 20 August 1927, AHQP-LTZ-11 (in Fren h), Ehrenfest to Lorentz 18 August 1927, AHQP-EHR-23 (in German). 64 Telegramme Lorentz to Lefébure, 19 O tober 1927, IIPCS 2541 (in Fren h), with Lefébure's note: `Oui'. 65 Lorentz to Brillouin, 15 De ember 1926, AHQP-LTZ-12 (in Fren h), Lorentz to Ehrenfest, 18 January 1927, AHQP-EHR-23 (in Dut h), Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German); ompare various other replies to Lorentz's invitation, in AHQP-LTZ-11, AHQP-LTZ-12 and AHQP-LTZ-13: Brillouin, 8 January 1927 (in Fren h), de Broglie, 8 January 1927 (in Fren h), W. L. Bragg, 7 February 1927 (in English), Wilson, 11 February 1927 (in English), Kramers, 14 February 1927 (in German), Debye, 24 February 1927 (in Dut h), Compton, 3 April 1927 (in English) [late be ause he had been away for two months `in the Orient'℄. 66 Lorentz to Lefébure, 21 May 1927, IIPCS 2521A (in Fren h). 67 Deslandres to Lorentz, 19 July 1927, AHQP-LTZ-11 (in Fren h). 68 See the letters of a
eptan e to Lorentz: De Donder, 8 July 1927, AHQP-LTZ-11, Henriot, 10 July 1927, AHQP-LTZ-12, Pi
ard, 2 O tober 1927, AHQP-LTZ-12 (all in Fren h). 69 W. H. Bragg to Lorentz, 12 and 17 July 1927, AHQP-LTZ-11 (in English). 70 Copies in AHQP-LTZ-12 and IIPCS 2543. Various replies: IIPCS 254451, 25536, 2558, 25603. 71 Cf. Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 72 Compare the answers by Bragg, 27 June 1927, and by Compton, 7 July 1927, both AHQP-LTZ-11 (in English), and the detailed ones by de Broglie, 27 June 1927, AHQP-LTZ-11 (in Fren h) and by Born, 23 June 1927, AHQP-LTZ-11 (in German); also S hrödinger to Lorentz, 23 June 1927, AHQP-LTZ-13 (original with S hrödinger's orre tions), and AHQP-41 se tion 9 ( arbon opy) (in German). 73 Mi ro lmed in AHQP-RDN, do uments M-0059 (Bragg, atalogued as `unidenti ed author'), M-0309 (Born and Heisenberg, with seven pages of notes by Ri hardson) and M-1354 (S hrödinger). 74 See in parti ular the already quoted Lorentz to Lefébure, 27 August 1927, IIPCS 2532A/B (in Fren h), as well as Lorentz to Lefébure, 23 September 1927, IIPCS 2523A/B, and 4 O tober 1927, IIPCS 2528 (both in Fren h), Gauthiers-Villars to Lefébure, 16 September 1927, IIPCS 2755 (in Fren h), Vers haelt to Lorentz, 6 O tober 1927, AHQP-LTZ-13 (in Dut h), de Broglie to Lorentz, 29 August 1927, AHQP-LTZ-8, and 11 O tober 1927, AHQP-LTZ-11 (both in Fren h), and Vers haelt to Lefébure, 15 O tober 1927, IIPCS 2756 (in Fren h). 75 Lorentz to Ehrenfest, 13 O tober 1927, AHQP-EHR-23 (in Dut h). See also Born to Lorentz, 11 O tober 1927 AHQP-LTZ-11 (in German), de
Notes to pp. 1924
29
Broglie to Lorentz, 11 O tober 1927, AHQP-LTZ-11 (in Fren h), Plan k to Lefébure, 17 O tober 1927, IIPCS 2558, and Ri hardson to Lefébure, IIPCS 2561. 76 `Une rise dans la physique moderne I & II', IIPCS 275051 (in Fren h). 77 Lorentz to Lefébure, 29 September 1927, IIPCS 2523A/B, Brillouin to Lorentz, 11 O tober 1927, AHQP-LTZ-11, Fondation Solvay to Lefébure, IIPCS 2582. Invitation: IIPCS 2615 and 2619, AHPQ-LTZ-8. Replies: IIPCS 2617 and 2618. (All do uments in Fren h.) 78 IIPCS 2530A/B, 2537. 79 IIPCS 2533, 2586A/B/C/D/E (the proposed menus from the Taverne Royale), 2587A/B. 80 Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 81 IIPCS 2340. 82 For the time and pla e of the rst session, see IIPCS 2523A/B. For the re eption at ULB, see IIPCS 2540 and 2629. There is a seating plan for the lun h with the royal ouple, IIPCS 2627. For the dinner with Armand Solvay, see IIPCS 2533, 2624 and 2625. See also Pelseneer, pp. 4950. 83 See Kramers to Vers haelt, 23 Mar h 1928, AHQP-28 (in Dut h), and Pelseneer, p. 50. 84 IIPCS 2621. 85 See AHQP-LTZ-12 (draft), and presumably IIPCS 2667. 86 S hrödinger to Lorentz, 23 June 1927, AHQP-LTZ-13 and AHQP-41, se tion 9 (in German), Lorentz to S hrödinger, 8 July 1927, AHPQ-41, se tion 9 (in German), Born to Lorentz, 23 June 1927, AHQP-LTZ-11 (in German). 87 Ehrenfest to Goudsmit, Uhlenbe k and Dieke, 3 November 1927, AHQP-61 (in German). 88 W. H. Bragg to Lorentz, 17 July [1927℄, AHQP-LTZ-11 (in English), Plan k to Lorentz, 2 February 1927, AHQP-LTZ-8 (in German). 89 See Pelseneer, pp. 5051. 90 Ehrenfest to Kramers, 6 November 1927, AHQP-9, se tion 10 (in German). The words `bekommt man' [?℄ are very faint. 91 Lorentz to Lefébure, 30 De ember 1927, IIPCS 2670A/B (in Fren h). 92 Vers haelt to Lefébure, 6 January 1928, IIPCS 2609, 2 Mar h 1928, IIPCS 2610 (both in Fren h). 93 Lefébure to Lorentz, [after 27 August 1927℄, IIPCS 2524 (in Fren h), Bohr to Kramers, 17 February 1928, AHQP-BSC-13 (in Danish), Brillouin to Lorentz, 31 De ember 1927, AHQP-LTZ-8 (in Fren h), and Kramers to Vers haelt, 28 Mar h 1928, AHQP-28, se tion 4 (in Dut h). 94 Mi ro lmed in AHQP-BMSS-11, se tion 5. 95 Bohr to Kramers, 17 and 27 February 1928, AHQP-BSC-13 (both in Danish). 96 Kramers to Vers haelt, 28 Mar h 1928, AHQP-28, se tion 4 (in Dut h). 97 In luded with the opy of Born and Heisenberg's report in AHQP-RDN, do ument M-0309. 98 Gauthier-Villars to Vers haelt, 6 De ember 1928, IIPCS 2762 (in Fren h), and Vers haelt to Lefébure, 11 [De ember℄ 1928, IIPCS 2761 (in Fren h). 99 IIPCS 2671 (in English), IIPCS 2672 (in German). 100 Quoted in Pelseneer, p. 50.
2 De Broglie's pilot-wave theory
2.1 Ba kground At a time when no single known fa t supported this theory, Louis de Broglie asserted that a stream of ele trons whi h passed through a very small hole in an opaque s reen must exhibit the same phenomena as a light ray under the same onditions. Prof. C. W. Oseen, Chairman of the Nobel Committee for Physi s, presentation spee h, 12 De ember 1929 (Oseen 1999)
In September 1923, Prin e Louis de Brogliea made one of the most astonishing predi tions in the history of theoreti al physi s: that material bodies would exhibit the wave-like phenomena of dira tion and interferen e upon passing through su iently narrow slits. Like Einstein's predi tion of the de e tion of light by the sun, whi h was based on a reinterpretation of gravitational for e in terms of geometry, de Broglie's predi tion of the de e tion of ele tron paths by narrow slits was made on the basis of a fundamental reappraisal of the nature of for es and of dynami s. De Broglie had proposed that Newton's rst law of motion be abandoned, and repla ed by a new postulate, a
ording to whi h a freely moving body follows a traje tory that is orthogonal to the surfa es of equal phase of an asso iated guiding wave. The resulting `de Broglian dynami s' or pilot-wave theory as de Broglie later alled it was a new approa h to the theory of motion, as radi al as Einstein's interpretation a The de Broglies had ome to Fran e from Italy in the seventeenth entury, the original name `Broglia' eventually being hanged to de Broglie. On his father's side, de Broglie's an estors in luded dukes, prin es, ambassadors, and marshals of Fran e. Nye (1997) onsiders the on i t between de Broglie's pursuit of s ien e and the expe tations of his aristo rati family. For a biography of de Broglie, see Lo hak (1992).
30
2.1 Ba kground
31
of the traje tories of falling bodies as geodesi s of a urved spa etime, and as far-rea hing in its impli ations. In 1929 de Broglie re eived the Nobel Prize, `for his dis overy of the wave nature of ele trons'. Strangely enough, however, even though de Broglie's predi tion was
on rmed experimentally a few years later, for most of the twentieth
entury single-parti le dira tion and interferen e were routinely ited as eviden e against de Broglie's ideas: even today, some textbooks on quantum me hani s assert that su h interferen e demonstrates that parti le traje tories annot exist in the quantum domain (see se tion 6.1). It is as if the de e tion of light by the sun had ome to be widely regarded as eviden e against Einstein's general theory of relativity. This remarkable misunderstanding illustrates the extent to whi h de Broglie's work in the 1920s has been underestimated, misrepresented, and indeed largely ignored, not only by physi ists but also by historians. De Broglie's PhD thesis of 1924 is of ourse re ognised as a landmark in the history of quantum theory. But what is usually remembered about that thesis is the proposed extension of Einstein's wave-parti le duality from light to matter, with the formulas E = hν and p = h/λ (relating energy and momentum to frequen y and wavelength) being applied to ele trons or `matter waves'. Usually, little attention is paid to the fa t that a entral theme of de Broglie's thesis was the onstru tion of a new form of dynami s, in whi h lassi al (Newtonian or Einsteinian) laws are abandoned, and repla ed by new laws a
ording to whi h parti le velo ities are determined by guiding waves, in a spe i manner that uni es the variational prin iples of Maupertuis and Fermat. Nor, indeed, have historians paid mu h attention to de Broglie's later and more
omplete form of pilot-wave dynami s, whi h he arrived at in a paper published in May 1927 in Journal de Physique, and whi h he then presented in O tober 1927 at the fth Solvay onferen e. Unlike the other main ontributors to quantum theory, de Broglie worked in relative isolation, having little onta t with the prin ipal resear h entres in Berlin, Copenhagen, Göttingen, Cambridge and Muni h. While Bohr, Heisenberg, Born, S hrödinger, Pauli and others visited ea h other frequently and orresponded regularly, de Broglie worked essentially alone in Paris.a In Fran e at the time, while pure a In his typed `Replies to Mr Kuhn's questions' in the Ar hive for the History of Quantum Physi s,1 de Broglie writes (p. 7): `Between 1919 and 1928, I worked very mu h in isolation ' (emphasis in the original). Regarding his Ph.D. thesis de Broglie re alled (p. 9): `I worked very mu h alone and almost without any ex hange of ideas'.
32
De Broglie's pilot-wave theory
mathemati s was well represented, there was very little a tivity in theoreti al physi s. In addition, after the rst world war, s ienti relations with Germany and Austria were interrupted.a All this seems to have suited de Broglie's rather solitary temperament. De Broglie's isolation, and the fa t that Fran e was outside the mainstream of theoreti al physi s, may a
ount in part for why so mu h of de Broglie's work went relatively unnoti ed at the time, and has remained largely ignored even to the present day. For some seventy years, the physi s ommunity tended to believe either that `hidden-variables' theories like de Broglie's were impossible, or that su h theories had been disproven experimentally. The situation
hanged onsiderably in the 1990s, with the publi ation of textbooks presenting quantum me hani s in the pilot-wave formulation (Bohm and Hiley 1993; Holland 1993). Pilot-wave theory as originated by de Broglie in 1927, and elaborated by Bohm 25 years later (Bohm 1952a,b) is now a
epted as an alternative (if little used) formulation of quantum theory. Fo ussing for simpli ity on the nonrelativisti quantum theory of a system of N (spinless) parti les with 3-ve tor positions xi (i = 1, 2, ..., N ), it is now generally agreed that, with appropriate initial onditions, quantum physi s may be a
ounted for by the deterministi dynami s de ned by two dierential equations, the S hrödinger equation N
i~
∂Ψ X ~2 2 − ∇ Ψ+VΨ = ∂t 2mi i i=1
(1)
for a `pilot wave' Ψ(x1 , x2 , ..., xN , t) in on guration spa e, and the de Broglie guidan e equation mi
dxi = ∇i S dt
(2)
for parti le traje tories xi (t), where the phase S(x1 , x2 , ..., xN , t) is lo ally de ned by S = ~ Im ln Ψ (so that Ψ = |Ψ| e(i/~)S ). This, as we shall see, is how de Broglie presented his dynami s in 1927. Bohm's presentation of 1952 was somewhat dierent. If one takes the time derivative of (2), then using (1) one obtains Newton's law of motion for a
eleration mi x ¨i = −∇i (V + Q) ,
(3)
a For more details on de Broglie's situation in Fran e at the time, see Mehra and Re henberg (1982a, pp. 57884).
2.1 Ba kground
33
where Q≡−
X ℏ2 ∇2 |Ψ| i 2m |Ψ| i i
(4)
is the `quantum potential'. Bohm regarded (3) as the law of motion, with (2) added as a onstraint on the initial momenta, a onstraint that Bohm thought ould be relaxed (see se tion 11.1). For de Broglie, in ontrast, the law of motion (2) for velo ity had a fundamental status, and for him represented the uni ation of the prin iples of Maupertuis and Fermat. One should then distinguish between de Broglie's rst-order (velo itybased) dynami s of 1927, and Bohm's se ond-order (a
eleration-based) dynami s of 1952. In this hapter, we shall be on erned with the histori al origins of de Broglie's 1927 dynami s de ned by (1) and (2). Some authors have referred to this dynami s as `Bohmian me hani s'. Su h terminology is misleading: it disregards de Broglie's priority, and misses de Broglie's physi al motivations for re asting dynami s in terms of velo ity; it also misrepresents Bohm's 1952 formulation, whi h was based on (1) and (3). These and other histori al mis on eptions on erning de Broglie-Bohm theory will be addressed in se tion 11.1. The two equations (1), (2) de ne a deterministi (de Broglian or pilotwave) dynami s for a single multiparti le system: given an initial wave fun tion Ψ(x1 , x2 , ..., xN , 0) at t = 0, (1) determines Ψ(x1 , x2 , ..., xN , t) at all times t; and given an initial on guration (x1 (0), x2 (0), ..., xN (0)), (2) then determines the traje tory (x1 (t), x2 (t), ..., xN (t)). For an ensemble of systems with the same initial wave fun tion Ψ(x1 , x2 , ..., xN , 0), and with initial on gurations (x1 (0), x2 (0), ..., xN (0)) distributed a
ording to the Born rule 2
P (x1 , x2 , ..., xN , 0) = |Ψ(x1 , x2 , ..., xN , 0)| ,
(5)
the statisti al distribution of out omes of quantum measurements will agree with the predi tions of standard quantum theory. This is shown by treating the measuring apparatus, together with the system being measured, as a single multiparti le system obeying de Broglian dynami s, so that (x1 , x2 , ..., xN ) de nes the `pointer position' of the apparatus as well as the on guration of the measured system. Given the initial
ondition (5) for any multiparti le system, the statisti al distribution of parti le positions at later times will also agree with the Born rule 2 P = |Ψ| . Thus, the statisti al distribution of pointer positions in any experiment will agree with the predi tions of quantum theory, yielding
34
De Broglie's pilot-wave theory
the orre t statisti al distribution of out omes for standard quantum measurements. In his 1927 Solvay report, de Broglie gave some simple appli ations of pilot-wave theory, with the assumed initial ondition (5). He applied the theory to single-photon interferen e, to atomi transitions, and to the s attering (or dira tion) of ele trons by a rystal latti e. But a detailed demonstration of equivalen e to quantum theory, and in parti ular a pilot-wave a
ount of the general quantum theory of measurement, was not provided until the work of Bohm in 1952. How did de Broglie ome to propose this theory in 1927? In this
hapter, we tra e de Broglie's work in this dire tion, from his early work leading to his do toral thesis of 1924 (de Broglie 1924e, 1925), to his ru ial paper of 1927 published in Journal de Physique (de Broglie 1927b), and ulminating in his presentation of pilot-wave theory at the fth Solvay onferen e. We examine in detail how de Broglie arrived at this new form of parti le dynami s, and what his attitude towards it was. Later, in hapter 10, we shall onsider some of the dis ussions of de Broglie's theory that took pla e at the onferen e, in parti ular the famous (and widely misunderstood) lash between de Broglie and Pauli. De Broglie's dynami s has the striking feature that ele trons and photons are regarded as both parti les and waves. Like many s ienti ideas, this mingling of parti le-like and wave-like aspe ts had pre ursors. In Newton's Opti ks ( rst published in 1704), both wave-like and parti lelike properties are attributed to light. Newton's so- alled ` orpus ular' theory was formulated on the basis of extensive and detailed experiments ( arried out by Grimaldi, Hooke, and Newton himself) involving what we would now all interferen e and dira tion. A
ording to Newton, light orpus les or light `Rays' as he alled thema generate `Waves of Vibrations' in an `Aethereal Medium', mu h as a stone thrown into water generates water waves (Newton 1730; reprint, pp. 3479). In addition, Newton supposed that the waves in turn ae t the motion of the orpus les, whi h `may be alternately a
elerated and retarded by the Vibrations' (p. 348). In parti ular, Newton thought that the ee t of the medium on the motion of the orpus les was responsible for the phenomena of interferen e and dira tion. He writes, for example (p. 350): And doth not the gradual ondensation of this Medium extend to some dis-
a The opening de nition of the Opti ks de nes `Rays' of light as `its least Parts'.
2.1 Ba kground
35
tan e from the Bodies, and thereby ause the In exions of the Rays of Light, whi h pass by the edges of dense Bodies, at some distan e from the Bodies?
Newton understood that, for dira tion to o
ur, the motion of the light
orpus les would have to be ae ted at a distan e by the dira ting body `Do not Bodies a t upon Light at a distan e, and by their a tion bend its Rays .... ?' (p. 339) and his proposed me hanism involved waves in an inhomogeneous ether. Further, a
ording to Newton, to a
ount for the oloured fringes that had been observed by Grimaldi in the dira tion of white light by opaque bodies, the orpus les would have to exe ute an os illatory motion `like that of an Eel' (p. 339): Are not the Rays of Light in passing by the edges and sides of Bodies, bent several times ba kwards and forwards, with a motion like that of an Eel? And do not the three Fringes of olour'd Light above-mention'd arise from three su h bendings?
For Newton, of ourse, su h non-re tilinear motion ould be aused only by a for e emanating from the dira ting body. It is interesting to note that, in the general dis ussion at the fth Solvay onferen e (p. 510), de Broglie ommented on this very point, with referen e to the `emission' (or orpus ular) theory, and pointed out that if pilot-wave dynami s were written in terms of a
eleration (as done later by Bohm) then just su h for es appeared: In the orpus ular on eption of light, the existen e of dira tion phenomena o
uring at the edge of a s reen requires us to assume that, in this ase, the traje tory of the photons is urved. The supporters of the emission theory said that the edge of the s reen exerts a for e on the orpus le. Now, if in the new me hani s as I develop it, one writes the Lagrange equations for the photon, one sees appear on the right-hand side of these equations a term .... [that℄ .... represents a sort of for e of a new kind, whi h exists only .... where there is interferen e. It is this for e that will urve the traje tory of the photon when its wave
ψ
is dira ted by the edge of a s reen.
The striking similarity between Newton's qualitative ideas and pilotwave theory has also been noted by Berry, who remarks that during interferen e or dira tion the de Broglie-Bohm traje tories indeed `wriggle like an eel' (Berry 1997, p. 42), in some sense vindi ating Newton. A mathemati al pre ursor to de Broglian dynami s is found in the early nineteenth entury, in Hamilton's formulation of geometri al opti s and parti le me hani s. As de Broglie points out in his Solvay report (pp. 377 and 383), Hamilton's theory is in fa t the short-wavelength limit of pilot-wave dynami s: for in that limit, the phase of the wave fun tion
36
De Broglie's pilot-wave theory
obeys the Hamilton-Ja obi equation, and de Broglie's traje tories redu e to those of lassi al me hani s. A physi al theory of light as both parti les and waves in ee t a revival of Newton's views emerged again with Einstein in 1905. It is less well known that, after 1905, Einstein tried to onstru t theories of lo alised light quanta oupled to ve tor elds in 3-spa e. As we shall see in hapter 9, Einstein's ideas in this vein show some resemblan e to de Broglie's but also dier from them. It should also be mentioned that, in the autumn of 1923 (the same year in whi h de Broglie rst elaborated his ideas), Slater tried to develop a theory in whi h the motion of photons was guided by the ele tromagneti eld. It appears that Slater rst attempted to onstru t a deterministi theory, but had trouble de ning an appropriate velo ity ve tor; he then
ame to the on lusion that photons and the ele tromagneti eld were related only statisti ally, with the photon probability density being given by the intensity of the eld. After dis ussing his ideas with Bohr and Kramers in 1924, the photons were removed from the theory, apparently against Slater's wishes (Mehra and Re henberg 1982a, pp. 5426). Note that, while de Broglie applied his theory to photons, he made it lear (for example in the general dis ussion, p. 509) that in his theory the guiding `ψ -wave' was distin t from the ele tromagneti eld. In the ase of light, then, the idea of ombining both parti le-like and wave-like aspe ts was an old one, going ba k indeed to Newton. In the
ase of ordinary matter, however, de Broglie seems to have been the rst to develop a physi al theory of this form. It is sometimes laimed that, for the ase of ele trons, ideas similar to de Broglie's were put forward by Madelung in 1926. What Madelung proposed, however, was to regard an ele tron with mass m and wave fun tion ψ not as a pointlike parti le within the wave, but as a ontinuous uid spread over spa e with mass density m |ψ|2 (Madelung 1926a,b). In this `hydrodynami al' interpretation, mathemati ally the uid velo ity
oin ides with de Broglie's velo ity eld; but physi ally, Madelung's theory seems more akin to S hrödinger's theory than to de Broglie's. Finally, before we examine de Broglie's work, we note what appears to be a re urring histori al opposition to dualisti physi al theories
ontaining both waves and parti les. In 180103, Thomas Young, who by his own a
ount regarded his theory as a development of Newton's ideas,a removed the orpus les from Newton's theory and produ ed a purely una See, for example, Bernard Cohen's prefa e to Newton's Opti ks (1730, reprint).
2.2 A new approa h to parti le dynami s: 192324
37
dulatory a
ount of light. In 1905, Einstein's dualist view of light was not taken seriously, and did not win widespread support until the dis overy of the Compton ee t in 1923. In 1924, Bohr and Kramers, who regarded the Bohr-Kramers-Slater theory as a development of Slater's original idea, insisted on removing the photons from Slater's theory of radiation.a And in 1926, S hrödinger, who regarded his work as a development of de Broglie's ideas, removed the traje tories from de Broglie's theory and produ ed a purely undulatory `wave me hani s'.b
2.2 A new approa h to parti le dynami s: 192324 In this se tion we show how de Broglie took his rst steps towards a new form of dynami s.a His aim was to explain the quantum phenomena known at the time in parti ular the Bohr-Sommerfeld quantisation of atomi energy levels, and the apparently dual nature of radiation by unifying the physi s of parti les with the physi s of waves. To a
omplish this, de Broglie began by extending Einstein's wave-parti le duality for light to all material bodies, by introdu ing a `phase wave' a
ompanying every material parti le. Then, inspired by the opti alme hani al analogy,b de Broglie proposed that Newton's rst law of motion should be abandoned, and repla ed by a new prin iple that uni ed Maupertuis' variational prin iple for me hani s with Fermat's variational prin iple for opti s. The result was a new form of dynami s in whi h the velo ity v of a parti le is determined by the gradient of the phase φ of an a
ompanying wave in ontrast with lassi al me hani s, where a
elerations are determined by for es. (Note that de Broglie's phase φ has a sign opposite to the phase S as we would normally de ne it now.) This new approa h to dynami s enabled de Broglie to obtain a wavelike explanation for the quantisation of atomi energy levels, to explain the observed interferen e of single photons, and to predi t for the rst time the new and unexpe ted phenomenon of the dira tion and interferen e of ele trons. a In 1925, Born and Jordan attempted to restore the photons, proposing a sto hasti theory reminis ent of Slater's original ideas; it appears that they were dissuaded from publi ation by Bohr. See Darrigol (1992, p. 253) and se tion 3.4.2. b Cf. se tion 4.5. a An insightful and general a
ount of de Broglie's early work, up to 1924, has been given by Darrigol (1993). b Possibly, de Broglie was also in uen ed by the philosopher Henri Bergson's writings
on erning time, ontinuity and motion (Feuer 1974, pp. 20614); though this is denied by Lo hak (1992).
38
De Broglie's pilot-wave theory
As we shall see, the theory proposed by de Broglie in 192324 was, in fa t, a simple form of pilot-wave dynami s, for the spe ial ase of independent parti les guided by waves in 3-spa e, and without a spe i wave equation.
2.2.1 First papers on pilot-wave theory (1923) De Broglie's earliest experien e of physi s was losely tied to experiment. During the rst world war he worked on wireless telegraphy, and after the war his rst papers on erned X-ray spe tros opy. In 1922 he published a paper treating bla kbody radiation as a gas of light quanta (de Broglie 1922). In this paper, de Broglie made the unusual assumption that photons had a very small but non-zero rest mass m0 . He was therefore now applying Einstein's relations E = hν and p = h/λ (relating energy and momentum to frequen y and wavelength) to massive parti les, even if these were still only photons. It seems that de Broglie made the assumption m0 6= 0 so that light quanta ould be treated in the same way as ordinary material parti les. It appears that this paper was the seed from whi h de Broglie's subsequent work grew.a A
ording to de Broglie's later re olle tions (L. de Broglie, AHQP interview, 7 January 1963, p. 1),2 his rst ideas on erning a pilot-wave theory of massive parti les arose as follows. During onversations on the subje t of X-rays with his older brother Mauri e de Broglie,b he be ame onvin ed that X-rays were both parti les and waves. Then, in the summer of 1923, de Broglie had the idea of extending this duality to ordinary matter, in parti ular to ele trons. He was drawn in this dire tion by onsideration of the opti al-me hani al analogy; further, the presen e of whole numbers in quantisation onditions suggested to him that waves must be involved. This last motivation was re alled by de Broglie (1999) in his Nobel le ture of 1929: .... the determination of the stable motions of the ele trons in the atom involves whole numbers, and so far the only phenomena in whi h whole numbers were involved in physi s were those of interferen e and of eigenvibrations. That
a In a olle tion of papers by de Broglie and Brillouin, published in 1928, a footnote added to de Broglie's 1922 paper on bla kbody radiation and light quanta remarks: `This paper .... was the origin of the ideas of the author on wave me hani s' (de Broglie and Brillouin 1928, p. 1). b Mauri e, the sixth du de Broglie, was a distinguished experimental physi ist, having done important work on the photoele tri ee t with X-rays experiments that were arried out in his private laboratory in Paris.
2.2 A new approa h to parti le dynami s: 192324
39
suggested the idea to me that ele trons themselves ould not be represented as simple orpus les either, but that a periodi ity had also to be assigned to them too.
De Broglie rst presented his new ideas in three notes published (in Fren h) in the Comptes Rendus of the A ademy of S ien es in Paris (de Broglie 1923a,b, ), and also in two papers published in English one in Nature (de Broglie 1923d), the other in the Philosophi al Magazine (de Broglie 1924a).a The ideas in these papers formed the basis for de Broglie's do toral thesis. The paper in the Philosophi al Magazine reads, in fa t, like a summary of mu h of the material in the thesis. Sin e the thesis provides a more systemati presentation, we shall give a detailed summary of it in the next subse tion; here, we give only a brief a
ount of the earlier papers, ex ept for the ru ial se ond paper, whose on eptual ontent warrants more detailed ommentary.b The rst ommuni ation (de Broglie 1923a), entitled `Waves and quanta', proposes that an `internal periodi phenomenon' should be asso iated with any massive parti le (in luding light quanta). In the rest frame of a parti le with rest mass m0 , the periodi phenomenon is assumed to have a frequen y ν0 = m0 c2 /h. In a frame where the parti le has uniform velo ity p v , de Broglie onsiders the two frequen ies ν and ν1 , where 2 ν = ν0 / 1 − v 2 /c2 is the frequen y p ν = mc /h asso iated p with the 2 2 relativisti mass in rease m = m0 / 1 − v /c and ν1 = ν0 1 − v 2 /c2 is the time-dilated frequen y. De Broglie shows that, be ause ν1 = ν(1− v 2 /c2 ), a ` titious' wave of frequen y ν and phase velo ity vph = c2 /v (propagating in the same dire tion as the parti le) will remain in phase with the internal os illation of frequen y ν1 . De Broglie then onsiders an atomi ele tron moving uniformly on a ir ular orbit. He proposes that orbits are stable only if the titious wave remains in phase with the internal os illation of the ele tron. From this ondition, de Broglie derives the Bohr-Sommerfeld quantisation ondition. The se ond ommuni ation (de Broglie 1923b), entitled `Light quanta, dira tion and interferen e', has a more on eptual tone. De Broglie begins by re alling his previous result, that a moving body must be asso iated with `a non-material sinusoidal wave'. He adds that the parti le velo ity v is equal to the group velo ity of the wave, whi h de Broglie here alls `the phase wave' be ause its phase at the lo ation of the a It seems possible that Comptes Rendus was not widely read by physi ists outside Fran e, but this ertainly was not true of Nature or the Philosophi al Magazine. b We do not always keep to de Broglie's original notation.
40
De Broglie's pilot-wave theory
parti le is equal to the phase of the internal os illation of the parti le. De Broglie then goes on to make some very signi ant observations about dira tion and the nature of the new dynami s that he is proposing. De Broglie asserts that dira tion phenomena prove that light quanta
annot always propagate in a straight line, even in what would normally be alled empty spa e. He draws the bold on lusion that Newton's rst law of motion (the `prin iple of inertia') must be abandoned (p. 549): The light quanta [atomes de lumière℄ whose existen e we assume do not always propagate in a straight line, as proved by the phenomena of dira tion. It then seems
ne essary
to modify the prin iple of inertia.
De Broglie then suggests repla ing Newton's rst law with a new postulate (p. 549): We propose to adopt the following postulate as the basis of the dynami s of the free material point: `At ea h point of its traje tory, a free moving body follows in a uniform motion the
ray
of its phase wave, that is (in an isotropi
medium), the normal to the surfa es of equal phase'.
The dira tion of light quanta is then explained sin e, as de Broglie notes, `if the moving body must pass through an opening whose dimensions are small ompared to the wavelength of the phase wave, in general its traje tory will urve like the ray of the dira ted wave'. In retrospe t, de Broglie's postulate for free parti les may be seen as a simpli ed form of the law of motion of what we now know as pilot-wave dynami s ex ept for the statement that the motion along a ray be `uniform' (that is, have onstant speed), whi h in pilot-wave theory is true only in spe ial ases.a De Broglie notes that his postulate respe ts
onservation of energy but not of momentum. And indeed, in pilot-wave theory the momentum of a `free' parti le is generally not onserved: in ee t (from the standpoint of Bohm's Newtonian formulation), the pilot wave or quantum potential a ts like an `external sour e' of momentum (and in general of energy too).b The abandonment of something as a From Bohm's se ond-order equation (3) applied to a single parti le, for timeindependent V it follows that d( 12 mv2 + V + Q)/dt = ∂Q/∂t (the usual energy
onservation formula with a time-dependent ontribution Q to the potential). In free spa e (V = 0), the speed v is onstant if and only if dQ/dt = ∂Q/∂t or v · ∇Q = 0 (so that the `quantum for e' does no work), whi h is true only in spe ial ases. b Again from (3), in free spa e the rate of hange of momentum p = mv (where v = ∇S/m) is dp/dt = −∇Q, whi h is generally non-zero. Further, in general (3) implies d( 21 mv2 + V )/dt = −v · ∇Q, so that the standard ( lassi al) expression for energy is onserved if and only if v · ∇Q = 0. If, on the other hand, one de nes 1 mv2 + V + Q as the `energy', it will be onserved if and only if ∂Q/∂t = 0, whi h 2
2.2 A new approa h to parti le dynami s: 192324
41
elementary as momentum onservation is ertainly a radi al step by any standards. On the other hand, if one is willing as de Broglie was to propose a fundamentally new approa h to the theory of motion, then the loss of lassi al onservation laws is not surprising, as these are really properties of lassi al equations of motion. De Broglie then makes a remarkable predi tion, that any moving body (not just light quanta) an undergo dira tion: .... any moving body ould in ertain ases be dira ted. A stream of ele trons passing through a small enough opening will show dira tion phenomena. It is in this dire tion that one should perhaps look for experimental on rmation of our ideas.
Next, de Broglie puts his proposals in a general on eptual and histori al perspe tive. Con erning the role of the phase wave, he writes (p. 549): We therefore on eive of the phase wave as guiding the movements of energy, and this is what an allow the synthesis of waves and quanta.
Here, for the rst time, de Broglie hara terises the phase wave as a `guiding' wave. De Broglie then remarks that, histori ally speaking, the theory of waves `went too far' by denying the dis ontinuous stru ture of radiation and `not far enough' by not playing a role in dynami s. For de Broglie, his proposal has a lear histori al signi an e (p. 549, itali s in the original): The new dynami s of the free material point is to the old dynami s (in luding that of Einstein) what wave opti s is to geometri al opti s. Upon re e tion one will see that the proposed synthesis appears as the logi al ulmination of the omparative development of dynami s and of opti s sin e the seventeenth
entury.
In the se ond part of this note, de Broglie onsiders the explanation of opti al interferen e fringes. He assumes that the probability for an atom to absorb or emit a light quantum is determined by `the resultant of one of the ve tors of the phase waves rossing ea h other there [se roisant sur lui℄' (pp. 54950). In Young's interferen e experiment, the light quanta passing through the two holes are dira ted, and the probability of them being dete ted behind the s reen will vary from point to point, depending on the `state of interferen e' of the phase waves. De Broglie is true only for spe ial ases (in parti ular for stationary states, sin e for these |Ψ| is time-independent).
42
De Broglie's pilot-wave theory
on ludes that there will be bright and dark fringes as predi ted by the wave theories, no matter how feeble the in ident light. This approa h to opti al interferen e in whi h interfering phase waves determine the probability for intera tion between photons and the atoms in the dete tion apparatus is elaborated in de Broglie's thesis (see below). Soon after ompleting his thesis (apparently), de Broglie abandoned this idea in favour of a simpler approa h, in whi h the interfering phase waves determine the number density of photon traje tories (see se tion 2.2.3). In de Broglie's third ommuni ation (de Broglie 1923 ), entitled `Quanta, the kineti theory of gases and Fermat's prin iple', part 1 onsiders the statisti al treatment of a gas of parti les a
ompanied by phase waves. De Broglie makes the following assumption: The state of the gas will then be stable only if the waves orresponding to all of the atoms form a system of stationary waves.
In other words, de Broglie onsiders the stationary modes, or standing waves, asso iated with a given spatial volume. He assumes that ea h mode ` an transport zero, one, two or several atoms', with probabilities determined by the Boltzmann fa tor.a A
ording to de Broglie, for a gas of nonrelativisti atoms his method yields the Maxwell distribution, while for a gas of photons it yields the Plan k distribution.b In part 2 of the same note, de Broglie shows how his new dynami al postulate amounts to a uni ation of Maupertuis' prin iple of least a tion with Fermat's prin iple of least time in opti s. Let us re all that, in the me hani al prin iple of Maupertuis for parti le traje tories, Z b δ mv · dx = 0 , (6) a
the ondition of stationarity determines the parti le paths. (In (6) the energy is xed on the varied paths; at the end points, ∆x = 0 but ∆t need not be zero.) While in the opti al prin iple of Fermat for light rays, Z b δ dφ = 0 , (7) a
the stationary line integral for the phase hange the stationary `opti al a As remarked by Pais (1982, pp. 4356), in this paper de Broglie `evaluated independently of Bose (and published before him) the density of radiation states in terms of parti le (photon) language'. b As shown by Darrigol (1993), de Broglie made some errors in his appli ation of the methods of statisti al me hani s.
2.2 A new approa h to parti le dynami s: 192324
43
path length' provides a ondition that determines the path of a ray
onne ting two points, in spa e (for the time-independent ase) or in spa etime. Now, a
ording to de Broglie's basi postulate: `The rays of the phase waves oin ide with the dynami ally possible traje tories' (p. 632). The rays are des ribed by Fermat's prin iple (for the ase of a dispersive medium), whi h de Broglie shows oin ides with Maupertuis' prin iple, as follows: writing the element of phase hange as dφ = 2πνdl/vph, where dl is an element of path and vph = c2 /v is the phase velo ity, and using the relation ν = E/h = mc2 /h, the element of phase
hange may be rewritten as (2π/h)mvdl, so that (7) oin ides with (6). As de Broglie puts it (p. 632): In this way the fundamental link that unites the two great prin iples of geometri al opti s and of dynami s is brought fully to light.
De Broglie remarks that some of the dynami ally possible traje tories will be `in resonan e with the phase wave', and that these orrespond to R Bohr's stable orbits, for whi h νdl/vph is a whole number. Soon afterwards, de Broglie introdu es a ovariant 4-ve tor formulation of his basi dynami al postulate (de Broglie 1924a,b). He de nes n), where n ˆ is a unit ve tor in the a 4-ve tor wµ = (ν/c, −(ν/vph )ˆ dire tion of a ray of the phase wave, and assumes it to be related to the energy-momentum 4-ve tor pµ = (E/c, −p) by pµ = hwµ . De Broglie notes that the identity of the prin iples of Maupertuis and Fermat then follows immediately. We shall dis uss this in more detail in the next subse tion.
2.2.2 Thesis (1924) He has lifted a orner of the great veil. Einstein, ommenting on de Broglie's thesis
a
De Broglie's do toral thesis (de Broglie 1924e) was mostly based on the above papers. It seems to have been ompleted in the summer of 1924, and was defended at the Sorbonne in November. The thesis was published early in 1925 in the Annales de Physique (de Broglie 1925).b a Letter to Langevin, 16 De ember 1924 (quoted in Darrigol 1993, p. 355). b An English translation of extra ts from de Broglie's thesis appears in Ludwig (1968). A omplete translation has been done by A. F. Kra klauer ( urrently online at http://www.ensmp.fr/a b/LDB-oeuvres/De_Broglie_Kra klauer.htm ). All translations here are ours.
44
De Broglie's pilot-wave theory
When writing his thesis, de Broglie was well aware that his theory had gaps. As he put it (p. 30):a .... the main aim of the present thesis is to present a more omplete a
ount of the new ideas that we have proposed, of the su
esses to whi h they have led, and also of the many gaps they ontain.
De Broglie begins his thesis with a histori al introdu tion. Newtonian me hani s, he notes, was eventually formulated in terms of the prin iple of least a tion, whi h was rst given by Maupertuis and then later in another form by Hamilton. As for the s ien e of light and opti s, the laws of geometri al opti s were eventually summarised by Fermat in terms of a prin iple whose form is reminis ent of the prin iple of least a tion. Newton tried to explain some of the phenomena of wave opti s in terms of his orpus ular theory, but the work of Young and Fresnel led to the rise of the wave theory of light, in parti ular the su
essful wave explanation of the re tilinear propagation of light (whi h had been so lear in the orpus ular or `emission' theory). On this, de Broglie
omments (p. 25): When two theories, based on ideas that seem entirely dierent, a
ount for the same experimental fa t with equal elegan e, one an always wonder if the opposition between the two points of view is truly real and is not due solely to an inadequa y of our eorts at synthesis.
This remark is, of ourse, a hint that the aim of the thesis is to ee t just su h a synthesis. De Broglie then turns to the rise of ele trodynami s, relativity, and the theory of energy quanta. He notes that Einstein's theory of the photoele tri ee t amounts to a revival of Newton's orpus ular theory. De Broglie then sket hes Bohr's 1913 theory of the atom, and goes on to point out that observations of the photoele tri ee t for X- and γ -rays seem to on rm the orpus ular hara ter of radiation. At the same time, the wave aspe t ontinues to be on rmed by the observed interferen e and dira tion of X-rays. Finally, de Broglie notes the very re ent orpus ular interpretation of Compton s attering. De Broglie on ludes his histori al introdu tion with a mention of his own re ent work (p. 30): ....
the moment seemed to have arrived to make an eort towards unifying
the orpus ular and wave points of view and to go a bit more deeply into the true meaning of the quanta. That is what we have done re ently ....
a Here and below, page referen es for de Broglie's thesis orrespond to the published version in Annales de Physique (de Broglie 1925).
2.2 A new approa h to parti le dynami s: 192324
45
De Broglie learly regarded his own work as a synthesis of earlier theories of dynami s and opti s, a synthesis in reasingly for ed upon us by a
umulating experimental eviden e. Chapter 1 of the thesis is entitled `The phase wave'. De Broglie begins by re alling the equivalen e of mass and energy implied by the theory of relativity. Turning to the problem of quanta, he remarks (pp. 323): It seems to us that the fundamental idea of the quantum theory is the impossibility of onsidering an isolated quantity of energy without asso iating a
ertain frequen y with it. This onne tion is expressed by what I shall all the quantum relation:
energy = h × frequency where
h
is Plan k's onstant.
To make sense of the quantum relation, de Broglie proposes that (p. 33) .... to ea h energy fragment of proper mass phenomenon of frequen y
ν0
m0
there is atta hed a periodi
su h that one has:
hν0 = m0 c2 ν0
being measured, of ourse, in the system tied to the energy fragment.
De Broglie asks if the periodi phenomenon must be assumed to be lo alised inside the energy fragment. He asserts that this is not at all ne essary, and that it will be seen to be `without doubt spread over an extensive region of spa e' (p. 34). De Broglie goes on to onsider p the apparent ontradi tion between the 2 ν = mc /h = ν / 1 − v 2 /c2 and the time-dilated frequen y frequen y 0 p 2 2 ν1 = ν0 1 − v /c . He proposes that the ontradi tion is resolved by the following `theorem of phase harmony' (p. 35): in a frame where the moving body has velo ity v , the periodi phenomenon tied to the moving body and with frequen y ν1 is always in phase with a wave of frequen y ν propagating in the same dire tion as the moving body with phase velo ity vph = c2 /v . This is shown by applying the Lorentz transformation to a rest-frame wave sin (ν0 t0 ), yielding a wave h i p sin ν0 t − vx/c2 / 1 − v 2 /c2 (8) p of frequen y ν = ν0 / 1 − v 2 /c2 and phase velo ity c2 /v . Regarding the nature of this wave de Broglie says that, be ause its velo ity is greater than c, it annot be a wave transporting energy: rather, `it represents the spatial distribution of the phases of a phenomenon; it is a phase wave ' (p. 36). De Broglie shows that the group velo ity of the phase
46
De Broglie's pilot-wave theory
wave is equal to the velo ity of the parti le. In the nal se tion of
hapter 1 (`The phase wave in spa etime'), he dis usses the appearan e of surfa es of onstant phase for dierently moving observers, from a spa etime perspe tive. Chapter 2 is entitled `Maupertuis' prin iple and Fermat's prin iple'. The aim is to generalise the results of the rst hapter to non-uniform, non-re tilinear motion. In the introdu tion to hapter 2 de Broglie writes (p. 45): Guided by the idea of a deep unity between the prin iple of least a tion and that of Fermat, from the beginning of my investigations on this subje t I was led to
assume
that, for a given value of the total energy of the moving body
and therefore of the frequen y of its phase wave, the dynami ally possible traje tories of the one oin ided with the possible rays of the other.
De Broglie dis usses the prin iple of least a tion, in the dierent forms given by Hamilton and by Maupertuis, and also for relativisti parti les in an external ele tromagneti eld. He writes Hamilton's prin iple as Z Q pµ dxµ = 0 δ (9) P
(µ = 0, 1, 2, 3, with dx0 = cdt), where P , Q are points in spa etime and pµ is the anoni al energy-momentum 4-ve tor, and notes that if p0 is
onstant the prin iple be omes Z B δ pi dxi = 0 (10) A
(i = 1, 2, 3), where A, B are the orresponding points in spa e that is, Hamilton's prin iple redu es to Maupertuis' prin iple. De Broglie then dis usses wave propagation and Fermat's prin iple from a spa etime perspe tive. He onsiders a sinusoidal fun tion sin φ, where the phase φ has a spa etime-dependent dierential dφ, and writes the variational prin iple for the ray in spa etime in the Hamiltonian form Z Q δ (11) dφ = 0 . P
De Broglie then introdu es a 4-ve tor eld wµ on spa etime, de ned by dφ = 2πwµ dxµ ,
(12)
where the wµ are generally fun tions on spa etime. (Of ourse, this implies that 2πwµ = ∂µ φ, though de Broglie does not write this expli itly.) De Broglie also notes that dφ = 2π(νdt − (ν/vph )dl) and
2.2 A new approa h to parti le dynami s: 192324
47
wµ = (ν/c, −(ν/vph )ˆ n), where n ˆ is a unit ve tor in the dire tion of propagation; and that if ν is onstant, the prin iple in the Hamiltonian form Z Q δ wµ dxµ = 0 (13) P
redu es to the prin iple in the Maupertuisian form Z B wi dxi = 0 , δ
(14)
A
or
δ
Z
B A
ν dl = 0 , vph
(15)
whi h is Fermat's prin iple. De Broglie then dis usses an `extension of the quantum relation' (that is, an extension of E = hν ). He states that the two 4-ve tors pµ and wµ play perfe tly symmetri al roles in the motion of a parti le and in the propagation of a wave. Writing the `quantum relation' E = hν as w0 = h1 p0 , de Broglie proposes the generalisation wµ =
1 pµ , h
(16)
so that dφ = 2πwµ dxµ =
2π pµ dxµ . h
Fermat's prin iple then be omes Z B pi dxi = 0 , δ
(17)
(18)
A
whi h is the same as Maupertuis' prin iple. Thus, de Broglie arrives at the following statement (p. 56): Fermat's prin iple applied to the phase wave is identi al to Maupertuis' prin iple applied to the moving body; the dynami ally possible traje tories of the moving body are identi al to the possible rays of the wave.
He adds that (p. 56): We think that this idea of a deep relationship between the two great prin iples of Geometri al Opti s and Dynami s ould be a valuable guide in realising the synthesis of waves and quanta.
48
De Broglie's pilot-wave theory
De Broglie then dis usses some parti ular ases: the free parti le, a parti le in an ele trostati eld, and a parti le in a general ele tromagneti eld. He al ulates the phase velo ity, whi h depends on the ele tromagneti potentials. He notes that the propagation of a phase wave in an external eld depends on the harge and mass of the moving body. And he shows that the group velo ity along a ray is still equal to the velo ity of the moving body. For the ase of an ele tron of harge e and velo ity v in an ele trostati potential ϕ, de Broglie writes down the following expressions for the frequen y ν and phase velo ity vph of the phase wave (p. 57): ν = (mc2 + eϕ)/h , vph = (mc2 + eϕ)/mv (19) p (where again m = m0 / 1 − v 2 /c2 ). He shows that vph may be rewritten as the free value c2 /v multiplied by a fa tor hν/(hν − eϕ) that depends on the potential ϕ. The expressions (19) formed the starting point for S hrödinger's work on the wave equation for de Broglie's phase waves (as re onstru ted by Mehra and Re henberg (1987, pp. 4235), see se tion 2.3). While de Broglie does not expli itly say so in his thesis, note that from the de nition (12) of wµ , the generalised quantum relation (16) may be written in the form pµ = ℏ∂µ φ .
(20)
This is what we would now all a relativisti guidan e equation, giving the velo ity of a parti le in terms of the gradient of the phase of a pilot wave (where here de Broglie de nes the phase φ to be dimensionless). In other words, the extended quantum relation is a rst-order equation of motion. In the presen e of an ele tromagneti eld, the anoni al momentum pµ ontains the 4-ve tor potential. For a free parti le, with pµ = (E/c, −p), the guidan e equation has omponents ˙ p = −ℏ∇φ , E = ℏφ,
where the spatial omponents may also be written as m0 v mv = p = −ℏ∇φ . 1 − v 2 /c2
(21)
(22)
For a plane wave of phase φ = ωt − k · x, we have E = ℏω, p = ℏk .
(23)
Thus, de Broglie's uni ation of the prin iples of Maupertuis and
2.2 A new approa h to parti le dynami s: 192324
49
Fermat amounts to a new dynami al law, (16) or (20), in whi h the phase of a guiding wave determines the parti le velo ity. This new law of motion is the essen e of de Broglie's new, rst-order dynami s. Chapter 3 of de Broglie's thesis is entitled `The quantum onditions for the stability of orbits'. De Broglie reviews Bohr's ondition for ir ular orbits, a
ording to whi h the angular R 2πmomentum of the ele tron must be a multiple of ℏ, or equivalently 0 pθ dθ = Hnh (pθ onjugate to θ). He also reviews Sommerfeld's generalisation, H P3 pi dqi = ni h (integral ni ) and Einstein's invariant formulation i=1 pi dqi = nh (integral n). De Broglie then provides an explanation for Einstein's ondition. The traje tory of the moving body oin ides with one of the rays of its phase wave, and the phase wave moves along the traje tory with a onstant frequen y (be ause the total energy is onstant) and with a variable speed whose value has been al ulated. To have a stable orbit, laims de Broglie, the length l of the orbit must be in `resonan e' with the wave: thus l = nλ in the ase of onstant wavelength, and H (ν/vph )dl = n (n integral) generally. De Broglie notes that this is pre isely the integral appearing in Fermat's prin iple, whi h has been shown to be equal to the integral giving the Maupertuisian a tion divided by h. The resonan e ondition is then identi al to the required stability
ondition. For the simple ase of ir ular orbits in the Bohr atom de Broglie shows, using H vph = νλ and h/λ = m0 v , that the resonan e
ondition be omes m0 vdl = nh or m0 ωR2 = nℏ (with v = ωR), as originally given by Bohr.a (Note that the simple argument ommonly found in textbooks, about the tting of whole numbers of wavelengths along a Bohr orbit, originates in this work of de Broglie's.) De Broglie thought that his explanation of the stability or quantisation
onditions onstituted important eviden e for his ideas. As he puts it (p. 65): This beautiful result, whose demonstration is so immediate when one has a
epted the ideas of the pre eding hapter, is the best justi ation we an give for our way of atta king the problem of quanta.
Certainly, de Broglie had a hieved a on rete realisation of his initial intuition that quantisation onditions for atomi energy levels ould arise from the properties of waves. In his hapter 4, de Broglie onsiders the two-body problem, in para De Broglie also laims to generalise his results from losed orbits to quasi-periodi (or multi-periodi ) motion: however, as shown by Darrigol (1993), de Broglie's derivation is faulty.
50
De Broglie's pilot-wave theory
ti ular the hydrogen atom. He expresses on ern over how to de ne the proper masses, taking into a
ount the intera tion energy. He dis usses the quantisation onditions for hydrogen from a two-body point of view: he has two phase waves, one for the ele tron and one for the nu leus. The subje t of hapter 5 is light quanta. De Broglie suggests that the
lassi al (ele tromagneti ) wave distribution in spa e is some sort of time average over the true distribution of phase waves. His light quantum is assigned a very small proper mass: the velo ity v of the quantum, and the phase velo ity c2 /v of the a
ompanying phase wave, are then both very lose to c. De Broglie points out that radiation is sometimes observed to violate re tilinear propagation: a light wave striking the edge of a s reen dira ts into the geometri al shadow, and rays passing lose to the s reen deviate from a straight line. De Broglie notes the two histori al explanations for this phenomenon on the one hand the explanation for dira tion given by the wave theory, and on the other the explanation given by Newton in his emission theory: `Newton assumed [the existen e of℄ a for e exerted by the edge of the s reen on the orpus le' (p. 80). De Broglie asserts that he an now give a uni ed explanation for dira tion, by abandoning Newton's rst law of motion (p. 80): .... the ray of the wave would urve as predi ted by the theory of waves, and the moving body, for whi h the prin iple of inertia would no longer be valid, would suer the same deviation as the ray with whi h its motion is bound up [solidaire℄ ....
De Broglie's words here deserve emphasis. As is also very lear in his se ond paper of the pre eding year (see se tion 2.2.1), de Broglie regards his explanation of parti le dira tion as based on a new form of dynami s in whi h Newton's rst law the prin iple that a free body will always move uniformly in a straight line is abandoned. At the same time, de Broglie re ognises that one an always adopt a lassi al-me hani al viewpoint if one wishes (pp. 8081): .... perhaps one ould say that the wall exerts a for e on it [the moving body℄ if one takes the urvature of the traje tory as a riterion for the existen e of a for e.
Here, de Broglie re ognises that one may still think in Newtonian terms, if one ontinues to identify a
eleration as indi ative of the presen e of a for e. Similarly, as we shall see, in 1927 de Broglie notes that his pilotwave dynami s may if one wishes be written in Newtonian form with a quantum potential. But de Broglie's preferred approa h, throughout his
2.2 A new approa h to parti le dynami s: 192324
51
work in the period 192327, is to abandon Newton's rst law and base his dynami s on velo ity rather than on a
eleration. After onsidering the Doppler ee t, re e tion by a moving mirror, and radiation pressure, all from a photon viewpoint, de Broglie turns to the phenomena of wave opti s, noting that (p. 86): The stumbling blo k of the theory of light quanta is the explanation of the phenomena that onstitute wave opti s.
Here it be omes apparent that, despite his understanding of how nonre tilinear parti le traje tories arise during dira tion and interferen e, de Broglie is not sure of the details of how to explain the observed bright and dark fringes in dira tion and interferen e experiments with light. In parti ular, de Broglie did not have a pre ise theory of the assumed statisti al relationship between his phase waves and the ele tromagneti eld. Even so, he went on to make what he alled `vague suggestions' (p. 87) towards a detailed theory of opti al interferen e. De Broglie's idea was that the phase waves would determine the probability for the light quanta to intera t with the atoms onstituting the equipment used to observe the radiation, in su h a way as to a
ount for the observed fringes (p. 88): .... the probability of rea tions between atoms of matter and atoms of light is at ea h point tied to the resultant (or rather to the mean value of this) of one of the ve tors hara terising the phase wave; where this resultant vanishes the light is undete table; there is interferen e. One then on eives that an atom of light traversing a region where the phase waves interfere will be able to be absorbed by matter at ertain points and not at others. This is the still very qualitative prin iple of an explanation of interferen e .... .
As we shall see in the next se tion, after ompleting his thesis de Broglie arrived at a simpler explanation of opti al interferen e fringes. The nal se tion of hapter 5 onsiders the explanation of Bohr's frequen y ondition hν = E1 − E2 for the light emitted by an atomi transition from energy state E1 to energy state E2 . De Broglie derives this from the assumption that ea h transition involves the emission of a single light quantum of energy E = hν (together with the assumption of energy onservation). De Broglie's hapter 6 dis usses the s attering of X- and γ -rays. In his hapter 7, de Broglie turns to statisti al me hani s, and shows how the on ept of statisti al equilibrium is to be modi ed in the presen e of phase waves. If ea h parti le or atom in a gas is a
ompanied by a phase wave, then a box of gas will be ` riss- rossed in all dire tions'
52
De Broglie's pilot-wave theory
(p. 110) by the waves. De Broglie nds it natural to assume that the only stable phase waves in the box will be those that form stationary or standing waves, and that only these will be relevant to thermodynami equilibrium. He illustrates his idea with a simple example of mole ules moving in one dimension, on ned to an interval of length l. In the nonrelativisti limit, ea h phase wave has a wavelength λ = h/m0 v and the `resonan e ondition' is l = nλ with n integral. Writing v0 = h/m0 l, one then has v = nv0 . As de Broglie notes (p. 112): `The speed will then be able to take only values equal to integer multiples of v0 '. (This is, of
ourse, the well-known quantisation of momentum for parti les on ned to a box.) De Broglie then argues that a velo ity element δv orresponds to a number δn = (m0 l/h)δv of states of a mole ule ( ompatible with the existen e of stationary phase waves), so that an element m0 δxδv of phase spa e volume orresponds to a number m0 δxδv/h of possible states. Generalising to three dimensions, de Broglie is led to take the element of phase spa e volume divided by h3 as the measure of the number of possible states of a mole ule, as assumed by Plan k. De Broglie then turns to the photon gas, for whi h he obtains Wien's law. He laims that, in order to get the Plan k law, the following further hypothesis is required (p. 116): If two or several atoms [of light℄ have phase waves that are exa tly superposed, of whi h one an therefore say that they are transported by the same wave, their motions an no longer be onsidered as entirely independent and these atoms an no longer be treated as separate units in al ulating the probabilities.
In de Broglie's approa h, the stationary phase waves play the role of the elementary obje ts of statisti al me hani s. De Broglie de nes stationary waves as a superposition of two waves of the form i i x x sin h sin h and , (24) 2π νt + + φ0 2π νt − + φ0 cos cos λ λ
where φ0 an take any value from 0 to 1 and ν takes one of the allowed values. Ea h elementary wave an arry any number 0, 1, 2, ... of atoms, and the probability of arrying n atoms is given by the Boltzmann fa tor e−nhν/kT . Applying this method to a gas of light quanta, de Broglie
laims to derive the Plan k distribution.a De Broglie's thesis ends with a summary and on lusions (pp. 1258). The seeds of the problem of quanta have been shown, he laims, to be a Again, as shown by Darrigol (1993), de Broglie's appli ation of statisti al me hani s ontains some errors.
2.2 A new approa h to parti le dynami s: 192324
53
ontained in the histori al `parallelism of the orpus ular and wave-like
on eptions of radiation'. He has postulated a periodi phenomenon asso iated with ea h energy fragment, and shown how relativity requires us to asso iate a phase wave with every uniformly moving body. For the ase of non-uniform motion, Maupertuis' prin iple and Fermat's prin iple ` ould well be two aspe ts of a single law', and this new approa h to dynami s led to an extension of the quantum relation, giving the speed of a phase wave in an ele tromagneti eld. The most important onsequen e is the interpretation of the quantum onditions for atomi orbits in terms of a resonan e of the phase wave along the traje tories: `this is the rst physi ally plausible explanation proposed for the Bohr-Sommerfeld stability onditions'. A `qualitative theory of interferen e' has been suggested. The phase wave has been introdu ed into statisti al me hani s, yielding a derivation of Plan k's phase volume element, and of the bla kbody spe trum. De Broglie has, he laims, perhaps ontributed to a uni ation of the opposing on eptions of waves and parti les, in whi h the dynami s of the material point is understood in terms of wave propagation. He adds that the ideas need further development: rst of all, a new ele tromagneti theory is required, that takes into a
ount the dis ontinuous stru ture of radiation and the physi al nature of phase waves, with Maxwell's theory emerging as a statisti al approximation. The nal paragraph of de Broglie's thesis emphasises the in ompleteness of his theory at the time: I have deliberately left rather vague the de nition of the phase wave, and of the periodi phenomenon of whi h it must in some sense be the translation, as well as that of the light quantum. The present theory should therefore be
onsidered as one whose physi al ontent is not entirely spe i ed, rather than as a onsistent and de nitively onstituted do trine.
As de Broglie's on luding paragraph makes lear, his theory of 1924 was rather abstra t. There was no spe i ed basis for the phase waves (they were ertainly not regarded as `material' waves); nor was any parti ular wave equation suggested. It should also be noted that at this time de Broglie's waves were real-valued fun tions of spa e and time, of the form ∝ sin(ωt − k · x), with a real os illating amplitude. They were not omplex waves ∝ ei(k·x−ωt) of uniform amplitude. Thus, de Broglie's `phase waves' had an os illating amplitude as well as a phase. (De Broglie seems to have alled them `phase waves' only be ause of his theorem of phase harmony.) Note also that, in his treatment of parti les
54
De Broglie's pilot-wave theory
in a box, de Broglie superposes waves propagating in opposite dire tions, yielding stationary waves whose amplitudes os illate in time. In his thesis de Broglie does not expli itly dis uss dira tion or interferen e experiments with ele trons, even though in his se ond ommuni ation of 1923 (de Broglie 1923b) he had suggested ele tron dira tion as an experimental test. A
ording to de Broglie's later re olle tions (L. de Broglie, AHQP Interview, 7 January 1963, p. 6),3 at his thesis defen e on 25 November 1924: Mr Jean Perrin, who haired the ommittee, asked me if my ideas ould lead to experimental on rmation. I replied that yes they ould, and I mentioned the dira tion of ele trons by rystals. Soon afterwards, I advised Mr Dauvillier .... to try the experiment, but, absorbed by other resear h, he did not do it. I do not know if he believed, or if he said to himself that it was perhaps very un ertain, that he was going to go to a lot of trouble for nothing it's possible. .... But the following year it was dis overed in Ameri a by Davisson and Germer.
2.2.3 Opti al interferen e fringes: November 1924 On 17 November 1924, just a few days before de Broglie defended his thesis, a further ommuni ation of de Broglie's was presented to the A ademy of S ien es: entitled `On the dynami s of the light quantum and interferen e', and published in the Comptes Rendus, this short note gave a new and improved a
ount of opti al interferen e in terms of light quanta (de Broglie 1924d). De Broglie began his note by re alling his unsatisfa tory dis ussion of opti al interferen e in his re ent work on the quantum theory (p. 1039): .... I had not rea hed a truly satisfying explanation for the phenomena of wave opti s whi h, in prin iple, all ome down to interferen e. I limited myself to putting forward a ertain onne tion between the state of interferen e of the waves and the probability for the absorption of light quanta by matter. This viewpoint now seems to me a bit arti ial and I tend towards adopting another, more in harmony with the broad outlines of my theory itself.
As we have seen, in his thesis de Broglie was unsure about how to a
ount for the bright and dark fringes observed in opti al interferen e experiments. He did not have a theory of the ele tromagneti eld, whi h he assumed emerged as some sort of average over his phase waves. To a
ount for opti al fringes, he had suggested that the phase waves somehow determined the probability for intera tions between photons and the atoms in the apparatus. Now, after ompleting his thesis, he
2.2 A new approa h to parti le dynami s: 192324
55
felt he had a better explanation, that was based purely on the spatial distribution of the photon traje tories. De Broglie's note ontinues by outlining his `new dynami s', in whi h the energy-momentum 4-ve tor of every material point is proportional to the ` hara teristi ' 4-ve tor of an asso iated wave, even when the wave undergoes interferen e or dira tion. He then gives his new view of interferen e fringes (p. 1040): The rays predi ted by the wave theories would then be in every ase the possible traje tories of the quantum. In the phenomena of interferen e, the rays be ome on entrated in those regions alled `bright fringes' and be ome diluted in those regions alled `dark fringes'.
In my rst explanation of
interferen e, the dark fringes were dark be ause the a tion of fragments of light on matter was zero there; in my urrent explanation, these fringes are dark be ause the number of quanta passing through them is small or zero.
Here, then, de Broglie explains bright and dark fringes simply in terms of a high or low density of photon traje tories in the orresponding regions. When de Broglie speaks of the traje tories being on entrated and diluted in regions of bright and dark fringes respe tively, he presumably had in mind that the number density of parti les in an interferen e zone should be proportional to the lassi al wave intensity, though he does not say this expli itly. We an dis ern the essen e of the more pre ise and omplete explanation of opti al interferen e given by de Broglie three years later in his Solvay report: there, de Broglie has the same photon traje tories, with a number density spe i ed as proportional to the amplitude-squared of the guiding wave (see pp. 384 f., and our dis ussion in se tion 6.1.1). In his note of November 1924, de Broglie goes on to illustrate his proposal for the ase of Young's interferen e experiment with two pinholes a ting as point sour es. De Broglie ites the well-known fa ts that in this ase the surfa es of equal phase are ellipsoids with the pinholes as fo i, and that the rays (whi h are normal to the ellipsoidal surfa es) are on entrated on hyperboloids of onstru tive interferen e where the
lassi al intensity has maxima. He then notes (p. 1040): Let let
r1 and r2 be the distan es from a point in spa e ψ be the fun tion 12 (r1 + r2 ), whi h is onstant on
to the two holes and ea h surfa e of equal
phase. One easily shows that the phase velo ity of the waves along a ray is equal to the value it would have in the ase of free propagation divided by the derivative of
ψ
taken along the ray; as for the speed of the quantum, it will
be equal to the speed of free motion multiplied by the same derivative.
De Broglie gives no further details, but these are easily re onstru ted.
56
De Broglie's pilot-wave theory
For an in ident beam of wavelength λ = 2π/k , ea h pinhole a ts as a sour e of a spheri al wave of wavelength λ, yielding a resultant wave proportional to the real part of ei(kr1 −ωt) ei(kr2 −ωt) + . r1 r2
(25)
If the pinholes have a separation d, then at large distan es (r1 , r2 >> d) from the s reen the resultant wave may be approximated as ei(kr−ωt) kd 2 (26) cos θ r 2 where θ is the (small) angular deviation from the normal to the s reen. The amplitude shows the well-known interferen e pattern. As for the phase φ = kr − ωt, with r = 12 (r1 + r2 ), the surfa es of equal phase are indeed the well-known ellipsoids. Further, the phase velo ity is given by vph =
|∂φ/∂t| ω 1 c = = , |∇φ| k |∇r| |∇r|
(27)
while de Broglie's parti le velo ity given by (22) has magnitude v=
ℏ |∇φ| k ℏ |∇φ| = = c2 |∇r| = c |∇r| 2 m (ℏω/c ) ω
(28)
(where m is the relativisti photon mass), in agreement with de Broglie's assertions. At the end of his note, de Broglie omments that this method may be applied to the study of s attering. In November 1924, then, de Broglie understood how interfering phase waves would ae t photon traje tories, ausing them to bun h together in regions oin iding with the observed bright fringes. Note that in this paper de Broglie treats his phase waves as if they were a dire t representation of the ele tromagneti eld. In his dis ussion of Young's interferen e experiment, he has phase waves emerging from the two holes and interfering, and he identi es the interferen e fringes of his phase waves with opti al interferen e fringes. However, he seems quite aware that this is a simpli ation,a remarking that `the whole theory will be ome truly lear only if one manages to de ne the stru ture of the light wave'. De Broglie is still not sure about the pre ise relationship between his phase waves and light waves, a situation that persists even until the a De Broglie may have thought of this as analogous to s alar wave opti s, whi h predi ts the orre t opti al interferen e fringes by treating the ele tromagneti eld simply as a s alar wave.
2.3 Towards a omplete pilot-wave dynami s: 192527
57
fth Solvay onferen e: there, while he gives (pp. 384 f.) a pre ise a
ount of opti al interferen e in his report, he points out (p. 509) that the onne tion between his guiding wave and the ele tromagneti eld is still unknown.
2.3 Towards a omplete pilot-wave dynami s: 192527 On 16 De ember 1924, Einstein wrote to Lorentz:a A younger brother of the de Broglie known to us [Mauri e de Broglie℄ has made a very interesting attempt to interpret the Bohr-Sommerfeld quantization rules (Paris Dissertation, 1924).
I believe that it is the rst feeble ray of light
to illuminate this, the worst of our physi al riddles.
I have also dis overed
something that supports his onstru tion.
What Einstein had dis overed, in support of de Broglie's ideas, appeared in the se ond of his famous papers on the quantum theory of the ideal gas (Einstein 1925a). Einstein showed that the u tuations asso iated with the new Bose-Einstein statisti s ontained two distin t terms that
ould be interpreted as parti le-like and wave-like ontributions just as Einstein had shown, many years earlier, for bla kbody radiation. Einstein argued that the wave-like ontribution should be interpreted in terms of de Broglie's matter waves, and he ited de Broglie's thesis. It was largely through this paper by Einstein that de Broglie's work be ame known outside Fran e. In the same paper, Einstein suggested that a mole ular beam would undergo dira tion through a su iently small aperture. De Broglie had already made a similar suggestion for ele trons, in his se ond ommuni ation to the Comptes Rendus (de Broglie 1923b). Even so, in their report at the fth Solvay onferen e, Born and Heisenberg state that in his gas theory paper Einstein `dedu ed from de Broglie's daring theory the possibility of dira tion of material parti les' (p. 426), giving the in orre t impression that Einstein had been the rst to see this onsequen e of de Broglie's theory. It seems likely that Born and Heisenberg did not noti e de Broglie's early papers in the Comptes Rendus. In 1925 Elsasser a student of Born's in Göttingen read de Broglie's thesis. Like most others outside Fran e, Elsasser had heard about de Broglie's thesis through Einstein's gas theory papers. Elsasser suspe ted that two observed experimental anomalies ould be explained a Quoted in Mehra and Re henberg (1982a, p. 604).
58
De Broglie's pilot-wave theory
by de Broglie's new dynami s. First, the Ramsauer ee t the surprisingly large mean free path of low-velo ity ele trons in gases whi h Elsasser thought ould be explained by ele tron interferen e. Se ond, the intensity maxima observed by Davisson and Kunsman at ertain angles of re e tion of ele trons from metal surfa es, whi h had been assumed to be aused by atomi shell stru ture, and whi h Elsasser thought were
aused by ele tron dira tion. Elsasser published a short note sket hing these ideas in Die Naturwissens haften (Elsasser 1925). Elsasser then tried to design an experiment to test the ideas further, with low-velo ity ele trons, but never arried it out. A
ording to Heisenberg's later re olle tion, Elsasser's supervisor Born was s epti al about the reality of matter waves, be ause they seemed in on i t with the observed parti le tra ks in loud hambers.a On 3 November 1925, S hrödinger wrote to Einstein: `A few days ago I read with the greatest interest the ingenious thesis of Louis de Broglie .... '.a S hrödinger too had be ome interested in de Broglie's thesis by reading Einstein's gas theory papers, and he set about trying to nd the wave equation for de Broglie's phase waves. As we have seen (se tion 2.2.2), in his thesis de Broglie had shown that, in an ele trostati potential ϕ, the phase wave of an ele tron of harge e and velo ity v would have (see equation (19)) frequen y ν = p (mc2 + V )/h and phase 2 velo ity vph = (mc + V )/mv , where m = m0 / 1 − v 2 /c2 and V = eϕ is the potential energy. These expressions for ν and vph , given by de Broglie, formed the starting point for S hrödinger's work on the wave equation. S hrödinger took de Broglie's formulas for ν and vph and applied them to the hydrogen atom, with a Coulomb eld ϕ = −e/r.b Using the formula for ν to eliminate v , S hrödinger rewrote the expression for the phase velo ity vph purely in terms of the frequen y ν and the ele tronproton distan e r: vph =
hν q m0 c
1 (hν/m0
c2
− V /m0
2 c2 )
(29) −1
a For this and further details on erning Elsasser, see Mehra and Re henberg (1982a, pp. 6247). See also the dis ussion in se tion 3.4.2. a Quoted in Mehra and Re henberg (1987, p. 412). b Here we follow the analysis by Mehra and Re henberg (1987, pp. 4235) of what they all S hrödinger's `earliest preserved [unpublished℄ manus ript on wave me hani s'. Similar reasoning is found in a letter from Pauli to Jordan of 12 April 1926 (Pauli 1979, pp. 31520).
2.3 Towards a omplete pilot-wave dynami s: 192527
59
(where V = −e2 /r). Then, writing de Broglie's phase wave as ψ = ψ(x, t), he took the equation for ψ to be the usual wave equation ∇2 ψ =
1 ∂2ψ 2 ∂t2 vph
(30)
with phase velo ity vph . Assuming ψ to have a time dependen e ∝ e−2πiνt , S hrödinger then obtained the time-independent equation ∇2 ψ = −
4π 2 ν 2 ψ 2 vph
(31)
with vph given by (29). This was S hrödinger's original (relativisti ) equation for the energy states of the hydrogen atom. As is well known, S hrödinger found that the energy levels predi ted by (31) that is, the eigenvalues hν disagreed with experiment. He then adopted a nonrelativisti approximation, and found that this yielded the orre t energy levels for the low-energy limit. It is in fa t straightforward to obtain the orre t nonrelativisti limit. Writing E = hν − m0 c2 , in the low-energy limit |E| /m0 c2 << 1 and |V | /m0 c2 << 1, we have ν2 2m0 2 = h2 (E − V ) , vph
(32)
so that (31) redu es to what we now know as the nonrelativisti timeindependent S hrödinger equation for a single parti le in a potential V : −
ℏ2 2 ∇ ψ + V ψ = Eψ . 2m0
(33)
Histori ally speaking, then, S hrödinger's original equation (31) for stationary states amounted to a mathemati al trans ription into the language of the standard wave equation (30) of de Broglie's expressions for the frequen y and phase velo ity of an ele tron wave. By studying the eigenvalue problem of this equation in its nonrelativisti form (33), S hrödinger was able to show that the eigenvalues agreed remarkably well with the observed features of the hydrogen spe trum. Thus, by adopting the formalism of a wave equation, S hrödinger transformed de Broglie's elementary derivation of the quantisation of energy levels into a rigorous and powerful te hnique. The time-dependent S hrödinger equation iℏ
ℏ2 2 ∂ψ ∇ ψ+Vψ , =− ∂t 2m0
(34)
60
De Broglie's pilot-wave theory
with a single time derivative, was eventually obtained by S hrödinger in his fourth paper on wave me hani s, ompleted in June 1926 (S hrödinger 1926g). The path taken by S hrödinger in these four published papers was rather tortuous. In the fourth paper, he a tually began by
onsidering a wave equation that was of se ond order in time and of fourth order in the spatial derivatives. However, he eventually settled on (34), deriving it by the following argument: for a wave with time dependen e ∝ e−(i/ℏ)Et , one may write the term Eψ in (33) as iℏ∂ψ/∂t and thus obtain (34), whi h must be valid for any E and therefore for any ψ that an be expanded as a Fourier time series.a The pri e paid for having an equation that was of only rst order in time was that ψ had to be omplex. In retrospe t, the time-dependent S hrödinger equation for a free parti le ((34) with V = 0) may be immediately derived as the simplest wave equation obeyed by a omplex plane wave ei(k·x−ωt) with the nonrelativisti dispersion relation ℏω = (ℏk)2 /2m (that is, E = p2 /2m
ombined with de Broglie's relations E = ℏω and p = ℏk ). This derivation is, in fa t, often found in textbooks. Not only did S hrödinger adopt de Broglie's idea that quantised energy levels ould be explained in terms of waves, he also took up de Broglie's
onvi tion that lassi al me hani s was merely the short-wavelength limit of a broader theory. Thus in his se ond paper on wave me hani s S hrödinger (1926 , p. 497) wrote: Maybe our lassi al me hani s is the
omplete
analogue of geometri al opti s
and as su h is wrong, not in agreement with reality; it fails as soon as the radii of urvature and the dimensions of the traje tory are no longer large
ompared to a ertain wavelength, whi h has a real meaning in the q -spa e. a Then one has to look for an `undulatory me hani s' and the most natural path towards this is surely the wave-theoreti al elaboration of the Hamiltonian pi ture.
S hrödinger used the opti al-me hani al analogy as a guide in the onstru tion of wave equations for atomi systems, parti ularly in his se ond paper. S hrödinger tended to think of the opti al-me hani al analogy in a A tually, S hrödinger onsidered the time dependen e ∝ e±(i/ℏ)Et , so that Eψ = ±iℏ∂ψ/∂t, leading to two possible wave equations diering by the sign of i. He wrote: `We shall require that the omplex wave fun tion ψ satisfy one of these two equations. Sin e at the same time the omplex onjugate fun tion ψ¯ satis es the other equation, one may onsider the real part of ψ as a real wave fun tion (if one needs it)' (S hrödinger 1926g, p. 112, original itali s). a Here S hrödinger adds a footnote: `Cf. also A. Einstein, Berl. Ber., pp. 9 . (1925)'.
2.3 Towards a omplete pilot-wave dynami s: 192527
61
terms of the Hamilton-Ja obi equation and the equation of geometri al opti s, whereas de Broglie tended to think of it in terms of the prin iples of Maupertuis and Fermat. These are of ourse two dierent ways of drawing the same analogy. While S hrödinger took up and developed many of de Broglie's ideas, he did not a
ept de Broglie's view that the parti le was lo alised within an extended wave. In ee t, S hrödinger removed the parti le traje tories from de Broglie's theory, and worked only with the extended (non-singular) waves. (For a detailed dis ussion of S hrödinger's work, see hapter 4.) In the period 192526, then, many of the ideas in de Broglie's thesis were taken up and developed by other workers, espe ially S hrödinger. But what of de Broglie himself? Unlike in his earlier work, during this period de Broglie onsidered spe i equations for his waves. Like S hrödinger, he took standard relativisti wave equations as his starting point. Unlike S hrödinger, however, de Broglie was guided by the following two ideas. First, that parti les are really small singular regions, of very large wave amplitude, within an extended wave. Se ond, that the motion of the parti les or singularities must in some sense satisfy the
ondition (35)
Maupertuis ≡ Fermat ,
so as to bring about a synthesis of the dynami s of parti les with the theory of waves, along the lines already sket hed in de Broglie's thesis. This work ame to a head in a remarkable paper published in May 1927, to whi h we now turn.
2.3.1 `Stru ture': Journal de Physique, May 1927 In the last number of the Journal de Physique, a paper by de Broglie has appeared .... de Broglie attempts here to re on ile the full determinism of physi al pro esses with the dualism between waves and orpus les .... even if this paper by de Broglie is o the mark (and I hope that a tually), still it is very ri h in ideas and very sharp, and on a mu h higher level than the hildish papers by S hrödinger, who even today still thinks he may .... abolish material points. Pauli, letter to Bohr, 6 August 1927 (Pauli 1979, pp. 4045)
What we now know as pilot-wave theory rst appears in a paper by de
62
De Broglie's pilot-wave theory
Broglie (1927b) entitled `Wave me hani s and the atomi stru ture of matter and of radiation', whi h was published in Journal de Physique in May 1927, and whi h we shall dis uss in detail in this se tion.a We shall refer to this ru ial paper as `Stru ture' for short. In `Stru ture', de Broglie presents a theory of parti les as moving singularities. It is argued, on the basis of ertain assumptions, that the equations of what we now all pilot-wave dynami s will emerge from this theory. At the end of the paper de Broglie proposes, as a provisional theory, simply taking the equations of pilot-wave dynami s as given, without trying to derive them from something deeper. It is this last, provisional theory that de Broglie presents a few months later at the fth Solvay onferen e. For histori al ompleteness we should point out that, a
ording to a footnote to the introdu tion, `Stru ture' was `the development of two notes' published earlier in Comptes Rendus (de Broglie 1926, 1927a). The rst, of 28 August 1926, onsiders a model of photons as moving singularities: arguments are given (similar to those found in `Stru ture') leading to the usual velo ity formula, and to the on lusion that the probability density of photons is proportional to the lassi al wave intensity. The se ond note, of 31 January 1927, sket hes analogous ideas for material bodies in an external potential, with the probability density now proportional to the amplitude-squared of the wave fun tion. Unlike `Stru ture', neither of these notes ontains the suggestion that pilot-wave theory may be adopted as a provisional view. We now turn to a detailed analysis of de Broglie's `Stru ture' paper. De Broglie rst onsiders a `material point of proper mass m0 ' moving in free spa e with a onstant velo ity v and represented by a wave u(x, t) satisfying what we would now all the Klein-Gordon equation ∇2 u −
1 ∂2u 4π 2 ν02 = u, 2 2 c ∂t c2
(36)
with ν0 = m0 c2 /h. De Broglie onsiders solutions of the form u(x, t) = f (x − vt) cos
2π φ(x, t) , h
(37)
where the amplitude f is singular at the lo ation x = vt of the moving a Again, we do not always follow de Broglie's notation. Note also that de Broglie moves ba k and forth between solutions of the form f cos φ and solutions `written in omplex form' f eiφ .
2.3 Towards a omplete pilot-wave dynami s: 192527
63
body and the phasea hν0 v · x φ(x, t) = p t− 2 c 1 − v 2 /c2
(38)
is equal to the lassi al Hamiltonian a tion of the parti le. A single parti le is represented by a wave (37), where the moving singularity has velo ity v and the phase φ(x, t) is extended over all spa e. De Broglie then onsiders an ensemble (or ` loud') of similar free parti les with no mutual intera tion, all having the same velo ity v, and with singular amplitudes f entred at dierent points in spa e. A
ording to de Broglie, this ensemble of moving singularities may be represented by a ontinuous solution of the same wave equation (36), of the form 2π Ψ(x, t) = a cos φ(x, t) , (39) h where a is a onstant and φ is given by (38). This ontinuous solution is said to ` orrespond to' the singular solution (37). (De Broglie points out that the ontinuous solutions Ψ are the same as those onsidered by S hrödinger.) The number density of parti les in the ensemble is taken to have a
onstant value ρ, whi h may be written as ρ = Ka2
(40)
for some onstant K . Thus, de Broglie notes, (39) gives the `distribution of phases in the loud of points' as well as the `density of the loud'; and for a single parti le with known velo ity and unknown position, a2 d3 x will measure the probability for the parti le to be in a volume element d3 x. Having dis ussed the free parti le, de Broglie moves on to the ase of a single parti le in a stati external potential V (x). The wave u `written in omplex form' now satis es what we would all the Klein-Gordon equation in an external potentiala V2 2i ∂u 1 1 ∂2u 2 2 2 (41) − 2 m0 c − 2 u = 0 . ∇ u − 2 2 + 2V c ∂t ℏc ∂t ℏ c De Broglie assumes that the parti le begins in free spa e, represented by a singular wave (37), and then enters a region where V 6= 0, into whi h a For simpli ity we ignore an arbitrary onstant in the phase, whi h was in luded by de Broglie. a This follows from (36) by the substitution iℏ∂/∂t → iℏ∂/∂t + V .
64
De Broglie's pilot-wave theory
the free solution (37) must be extended. De Broglie writes (37) `in
omplex form', substitutes into (41), and takes the real and imaginary parts, yielding two oupled partial dierential equations for f and φ. Writing φ(x, t) = Et − φ1 (x)
(42)
(where E = hν is onstant), one of the said partial dierential equations be omes f 1 ℏ2 (43) = (∇φ1 )2 − 2 (E − V )2 + m20 c2 f c (where ≡ ∇2 − (1/c2 )∂ 2 /∂t2 ). De Broglie notes that if the left-hand side is negligible, (43) redu es to the `Ja obi equation' for relativisti dynami s in a stati potential, so that φ1 redu es to the `Ja obi fun tion'. Deviations from lassi al me hani s o
ur, de Broglie notes, when f is non-zero. De Broglie now remarks that, in the lassi al limit, the velo ity of the parti le has the same dire tion as the ve tor ∇φ1 ; he then expli itly assumes that this is still true in the general ase. Here, the identity (35) of the prin iples of Maupertuis and Fermat is being invoked. With this assumption, de Broglie shows that the singularity must have velo ity v=
c2 ∇φ1 , E−V
(44)
and he adds that in the nonrelativisti approximation, E − V ≈ m0 c2 , so that 1 ∇φ1 . v= (45) m0 De Broglie remarks that, in the lassi al approximation, φ1 be omes the `Ja obi fun tion' and that (44) agrees with the relativisti relation between velo ity and momentum. He adds (p. 230): The aim of the pre eding arguments is to make it plausible that the relation [(44)℄ is stri tly valid in the new me hani s.
The importan e of the relation (44) for de Broglie is, of ourse, that it embodies the identity of the prin iples of Maupertuis and Fermat. As in the free ase, de Broglie then goes on to onsider an ensemble of su h parti les, with ea h parti le represented by a moving singularity. He assumes that the phase fun tion φ1 is the same for all the parti les, whose velo ities are then given by (44). The (time-independent) parti le
2.3 Towards a omplete pilot-wave dynami s: 192527
65
density ρ(x) must then satisfy the ontinuity equation (46)
∇ · (ρv) = 0 .
On e again, de Broglie introdu es a representation of the ensemble by a ontinuous solution of (41). In the presen e of the potential V , the solution is taken to have the form 2π ′ 1 ′ Ψ(x, t) = a(x) cos (47) φ (x, t) = a(x) cos 2π νt − φ1 (x) . h h Again, writing Ψ `in omplex form', substituting into (41) and taking real and imaginary parts, de Broglie obtains two oupled partial dierential equations, now in a and φ′1 . One of these reads ℏ2
∇2 a 1 2 = (∇φ′1 ) − 2 (E − V )2 + m20 c2 . a c
(48)
In this ase de Broglie notes that if the left-hand side is negligible, (48) redu es to the equation of geometri al opti s asso iated with the wave equation (41). Comparing (43) and (48), de Broglie notes that when the left-hand sides are negligible, one re overs both lassi al me hani s (for a moving singularity) and geometri al opti s (for a ontinuous wave). In this limit, the fun tions φ1 and φ′1 are identi al, being both equal to the `Ja obi fun tion'. At this point de Broglie makes a ru ial assumption. He proposes to assume, as a hypothesis, that φ1 and φ′1 are always equal, regardless of any approximation (p. 2312): We now make the essential hypothesis that
φ1
and
φ′1
are still identi al when
[the left-hand sides of (43) and (48)℄ an no longer be negle ted. Obviously, this requires that one have:
f ∇2 a = . a f We shall refer to this postulate by the name of `prin iple of the double solution', be ause it implies the existen e of two sinusoidal solutions of equation [(41)℄ having the same phase fa tor, the one onsisting of a point-like singularity and the other having, on the ontrary, a ontinuous amplitude.
This hypothesis expresses, in a more on rete form, de Broglie's earlier idea of `phase harmony' between the internal os illation of a parti le and the os illation of an a
ompanying extended wave: the ondition that φ1 and φ′1 should oin ide amounts to a phase harmony between the
66
De Broglie's pilot-wave theory
singular u-wave representing the point-like parti le and the ontinuous Ψ-wave. Given this ondition, de Broglie dedu es that the ratio ρ/a2 (E − V ) is
onstant along the parti le traje tories. Sin e ea h parti le is assumed to begin in free spa e, where V = 0 and (a
ording to (40)) ρ = Ka2 , de Broglie dedu es that in general V (x) , ρ(x) = Ka2 (x) 1 − (49) E where K is a onstant. The ontinuous wave then gives the ensemble density at ea h point. In the nonrelativisti approximation, this be omes ρ(x) = Ka2 (x) .
(50)
De Broglie notes that ea h possible initial position for the parti le gives rise to a possible traje tory, and that an ensemble of initial positions gives rise to an ensemble of motions. Again, de Broglie takes ρ(x)d3 x as the probability for a single parti le to be in a volume element d3 x. This probability is, as he notes, given in terms of Ψ by (49) or (50). De Broglie adds that Ψ also determines the traje tories (p. 232): The form of the traje tories is moreover equally determined by knowledge of the ontinuous wave, sin e these traje tories are orthogonal to the surfa es of equal phase.
Here we see the rst suggestion that Ψ itself may be regarded as determining the traje tories, an idea that de Broglie proposes more fully at the end of the paper. After sket hing how the above ould be used to al ulate probabilities for ele tron s attering o a xed potential, de Broglie goes on to generalise his results to the ase of a parti le of harge e in a time-dependent ele tromagneti eld (V(x, t), A(x, t)). He writes down the orresponding Klein-Gordon equation, and on e again onsiders solutions of the form (37) with a moving singularity, following the same pro edure as before. Be ause of the time-dependen e of the potentials, φ no longer takes the form (42). Instead of (44), the velo ity of the singularity is now found to be ∇φ + ec A , v = −c2 (51) φ˙ − eV or, in the nonrelativisti approximation, e 1 ∇φ + A . v=− m0 c
(52)
2.3 Towards a omplete pilot-wave dynami s: 192527
67
On e again, de Broglie then onsiders an ensemble of su h moving singularities, with the same phase fun tion φ(x, t). The density ρ(x, t) now obeys a ontinuity equation ∂ρ (53) + ∇ · (ρv) = 0 , ∂t with velo ity eld v given by (51). Again, de Broglie pro eeds to represent the motion of the ensemble by means of a ontinuous solution Ψ of the wave equation, this time of the form 2π ′ φ (x, t) , (54) h with a time-dependent amplitude and a phase no longer linear in t. And again, de Broglie assumes the prin iple of the double solution, that the phase fun tions φ(x, t) and φ′ (x, t) are the same. From this, de Broglie ˙ is onstant along parti le traje tories. dedu es that the ratio ρ/a2 (φ−eV) Using on e more the expression ρ = Ka2 for free spa e, where V = 0 and φ˙ = E0 , de Broglie argues that in general K 2 a (x, t) φ˙ − eV = K ′ a2 φ˙ − eV , ρ(x, t) = (55) E0 Ψ(x, t) = a(x, t) cos
and that in the nonrelativisti limit ρ is still proportional to a2 . As before, the ensemble may be regarded as omposed of all the possible positions of a single parti le, of whi h only the initial velo ity is known, and the probability that the parti le is in the volume element d3 x at time t is equal to ρ(x, t)d3 x and is given in terms of Ψ by (55). De Broglie now shows how the above results for the motion of a parti le in a potential V = eV (ignoring for simpli ity the ve tor potential A) may be obtained from lassi al me hani s, with a Lagrangian of the standard form p L = −M0 c2 1 − v 2 /c2 − V , (56) by assuming that the parti le has a variable proper mass r ℏ2 a M0 (x, t) = m20 − 2 (57) , c a where a is the amplitude of the ontinuous Ψ wave. Further, he onsiders this point of view in the nonrelativisti limit. Writing M0 (x, t) = m0 + ε(x, t) with ε small, the Lagrangian takes the approximate form 1 L = −m0 c2 + m0 v 2 − εc2 − V . 2 As de Broglie remarks (p. 237):
(58)
68
De Broglie's pilot-wave theory
Everything then takes pla e as if there existed, in addition to εc2 .
V,
a potential
energy term
This extra term oin ides, of ourse, with Bohm's quantum potential Q (equation (4)), sin e to lowest order εc2 = −
ℏ2 a ¨ ℏ2 ∇2 a ℏ2 a =− + 2 2m0 a 2m0 a 2m0 c a
(59)
and the se ond term is negligible in the nonrelativisti limit (c → ∞). As we have already repeatedly remarked, for de Broglie the guidan e equation v ∝ ∇φ for velo ity is the fundamental equation of motion, expressing as it does the identity of the prin iples of Maupertuis and Fermat. Even so, de Broglie points out that one an write the dynami s of the parti le in lassi al (Newtonian or Einsteinian) terms, provided one in ludes a variable proper mass or, in the nonrelativisti limit, an additional quantum potential. It is this latter, Newtonian formulation that Bohm proposes in 1952. De Broglie now turns to the ase of a many-body system, onsisting of N parti les with proper masses m1 , m2 , ..., mN . In the nonrelativisti approximation, if the system has total (Newtonian) energy E and potential energy V (x1 , ..., xN ), then following S hrödinger one may onsider the propagation of a wave u in the 3N -dimensional on guration spa e, satisfying the wave equation N X i=1
−
ℏ2 2 ∇ u + V u = Eu 2mi i
(60)
(the time-independent S hrödinger equation). De Broglie remarks that this seems natural, be ause (60) is the obvious generalisation of the one-body S hrödinger equation, whi h follows from the nonrelativisti limit of de Broglie's wave equation (41). However, de Broglie riti ises S hrödinger's interpretation, a
ording to whi h a parti le is identi ed with an extended, non-singular wave pa ket, having no pre ise position or traje tory. De Broglie obje ts that, without well-de ned parti le positions, the oordinates x1 , ..., xN used to onstru t the on guration spa e would have no meaning. Further, de Broglie asserts that on guration spa e is `purely abstra t', and that a wave propagating in this spa e annot be a physi al wave: instead, the physi al pi ture of the system must involve N waves propagating in 3-spa e. `What then', asks de Broglie (p. 238), `is the true meaning of the S hrödinger equation?' To answer this question, de Broglie onsiders,
2.3 Towards a omplete pilot-wave dynami s: 192527
69
for simpli ity, the ase of two parti les, whi h for him are singularities of a wave-like phenomenon in 3-spa e. Negle ting the ve tor potential, the two singular waves u1 (x, t) and u2 (x, t) satisfy the oupled equations 2i V2 ∂u1 1 u1 + 2 V1 − 2 m21 c2 − 12 u1 = 0 , ℏc ∂t ℏ c (61) 2i V2 ∂u2 1 u2 + 2 V2 − 2 m22 c2 − 22 u2 = 0 , ℏc ∂t ℏ c with potentials V1 = V (|x − x2 |)
, V2 = V (|x − x1 |) ,
(62)
where x1 , x2 are the positions of the two parti les (or singularities). The propagation of ea h wave then depends, through V , on the position of the singularity in the other wave. De Broglie is now fa ed with the formidable problem of solving the simultaneous partial dierential equations (61), in order to obtain the motions of the two singularities. Not surprisingly, de Broglie does not attempt to arry through su h a solution. Instead he notes that, in
lassi al me hani s in the nonrelativisti limit, it is possible to nd a `Ja obi fun tion' φ(x1 , x2 ) for the two-parti le system, su h that the parti le momenta are given by m1 v1 = −∇1 φ ,
m2 v2 = −∇2 φ .
(63)
De Broglie then asks (p. 238): `Can the new me hani s .... de ne su h a fun tion φ?' De Broglie is asking whether, if one ould solve the oupled equations (61), the resulting motions of the singularities would (in the nonrelativisti limit) satisfy the guidan e equations (63) for some fun tion φ(x1 , x2 ). He then asserts, on the basis of an in orre t argument (to whi h we shall return in a moment), that this will indeed be the ase. Having on luded that a fun tion φ(x1 , x2 ) generating the motions of the singularities will in fa t exist, he goes on to identify φ(x1 , x2 ) with the phase of a ontinuous solution Ψ(x1 , x2 , t) of the S hrödinger equation (60) for the two-parti le system. While de Broglie does not say so expli itly, in ee t he assumes that there exists a ontinuous solution of (60) whose phase oin ides with the fun tion φ(x1 , x2 ) that generates the motions of the singularities via (63). This seems to be de Broglie's answer to the question he raises, as to the `true meaning' of the S hrödinger equation: solving the S hrödinger equation (60) in on guration spa e provides
70
De Broglie's pilot-wave theory
an ee tive means of obtaining the motions of the singularities without having to solve the oupled partial dierential equations (61) in 3-spa e. De Broglie's in orre t argument for the existen e of an appropriate fun tion φ, derived from the equations (61), pro eeds as follows. He rst imagines that the motion of singularity 2 is already known, so that the problem of motion for singularity 1 redu es to a ase that has already been dis ussed, of a single parti le in a given time-dependent potential. For this ase, it has already been shown that the ensemble of possible traje tories may be represented by a ontinuous wave a1 (x, t)e(i/ℏ)φ1 (x,t) .
(64)
Further, as was shown earlier, the equations of motion for the parti le may be written in Lagrangian form with an additional potential ε1 (x1 , t)c2 . Similarly, onsidering the motion of singularity 1 as known, the motion of singularity 2 an be des ribed by a Lagrangian with an additional potential ε2 (x2 , t)c2 . The two sets of ( lassi al) equations of motion an be derived from a single Lagrangian, de Broglie argues, only if ε1 and ε2 redu e to a single fun tion ε(|x1 − x2 |) of the interparti le separation. `If we assume this', says de Broglie, the total Lagrangian for the system will be L=
1 1 m1 v12 + m2 v22 − V (|x1 − x2 |) − ε(|x1 − x2 |)c2 , 2 2
(65)
and one will be able to dedu e, in the usual fashion, the existen e of a fun tion φ(x1 , x2 ) satisfying the equations (63). It is not entirely lear from the text if de Broglie is really onvin ed that ε1 and ε2 will indeed redu e to a single fun tion ε(|x1 − x2 |). With hindsight one sees, in fa t, that this will not usually be the ase. For it amounts to requiring that the quantum potential Q(x1 , x2 ) for two parti les should take the form Q(x1 , x2 ) = Q(|x1 − x2 |), whi h is generally false.a After giving his in orre t argument for the existen e of φ, de Broglie points out that, sin e the parti les have de nite traje tories, it is meaningful to onsider their six-dimensional on guration spa e, and to represent the two parti les by a single point in this spa e, with velo ity
omponents given by (63). If the initial velo ity omponents are given, but the initial positions of the two parti les are not, one may onsider an ensemble of representative points (x1 , x2 ), whose density ρ(x1 , x2 ) in a One might also question the meaning of de Broglie's wave fun tion (64) for parti le 1, in the ontext of a system of two intera ting parti les, whi h must have an entangled wave fun tion Ψ(x1 , x2 , t).
2.3 Towards a omplete pilot-wave dynami s: 192527
71
on guration spa e satis es the ontinuity equation, ∇ · (ρv) = 0 ,
(66)
where v is the (six-dimensional) velo ity eld given by (63). De Broglie then onsiders, as we have said, a ontinuous solution of the S hrödinger equation (60), of the form Ψ(x1 , x2 , t) = A(x1 , x2 )e(i/ℏ)φ ,
(67)
with the ta it assumption that the phase φ is the same fun tion φ whose existen e has been (apparently) established by the above (in orre t) argument. He then shows, from (60), that A2 satis es the on gurationspa e ontinuity equation (66). De Broglie on ludes that A2 d3 x1 d3 x2 is the probability for the representative point (x1 , x2 ) to be present in a volume element d3 x1 d3 x2 of on guration spa e. Here, then, de Broglie has arrived at a higher-dimensional analogue of his previous results: the wave fun tion Ψ in on guration spa e determines the ensemble probability density through its amplitude, as well as the motion of a single system through its phase. De Broglie ends this se tion by remarking that it seems di ult to nd an equation playing a role similar to (60) outside of the nonrelativisti approximation, and that Fermi's al ulation on erning the s attering of ele trons by a rotator may be regarded as illustrating the above. From the point of view of the history of pilot-wave theory, the most important se tion of de Broglie's paper now follows. Entitled `The pilot wave' (`L'onde pilote'), it begins by re alling the results obtained in the
ase of a single parti le in a time-dependent potential, whi h de Broglie states may be summarised by `the two fundamental formulas' (51) and (55) for the parti le velo ity and probability density respe tively. De Broglie notes that he arrived at the velo ity formula (51) by invoking the prin iple of the double solution a prin iple that is valid in free spa e but whi h `remains a hypothesis in the general ase' (p. 241). At this point, de Broglie makes a remarkable suggestion: that instead of trying to derive the velo ity eld (51) from an underlying theory, one
ould simply take it as a postulate, and regard Ψ as a physi ally-real `pilot wave' guiding the motion of the parti le. To quote de Broglie (p. 241): But if one does not wish to invoke the prin iple of the double solution, it is a
eptable to adopt the following point of view:
one will assume the
existen e, as distin t realities, of the material point and of the ontinuous wave represented by the fun tion
Ψ,
and one will take it as a postulate that
72
De Broglie's pilot-wave theory
the motion of the point is determined as a fun tion of the phase of the wave by the equation [(51)℄. One then on eives the ontinuous wave as guiding the motion of the parti le. It is a pilot wave.
This is the rst appearan e in the literature of what we now know as pilot-wave or de Broglian dynami s (albeit stated expli itly for only a single parti le). The pilot wave Ψ satisfying the S hrödinger equation, and the material point, are regarded as `distin t realities', with the former guiding the motion of the latter a
ording to the velo ity law (51). De Broglie made it lear, however, that he thought su h a dynami s
ould be only provisional (p. 241): By thus taking [(51)℄ as a postulate, one avoids having to justify it by the prin iple of the double solution; but this an only be, I believe, a provisional attitude. No doubt one will indeed have to
rein orporate
the parti le into the
wave-like phenomenon, and one will probably be led ba k to ideas analogous to those that have been developed above.
As we shall dis uss in the next se tion, de Broglie's proposal of pilotwave theory as a provisional measure has striking analogues in the early history of Newtonian gravity and of Maxwellian ele tromagneti theory. De Broglie goes on to point out two important appli ations of the formulas (51) and (55). First, applying the theory to light, a
ording to (55) `the density of the photons is proportional to the square of the amplitude' of the guiding wave Ψ, yielding agreement with the predi tions of wave opti s. Se ond, an ensemble of hydrogen atoms with a de nite state Ψ will have a mean ele troni harge density proportional 2 to |Ψ| , as assumed by S hrödinger. De Broglie's `Stru ture' paper ends with some remarks on the pressure exerted by parti les on a wall (for example of a box of gas). He notes that a
ording to (51), be ause of the interferen e between the in ident and re e ted waves, the parti les will not a tually strike the wall, raising the question of how the pressure is produ ed. De Broglie laims that the pressure omes from stresses in the interferen e zone, as appear in an expression obtained by S hrödinger (1927 ) for the stress-energy tensor asso iated with the wave Ψ.
2.3.2 Signi an e of de Broglie's `Stru ture' paper The ex eptional quality of de Broglie's `Stru ture' paper in Journal de Physique was noted by Pauli, in a letter to Bohr dated 6 August
2.3 Towards a omplete pilot-wave dynami s: 192527
73
1927, already quoted as the epigraph to the last subse tion: `.... it is very ri h in ideas and very sharp, and on a mu h higher level than the hildish papers by S hrödinger ....' (Pauli 1979, pp. 4045). In the same letter, Pauli suggested that Bohr would have to refer to de Broglie's paper in his Como le ture (in whi h Bohr developed the idea of omplementarity between waves and parti les); and in fa t, in what was to remain an unpublished version of the Como le ture, Bohr did take an expli it stand against de Broglie's ideas (Bohr 1985, pp. 8998). Bohr hara terised de Broglie's work as attempting `to re on ile the two apparently ontradi tory sides of the phenomena by regarding the individual parti les or light quanta as singularities in the wave eld'; further, Bohr suggested that de Broglie's view rested upon `the on epts of lassi al physi s' and was therefore not `suited to help us over the fundamental di ulties' (Bohr 1985, p. 92). On the whole, though, this paper by de Broglie has been essentially ignored, by both physi ists and historians. De Broglie's `Stru ture' paper may be summed up as follows. De Broglie has a model of parti les as singularities of 3-spa e waves ui , in whi h the motion of the individual parti les, as well as the ensemble distribution of the parti les, are determined by a ontinuous wave fun tion Ψ. (We emphasise that de Broglie has both the wave fun tion Ψ and the singular u-waves.) The result is a rst-order theory of motion, based on the velo ity law mi vi = ∇i φ, in whi h the prin iples of Maupertuis and Fermat are uni ed. Then, at the end of the paper, de Broglie re ognises that his singularity model of parti les an be dropped, and that the results he has obtained the formulas for velo ity and probability density in terms of Ψ an be simply postulated, yielding pilot-wave theory as a provisional measure. A few months later, in O tober 1927, de Broglie presented this pilotwave theory at the fth Solvay onferen e, for a nonrelativisti system of N parti les guided by a wave fun tion Ψ in on guration spa e. Before dis ussing de Broglie's Solvay report, however, it is worth pausing to
onsider the role played by the `Stru ture' paper in de Broglie's thinking. From a histori al point of view, the signi an e of de Broglie's `Stru ture' paper inevitably depends on the signi an e one as ribes to the provisional pilot-wave theory arrived at in that paper. Given what we know today that pilot-wave theory provides a onsistent a
ount of quantum phenomena de Broglie's `Stru ture' paper now seems
onsiderably more signi ant than it has seemed in the past. From the point of view of pilot-wave theory as we know it today, the
74
De Broglie's pilot-wave theory
singular u-waves played a similar role for de Broglie as the material ether did for Maxwell in ele tromagneti theory. In both ases, there was a on eptual s aolding that was used to build a new theory, and that ould be dropped on e the results had been arrived at. De Broglie re ognised at the end of his paper that, if one took pilot-wave dynami s as a provisional theory, then the s aolding he had used to onstru t this theory ould be dropped. At the same time, de Broglie insisted that taking su h a step was indeed only provisional, and that an underlying theory was still needed, probably along the lines he had been pursuing. Similar situations have arisen before in the history of s ien e. Abstra ting away the details of a model based on an older theory sometimes results in a new theory in its own right, involving new on epts, where, however, the author regards the new theory as only a provisional measure, and expe ts that the model based on the older theory will eventually provide a proper basis for the provisional theory. Thus, for example, Newton tried to explain gravitation on the basis of a tion by onta t, involving a material medium lling spa e the same `Aethereal Medium', in fa t, as he thought responsible for the interferen e and dira tion of light (Newton 1730; reprint, pp. 35053). Newton regarded his theory of gravitation, with a tion at a distan e, as merely a provisional and phenomenologi al theory, that would later nd an explanation in terms of ontiguous a tion. Eventually, however, the on ept of `gravitational a tion at a distan e' be ame widely a
epted in its own right (though it was later to be overthrown, of ourse, by general relativity). Similarly, Maxwell used me hani al models of an ether to develop his theory of the ele tromagneti eld. Maxwell himself may or may not have re ognised that this s aolding ould be dropped (Hendry 1986). But ertainly, many of his immediate followers did not: they regarded working only with Maxwell's equations as provisional and phenomenologi al, pending the development of an underlying me hani al model of an ether. Again, the on ept of `ele tromagneti eld' eventually be ame widely a
epted as a physi al entity in its own right. In the ase of de Broglie in 1927, in his `Stru ture' paper (and subsequently at the fth Solvay onferen e), he arrived at the new on ept of `pilot wave in on guration spa e', an entirely new kind of physi al entity that, a
ording to de Broglie's (provisional) theory, guides the motion of material systems.
2.4 1927 Solvay report: the new dynami s of quanta
75
2.4 1927 Solvay report: the new dynami s of quanta We now turn to a (brief) summary of and ommentary on de Broglie's report at the fth Solvay onferen e.a As we have noted, the theory presented in this report is pilot-wave theory as we know it today, for a nonrelativisti many-body system, with a guiding wave in on guration spa e that determines the parti le velo ities a
ording to de Broglie's basi law of motion. De Broglie's ideas about parti les as singularities of 3-spa e waves are mentioned only brie y. The theory de Broglie presents in Brussels in O tober 1927 is, indeed, just the provisional theory he proposed a few months earlier at the end of his `Stru ture' paper (though now expli itly applied to many-body systems as well). De Broglie begins part I of his report by reviewing the results obtained in his do toral thesis. The energy E and momentum p of a parti le are determined by the phase φ of an asso iated wave:a E = ∂φ/∂t ,
p = −∇φ .
(68)
(These are the relativisti guidan e equations of de Broglie's early pilotwave theory of 192324.) It follows that, as de Broglie remarks, `the prin iples of least a tion and of Fermat are identi al' (p. 376). Quantisation onditions appear in a natural way, and there are far-rea hing impli ations for statisti al me hani s. De Broglie ends the review of his early work with some general remarks. He points out (p. 377) that he `has always assumed that the material point o
upies a well-de ned position in spa e', and he asserts that as a result the wave amplitude must be singular, or take very large values in a small region, somewhere within the extended wave. Signi antly, de Broglie adds: `But, in fa t, the form of the amplitude plays no role in the results reviewed above'. This seems to be a rst hint that the a tual results, su h as (68), do not depend on the details of any underlying model of the parti les as moving singularities. De Broglie then outlines the work of S hrödinger. He notes that S hrödinger's wave equation is onstru ted in order that the phase φ of the wave fun tion 2π Ψ = a cos (69) φ h be a solution of the Hamilton-Ja obi equation in the geometri al-opti s limit. He points out that S hrödinger identi es parti les with lo alised a Again, our notation sometimes departs from de Broglie's. a Again, de Broglie's phase φ has a sign opposite to the phase S as we would normally de ne it now.
76
De Broglie's pilot-wave theory
wave pa kets instead of with a small on entration within an extended wave, and that for a many-body system S hrödinger has a wave Ψ propagating in on guration spa e. For both the one-body and many-body
ases, de Broglie writes down the time-independent S hrödinger equation only, with a stati potential energy fun tion. As we shall see, de Broglie in fa t onsiders non-stationary wave fun tions as well. For the one-body
ase, de Broglie also writes down a relativisti , and time-dependent, equation what we now know as the Klein-Gordon equation in an external ele tromagneti eld. De Broglie then raises two on eptual di ulties with S hrödinger's work (similar in spirit to the obje tions he raises in `Stru ture'): (1) He questions how one an meaningfully onstru t a on guration spa e without a real on guration, asserting that it `seems a little paradoxi al to onstru t a on guration spa e with the oordinates of points that do not exist' (p. 380). (2) He laims that the physi al meaning of the wave Ψ annot be ompared with that of an ordinary wave in 3-spa e, be ause the number of dimensions of the abstra t spa e on whi h Ψ is de ned is determined by the number of degrees of freedom of the system. (This se ond point seems indeed a very ee tive way to make lear that Ψ is quite dierent from a onventional wave or eld on 3-spa e.) Part I of the report ends with some remarks on Born's statisti al interpretation, whi h de Broglie asserts is analogous to his own. In part II of his report, de Broglie presents his own interpretation of the wave fun tion. Pilot-wave theory is learly formulated, rst for a single parti le, and then for a system of N parti les. Several appli ations are outlined. It is interesting to see how de Broglie motivates his theory and ompares it with the ontenders. De Broglie begins by asking what the relationship is between parti les and the wave Ψ. He rst onsiders a single relativisti parti le in an external ele tromagneti eld with potentials (V , A). De Broglie notes that, in the lassi al limit, the phase φ of Ψ obeys the Hamilton-Ja obi equation, and the velo ity of the parti le is given by v = −c2
∇φ + ec A φ˙ − eV
(70)
(the same formula (51) dis ussed in `Stru ture'). De Broglie then proposes that this velo ity formula is valid even outside the lassi al limit (p. 383, itali s in the original): We propose to assume by indu tion that this formula is still valid when the old Me hani s is no longer su ient, that is to say when [φ℄ is no longer a solution
2.4 1927 Solvay report: the new dynami s of quanta
77
of the Ja obi equation. If one a
epts this hypothesis, whi h appears justi ed by its onsequen es, the formula [(70)℄ ompletely determines the motion of the orpus le
as soon as one is given its position at an initial instant.
In
other words, the fun tion [φ℄, just like the Ja obi fun tion of whi h it is the generalisation, determines a whole lass of motions, and to know whi h of these motions is a tually des ribed it su es to know the initial position.
We emphasise that there is no appeal to singular u-waves anywhere in de Broglie's report. No use is made of the prin iple of the double solution. As he had suggested at the end of his `Stru ture' paper a few months earlier, de Broglie simply postulates the basi equations of pilot-wave dynami s, without trying to derive them from anything else. To motivate the guidan e equation, de Broglie simply generalises the lassi al Hamilton-Ja obi velo ity formula to the non- lassi al domain: he assumes `by indu tion' that the formula holds even outside the
lassi al limit. And de Broglie is quite expli it that he is proposing a rst-order theory of motion, based on velo ities: given the wave fun tion, the initial position alone determines the traje tory. So far, then, for a single parti le de Broglie has a (relativisti ) wave fun tion Ψ = a cos 2π h φ whose phase φ determines the parti le velo ity by the guidan e equation (70). De Broglie now goes on to point out that, for an ensemble of parti les guided by the velo ity eld (70), the distribution Ka2 φ˙ − eV (71)
is preserved in time (that is, equivariant). He on ludes that, if the initial position of the parti le is ignored, then the probability for the parti le to be present (at time t) in a spatial volume dτ is πdτ = Ka2 φ˙ − eV dτ . (72)
(This is the same probability formula (55) arrived at in `Stru ture'.) De Broglie's expression `probabilité de présen e' makes it lear that we have to do with a probability for the ele tron being somewhere, and not merely with a probability for an experimenter nding the ele tron somewhere f. Bell (1990, p. 29). De Broglie simply assumes that the equivariant distribution is the orre t probability measure for a parti le of unknown position. In fa t, this distribution is only an equilibrium distribution, analogous to thermal equilibrium in lassi al statisti al me hani s (see se tion 8.2). De Broglie sums up his results so far (p. 383):
78
De Broglie's pilot-wave theory
In brief, in our hypotheses, ea h wave
Ψ
determines a ` lass of motions', and
ea h one of these motions is governed by equation [(70)℄ when one knows the initial position of the orpus le. If one ignores this initial position, the formula [(72)℄ gives the probability for the presen e of the orpus le in the element of volume
dτ
at the instant
t.
Ψ then appears probability wave.
The wave
(Führungsfeld of Mr Born) and a
as both a
pilot wave
There are, de Broglie adds, no grounds for abandoning determinism, and in this his theory diers from that of Born. De Broglie adds that, in the nonrelativisti approximation (where φ˙ − eV ≈ m0 c2 ), the guidan e and probability formulas (70) and (72) redu e to 1 e v=− (73) ∇φ + A m0 c and
π = const · a2 .
(74)
These are the standard pilot-wave equations for a single parti le (in an external ele tromagneti eld). De Broglie then remarks on how the above formulas may be applied to the s attering of a single parti le by a xed potential. The ensemble of in ident parti les may be represented by a plane wave with a uniform probability distribution. Upon entering a region of non-zero eld, the behaviour of the wave fun tion may be al ulated using Born's perturbation theory (for example, for the Rutherford s attering of an ele tron by an atomi nu leus). De Broglie draws an analogy between the s attering of the wave fun tion Ψ and the lassi al s attering of light by a refra ting medium. As in his `Stru ture' paper, de Broglie remarks in passing that, for a relativisti parti le governed by the guidan e equation (70), one may write down the equations of lassi al dynami s with a variable rest mass M0 given by equation (57). As we have already noted, in the nonrelativisti approximation this yields an additional potential energy term, whi h is pre isely Bohm's `quantum potential'. And, on e again, we emphasise that for de Broglie the equation v ∝ ∇φ for velo ity is the fundamental equation of motion, expressing the identity of the prin iples of Maupertuis and Fermat: de Broglie merely points out that, if one wishes, the dynami s an be written in lassi al terms (as Bohm did in 1952). De Broglie then turns to the interpretation of interferen e and dira tion, for the ase of photons. Here, the guiding wave Ψ is similar to
2.4 1927 Solvay report: the new dynami s of quanta
79
but not the same as a light wave.a De Broglie onsiders s attering by xed obsta les, in whi h ase the guiding wave may be taken to have a
onstant frequen y ν . The relativisti equations (70), (72) then be ome v=−
c2 ∇φ, hν
(75)
π = const · a2 .
De Broglie points out that the se ond equation predi ts the well-known interferen e and dira tion patterns. He argues that the results will be the same, whether the experiment is done with an intense beam over a short time or with a feeble beam over a long time. De Broglie remarks, as he did at the end of `Stru ture', on the stresses appearing in S hrödinger's expression for the stress-energy tensor of the wave Ψ. A
ording to de Broglie, these stresses provide an explanation for the pressure exerted by light re e ting on a mirror despite the fa t that, a
ording to (70), the photons never a tually strike the surfa e of the mirror (as shown expli itly by Brillouin in the dis ussion after de Broglie's report). De Broglie then turns to the generalisation of the above dynami s to a (nonrelativisti ) many-body system. Remarkably, many historians and ommentators have not noti ed this proposal by de Broglie of a many-body dynami s in on guration spa e, a proposal that is usually attributed to Bohm ( f. se tion 11.1.) De Broglie begins by pointing out how the two di ulties he has raised against S hrödinger's wave me hani s might be solved. First, if a real on guration exists at all times, one an meaningfully onstru t the
on guration spa e. As for the se ond di ulty, regarding the meaning of a wave in the abstra t on guration spa e, de Broglie makes the following preliminary remark (p. 386): It appears to us ertain that if one wants to of a system of
N
in spa e, ea h of the
N −1
physi ally
represent the evolution
orpus les, one must onsider the propagation of
N
N
waves
propagations being determined by the a tion of the
orpus les onne ted to the other waves.
This seems a lear referen e to the theory of intera ting singular u-waves dis ussed a few months earlier in de Broglie's `Stru ture' paper.a (The a De Broglie did not identify the photoni pilot wave with the ele tromagneti eld. In the general dis ussion, on p. 509, de Broglie expli itly states that in his theory the wave Ψ for the ase of photons is distin t from the ele tromagneti eld. a It might be thought that here de Broglie has in mind the fa t that N moving parti les may be asso iated with N velo ity elds in 3-spa e, whi h may be asso iated with N (non-singular) guiding waves in 3-spa e. From a pilot-wave perspe tive, su h N guiding waves may be identi ed with the N ` onditional' wave
80
De Broglie's pilot-wave theory
propagation of ea h wave must depend on the motion of the other N − 1 parti les, of ourse, in order to a
ount for intera tions between the parti les.) As we saw above (se tion 2.3.1), in `Stru ture' de Broglie expli itly onsidered the ase of two parti les, whi h were represented by two singular waves u1 (x, t) and u2 (x, t) satisfying a pair of oupled partial dierential equations (61), where the equation for ea h wave
ontained a potential depending on the position of the singularity in the other wave. Instead of trying to solve the equations, de Broglie assumed that the resulting velo ities of the moving singularities ould be written as the gradient of a fun tion φ(x1 , x2 ), whi h ould be identi ed with the phase of a solution Ψ(x1 , x2 , t) of the (time-independent) S hrödinger equation. In `Stru ture', then, the wave fun tion in on guration spa e appeared as an ee tive des ription of the motions of the parti les (or singularities). An e ho of this view is dis ernible in the Solvay report, whi h ontinues with the following justi ation for introdu ing a guiding wave in on guration spa e (p. 386): Nevertheless, if one fo usses one's attention only on the orpus les, one an represent their states by a point in on guration spa e, and one an try to relate the motion of this representative point to the propagation of a titious wave
Ψ
in on guration spa e.
De Broglie seems to be saying that, if one is on erned only with a su
in t mathemati al a
ount of parti le motion (as opposed to a physi al representation), then this an be obtained by introdu ing a guiding wave in on guration spa e. Certainly, the dynami s is mu h simpler with a single, autonomous wave Ψ in on guration spa e. As in his `Stru ture' paper, this seems to be de Broglie's explanation for S hrödinger's otherwise mysterious on guration-spa e wave. When de Broglie states that the wave Ψ is ` titious', he presumably means that it has only mathemati al, and not physi al, signi an e.a It provides a onvenient mathemati al a
ount of parti le motion, pending fun tions ψi (xi , t) ≡ Ψ(X1 , X2 , ..., xi , ..., XN , t), where xi (i = 1, 2, ..., N ) ranges over all positions in 3-spa e and Xj is the a tual position of the j th parti le. Ea h ψi (xi , t) de nes a wave in 3-spa e that determines the velo ity of the ith parti le (through the gradient of its phase); and ea h ψi (xi , t) depends on the positions of the other N − 1 parti les. However, given the ontext (in parti ular the re ent publi ation of `Stru ture'), it seems lear that here de Broglie's N waves are the singular u-waves of his double-solution theory. a This is somewhat in ontrast with de Broglie's introdu tion of the pilot wave at the end of `Stru ture', where he refers to the parti le and the guiding wave as `distin t realities'. However, there de Broglie expli itly proposed pilot-wave theory for a single parti le only, and it is likely that while he was omfortable with the idea of a physi ally real pilot wave in 3-spa e (for the one-body ase), he ould not regard a pilot wave in on guration spa e as having more than mathemati al signi an e.
2.4 1927 Solvay report: the new dynami s of quanta
81
a full physi al des ription by a more detailed theory. As we dis ussed in se tion 2.3.2, the provisional introdu tion of a `mathemati al' des ription, pending the development of a proper `physi al' model, has distinguished pre edents in the history of Newtonian gravity and Maxwellian ele trodynami s. Having motivated the introdu tion of a guiding wave Ψ in on guration spa e, de Broglie suggests (p. 386) that Ψ ....
plays for the representative point of the system in on guration spa e
the same role of pilot wave and of probability wave that the wave
Ψ
plays in
ordinary spa e in the ase of a single material point.
Thus, in the nonrelativisti approximation, de Broglie onsiders N parti les with positions x1 , x2 , ..., xN . He states that the wave Ψ determines the velo ity of the representative point in on guration spa e by the formula 1 vk = − (76) ∇k φ mk (where mk is the mass of the k th parti le). As in the ase of a single parti le, notes de Broglie, the probability for the system to be present in a volume element dτ of on guration spa e is πdτ = const · a2 dτ .
(77)
These are the standard pilot-wave equations for a many-body system, repla ing the single-parti le formulas (73) and (74). De Broglie remarks that (77) seems to agree with Born's results for ele tron s attering by an atom and with Fermi's for s attering by a rotator. Finally, at the end of his se tion on the many-body ase, de Broglie notes that it seems di ult to onstru t a wave Ψ that an generate the motion of a relativisti many-body system (in ontrast with the relativisti one-body ase), a point that de Broglie had already noted in `Stru ture'. From the very beginning, then, it was re ognised that it would be di ult to formulate a fundamentally Lorentz-invariant pilotwave theory for a many-body system, a situation that persists to this day. Thus, in his report at the fth Solvay onferen e, de Broglie arrived at pilot-wave theory for a many-body system, with a guiding wave in
on guration spa e. Judging by de Broglie's omments about a physi al representation requiring N waves in 3-spa e, and his hara terisation of Ψ as a ` titious' wave, it seems lear that he still regarded his new
82
De Broglie's pilot-wave theory
dynami s as an ee tive, mathemati al theory only (just as he had a few months earlier in his `Stru ture' paper). De Broglie's report then moves on to sket h some appli ations to atomi theory. For stationary states of hydrogen, de Broglie notes that the ele tron motion is ir ular, ex ept in the ase of magneti quantum number m = 0 for whi h the ele tron is at rest. (Note that de Broglie expresses no on ern that the ele tron is predi ted to be motionless in the ground state of hydrogen. This result may seem puzzling lassi ally, just as Bohr's quantised atomi orbits seem puzzling lassi ally: but both are natural onsequen es of de Broglie's non- lassi al dynami s.) De Broglie also points out that one an al ulate the ele tron velo ity during an atomi transition i → j , so that su h transitions an be visualised. In this example, de Broglie's guiding wave is a solution of the time-dependent S hrödinger equation, with a time-dependent amplitude as well as phase. (The atomi wave fun tion is taken to have the form Ψ = ci Ψi + cj Ψj , where Ψi , Ψj are eigenfun tions orresponding to the atomi states i, j , and ci , cj are fun tions of time.) Clearly, then, de Broglie applied his pilot-wave dynami s not just to stationary states, but to quite general wave fun tions as he had also done (in the ase of a single parti le) in `Stru ture'. De Broglie then outlines how, in his theory, one an obtain expressions for the mean harge and urrent density for an ensemble of atoms. These expressions are the same as those used by S hrödinger and others. De Broglie remarks that, `denoting by Ψ the wave written in omplex form, and by [Ψ∗ ℄ the onjugate fun tion', in the nonrelativisti limit the
harge density is proportional to |Ψ|2 : the ele tri dipole moment then
ontains the orre t transition frequen ies. By using these expressions as sour es in Maxwell's equations, says de Broglie, one an orre tly predi t the mean energy radiated by an atom. This is just semi lassi al radiation theory, whi h is still widely used today in quantum opti s.a Part II of de Broglie's report ends with some general remarks. First and foremost, de Broglie makes it lear that he regards pilot-wave theory as only provisional (p. 390): So far we have onsidered the orpus les as `exterior' to the wave motion being only determined by the propagation of the wave.
Ψ,
their
This is, no
doubt, only a provisional point of view: a true theory of the atomi stru ture of matter and radiation should, it seems to us,
in orporate
the orpus les in
the wave phenomenon by onsidering singular solutions of the wave equations.
a Cf. se tion 4.4.
2.4 1927 Solvay report: the new dynami s of quanta
83
De Broglie goes on to suggest that in a deeper theory one ould `show that there exists a orresponden e between the singular waves and the waves Ψ, su h that the motion of the singularities is onne ted to the propagation of the waves Ψ', just as he had suggested in `Stru ture'. Part II ends by noting the in omplete state of the theory with respe t to the ele tromagneti eld and ele tron spin. De Broglie's nal part III ontains a lengthy dis ussion and review of re ent experiments involving the dira tion, interferen e and s attering of ele trons (as had been requested by Lorentz4 ). He regards the results as eviden e for his `new Dynami s'. For the ase of the dira tion of ele trons by a rystal latti e, de Broglie points out that the s attered wave fun tion Ψ has maxima in ertain dire tions, and notes that a
ording to his theory the ele trons should be preferentially s attered in these dire tions (p. 392): Be ause of the role of pilot wave played by the wave
Ψ,
one must then observe
a sele tive s attering of the ele trons in these dire tions.
What de Broglie is (brie y) des ribing here is the separation of the in ident wave fun tion Ψ into distin t (non-overlapping) emerging beams, with ea h outgoing ele tron o
upying one beam, and with an ensemble of ele trons being distributed among the emerging beams a
ording to the Born rule. This is relevant to a proper understanding of the de Broglie-Pauli en ounter, dis ussed in se tion 10.2. Con erning the s attering maxima observed re ently by Davisson and Germer, for ele trons in ident on a rystal, de Broglie remarks (p. 395): There is dire t numeri al on rmation of the formulas of the new Dynami s .... .
De Broglie also dis usses the inelasti s attering of ele trons by atoms. A
ording to Born's al ulations, one should observe maxima and minima in the angular dependen e of the dierential s attering ross se tion. A
ording to de Broglie, it is premature to speak of an agreement with experiment. The results of Dymond, for the inelasti s attering of ele trons by helium atoms, do however show maxima in the ross se tion, in qualitative agreement with the predi tions. De Broglie makes it quite
lear that, at the time, theory was far behind experiment. The dis ussion following de Broglie's report was extensive, detailed, and varied. A number of parti ipants raised queries, and de Broglie replied to most of them. Some of this dis ussion will be onsidered in
84
De Broglie's pilot-wave theory
detail in hapters 10 and 11. Here, we limit ourselves to a brief summary of the questions raised. Lorentz asked how, in the simple pilot-wave theory of 1924, de Broglie derived quantisation onditions for the ase of multiperiodi atomi orbits. Born questioned the validity of de Broglie's guidan e equation for an elasti ollision, while Pauli suggested that the key idea behind de Broglie's theory was the asso iation of parti le traje tories with a lo ally
onserved urrent. S hrödinger raised the question of an alternative velo ity eld dierent from that assumed by de Broglie, while Kramers raised the question of how the Maxwell energy-momentum tensor ould arise from independently moving photons. Lorentz, Ehrenfest and S hrödinger asked about the properties of ele tron orbits in hydrogen. Brillouin dis ussed at length the simple example of a photon olliding with a mirror: in his Fig. 2, the in ident and emergent photon traje tories are lo ated inside pa kets of limited extent (a point relevant for the de Broglie-Pauli en ounter in the general dis ussion). Finally, Lorentz
onsidered, in lassi al Maxwell theory, the near- eld attra tive stress between two prisms, and laimed that this `negative pressure' ould not be produ ed by the motion of orpus les (photons).
2.5 Signi an e of de Broglie's work from 1923 to 1927 In his papers and thesis of 192324, de Broglie proposed a simple form of a new, non- lassi al dynami s of parti les with velo ities determined by the phase gradient of abstra t waves in 3-spa e. De Broglie onstru ted the dynami s in su h a way as to unify the me hani al prin iple of Maupertuis with the opti al prin iple of Fermat. He showed that it gave an a
ount of some simple quantum phenomena, in luding single-parti le interferen e and quantised atomi energy levels. As we have seen, the s ope and ambition of de Broglie's do toral thesis went far beyond a mere extension of the relations E = hν , p = h/λ from photons to other parti les. Yet, the thesis is usually remembered solely for this idea. The depth and inner logi of de Broglie's thinking is not usually appre iated, neither by physi ists nor by historians. An ex eption is Darrigol (1993), who on this very point writes (pp. 3034): For one who only knows of de Broglie's relation
λ = h/p,
two explanations
of his originality oer themselves. The rst has him as a lu ky dreamer who hit upon a great idea amidst a foolish play with analogies and formulas. The se ond has him a providential deep thinker, who dedu ed unsuspe ted onne tions by rationally ombining distant on epts. The rst explanation is the
2.5 Signi an e of de Broglie's work from 1923 to 1927
85
most popular, though rarely expressed in print. .... The se ond explanation of Louis de Broglie's originality, though also extreme, is ertainly loser to the truth. Anyone who has read de Broglie's thesis annot help admiring the unity and inner onsisten y of his views, the inspired use of general prin iples, and a ne essary reserve.
In 1926, starting from de Broglie's expressions for the frequen y and phase velo ity of an ele tron wave in an external potential, S hrödinger found the (nonrelativisti ) wave equation for de Broglie's waves. It was de Broglie's work, beginning in 1923, that initiated the notion of a wave fun tion for material parti les. And it was de Broglie's view of the signi an e of the opti al-me hani al analogy, and of the role waves
ould play in bringing about the existen e of integer-valued quantum numbers, that formed the basis for S hrödinger's development of the wave equation and the asso iated eigenvalue problem. In 1927, de Broglie proposed what we now know as pilot-wave or de Broglian dynami s for a many-body system, with a guiding wave in
on guration spa e. De Broglie regarded this theory as provisional: he thought it should emerge as an ee tive theory, from a more fundamental theory in whi h parti les are represented by singularities of 3-spa e waves. Even so, he did propose the theory, in the rst-order form most
ommonly used today. Further, as we have seen, de Broglie's view of the pilot wave as merely phenomenologi al is strikingly reminis ent of (for example) late-nineteenth- entury views of the ele tromagneti eld. In retrospe t, one may regard de Broglie as having unwittingly arrived at a new and fundamental on ept, that of a pilot wave in on guration spa e. De Broglie was unable to show that his new dynami s a
ounted for all quantum phenomena. In parti ular, as we shall dis uss in hapters 10 and 11, de Broglie did not understand how to des ribe the pro ess of measurement of arbitrary quantum observables in pilot-wave theory: as shown in detail by Bohm in 1952, this requires an appli ation of de Broglie's dynami s to the measuring devi e itself. De Broglie did, however, possess the fundamental dynami s in omplete form. Furthermore, de Broglie did understand how his theory a
ounted for single-parti le interferen e, for the dire ted s attering asso iated with rystal dira tion, and for ele tron s attering by atoms (for the latter, see se tion 10.2). Clearly, in 1927, many appli ations of pilot-wave theory remained to be developed; just as, indeed, many appli ations of quantum theory in luding the general quantum theory of measurement remained to be developed and lari ed.
86
De Broglie's pilot-wave theory
In retrospe t, de Broglian dynami s seems as radi al as and indeed somewhat reminis ent of Einstein's theory of gravity. A
ording to Einstein, there is no gravitational for e, and a freely falling body follows the straightest path in a urved spa etime. A
ording to de Broglie, a massive body undergoing dira tion and following a urvilinear path is not a ted upon by a Newtonian for e: it is following the ray of a guiding wave. De Broglie's abandonment of Newton's rst law of motion in 1923, and the adoption of a dynami s based on velo ity rather than a
eleration, amounts to a far-rea hing departure from lassi al me hani s and (arguably) from lassi al kinemati s too with impli ations for the stru ture of spa etime that have perhaps not been understood (Valentini 1997). Certainly, the extent to whi h de Broglie's dynami s departs from
lassi al ideas was unfortunately obs ured by Bohm's presentation of it, in 1952, in terms of a
eleration and a pseudo-Newtonian quantum potential, a formulation that today seems arti ial and inelegant ompared with de Broglie's (mu h as the rewriting of general relativity as a eld theory on at spa etime seems unnatural and hardly illuminating). The fundamentally se ond-order nature of lassi al physi s is today embodied in the formalism of Hamiltonian dynami s in phase spa e. In ontrast, de Broglie's rst-order approa h to the theory of motion seems more naturally ast in terms of a dynami s in on guration spa e. Regardless of how one may wish to interpret it, by any standards de Broglie's work from 1923 to 1927 shows a remarkable progression of thought, beginning from early intuitions and simple models of the relationship between parti les and waves, and ending (with S hrödinger's help) with a omplete and new form of dynami s for nonrelativisti systems a deterministi dynami s that was later shown by Bohm to be empiri ally equivalent to quantum theory (given a Born-rule distribution of initial parti le positions). The inner logi of de Broglie's work in this period, his drive to unite the physi s of parti les with the physi s of waves by unifying the prin iples of Maupertuis and Fermat, his wave-like explanation for quantised energy levels, his predi tion of ele tron dira tion, his explanation for single-parti le interferen e, his attempts to onstru t a eld-theoreti al pi ture of parti les as moving singularities, and his eventual proposal of pilot-wave dynami s as a provisional theory all this ompels admiration, all the more for being largely unknown and unappre iated. Today, pilot-wave theory is often hara terised as simply adding parti le traje tories to the S hrödinger equation. An understanding of de Broglie's thought from 1923 to 1927, and of the role it played in
2.5 Signi an e of de Broglie's work from 1923 to 1927
87
S hrödinger's work, shows the gross ina
ura y of this hara terisation: after all, it was a tually S hrödinger who removed the traje tories from de Broglie's theory ( f. se tion 11.1). It is di ult to avoid the on lusion that de Broglie's stature as a major ontributor to quantum theory has suered unduly from the ir umstan e that, for most of the twentieth entury, the theory proposed by him was in orre tly regarded as untenable or in onsistent with experiment. Regardless of whether or not de Broglie's pilot-wave theory is loser to the truth than other interpretations, the fa t that it is a onsistent and viable approa h to quantum physi s whi h has no measurement problem, whi h shows that obje tivity and determinism are not in ompatible with quantum physi s, and whi h stimulated Bell to develop his famous inequalities ne essarily entails a reappraisal of de Broglie's pla e in the history of twentieth- entury physi s.
88
Notes to pp. 3183
Ar hival notes 1 AHQP-OHI, Louis de Broglie, `Replies to Mr Kuhn's questions'. 2 AHQP-OHI, Louis de Broglie, session 1 (tape 43a), 7 January 1963, Paris; 0.75 h, in Fren h, 13 pp.; by T. S. Kuhn, T. Kahan and A. George.
3
Ibid.
4 Cf. de Broglie to Lorentz, 27 June 1927, AHQP-LTZ-11 (in Fren h).
3 From matrix me hani s to quantum me hani s
The report by Born and Heisenberg on `quantum me hani s' may seem surprisingly di ult to the modern reader. This is partly be ause Born and Heisenberg are des ribing various stages of development of the theory that are quite dierent from today's quantum me hani s. Among these, it should be noted in parti ular that the theory developed by Heisenberg, Born and Jordan in the years 192526 and known today as matrix me hani s (Heisenberg 1925b [1℄, Born and Jordan 1925 [2℄, Born, Heisenberg and Jordan 1926 [4℄)a diers from standard quantum me hani s in several important respe ts. At the same time, the interpretation of the theory (the topi of se tion II of the report) also appears to have undergone important modi ations, in parti ular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between dierent stationary states). Then the notion of the wave fun tion is introdu ed and related to probabilities for the stationary states. At a later stage probabilisti notions (in parti ular, what one now alls transition probabilities) are extended to arbitrary observables, but it remains somewhat un lear whether the wave fun tion itself should be regarded as a fundamental entity or merely as an ee tive one. This may re e t the dierent routes followed by Born and by Heisenberg in the development of their ideas. The ommon position presented by Born and Heisenberg emphasises the probabilisti aspe t of the theory as fundamental, and the on lusion of the report expresses strong on den e in the resulting pi ture. The two main se tions of this hapter, se tion 3.3 and se tion 3.4, will be devoted, respe tively, to providing more details on the various a Throughout this hapter, numbers in square bra kets refer to entries in the original bibliography at the end of Born and Heisenberg's report.
89
90
From matrix to quantum me hani s
stages of development of the theory, and to disentangling various threads of interpretation that appear to be present in Born and Heisenberg's report. Before that, we provide a summary (se tion 3.1) and a few remarks on the authorship and writing of the various se tions of the report (se tion 3.2).
3.1 Summary of Born and Heisenberg's report Born and Heisenberg's report has four se tions (together with an introdu tion and on lusion): I on formalism, II on interpretation, III on axiomati formulations and on un ertainty, and IV on appli ations. The formalism that is des ribed is initially that of matrix me hani s, whi h is then extended beyond the original framework (among other things, in order to make the onne tion with S hrödinger's wave me hani s). Then, further developments of matrix me hani s are sket hed. These allow one to in orporate a `statisti al' interpretation. After a brief dis ussion of Jordan's (1927b, [39℄) axiomati formulation, the un ertainty relations are used to justify the statisti al element of the interpretation. A few appli ations of spe ial interest and some brief nal remarks on lude the report. As we dis uss below, Born drafted se tions I and II, and Heisenberg drafted the introdu tion, se tions III and IV and the on lusion. Born and Heisenberg's introdu tion stresses the ontinuity of quantum me hani s with the old quantum theory of Plan k, Einstein and Bohr, and tou hes brie y on su h themes as dis ontinuity, observability in prin iple, `Ans hauli hkeit' (for whi h see se tion 4.6 below) and the statisti al element in quantum me hani s. Se tion I, `The mathemati al methods of quantum me hani s', rst sket hes matrix me hani s roughly as developed in the `three-man paper' by Born, Heisenberg and Jordan (1926 [4℄): the basi framework of position and momentum matri es and of the anoni al equations, the perturbation theory, and the onne tion with the theory of quadrati forms (the latter leading to both dis rete and ontinuous spe tra).a Next, Born and Heisenberg des ribe two generalisations of matrix me hani s: Dira 's (1926a [7℄) q-number theory and what they refer to as Born and Wiener's (1926a [21℄, 1926b) operator al ulus. As a a Note that Born and Heisenberg use the term `quantum me hani s' to refer also to matrix me hani s, in keeping with the terminology of the original papers. See also Mehra and Re henberg (1982 , fn. 72 on pp. 612). Heisenberg expresses his dislike for the term `matrix physi s' in Heisenberg to Pauli, 16 November 1925 (Pauli, 1979, pp. 2556).
3.1 Summary of Born and Heisenberg's report
91
matter of fa t, they already sket h von Neumann's representation of physi al quantities by operators in Hilbert spa e. This is identi ed as a mathemati ally rigorous version of the transformation theory of Dira (1927a [38℄) and Jordan (1927b, [39℄).a Born and Heisenberg note that, if one takes as the Hilbert spa e the appropriate fun tion spa e, the problem of diagonalising the Hamiltonian operator leads to the timeindependent S hrödinger equation. Thus, in this (formal) sense, the S hrödinger theory is a spe ial ase of the operator version of quantum me hani s. The eigenfun tions of the Hamiltonian an be asso iated with the unitary transformation that diagonalises it. Matrix me hani s, too, is a spe ial ase of this more general formalism, if one takes the (dis rete) energy eigenstates as the basis for the matrix representation. Se tion II on `Physi al interpretation' is probably the most striking. It begins by stating that matrix me hani s des ribes neither the a tual state of a system, nor when hanges in the a tual state take pla e, suggesting that this is related to the idea that matrix me hani s des ribes only losed systems. In order to obtain some des ription of
hange, one must onsider open systems. Two methods for doing this are introdu ed. First of all, following Heisenberg (1926b [35℄) and Jordan (1927a [36℄), one an onsider the matrix me hani al des ription of two oupled systems that are in resonan e. One an show that the resulting des ription an be interpreted in terms of quantum jumps between the energy levels of the two systems, with an expli it expression for the transition probabilities. The se ond method uses the generalised formalism of se tion I, in whi h time dependen e an be introdu ed expli itly via the time-dependent S hrödinger equation. Again, expressions an be found that an be interpreted as transition probabilities, and similarly the squared modulus of the wave fun tion's oe ients in the energy basis is interpreted as the probability for the o
urren e of the orresponding stationary state. Born and Heisenberg introdu e the notion of interferen e of probabilities. Only at this stage are experiments mentioned as playing a on eptually ru ial role, namely in an argument why su h interferen e `does not represent a ontradi tion with the rules of the probability al ulus' (p. 425). Interferen e is then related to the wave theory of de Broglie and S hrödinger, and the interpretation of a See below, se tions 3.3.3 and 3.4.1, for some dis ussion of Born and Wiener's work, and se tion 3.4.5 for the transformation theory (mainly Dira 's). Note that von Neumann had published a series of papers on the Hilbert-spa e formulation of quantum me hani s before his well-known treatise of 1932. It is to two of these that Born and Heisenberg refer, Hilbert, von Neumann and Nordheim (1928 [41℄, submitted April 1927) and von Neumann (1927 [42℄).
92
From matrix to quantum me hani s
the squared modulus of the wave fun tion as a probability density for position is introdu ed. Born and Heisenberg also de ne the `relative' (i.e. onditional) position density given an energy value, as the squared modulus of the amplitude of a stationary state. Finally, in the ontext of Dira 's and Jordan's transformation theory, the notions of transition probability (for a single quantity) and of onditional probability (for a pair of quantities) are generalised to arbitrary physi al quantities. Se tion III opens with a on ise exposition of Jordan's (1927b, [39℄) axioms for quantum me hani s, whi h take the notion of probability amplitude as primary. Born and Heisenberg point out some formal drawba ks, su h as the use of δ -fun tions and the presen e of unobservable phases in the probability amplitudes. They note that su h drawba ks have been over ome by the formulation of von Neumann (whi h they do not go on to dis uss further). Born and Heisenberg then pro eed to dis uss in parti ular whether the statisti al element in the theory an be re on iled with ma ros opi determinism. To this ee t they rst justify the ne essity of using probabilisti notions by appeal to the notion of un ertainty (dis ussed using the example of dira tion of light by a single slit); then they point out that, while for instan e in ases of dira tion the laws of propagation of the probabilities in quantum me hani s are very dierent from the lassi al evolution of a probability density, there are ases where the two oin ide to a very good approximation; they write that this justi es the lassi al treatment of α- and β -parti les in a
loud hamber. Se tion IV dis usses brie y some appli ations, hosen for their ` lose relation to questions of prin iple' (p. 434). Mostly, these appli ations highlight the importan e of the generalised formalism (introdu ed in se tion I), by going beyond matrix me hani s proper or wave me hani s proper. The rst example is that of spin, whi h, requiring nite matri es, is taken to be a problemati on ept for wave me hani s (but not for matrix me hani s). The main example is given by the dis ussion of identi al parti les, the Pauli prin iple and quantum statisti s. Born and Heisenberg note that the hoi e of whether the wave fun tion should be symmetri or antisymmetri appears to be arbitrary. They note, moreover, that the appropriate hoi e arises naturally if one quantises the normal modes of a bla k body, leading to Bose-Einstein statisti s, or, if one adopts Jordan's (1927d [54℄) quantisation pro edure, to FermiDira statisti s.a Finally, Born and Heisenberg omment on Dira 's a The general dis ussion returns to these issues in more detail; see for instan e Dira 's riti al remarks on Jordan's pro edure, p. 502.
3.2 Writing of the report
93
quantum ele trodynami s (Dira 1927b, [51,52℄), noting in parti ular that it yields the transition probabilities for spontaneous emission. The on lusion dis usses quantum me hani s as a ` losed theory' (see se tion 3.4.7 below) and addresses the question of whether indeterminism in quantum me hani s is fundamental. In parti ular, Born and Heisenberg state that the assumption of indeterminism agrees with experien e and that the treatment of ele trodynami s will not modify this state of aairs. They on lude by noting that the existing problems
on ern rather the development of a fully relativisti theory of quantum ele trodynami s, but that there is progress also in this dire tion. The dis ussion is omparatively brief. Dira des ribes the analogy between the matrix method and the Hamilton-Ja obi theory. Then Lorentz makes a few remarks, in parti ular emphasising that the freedom to hoose the phases in the matri es q and p, (2) and (5) below, is not limited to a dierent hoi e of the time origin, a possibility already noted by Heisenberg (1925b [1℄), but extends to arbitrary phase fa tors of the form ei(δm −δn ) , as noted by Born and Jordan (1925 [2℄).a A
ording to Lorentz, this fa t suggests that the `true os illators' are asso iated with the dierent stationary states rather than with the spe tral frequen ies.1 The last few and brief ontributions address the question of the phases.b
3.2 Writing of the report The respe tive ontributions by Born and by Heisenberg to the report be ome lear from their orresponden e with Lorentz. As mentioned in
hapter 1, Lorentz had originally planned to have a report on quantum me hani s from either Born or Heisenberg, leaving to them the hoi e of who was to write it. In reply, Born and Heisenberg suggested that they would provide a joint report.2 Heisenberg was going to visit Born in Göttingen in early July 1927, on whi h o
asion they would de ide on the stru ture of the report and divide up the work, planning to meet again in August to nish it. The report (as requested by Lorentz) would in lude the results by Dira , who was in Göttingen at the time.c As a This orresponds of ourse to the hoi e of a phase fa tor for ea h energy state. b As mentioned in footnote on p. 23, Bohr suggested to omit this entire dis ussion from the published pro eedings.
Dira was in Göttingen from February 1927 to the end of June, after having spent September to February in Copenhagen. Dira 's paper on transformation theory (1927a [38℄) and his paper on emission and absorption (Dira 1927b [51℄) were written in Copenhagen, while the paper on dispersion (Dira 1927 [52℄) was written in Göttingen. Compare Kragh (1990, h. 2. pp. 37 .).
94
From matrix to quantum me hani s
Born explains, the report would `above all emphasise the viewpoints that are presumably taken less into a
ount by S hrödinger, namely the statisti al on eption of quantum me hani s'. Either author, or both, ould orally present the report at the onferen e (Heisenberg, in a parallel letter, suggested that Born should be the appropriate hoi e, sin e he was the senior s ientist3 ); Born and Heisenberg ould also give additional explanations in English. Born did not speak Fren h, nor (Born thought) did Heisenberg. In a later letter,4 Born reports that Heisenberg had sent him at the end of July the draft of introdu tion, on lusion and se tions III and IV, upon whi h Born had written se tions I and II, reworked Heisenberg's draft, and sent everything ba k to Heisenberg. Heisenberg in turn had made some further small hanges. Due to illness and a small operation, Born had been unable to go to Muni h in mid-August, so that he had not seen the hanges. He trusted, however, that he would agree with the details of Heisenberg's nal version.a Born also mentions that Heisenberg and himself would like to go through the text again, at least at proof stage.b The presentation would be split between Born (introdu tion and se tions I and II) and Heisenberg (se tions III and IV and on lusion). Born on ludes with the following words: `It would be parti ularly important for us to ome to an agreement with S hrödinger regarding the physi al interpretation of the quantum formalisms'.
3.3 Formalism
3.3.1 Before matrix me hani s In the old Bohr-Sommerfeld theory, ele tron orbits are des ribed as
lassi al Kepler orbits subje t to additional onstraints (the `quantum
onditions'), yielding dis rete stationary states. Su h a pro edure, in Born and Heisenberg's introdu tion, is riti ised as arti ial.c An atom is assumed to `jump' between its various stationary states, and energy dieren es between stationary states are related to the spe tral frequen ies via Bohr's frequen y ondition. Spe tral frequen ies and a Heisenberg had sent the nal text to Lorentz on 24 August.5 b The published version and the types ript show only minor dis repan ies, and many of these are learly mistakes in the published version (detailed in our endnotes to the report). This and the fa t that Born and Heisenberg appear not to have spoken Fren h suggests that there was in fa t no proofreading on their part.
The remark is rather brief, but note that Born and Heisenberg seem to onsider the introdu tion of photons into the lassi al ele tromagneti theory (a orpus ular dis ontinuity) to be as arti ial as the introdu tion of dis rete stationary states into lassi al me hani s.
3.3 Formalism
95
orbital frequen ies, however, appear to be quite unrelated. This is the
ru ial `radiation problem' of the old quantum theory (see also below, se tion 4.4). The BKS theory of radiationa in ludes a rather dierent pi ture of the atom, and arguably provides a solution to the radiation problem just mentioned. In the BKS theory, one asso iates to ea h atom a set of `virtual' os illators, with frequen ies equal to the spe tral frequen ies. Spe i ally, when the atom is in the stationary state n, the os illators that are ex ited are those with frequen ies orresponding to transitions from the energy En to both lower and higher energy levels. These os illators produ e a virtual radiation eld, whi h propagates ( lassi ally) a
ording to Maxwell's equations and intera ts (non- lassi ally) with the virtual os illators, in parti ular in uen ing the probabilities for indu ed emission and absorption in other atoms and determining the probabilities for spontaneous emission in the atom itself. The a tual emission or absorption of energy is asso iated with the orresponding transition between stationary states. Note that an emission in one atom is not onne ted dire tly to an absorption in another, so that energy and momentum are onserved only statisti ally. Several important results derived by developing orresponden e arguments into translation rules from lassi al to quantum expressions were formulated within the ontext of the BKS theory, in parti ular Kramers' (1924) dispersion theory. Other examples were Born's (1924) perturbation formula, Kramers and Heisenberg's (1925) joint paper on dispersion and Heisenberg's (1925a) paper on polarisation of uores en e radiation.b The same arguments also led to a natural reformulation of the quantum onditions independent of the old ` lassi al models' (Kuhn 1925, Thomas 1925). As rst argued by Pauli (1925), su h results were entirely independent of the presumed me hanism of radiation.c Indeed, after the demise of the BKS theory following the experimental veri ation of the onservation laws in the Bothe-Geiger experiments, these results and the orresponding te hniques of symboli translation formed the basis for Heisenberg's formulation of matrix me hani s in his famous paper of 1925 (Heisenberg 1925b [1℄). Heisenberg's stroke of genius was to give up altogether the kinemati al des ription in terms of spatial oordinates, while retaining the lassi al form of the equations a See also the des ription in se tion 1.3. b See Darrigol (1992, pp. 22446) and Mehra and Re henberg (1982b, se tions III.4 and III.5).
Cf. Darrigol (1992, pp. 2445). Cf. also Jordan (1927e [63℄), among others.
96
From matrix to quantum me hani s
of motion, to be solved under suitably reformulated quantum onditions (those of Kuhn and Thomas). While Heisenberg had dropped both the me hanism of radiation and the older ele troni orbits, there were points of ontinuity with the BKS theory and with the old quantum theory; in parti ular, as we shall see, the new variables were at least formally related to the virtual os illators, and the pi ture of quantum jumps was retained, at least for the time being.
3.3.2 Matrix me hani s Heisenberg's original paper is rather dierent from the more de nitive presentations of matrix me hani s: the equations are given in Newtonian (not Hamiltonian) form, energy onservation is not established to all orders, and, as is well known, Heisenberg at rst had not re ognised that his theory used the mathemati al ma hinery of matri es. Like Born and Heisenberg in their report, we shall a
ordingly follow in this se tion mainly the papers by Born and Jordan (1925 [2℄) and Born, Heisenberg and Jordan (1926 [4℄), as well as the book by Born (1926d,e [58℄), supplementing the report with more details when useful. Kinemati ally, the des ription of `motion' in matrix me hani s generalises that given by the set of the omponents of a lassi al Fourier series, qn e2πinνt
(1)
(and thus generalises the idea of a periodi motion). As frequen ies, one
hooses, instead of ν(n) = nν , the transition frequen ies of the system under onsideration, whi h are required to obey Ritz's ombination prin iple, νij = Ti − Tj (relating the frequen ies to the spe tros opi terms Ti = Ei /h, or, in Born and Heisenberg's notation, Ti = Wi /h). The omponents thus de ned form a doubly in nite array: q11 e2πiν11 t q12 e2πiν12 t . . . 2πiν21 t q22 e2πiν22 t . . . q = q21 e (2) . .. .. .. . . .
From the ombination prin iple for the νij , it follows that νij = −νji . Further, it is required of the (generally omplex) amplitudes that ∗ qji = qij ,
(3)
by analogy with a lassi al Fourier series, so that the array is in fa t a Hermitian matrix.
3.3 Formalism
97
In modern notation, the above matrix onsists of the elements of the position observable (in Heisenberg pi ture) in the energy basis. For a
losed system with Hamiltonian H , these indeed take the form hEi | Q(t) |Ej i = hEi | e(i/~)Ht Q(0)e−(i/~)Ht |Ej i = hEi | Q(0) |Ej i e(i/~)(Ei −Ej )t .
(4)
Along with the position matrix q , one onsiders also a momentum matrix: p11 e2πiν11 t p12 e2πiν12 t . . . 2πiν21 t p22 e2πiν22 t . . . p = p21 e (5) , .. .. .. . . .
as well as all other matrix quantities that an be obtained as fun tions of q and p, de ned as polynomials or as power series (leaving aside questions of onvergen e). Again from the ombination prin iple, it follows that under multipli ation of two su h matri es, one obtains another matrix of the same form (i.e. with the same time-dependent phases). Thus one justi es taking these two-dimensional arrays as matri es. The most general physi al quantity in matrix me hani s is a matrix whose elements (in modern terminology) are the elements of an arbitrary Heisenbergpi ture observable in the energy basis. The values of these matri es (in parti ular the amplitudes qij and the frequen ies νij ) will have to be determined by solving the equations of motion under some suitable quantum onditions. Spe i ally, the equations of motion are postulated to be the analogues of the lassi al Hamiltonian equations: dq ∂H = , dt ∂p
dp ∂H =− , dt ∂q
(6)
where H is a suitable matrix fun tion of q and p. The quantum onditions are formulated as pq − qp =
h ·1 , 2πi
(7)
where 1 is the identity matrix. In the limit of large quantum numbers, (7) orresponds to the `old' quantum onditions. Of ourse, dierentiation with respe t to t and with respe t to matrix arguments need to be suitably de ned. The former is de ned elementwise as (8) f˙ij = 2πiνij fij ,
98
From matrix to quantum me hani s
whi h an be written equivalently as 2πi f˙ = (9) (W f − f W ) , h where W is the diagonal matrix with Wij = Wi δij . Dierentiation with respe t to matrix arguments is de ned (in slightly modernised form) as f (x + α1) − f (x) df (10) = lim , dx α→0 α where the limit is also understood elementwise.a On e the pre eding s heme has been set up, the next question is how to go about solving the equations of motion. First, for the spe ial ase of dierentiation with respe t to q or p, one shows that f q − qf =
h ∂f 2πi ∂p
(11)
pf − f p =
h ∂f 2πi ∂q
(12)
and
(by indu tion on the form of f and using (7)). From this and from (9), one an easily show that the equations of motion are equivalent to W q − qW = Hq − qH
(13)
W p − pW = Hp − pH .
(14)
and Thus, W − H ommutes with both q and p, and therefore also with H . It follows that W H − HW = 0 ,
(15)
H˙ = 0 ,
(16)
whi h by (9) means that that is, energy is onserved. In the non-degenerate ase, it follows that H is a diagonal matrix. Denoting its diagonal terms by Hi , (13) yields further the Bohr frequen y ondition, Hi − Hj = Wi − Wj = hνij ,
(17)
and xing an arbitrary onstant one an set H = W . The above proof shows also that onversely, (16), (17) and the ommutation relations a This de nition was agreed upon after some debate. See Mehra and Re henberg (1982 , pp. 6971 and 97100).
3.3 Formalism
99
imply the equations of motion. Therefore, the problem of solving the equations of motion essentially redu es to that of diagonalising the energy matrix. This dis ussion leads also to the notion of a ` anoni al transformation' (a transformation that leaves the equations of motion invariant) as a transformation that preserves the ommutation relations. Obviously any transformation of the form Q = SqS −1 ,
P = SpS −1
(18)
(with S invertible) will be su h a transformation, and it was onje tured that these were the most general anoni al transformations. At this stage, S is required only to be invertible; indeed, the notion of unitarity has not been introdu ed yet. If H is a suitably symmetrised fun tion of q and p, however, it will also be a Hermitian matrix, and the problem of diagonalising it an be solved using the theory of quadrati forms of in nitely many variables (assuming that the theory as known at the time extends also to unbounded quadrati forms). In parti ular, it will be possible to diagonalise H using a unitary S , whi h also guarantees that the new oordinates Q and P will be Hermitian. What does su h a solution to the equations of motion yield? In the rst pla e, one obtains the values of the frequen ies νij and of the energies Wi (along with the values of other onserved quantities). In the se ond pla e, based on orresponden e arguments, one an identify the diagonal elements of a matrix with `time average' in the respe tive stationary states,a and relate the squared amplitudes |qij |2 to the intensities of spontaneous emission, or equivalently to the orresponding transition probabilities between stationary states.b This, however, is not an argument based on rst prin iples. Indeed, as Born and Heisenberg repeatedly stress, matrix me hani s in the above form des ribes a losed,
onservative system, in whi h no hange should take pla e. As Born and Heisenberg note in se tion II, a tual transfer of energy is to be expe ted only when the atom is oupled to some other system, spe i ally the radiation eld. Born and Jordan's paper (1925 [2℄) and Born, Heisenberg and Jordan's paper (1926 [4℄) in lude a rst attempt at treating the radiation eld quantum me hani ally and at justifying the assumed interpretation of the squared amplitudes. A satisfa tory treatment of a The on ept of `time averages' an also be thought of as related to the BKS theory, as remarked by Heisenberg himself in a letter to Einstein of February 1926.6 b The determination of intensities had been a pressing problem sin e the work by Born (1924) and the `Utre ht observations' referred to in the report (p. 412). See Darrigol (1992, pp. 2345).
100
From matrix to quantum me hani s
spontaneous radiation was later given by Dira (1927b [51℄), as also mentioned in the report.a In the third pla e, by setting up relations between the amplitudes qij and the matrix elements of other onserved quantities su h as angular momentum, one is able to identify whi h transitions are possible between states with the orresponding quantum numbers (derivation of `sele tion rules'). Expe tation values, other than in the form of time averages for the stationary states, are la king. It should be lear that matrix me hani s in its histori al formulation is not to be identi ed with today's quantum me hani s in the Heisenberg pi ture. The matri es allowed as solutions in matrix me hani s are basis-dependent. Moreover, quite apart from the fa t that solutions to the equations of motion are hard to nd,a the results obtained are relatively modest. In parti ular, there are no general expe tation values for physi al quantities. In dis ussing both the mathemati al formalism and the physi al interpretation, Born and Heisenberg suggest that the original matrix theory is inadequate and in need of extension.
3.3.3 Formal extensions of matrix me hani s As presented in the report, the hief formal rather than interpretational di ulty for matrix me hani s is the failure to des ribe aperiodi quantities. As we have seen, the matri es were understood as a generalisation of lassi al Fourier series. Stri tly speaking, if one follows this analogy, a periodi quantity is represented by a dis rete (if doubly in nite) matrix, and one an envisage representing aperiodi quantities by ontinuous matri es analogously to the representation of lassi al quantities by Fourier integrals. Indeed, already Heisenberg (1925b [1℄) points out that his quadrati arrays would have in general both a periodi part (dis rete) and an aperiodi part ( ontinuous). The onne tion between matri es and quadrati forms showed in fa t that matri es in general had also a ontinous spe trum (if the theory extended to the unbounded ase). Still, ontinuous matri es are unwieldy, and in ertain ases the matrix elements (whi h are always evaluated in the energy basis) will be ome singular, as when trying to des ribe a free parti le. (This is not at all surprising, onsidering that also in the lassi al analogy the Fourier integral of the fun tion at + b fails to onverge.) a Note that by this time Dira (1927a [38℄) had already introdu ed a probabilisti interpretation based on the transformation theory. a In parti ular, one was unable to introdu e a tion-angle variables to solve the Hamiltonian equations as in lassi al me hani s; f. Born and Heisenberg's remarks on `aperiodi motions'.
3.3 Formalism
101
This problem was addressed by generalising the notion of a matrix to that of a q-number (Dira 1926a [7℄) and to that of an operator (Born and Wiener 1926a [21℄, 1926b). Dira 's q-numbers are abstra t obje ts that are assumed to form a non ommutative algebra, while Born and Wiener's operators are hara terised not by their elements in some basis, but by their a tion on a spa e of fun tions. In both approa hes, one an deal with aperiodi quantities and quantities with a singular matrix representation. Born and Wiener's approa h is the lesser-known of the two, and shall therefore be brie y sket hed.a The operators in question are linear operators a ting on a spa e of fun tions x(t) (whi h are not given any spe i physi al interpretation). These fun tions are understood as generalising the fun tions having the form x(t) =
∞ X
xn e
2πi h Wn t
.
(19)
n=1
That is, they generalise the fun tions that an be written in a Fourier-like series with frequen ies equal to the spe tral frequen ies. Therefore, the operators generalise the matri es that a t on in nite-dimensional ve tors indexed by the energy values, i.e. they generalise the matri es in the energy basis. By using these operators, Born and Wiener free themselves from the need to operate with the matrix elements, and they are able to solve the equations of motion expli itly for systems more general than those treated in matrix me hani s until then. Their main example is the (aperiodi ) one-dimensional free parti le. Instead of presenting this formalism, however, in the report Born and Heisenberg present dire tly von Neumann's formalism of operators on Hilbert spa e (evidently but ta itly onsidering Born and Wiener's formalism as its natural pre ursor). They note that in this formalism it is possible to onsider matri es in arbitrary, even ontinuous bases, and they use this fa t to make the onne tion with S hrödinger's theory. In parti ular, they point out that solving the (time-independent) S hrödinger equation is equivalent to diagonalising the Hamiltonian quadrati form, in the sense that the set of S hrödinger eigenfun tions ϕ(q, W ) (wave fun tions in q indexed by W ) yields the transformation matrix S from the position basis to the energy basis. Thus one an use the familiar methods of partial dierential equations to diagonalise H .b a Born and Wiener published two very similar papers on their operator theory, one in German, referred to in the report (1926a [21℄), and one in English (1926b). b Cf. below, se tion 3.4.5. The onne tion between matrix and wave me hani s was
102
From matrix to quantum me hani s
Despite providing very useful formal extensions of matrix me hani s, the q-numbers and the operators (at least as presented in se tion I of the report) do not provide further insights into the physi al interpretation of the theory. In the following se tion II, on `Physi al interpretation', Born and Heisenberg dis uss the problem of des ribing a tual states and pro esses in matrix me hani s, suitably extended, and the surprising rami ations of this problem.
3.4 Interpretation Born and Heisenberg's se tion II begins with the following statement (p. 420): The most noti eable defe t of the original matrix me hani s onsists in the fa t that at rst it appears to give information not about a tual phenomena, but rather only about possible states and pro esses. It allows one to al ulate the possible stationary states of a system; further it makes a statement about the nature of the harmoni os illation that an manifest itself as a light wave in a quantum jump. But it says nothing about when a given state is present, or when a hange is to be expe ted.
The reason for this is lear:
matrix
me hani s deals only with losed periodi systems, and in these there are indeed no hanges. In order to have true pro esses, as long as one remains in the domain of matrix me hani s, one must dire t one's attention to a
part
of
the system; this is no longer losed and enters into intera tion with the rest of the system. The question is what matrix me hani s an tell us about this.
(This is again one of the se tions originally drafted by Born.) As raised here, the question to be addressed is how to in orporate into matrix me hani s the (a tual) state of a system, and the time development of su h a state. The dis ussion given in the report may give rise to some onfusion, be ause it arguably ontains at least two, if not three, disparate approa hes to what is a state in quantum me hani s. The rst, re e ted in the above quotation, is the idea that a state of a system is always a stationary state, whi h stems from Bohr's quantum theory and whi h appears to have lived on through the BKS phase until well into the development of matrix me hani s, indeed at least as late as Heisenberg's dis overed independently by S hrödinger (1926d), E kart (1926 [22℄) and Pauli (Pauli to Jordan, 12 April 1926, in Pauli 1979, pp. 31520). For a des ription of the various ontributions, in luding the work of Lan zos (1926 [23℄), see Mehra and Re henberg (1987, pp. 63684). Muller (1997) argues that matrix me hani s and wave me hani s, as formulated and understood at the time, were nevertheless inequivalent theories. For on rete examples in whi h the two theories were indeed
onsidered to yield dierent predi tions, see below p. 143 (in luding the footnote) and the dis ussion on p. 470.
3.4 Interpretation
103
paper on resonan e (Heisenberg 1926b [35℄). The se ond is the idea that the state of a system is given by its wave fun tion, but it is bound up with the question of whether the latter should be seen as a `spread-out' entity, a `guiding eld', a `statisti al state' or something else.a Born's papers on ollisions (Born 1926a,b [30℄) an be said to ontain elements of both these approa hes. Yet a third approa h may well be present in the report, an approa h in whi h the notion of state would be purely an ee tive one. Some pronoun ements by Heisenberg, in orresponden e and in the un ertainty paper (Heisenberg 1927 [46℄), may support this further (tentative) suggestion. It seems to us that Born and Heisenberg's statements be ome learer if one is aware of the dierent ba kgrounds to their dis ussion. A
ordingly, we shall dis uss in turn, brie y, various developments that appear to have fed into their on eption of quantum me hani s, in parti ular previous work by Born and Wiener on generalising matrix me hani s (se tion 3.4.1), by Born on guiding elds (se tion 3.4.2), by Bohr as well as famously by Born on ollision pro esses (again se tion 3.4.2 and se tion 3.4.3), by Heisenberg on atoms in resonan e (se tion 3.4.4) and by Dira on the transformation theory (se tion 3.4.5). We shall then dis uss the treatment of interpretational issues given in the report in se tions 3.4.6 and 3.4.7. The latter in ludes some brief omments on the notion of a ` losed theory', whi h is prominent in some of Heisenberg's later writings (e.g. Heisenberg 1948), and whi h appears to be used here for the rst time. We do not laim to have settled the interpretational issues, and will return to several of them in Part II. Overall, the `physi al interpretation' of the report requires areful reading and assessment.
3.4.1 Matrix me hani s, Born and Wiener As we have seen, in the old quantum theory the only states allowed for atomi systems are stationary states, understood in terms of lassi al orbits subje t to the quantum onditions of Bohr and Sommerfeld. The same is true in the BKS theory, where stationary states be ome more abstra t and are represented by the olle tion of virtual os illators orresponding to the transitions of the atom from a given energy level. In both theories, dis ontinuous transitions between the stationary states, so- alled quantum jumps, are assumed to o
ur. Although in today's quantum theory one usually still talks about dis ontinuous transitions, a Cf. the next hapter.
104
From matrix to quantum me hani s
these are asso iated with the ollapse postulate and with the on ept of a measurement. The states performing these transitions are not ne essarily stationary states, i.e. eigenstates of energy (whether one des ribes them dynami ally in the S hrödinger pi ture or stati ally in the Heisenberg pi ture). This modern notion of state is strikingly absent also in matrix me hani s, as we have seen it formulated in the original papers. Indeed, the interpretation of the theory is still ostensibly in terms of stationary states and quantum jumps, but the formalism itself ontains only matri es, whi h an at most be seen as a olle tive representation of all stationary states, in the following sense. The matrix (2) in orporates the os illations orresponding to all possible transitions of the system. Ea h matrix element is formally analogous to a virtual os illator with frequen y νij = (Ei − Ej )/h ( orresponding to an atomi transition Ei → Ej ). Therefore, ea h row (or olumn) ontains all the frequen ies
orresponding to the transitions from (or to) a given energy level, mu h like a stationary state in the BKS theory. As in the above quotation, the matrix an thus be seen as the olle tive representation of all the stationary states of a losed system.a This analysis is further supported by examining Born and Wiener's work, in whi h the pi ture of stationary states as the rows of the position matrix (now the position operator) be omes even more expli it. While the fun tions x(t) remain uninterpreted, Born and Wiener appear to asso iate the `rows' or, rather, the ` olumns' of their operators with (stationary) states of motion. This is lear from the following. Born and Wiener introdu e a notion of ` olumn sum', that is, a generalisation of the sum of the elements in the olumn of a matrix. In dis ussing their main example, the free parti le, they then show that, for the position operator, this generalised olumn sum q(t, W ) takes the form s 2W + F (W ) , q(t, W ) = t (20) µ where µ is the mass of the parti le and F (W ) is a omplex-valued expression independent of t. They expli itly draw the on lusion that at least the real part of the generalised olumn sum represents a lassi al inertial motion with the energy W . a A somewhat similar idea appears to be expressed by Dira at the beginning of the dis ussion after the report (p. 442), where he emphasises the parallel between matrix me hani s and lassi al Hamiton-Ja obi theory, with the latter also des ribing not single traje tories but whole families of traje tories.
3.4 Interpretation
105
Note that this indi ates not only that Born and Wiener asso iate the state of a system with a olumn of the position operator. It also suggests that, in their view, at least a limited spatio-temporal pi ture of parti le traje tories in the absen e of intera tions is possible (analogously to the earlier limited use of spatio-temporal pi tures in des ribing atomi states in the absen e of transitions).
3.4.2 Born and Jordan on guiding elds, Bohr on ollisions The early history of the guiding eld idea, in onne tion with Einstein and with the BKS theory (in the ase of photons), and in onne tion with de Broglie (in the ase of both photons and material parti les), is dis ussed mainly in hapters 9 and 2, respe tively. Slater's original intention was in fa t to have the virtual elds to be guiding elds for the photons, whi h were to arry energy and momentum, but this aspe t was not in orporated in the BKS theory. However, after this theory was reje ted pre isely be ause of the results on energy and momentum
onservation in individual pro esses (detailed, for instan e, in Compton's report), Slater's original idea was eetingly revived. On 24 April 1925, after learning from Fran k that Bohr had in fa t given up the BKS theory, Born sent Bohr the des ription of su h a proposal (whi h, as he wrote, he had been working on for some weeks with Jordan). A manus ript by Born and Jordan followed, entitled `Zur Strahlungstheorie', whi h is found today in the Bohr ar hives. a The proposal ombines the BKS idea of emission of waves while the atom is in a stationary state with the emission of a light quantum during an instantaneous quantum jump, in order to give a spa etime pi ture of radiation. The light quantum thus follows the rear end of the wave. It
an be s attered or absorbed by other atoms (both pro esses depending on the dipole moment of the appropriate virtual os illators in the atoms), in the latter ase leaving a `dead' wave that has no further physi al ee ts. Born and Jordan had applied this pi ture with some su
ess to a few simple examples, and were intending to publish the idea in Naturwissens haften, provided Bohr or Kramers did not nd fault with it (Bohr 1984, pp. 845 and 30810). Bohr replied on 1 May, after re eipt of both the letter and the manus ript,
riti ising the proposal, on the grounds, rst, that the proposed me hanism did not guarantee that the traje tories of the light quanta would a Cf. Darrigol (1992, p. 253). We are espe ially grateful to Olivier Darrigol for helpful orresponden e on this matter.
106
From matrix to quantum me hani s
oin ide with the propagation of the wave, and se ond, that the ross se tion for the absorption ought to be a onstant in order for the parti le number density to be proportional to the intensity of the wave. Bohr reiterated the beliefs he had expressed to Fran k: that the oupling between state transitions in dierent atoms ex luded a des ription that used ans hauli h pi tures, and that he suspe ted the same on lusion to be likely in the ase of ollision phenomena (Bohr 1984, pp. 85 and 31011).a Bohr's work on ollisions, to whi h he alluded here, was an extension of the BKS idea of merely statisti al onservation laws (Bohr 1925).a The idea, as paraphrased by Born, was `to regard the eld of the parti le passing by in the same way as the eld of a light wave; thus, it only produ es a probability for the absorption of energy, and this [absorption℄ only o
urs when the ollision lasts su iently long (the parti le passes slowly)' (Born to Bohr, 15 January 1925, quoted in Bohr 1984, p. 73).b However, the Ramsauer ee t the anomalously low ross se tion of atoms of ertain gases for slow ele trons ould not be a
ommodated in Bohr's s heme. Bohr therefore was developing doubts about statisti al onservation laws (and further, about the feasibility altogether of a spa etime pi ture of ollisionsc ), even as he was submitting his paper on ollisions,d that is, even before the results of the Bothe-Geiger experiments were on rmed in April 1925. Bohr's paper was published only after Bohr in luded an addendum in July 1925, whi h draws the
onsequen es from both the Bothe-Geiger experiments and the di ulties with the Ramsauer ee t. In the same month of July, an explanation a See Bohr to Fran k, 21 April 1925, in Bohr (1984, pp. 35051). a Cf. also Darrigol (1992, pp. 24951) and pp. 8993 in Stolzenburg's introdu tion to Part I of Bohr (1984). b Another olourful paraphrase is in Bohr to Fran k, 30 Mar h 1925: `If two atoms have the possibility of settling their mutual a
ount, it is, of ourse, simplest that they do so. However, when the invoi es annot be submitted simultaneously, they must be satis ed with a running a
ount' (quoted in Bohr 1984, p. 74).
One of the most striking expressions of this is a passage in Bohr to Heisenberg, 18 April 1925: `Stimulated espe ially by talks with Pauli, I am for ing myself these days with all my strength to familiarise myself with the mysti ism of nature and am attempting to prepare myself for all eventualities, indeed even for the assumption of a oupling of quantum pro esses in separated atoms. However, the osts of this assumption are so great that they annot be estimated within the ordinary spa etime des ription' (Bohr 1984, pp. 36061). For other qualms about su h `quantum nonlo ality', f. also Jordan's habilitation le ture (1927f [62℄), in whi h Jordan onsiders the idea of mi ros opi indeterminism to be omprehensible only if the elementary random events are independent. (This le ture was translated into English by Oppenheimer and published in Nature as Jordan (1927g).) d The paper was re eived by Zeits hrift für Physik on 30 Mar h 1925; on that very day Bohr was expressing his doubts in a letter to Fran k (Bohr 1984, pp. 34850).
3.4 Interpretation
107
for the Ramsauer ee t was suggested by Elsasser (1925) in Göttingen, on the basis of de Broglie's matter waves.
3.4.3 Born's ollision papers The above provides a useful ba kdrop for dis ussing Born's own work on ollisions (Born 1926a,b [30℄),a whi h treats ollision problems on the basis of S hrödinger's wave me hani s.b This work also re e ts the idea that the states of a system are stationary states undergoing transitions. Indeed, Born presents the problem as that of in luding in matrix me hani s a des ription of the transitions between stationary states. The ase of ollisions, say between an atom and an ele tron, is hosen as the simplest for treating this problem (while still leading to interesting predi tions), be ause it is natural to expe t that in this ase the ombined system is asymptoti ally in a stationary state for the atom and a state of uniform translational motion for the ele tron. (Note that this is onne ted to the treatment of the free parti le by Born and Wiener.) If one an manage to des ribe the asymptoti behaviour of the ombined system mathemati ally, this will give on rete indi ations as to the transitions between the initial and nal asymptoti states. Born managed to nd the solution spe i ally by wave me hani al methods.c Note that Born onsiders two on eptually distin t obje ts, the wave fun tion on the one hand and the stationary states of the atom and the ele tron on the other, the onne tion between them being that the wave fun tion de nes a probability distribution over the stationary states. (Note also that he reserves the word `state' only for the stationary states.) Born solves by perturbation methods the time-independent S hrödinger equation for the ombined system of atom and ele tron under the ondition that asymptoti ally for z → ∞, the solution has the form 2π ψn (q)e λ z (a produ t of the n-th eigenstate of the atom with a plane wave oming from the z -dire tion), with energy τ . Born's solution has a For a modern dis ussion of ollisions, f. se tion 10.1. b The Ramsauer ee t, however, is ex luded from Born's dis ussion (1926b [30℄, footnote on p. 824). A few passages might be seen to refer to Born's ex hange with Bohr, in parti ular the remark that, at the pri e of dropping ausality, the usual spa etime pi ture an be maintained (1926b [30℄, p. 826).
Cf. Born to S hrödinger, 16 May 1927: `the simple possibility of treating with it aperiodi pro esses ( ollisions) made me rst believe that your on eption was superior' (quoted in Mehra and Re henberg 2000, p. 135).
108
From matrix to quantum me hani s
the form: XZ m
τ
αx+βy+γz>0
Φτnm (α, β, γ)ψm (q)eknm (αx+βy+γz+δ) dω ,
(21)
τ equals where the energy orresponding to the wave number knm τ Enm = hνnm + τ ,
(22)
the νnm being the transition frequen ies of the atom. The omponents of the superposition an thus be asso iated with various, generally inelasti , ollisions in whi h energy is onserved, and Born interprets |Φλnm (α, β, γ)|2 as the probability for the atom to be in the stationary state m and the ele tron to be s attered in the dire tion (α, β, γ).a But now, ru ially, sin e the initial wave fun tion orresponds to a fully determined stationary state and inertial motion, this probability is also the probability for a quantum jump from the given initial state to the given nal state, i.e. a transition probability. This idea is linked to that of a guiding eld. The link is made expli itly at the beginning of Born's se ond paper (whi h in ludes the details of the derivation and some quantitative predi tions). While Born judges that in the ontext of opti s one ought to wait until the development of a proper quantum ele trodynami s, in the ontext of the quantum me hani s of material parti les the guiding eld idea an be applied already, using the de Broglie-S hrödinger waves as guiding elds. The traje tories of material parti les, however, are determined by the guiding eld merely probabilisti ally (1926b [30℄, pp. 8034). In his
on lusion, Born regards the pi ture of the guiding eld as fundamentally indeterministi . A deterministi ompletion, if possible, would not have any pra ti al use. Born also expresses the hope that the `laws of motion for light quanta' will nd a similar treatment to the one given for ele trons, and refers to the di ulties `so far' of pursuing a guiding eld approa h in opti s (pp. 8267).b a A statisti al interpretation for the modulus squared of the oe ients of the wave fun tion in the energy basis was introdu ed also by Dira (1926 [37℄), at the same time as and presumably independently of Born ( f. Darrigol 1992, p. 333). Note further that, even though it may be tempting to assume that ea h traje tory pro eeds from the s attering entre, stri tly speaking to ea h stationary state of the free parti le orresponds a whole family of inertial traje tories. (Similarly, in the ase treated by Born and Wiener, ea h generalised olumn sum asso iated with a stationary state orresponds to two dierent inertial traje tories, depending on the sign of the square root in (20).) b Born's views on quantum me hani s from this period are also presented in Born (1927), an expanded version (published Mar h 1927) of a talk given by Born in August 1926.
3.4 Interpretation
109
3.4.4 Heisenberg on energy u tuations As dis ussed in the next hapter, Heisenberg in parti ular among matrix physi ists was opposed to S hrödinger's attempt to re ast and reinterpret quantum theory on the basis of ontinuous wave fun tions. S hrödinger's wave fun tions were meant from the start as des riptions of individual states of a physi al system. Even though in general they are abstra t fun tions (on on guration spa e), they an provide a pi ture of Bohr's stationary states. Furthermore, the solution of the timedependent S hrödinger equation appears to provide a generalisation of the state of a system as it evolves in time. Heisenberg appears to have been disturbed initially also by Born's use of S hrödinger's theory in the treatment of ollisions, an attitude re e ted in parti ular in his orresponden e with Pauli.a In this onne tion, the fa t that Pauli in his letter of 19 O tober (Pauli 1979, pp. 3409) was in ee t able to sket h how one ould reinterpret Born's results in terms of matrix elements, must have been of parti ular signi an e: `Your al ulations have given me again great hope, be ause they show that Born's somewhat dogmati viewpoint of the probability waves is only one of many possible s hemes'.b A few days later, Heisenberg sent Pauli the manus ript of his paper on u tuation phenomena (Heisenberg 1926b [35℄), in whi h he developed
onsiderations similar to Pauli's in the ontext of a hara teristi example, that of two atoms in resonan e. By fo ussing on the subsystems of a losed system, Heisenberg was able to derive expressions for (transition) probabilities within matrix me hani s proper, without having to introdu e the wave fun tion as an external aid. A very similar result was derived at the same time by Jordan (1927 [36℄), using two systems with a single energy dieren e in ommon. We shall now sket h Heisenberg's reasoning. We adapt the presentation of the argument given by Heisenberg in The Physi al Prin iples of the Quantum Theory (Heisenberg 1930b, pp. 1427),c whi h is more general than the one given in the paper and learer than the one given in Born and Heisenberg's report. Take two systems, 1 and 2, that are in resonan e. Consider, to begin with, that the frequen y of the transition n1 → m1 in system a `One senten e [of Born's paper℄ reminded me vividly of a hapter in the Christian
reed: An ele tron is a plane wave... ' (Heisenberg to Pauli 28 July 1926, in Pauli 1979, p. 338, original emphasis). b Heisenberg to Pauli, 28 O tober 1926, in Pauli (1979, p. 350).
This very remarkable book is an expanded English edition of Heisenberg (1930a).
110
From matrix to quantum me hani s
1 orresponds to exa tly one transition frequen y in system 2, say ν (1) (n1 m1 ) = ν (2) (n2 m2 ) .
(23)
If the systems are un oupled, the ombined system has the degenerate eigenvalue of energy (1) (2) . + Wn(2) = Wm + Wm Wn(1) 2 1 2 1
(24)
If we ouple weakly the two systems, the degenera y will be lifted. Let us label the new eigenstates of the ombined system by a and b. We
an now onsider the matrix S that transforms the basis of eigenstates of energy of the oupled system to the (produ t) basis of eigenstates of energy of the un oupled systems. In parti ular we an onsider the submatrix Sn1 m2 ,a Sn1 m2 ,b . (25) Sn2 m1 ,a Sn2 m1 ,b Choose one of the stationary states of the ombined system, say a. If the ombined system is in the state a, what an one say about the energy of the subsystems, for instan e H (1) ? Heisenberg's answer, in the terminology and notation of his book (1930b), is that in the state a, the time average of H (1) (whi h is no longer a diagonal matrix, thus no longer time-independent), or of any fun tion f (H (1) ), is (1) )|Sm1 n2 ,a |2 . )|Sn1 m2 ,a |2 + f (Wm f (H (1) ) = f (Wn(1) 1 1
(26)
Sin e f is arbitrary, Heisenberg on ludes that |Sn1 m2 ,a |2 is the probability that the state n1 has remained the same (and the state m2 has remained the same), and |Sm1 n2 ,a |2 is the probability that the state n1 has jumped to the state m1 (and m2 has jumped to n2 ). The asso iated transfer of energy between the systems appears to be instantaneous, in that a quantum jump in one system (from a higher to a lower energy level) is a
ompanied by a orresponding jump in the other system (from a lower to a higher energy level). The paper merely mentions, without elaborating further, that a light quantum (or better a `sound quantum') is ex hanged over and over again between the two systems. The pi ture thus avoids the non- onservation of energy of the BKS theory, at the pri e of what appears to be an expli it orrelation at a distan e. In modern terms, Heisenberg has al ulated the expe tation value of
3.4 Interpretation
111
the observable f (H (1) ) ⊗ 1 in the state a: ha|f (H (1) ) ⊗ 1|ai =
ha |n1 m2 ihn1 m2 | f (H (1) ) ⊗ 1 |n1 m2 ihn1 m2 | ai+
ha |m1 n2 ihm1 n2 | f (H (1) ) ⊗ 1 |m1 n2 ihm1 n2 | ai = hn1 |f (H
(1)
hm1 |f (H
(27)
2
)|n1 i|hn1 m2 |ai| +
(1)
)|m1 i|hm1 n2 |ai|2 .
Note, however, that rather than fo ussing on the idea that |Sn1 m2 ,a |2 = |hn1 m2 |ai|2
(28)
is a onditional probability (in fa t what we would today all a transition probability), Heisenberg is still fo ussing on the transitions between the stationary states of the subsystems, as in Born's work on ollisions.
3.4.5 Transformation theory Less than three weeks after ompleting his draft on u tuation phenomena, we nd Heisenberg reporting to Pauli about Dira 's transformation theory, whi h generalises pre isely the formal expression of a onditional probability given by Heisenberg in terms of the transformation matrix: `Here [in Copenhagen℄ we have also been thinking more about the question of the meaning of the transformation fun tion S and Dira has a hieved an extraordinarily broad generalisation of this assumption from my note on u tuations' (Pauli 1979, p. 357, original emphasis). Dira indeed presents his results in his paper, signi antly titled `The physi al interpretation of quantum dynami s' (1927a [38℄), as a generalisation of Heisenberg's approa h. The main goal of the paper is the following.a Take any pair of onjugate matrix quantities ξ and η , and any ` onstant of integration' g(ξ, η).b Given a value of ξ as a -number, nd the fra tion of η -spa e for whi h g lies between any two numeri al values. If η is assumed to be distributed uniformly,c this result will yield the frequen y of the given values of g in an ensemble of systems. a In our presentation we shall partly follow the analysis by Darrigol (1992, pp. 33745). b This term is meant to in lude any value of a dynami al quantity at a spe i ed time t = t0 (Dira 1927 [38℄, p. 623, footnote).
This assumption may sound strange, espe ially sin e ξ is assumed to have a de nite value and ξ and η are anoni ally onjugate. Cf. however Dira 's remarks on interpretation below.
112
From matrix to quantum me hani s
Equivalently, we an state Dira 's goal as that of nding the expe tation value (or more pre isely, the η -average) of any xed-time observable g , given a ertain value of ξ . The main part of the paper is devoted to developing a `transformation theory' that will allow Dira to write the quantity g not in the usual energy representation but in an arbitrary ξ -representation. Dira then suggests taking the η -averaged value of g , for ξ taking the -number value ξ ′ , as given by the diagonal element gξ′ ξ′ of g in this representation. This is in fa t a natural if `extremely broad' generalisation, to an arbitrary pair of onjugate quantities (ξ, η), of the assumption that the diagonal elements of a matrix in the energy representation (su h as Heisenberg's f (H (1) ) from the previous se tion) are time averages, although the justi ation Dira gives for this assumption is merely that `the diagonal elements .... ertainly would [determine the average values℄ in the limiting ase of large quantum numbers' (Dira 1927 [38℄, p. 637). Dira writes the elements of a general transformation matrix between two omplete sets of variables ξ and α (whether dis rete or ontinuous) as (ξ ′ /α′ ) (what we would now write hξ ′ |α′ i), so that the matrix elements transform as Z gξ′ ξ′′ = (ξ ′ /α′ )gα′ α′′ (α′′ /ξ ′′ )dα′ dα′′ (29) (Dira 's notation is meant to in lude the possibility of dis rete sums). His main analyti tool is the manipulation of δ -fun tions and their derivatives. Dira shows in parti ular that for the quantity ξ itself, ξξ′ ξ′′ = ξ ′ δ(ξ ′ − ξ ′′ ) ,
(30)
and that for the quantity η anoni ally onjugate to ξ , ηξ′ ξ′′ = −i~δ ′ (ξ ′ − ξ ′′ ) .
(31)
In a mixed representation, one has ξξ′ α′ = ξ ′ (ξ ′ /α′ ) ,
(32)
and ηξ′ α′ = −i~
∂ ′ ′ (ξ /α ) , ∂ξ ′
(33)
from whi h follows ∂ gξ′ α′ = g ξ, −i~ ′ (ξ ′ /α′ ) ∂ξ
for arbitrary g = g(ξ, η).
(34)
3.4 Interpretation
113
Choosing α su h that g is diagonal in the α-representation, one has gξ′ α′ = gα′ (ξ ′ /α′ ) ,
(35)
where the gα′ are the eigenvalues of g . Therefore, (34) be omes a dierential equation for the (ξ ′ /α′ ) (seen as fun tions of ξ ′ ), whi h generalises the time-independent S hrödinger equation.a On e this equation is solved, one ould obtain the desired gξ′ ξ′ from the gξ′ α′ by the appropriate transformation. The way Dira states his nal result, however, is by onsidering the matrix δ(g − g ′ ). The numeri al fun tion δ(g − g ′ ), when integrated, Z b δ(g − g ′ )dg ′ , (36) a
yields the hara teristi fun tion of the set a < g < b. Therefore, in Dira 's proposed interpretation, the diagonal elements of the orresponding matrix in the ξ -representation yield, for ea h value ξ = ξ ′ , the fra tion of the η -spa e for whi h a < g < b. That is, if η is assumed to be distributed uniformly, these diagonal elements yield the onditional probability for a < g < b given ξ = ξ ′ . Thus, the diagonal elements of the matrix δ(g − g ′ ) yield the orresponding onditional probability density for g given ξ = ξ ′ . But now, e.g. sin e for any fun tion f (g) one has Z f (g)ξ′ ξ′′ = (ξ ′ /g ′′ )f (g ′′ )(g ′′ /ξ ′′ )dg ′′ , (37) one has in parti ular Z δ(g − g ′ )ξ′ ξ′ = (ξ ′ /g ′′ )δ(g ′′ − g ′ )(g ′′ /ξ ′ )dg ′′ = (ξ ′ /g ′ )(g ′ /ξ ′ ) .
(38)
Therefore, the onditional probability density for g given ξ = ξ ′ is equal to |(g ′ /ξ ′ )|2 , a result that Dira illustrates by dis ussing Heisenberg's example of transition probabilities in resonant atoms and Born's ollision problem. In parallel with Dira 's development of transformation theory, Jordan (1927b, [39℄) also arrived at a similar theory, following on dire tly from his paper on quantum jumps (1927a [36℄) and from his earlier work on anoni al transformations (1926a,b), to whi h Dira also makes an expli it onne tion. Although Born and Heisenberg state in the report (p. 430) that the two methods are equivalent, Darrigol (1992, pp. 3434) a Dira also gives a generalisation of the time-dependent S hrödinger equation.
114
From matrix to quantum me hani s
points to some subtle dieren es, whi h are also related to Dira 's riti ism in the general dis ussion of Jordan's introdu tion of anti ommuting elds (p. 502). Dira also notes that his theory generalises the work by Lan zos (1926 [23℄). The development of transformation theory from the idea of anoni al transformations led further towards the realisation that quantum me hani al operators a t on a Hilbert spa e and that the natural transformations are in fa t unitary.a Dira on ludes his paper with an intriguing suggestion of .... a point of view for regarding quantum phenomena rather dierent from the usual ones. One an suppose that the initial state of a system determines de nitely the state of the system at any subsequent time.
If, however, one
des ribes the state of the system at an arbitrary time by giving numeri al values to the o-ordinates and momenta, then one annot a tually set up a one-one orresponden e between the values of these o-ordinates and momenta initially and their values at a subsequent time. All the same one an obtain a good deal of information (of the nature of averages) about the values at the subsequent time onsidered as fun tions of the initial values. The notion of probabilities does not enter into the ultimate des ription of me hani al pro esses: only when one is given some information that involves a probability (e.g., that all points in
η -spa e
are equally probable for representing the
system) an one dedu e results that involve probabilities. (Dira 1927a [38℄, p. 641)
Here Dira does not impute indeterminism to nature itself (the matrix equations are after all deterministi ), but instead apparently identi es the sour e of the statisti al element in the hoi e of probabilisti initial data.a A
ording to Heisenberg, however, and despite the generality of the results and the `extraordinary progress' obtained (Pauli 1979, p. 358), Dira 's transformation theory did not resolve the question of the meaning of quantum me hani s. As Darrigol (1992, p. 344) emphasises, there is no notion of state ve tor in Dira 's paper (the well-known bras and kets do not appear yet). C-number values, and probability distribua See La ki (2004), who gives details also of London's (1926a,b) ontributions to this development. Note also the onne tion between transformation theory and the work on the `equivalen e' between matrix me hani s and wave me hani s. a This should be ompared to the general dis ussion, in whi h Dira (a) talks of `an irrevo able hoi e of nature' (p. 495) in relation to the out omes of an experiment, (b) uses expli itly (perhaps for the rst time) the notion of the state ve tor, when he a rms that `[a℄
ording to quantum me hani s the state of the world at any time is des ribable by a wave fun tion ψ, whi h normally varies a
ording to a ausal law, so that its initial value determines its value at any later time' (p. 495), and in whi h ( ) he des ribes the initial data taken for quantum me hani al al ulations as des ribing `a ts of freewill', namely `the disturban es that an experimenter applies to a system to observe it' (p. 494); see also se tion 8.2.
3.4 Interpretation
115
tions over -number values, now refer to arbitrary quantities or pairs of quantities. As Heisenberg wrote: `there are too many -numbers in all our utteran es used to des ribe a fa t' (Pauli 1979, p. 359). Cru ially, however, the energy variable and the stationary states no longer played a privileged role.
3.4.6 Development of the `statisti al view' in the report In the report, Born and Heisenberg appear to understand Born's ollision papers (1926a,b [30℄) on the one hand and the papers by Heisenberg (1926b [35℄) and Jordan (1927a [36℄) on the other broadly in the same way, as seeking to obtain `information .... about a tual phenomena', by `dire t[ing℄ one's attention to a part of the system' (p. 420 of the report). And indeed, by onsidering oupled systems all of these papers manage to derive quantitative expressions for the probabilities of quantum jumps between energy eigenstates. Heisenberg's setting is the one hosen in the report, and sin e Heisenberg's treatment of intera ting systems does not use the formalism of wave me hani s, this hoi e may be intended to make the point that matrix me hani s an indeed a
ount for time-dependent phenomena without the aid of wave me hani s. The form of Heisenberg's result (26) as given in the report is in terms of the expe ted deviation of the value of energy from a given initial value, for instan e n1 :a )= δfn1 =f (H (1) ) − f (Wn(1) 1
)}|Sn1 m2 ,a |2 + ) − f (Wn(1) {f (Wn(1) 1 1 (1) ) {f (Wm 1
−
)}|Sm1 n2 ,a |2 f (Wn(1) 1
(39)
.
If we write Φn1 m1 for the probability of the transition n1 → m1 in system 1, this be omes equation (20) of the report, ex ept that Born and Heisenberg label the matrix elements Sn1 m2 ,a and Sm1 n2 ,a , respe tively, by the transitions n1 m2 → n1 m2 and n1 m2 → m1 n2 , that is, as Sn1 m2 ,n1 m2 and Sn1 m2 ,m1 n2 , omitting referen e to the state a.b One further dieren e between our des ription above and the one given in the report is that Born and Heisenberg are treating the ase in whi h the transition n1 → m1 in system 1 may resonate with more than one a Note that |Sn1 m2 ,a |2 + |Sm1 n2 ,a |2 = 1, be ause a is normalised. b In Heisenberg's paper, the matrix (25) orresponds to a 45-degree rotation, so the probabilities are independent of the hoi e of a or b.
116
From matrix to quantum me hani s
transition in system 2. In this ase, the total transition probability Φn1 m1 is no longer equal to |Sn1 m2 ,m1 n2 |2 but to X |Sn1 m2 ,m1 n2 |2 . Φn1 m1 = (40) m2 n2
Thus we also have equation (21) of the report. It is only after this matrix me hani al dis ussion that Born and Heisenberg introdu e the time-dependent S hrödinger equation, as a more ` onvenient' formalism for `thinking of the system under onsideration as oupled to another one and negle ting the rea tion on the latter' (p. 423). This suggests that Born and Heisenberg may onsider the timedependent S hrödinger equation only as an ee tive des ription. This impression is reinfor ed by omparing with Heisenberg's book (1930b, pp. 14850) where, after the above derivation, Heisenberg ontinues with a more general derivation of time-dependent probabilities, whi h he then relates to the usual time-dependent S hrödinger equation. Nevertheless, Born and Heisenberg use the wave fun tion throughout the ensuing dis ussion of probabilities, noting that this formalism `leads to a further development of the statisti al view', by whi h they mean in parti ular the idea of interferen e of probabilities.a First of all Born and Heisenberg relate the wave fun tion to probabilities. They take the time-dependent transformation matrix S(t) given by the unitary evolution. For the oe ients of the wave fun tion in the energy basis one has (equation (25) in the report): X cn (t) = (41) Snm (t)cm (0) . m
If now all cm (0) ex ept one (say, ck (0)) are zero, from the assumption that a system is always in a stationary state it is natural to on lude that the |Snk (t)|2 are the probabilities for transitions to the respe tive energy states (`transition probabilties'), and the |cn (t)|2 are the resulting probabilities for the stationary states (`state probabilities'). In support of this interpretation (whi h is the same as in Born's ollision papers), the report quotes in parti ular Born's paper on the adiabati prin iple (1926 [34℄), that is, Born's proof that in the adiabati ase the transition probabilities between dierent states tend to zero, in a
ordan e with Ehrenfest's (1917) prin iple.b a On these matters f. also se tion 8.1. b Ehrenfest had the idea that sin e quantised variables annot hange by arbitrarily small amounts, they should remain onstant under adiabati perturbations. This
3.4 Interpretation
117
Then Born and Heisenberg ome to dis ussing interferen e.a This is also the rst time in the presentation of the physi al interpretation of the theory that measurements enter the pi ture. Born and Heisenberg note that if ck is not the only non-zero oe ient at t = 0, then (41) does not imply X |cn (t)|2 = (42) |Snm (t)|2 |cm (0)|2 , m
but that instead one has
|cn (t)|2 = |
X
Snm (t)cm (0)|2 .
(43)
m
The passage immediately following this is both remarkable and, in our opinion, very signi ant (p. 425):a it should be noted that this `interferen e' does not represent a ontradi tion with the rules of the probability al ulus, that is, with the assumption that 2 the |Snk | are quite usual probabilities. In fa t, the omposition rule [(42)℄
follows from the on ept of probability for the problem treated here when and |cn |2 of the atoms in
only when the relative number, that is, the probability the state phases
n,
γn
has been
established
beforehand
experimentally.
In this ase the
are unknown in prin iple, so that [(43)℄ then naturally goes over to
[(42)℄ .... .
(The passage ends with a referen e to Heisenberg's un ertainty paper.) How do Born and Heisenberg propose to resolve this apparent ontradi tion? It would make sense to say that the |Snk (t)|2 annot be taken in general as probabilities for quantum jumps, be ause the derivation of |Snk (t)|2 as a transition probability works only in a spe ial ase (that is, presumably, if the energy at t = 0 has in fa t been measured). On this reading, there might on eivably exist some quite dierent transition led him to formulate the prin iple stating that the lassi al variables to be quantised are the adiabati invariants of the system. Cf. Born (1969, pp. 113 .). a Born and Heisenberg give redit to Pauli for the notion of interferen e of probabilities (p. 424). Note that Pauli ontributed signi antly to the development of the `statisti al view', albeit mainly in orresponden e and dis ussion. Contrary to what is ommonly assumed, the idea of a probability density for position is not ontained in Born's ollision papers, but appears in fa t in Pauli's letter to Heisenberg of 19 O tober 1926 (Pauli 1979, p. 3409), together with the idea of the orresponding momentum density, and in print in a footnote of Pauli's paper on gas degenera y and paramagnetism (1927 [44℄). (See also Heisenberg to Pauli, 28 O tober 1926, in Pauli 1979, pp. 34052.) Jordan (1927b [39℄), in his se ond paper on the transformation theory, even gives redit to Pauli for the introdu tion of arbitrary transition probabilities and amplitudes. a For further dis ussion, see se tion 6.1.2.
118
From matrix to quantum me hani s
probabilities, whi h lie outside the s ope of quantum me hani s and are presumbly of no pra ti al value (like a deterministi ompletion in the
ase of ollision pro esses).a However, this reading does not seem to t what Born and Heisenberg a tually say. Their suggestion seems to be that the |Snk (t)|2 are indeed always transition probabilities, but that the |cn |2 are not always state probabilities: the |cn |2 will be state probabilities if and only if the energies have been measured (non-sele tively). This seems analogous to Heisenberg's (1927 [46℄, p. 197) idea in the un ertainty paper that the `law of ausality' is inappli able be ause it is impossible in prin iple to know the present with su ient a
ura y (i.e. the ante edent of the law of ausality fails).a Born and Heisenberg swiftly move on to generalising the dis ussion to the ase of arbitrary observables, on the basis of Dira 's and Jordan's transformation theory (Dira 1927a [38℄, Jordan 1927b, [39℄). They introdu e the interpretation of |ϕ|2 as a position density, and onsider in parti ular the density |ϕ(q ′ , W ′ )|2 de ned by the stationary state ϕ(q ′ , W ′ ) with the energy W ′ , or in modern notation, |ϕ(q ′ , W ′ )|2 = |ϕW ′ (q ′ )|2 = |hq ′ |W ′ i|2 .
(44)
This is immediately generalised to arbitrary pairs of observables q and Q with values q ′ and Q′ : |ϕ(q ′ , Q′ )|2 = |hq ′ |Q′ i|2 .
(45)
Born and Heisenberg all this a `relative state probability', reserving the term `transition probability' for the ase of a single observable evolving in time (or depending on some external parameter), always with the proviso that in general one should expe t interferen e of the orresponding `transition amplitudes'. Note that the physi al interpretation of this generalisation makes sense only if one takes over from the above the idea that probabilities su h as a That su h probabilities an be de ned (albeit non-uniquely), leading to well-de ned sto hasti pro esses for the quantum jumps, was shown expli itly by Bell (1984). a It appears not to be well known that the last 20 pages of the original types ript of Heisenberg's un ertainty paper are ontained in AHQP, mis atalogued as an 'in omplete and unpublished paper (pp. 12-31)' by Kramers.7 The types ript
ontains slight textual variants (as ompared with the published version) and manus ript orre tions in what appears to be Heisenberg's hand, but does not in lude the famous addendum in proof in response to Bohr's riti ism ( f. p. 146 below). On Heisenberg's treatment of the `law of ausality', see also Beller (1999, pp. 11013).
3.4 Interpretation
119
|ϕ|2 are well-de ned only upon measurement, or more pre isely, that a tual frequen ies upon measurement will be given by the expression |ϕ|2 . Born and Heisenberg's terminology, however, is somewhat ambiguous.a A major on eptual shift appears to be taking pla e, whi h may be easy to miss. Do quantum jumps still o
ur whenever two systems intera t, or do they now o
ur only between measurements? Indeed, are systems always in stationary states, as has been expli itly assumed until now, or only when the energy is measured? Heisenberg's un ertainty paper (on pp. 1901), as well as the orresponden e with Pauli (Heisenberg to Pauli, 23 February 1927, in Pauli 1979, pp. 37682) both mention expli itly the loss of a privileged status for stationary states. It seems that, even though we are not expli itly told so, the pi ture of quantum jumps (that is, of probabilisti transitions between possessed values of energy) is shifting to that of probabilisti transitions from one measurement to the next. The idea that frequen ies are well-de ned only upon measurement appears to play the same role as von Neumann's proje tion postulate. As dis ussed in more detail in hapter 6, however, it is far from lear whether that is what Born and Heisenberg have in mind. A similar notion is introdu ed in Heisenberg's un ertainty paper (1927 [46℄, p. 186), but again in terms that are `somewhat mysti al' (Pauli to Bohr, 17 O tober 1927, in Pauli 1979, p. 411). It appears expli itly in the pro eedings only in the general dis ussion, in Born's main ontribution (p. 483) and in the intriguing ex hange between Dira and Heisenberg (pp. 493 .). In hapter 6 and se tion 8.1, we shall return to Born and Heisenberg's view of interferen e and to the question of whether, a
ording to them, the ollapse of the wave fun tion and the time-dependent S hrödinger evolution are at all fundamental pro esses.
3.4.7 Justi ation and overall on lusions The following se tion III (drafted by Heisenberg) presents Jordan's axiomati formulation of quantum me hani s (Jordan 1927b, [39℄),a and justi es the ne essity of a statisti al view in the ontext of Heisenberg's notion of un ertainty (Heisenberg 1927 [46℄). Born and Heisenberg a In one paragraph they refer to `the probability that for given energy W ′ the
oordinate q ′ is in some given element dq ′ ', whereas in the next they refer to `the probability, given q ′ , to nd the value of Q′ in dQ′ ' (p. 427; itali s added). a This formulation, whi h is how Jordan presents his transformation theory, was expli itly intended as a generalisation of the formalisms of matrix me hani s, wave me hani s, q-number theory and of Born and Wiener's original operator formalism.
120
From matrix to quantum me hani s
argue as follows. Even in lassi al me hani s, if ertain quantities (for instan e the phases of the motion) were known only with a ertain impre ision, the future evolution of the system would be only statisti ally
onstrained. Now, the un ertainty relations prevent one from determining the values of all physi al quantities, providing a fundamental limit of pre ision. In addition, quantum me hani s pres ibes dierent laws for the time evolution of the statisti al onstraints. Impre ise initial
onditions an be des ribed by hoosing ertain `probability fun tions' (this is the losest Born and Heisenberg ome to dis ussing the `redu tion of the wave pa ket' as presented in the un ertainty paper), and `the quantum me hani al laws determine the hange (wave-like propagation) of these probability fun tions' (p. 432). Born and Heisenberg laim that dis ussion of the ases in whi h these laws oin ide to a very good approximation with the lassi al evolution of a probability density justi es the lassi al treatment of α- and β -parti le traje tories in a
loud hamber.a They thus maintain that the statisti al element in the theory an be re on iled with ma ros opi determinism.b Se tion III arguably addresses the dual task set in the introdu tion of ensuring that quantum me hani s is ` onsistent in itself' and of showing that quantum me hani s an be taken to `predi t unambiguously the results for all experiments on eivable in its domain' (p. 409). This task appears to be related to two on eptual desiderata, that the theory be intuitive (ans hauli h) and losed (abges hlossen). These are tou hed upon brie y in the report, espe ially in the introdu tion and on lusion, but are important both in the debate with S hrödinger (see se tion 4.6) and in some of Heisenberg's later writings (in parti ular, Heisenberg 1948). The report appears to be the rst instan e in whi h Heisenberg uses the on ept of a losed theory.a a Born and Heisenberg's remarks about dierent laws of propagation of the probabilities may refer to the onditions under whi h (43) redu es to (42), whi h would arguably be an early example of de oheren e onsiderations. However, the remark is too brief, and Born and Heisenberg may be merely omparing the spreading of the quantum probabilities with that of a Liouville distribution. For a modern treatment of the latter omparison, see Ballentine (2003). b For further dis ussion of these issues see se tions 6.2.1 and 6.4. a Compare also S heibe (1993) on the on ept of losed theories in Heisenberg's thought. The origin of this on ept has also been tra ed to two earlier papers (Heisenberg 1926a [28℄ and 1926 [60℄); see for instan e Chevalley (1988). However, in the rst paper there is no mention of losed theories, only of losed systems of terms (symmetri and antisymmetri ), a point also repeated in the se ond paper. In the latter, Heisenberg mentions the need to introdu e equations for the matrix variables in order to obtain a ` losed theory', but this does not seem to be the same use of the term as in the report. We wish to thank also Gregor S hiemann for orresponden e and referen es on this topi .
3.4 Interpretation
121
As de ned in the report, a losed theory is one that has a hieved a de nitive form, and is no longer liable to modi ation, either in its mathemati al formulation or in its physi al meaning. This is made more pre ise in later presentations (e.g. Heisenberg 1948), in whi h Heisenberg in ludes the appli ability of the on epts of a theory in the analysis. All
losed theories possess a spe i domain of appli ation, within whi h they are and will always remain orre t. Indeed, their on epts always remain part of the s ienti language and are onstitutive of our physi al understanding of the world. In the report, quantum me hani s (without the in lusion of ele trodynami s) is indeed taken to be a losed theory,a so that dierent assumptions about the physi al meaning of quantum me hani s (su h as S hrödinger's idea of taking |ϕ|2 to be a harge densityb ), would lead to ontradi tions with experien e. Thus, the report ends on a note of utmost on den e.
a Heisenberg (1948) lists Newtonian me hani s, Maxwellian ele trodynami s and spe ial-relativisti physi s, thermodynami s, and nonrelativisti quantum me hani s as the four main examples of su h theories. b Des ribed in more detail in se tion 4.4.
122
Notes to pp. 93118
Ar hival notes 1 Cf. also Lorentz to Ehrenfest, 4 July 1927, AHQP-EHR-23 (in Dut h). 2 This and the following remarks are based on Born to Lorentz, 23 June 1927, AHQP-LTZ-11 (in German). 3 Heisenberg to Lorentz, 23 June 1927, AHQP-LTZ-12 (in German). 4 Born to Lorentz, 29 August 1927, AHQP-LTZ-11 (in German). 5 Heisenberg to Lorentz, 24 August 1927, AHQP-LTZ-12 (in German). 6 Heisenberg to Einstein, 18 February 1926, AEA 12-172.00 (in German). 7 AHQP-28 (H. A. Kramers, notes and drafts 192652), se tion 6.
4 S hrödinger's wave me hani s
S hrödinger's work on wave me hani s in 1926 appears to have been driven by the idea that one ould give a purely wave-theoreti al des ription of matter. Key elements in this pi ture were the idea of parti les as wave pa kets (se tion 4.3) and the possible impli ations for the problem of radiation (se tion 4.4). This pure wave theory, in ontrast to de Broglie's work, did away with the idea of point parti les altogether (se tion 4.5). The main on i t, however, was between S hrödinger and the proponents of quantum me hani s (se tion 4.6), both in its form at the time of S hrödinger's papers and in its further developments as sket hed in the previous hapter. For referen e, we provide a brief hronology of S hrödinger's writings relating to wave me hani s up to the Solvay onferen e: • Paper on Einstein's gas theory, submitted 15 De ember 1925, published 1 Mar h 1926 (S hrödinger 1926a). • First paper on quantisation, submitted 27 January 1926, addendum in proof 28 February 1926, published 13 Mar h 1926 (S hrödinger 1926b). • Se ond paper on quantisation, submitted 23 February 1926, published 6 April 1926 (S hrödinger 1926 ). • Paper on the relation between wave and matrix me hani s (`equivalen e paper'), submitted 18 Mar h 1926, published 4 May 1926 (S hrödinger 1926d). • Paper on mi ro- and ma rome hani s ( oherent states for the harmoni os illator), published 9 July 1926 (S hrödinger 1926e). • Third paper on quantisation, submitted 10 May 1926, published 13 July 1926 (S hrödinger 1926f).
123
124
S hrödinger's wave me hani s
• Fourth paper on quantisation, submitted 21 June 1926, published 5 September 1926 (S hrödinger 1926g). • Review paper in English for the Physi al Review, submitted 3 September 1926, published De ember 1926 (S hrödinger 1926h). • Prefa e to the rst edition of Abhandlungen zur Wellenme hanik, dated November 1926 (S hrödinger 1926i). • Paper on the Compton ee t in wave me hani s, submitted 30 November 1926, published 10 January 1927 (S hrödinger 1927a). • Paper on the energy-momentum tensor, submitted 10 De ember 1926, published 10 January 1927 (S hrödinger 1927b). • Paper on energy ex hange in wave me hani s, submitted 10 June 1927, published 9 August 1927 (S hrödinger 1927 ).
We shall now dis uss the above points in turn, after a brief dis ussion of the planning of S hrödinger's report for the onferen e (se tion 4.1) and a summary of the report itself (se tion 4.2).
4.1 Planning of S hrödinger's report As reported in hapter 1, the s ienti ommittee of the Solvay institute met in Brussels on 1 and 2 April 1926 to plan the fth Solvay onferen e. Lorentz had asked Ehrenfest to suggest some further names of possible parti ipants, and it is in Ehrenfest's letter of 30 Mar h1 that S hrödinger's name is rst mentioned in onne tion with the onferen e.a On this o
asion, Ehrenfest suggested S hrödinger on the basis of a paper in whi h S hrödinger proposed an expression for the broadening of spe tral lines due to the Doppler ee t, and whi h applied the onservation laws to phenomena involving single light quanta (S hrödinger 1922).b Evidently, neither Lorentz nor Ehrenfest were yet aware of S hrödinger's work on wave me hani s. In the meantime, S hrödinger sent to Lorentz the proof sheets of his rst two papers on quantisation, also on 30 Mar h,2 thus initiating his well-known orresponden e with Lorentz on wave me hani s.c a S hrödinger had already been a parti ipant in the fourth Solvay onferen e, though not a speaker; f. Moore (1989, pp. 1578). b As S hrödinger points out, the al ulated broadening of the spe tral lines is small
ompared to that expe ted on the basis of the thermal agitation of the radiating gas, otherwise the ee t ould be used as a test of the light quantum hypothesis.
Most of this orresponden e is translated in Przibram (1967). In the letter of 30 Mar h, not in luded there, S hrödinger suggests reading the se ond paper on quantisation before the rst, whi h should be seen rather an as example of an appli ation. Also, he writes that the variational prin iple of the rst paper is
4.1 Planning of S hrödinger's report
125
A
ordingly, already in a report of 8 April 1926 to the administrative
ommission,3 S hrödinger is listed as a possible substitute for Heisenberg for a le ture on the `adaptation of the foundations of dynami s to the quantum theory'. (It is unlikely, however, that the papers rea hed Lorentz before the meeting in Brussels.a ) In January 1927 then, as most of the other parti ipants, S hrödinger was invited to the fth Solvay
onferen e.5 A few weeks later, S hrödinger had the opportunity to dis uss personally with Lorentz the plans for the report `under the beautiful palms of Pasadena', as he re alls in a letter of June 1927. In the same letter, we nd a useful sket h of the theme and fo us of S hrödinger's report; we also gather that S hrödinger was wary of the potential for a onfrontation in Brussels:6 ....
I nurtured the quiet hope you would yet return to your rst plan and
entrust only Messrs [d℄e Broglie and Heisenberg with reports on the new me hani s. But now you have de ided otherwise and I will of ourse happily perform my duty. Yet I fear that the `matri ians' (as Mr Ehrenfest used to say) will feel disadvantaged. Should it ome to dierent views, whi h might after all urge on the ommittee the wish to limit the reports to two, you know, dear Professor, that I shall always happily remit my harge into your hands.
A
ording to S hrödinger's sket h, the report is to stress points of prin iple, rather than the (by then many) appli ations of the theory. First of all, one has to distinguish learly between two wave-me hani al theories: a theory of waves in spa e and time (whi h however runs into di ulties espe ially with the many-ele tron problem), and the highly su
essful theory of waves in on guration spa e (whi h however is not relativisti ). A di ulty of prin iple to be dis ussed in the ontext of the spa etime theory is the possibility of developing an intera ting theory, whi h seems to require distinguishing between the elds generated by dierent parti les, and whether this an be done in a sensible way, or perhaps be avoided.a In the ontext of the on guration-spa e theory, the main question is how to interpret the wave fun tion. S hrödinger given a sensible formulation only in the addendum in proof. Finally, he mentions the paper on Einstein's gas theory in the Physikalis he Zeits hrift (1926a) as a kind of preparatory work. Note that Lorentz on 27 May thanks S hrödinger for the proof sheets of three papers rather than the two mentioned in S hrödinger's letter. This third paper is learly the equivalen e paper (1926d), and was presumably sent separately ( f. Przibram 1967, pp. 43 and 556). a Lorentz had written to Ehrenfest on 29 Mar h from Paris, where he had another meeting, and appears to have travelled to Brussels dire tly from there.4 a Cf. p. 461 of the report.
126
S hrödinger's wave me hani s
mentions the widespread view that the wave des ribes only ensembles, as well as his own `preferred interpretation as a real des ription of the individual system, whi h thereby be omes a kind of mollus '. In the letter (but not in the report), he is expli it about some of his misgivings about the ensemble view (as well as about the di ulties with his own preferred understanding, namely the `failure of the ele trons to stay together and similar'). Indeed, he points out that the S hrödinger equation is time-symmetri (if one in ludes omplex onjugation), while experien e tea hes us that the statisti al behaviour of ensembles annot be des ribed time-symmetri ally. Also, insisten e on a statisti al interpretation leads to `mysti al' al ulations with amplitudes and thus to problems with the laws of probability.7
4.2 Summary of the report The eventual form of S hrödinger's report follows roughly the sket h given above, with an introdu tion, followed by three main se tions, respe tively on the on guration-spa e theory, on the spa etime theory and on the many-ele tron problem. Introdu tion. S hrödinger draws the distin tion between the spa etime theory (four-dimensional) and the on guration-spa e theory (multidimensional). He states that the use of on guration spa e is a mathemati al way for des ribing what are in fa t events in spa e and time. However, it is the multi-dimensional theory that is the most su
essful and has proved to be a powerful analyti tool in relation to Heisenberg and Born's matrix me hani s. The multi-dimensional point of view has not been re on iled yet with the four-dimensional one.a I. Multi-dimensional theory. S hrödinger sket hes a derivation of his time-independent wave equation, noting that it reprodu es or improves on the results of Bohr's quantum theory. He also notes that the stationary states allow one to al ulate the transition probabilities en ountered in matrix me hani s. If one wishes to onsistently develop a formalism in whi h there are only dis ontinuous transitions, he suggests one should take seriously the idea that the transitions do not o
ur against a ontinuous time ba kground; the appearen e of a a As be omes lear in the dis ussion, S hrödinger thinks that the multi-dimensional theory may prove indispensable, so that one should a
ept the notion of a ψfun tion on on guration spa e and try instead to understand its physi al meaning in terms of its manifestation in spa e and time. (See the dis ussion in se tion 4.4 below.)
4.2 Summary of the report
127
ontinuous time parameter would be purely statisti al, so to speak.a As an alternative, he suggests interpreting the time-independent equation as arising from a time-dependent one from whi h the time variable is eliminated by assuming a stationary solution. He thus arrives to the des ription of a quantum system in terms of a time-dependent wave fun tion on on guration spa e. He then asks what the meaning of this wave fun tion is: `how does the system des ribed by it really look like in three dimensions?' (p. 454, S hrödinger's emphasis). He brie y mentions the view that the ψ -fun tion des ribes an ensemble of systems, whi h Born and Heisenberg are going to dis uss. S hrödinger instead nds it useful (if perhaps `a bit naive') to imagine an individual system as ontinuously lling the whole of spa e somehow weighted by |ψ|2 (as he further lari es in the dis ussion). S hrödinger then arefully spells out that the spatial density resulting from the on guration spa e density is not a lassi al harge density, in the sense that the a tion on the parti les by external elds and the intera tion between the parti les are already des ribed by the potentials in the wave equation, and that it is in onsistent to assume that this spatial density is also a ted upon in the manner of a lassi al harge density. Instead, it is possible to interpret it as a harge density (with some quali ations, some of whi h are spelled out only in the dis ussion) for the purpose of al ulating the ( lassi al) radiation eld, thus yielding a partial vindi ation of the idea of spatial densities. This, however, must be an approximation, sin e the observation of su h emitted radiation is itself an intera tion between the emitting atom and some other absorbing atom or mole ule, to be des ribed again by the appropriate potentials in the wave equation. II. Four-dimensional theory. S hrödinger shows that the time-dependent wave equation for a single parti le is a nonrelativisti approximation (with subtra tion of a rest frequen y) to the wave equation for the de Broglie phase wave of the parti le. The latter an be made manifestly relativisti by in luding ve tor potentials. (In modern terminology, this is the Klein-Gordon equation.) If one ouples the Maxwell eld to it, the same spatial densities dis ussed in se tion I appear as harge (and
urrent) densities. However, in the appli ation to the ele tron in the hydrogen atom, it be omes apparent that adding the self- eld of the ele tron to the (external) eld of the nu leus yields the wrong results.a Thus S hrödinger argues that if one hopes to develop a spa etime theory of intera ting parti les, it will be ne essary to onsider not just the a Cf. the dis ussion in h. 8.1. a Cf. S hrödinger (1927b), as mentioned in se tion 4.6 below.
128
S hrödinger's wave me hani s
overall eld generated by the parti les, but to distinguish between the (spatially overlapping) elds generated by ea h individual parti le, ea h eld a ting only on the other parti les of the system. Finally, he notes that the Klein-Gordon equation needs to be modi ed in order to des ribe spin ee ts, and that it may be possible to do so by onsidering a ve torial instead of a s alar ψ . III. The many-ele tron problem. S hrödinger returns to the multidimensional theory and its treatment, by approximation, of the manyele tron atom. His interest in this spe i example relates to the question of whether this multi-dimensional system an be understood in spa etime terms. The treatment rst negle ts the intera tion potentials between the ele trons, and as a rst approximation takes produ ts ψk1 . . . ψkn of the single-ele tron wave fun tions as solutions. One then expands the solution of the full equation in terms of the produ t wave fun tions. The time-dependent oe ients ak1 ...kn (t) in this expansion
an then be al ulated approximately if the intera tion between the ele trons is small. S hrödinger shows that before any approximation the oe ients in the equations for the ak1 ...kn (t) depend only on potentials al ulated from the spatial harge densities asso iated with the ψki . Thus, although the solution to the full equation is a fun tion on
on guration spa e, it is determined by purely spatial harge densities. A
ording to S hrödinger, this reinfor es the hope of providing a spatial interpretation of the wave fun tion. A sket h of the approximation method on ludes this se tion and the report.
4.3 Parti les as wave pa kets The idea of parti les as wave pa kets is ru ial to the development of S hrödinger's ideas and appears to provide one of the main motivations, at least initially, behind the idea of a des ription of matter purely in terms of waves. S hrödinger's earliest spe ulation about wave pa kets (for both material parti les and light quanta) is found in his paper on Einstein's gas statisti s (1926a), in se tion 5, `On the possibility of representing mole ules or light quanta through interferen e of phase waves'. As he explains, S hrödinger nds it un anny that in de Broglie's theory one should onsider the phase waves of the orpus les to be plane waves, sin e it is lear that by appropriate superposition of dierent plane waves one
an onstru t a `signal', whi h following Debye (1909) and von Laue
4.3 Parti les as wave pa kets
129
(1914, se tion 2) an be onstrained to a small spatial volume. He then
ontinues: On the other hand, it is of ourse
not
to be a hieved by the lassi al wave
laws, that the onstru ted `model of a light quantum' whi h by the way extends indeed for stays together.
many
wavelengths in every dire tion also
permanently
Rather, it spreads itself out [zerstreut si h℄ over ever larger
volumes after passing through a fo al point. If one ould avoid this last on lusion by a quantum theoreti al modi ation of the lassi al wave laws, then a way to deliveran e from the light quantum dilemma would appear to be paved [angebahnt℄. (1926a, p. 101)
Wave pa kets are rst dis ussed at length in the se ond paper on quantisation (1926 ): after des ribing the opti al-me hani al analogy, S hrödinger dis usses how one an onstru t wave pa kets (using the analogues of the opti al onstru tions by Debye and von Laue), and then shows that the entroid of su h a wave pa ket follows the lassi al equations of motion. S hrödinger onje tures that material points are in fa t des ribed by wave pa kets of small dimensions. He notes also that, for systems moving along very small orbits, the pa ket will be spread out, so that the idea of the traje tory or of the position of the ele tron inside the atom loses its meaning. In fa t, the main problem that S hrödinger was to fa e with regard to wave pa kets turned out to be that spreading of wave pa kets is a mu h more generi feature than he imagined at rst. As mentioned, S hrödinger sent to Lorentz the proofs of his rst two papers on quantisation (1926b, ) on 30 Mar h 1926. In his reply, among many other things, Lorentz dis ussed expli itly the idea of wave pa kets, indeed doubting that they would stay together (Przibram 1967, pp. 478).a S hrödinger ommented on this both in his reply of 6 June (Przibram 1967, pp. 5566) and in an earlier letter of 31 May to Plan k (Przibram 1967, pp. 811). In the latter, he admits that there will always exist spread-out states, be ause of linearity, but still hopes it will be possible to onstru t pa kets that stay together for hydrogen orbits of high quantum number. As he notes in the reply to Lorentz, this would imply that there is no general identi ation between hydrogen eigenstates and Bohr orbits, sin e a Bohr orbit of high quantum number would be represented by a wave pa ket rather than a stationary wave. a As Lorentz remarks, the alternative would be `to dissolve the ele tron ompletely .... and to repla e it by a system of waves' (p. 48), whi h, however, would make it di ult to understand phenomena su h as the photoele tri ee t. The latter was in fa t one of the riti isms levelled at S hrödinger's theory by Heisenberg (see below, se tion 4.6).
130
S hrödinger's wave me hani s
(For an ele tron in a hydrogen orbit of low quantum number, S hrödinger did not envisage an orbiting pa ket but indeed a spread-out ele tron.) With the reply to Lorentz, S hrödinger further sent his paper on mi roand ma rome hani s (1926e), in whi h he showed that for the harmoni os illator, wave pa kets do stay together. He hoped that the result would generalise to all quasi-periodi motions (admitting that maybe there would be `dissolution' for a free ele tron). However, on 19 June Lorentz sent S hrödinger a al ulation showing that wave pa kets on a high hydrogen orbit would indeed spread out ( f. Przibram 1967, pp. 6971, where the details of the al ulation, however, are omitted). The relevan e of high hydrogen orbits is, of ourse, the role they play in Bohr's orresponden e prin iple. The fa t that wave pa kets along su h orbits do not stay together was thus a blow for any hopes S hrödinger might have had of explaining the transition from mi ro- to ma rome hani s along the lines of the orresponden e prin iple. A further blow to the idea of wave pa kets must have ome with Born's papers on ollision theory during the summer of 1926 (Born 1926a,b; f. se tion 3.4.3 above). In the equivalen e paper, S hrödinger had in luded a remark about s attering, for whi h, he wrote, it is `indeed ne essary to understand learly the ontinuous transition between the ma ros opi ans hauli h me hani s and the mi rome hani s of the atom' (1926d, p. 753; see below, se tion 4.6, for the notion of Ans hauli hkeit). Given that at the time S hrödinger thought that ele trons on high quantum orbits should be des ribed by wave pa kets, this remark may indi ate that S hrödinger also thought that s attering should involve a de e tion of wave pa kets, whi h would move asymptoti ally in straight lines.a Born's work on ollisions in the summer of 1926 made essential use of wave me hani s, but suggested a very dierent pi ture of s attering. Born expli itly understood his work as providing an alternative interpretation both to his earlier views on matrix me hani s and to S hrödinger's views. In turn, S hrödinger appeared to be s epti al of Born's suggestions, writing on 25 August to Wilhelm Wien: Zeits h. f. Phys. I know more or waves must be stri tly ausally determined through eld laws, the wavefun tions on the other hand have only the meaning of probabilities for the a tual motions of light- or material-parti les. I believe
From an oprint of Born's last work in the less how he thinks of things: the
that Born thereby overlooks that ....
it would depend on the taste of the
a If thus was indeed S hrödinger's intuition, it may seem quite remarkable. On the other hand, so is the fa t that S hrödinger does not seem to dis uss dira tion of material parti les, whi h seems equally problemati for the idea of wave pa kets.
4.3 Parti les as wave pa kets
131
observer whi h he now wishes to regard as real, the parti le or the guiding eld. (Quoted in Moore 1989, p. 225)
A few days later, S hrödinger submitted a review paper on `undulatory me hani s' to the Physi al Review (1926h). In it, he quali ed rather strongly the idea that `material points onsist of, or are nothing but, wave systems' (p. 1049). Indeed, he ontinued (pp. 104950): This extreme on eption may be wrong, indeed it does not oer as yet the slightest explanation of why su h wave-systems seem to be realized in nature as orrespond to mass-points of de nite mass and harge.
....
a thorough
orrelation of all features of physi al phenomena an probably be aorded only by a harmonious union of these two extremes.
During the general dis ussion at the Solvay onferen e, S hrödinger summarised the situation with the following words (this volume, p. 519): The original pi ture was this, that what moves is in reality not a point but a domain of ex itation of nite dimensions ....
.
One has sin e found that
the naive identi ation of an ele tron, moving on a ma ros opi orbit, with a wave pa ket en ounters di ulties and so annot be a
epted to the letter. The main di ulty is this, that with ertainty the wave pa ket spreads in all dire tions when it strikes an obsta le, an atom for example.
We know
today, from the interferen e experiments with athode rays by Davisson and Germer, that this is part of the truth, while on the other hand the Wilson
loud hamber experiments have shown that there must be something that
ontinues to des ribe a well-de ned traje tory after the ollision with the obsta le. I regard the ompromise proposed from dierent sides, whi h onsists of assuming a ombination of waves and point ele trons, as simply a provisional manner of resolving the di ulty.
The problem of the relation between mi ro- and ma rophysi s is onne ted of ourse to the linearity of the wave equation, whi h appears to lead dire tly to highly non lassi al states (witness S hrödinger's famous ` at' example, S hrödinger 1935).a One might ask whether S hrödinger himself onsidered the idea of a nonlinear wave equation. In this
onne tion, a few remarks by S hrödinger may be worth investigating further. One expli it, if early, referen e to nonlinearity is ontained in the letter of 31 May to Plan k, where, after noti ing that linearity for es the existen e of non- lassi al states, S hrödinger indeed spe ulates that the equations might be only approximately linear (Przibram 1967, p. 10). In the orresponden e with Lorentz, the question of nonlinearity arises a Note, on the other hand, that S hulman (1997) shows there are ` lassi al' solutions to the linear equation in quite realisti models of oupling between mi ro- and ma rosystems (measurements), provided one requires them to satify appropriate boundary onditions at both the initial and the nal time.
132
S hrödinger's wave me hani s
in other ontexts. In the ontext of the problem of radiation, it appears at rst to be ne essary for ombination tones to arise (pp. 4950). In the
ontext of radiation rea tion, S hrödinger writes that the ex hange with Lorentz has onvin ed him of the ne essity of nonlinear terms (p. 62). In print, S hrödinger mentioned the possibility of a nonlinear term in order to in lude radiaton rea tion in his fourth paper on quantisation (1926g, p. 130).a Finally, a few years later, in his se ond paper on entanglement (1936, pp. 4512), S hrödinger mentioned the possibility of spontaneous de ay of entanglement at spatial separation, whi h would have meant a yet untested violation of the S hrödinger equation. Modern ollapse theories, su h as those by Ghirardi, Rimini and Weber (1986) or by Pearle (1976, 1979, 1989), modify the S hrödinger equation sto hasti ally, in a way that su
essfully ountera ts spreading with in reasing s ale of the system. S hrödinger's strategy based on wave pa kets thus appears to be viable at least if one a
epts sto hasti modi ations to S hrödinger's equation (as S hrödinger was not ne essarily likely to do). Note that while ru ial, the failure of the straightforward idea of wave pa kets for representing the ma ros opi , or lassi al, regime of the theory is distin t from the question of whether S hrödinger's wave pi ture ould adequately des ribe what appeared to be other examples or lear indi ations of parti ulate or `dis ontinuous' behaviour, su h as en ountered in the photoele tri ee t. This question was parti ularly important in the dialogue with Heisenberg and Bohr (se tion 4.6). We shall also see that S hrödinger ontinued to explore how ontinuous waves might provide des riptions of apparently parti ulate or dis ontinuous quantum phenomena, su h as the Compton ee t, quantum jumps, bla kbody radiation and even the photoele tri ee t (se tion 4.6.3). a An expli it proposal for in luding radiation rea tion through a nonlinear term is due to Fermi (1927). Classi ally, if one in ludes radiation rea tion, one has the third-order nonrelativisti Abraham-Lorentz equation ...
x ¨ − τ x = F/m ,
(1)
where F is the externally applied for e and τ = (2/3)e2 /mc3 (e is the ele tron
harge, m is the mass). Fermi (1927) proposed modifying the S hrödinger equation as follows: i
3 ∂Ψ ˆ − mτ Ψx· d = HΨ ∂t dt3
Z
d3 x′ Ψ∗ x′ Ψ . 3
(2)
d That is, he added an extra `potential' −mτ x· dt 3 hxi. Rederiving the Ehrenfest theorem, one nds that hxi then obeys the above Abraham-Lorentz equation.
4.4 The problem of radiation
133
4.4 The problem of radiation In Bohr's theory of the atom, the frequen y of emitted light orresponded not to the frequen y of os illation of an ele tron on a Bohr orbit, but to the term dieren e between two Bohr orbits. No known me hanism
ould explain the dieren e between the frequen y of os illation and the frequen y of emission. Bohr's theory simply postulated quantum jumps Ei → Ej between the stationary states of energy Ei and Ej , a
ompanied by emission (or absorption) of light of the orresponding E −E frequen y νij = i h j . In this respe t, the Bohr-Kramers-Slater (BKS) theory had the advantage of postulating a olle tion of virtual os illators with the observed frequen ies. This was also, in a sense, that aspe t of the BKS theory that survived into Heisenberg's matrix me hani s.a The idea that wave me hani s ould provide a ontinuous des ription of the radiation pro ess (as opposed to the pi ture of quantum jumps) appears to have been also one of the main bones of ontention between S hrödinger and the Copenhagen-Göttingen physi ists. As S hrödinger wrote to Lorentz on 6 June 1926: The frequen y dis repan y in the Bohr model, on the other hand, seems to me, (and has indeed seemed to me sin e 1914), to be something so
monstrous,
that I should like to hara terize the ex itation of light in this way as really a almost in on eivable. (Przibram 1967, p. 61)
The rea tion by S hrödinger to the BKS theory instead was quite enthusiasti . In part, S hrödinger was well predisposed towards the possibility that energy and momentum onservation be only statisti ally valid.b In part (as argued by de Regt, 1997), this was pre isely be ause the BKS theory provided a me hanism for radiation, in fa t a very ans hauli h me hanism, albeit `virtual'. S hrödinger published a paper on the BKS theory, ontaining an estimate of its energy u tuations (1924b).c In 1926, with the development of wave me hani s, S hrödinger saw a new possibility of on eiving a me hanism for radiation: the superposia See above se tion 1.3 (p. 13) and hapter 3, espe ially se tion 3.3.1. a The emphasis here is strong, but f. the ontext of this passage in S hrödinger's letter. b Cf. S hrödinger to Pauli, 8 November 1922, in Pauli (1979, pp. 6971). The idea of abandoning exa t onservation laws seems to be derived from S hrödinger's tea her Exner (ibid., and Moore 1989, pp. 1524). S hrödinger publi ly stated this view and allegian e in his inaugural le ture at the University of Züri h (9 De ember 1922), published as S hrödinger (1929a).
Cf. Darrigol (1992, pp. 2478) on the problem of the inde nite growth of the energy u tuations with time (as raised in parti ular by Einstein). A
ording to S hrödinger, an isolated system would behave in this way, but the problem would disappear for a system oupled to an in nite thermal bath.
134
S hrödinger's wave me hani s
tion of two waves would involve two frequen ies, and emitted radiation
ould be understood as some kind of `dieren e tone'. In his rst paper on quantisation (1926b), S hrödinger states that this pi ture would be `mu h more pleasing [um vieles sympathis her℄' than the one of quantum jumps (p. 375), but the idea is rather sket hy: S hrödinger spe ulates that the energies of the dierent eigenstates all share a large onstant term, and that if the square of the frequen y is proportional to mc2 + E , then the frequen y dieren es (and therefore the beat frequen ies) are approximately given by the hydrogen term dieren es. The se ond paper (1926 ) refers to radiation only in passing. Commenting on these papers in his letter to S hrödinger of 27 May, Lorentz pointed out that while beats would arise if the time-dependent wave equation (whi h S hrödinger did not have yet) were linear, they would still not produ e radiation by any known me hanism. Combination tones would arise if the wave equation were nonlinear (Przibram 1967, pp. 4950). As be omes lear in the following letters between S hrödinger and Lorentz, on e the harge density of a parti le is asso iated with a quadrati fun tion of ψ , su h as |ψ|2 , `dieren e tones' in the os illating harge density arise regardless of whether the wave equation is linear or nonlinear. This idea is still the basis of today's semi lassi al radiation theory (often used in quantum opti s), that is, the determination of lassi al ele tromagneti radiation from the urrent asso iated with a harge density proportional to |ψ|2 (for a non-stationary ψ ).a S hrödinger arrived at this result through his work onne ting wave me hani s and matrix me hani s (his `equivalen e paper', 1926d). In fa t, S hrödinger showed how to express the elements of Heisenberg's matri es wave me hani ally, in parti ular the elements of the dipole moment matrix, whi h (by orresponden e arguments) were interpreted as proportional to the radiation intensities. S hrödinger now suggested it might be possible `to give an extraordinarily ans hauli h interpretation of the intensity and polarisation of radiation' (1926d, p. 755), by de ning an appropriate
harge density in terms of ψ . In this paper, he suggested as yet, for a ∗ single ele tron, to use (the real part of) the quadrati fun tion ψ ∂ψ ∂t . The third paper on quantisation (1926f) is on erned with perturbation theory and its appli ation to the Stark ee t, but S hrödinger notes in an addendum in proof (footnote on p. 476) that the orre t a This method is tou hed on previously. Pauli uses it (for the ase of one parti le) in al ulating the s attered radiation in the Compton ee t during the dis ussion after Compton's talk (p. 365).
4.4 The problem of radiation
135
harge density is given by |ψ|2 . In the letter to Lorentz of 6 June 1926, he explains in detail how this gives rise to a sensible notion of harge density also for several parti les, ea h ontribution being obtained by integrating over the other parti les (Przibram 1967, p. 56). This idea is then used and dis ussed in print (as a `heuristi hypothesis') in the fourth paper on quantisation (1926g, p. 118, S hrödinger's itali s), where the wave fun tion is also expli itly interpreted in terms of a superposition of all lassi alR on gurations of a system weighted by |ψ|2 , and the time
onstan y of |ψ|2 is derived (pp. 1356).a This is also the pi ture of the wave fun tion given by S hrödinger in the Solvay report (pp. 454 .). S hrödinger dis reetly skips dis ussing the view of the wave fun tion as des ribing only an ensemble. S hrödinger's on ern in interpreting the wave fun tion is to understand its manifestation in spa etime. This on ern also motivates S hrödinger's dis ussion of the many-ele tron atoma , where he stresses that the spatial
harge distributions of the single (non-intera ting) ele trons already determine the wave fun tion of the intera ting ele trons. In the report, S hrödinger starts by rephrasing the question of the meaning of the wave fun tion as that of how a system des ribed by a ertain (multi-parti le) wave fun tion looks like in three dimensions. He des ribes this as taking all possible on gurations of the lassi al system simultaneously and weighting them a
ording to |ψ|2 . To this pi ture of the system as a `snapshot' (as he alls it in the report) or as a `mollus ' (as he had written to Lorentz) S hrödinger then asso iates the
orresponding ele tri harge density in 3-spa e. He is areful to state, however, that this is not an ele tri harge in the usual sense. For one thing, the ele tromagneti eld does not exert for es on it. While this
harge does des ribe the sour es of the eld in the semi- lassi al theory, this oupling of the eld to the harges is des ribed as `provisional', rst, be ause of the problem of radiation rea tion (whi h is not taken into a
ount in the S hrödinger equation), and se ond, be ause within a
losed system it is in onsistent to model the intera tion between dierent
harged parti les using the semi- lassi al eld. In parti ular, the observation of emitted radiation is in prin iple again a quantum me hani al intera tion, and should be des ribed by orresponding potentials in the equation for the total system. a The latter question had also been raised by Lorentz (Przibram 1967, p. 71) and answered in S hrödinger's next letter, not in luded in Przibram's olle tion.8 a Previously unpublished but deriving from methods used in S hrödinger's paper on energy ex hange (1927 ).
136
S hrödinger's wave me hani s
The above pi ture of the wave fun tion and the question of regarding S hrödinger's formal harge density as a sour e of lassi al ele tromagneti radiation evidently raised many questions. (S hrödinger introdu es the dis ussion after his report by the remark: `It would seem that my des ription in terms of a snapshot was not very fortunate, sin e it has been misunderstood', p. 469.) It also eli ited the most dis ussion following S hrödinger's report.a A di ulty that is spelled out only in the dis ussion ( ontributions by Bohr and by S hrödinger, p. 469), is that using S hrödinger's dipole moment (equation (13) in S hrödinger's talk) to al ulate the radiation does not dire tly yield the orre t intensities. As pointed out in the P talk, if one evaluates the dipole moment for Ra superposition k ck ψk , one obtains terms ontaining the integrals −e qψk ψl∗ dτ . These are the matrix elements of the dipole moment matrix in matrix me hani s, and they an be used within ertain limits to al ulate the emitted radiation (in parti ular, as S hrödinger points out, vanishing of the integral implies vanishing of the orresponding spe tral line). However, these integrals appear with the oe ients ck c∗l , whereas both a
ording to Bohr's old quantum theory and to experiment, the intensity of radiation should not depend on the oe ient of the ` nal state', say ψl . Thus, the use of (13) as a lassi al dipole moment in the al ulation of emitted radiation does not in general yield the orre t intensities. Bohr also drew attention to the fa t that by the time of the Solvay
onferen e, Dira (1927b, ) had already published his treatment of the intera tion of the (S hrödinger) ele tron with the quantised ele tromagneti eld. S hrödinger replied that he was aware of Dira 's work, but had the same misgivings with q-numbers as he had with matri es: the la k of `physi al meaning', without whi h he thought the further development of a relativisti theory would be di ult.a a The rest of the dis ussion in ludes a few te hni al questions and omments ( ontributions by Fowler and De Donder, Born's report on some numeri al work on perturbation theory), and some dis ussion of the `three-dimensionality' of the many-ele tron atom. Further dis ussion of the meaning of S hrödinger's harge densities (and of the meaning of the S hrödinger wave in the ontext of the transformation theory of matrix me hani s) took pla e in the general dis ussion, espe ially in ontributions by Dira and by Kramers (pp. 492 and 496). a Note that in his treatment of the emission of radiation in The Physi al Prin iples of the Quantum Theory (Heisenberg 1930b, pp. 824), Heisenberg seems to follow and expand on the dis ussion of S hrödinger's report. Indeed, Heisenberg rst des ribes two related methods for al ulating the radiation, based respe tively on
al ulating the matrix element of the dipole moment of the atom (justi ed via the
orresponden e prin iple), and on al ulating the dipole moment of S hrödinger's `virtual harges' (as he alls them). He then explains pre isely the di ulty with
4.5 S hrödinger and de Broglie
137
If the interpretation of the wave fun tion as a harge density raises problems of prin iple (as Born and Heisenberg also stress in their report, p. 437), what is the point of suggesting su h an interpretation? One possible way of understanding S hrödinger's intentions is to say that he is proposing an ans hauli h image of the wave fun tion in terms of the spatial density it de nes, without in general spe ifying the dynami al role played by this density. Under ertain ir umstan es, then, this spatial density takes on the dynami al role of a harge density, in parti ular as a sour e of radiation. This point of view does not resolve the other problems onne ted with the wave on eption of matter (spreading of the wave pa ket, S hrödinger's at, mi ro-ma ro question), but it oers a platform from whi h to work towards their possible resolution. This of ourse may be an overinterpretation of S hrödinger's position, but it ts approximately with later developments that take S hrödinger's wave
on eption seriously, e.g. Bell's (1990) idea of |ψ|2 as `density of stu' (in on guration spa e).
4.5 S hrödinger and de Broglie Thus opens S hrödinger's review paper on wave me hani s for the Physi al Review (1926h, p. 1049, referen es omitted): The theory whi h is reported in the following pages is based on the very interesting and fundamental resear hes of L. de Broglie on what he alled `phase waves' (`ondes de phase') and thought to be asso iated with the motion a of material points, espe ially with the motion of an ele tron or [photon℄. The point of view taken here, whi h was rst published in a series of German papers, is rather that material points onsist of, or are nothing but, wavesystems.
This passage both illustrates the well-known fa t that S hrödinger arrived at his wave me hani s by developing further the ideas of de Broglie starting with his paper on gas statisti s (S hrödinger 1926a) and emphasises the main on eptual dieren e between de Broglie's and S hrödinger's approa hes. While some details of the relation between S hrödinger's and de Broglie's work are dis ussed in se tion 2.3, in this se tion we wish to raise the question of why S hrödinger should have developed su h a dierent the latter dis ussed here by Bohr and S hrödinger, and goes on to sket h (a variant of) Dira 's treatment of the problem. a The text here reads `proton', whi h is very likely a misprint, sin e de Broglie's work indeed fo ussed on ele trons and photons.
138
S hrödinger's wave me hani s
pi ture of wave me hani s. Indeed, although S hrödinger quotes de Broglie as the redis overer of Hamilton's opti al-me hani al analogy (1926h, footnote 3 on p. 1052), the two authors apply the analogy in opposite dire tions: de Broglie treats even the photon as a material parti le with a traje tory, while S hrödinger treats even the ele tron as a pure wave.a As S hrödinger writes in his rst paper on quantisation, it was in parti ular `re e tion on the spatial distribution' of de Broglie's phase waves that gave the impulse for the development of his own theory of wave me hani s (1926b, p. 372). We gain some insight as to what this refers to from a letter from S hrödinger to Landé of 16 November 1925: I have tried in vain to make for myself a pi ture of the phase wave of the ele tron in the Kepler orbit. Closely neighbouring Kepler ellipses are onsidered as `rays'. This, however, gives horrible ` austi s' or the like for the wave fronts. (Quoted in Moore 1989, p. 192)
Thus, one reason for S hrödinger to abandon the idea of traje tories in favour of the pure wave theory might have been that well-behaved traje tories seemed to be in ompatible with well-behaved waves. And indeed, in his presentations of the opti al-me hani al analogy, S hrödinger states that outside of the geometri limit the notion of `ray' be omes meaningless (1926b, pp. 495 and 507, 1926h, pp. 10523). This also seems to be at issue in the ex hange between S hrödinger and Lorentz during the dis ussion of de Broglie's report, where S hrödinger points out that in ases of degenera y, any arbitrary linear ombination of solutions is allowed, and de Broglie's theory would predi t very ompli ated orbits (p. 402). There are other aspe ts that ould on eivably provide further reasons for S hrödinger's de nite abandoning of the traje tories. One possibility is that S hrödinger pi ked up from de Broglie spe i ally the idea of parti les as singularities of the wave,a and was happy to relax it to the a As is lear from de Broglie's thesis and as mentioned by de Broglie himself in his report (p. 377), de Broglie had always assumed the pi ture of traje tories. The above quotation in any ase makes it lear that this was S hrödinger's own reading of de Broglie. Already as early as January 1926, S hrödinger writes in his gas theory paper about `the de Broglie-Einstein undulatory theory of orpus les in motion, a
ording to whi h the latter are no more than a kind of foam rest [S hammkaum℄ on the wave radiation that onstitutes the world ba kground [Weltgrund℄' (1926a, p. 95). Other indi ations that this was well-known are Lorentz's omments to S hrödinger on the onstru tion of ele tron orbits in his letter of 19 June 1926 oming ` lose to de Broglie's arguments' (Przibram 1967, p. 74) and Pauli's referen e to Einstein's and de Broglie's `moving point masses' in his letter to Jordan of 12 April 1926 (Pauli 1979, p. 316). a Cf. S hrödinger (1926a, p. 99): `The universal radiation, as signals or perhaps singularities of whi h the parti les are meant to o
ur, is thus something quite
4.6 The on i t with matrix me hani s
139
idea of wave pa kets. Indeed, as we have seen above, S hrödinger pla ed great emphasis on the notion of a wave pa ket, and if it had been an adequate notion, there would have been no need for a separate notion of a orpus le in order to explain the parti ulate aspe ts of matter. We have also seen that S hrödinger was a utely aware of the radiation problem, namely of the dis repan y between the orbital frequen y of the ele tron and the frequen y of the emitted radiation. While de Broglie in his thesis was able to derive the Bohr orbits from wave onsiderations, this would in no way seem to alleviate the problem: the frequen y of revolution of the ele tron in de Broglie's theory was the same as in the Bohr model, and would lead to the wrong frequen y of radiation if the ele tron was treated as a lassi al sour e. (In the ase of degenera y noted above, the situation would be even more ompli ated.)a Finally, S hrödinger appeared to be riti al of proposals ombining waves and parti les, for instan e as appeared to be done by Born in his
ollision papers (see se tion 4.3 above). Su h misgivings ould easily have applied also to de Broglie's theory.
4.6 The on i t with matrix me hani s In the early dis ussions on the meaning of quantum theory, the notion of `Ans hauli hkeit' resurfa es time and again. The verb `ans hauen' means `to look at', and `ans hauli h', whi h means ` lear', `vivid' or `intuitive', has visual onnotations that the English word `intuitive' la ks. Ans hauli hkeit of a physi al theory is thus a quality of ready
omprehensibility that may (but need notb ) in lude a strong omponent of literal pi turability. For S hrödinger, at any rate, it seems that the possibility of grasping a theory through some kind of spatio-temporal intuition was a key omponent of physi al understanding.c It is lear, however, that the on i t between wave me hani s and matrix or quantum me hani s was not, or not only, a philosophi al issue, say about the validity of the on epts of spa etime and ausality,d and a b
d
essentially more ompli ated than for instan e the wave radiation of Maxwell's theory .... ' Note, however, that de Broglie (1924 ) had dis ussed the solution of this problem in the orresponden e limit (i.e. for the ase of high quantum numbers). Cf. below Heisenberg's use of the term in the un ertainty paper. A good dis ussion of the role the notion of Ans hauli hkeit played for S hrödinger is given by de Regt (1997, 2001), who argues that S hrödinger's requirement of Ans hauli hkeit is derived indire tly from Boltzmann, and is essentially a methodologi al requirement (as opposed to being a ommitment to realism
f. also the introdu tion by Bitbol in S hrödinger 1995, p. 4 fn. 10). Note that Kant's on eption of spa e and time is formulated in terms of what he
140
S hrödinger's wave me hani s
that, in the minds of the parties involved, these issues were onne ted with spe i ally s ienti questions.a The list of these questions is extensive. In his `equivalen e' paper (1926d), S hrödinger states that mathemati al equivalen e is not the same as physi al equivalen e, in the sense that two theories an oer quite dierent possibilities for generalisation and further development.a He thinks of two problems in parti ular: rst, the problem of s attering (as mentioned in se tion 4.3 above), se ond, the problem of radiation (dis ussed in se tion 4.4), in onne tion with whi h he then des ribes the idea of the vibrating harge density. The radiation problem in turn links to further issues at the heart of the debate with Heisenberg and Bohr in parti ular, on whether there are quantum jumps or whether the pro ess of radiation and the atomi transitions an be des ribed as
ontinuous pro esses in spa e and time. In a sense, neither the issue of s attering nor that of radiation resolved the debate in favour of either theory: both theories were modi ed or reinterpreted in the ourse of these developments, even though the result (statisti al interpretation, Copenhagen interpretation) was not to S hrödinger's taste. We shall now follow how the on i t evolved, from the beginnings to the time of the Solvay onferen e, sin e both sides developed onsiderably during the ru ial period between S hrödinger's rst papers and the onferen e.
4.6.1 Early days At the time of S hrödinger's rst two papers on quantisation, as we have seen in the previous hapter, matrix me hani s was a theory that reje ted the notion of ele tron orbits, indeed the very possibility of a spa etime des ription, substituting the lassi al kinemati al quantities with matrix quantities; but it kept the postulate of stationary states and of quantum jumps between these states. Matrix me hani s did not des ribe the stationary states individually, only olle tively, and allowed one to al ulate only transition probabilities for the jumps. S hrödinger's rst paper on quantisation (1926b) ontains only a few
omments on the possible ontinuous pi ture of atomi transitions, as
alls the `Ans hauungsformen' (the `forms of intuition'), so that the dis ussion has indeed strong philosophi al overtones. a This point has been re ently argued also by Perovi (2006). a Similarly, in the dis ussion after his own report, S hrödinger insists that nding a physi al interpretation of the theory is `indispensable for the further development of the theory' (p. 473).
4.6 The on i t with matrix me hani s
141
opposed to quantum jumps. (But of ourse, at this stage, he has very little to say about the problem of radiation, in parti ular nothing about intensities.) The rst time he omments expli itly on the dieren es between wave me hani s and quantum me hani s is in his se ond paper. There he writes that wave me hani s oers a way to interpret the onvi tion, more and more oming to the fore today, that should deny real signi an e to the
se ond:
phase
rst:
one
of the ele tron motions in the atom;
that one may not even laim that the ele tron at a ertain time is
on one parti ular of the quantum traje tories distinguished by the quantum onditions; third: the true laws of quantum me hani s onsist not in de nite pres riptions for the individual traje tory, rather these true laws lo ated
relate through equations the elements of the whole manifold of traje tories of a system through, so that apparently a ertain intera tion between the dierent traje tories obtains. (1926 , pp. 508)
As S hrödinger pro eeds to say, these laims are in ontradi tion with the ideas of ele tron position and ele tron orbit, but should not be taken as for ing a omplete surrender of spatio-temporal ideas. At the time, the mathemati al relation between wave me hani s and matrix me hani s was not yet lari ed, but S hrödinger hopes there would be a well-de ned mathemati al relation between the two, whi h ould then
omplement ea h other. A
ording to S hrödinger, Heisenberg's theory yields the line intensities, his own oers the possibility of bridging the mi ro-ma ro gap.a Personally, he nds the on eption of emitted frequen ies as `beats' parti ularly attra tive and believes it will provide an ans hauli h understanding of the intensity formulas (1926 , pp. 51314). Within weeks, the relation between the two theories was lari ed independently by S hrödinger (1926d), by E kart (1926) and by Pauli, in his remarkable letter to Jordan of 12 April 1926 (Pauli 1979, pp. 31520). This is one of the rst do umented rea tions to S hrödinger's new work from a physi ist of the Copenhagen-Göttingen s hool.b In it, Pauli emphasises in fa t that in S hrödinger's theory there are no ele tron orbits, sin e traje tories are a on ept belonging to the geometri limit of the theory. Pauli seems to say that insofar as S hrödinger provides a a S hrödinger's olle ted works (1984) reprodu e this and other papers from Erwin and Anny S hrödinger's own opy of the se ond edition of Abhandlungen zur Wellenme hanik (S hrödinger 1928). In this opy of the book, the passage about the mi ro-ma ro bridge is underlined. b Ehrenfest informed Lorentz of the `equivalen e' result (and of Klein's (1926) theory) on 5 May 1926, when Kramers reported them in the olloquium at Leiden. Ehrenfest also mentioned the relation of this result to Lan zos's (1927) work.9 As noted above (footnote on p. 125), by 27 May Lorentz had re eived a opy of the paper dire tly from S hrödinger.
142
S hrödinger's wave me hani s
des ription of individual stationary states, his theory is not in on i t with matrix me hani s, sin e it also does not ontain the on ept of ele tron orbits.a The tone of S hrödinger's remarks and of the omments they eli it
hanges with the equivalen e paper. In a mu h-quoted footnote, S hrödinger says he had known of Heisenberg's theory but was `s ared away, not to say repelled', by the ompli ated algebrai methods and the la k of Ans hauli hkeit (1926d, p. 735). On 8 June 1926, Heisenberg sends Pauli an equally notorious (but often mis-quoted) omment: The more I re e t on the physi al part of S hrödinger's theory, the more disgusting [abs heuli h℄ I nd it. Imagine the rotating ele tron, whose harge is distributed over the whole spa e with its axis in a fourth and fth dimension.
What S hrödinger writes of the Ans hauli hkeit of his theory would [be ℄
an appropriate...
s ar ely
in other words I nd it poppy o k [Mist℄. (Pauli 1979,
p. 328)
It seems lear that, a
ording to Heisenberg, it is S hrödinger's laim of Ans hauli hkeit for his theory that is ludi rous, presumably partly be ause of the spread-out ele tron and partly be ause the waves are in
on guration spa e.a
4.6.2 From Muni h to Copenhagen In the summer of 1926, S hrödinger gave a series of talks on wave me hani s in various German universities, in parti ular, on 21 and 23 July, he talked in Muni h at the invitation of Sommerfeld and of Wien.b The des ription of a me hanism for radiation eli ited enthusiasti omments by Wien, but riti ism from Sommerfeld and from Heisenberg. The dis ussion was apparently heated, and eventually identi ed a ru ial a Pauli knew very early about S hrödinger's paper, having been informed by Sommerfeld, to whom at S hrödinger's request a opy of the paper had been forwarded by Wien, the editor of Annalen der Physik ( f. Pauli 1979, pp. 278 and 293). Pauli's analysis in the letter of 12 April appears to be based only on S hrödinger's rst paper on quantisation, although at least the o ial publi ation date of the se ond paper was 6 April 1926. See also the letters between Pauli and S hrödinger reprodu ed in Pauli (1979). a Alternatively, with a `fourth and fth dimension', Heisenberg might on eivably be referring to Klein's ve-dimensional extension of S hrödinger's equation (Klein 1926). As one example, here is how Moore (1989, p. 221) quotes the passage: `The more I think of the physi al part of the S hrödinger theory, the more abominable I nd it. What S hrödinger writes about Ans hauli hkeit makes s ar ely any sense, in other words I think it is bullshit [Mist℄'. b Here we follow mainly the a
ount given by Heisenberg (1946).
4.6 The on i t with matrix me hani s
143
experiment that would de ide between the idea of the ontinuous me hanism of radiation envisaged in wave me hani s and the idea of quantum jumps. This was in oherent s attering, i.e. the Raman ee t (at that time neither observed nor thus named). A
ording to the quantum predi tion (Smekal 1923, Kramers and Heisenberg 1925), in oherent s attering would exist also for atoms in the ground state, be ause the atom ould be ex ited by the in ident light. A
ording to S hrödinger instead, the ee t was due to indu ed vibrations for the ase in whi h at least two atomi frequen ies were already present, and so would not o
ur in the ground state.a Apparently, Sommerfeld and Heisenberg were `prepared to enter a bet for its existen e, while the experimental physi ists were against it and S hrödinger took a more wait-and-see attitude' (Heisenberg 1946, p. 5). In a letter to Pauli of 28 July, Heisenberg gives other spe i riti isms of S hrödinger, for throwing overboard `everything quantum theoreti al: namely photoele tri ee t, Fran k ollisions, Stern-Gerla h ee t et .' (Pauli 1979, p. 338). As a matter of fa t, in the letter to Wien of 25 August 1926, quoted above (se tion 4.3), S hrödinger admitted that he had great on eptual di ulties with the photoele tri ee t (but see below the dis ussion of S hrödinger, 1929b). Heisenberg further mentioned to Pauli that, together with S hrödinger and Wien, he had dis ussed Wien's experiments on the de ay of lumines en e (Wien 1923,
h. XX). This was another point where S hrödinger thought wave me hani s proved superior to matrix me hani s, and Heisenberg en ouraged Pauli to al ulate and publish the damping oe ients for the hydrogen spe trum.a Heisenberg mentions similar riti isms in his talk on `Quantum me hani s', given at the 89th meeting of German S ientists and Physi ians in Düsseldorf on 23 September 1926 and published in the issue of 5 November of Naturwissens haften (Heisenberg 1926 ). This talk ould be seen as a publi response to the laims that the return to a ` ontinuum theory' was possible. In it, Heisenberg addresses in parti ular the problem of Ans hauli hkeit: a
ording to Heisenberg, the usual notions a Note the implied inequivalen e of wave and matrix me hani s (despite the re ent `equivalen e proofs'). Another su h possibility of experimental inequivalen e is mentioned in the dis ussion after S hrödinger's report, with regard to the quadrupole radiation of the atom (p. 470). Cf. also Muller (1997) on the inequivalen e of the two theories. a Cf. Born and Heisenberg's report (pp. 434 and 436), where spin is onsidered to be problemati for wave me hani s, and where it is expli itly stated that Dira (1927 ) provides an explanation for the de ay experiments.
144
S hrödinger's wave me hani s
of spa e and time, and in parti ular their appli ation to physi s with the idea that spa e and matter are in prin iple ontinuously divisible, turn out to be mistaken, rst of all due to the Unans hauli hkeit of the orpus ular nature of matter, then through the theoreti al and experimental
onsiderations leading to the idea of stationary states and quantum jumps (Bohr, Fran k-Hertz, Stern-Gerla h), and nally through onsideration of radiation phenomena (Plan k's radiation formula, Einstein's light quantum, the Compton ee t and the Bothe-Geiger experiments). This issue, it is laimed, also relates losely to the question of the degree of `reality' to be as ribed to material parti les or light quanta. Quantum me hani s in its development had thus rst of all to free itself from notions of Ans hauli hkeit, in order to set up a new kind of kinemati s and me hani s. In dis ussing the wave theory, Heisenberg onsiders rst de Broglie and Einstein as having developed wave-parti le dualism for matter and having suggested the possibility of interferen e for an ensemble [S har℄ of parti les. He then explains that S hrödinger found a dierential equation for the matter waves that reprodu es the eigenvalue problem of quantum (i.e. matrix) me hani s. However, a
ording to Heisenberg, the S hrödinger theory fails to provide the link with de Broglie's ideas, that is, it fails to provide the analogy with light waves in ordinary spa e, be ause of the need to onsider waves in on guration spa e; the latter therefore have only a formal signi an e. Heisenberg refers to the laim that on the basis of S hrödinger's theory one may be able to return to `a purely ontinuous des ription of the quantum theoreti al phenomena', and ontinues (Heisenberg 1926 , p. 992): In developing onsequently this point of view one leaves in fa t the ground of de Broglie's theory, thus of Q.M. and indeed of all quantum theory and arrives in my opinion at a omplete ontradi tion with experien e (law of bla kbody radiation; dispersion theory). This route is thus not viable. The a tual reality of the de Broglie waves lies rather in the interferen e phenomena mentioned above, whi h defy any interpretation on the basis of lassi al on epts. The extraordinary physi al signi an e of S hrödinger's results lies in the realisation that an ans hauli h interpretation of the quantum me hani al formulas
ontains both typi al features of a orpus ular theory and typi al features of a wave theory.
After dis ussing quantum statisti s (`whi h in any ase represents a very bizarre further limiting of the reality of the orpus les', p. 992), Heisenberg on ludes (p. 994): In our ans hauli h interpretation of the physi al pro esses and mathemati al formulas there is a dualism between wave theory and orpus ular theory su h
4.6 The on i t with matrix me hani s
145
that many phenomena are des ribed most naturally by a wave theory of light as well as of matter, in parti ular interferen e and dira tion phenomena, while other phenomena in turn an be interpreted only on the basis of the
orpus ular theory. .... The ontradi tions of the ans hauli h interpretations of dierent phenomena ontained in the urrent s heme are ompletely unsatisfa tory. For a ontradi tion-free ans hauli h interpretation of the experiments, whi h in themselves are indeed ontradi tion-free, some essential trait in our pi ture of the stru ture of matter is urrently still missing.
After the summer, S hrödinger visited Copenhagen for an intense round of dis ussions. A
ording to Heisenberg's re onstru tion in Der Teil und das Ganze (Heisenberg 1969), S hrödinger argued with Bohr pre isely about the ne essity of nding a me hanism for radiation, while Bohr insisted that quantum jumps were ne essary for the derivation of Plan k's radiation law, as well as being dire tly observable in experiments. Heisenberg (1946, p. 6) states that the dis ussion ended with the re ognition `that an interpretation of wave me hani s without quantum jumps was impossible and that the mentioned ru ial experiment [i.e. the Raman ee t℄ in any ase would turn out in favour of the quantum jumps'.a S hrödinger in turn admitted his di ulties. In a letter to Wien of 21 O tober 1926 (quoted in Pauli 1979, p. 339), he wrote: `It is quite
ertain that the position of ans hauli h images, whi h de Broglie and I take, has not nearly been developed far enough to a
ount even just for the most important fa ts. And it is downright probable that here and there a wrong path has been taken that needs to be abandoned'. And,
ommenting in his prefa e (dated November 1926) to the rst edition of Abhandlungen zur Wellenme hanik (1926i) on the fa t that the papers were being reprinted un hanged, he invoked `the impossibility at the
urrent stage of giving an essentially more satisfa tory or even de nitive new presentation'. In the meantime, both Bohr and Heisenberg were working to nd their own satisfa tory interpretation of the theory.b In parti ular as regards Heisenberg, the orresponden e with Pauli is very telling.c First of all one nds the dis ussion of Born's ollision papers (Born 1926a,b), whi h led to Heisenberg's paper on u tuation phenomena (Heisenberg 1926b). a The well-known quotation `If this damned quantum jumping is indeed to stay, then I regret having worked on this subje t at all' is reported both in Heisenberg (1946) and Heisenberg (1969). b See also se tion 3.4 above.
As noted in Pauli (1979, p. 339 and fn. 3 on p. 340), however, most of Pauli's letters to Heisenberg from this period appear to have been destroyed in the war.
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S hrödinger's wave me hani s
As already mentioned in se tion 3.4.5, Heisenberg then reports to Pauli about Dira 's transformation theory (Dira 1927a), whi h, however, is seen only as an extraordinary formal development. The next letters from Heisenberg to Pauli are from February 1927 and in lude on 23 February the sket h of the ideas of the un ertainty paper (Pauli 1979, pp. 37681). This, as was Bohr's simultaneous development of the idea of
omplementarity, was meant to provide the ans hauli h pi ture that was still missing. Indeed, in the un ertainty paper, Heisenberg (1927, p. 172) formulates Ans hauli hkeit as the possibility of arriving at qualitative predi tions in simple ases together with formal onsisten y. With the formulation and appli ation of the un ertainty relations, he was then satis ed to have found su h an interpretation. In a sense, however, the un ertainty paper was also the end of the original notion of `quantum jumps' as transitions between stationary states of a system. Indeed, as had to be the ase given the generalisation of transition probabilities to arbitrary pairs of observables (and as was implied in the Heisenberg-Pauli orresponden e), the privileged role of stationary states had to give way. It is somewhat ironi that Heisenberg was to give up quantum jumps only a few months after the dis ussions with S hrödinger in Copenhagen; nevertheless, quantum me hani s thus wedded to probabilisti transitions between measurements was just as dis ontinuous as the pi ture of quantum jumps between stationary states that it repla ed, and, for S hrödinger, it was equally unsatisfa tory. Note that, for Bohr at least, wave aspe ts played a ru ial role in the resulting `Copenhagen' interpretation. Yet, just as in the ase of Born's (1926a,b) use of wave me hani s in the dis ussion of ollisions (see se tion 3.4.4), Heisenberg appears to have been onvin ed that the apparent wave aspe ts ould be interpreted entirely in matrix terms. This dieren e of opinion was re e ted in the sometimes tense dis ussions between Heisenberg and Bohr at the time, in parti ular on the topi of Heisenberg's treatment of the γ -ray mi ros ope in the un ertainty paper and the orresponding addendum in proof.a
4.6.3 Continuity and dis ontinuity Between late 1926 and the time of the Solvay onferen e, S hrödinger
ontinued to work along lines that brought out the attra tive features and sometimes the limitations of the wave pi ture. In late 1926 and a For more on this issue, see Beller (1999, pp. 714 and 13841) and Camilleri (2006). For Heisenberg's way out of the di ulty, see Heisenberg (1929, pp. 4945).
4.6 The on i t with matrix me hani s
147
early 1927, S hrödinger fo ussed on the `four-dimensional' theory, with his papers on the Compton ee t (1927a), and on the energy-momentum tensor (1927b), while he returned to the `many-dimensional' theory shortly before the Solvay onferen e, with his work on energy ex hange (1927 ) and the treatment of the many-ele tron atom presented in his report. The Compton ee t is of ourse a paradigmati example of a `dis ontinuous' phenomenon, but S hrödinger (1927a) gives it a wave me hani al treatment, by analogy with the lassi al ase of re e tion of a light wave when it en ounters a sound wave (Brillouin 1922).a As he remarks in the paper, and as remarked in the dis ussion after Compton's report, this treatment relies on the onsideration of stationary waves and does not dire tly des ribe an individual Compton ollision.b The paper on the energy-momentum tensor (1927b), following Gordon (1926), takes the Lagrangian approa h to deriving the Klein-Gordon equation, and varies also the ele tromagneti potentials, thus deriving the Maxwell equations as well. S hrödinger then onsiders in parti ular the energy-momentum tensor of the ombined Maxwell and KleinGordon elds. As he remarks (S hrödinger 1928, p. x), this is a beautiful formal development of the theory, but it heightens starkly the di ulties with the four-dimensional view, sin e one annot insert the ele tromagneti potentials thus obtained ba k into the Klein-Gordon equation. For instan e, in luding the self- eld of the ele tron in the treatment of the hydrogen atom would yield the wrong results, as S hrödinger also mentions in se tion II of his report. In this onne tion, S hrödinger states that: `The ex hange of energy and momentum between the ele tromagneti eld and matter does not in reality take pla e in a ontinuous way, as the [given℄ eldlike expression suggests' (1927b, p. 271).c A
ording to Heisenberg (1929), the further development of the four-dimensional theory was indeed purely formal, but provided a basis for the later development of quantum eld theory.d a S hrödinger had given a derivation of Brillouin's result by assuming that energy and momentum were ex hanged in the form of quanta (1924a). In the paper on the Compton ee t he now omments on how, in a sense, he is reversing his own earlier reasoning. b See in parti ular the remarks by Pauli (p. 364) and the dis ussion following S hrödinger's ontribution (p. 370). Cf. also the losing paragraph of Pauli's letter to S hrödinger of 12 De ember 1927 (Pauli 1979, p. 366).
See again the ex hange of letters between Pauli and S hrödinger in De ember 1926 (Pauli 1979, pp. 3648). d Heisenberg (1930b) in orporates into his view of quantum theory both the multi-dimensional theory and the four-dimensional theory of S hrödinger's report.
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S hrödinger's wave me hani s
The paper on energy ex hange (1927 ), S hrödinger's last paper before the Solvay onferen e, marks an attempt to meet the riti ism that wave me hani s annot a
ount for ru ial phenomena involving `quantum jumps'. Following Dira (1926 ), S hrödinger sket hes the method of the variation of onstants, whi h he is to use also in the dis ussion of the many-ele tron atom in his Solvay report. He then applies it to the system of two atoms in resonan e dis ussed by Heisenberg (1926b) (and by Jordan (1927a)). As dis ussed in se tion 3.4, Heisenberg uses this example to show how in matrix me hani s one an indeed des ribe
hange starting from rst prin iples, in parti ular how one an determine the probabilities for quantum jumps. Now S hrödinger turns the tables around and argues that the treatment of atoms in resonan e using wave me hani s shows how one an eliminate quantum jumps from the pi ture. He argues that the two atoms ex hange energies as if they were ex hanging de nite quanta. Indeed, he goes further and suggests that the idea of quantised energy itself should be reinterpreted in terms of wave frequen y,a and that resonan e phenomena are indeed the key to the `quantum postulates'. S hrödinger then pro eeds to formulate statisti al onsiderations about the distributions of the amplitudes of the two systems in resonan e, leading to the idea of the squares of the amplitudes as measures of the strength of ex itation of an eigenvalue. He then returns to resonan e onsiderations in the ase of a system
oupled to a heat bath, whi h he onsiders would su e in prin iple for the derivation of Plan k's radiation formula and of all the results of the `old quantum statisti s'. In the aftermath of the Solvay onferen e, although in pla es one nds S hrödinger at least temporarily espousing views mu h loser to those of the Copenhagen-Göttingen s hool ( f. Moore 1989, pp. 25051), S hrödinger ontinued to explore the possibilities of the wave pi ture.a The following is a telling example. In a short paper in Naturwissens haften, S hrödinger (1929b) proposes to illustrate `how the quantum theory in its newest phase again makes use of ontinuous spa etime fun tions, The latter is interpreted as a lassi al wave theory of matter that forms the ba kground for a se ond quantisation, and in ludes the ba krea tion of the self- eld via in lusion in the potential ( f. also above, fn. on p. 132). a S hrödinger had expressed the idea of energy as frequen y already in his letter to Wien of 25 August 1926: `What we all the energy of an individual ele tron is its frequen y. Basi ally it does not move with a ertain speed be ause it has re eived a ertain shove, but be ause a dispersion law holds for the waves of whi h it
onsists, as a onsequen e of whi h a wave pa ket of this frequen y has exa tly this speed of propagation' (as quoted in Moore 1989, p. 225). a Cf. also Bitbol's introdu tion in S hrödinger (1995), p. 5.
4.6 The on i t with matrix me hani s
149
indeed of properties of their form, to des ribe the state and behaviour of a system ....' (p. 487). S hrödinger quotes the example of how photo hemi al and photoele tri phenomena depend on the form of an impinging wave (i.e. on its Fourier de omposition), rather than on lo al properties of the wave at the point where it impinges on the relevant system. He argues that a ontinuous pi ture an be retained, but introdu es the idea (whi h, as he remarks, generalises without dif ulty to the ase of many-parti le wave fun tions) that the ru ial properties of wave fun tions are in fa t properties pertaining to the entire wave. S hrödinger may thus have been the rst to introdu e the idea of nonlo alisable properties, as part of the pri e to pay in order to pursue a wave pi ture of matter.
150
Notes to pp. 124141
Ar hival notes 1 Ehrenfest to Lorentz, 30 Mar h 1926, AHQP-LTZ-11 (in German). 2 S hrödinger to Lorentz, 30 Mar h 1926, AHQP-LTZ-8 (in German). 3 Vers haelt to Lefébure, 8 April 1926, IIPCS 2573 (in Fren h). 4 Lorentz to Einstein, 6 April 1926, AHQP-86 (in German). 5 Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 6 S hrödinger to Lorentz, 23 June 1927, AHQP-LTZ-13 (original with S hrödinger's orre tions) and AHQP-41, se tion 9 ( arbon opy) (in German). 7 See also S hrödinger to Lorentz, 16 July 1927, AHQP-LTZ-13 (in German). 8 S hrödinger to Lorentz, 23 June 1926, AHQP-LTZ-8 (in German). 9 Ehrenfest to Lorentz, 5 May 1926, AHQP-EHR-23 (in German).
Part II Quantum foundations and the 1927 Solvay
onferen e
5 Quantum theory and the measurement problem
5.1 What is quantum theory? For mu h of the twentieth entury, it was widely believed that the interpretation of quantum theory had been essentially settled by Bohr and Heisenberg in 1927. But not only were the `dissenters' of 1927 in parti ular de Broglie, Einstein, and S hrödinger un onvin ed at the time: similar dissenting points of view are not un ommon even today. What Popper alled `the s hism in physi s' (Popper 1982) never really healed. Soon after 1927 it be ame standard to assert that matters of interpretation had been dealt with, but the sense of puzzlement and paradox surrounding quantum theory never disappeared. As the entury wore on, many of the on erns and alternative viewpoints expressed in 1927 slowly but surely revived. In 1952, Bohm revived and extended de Broglie's theory (Bohm 1952a,b), and in 1993 the de Broglie-Bohm theory nally re eived textbook treatment as an alternative formulation of quantum theory (Bohm and Hiley 1993; Holland 1993). In 1957, Everett (1957) revived S hrödinger's view that the wave fun tion, and the wave fun tion alone, is real (albeit in a very novel sense), and the resulting `Everett' or `many-worlds' interpretation (DeWitt and Graham 1973) gradually won widespread support, espe ially among physi ists interested in quantum gravity and quantum
osmology. Theories even loser to S hrödinger's ideas ollapse theories, with ma ros opi obje ts regarded as wave pa kets whose spreading is prevented by sto hasti ollapse were developed from the 1970s onwards (Pearle 1976, 1979; Ghirardi, Rimini and Weber 1986). As for Einstein's on erns in 1927 about the nonlo ality of quantum theory (see hapter 7), re-expressed in the famous EPR paper of 1935, matters
ame to a head in 1964 with the publi ation of Bell's theorem (Bell 153
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Quantum theory and the measurement problem
1964). In the losing de ades of the twentieth entury, after many stringent experimental tests showed that Bell's inequality was violated by entangled quantum states, nonlo ality ame to be widely regarded as a entral fa t of the quantum world. Other on erns, voi ed by S hrödinger just a few years after the fth Solvay onferen e, also eventually played a entral role after de ades of obs urity. S hrödinger's ` at paradox' of 1935 ame to dominate dis ussions about the meaning of quantum theory. And the pe uliar `entanglement' that S hrödinger had highlighted as a key dieren e between lassi al and quantum physi s (S hrödinger 1935) eventually found its pla e as a entral on ept in quantum information theory: as well as being a matter of `philosophi al' on ern, entanglement ame to be seen as a physi al resour e to be exploited for te hnologi al purposes, and as a entral feature of quantum physi s that had been strangely under-appre iated for most of the twentieth entury. The interpretation of quantum theory is probably as ontroversial now as it ever has been. Many workers now re ognise that standard quantum theory entred as it is around the notion of `measurement' requires a lassi al ba kground ( ontaining ma ros opi measuring devi es), whi h an never be sharply de ned, and whi h in prin iple does not even exist. Even so, the operational approa h to the interpretation of quantum physi s is still being pursued by some, in terms of new axioms that onstrain the stru ture of quantum theory (Hardy 2001, 2002; Clifton, Bub and Halvorson 2003). On the other hand, those who do regard the ba kground problem as ru ial tend to assert that everything in the universe mi ros opi systems, ma ros opi equipment, and even human experimenters should in prin iple be des ribed in a uni ed manner, and that `measurement' pro esses must be regarded as physi al pro esses like any other. Approa hes of this type in lude: the Everett interpretation (Everett 1957), whi h is being subje ted to in reasing s rutiny at a foundational level (Saunders 1995, 1998; Deuts h 1999; Walla e 2003a,b); the pilot-wave theory of de Broglie and Bohm (de Broglie 1928; Bohm 1952a,b; Bohm and Hiley 1993; Holland 1993), whi h is being pursued and developed more than ever before (Cushing, Fine and Goldstein 1996; Pearle and Valentini 2006; Valentini 2007);
ollapse models (Pearle 1976, 1979, 1989; Ghirardi, Rimini and Weber 1986), whi h are being subje ted to ever more stringent experimental tests (Pearle and Valentini 2006); and theories of ` onsistent' or `de oherent' histories (Gri ths 1984, 2002; Gell-Mann and Hartle 1990; Hartle 1995; Omnès 1992, 1994).
5.2 The measurement problem today
155
Today, it is simply untenable to regard the views of Bohr and Heisenberg (whi h in any ase diered onsiderably from ea h other) as in any sense standard or anoni al. The meaning of quantum theory is today an open question, arguably as mu h as it was in O tober 1927.
5.2 The measurement problem today The problem of measurement and the observer is the problem of where the measurement begins and ends, and where the observer begins and ends. .... I think, that when you analyse this language that the physi ists have fallen into, that physi s is about the results of observations you nd that on analysis it evaporates, and nothing very lear is being said. J. S. Bell (1986, p. 48)
The re urring puzzlement over the meaning of quantum theory often
entres around a group of related on eptual questions that usually ome under the general heading of the `measurement problem'.
5.2.1 A fundamental ambiguity As normally presented in textbooks, quantum theory des ribes experiments in a way that is ertainly pra ti ally su
essful, but seemingly fundamentally ill-de ned. For it is usually impli itly or expli itly assumed that there is a lear boundary between mi ros opi quantum systems and ma ros opi lassi al apparatus, or that there is a lear dividing line between `mi ros opi inde niteness' and the de nite states of our lassi al ma ros opi realm. Yet, su h distin tions defy sharp and pre ise formulation. That quantum theory is therefore fundamentally ambiguous was argued with parti ular larity by Bell. For example (Bell 1986, p. 54): The formulations of quantum me hani s that you nd in the books involve dividing the world into an observer and an observed, and you are not told where that division omes on whi h side of my spe ta les it omes, for example or at whi h end of my opti nerve.
The problem being pointed to here is the la k of a pre ise boundary between the quantum system and the rest of the world (in luding the apparatus and the experimenter). A losely-related aspe t of the `measurement problem' is the need to explain what happens to the de nite states of the everyday ma ros opi
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Quantum theory and the measurement problem
domain as one goes to smaller s ales. Where does ma ros opi de niteness give way to mi ros opi inde niteness? Does the transition o
ur somewhere between pollen grains and ma romole ules, and if so, where? On whi h side of the line is a virus? Nor an quantum `inde niteness' or `fuzziness' be easily on ned to the atomi level. For ma ros opi obje ts are made of atoms, and so inevitably one is led to doubt whether ro ks, trees, or even the Moon, have de nite ma ros opi states, espe ially when observers are not present. And this in the fa e of remarkable developments in twentieth- entury astrophysi s and osmology, whi h have tra ed the origins of stars, galaxies, helium and the other elements, to times long before human observers existed. The notion of a `real state of aairs' is familiar from everyday experien e: for example, the lo ation and number of ma ros opi bodies in a laboratory. S ien e has shown that there is more to the real state of things than is immediately obvious (for example, the ele tromagneti eld). Further, it has been shown that the hara ter of the real state of things hanges with s ale: on large s ales we nd planets, stars and galaxies, while on small s ales we nd pollen grains, viruses, mole ules, and atoms. Nevertheless, at least outside of the quantum domain, the notion of `real state' remains. The ambiguity emphasised by Bell onsists of the la k of a sharp boundary between the ` lassi al' domain, in whi h `real state' is a valid on ept, and the `quantum domain', in whi h `real state' is not a valid on ept. Despite de ades of eort, this ambiguity remains unresolved within standard textbook quantum theory, and many riti s have been led to argue that the notion of real state should be extended, in some appropriate way, into the quantum domain. Thus, for example, Bell (1987, pp. 2930): Theoreti al physi ists live in a lassi al world, looking out into a quantumme hani al world. The latter we des ribe only subje tively, in terms of pro edures and results in our lassi al domain.
This subje tive des ription is
ee ted by means of quantum-me hani al state fun tions
ψ
world of ourse is des ribed quite dire tly `as it is'.
.... . The lassi al ....
Now nobody
knows just where the boundary between the lassi al and quantum domain is situated. ....
A possibility is that we nd exa tly where the boundary lies.
More plausible to me is that we will nd that there is no boundary. It is hard for me to envisage intelligible dis ourse about a world with no lassi al part no base of given events .... to be orrelated. On the other hand, it is easy to imagine that the lassi al domain ould be extended to over the whole.
5.2 The measurement problem today
157
While Bell goes on to argue in favour of adding extra (`hidden') parameters to the quantum formalism, for our purposes the key point being made here is the need to extend the notion of real state into the mi ros opi domain. This might indeed be a hieved by introdu ing hidden variables, or by other means (for example, the Everett approa h). Whatever form the theory may take, the real ma ros opi states onsidered in the rest of s ien e should be part of a uni ed des ription of mi ros opi and ma ros opi phenomena what Bell (1987, p. 30)
alled `a homogeneous a
ount of the world'. There are in fa t, as we have mentioned, several well-developed proposals for su h a uni ed or homogeneous (or `realist') a
ount of the world: the pilot-wave theory of de Broglie (1928) and Bohm (1952a,b); theories of dynami al wave-fun tion ollapse (Pearle 1976, 1979, 1989; Ghirardi, Rimini and Weber 1986); and the many-worlds interpretation of Everett (1957). The available proposals that have broad s ope assume that the wave fun tion is a real obje t that is part of the stru ture of an individual system. At the time of writing, it is not known if realist theories may be onstru ted without this feature.a
5.2.2 Measurement as a physi al pro ess: quantum theory `without observers' Another losely-related aspe t of the measurement problem is the question of how quantum theory may be applied to the pro ess of measurement itself. For it seems ines apable that it should be possible (in prin iple) to treat apparatus and observers as physi al systems, and to dis uss the pro ess of measurement in purely quantum-theoreti al terms. However, attempts to do so are notoriously ontroversial and apt to result in paradox and onfusion. For example, in the paradox of `Wigner's friend' (Wigner 1961), an experimenter A (Wigner) possesses a box ontaining an experimenter B (his friend) and a mi ros opi system S. Suppose S is initially in, for example, a superposition of energy states 1 |ψ0 i = √ (|E1 i + |E2 i) 2 a For example, a ording to the sto hasti hidden-variables theory of Fényes (1952) and Nelson (1966), the wave fun tion merely provides an emergent des ription of probabilities. However, despite appearan es, it seems that for te hni al reasons this theory is awed and does not really reprodu e quantum theory: the S hrödinger equation is obtained only for ex eptional (nodeless) wave fun tions (Wallstrom 1994; Pearle and Valentini 2006).
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Quantum theory and the measurement problem
and that the whole box is initially in a state |Ψ0 i = |B0 i⊗|ψ0 i (idealising B as initially in a pure state |B0 i). Let B perform an ideal energy measurement on S. If A does not arry out any measurement, then from the point of view of A the quantum state of the whole box evolves ontinuously (a
ording to the S hrödinger equation) into a superposition of states 1 |Ψ(t)i = √ (|B1 i ⊗ |E1 i + |B2 i ⊗ |E2 i) (1) 2 (where |Bi i is a state su h that B has found the energy value Ei ). Now, if A wished to, ould he (in prin iple) at later times, by appropriate experiments on the whole box, observe interferen e ee ts involving both bran hes of the superposition in (1)? If so, ould this be onsistent with the point of view of B, a
ording to whi h the energy measurement had a de nite result? One may well question whether the above s enario is realisti , even in prin iple (given the resour es in our universe). For example, one might question whether a box ontaining a human observer ould ever be su iently isolated for environmental de oheren e to be negligible. However, if the above `experimenter B' were repla ed by an automati devi e or ma hine, the s enario may indeed be ome realisti , depending on the possibility of isolating the box to su ient a
ura y.a Most s ientists agree that ma ros opi equipment is subje t to the laws of physi s, just like any other system, and that it should be possible to des ribe the operation of su h equipment purely in terms of the most fundamental theory available. There is somewhat less onsensus over the status of human experimenters as physi al systems. Some physi ists have suggested, in the ontext of quantum physi s, that human beings
annot be treated as just another physi al system, and that human
ons iousness plays a spe ial role. For example, Wigner (1961) on luded from his paradox that `the being with a ons iousness must have a dierent role in quantum me hani s than the inanimate measuring devi e', and that for a system ontaining a ons ious observer `the quantum me hani al equations of motion annot be linear'. Wigner's on lusion, that living beings violate quantum laws, seems in reasingly in redible given the impressive progress made in human biology and neuros ien e, in whi h the human organism in luding the brain is treated as (ultimately) a omplex ele tro- hemi al sysa It is perhaps worth remarking that, even if de oheren e has a role to play here, one has to realise what the problem is in order to understand whether and how de oheren e might ontribute to a solution ( f. Ba
iagaluppi 2005).
5.2 The measurement problem today
159
tem. There is no eviden e that human beings are able to violate, for example, the laws of gravity, or of thermodynami s, or basi prin iples of hemistry, and the on lusion that human beings in parti ular should be outside the domain of quantum laws seems di ult to a
ept. An alternative on lusion, of ourse, is that something is missing from orthodox quantum theory. Assuming, then, that human experimenters and their equipment may in prin iple be regarded as physi al systems subje t to the usual laws, their intera tion with mi rosystems ought to be analysable, and the pro ess of measurement ought to be treatable as a physi al pro ess like any other. One an then ask if, over an ensemble of similar experiments, it would be possible in prin iple for the external experimenter A to observe (at the statisti al level) interferen e ee ts asso iated with both terms in (1). To deny this possibility would be to laim (with Wigner) that a box ontaining a human being violates the laws of quantum theory. To a
ept the possibility would seem to imply that, at time t before experimenter A makes a measurement, there was (at least a
ording to A) no matter of fa t about the result of B's observation, notwithstanding the expli it supposition that B had indeed arried out an observation by time t. It is sometimes said that Wigner's paradox may be evaded by noting that, if the external experimenter A a tually performs an experiment on the whole box that reveals interferen e between the two bran hes of (1), then this operation will destroy the memory the internal experimenter B had of obtaining a parti ular result, so that there is no ontradi tion. But this misses the point. For while it is true that B will then not have any memory of having obtained a parti ular experimental result, the
ontradi tion remains with there being a purported matter of fa t (at time t) as to the result of B's observation, regardless of whether or not B has subsequently forgotten it. (Note that in this dis ussion we are talking about matters of fa t, not for mi rosystems, but for ma ros opi experimental results.) Let us examine the reasoning behind Wigner's paradox more losely. We take it that experimenter B agreed beforehand to enter the box and perform an energy measurement on the mi ros opi system S; and that it was further agreed that after su ient time had elapsed for B to perform the measurement, A would de ide whether or not to arry out an experiment showing interferen e between the two bran hes of (1). Considering an ensemble of similar experiments, the paradox onsists of
160
Quantum theory and the measurement problem
a ontradi tion between the following statements on erning the physi al state of B just before A de ides what to do: (I) There is no de nite state of B, be ause A an if he wishes perform measurements showing the presen e of interferen e between dierent states of B. (II) There is a de nite state of B, be ause B is a human experimenter like any other, and be ause instead of testing for interferen e A an simply ask B what he saw. The argument in (I) is the familiar one from standard quantum theory, applied to the unusual ase of a box ontaining an experimenter. The argument in (II) is unusual: it requires omment and elaboration. Be ause B is a human experimenter like any other, we are driven to onsider the theoreti al possibility that, in the distant future, some `super-experimenter' ould de ide to perform an interferen e experiment on a `box' ontaining us and our equipment. What would happen to our
urrent (ma ros opi ally-re orded) experimental fa ts on erning for example the out ome of a spin measurement performed in the laboratory? To be sure, our re ords of these experiments ould one day be erased, but it would be illogi al to suppose that the fa t of these experiments having been arried out (with de nite results) ould ever be hanged. Unless we a
ept (II), we are in danger of en ountering the paradox that fa ts about what we have done in the laboratory today might later turn out not to be fa ts. Further support for (II) omes from Wigner's original argument, whi h
entred around the assumed reality of other minds. (Wigner did not regard solipsism as worthy of serious onsideration.) From this assumption Wigner inferred that, whatever the ir umstan es, if one asks a `friend' what he saw, the answer given by the friend must have been, as Wigner put it, `already de ided in his mind, before I asked him'. But then, if A de ides not to perform an interferen e experiment on the box, and simply asks B what he saw, A is obliged to take B's answer as indi ative of the state of B's mind before A asked the question indeed, before A de ided on whether or not to perform an interferen e experiment. For Wigner, a superposition of the form (1) is una
eptable for a system
ontaining a human experimenter or `friend', be ause it implies that the friend `was in a state of suspended animation before he answered my question'. As Wigner presented it, the argument is based on the assumption that
5.2 The measurement problem today
161
a ons ious being will always have a de nite state of ons iousness. In orthodox quantum theory, of ourse, one might dismiss as `meaningless' the question of whether the friend's ons iousness ontained one impression or the other (B1 or B2 ) before he was asked. However, as Wigner put it, `to deny the existen e of the ons iousness of a friend to this extent is surely an unnatural attitude, approa hing solipsism'. Finally, on the topi of Wigner's paradox, it is important to emphasise the distin tion between `matters of fa t' on the one hand, and `memories' (true or false) on the other a distin tion that is omparable to the distin tion between fa ts and opinions, or between truth and belief, or between reality and appearan e, distin tions that form part and par el of the s ienti method. Thus, again, while an experimenter's memory of having obtained a ertain result might be erased in the future, the fa t that he on e obtained a ertain result will ne essarily remain a fa t: to assert otherwise would be a logi al ontradi tion. A further, more subtle motivation for treating measurement as a physi al pro ess omes from onsidering the very nature of `measurement'. As is well known to philosophers, and to experimental physi ists, the pro ess of measurement is `theory-laden'. That is, in order to know how to
arry out a measurement orre tly, or how to design a spe i measuring apparatus orre tly, some prior body of theory is required: in parti ular, one needs some understanding of how the equipment fun tions, and how it intera ts with the system being examined. For this reason, it is di ult to see how the pro ess of quantum measurement an be properly understood, without some prior body of theory that des ribes the equipment itself and its intera tion with the `system'. And sin e the equipment usually belongs to the de nite ma ros opi realm, and the `system' often does not, a proper understanding seems to require a `homogeneous a
ount of the world' as dis ussed above, that is, a theory in whi h an obje tive a
ount is provided not only of the ma ros opi apparatus, but also of the mi rosystem and its intera tion with the apparatus. A ommon on lusion, then, is that a oherent a
ount of quantum measurement requires that quantum theory be somehow extended from a theory of mi rosystems to a universal physi al theory with an unbounded domain of appli ation, with our everyday ma ros opi realism being somehow extended to the mi ros opi level. Given su h a well-de ned and universal physi al theory, whose subje t matter onsists of the real states of the world as a whole, it would be possible in prin iple to use the theory to analyse the pro ess of measurement as a physi al pro ess like
162
Quantum theory and the measurement problem
any other (just as, for example, lassi al ele trodynami s may be used to analyse the pro ess involving for es exerted by magneti elds by whi h an ammeter measures an ele tri urrent). Thus, for example, pilot-wave theory, or the Everett interpretation, or
ollapse models, may be applied to situations where a quantum measurement is taking pla e. If the theory provides an unambiguous a
ount of obje tive pro esses in general, it will provide an unambiguous a
ount of the quantum measurement pro ess in parti ular. The result is a quantum theory `without observers', in the sense that observers are physi al systems obeying the same laws as all other systems, and do not have to be added to the theory as extra-physi al elements. Con lusions as to what is a tually happening during quantum measurements will, of ourse, depend on the details of the theory. For example, onsider again Wigner's s enario above. In the Everett interpretation, B's observation within the box has two results, and there is no ontradi tion if the external experimenter A subsequently observes interferen e between them. In de Broglie-Bohm theory, B's observation has only one result sele ted by the a tual on guration, but even so the empty wave pa ket still exists in on guration spa e, and an in prin iple re-overlap with (and hen e interfere with) the o
upied pa ket if appropriate Hamiltonians are applied.
5.2.3 Quantum osmology Further losely-related questions, again broadly under the heading of the `measurement problem', on ern the des ription of the distant past before human beings and other life forms evolved on Earth, and indeed the des ription of the universe as a whole in epo hs before life existed. While the basi theoreti al foundations of big-bang osmology had already been laid by 1927 (through the work of Friedmann and Lemaître), at that time any suggestion of the need to provide a quantum-theoreti al a
ount of the early universe ould easily have been dismissed as being of no pra ti al or experimental import. By the 1980s, however, with the development of in ationary osmology (Guth 1981), the theoreti al question be ame a pra ti al one, with observational impli ations. A
ording to our urrent understanding, the small non-uniformities of temperature observed in the osmi mi rowave ba kground originated from lassi al density perturbations in the early (and approximately homogenous) universe (Padmanabhan 1993). And a
ording to in ationary theory, those early lassi al density perturbations originated from
5.2 The measurement problem today
163
quantum u tuations at even earlier times (Liddle and Lyth 2000). Here we have an example of a osmologi al theory in whi h a `quantum-to lassi al transition' o
urred long before life (or even galaxies) developed, and whose details have left an imprint on the sky that an be measured today. This is the measurement problem on a osmi s ale (Kiefer, Polarski and Starobinsky 1998; Perez, Sahlmann and Sudarsky 2006; Valentini 2006). But the tension between `Copenhagen' quantum theory and the requirements of osmology was felt long before osmology matured as an experimental s ien e. Thus, for example, in his pioneering work in the 1960s on quantum gravity, when it ame to applying the theory to a
losed universe DeWitt wrote (DeWitt 1967, p. 1131): The Copenhagen view depends on the assumed
a priori
existen e of a lassi al
level to whi h all questions of observation may ultimately be referred. Here, however, the whole universe is the obje t of inspe tion; there is no lassi al vantage point, and hen e the interpretation question must be re-argued from the beginning.
DeWitt went on to argue (pp. 11402) that, in the absen e of a lassi al level, the Everett interpretation should be adopted. A
ording to DeWitt (p. 1141): Everett's view of the world is a very natural one to adopt in the quantum theory of gravity, where one is a
ustomed to speak without embarrassment of the `wave fun tion of the universe'.
While DeWitt expresses a preferen e for the Everett interpretation, for our purposes the entral point being made is that, if the whole universe is treated as a quantum obje t, with no de nite ( lassi al) ba kground `to whi h all questions of observation may ultimately be referred', then the physi s be omes unintelligible unless some form of real state (or ontology) is as ribed to the quantum obje t. The Everett interpretation provides one way, among others, to do this.a Everett himself, in 1957, had already ited the quantum theory of
osmology as one of his main motivations for going beyond what he alled the ` onventional or external observation formulation of quantum me hani s' (Everett 1957, p. 454). Everett's general motivation was the need to des ribe the quantum physi s internal to an isolated system, in a Everett's original formulation was of ourse open to a number of riti isms, in parti ular on erning the notion of `world' and the idea of probability. Su h riti isms are arguably being met only through more re ent developments (Saunders 1995, 1998; Deuts h 1999; Walla e 2003a,b).
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Quantum theory and the measurement problem
parti ular one ontaining observers. A losed universe was a spe ial ase of su h a system, and one that arguably would have to be onsidered as the s ien e of osmology progressed (as has indeed proved to be the
ase). Thus Everett wrote (p. 455): How is one to apply the onventional formulation of quantum me hani s to the spa e-time geometry itself ?
The issue be omes espe ially a ute in the
ase of a losed universe. There is no pla e to stand outside the system to observe it. There is nothing outside it to produ e transitions from one state to another. .... No way is evident to apply the onventional formulation of quantum me hani s to a system that is not subje t to
external
observation.
The whole interpretive s heme of that formalism rests upon the notion of external observation.
The probabilities of the various possible out omes of
the observation are pres ribed ex lusively by Pro ess 1 [dis ontinuous wave fun tion ollapse℄.
In more re ent years, similar on erns have motivated the development of a `generalised quantum me hani s' based on ` onsistent' or `de oherent' histories (Gri ths 1984, 2002; Gell-Mann and Hartle 1990; Hartle 1995; Omnès 1992, 1994), an approa h that is also supposed to provide a quantum theory `without observers', and without a presumed lassi al ba kground, so as to be appli able to quantum osmology.
5.2.4 The measurement problem in `statisti al' interpretations of ψ The measurement problem is often posed simply as the problem of how to interpret a ma ros opi superposition of quantum states, su h as (pure) states of S hrödinger's at. This way of posing the measurement problem
an be misleading, however, as it usually rests on the impli it assumption (or suggestion) that the quantum wave fun tion ψ is a real physi al obje t identi able as a omplete des ription of an individual system. It might be that ψ is indeed a real obje t, but not a omplete des ription (as in pilot-wave theory). Or, ψ might not be a real obje t at all. Here we fo us on the latter possibility. The quantum wave fun tion ψ might be merely a mathemati al tool for al ulating and predi ting the measured frequen ies of out omes over an ensemble of similar experiments. In whi h ase, it would be immediately wrong to interpret a mathemati al superposition of terms in ψ as somehow orresponding to a physi al superposition of real states for individual systems, and the `measurement problem' in the limited sense just mentioned would be a pseudo-problem. This `statisti al in-
5.2 The measurement problem today
165
terpretation' of quantum theory has been hampioned in parti ular by Ballentine (1970). However, even in the statisti al interpretation, the `measurement problem' in the more general sense remains. For quantum theory is then an in omplete theory that refers only to ensembles, and simply does not fully des ribe individual quantum systems or their relation to real, individual ma ros opi states. The statisti al interpretation gives no a
ount of what happens, for example, when an individual ele tron is being measured: it talks only about the distribution of (ma ros opi allyregistered) measurement out omes over an ensemble of similar experiments. Nor does the statisti al interpretation provide any sharp delineation of the boundary between `ma ros opi ' obje ts with an individual (non-ensemble) des ription and `mi ros opi ' obje ts with no su h des ription.a In the statisti al interpretation, then, a solution of the measurement problem in the general sense will require the development of a omplete des ription of individual systems. This was Einstein's point of view (Einstein 1949, pp. 6712): The attempt to on eive the quantum-theoreti al des ription as the omplete des ription of the individual systems leads to unnatural theoreti al interpretations, whi h be ome immediately unne essary if one a
epts the interpretation that the des ription refers to ensembles of systems and not to individual systems. .... [I℄t appears unavoidable to look elsewhere for a omplete des ription of the individual system .... .
Einstein was arguably the founder of the statisti al interpretation.b It should be noted, however, that while Einstein's on lusion about the nature of ψ might turn out to be orre t, what seems to have been his main argument for this on lusion now appears to be wrong, in that it was based on what now appears to be a false premise the assumption of lo ality. For example, in a letter to his friend Mi hele Besso, dated 8 O tober 1952, Einstein argued that the `quantum state' ψ ould not be a omplete hara terisation of the `real state' of an individual system, on the following grounds: A system
S2 ,
S12 ,
with known fun tion
whi h at time
t
ψ12 ,
is omposed of subsystems
S1
and
are far away from ea h other. If one makes a ` omplete'
a There are dierent ways of onsidering a `statisti al' interpretation, depending on one's point of view on erning the nature of probability. In any ase, the `measurement problem' in the general sense still stands. b Cf. Ballentine (1972).
166
Quantum theory and the measurement problem
S1 , this an be done in dierent ways .... . From the meaand the ψ -fun tion ψ12 , one an determine .... the ψ -fun tion ψ2 of the se ond system. This will take on dierent forms, a
ording to the kind of measurement applied to S1 .
measurement on surement result
But this is in ontradi tion with assumption (1) [that the quantum state
hara terises the real state ompletely℄, Then in fa t the measurement on
S2 ,
S1
if one ex ludes a tion at a distan e.
an have no in uen e on the real state of
and therefore a
ording to (1) an have also no in uen e on the quantum
state of
S2
des ribed by
ψ2 .
(Einstein and Besso 1972, pp. 4878, emphasis
in the original)
Einstein's argument hinges on the fa t that, in a lo al physi s, the measurement made on S1 an have no ee t on the real state of S2 .a With the development of Bell's theorem, however, it seems to be beyond reasonable doubt that quantum physi s is not lo al (if one assumes the absen e of ba kwards ausation or of many worlds). For if lo ality is assumed, one may use the EPR argument to infer determinism for the out omes of quantum measurements at widely-separated wings of an entangled state.b Following further reasoning by Bell (1964), one may then show that any lo al and deterministi ompletion of quantum theory annot reprodu e quantum orrelations for all measurements on entangled states. Therefore, lo ality ontradi ts quantum theory.c Be ause the premise of Einstein's argument ontradi ts quantum theory, the argument annot be used to infer anything about quantum theory or about the nature of ψ . Thus, Einstein's argument does not establish the `statisti al' or `ensemble' nature of ψ .d a The notion of lo ality that Einstein uses here is, to be pre ise, a ombination of the prin iples of `separability' (that widely-separated systems have lo ally-de ned real states) and of `no a tion at a distan e'. Cf. Howard (1990). b `It is important to note that .... determinism .... in the EPR argument .... is not assumed but inferred [from lo ality℄. .... It is remarkably di ult to get this point a ross, that determinism is not a presupposition of the analysis' (Bell 1987, p. 143, itali s in the original).
See, however, Fine (1999) for a dissenting view. d Einstein's argument above has re ently been revived by Fu hs (2002) (who is, however, not expli it about the ompleteness or in ompleteness of quantum theory). Fu hs states (p. 9) that Einstein `was the rst person to say in absolutely unambiguous terms why the quantum state should be viewed as information .... . His argument was simply that a quantum-state assignment for a system an be for ed to go one way or the other by intera ting with a part of the world that should have no ausal onne tion with the system of interest'. Fu hs then quotes at length (p. 10) the above letter by Einstein. Later in the same paper, Fu hs writes (p. 39): `Re all what I viewed to be the most powerful argument for the quantum state's subje tivity the Einsteinian argument of [the above letter℄. Sin e [for entangled systems℄ we an toggle the quantum state from a distan e, it must not be something sitting over there, but rather something sitting over here: It an only be our information about the far-away system'. Again, the premise
5.2 The measurement problem today
167
of this Einsteinian argument lo ality is nowadays no longer reasonable (as of
ourse it was in 1952), and so the argument annot be used to infer the subje tive or epistemi nature of the quantum state.
6 Interferen e, superposition, and wave pa ket ollapse
6.1 Probability and interferen e A
ording to Feynman (1965, hap. 1, p. 1), single-parti le interferen e is `the only mystery' of quantum theory. Feynman onsidered an experiment in whi h parti les are red, one at a time, towards a s reen with two holes labelled 1 and 2. With both holes open, the distribution P12 of parti les at the ba kstop displays an os illatory pattern of bright and dark fringes. If P1 is the distribution with only hole 1 open, and P2 is the distribution with only hole 2 open, then experimentally it is found that P12 6= P1 + P2 . A
ording to the argument given by Feynman (as well as by many other authors), this result is inexpli able by ` lassi al' reasoning. By his presentation of the two-slit experiment (as well as by his development of the path-integral formulation of quantum theory), Feynman popularised the idea that the usual probability al ulus breaks down in the presen e of quantum interferen e, where it is probability amplitudes (and not probabilities themselves) that are to be added. As pointed out by Koopman (1955), and by Ballentine (1986), this argument is mistaken: the probability distributions at the ba kstop P12 , P1 and P2 are onditional probabilities with three distin t onditions (both slits open, one or other slit losed), and probability al ulus does not imply any relationship between these. Feynman's argument notwithstanding, standard probability al ulus is perfe tly onsistent with the two-slit experiment. In his in uential le tures on physi s, as well as asserting the breakdown of probability al ulus, Feynman laimed that no theory with parti le traje tories ould explain the two-slit experiment. This laim 168
6.1 Probability and interferen e
169
is still found in many textbooks.a From a histori al point of view, it is remarkable indeed that single-parti le interferen e ame to be widely regarded as in onsistent with any theory ontaining parti le traje tories: for as we have seen in hapter 2, in the ase of ele trons this phenomenon was in fa t rst predi ted (by de Broglie) on the basis of pre isely su h a theory. As we shall now dis uss, in his report at the fth Solvay onferen e de Broglie gave a lear and simple explanation for single-parti le interferen e on the basis of his pilot-wave theory; and the extensive dis ussions at the onferen e ontain no sign of any obje tion to the onsisten y of de Broglie's position on this point. As for S hrödinger's theory of wave me hani s, in whi h parti les were supposed to be onstru ted out of lo alised wave pa kets, in retrospe t it is di ult to see how single-parti le interferen e ould have been a
ounted for. It is then perhaps not surprising that, in Brussels in 1927, no spe i dis ussion of interferen e appears in S hrödinger's
ontributions. Born and Heisenberg, on the other hand, do dis uss interferen e in their report, from the point of view of their `quantum me hani s'. And, they do onsider the question of the appli ability of probability al ulus. The views they present are, interestingly enough, rather dierent from the views usually asso iated with quantum me hani s today. In parti ular, as we shall see below, a
ording to Born and Heisenberg there was (in a very spe i sense) no on i t between quantum interferen e and the ordinary probability al ulus. Interferen e was also onsidered in the general dis ussion, in parti ular by Dira and Heisenberg: this latter material will be dis ussed later, in se tion 6.3.
6.1.1 Interferen e in de Broglie's pilot-wave theory At the fth Solvay onferen e, the subje t of interferen e was addressed from a pilot-wave perspe tive by de Broglie in his report. In his se tion 5, `The interpretation of interferen e', de Broglie onsidered interferen e experiments with light of a given frequen y ν . For a guiding wave Ψ a For example, Shankar (1994) dis usses the two-slit experiment at length in his
hapter 3, and laims (p. 111) that the observed single-photon interferen e pattern ` ompletely rules out the possibility that photons move in well-de ned traje tories'. Further, a
ording to Shankar (p. 112): `It is now widely a
epted that all parti les are des ribed by probability amplitudes ψ(x), and that the assumption that they move in de nite traje tories is ruled out by experiment'.
170
Interferen e, superposition, and ollapse
of phase φ and amplitude a, de Broglie took the photon velo ity to be c2 given by v = − hν ∇φ, while the probability distribution was taken to be π = const·a2 . As de Broglie had pointed out, the latter distribution is preserved over time by the assumed motion of the photons. Therefore, the usual interferen e and dira tion patterns follow immediately. To quote de Broglie (p. 385): the bright and dark fringes predi ted by the new theory will oin ide with those predi ted by the old [that is, by lassi al wave opti s℄.
De Broglie also pointed out that his theory gave the orre t bright and dark fringes for photon interferen e experiments, regardless of whether the experiments were performed with an intense or a very feeble sour e. As he put it (p. 385): one an do an experiment of short duration with intense irradiation, or an experiment of long duration with feeble irradiation .... if the light quanta do not a t on ea h other the statisti al result must evidently be the same.
De Broglie's dis ussion here addresses pre isely the supposed di ulty highlighted mu h later by Feynman. It is noteworthy that a lear and simple answer to what Feynman thought was `the only mystery' of quantum me hani s was published as long ago as the 1920s. Even so, for the rest of the twentieth entury, the two-slit experiment was widely ited as proof of the non-existen e of parti le traje tories in the quantum domain. Su h traje tories were thought to imply the relation P12 = P1 + P2 , whi h is violated by experiment. As Feynman (1965, hap. 1, p. 6) put it, on the basis of this argument it should `undoubtedly' be on luded that: `It is not true that the ele trons go either through hole 1 or hole 2'. Feynman also suggested that, by 1965, there had been a long history of failures to explain interferen e in terms of traje tories: Many ideas have been on o ted to try to explain the urve for
P12
[that
is, the interferen e pattern℄ in terms of individual ele trons going around in
ompli ated ways through the holes. None of them has su
eeded. (Feynman 1965, hap. 1, p. 6)
Yet, de Broglie's onstru tion is so simple as to be almost trivial: the 2 quantum probability density |ψ| for parti le position obeys a ontinuity equation, with a lo al probability urrent; if the traje tories follow the ow lines of the quantum urrent then, by onstru tion, an in ident 2 distribution |ψ0 | of parti les will ne essarily evolve into a distribution
6.1 Probability and interferen e
171
2
|ψ| at the ba kstop with interferen e or dira tion, as the ase may be, depending on the potential in whi h the wave ψ evolves. Not only did Feynman laim, wrongly, that no one had ever su
eeded in explaining interferen e in terms of traje tories; he also gave an argument to the ee t that any su h explanation was impossible: Suppose we were to assume that inside the ele tron there is some kind of ma hinery that determines where it is going to end up. That ma hine must
also
determine whi h hole it is going to go through on its way. But .... what
is inside the ele tron should not be dependent .... upon whether we open or
lose one of the holes. So if an ele tron, before it starts, has already made up its mind (a) whi h hole it is going to use, and (b) where it is going to land, we should nd
P1
for those ele trons that have hosen hole 1,
P2
that have hosen hole 2,
and ne essarily
through the two holes.
There seems to be no way around this.
the sum
P1 + P2
for those
for those that arrive (Feynman
1965, hap. 1, p. 10)
Feynman's argument assumes that the motion of the ele tron is unae ted by opening or losing one of the holes. This assumption is violated in pilot-wave theory, where the form of the guiding wave behind the two-slit s reen does depend on whether or not both slits are open. A similar assumption is made in the dis ussion of the two-slit experiment by Heisenberg (1962), in hapter III of his book Physi s and Philosophy. Heisenberg onsiders single photons in ident on a s reen with two small holes and a photographi plate on the far side, and gives the familiar argument that the existen e of parti le traje tories implies the non-interfering result P1 + P2 . As Heisenberg puts it: If [a single photon℄ goes through the rst hole and is s attered there, its probability for being absorbed at a ertain point of the photographi plate
annot depend upon whether the se ond hole is losed or open. (Heisenberg 1962)
This assertion is denied by pilot-wave theory, whi h provides a simple
ounterexample to Heisenberg's on lusion that `the statement that any light quantum must have gone either through the rst or through the se ond hole is problemati and leads to ontradi tions'. Finally, we note that interferen e was also onsidered by Brillouin (pp. 402 .) in the dis ussion following de Broglie's report, for the ase of photons re e ted by a mirror. Brillouin drew a gure (p. 404), with a sket h of a photon traje tory passing through an interferen e region. To our knowledge, plots of traje tories in ases of interferen e did not appear again in the literature until the pioneering numeri al work by Philippidis, Dewdney and Hiley (1979).
172
Interferen e, superposition, and ollapse
6.1.2 Interferen e in the `quantum me hani s' of Born and Heisenberg The subje t of interferen e was onsidered by Born and Heisenberg, in their report on quantum me hani s (pp. 424 f.), for the ase of an atom initially in a superposition X |ψ(0)i = (1) cn (0) |ni n
of energy states |ni, with oe ients cn (0) = |cn (0)| eiγn and eigenvalues En . The S hrödinger equation implies a time evolution X cn (t) = (2) Snm (t)cm (0) m
with (in modern notation) Snm (t) = hn| U (t, 0) |mi, where U (t, 0) is the evolution operator. In the spe ial ase where cm (0) = δmk for some k , we 2 2 2 have |cn (t)| = |Snk (t)| , and Born and Heisenberg interpret |Snk (t)| as a transition probability. They also draw the on lusion that `the |cn (t)|2 must be the state probabilities' (p. 424). Born and Heisenberg seem to adopt a statisti al interpretation, a
ording to whi h the system is always in a de nite energy state, with 2 jump probabilities |Snk (t)| and o
upation (or `state') probabilities 2 |cn (t)| ( f. se tion 3.4.6). This is stated quite expli itly (p. 423): From the point of view of Bohr's theory a system an always be in only
one
quantum state. .... A
ording to Bohr's prin iples it makes no sense to say a system is simultaneously in several states. The only possible interpretation seems to be statisti al: the superposition of several eigensolutions expresses that through the perturbation the initial state an go over to any other quantum state .... .
Note that this is quite dierent from present-day quantum me hani s, in whi h a system des ribed by the superposition (1) would not normally be regarded as always o
upying only one energy state. At the same time, Born and Heisenberg re ognise a di ulty (p. 424): Here, however, one runs into a di ulty of prin iple that is of great importan e, as soon as one starts from an initial state for whi h not all the
cn (0)
ex ept one vanish.
The di ulty, of ourse, is that for an initial superposition the nal probability distribution is given by X 2 |cn (t)|2 = (3) Snm (t)cm (0) m
6.1 Probability and interferen e
173
|cn (t)|2 =
(4)
as opposed to X m
|Snm (t)|2 |cm (0)|2
whi h, as Born and Heisenberg remark, `one might suppose from the usual probability al ulus'. (In standard probability al ulus, of ourse, (4) expresses |cn (t)|2 as a sum over onditional transition probabilities 2 2 |Snm (t)| weighted by the initial population probabilities |cm (0)| .) While Born and Heisenberg refer to (3) as the `theorem of the interferen e of probabilities', they make the remarkable assertion that there is in fa t no ontradi tion with the usual rules of probability al ulus, and 2 that the |Snm | may still be regarded as ordinary probabilities. Further, and equally remarkably, it is laimed that in any ase where the state probabilities |cn |2 are experimentally established, the presen e of the unknown phases γn makes the interfering expression (3) redu e to the non-interfering expression (4) (p. 425): .... it should be noted that this `interferen e' does not represent a ontradi tion with the rules of the probability al ulus, that is, with the assumption that 2 the |Snk | are quite usual probabilities. In fa t, .... [(4)℄ follows from the
on ept of probability .... when and only when the relative number, that is, the |cn |2 of the atoms in the state n, has been established beforehand
probability
experimentally.
In this ase the phases
γn
are unknown in prin iple, so that
[(3)℄ then naturally goes over to [(4)℄.... .
Here, Born and Heisenberg refer to Heisenberg's (re ently-published) un ertainty paper, whi h ontains a similar laim. There, Heisenberg
onsiders a Stern-Gerla h atomi beam passing through two su
essive regions of eld inhomogeneous in the dire tion of the beam (so as to indu e transitions between energy states without separating the beam into omponents). If the input beam is in a de nite energy state then the beam emerging from the rst region will be in a superposition. The probability distribution for energy emerging from the se ond region will then ontain interferen e as in (3), where the `initial' superposition (1) is now the state emerging from the rst region. Heisenberg asserts that, if the energy of an atom is a tually measured between the two regions, then be ause of the resulting perturbation `the phase of the atom hanges by amounts that are in prin iple un ontrollable' (Heisenberg 1927, pp. 183 4), and averaging over the unknown phases in the nal superposition yields a non-interfering result. The same example, with the same phase randomisation argument, is also given by Heisenberg (1930b) in his book The Physi al Prin iples
174
Interferen e, superposition, and ollapse
of the Quantum Theory ( hapter IV, se tion 2), whi h was based on le tures delivered at Chi ago in 1929. Heisenberg asserts (p. 60) that an energy measurement for an atom in the intermediate region `will ne essarily alter the phase of the de Broglie wave of the atom in state m by an unknown amount of order of magnitude one', so that in applying the (analogue of the) interfering expression (3) ea h term in the sum `must thus be multiplied by the arbitrary fa tor exp (iϕm ) and then averaged over all values of ϕm '. From a modern perspe tive, this argument seems strange and unfamiliar, and indeed quite wrong. However, the argument makes rather more sense, if one re ognises that the `quantum me hani s' des ribed by Born and Heisenberg is not quantum me hani s as we usually know it today. In parti ular, the theory as they present it appears to ontain no notion of wave pa ket ollapse (or state ve tor redu tion). The argument given by Born and Heisenberg amounts to saying, in modern language, that if the energies of an atomi population have a tually been measured, then one will have a mixture of states of the superposed form (1), with randomly-distributed phases γn . Su h a mixture is indeed statisti ally equivalent to a mixture of energy states |ni with weights |cn (0)|2 , be ause the density operators are the same: ! X Y 1 Z |cn (0)| |cm (0)| ei(γn −γm ) |ni hm| = dγk 2π n,m k X 2 |cn (0)| |ni hn| . (5) n
However, from a modern point of view, if one did measure the atomi energies and nd the value En with frequen y pn , the resulting total ensemble would naturally be represented by a density operator ρ = P n pn |ni hn|, and there would seem to be no parti ular reason to rewrite this in terms of the alternative de omposition on the left-hand side of (5) (with |cn (0)|2 = pn and random phases eiγn ); though of ourse one ould if one wished to. What is more, an a tual in onsisten y would appear if having measured the atomi energies one sele ted a parti ular atom that was found to have energy Em : a subsequent and immediate energy measurement for this parti ular atom should again yield the result Em with ertainty, as is onsistent with the usual representation of the atom by the state |mi, and this ertainty would be in onsistent with what appears to be (at least in ee t) the proposed representation of the atom P by a state n |cn (0)| eiγn |ni with randomised phases γn . Indeed, for any
6.1 Probability and interferen e
175
subensemble omposed of the latter states, all the energy values present in the sum will be possible out omes of subsequent and immediate energy measurements.a This in onsisten y arises, however, if one applies the modern notion of state ve tor ollapse a notion that, upon lose examination, appears to be quite absent from the theory presented by Born and Heisenberg. Instead of applying the usual ollapse rule, Born and Heisenberg seem to interpret the quantities |Snm |2 as `quite usual' transition probabilities in all ir umstan es, even in the presen e of interferen e. On this view, then, an atom that has been found to have energy Em an be represented P by a state n |cn (0)| eiγn |ni with randomised phases γn , and the probability of obtaining a value En in an immediately su
essive measurement is given not by |cn (0)|2 (as would follow from the usual ollapse rule) 2 but by the transition probability |Snm | where the latter does indeed approa h δnm as the time interval between the two energy measurements tends to zero, so that the above ontradi tion does not in fa t arise. Considering now the whole atomi ensemble, if the energies of the atoms have indeed been measured, then using the |Snm |2 as transition 2 probabilities and the |cm (0)| as population probabilities, appli ation of the probability al ulus gives the non-interfering result (4). As noted by Born and Heisenberg, on their view exa tly the same result is obtained from the `interfering' expression (3), with random phases γm appearing in the oe ients cm (0) = |cm (0)| eiγm . If instead the atomi energies have not been measured, then, a
ording to Born and Heisenberg, the phases γm have not been randomised and the expression (3) does show interferen e, in ontradi tion with the noninterfering expression (4). How do Born and Heisenberg re on ile the breakdown of (4) with their laim that the ordinary probability al ulus still holds, with the |Snm |2 being quite ordinary probabilities? The answer seems to be that, if the energies have not been measured, then the population probabilities |cm (0)|2 are in some sense ill-de ned, so that the usual probability formula (4) simply annot be applied: `[(4)℄ follows from the on ept of probability .... when and only when .... the probability |cn |2 .... has been established beforehand experimentally'. Born and Heisenberg seem to take an `operational' view of the populaa Of ourse, if we do not subdivide the atomi ensemble on the basis of the measured energies, the total ensemble will be a mixture with density operator P 2 n |cn (0)|P|ni hn| and will indeed be indistinguishable from the proposed mixture iγn |ni with randomly-distributed phases. of states But there is n |cn (0)| e nothing to prevent an experimenter from sele ting atoms a
ording to their measured energies.
176
Interferen e, superposition, and ollapse
tion probabilities, in the sense that these are to be regarded as meaningful only when dire tly measured. And the ited argument in Heisenberg's un ertainty paper suggests that it is operationally impossible to have simultaneously well-de ned phase relations and population probabilities in the same experiment. This impossibility was presumably regarded as
omparable to the (operational) impossibility of having simultaneously a well-de ned position and momentum for a parti le. What seems to be at work here, then, is some form of un ertainty relation (or omplementarity) between population probabilities and phases: measurement of the former makes the latter ill-de ned, and vi e versa. Interferen e requires de nite phase relationships, whi h pre lude a well-de ned population probability, so that the ordinary probability al ulus annot be applied.a If instead the population probability has a tually been measured, then the phases are inde nite and averaging over them washes out any interferen e. The resulting viewpoint is ertainly remarkable. A
ording to Born 2 and Heisenberg, in the presen e of interferen e, the quantities |Snm |
ontinue to be quite ordinary (transition) probabilities, while the quanti2 ties |cm (0)| annot be regarded as population probabilities rendering the formula (4) inappli able. On this view, ordinary probability al ulus is not violated; it is simply wrong to assert that the |cm (0)|2 represent state probabilities in an interfering ase. One may well obje t to this point of view on the grounds that, even without measuring the energies, for a given preparation of the state (1) the oe ients cm (0) and hen e the values of |cm (0)|2 will be known (up to an overall phase). However, presumably, Born and Heisenberg would have had to assert that while the |cm (0)|2 always exist as mathemati al quantities, they annot be properly interpreted as population probabilities unless the energies have been measured dire tly. From a modern perspe tive, Born and Heisenberg's treatment of interferen e is surprising: in modern quantum me hani s, of ourse, in ases where interferen e o
urs the quantities |Snm |2 would not normally be interpreted as `quite usual' transition probabilities; while in ases where interferen e does not o
ur, the non-interfering result (4) (as applied a There is an analogy here with Heisenberg's view of ausality, expressed in his un ertainty paper, a
ording to whi h ausality annot be applied be ause its premiss is generally false: `.... in the sharp formulation of the law of ausality, If we know the present exa tly, we an al ulate the future, it is not the onsequent that is wrong, but the ante edent. We annot in prin iple get to know the present in all [its℄ determining data' (Heisenberg 1927, p. 197).
6.2 Ma ros opi superposition: Born's dis ussion of the loud hamber 177 here) would not normally be regarded as arising from the interfering result (3) through a pro ess of phase randomisation.
6.2 Ma ros opi superposition: Born's dis ussion of the loud
hamber Quantum theory is normally understood to allow the `superposition of distin t physi al states'. However, while `superposition' is well-de ned as a mathemati al term, it is hard to make sense of when applied to physi al states that is, when the omponents in a superposition are regarded as simultaneous physi al attributes of a single system. The need to understand su h `physi al superposition' seems parti ularly a ute when it is onsidered at the ma ros opi level. The di ulty here is losely related to the question of wave pa ket ollapse: how is a mathemati al superposition of ma ros opi ally-distin t states related to the de nite ma ros opi states seen in the laboratory? The Wilson loud hamber, as used to observe the tra ks of α-parti les, was dis ussed at length by Born in the general dis ussion. The loud
hamber illustrates the measurement problem rather well, and is a good example of how mi ros opi superpositions an be ome transferred to the ma ros opi domain. It also illustrates how extending the formal quantum des ription to the environment does not by itself alleviate the measurement problem (despite many laims to the ontrary, for example Zurek (1991)).a Remarkably, as we shall see, Born asserts that wave pa ket ollapse is not required to dis uss the loud hamber. The me hanism of the loud hamber is well known. The α-parti les pass through a supersaturated vapour. The passage of the parti les
auses ionisation, and the vapour ondenses around the ions, resulting in the formation of tiny droplets. The droplets s atter light, making the parti le tra ks visible. If the emission of an α-parti le is undire ted, so that the emitted wave fun tion is approximately spheri al, how does one a
ount for the approximately straight parti le tra k revealed by the loud hamber? In the general dis ussion, Born attributes this question to Einstein, and asserts that to answer it (p. 483) ....
one must appeal to the notion of `redu tion of the probability pa ket'
developed by Heisenberg.
a For a summary of riti isms of environmental de oheren e as a solution to the measurement problem, see Ba
iagaluppi (2005).
178
Interferen e, superposition, and ollapse
This notion appears in Heisenberg's un ertainty paper, whi h had been published in May of 1927. In se tion 3, entitled `The transition from mi ro- to ma rome hani s', Heisenberg had des ribed how a lassi al ele tron orbit ` omes into being' through repeated observation of the ele tron position, using light of wavelength λ. A
ording to Heisenberg, the result of ea h observation an be hara terised by a probability pa ket of width λ, where the pa ket spreads freely until the next observation: `Every determination of position redu es therefore the wave pa ket ba k to its original size λ' (Heisenberg 1927, p. 186). In the ase of the loud hamber, the ollapse of the wave pa ket is applied repeatedly to the α-parti le alone. Upon produ ing visible ionisation, the wave pa ket of the α-parti le ollapses, and then starts to spread again, until further visible ionisation is produ ed, whereupon
ollapse o
urs again, and so on. The probability for the resulting `traje tory' is on entrated along straight lines, a
ounting for the observed tra k in the loud hamber. As Born puts it (p. 483): The des ription of the emission by a spheri al wave is valid only for as long as one does not observe ionisation; as soon as su h ionisation is shown by the appearan e of loud droplets, in order to des ribe what happens afterwards one must `redu e' the wave pa ket in the immediate vi inity of the drops. One thus obtains a wave pa ket in the form of a ray, whi h orresponds to the
orpus ular hara ter of the phenomenon.
Here, the loud hamber itself the ionisation, and the formation of droplets from the vapour is treated as if it were an external ` lassi al apparatus': only the α-parti le appears in the wave fun tion.
6.2.1 Quantum me hani s without wave pa ket ollapse? Born goes on to onsider if wave pa ket redu tion an be avoided by treating the atoms of the loud hamber, along with the α-parti le, as a single system des ribed by quantum theory, a suggestion that he attributes to Pauli (p. 483): Mr Pauli has asked me if it is not possible to des ribe the pro ess without the redu tion of wave pa kets, by resorting to a multi-dimensional spa e, whose number of dimensions is three times the number of all the parti les present .... . This is in fa t possible .... but this does not lead us further as regards the fundamental questions.
Remarkably, Born laims that a treatment without redu tion is `in fa t' possible, and goes on to illustrate how, in his opinion, this an be
6.2 Ma ros opi superposition
179
done. As we shall see, Born seems to make use of a ` lassi al' probability redu tion only, without any redu tion for the on guration-spa e wave pa ket. As for Born's referen e to Pauli, around the time of the Solvay onferen e Pauli believed that wave pa ket redu tion was needed only for des ribing subsystems. This is lear from a letter he wrote to Bohr, on 17 O tober 1927 (one week before the Solvay meeting began), in whi h Pauli omments on wave pa ket redu tion (Pauli 1979, p. 411): This is pre isely a point that was not quite satisfa tory in Heisenberg [that is, in the un ertainty paper℄; there the `redu tion of the pa kets' seemed a bit mysti al. Now however, it is to be stressed that at rst su h redu tions are not ne essary if one
in ludes
in the system all means of measurement. But
in order to des ribe observational results theoreti ally at all, one has to ask what one an say about just a
part
of the whole system. And then from the
omplete solution one sees immediately that, in many ases (of ourse not always), leaving out the means of observation an be formally repla ed by su h redu tions.
Thus, at that time, Pauli thought that the redu tion was a formality asso iated with an ee tive des ription of subsystems alone, and that if the apparatus were in luded in the system then redu tion would not be needed at all. Born, then, presents a multi-dimensional treatment, in whi h atoms in the loud hamber are des ribed by quantum theory on the same footing as the α-parti le. Born onsiders the simple ase of a model loud
hamber onsisting of just two atoms in one spatial dimension. There are two ases, one with both atoms on the same side of the origin (where the α-parti le is emitted), and the other with the atoms on opposite sides of the origin. The two `tubes' in Born's diagram (see his gure) represent, for the two ases, the time development of the total (lo alised) pa ket in 3-dimensional on guration spa e. The oordinate x0 of the α-parti le is perpendi ular to the page, while x1 , x2 are the oordinates of the two atoms. In ase I, the initial state is lo alised at x0 = 0, x1 , x2 > 0; in ase II it is lo alised at x0 = 0, x1 > 0, x2 < 0. Ea h initial pa ket separates into two pa kets moving in opposite dire tions along the x0 -axis. In the rst ase the two ollisions (indi ated by kinks in the traje tory of the pa keta ) take pla e on the same side of the origin; in the se ond ase they take pla e on opposite sides. Note that, in both ases, both bran hes of a Note that the motion o
urs in one spatial dimension; a kink in the on gurationspa e traje tory shows that the orresponding atom has undergone a small spatial displa ement.
180
Interferen e, superposition, and ollapse
the wave pa ket moving in opposite dire tions are shown; that is, the omplete (`un ollapsed') pa kets are shown in the gure. Born remarks (p. 486) that: To the `redu tion' of the wave pa ket orresponds the hoi e of one of the two dire tions of propagation
+x0 , −x0 ,
whi h one must take as soon as it is
established that one of the two points 1 and 2 is hit, that is to say, that the traje tory of the pa ket has re eived a kink.
Here Born seems to be saying that, instead of redu tion, what takes pla e is a hoi e of dire tion of propagation. But propagation of what? Born presumably does not mean a hoi e of dire tion of propagation of the wave pa ket, for that would amount to wave pa ket redu tion, whi h Born at the outset has laimed is unne essary in a multi-dimensional treatment. (And indeed, his gure shows the wave pa ket propagating in both dire tions.) Instead, Born seems to be referring to a hoi e in the dire tion of propagation of the system (whi h we would represent by a point in on guration spa e). The wave pa ket spreads in both dire tions, and determines the probabilities for the dierent possible motions of the system. On e the dire tion of motion of the system is established, by the o
urren e of ollisions, an ordinary (` lassi al') redu tion of the probability distribution o
urs while the wave pa ket itself is un hanged. In other words, the probabilities are updated but the wave pa ket does not ollapse. This, at least, appears to be Born's point of view. This may seem a pe uliar interpretation ordinary probabilisti
ollapse without wave pa ket ollapse but it is perhaps related to the intuitive thinking behind Born's famous ollision papers of the previous year (Born 1926a,b) (papers in whi h probabilities for results of
ollisions were identi ed with squares of s attering amplitudes for the wave fun tiona ). Born drew an analogy with Einstein's notion of a `ghost eld' that determines probabilities for photons (see hapter 9). As Born put it: In this, I start from a remark by Einstein on the relationship between the wave eld and light quanta; he said, for instan e, that the waves are there only to show the orpus ular light quanta the way, and in this sense he talked of a `ghost eld'.
This determines the probability for a light quantum, the
arrier of energy and momentum, to take a parti ular path; the eld itself, however, possesses no energy and no momentum.
....
analogy between the light quantum and the ele tron ....
Given the perfe t one will think of
formulating the laws of motion of ele trons in a similar way.
a Cf. the dis ussion in se tion 3.4.3.
And here it
6.2 Ma ros opi superposition
181
is natural to onsider the de Broglie-S hrödinger waves as the `ghost eld', or better, `guiding eld' .... [whi h℄ propagates a
ording to the S hrödinger equation.
Momentum and energy, however, are transferred as if orpus les
(ele trons) were a tually ying around. The traje tories of these orpus les are determined only insofar as they are onstrained by the onservation of energy and momentum; furthermore, only a probability for taking a ertain path is determined by the distribution of values of the fun tion
ψ.
(Born
1926b, pp. 8034)
Here, Born seems to be suggesting that there are sto hasti traje tories for ele trons, with probabilities for paths determined by the wave fun tion ψ . It is not lear, though, whether the ψ eld is to be regarded as a physi al eld asso iated with individual systems (as the ele tromagneti eld usually is), or whether it is to be regarded merely as relating to an ensemble. If the former, then it might make sense to apply ollapse to the probabilities without applying ollapse to ψ itself. For example, this ould happen in a sto hasti version of de Broglie-Bohm theory: ψ
ould be a physi al eld evolving at all times by the S hrödinger equation (hen e never ollapsing), and instead of generating deterministi parti le traje tories (as in standard de Broglie-Bohm theory) ψ ould generate probabilisti motions only. Further eviden e that Born was indeed thinking along su h lines omes from an unpublished manus ript by Born and Jordan, written in 1925, in whi h they propose a sto hasti theory of photon traje tories with probabilities determined by the ele tromagneti eld (as originally envisaged by Slater) see Darrigol (1992, p. 253) and se tion 3.4.2.a Thus, in his dis ussion of the loud hamber, when Born spoke of `the
hoi e of one of the two dire tions of propagation', he may indeed have been referring to the possible dire tions of propagation of the system
on guration, with the ψ eld remaining in a superposition. On this reading, wave pa ket redu tion for the α-parti le would be only an ee tive des ription, whi h properly orresponds to the bran hing of the total wave fun tion together with a random hoi e of traje tory (in multi-dimensional on guration spa e). Unfortunately, however, Born's intentions are not entirely lear: whether this really is what Born had in mind, in 1926 or 1927, is di ult to say. Certainly, in the general a In this onne tion it is interesting to note the following passage from Born's book Atomi Physi s (Born 1969), whi h was rst published in German in 1933: `A me hani al pro ess is therefore a
ompanied by a wave pro ess, the guiding wave, des ribed by S hrödinger's equation, the signi an e of whi h is that it gives the probability of a de nite ourse of the me hani al pro ess'. Born's referen e to the wave fun tion as `the guiding wave' shows the lingering in uen e of the ideas that had inspired him in 1926.
182
Interferen e, superposition, and ollapse
dis ussion at the fth Solvay onferen e, Born did maintain that ψ does not really ollapse (in this multi-dimensional treatment), so it is di ult to see how he ould have thought that ψ gave merely a probability distribution over an ensemble. The laim that the wave fun tion ψ does not ollapse is also found, in ee t, in se tion II of the report that Born and Heisenberg gave on quantum me hani s. As we saw in se tion 6.1.2, in their dis ussion of interferen e, an energy measurement is taken to indu e a randomisation of the phases appearing in a superposition of energy states, instead of the usual ollapse to an energy eigenstate. In their example, after an energy measurement all of the omponents of the superposition are still present, and the phase relations between them are randomised. In Born's example of the loud hamber, it seems that, here too, all of the omponents of the wave fun tion are still present at the end of a measurement. Nothing is said, however, about the relative phases of the
omponents, and we do not know whether or not Born had in mind a similar phase randomisation in this ase also.
6.3 Dira and Heisenberg: interferen e, state redu tion, and delayed hoi e Another striking feature of quantum theory, as normally understood, is `interferen e between alternative histories'. Like superposition, interferen e is mathemati ally well-de ned but its physi al meaning is ambiguous, and it has long been onsidered one of the main mysteries of quantum theory (as we saw in se tion 6.1). The question of when interferen e an or annot take pla e is intimately bound up with the measurement problem, in parti ular, with the question of when or how de nite out omes emerge from quantum experiments, and with the question of the boundary between the quantum and lassi al domains. In modern times, one of the most puzzling aspe ts of interferen e was emphasised by Wheeler (1978). In his `delayed- hoi e' experiment, it appears that the existen e or non-existen e of interfering histories in the past is determined by an experimental hoi e made in the present. One version of Wheeler's experiment a `delayed- hoi e double-slit experiment' with single photons is shown in Fig. 6.1. A single photon is in ident on a s reen with two slits. The waves emerging from the slits are fo ussed (by o- entred lenses) so as to ross ea h other as shown. The insertion of a photographi plate in the interferen e region would seem (from the interferen e pattern) to imply the past existen e of
6.3 Dira and Heisenberg
183
Fig. 6.1. Delayed- hoi e double-slit experiment.
interfering traje tories passing through both slits. On the other hand, if no su h plate is inserted, then a dete tion at P or P seems to imply that the parti le passed through the bottom or top slit respe tively.a Sin e the plate ould have been inserted long after the photon ompleted most of its journey (or journeys), it appears that an experimental hoi e now
an ae t whether or not there was a de nite photon path in the past. A similar point arose in 1927 in the general dis ussion. Dira expounded his view that quantum out omes o
ur when nature makes a hoi e. Heisenberg replied that this ould not be, be ause of the possibility of observing interferen e later on by hoosing an appropriate experimental arrangement, leading Heisenberg to on lude that out omes o
ur when a hoi e is made (or brought about) not by nature but by the observer. Heisenberg's view here bears some resemblan e to Wheeler's. Dira , in
ontrast, seems to say on the one hand that sto hasti ollapse of the wave pa ket o
urs for mi ros opi systems, while on the other hand a This inferen e is ommonly made, usually without expli it justi ation. Some authors appeal to onservation of momentum for a free parti le, but it is not lear how su h an argument ould be made pre ise after all there are no parti le traje tories in standard quantum theory. In de Broglie-Bohm theory, the inferen e is a tually wrong: parti les dete ted at P or P ome from the top or bottom slits respe tively (Bell 1980; Bell 1987, hap. 14).
184
Interferen e, superposition, and ollapse
that if the experiment is hosen so as to allow interferen e then su h
ollapse is postponed. Here is how Dira expresses it (p. 495): A
ording to quantum me hani s the state of the world at any time is des ribable by a wave fun tion
ψ,
whi h normally varies a
ording to a ausal law,
so that its initial value determines its value at any later time. It may however happen that at a ertain time
t1 , ψ ψ=
an be expanded in the form
X
cn ψn ,
n
where the
ψn 's
are wave fun tions of su h a nature that they annot interfere
with one another at any time subsequent to
t1 .
If su h is the ase, then the
world at times later than t1 will be des ribed not by The parti ular
ψn
ψ
but by one of the
ψn 's.
that it shall be must be regarded as hosen by nature.
Note rst of all that Dira regards ψ as des ribing the state `of the world' presumably the whole world. Then, in ir umstan es where ψ may be expanded in terms of non-interfering states ψn , the world is subsequently des ribed by one of the ψn (the hoi e being made by nature, the probability for ψn being |cn |2 ). Dira does not elaborate on pre isely when or why a de omposition into non-interfering states should exist, nor does he address the question of whether su h a de omposition is likely to be unique. Su h questions, of ourse, go to the heart of the measurement problem, and are lively topi s of urrent resear h. It is interesting that, in Dira 's view (apparently), there are ir umstan es in whi h interferen e is ompletely and irreversibly destroyed. For him, the parti ular ψn results from (p. 495): an irrevo able hoi e of nature, whi h must ae t the whole of the future
ourse of events.
It seems that a
ording to Dira , on e nature makes a hoi e of one bran h, interferen e with the other bran hes is impossible for the whole of the future. A de nite ollapse has o
urred, after whi h interferen e between the alternative out omes is no longer possible, even in prin iple. This view learly violates the S hrödinger equation as applied to the whole world: as Dira states, the wave fun tion ψ of the world `normally' evolves a
ording to a ausal law, but not always. But Dira goes further, and re ognises that there are ir umstan es where the hoi e made by nature annot have o
urred at the point where it might have been expe ted. Dira onsiders the spe i example of the s attering of an ele tron. He rst notes that, after the s attering, one must take the wave fun tion to be not the whole s attered wave
6.3 Dira and Heisenberg
185
but a pa ket moving in a spe i dire tion (that is, one of the ψn ). He
laims (p. 495) that one ould infer that nature had hosen this spe i dire tion: From the results of an experiment, by tra ing ba k a hain of ausally onne ted events one ould determine in whi h dire tion the ele tron was s attered and one would thus infer that nature had hosen this dire tion.
This is illustrated in Fig. 6.2(a). If the ele tron is dete ted at P, for example, one may arguably infer a orresponding hoi e of dire tion at the time of s attering. (Note that Figs. 6.2(a)( ) are ours.) On the other hand, Dira goes on to make the following observation (p. 495): If, now, one arranged a mirror to re e t the ele tron wave s attered in one dire tion
d1
so as to make it interfere with the ele tron wave s attered in
another dire tion
d2 ,
one would not be able to distinguish between the ase
when the ele tron is s attered in the dire tion the dire tion
d1
and re e ted ba k into
d2 .
d2
and when it is s attered in
One would then not be able to
tra e ba k the hain of ausal events so far, and one would not be able to say that nature had hosen a dire tion as soon as the ollision o
urred, but only [that℄ at a later time nature hose where the ele tron should appear.
Dira 's modi ed s enario is sket hed in Fig. 6.2(b). The presen e of the mirror leads to interferen e, at Q, between parts of the ele tron wave s attered in dierent dire tions d1 , d2 . And Dira 's interpretation is that this interferen e is intimately related to the fa t that an experimenter observing the outgoing ele tron in a dire tion d2 `would not be able to distinguish between the ase when the ele tron is s attered in the dire tion d2 and when it is s attered in the dire tion d1 and re e ted ba k into d2 '. The experimenter would not be able to `tra e ba k the
hain of ausal events' to the point where he ould say that `nature had
hosen a dire tion as soon as the ollision o
urred'. For Dira , in this
ase, nature did not make a hoi e at the time of the ollision, and only later nature ` hose where the ele tron should appear'. What Dira des ribes here is pre isely the viewpoint popularised by Feynman in his famous le tures (Feynman 1965), a
ording to whi h if a pro ess o
urs by dierent routes that are subsequently indistinguishable (in the sense that afterwards an experimenter is in prin iple unable to tell whi h route was taken) then the probability amplitudes for the dierent routes are to be added; whereas if the dierent routes are subsequently distinguishable in prin iple, then the probabilities are to be added.a a Note that Feynman's path-integral formulation of quantum theory developed in
186
Interferen e, superposition, and ollapse
Fig. 6.2. Re onstru tion of s attering s enarios dis ussed by Dira (Figs. (a), (b)) and Heisenberg (Fig. ( )).
Dira 's presentation of the s attering experiment with the mirror ends with the statement (p. 496): The interferen e between the
ψn 's
ompels nature to postpone her hoi e.
his PhD thesis and elsewhere (Feynman 1942, 1948) was anti ipated by Dira (1933). Feynman's thesis and Dira 's paper are reprinted in Brown (2005).
6.3 Dira and Heisenberg
187
In his manus ript, a an elled version of the senten e begins with `Thus a possibility of interferen e between .... ', while another an elled version begins as `Thus the existen e of .... ' (itali s added). Possibly, Dira hesitated here be ause he saw that the mirror ould be added by the experimenter after the s attering had taken pla e, leading to di ulties with his view that without the mirror nature makes a hoi e at the time of s attering. For if, in the absen e of the mirror, nature indeed makes a hoi e at the time of s attering, how ould this hoi e be undone by subsequent addition of the mirror? Whether Dira really foresaw this di ulty is hard to say. In any ase, pre isely this point was made by Heisenberg, and Dira 's hesitation here ertainly re e ts a deep di ulty that lies at the heart of the measurement problem. Heisenberg makes his point with disarming simpli ity (p. 497): I do not agree with Mr Dira when he says that, in the des ribed experiment, nature makes a hoi e.
Even if you pla e yourself very far away from your
s attering material, and if you measure after a very long time, you are able to obtain interferen e by taking two mirrors. If nature had made a hoi e, it would be di ult to imagine how the interferen e is produ ed.
What Heisenberg had in mind seems to have been something like the set-up shown in Fig. 6.2( ), where a pair of mirrors is pla ed far away from the s attering region, ausing dierent parts of the s attered wave to re-overlap and interfere at R. A
ording to Dira 's a
ount, in the absen e of any mirrors (Fig. 6.2(a)), upon dete tion of the parti le one might say that nature hose a spe i dire tion at the time of s attering. Heisenberg points out that, by pla ing mirrors far away (and removing the dete tors at P and P), interferen e may be observed a long time after the s attering took pla e. While Heisenberg does not mention it expli itly, in this example the
hoi e between `whi h-way information' on the one hand, or interfering paths on the other, may be made long after the parti le has ompleted most of its journey (or journeys), just as in Wheeler's delayed- hoi e experiment. Dira 's set-up with no mirrors at all provides whi h-way information, sin e dete tion of the parti le at a point far away may be interpreted as providing information on the dire tion hosen at the time of s attering (Fig. 6.2(a)). Heisenberg's modi ation, with the two mirrors, demonstrates interferen e between alternative paths starting from the s attering region (Fig. 6.2( )). Unlike Wheeler, however, Heisenberg does not expli itly emphasise that the hoi e of whether or not to add the mirrors ould be made at the last moment, long after the s attering
188
Interferen e, superposition, and ollapse
takes pla e. On the other hand, Heisenberg does emphasise that the measurement with the mirrors ould be done `very far away' and `after a very long time', and notes the ontradi tion with nature having made a hoi e at the time of s attering. Thus, Heisenberg's remarks arguably
ontain the essen e of Wheeler's delayed- hoi e experiment. Heisenberg goes on to say (p. 497) that, instead of nature making a
hoi e, I should rather say, as I did in my last paper, that the
observer himself
makes
the hoi e, be ause it is only at the moment where the observation is made that the ` hoi e' has be ome a physi al reality and that the phase relationship in the waves, the power of interferen e, is destroyed.
From the hronology of Heisenberg's publi ations, here he must be referring to his un ertainty paper (published in May 1927), in whi h he writes that `all per eiving is a hoi e from a plenitude of possibilities' (Heisenberg 1927, p. 197). Heisenberg's statement above that the observer `makes' the hoi e seems to be meant in the sense of the observer `bringing about' the hoi e. Thus it would seem that, for Heisenberg, a de nite out ome o
urs and there is no longer any possibility of interferen e only when an experimenter makes an observation. Similar views have been expressed by Wheeler (1986). One may, however, obje t to this viewpoint, on the grounds that there is no reason in prin iple why a more advan ed being ould not observe interferen e between the alternative states of the dete tor registering interferen e, or, between the alternative states of the human observer wat hing the dete tor. (Cf. the dis ussion of Wigner's paradox in se tion 5.2.2.) After all, the dete tor is ertainly just another physi al system, built out of atoms. And as far as we an tell, human observers
an likewise be treated as physi al systems built out of atoms. To say that `the power of interferen e' is `destroyed' when and only when a human observer intervenes is to make a remarkable assertion to the ee t that human beings, unlike any other physi al systems, have spe ial properties by virtue of whi h they annot be treated by ordinary physi al laws but generate deviations from those laws. As we have already mentioned, there is no eviden e that human beings are able to violate, for example, the laws of gravity or of thermodynami s, and it would be remarkable if they were indeed able to violate the laws of quantum physi s. It is interesting to note that, while for Heisenberg the human observer seems to play a ru ial role at the end of a quantum experiment, for
6.3 Further remarks on Born and Heisenberg
189
Dira the human observer and his `freewill' seems to play a ru ial role at the beginning, in the preparation stage. For as Dira puts it (p. 494, Dira 's itali s): `The disturban es that an experimenter applies to a system to observe it are dire tly under his ontrol, and are a ts of freewill by him. It is only the numbers that des ribe these a ts of freewill that an be taken as initial numbers for a al ulation in the quantum theory'. (Cf. the dis ussion about determinism in se tion 8.2.) Returning to Dira 's view of quantum out omes, Heisenberg's obje tion ertainly auses a di ulty. If a hoi e or ollapse to a parti ular ψn really does o
ur around the time of s attering, then a `delayed interferen e experiment' of the form des ribed by Heisenberg should show no interferen e, and Dira 's view would amount to a violation of the quantum formalism along the lines of dynami al models of wave fun tion ollapse (Pearle 1976, 1979, 1989; Ghirardi, Rimini and Weber 1986). And Dira 's aveats on erning the possibility of tra ing ba k a hain of ausal events do not lead to a really satisfa tory position either. As in Feynman's view that interferen e o
urs only for paths that are subsequently indistinguishable, the question is begged as to the pre ise de nition of subsequently distinguishable or subsequently indistinguishable paths: for in a delayed- hoi e set-up, it appears to be at the later whim of the experimenter to de ide whether ertain paths taken in the past are subsequently distinguishable or not. This pro edure
orre tly predi ts the experimental results (or statisti s thereof), but it has the pe uliar onsequen e that whether or not there is a matter of fa t about the past depends on what the experimenter does in the present. Finally, as dis ussed in se tion 6.1.1, interferen e was onsidered from a pilot-wave perspe tive by de Broglie in his report and by Brillouin in the dis ussion that followed. De Broglie did not omment, however, on the ex hange between Dira and Heisenberg. From a modern point of view it is lear that, in his theory, the parti le traje tory does take one parti ular route after a s attering pro ess, while at the same time there are portions of the s attered wave travelling along the alternative routes. An `empty' part of the wave an be subsequently re e ted by a mirror, and if the re e ted wave later reoverlaps with the part of the wave arrying the parti le, then in the interferen e zone the parti le is indeed ae ted by both omponents. Similarly, de Broglie's theory provides a straightforward a
ount of Wheeler's delayed- hoi e experiment, without present a tions in uen ing the past in any way (Bell 1980; Bell 1987, hap. 14; Bohm, Dewdney and Hiley 1985).
190
Interferen e, superposition, and ollapse
6.4 Further remarks on Born and Heisenberg's quantum me hani s As we saw in se tion 6.1.2, Born and Heisenberg's report ontains some remarkable omments about the nature of interferen e (in se tion II, `Physi al interpretation'). These omments are perhaps related to a
on eptual transition that seems to o
ur at around this point in their presentation. In the earlier part of their se tion II, Born and Heisenberg des ribe a theory in whi h probabilisti transitions o
ur between possessed values of energy; while later in the same se tion, in their dis ussion of arbitrary observables, they emphasise probabilisti transitions from one measurement to the next (still noting the presen e of interferen e). Earlier in that se tion they expli itly assert that a system always o
upies a de nite energy state at any one time, while in the later treatment of arbitrary observables nothing is said about whether a system always possesses de nite values or not. This is perhaps not surprising, given that the dis ussion of interferen e (for the ase of energy measurements) 2 2 made it lear that taking the quantities |cn (t)| = |hn |ψ(t)i| to be population probabilities for energies En led to a di ulty in the presen e of interferen e. As we saw in se tion 6.1.2, Born and Heisenberg resolved the di ulty by asserting that unmeasured population probabilities are somehow not appli able or meaningful. This does not seem onsistent with the view they expressed earlier, that atoms always have de nite energy states even when these are not measured. How ould an ensemble of atoms have de nite energy states, without the energy distribution being meaningful? Consideration of interferen e, then, was likely to for e a shift away from the view that atoms are always in de nite stationary states. Later, in his book of 1930, Heisenberg did in fa t expli itly deny that atoms are always in su h states. Considering again the example from his un ertainty paper, of atoms passing through two su
essive regions of inhomogeneous eld (see se tion 6.1.2), Heisenberg notes that if the energies are not a tually measured in the intermediate region, then, be ause of the resulting `interferen e of probabilities', it is not reasonable to speak of the atom as having been in a stationary state between
F1
and
F2
[that is, in the intermediate region℄. (Heisenberg 1930b,
p. 61)
As we have already noted, again in se tion 6.1.2, Born and Heisenberg's dis ussion of interferen e seems to dispense with the standard
ollapse postulate for quantum states. Upon performing an energy mea-
6.4 Further remarks on Born and Heisenberg
191
surement, instead of the usual ollapse to a single energy eigenstate |ni, P Born and Heisenberg in ee t repla e the superposition n |cn (0)| eiγn |ni by a similar expression with randomised phases γn . And the justi ation given for this appears to be some form of un ertainty relation or omplementarity between population probabilities and phases: if the former have been measured, then the latter are ill-de ned, and vi e versa. On this view, the de nite phase relationships asso iated with interferen e pre lude the possibility of speaking of a well-de ned population probability, so that the usual formulas of probability al ulus annot be properly applied; on the other hand, if the population probability has been measured experimentally, then the phases are ill-de ned, and averaging over the random phases destroys interferen e. One ru ial point is not entirely lear, however. Was the phase randomisation thought to o
ur only upon measurement of energy, or upon measurement of any arbitrary observable? The phase randomisation expli itly appealed to by Born and Heisenberg takes the following form: for a quantum state X X |ψ(t)i = (6) |Ei hE |ψ(t)i = |Ei hE |ψ(0)i e−iEt E
E
an energy measurement indu es a random hange in ea h phase fa tor e−iEt → e−iEt eiγ(E) , where ea h γ(E) is random on the unit ir le. This pro edure might be generalised to, for example, measurements of position, as follows: for a state ! X X X |ψi ∝ (7) |xi hx |ψi ∝ |pi e−ip·x hx |ψi x
x
p
(writing as if x and p were dis rete, for simpli ity) one might suppose that a position measurement indu es a random hange e−ip·x → e−ip·x eiγ(x) , resulting in a state X (8) |xi hx |ψi eiγ(x) x
with random relative phases. Averaging over the random phases γ(x) would then destroy interferen e between dierent positions, just as in the ase of energy measurements. However, we have found no lear eviden e that Born or Heisenberg
onsidered any su h generalisation. It then seems possible that the phase randomisation argument for the suppression of interferen e was to be applied to the ase of energy measurements only. On the other hand,
192
Interferen e, superposition, and ollapse
there is a suggestive remark by Heisenberg in the general dis ussion (p. 497), quoted in the last se tion. When expressing his view that de nite out omes o
ur only when an experimenter makes an observation, Heisenberg refers to an example where position measurements are made at the end of a s attering pro ess (see Fig. 6.2( )), and he states that it is only when the observation is made that `the phase relationship in the waves, the power of interferen e, is destroyed'. This might be read as suggesting that the waves ontinue to exist, but may or may not have the apa ity to interfere depending on whether or not the phase relations have been randomised by the position measurement. If Heisenberg did take su h a view, his use of wave pa ket redu tion for position measurements in the un ertainty paper would have to be interpreted as some sort of ee tive des ription. Even in the ase of energy measurements, the status of the phase randomisation argument is not lear. After all, Born and Heisenberg assert that the time-dependent S hrödinger equation itself (whi h they use to dis uss interferen e) is only phenomenologi al, and appli able to subsystems only. Fundamentally, they have a time-independent theory for a losed system. Presumably, the phase randomisation for energy measurements was also seen as phenomenologi al only, with the measured system being treated as a subsystem. As we saw in se tion 6.3, in the general dis ussion Dira des ribes what is re ognisably the pro ess of wave pa ket redu tion. Born and Heisenberg, in ontrast, seem to speak only of the ordinary redu tion (or onditionalisation) of `probability fun tions' as it appears in standard probability al ulus. As they put it, near the end of se tion III of their report (p. 432): the result of ea h measurement an be expressed by the hoi e of appropriate initial values for probability fun tions .... . Ea h new experiment repla es the probability fun tions valid until now with new ones, whi h orrespond to the result of the observation .... .
On the other hand, Born and Heisenberg's ontributions at the Solvay
onferen e do not seem su iently lear or omplete to warrant de nite
on lusions as to what they believed on erning the pre ise relationship between probabilities and the wave fun tion; sometimes it is un lear whether or not they mean to draw a distin tion between probability distributions on the one hand and wave fun tions on the other. It may be that, in O tober 1927, Born and Heisenberg had in some respe ts not yet rea hed a de nitive point of view, perhaps partly be ause
6.4 Further remarks on Born and Heisenberg
193
of the dierent perspe tives that Born and Heisenberg ea h brought to the subje t. Born's re ent thinking (in 1926) had been in uen ed by Einstein's idea of a guiding eld, while Heisenberg's re ent thinking (in his un ertainty paper) had been in uen ed by the operational approa h to physi s. Con erning the question of wave pa ket ollapse, it should also be remembered that Pauli seems to have played an important role in Born and Heisenberg's thinking at the time. In parti ular, as we saw in Born's dis ussion of the loud hamber (se tion 6.2.1), Pauli had been riti al of Heisenberg's use of the redu tion of the wave pa ket in the un ertainty paper (in a dis ussion of lassi al ele tron orbits), and Born who by his own a
ount was following Pauli's suggestion tried to show that su h redu tion was unne essary.
7 Lo ality and in ompleteness
7.1 Einstein's 1927 argument for in ompleteness A huge literature arose out of the famous `EPR' paper by Einstein, Podolsky and Rosen (1935), entitled `Can quantum-me hani al des ription of physi al reality be onsidered omplete?'. The EPR paper argued, on the basis of (among other things) the absen e of a tion at a distan e, that quantum theory must be in omplete.a It is less wellknown that a mu h simpler argument, leading to the same on lusion, was presented by Einstein eight years earlier in the general dis ussion at the fth Solvay onferen e (pp. 486 .). Einstein ompares and ontrasts two views about the nature of the wave fun tion ψ , for the spe i ase of a single ele tron. A
ording to view I, ψ represents an ensemble (or ` loud') of ele trons; while a
ording to view II, ψ is a omplete des ription of an individual ele tron. Einstein argues that view II is in ompatible with lo ality, and that to avoid this, in addition to ψ there should exist a lo alised parti le (along the lines of de Broglie's theory). Thus, a
ording to this reasoning, if one assumes lo ality, then quantum theory (as normally understood today) is in omplete. The on lusion of Einstein's argument in 1927 is the same as that of EPR in 1935, even if the form of the argument is rather dierent. Einstein onsiders ele trons striking a s reen with a small hole that dira ts the ele tron wave, whi h on the far side of the s reen spreads out uniformly in all dire tions and strikes a photographi lm in the shape of a hemisphere with large radius (see Einstein's gure). Einstein's argument against view II is then as follows: a Note that, as pointed out by Fine (1986) and dis ussed further by Howard (1990), the logi al stru ture of the EPR paper (whi h was a tually written by Podolsky) is more ompli ated and less dire t than Einstein had intended.
194
7.1 Einstein's 1927 argument for in ompleteness If
|ψ|2
195
were simply regarded as the probability that at a ertain point a given
the same elementary in two or several pla es on the s reen. But the 2 to whi h |ψ| expresses the probability that this
parti le is found at a given time, it ould happen that pro ess produ es an a tion interpretation, a
ording
parti le is found at a given point, assumes an entirely pe uliar me hanism of a tion at a distan e, whi h prevents the wave ontinuously distributed in spa e from produ ing an a tion in
two
pla es on the s reen.
The key point here is that, if there is no a tion at a distan e, and if the extended eld ψ is indeed a omplete des ription of the physi al situation, then if the ele tron is dete ted at a point P on the lm, it
ould happen that the ele tron is also dete ted at another point Q, or 2 indeed at any point where |ψ| is non-zero. Upon dete tion at P , it appears that a `me hanism of a tion at a distan e' prevents dete tion elsewhere. Einstein's argument is so on ise that its point is easily missed, and one might well dismiss it as arising from an elementary onfusion about the nature of probability. (Indeed, Bohr omments that he does not `understand what pre isely is the point' Einstein is making.) For example, it might be thought that, sin e we are talking about a probability distribution for just one parti le, it is a matter of pure logi that only one dete tion an o
ur.a But this would be to beg the question on erning the nature of ψ . Einstein's wording above attempts to onvey a distin tion between probability for a `given' parti le (leading to the possibility of multiple dete tions) and probability for `this' parti le (leading to single dete tion only). The wording is not su h as to onvey the distin tion very learly, perhaps indi ating an inadequate translation of Einstein's German into Fren h.b But from the ontext, the words `probability that this parti le is found' are learly being used to express the assumption that in this ase ψ indeed expresses the probability for just one parti le dete tion. As shown by Hardy (1995), Einstein's argument may be readily put into the same rigorous form as the later EPR argument. (See Norsen (2005) for a areful and extensive dis ussion.) Hardy simpli es Einstein's example, and onsiders a single parti le in ident on a beam splitter (Fig. 7.1), so that there are only two points P1 , P2 at whi h the parti le might be dete ted. One may then adopt the following su ient
ondition, given by EPR, for the existen e of an element of reality: a Cf. Shimony (2005). b Unfortunately, the full German text of Einstein's ontribution to the general dis ussion seems to have been lost; the Einstein ar hives ontain only a fragment,
onsisting of just the rst four paragraphs (AEA 16-617.00).
196
Lo ality and in ompleteness
If, without in any way disturbing a system, we an predi t with ertainty (i.e., with probability equal to unity) the value of a physi al quantity, then there exists an element of physi al reality orresponding to this physi al quantity. (Einstein, Podolsky and Rosen 1935, p. 777)
Now, if a dete tor is pla ed at P1 , either it will re or it will not. In either
ase, from the state of the dete tor at P1 one ould dedu e with ertainty whether or not a dete tor pla ed at P2 would re. Su h dedu tions ould be made for any individual run of the experiment. Even though the out ome at P1 annot be predi ted in advan e, in ea h ase the out ome allows us to infer the existen e of a de nite element of reality at P2 . If lo ality holds, an element of reality at P2 annot be ae ted by the presen e or absen e of a dete tor at P1 . Therefore, even if no dete tor is pla ed at P1 , there must still be an element of reality at P2 orresponding to dete tion or no dete tion at P2 . Sin e ψ is a superposition of dete tion and no dete tion at P2 , ψ ontains nothing orresponding to the dedu ed element of reality at P2 . Therefore, ψ is not a omplete des ription of a single parti le.a Thus, `the essential points in the EPR argument had already been made by Einstein some eight years earlier at the fth Solvay onferen e' (Hardy 1995, p. 600). Einstein on ludes that: In my opinion, one an remove this obje tion only in the following way, that one does not des ribe the pro ess solely by the S hrödinger wave, but that at the same time one lo alises the parti le during the propagation.
I think
that Mr de Broglie is right to sear h in this dire tion. If one works solely 2 with the S hrödinger waves, interpretation II of |ψ| implies to my mind a
ontradi tion with the postulate of relativity.
In other words, for Einstein, a tion at a distan e an be avoided only by admitting that the wave fun tion is in omplete. A
ording to Einstein's argument, quantum theory is either nonlo al or in omplete. For the rest of his life, Einstein ontinued to believe that lo ality was a fundamental prin iple of physi s, and so he adhered to the view that quantum theory must be in omplete. However, further reasoning by Bell (1964) showed that any ompletion of quantum theory would still require nonlo ality, in order to reprodu e the details of quantum
orrelations for entangled states (assuming the absen e of ba kwards
ausation or of many worldsb). It then appears that, whether omplete a Note that in this argument the in ompleteness of quantum theory is inferred from the assumption of lo ality. Cf. Bell (1987, p. 143). b Bell's argument assumes that there is no ommon ause between the hidden
7.1 Einstein's 1927 argument for in ompleteness
197
Fig. 7.1. Hardy's simpli ed version of Einstein's argument.
or in omplete, quantum theory is ne essarily nonlo al, a on lusion that would surely have been deeply disturbing to Einstein. It is ironi that Einstein's (and EPR's) argument started out by holding steadfast to lo ality and dedu ing that quantum theory is in omplete. But then the argument, as arried further by Bell, led to a ontradi tion between lo ality and quantum orrelations, so that in the end one fails to establish in ompleteness and instead establishes nonlo ality (with ompleteness or in ompleteness remaining an open question).
variables (de ned at the time of preparation) and the settings of the measuring apparatus. It also assumes that there is no ba kwards ausation, so that the hidden variables are not ae ted by the future out omes or apparatus settings. Further, the derivation of the Bell inequalities assumes that a quantum measurement has only one out ome, and therefore does not apply in the many-worlds interpretation.
198
Lo ality and in ompleteness
7.2 A pre ursor: Einstein at Salzburg in 1909 In September 1909, at a meeting in Salzburg, Einstein gave a le ture entitled `On the development of our views on erning the nature and
onstitution of radiation' (Einstein 1909). Einstein summarised what he saw as eviden e for the dual nature of radiation: he held that light had both parti le and wave aspe ts, and argued that lassi al ele tromagneti theory would have to be abandoned. It seems to have gone unnoti ed that one of Einstein's arguments at Salzburg was essentially the same as the argument he presented at the 1927 Solvay onferen e (though applied to light quanta instead of to ele trons). Einstein began his le ture by noting that the phenomena of interferen e and dira tion make it plain that, at least in some respe ts, light behaves like a wave. He then went on to des ribe how, in other respe ts, light behaves as if it onsisted of parti les. In experiments involving the photoele tri ee t, it had been found that the velo ity of the photoele trons was independent of the radiation intensity. A
ording to Einstein, this was more onsistent with `Newton's emission theory of light' than with the wave theory. Einstein also dis ussed pressure u tuations in bla kbody radiation, and showed that these ontained two terms, whi h ould be naturally identi ed as ontributions from parti le-like and wave-like aspe ts of the radiation. Of spe ial interest here is another argument Einstein gave for the existen e of lo alised light quanta. Einstein onsidered a beam of ele trons (`primary athode rays') in ident upon a metal plate P1 and produ ing X-rays (see Fig. 7.2). These X-rays, in turn, strike a se ond metal plate P2 leading to the produ tion of ele trons (`se ondary athode rays') from P2 . Experimentally, it had been found that the velo ity of the se ondary ele trons had the same order of magnitude as the velo ity of the primary ele trons. Further, the available eviden e suggested that the velo ity of the se ondary ele trons did not depend at all on the distan e between the plates P1 and P2 , or on the intensity of the primary ele tron beam, but only on the velo ity of the primary ele trons. Assuming this to be stri tly true, Einstein then asked what would happen if the primary intensity were so small, or the area of the plate P1 so small, that one ould onsider just one ele tron striking the plate, as in Fig. 7.2. A
ording to Einstein, we will have to assume that on ele tron on
P1 )
P2
(as a result of the impinging of the above
either nothing is being produ ed or that a se ondary emission
of an ele tron o
urs on it with a velo ity of the same order of magnitude as of the ele tron impinging on
P1 .
In other words, the elementary radiation
pro ess seems to pro eed su h that it does not, as the wave theory would
7.2 A pre ursor: Einstein at Salzburg in 1909
199
Fig. 7.2. Figure based on Einstein's 1909 argument for the existen e of lo alised light quanta. Assuming the prin iple of lo al a tion, the delo alised X-ray wave an produ e an ele tron (of energy omparable to that of the primary ele tron) in a small region of the se ond plate only if, in addition to the wave, there is a lo alised energy fragment propagating in spa e from to
P1
P2 .
require, distribute and s atter the energy of the primary ele tron in a spheri al wave propagating in all dire tions. Rather, it seems that at least a large part of this energy is available at some lo ation of
P2
or somewhere else. (Einstein
1909, English translation, p. 388)
Einstein's argument, then, is that a
ording to the wave theory the point of emission of the X-ray from the rst plate must be the sour e of waves spreading out in spa e, waves whose amplitude will spread over the region o
upied by the se ond plate. And yet, in the se ond plate, all the energy of the X-ray be omes on entrated in the vi inity of a single point, leading to the produ tion of an ele tron with velo ity omparable to that of the primary ele tron.a Einstein on luded from this that, in addition to the wave spreading from the point of emission, there seems also to be a lo alised energy fragment propagating from the point of a Cf. Compton's report, p. 339: `It is learly impossible that all the energy of an X-ray pulse whi h has spread out in a spheri al wave should spend itself on this [small region℄'.
200
Lo ality and in ompleteness
emission of the X-ray wave to the point of produ tion of the se ondary ele tron. As Einstein put it: `the elementary pro ess of radiation seems to be dire ted '. Now, Einstein's argument of 1909 impli itly assumes a prin iple of lo al a tion, similar to that expli itly assumed in his published ritique of quantum theory at the 1927 Solvay onferen e. Be ause the distan e between the plates P1 and P2 an be arbitrarily large, the wave impinging on P2 an be spread over an arbitrarily large area. The produ tion of an ele tron in a highly lo alised region of P2 an then be a
ounted for only if, in addition to the delo alised wave, there is a lo alised energy fragment propagating through spa e for otherwise, there would have to be some me hanism by means of whi h energy spread out over arbitrarily large regions of spa e suddenly be omes on entrated in the neighbourhood of a single point. It should be quite lear, then, that Einstein's 1927 argument for the existen e of lo alised ele trons (a
ompanying de Broglie-S hrödinger waves) was identi al in form to one of his 1909 arguments for the existen e of lo alised light quanta (a
ompanying ele tromagneti waves). In his 1927 argument, the small hole in the s reen (see his gure) a ts as a sour e for an ele tron wave, whi h spreads over the area of the photographi lm just as, in the 1909 argument, the point where the primary ele tron strikes the rst plate a ts as a sour e for an X-ray wave, whi h spreads over the area of the se ond plate. Both arguments depend ru ially on the assumption (impli it in 1909, expli it in 1927) that there is no a tion at a distan e. As we shall dis uss further in hapter 9, by 1927 Einstein had already spent over twenty years trying to re on ile lo alised energy quanta whi h he had postulated in 1905 with the wave aspe t of radiation. And for mu h of that time, he had been more or less alone in his belief in the existen e of su h quanta. It is then perhaps not so surprising that at the 1927 Solvay meeting Einstein was able to raise su h a penetrating
ritique of the view that the wave fun tion is a omplete des ription of a single ele tron: from his long and largely solitary experien e pondering the wave-parti le duality of light, Einstein ould immediately see that, in the analogous ase of ele tron waves, the prin iple of lo al a tion entailed the existen e of lo alised parti les moving through spa e, in addition to the wave fun tion. The meeting in Salzburg took pla e four years before Bohr published his model of the atom. After 1913, one might have simply rephrased Einstein's argument in terms of atomi transitions. Consider an atom
7.2 A pre ursor: Einstein at Salzburg in 1909
201
A that makes a transition from an initial stationary state with energy Ei to a nal stationary state with lower energy Ef . At a later time, the energy Ei − Ef lost by atom A may be wholly absorbed by an arbitrarily distant atom B, if there exists an appropriate transition from the initial state of B to a nal state orresponding to an energy in rease Ei − Ef . This pro ess may seem unmysterious, if one imagines atom A emitting a photon, or `lo alised energy quantum', whi h somehow propagates through spa e from A to B. However, if one tries to make do without the photon on ept, and represents the ele tromagneti eld in terms of ( lassi al) waves only whi h spread out in all dire tions from A then it is hard to understand how the energy lost by A may be wholly transferred to B: instead, one would expe t the energy to spread out in spa e like the waves themselves, so that the energy density be omes diluted. We have laboured this point be ause the power of Einstein's simple argument seems to have been generally missed, not only in 1909, but also in 1927, and for de ades afterwards. Indeed, it appears that Einstein's point did not start to be ome widely appre iated until the late twentieth
entury (see, again, Norsen (2005)). In retrospe t, it seems quite puzzling that Einstein's simple argument should have taken so long to be understood. A perhaps related puzzle, emphasised by Pais (1982, pp. 3826), is why Einstein's light-quantum hypothesis itself should have been largely ignored by so many physi ists until the advent of the Compton ee t in 1923. Even after Millikan's experimental on rmation of Einstein's photoele tri equation in 1916, `almost no one but Einstein himself would have anything to do with light-quanta' (Pais 1982, p. 386).a We do not wish to suggest, of ourse, that Einstein's lo ality argument should today be regarded as establishing the existen e of lo alised photons: for the impli it premise of Einstein's argument the prin iple of lo ality today seems to be ruled out by Bell's theorem. Our point, rather, is that prior to Bell (and ertainly in 1909) Einstein's arguments were indeed ompelling and should have been taken more seriously.
a Though a
ording to Brillouin's re olle tions of 1962, the situation was rather dierent in Fran e, where Einstein's light quantum was a
epted (by Langevin, Perrin and Marie Curie) mu h earlier than it was elsewhere (Mehra and Re henberg 1982a, p. 580).
202
Lo ality and in ompleteness
7.3 More on nonlo ality and relativity At the end of his long ontribution to the general dis ussion (in whi h he argued for the in ompleteness of quantum theory), Einstein obje ted to the multi-dimensional representation in on guration spa e, on the grounds that (p. 488) .... the feature of for es of a ting only at small
spatial
distan es nds a less
natural expression in on guration spa e than in the spa e of three or four dimensions.
As Einstein himself stated, he was here adding another argument against what he alled view II (the view that ψ is a omplete des ription of an individual system), a view that he laimed is `essentially tied to a multi-dimensional representation ( on guration spa e)'. Einstein's point seems to be that, if physi s is fundamentally grounded in on guration spa e, there will be no reason to expe t physi s to be
hara terised by lo al a tion. This obje tion should be seen in the
ontext of Einstein's on erns, in the period 192627, over the nonseparability of S hrödinger's wave me hani s for many-body systems (Howard 1990, pp. 8391; f. se tion 12.2). A ertain form of lassi al lo ality survives, of ourse, in modern quantum theory and quantum eld theory, in the stru ture of the Hamiltonian or Lagrangian, a stru ture that ensures the absen e of ontrollable nonlo al signals at the statisti al level. But even so, we understand today that, in fa t, quantum physi s is hara terised by nonlo ality. And the nonlo ality may indeed be tra ed to the fa t that, unlike lassi al theory, quantum theory is not grounded in ordinary three-dimensional spa e. The setting for standard quantum theory is Hilbert spa e, whose tensor-produ t stru ture allows for entanglement and asso iated nonlo al ee ts. In the pilot-wave formulation of quantum theory, the setting is on guration spa e (in whi h the pilot wave propagates), and in general the motions of spatially-separated parti les are nonlo ally onne ted. Further, Bell's theorem shows that, if we leave aside ba kwards ausation or many worlds, then quantum theory is in some sense nonlo al under any interpretation or formulation. As Ballentine on e pointed out, while dis ussing the signi an e of Bell's theorem: Perhaps what is needed is not an explanation of nonlo ality, but an explanation of lo ality. Why, if lo ality is not true, does it work so well in so many dierent
ontexts? (Ballentine 1987, pp. 7867)
Einstein's fear, that there would be di ulties with lo ality in quantum physi s, has ertainly been borne out by subsequent developments. In
7.3 More on nonlo ality and relativity
203
standard quantum theory, there appears to be a pea eful but uneasy ` oexisten e' with relativity. While from a pilot-wave (or more generally, from a deterministi hidden-variables) point of view, statisti al lo ality appears as an a
idental feature of the `quantum equilibrium' state (Valentini 1991b, 2002a). In his main ontribution to the general dis ussion Dira (p. 492) also notes that `the general theory of the wave fun tion in many-dimensional spa e ne essarily involves the abandonment of relativity', but he suggests that this problem might be solved by `quantising 3-dimensional waves' (that is, by what we would now all quantum eld theory). And de Broglie in his report, when onsidering the pilot-wave dynami s of manybody systems, notes that unlike in the ase of a single parti le `it does not appear easy to nd a wave Ψ that would de ne the motion of the system taking Relativity into a
ount' (p. 387), a di ulty that has persisted in pilot-wave theory right up to the present day (see, for example, Berndl et al. (1996)). In 1927, then, there was a fairly broad re ognition that the fundamental use of on guration spa e did not bode well for onsisten y with relativity.
8 Time, determinism, and the spa etime framework
8.1 Time in quantum theory By 1920, the spe ta ular on rmation of general relativity, during the solar e lipse of 1919, had made Einstein a household name. Not only did relativity theory (both spe ial and general) upset the long-re eived Newtonian ideas of spa e and time, it also stimulated a widespread `operationalist' attitude to physi al theories. Physi al quantities ame to be seen as inextri ably interwoven with our means of measuring them, in the sense that any limits on our means of measurement were taken to imply limits on the de nability, or `meaningfulness', of the physi al quantities themselves. In parti ular, Einstein's relativity paper of 1905 with its operational analysis of simultaneity ame to be widely regarded as a model for the new operationalist approa h to physi s. Not surprisingly, then, as the puzzles ontinued to emerge from atomi experiments, in the 1920s a number of workers suggested that the
on epts of spa e and time would require still further revision in the atomi domain. Thus, Campbell (1921, 1926) suggested that the puzzles in atomi physi s ould be removed if the on ept of time was given a purely statisti al signi an e: `time, like temperature, is a purely statisti al on eption, having no meaning ex ept as applied to statisti al aggregates' (quoted in Beller 1999, p. 97). In an operationalist vein, Campbell onsidered ` lo ks' based on (random) radioa tive de ays. He suggested that it might be possible to onstru t a theory that did not involve time at all, and in whi h `all the experiments on whi h the prevailing temporal on eptions are based an be des ribed in terms of statisti s' (quoted in Beller 1999, p. 98). On the other hand, Senftleben (1923) asserted that Plan k's onstant h set limits to the de nability of the on epts of spa e and time, and on luded that spa etime must be 204
8.1 Time in quantum theory
205
dis ontinuous. A
ording to Beller (1999, pp. 96101), both Campbell and Senftleben had a signi ant in uen e on Heisenberg in his formulation of the un ertainty prin iple.a Certainly, in a letter to Pauli of 28 O tober 1926, Heisenberg expresses views very similar to Campbell's (Pauli 1979, p. 350): I have for all that a hope in a later solution of more or less the following kind (but one should not say something like this aloud): that spa e and time are really only statisti al on epts, su h as, say, temperature, pressure et . a gas.
I mean that spatial and temporal on epts are meaningless for
in
one
orpus le and that they make more and more sense the more parti les are present. I often try to get further in this dire tion, but until now it will not work.
Be that as it may, at the 1927 Solvay onferen e, in their report on quantum me hani s, Born and Heisenberg seem to express the remarkable view that temporal hanges do not o
ur at all for losed systems, and that the time-dependent S hrödinger equation emerges only as an ee tive and approximate des ription for subsystems. How these views related to Campbell's, or indeed if they did at all, is not lear. In their se tion II, `Physi al interpretation', Born and Heisenberg begin with the following statement (p. 420): The most noti eable defe t of the original matrix me hani s onsists in the fa t that at rst it appears to give information not about a tual phenomena, but rather only about possible states and pro esses. .... it says nothing about when a given state is present ....
matrix me hani s deals only with losed
periodi systems, and in these there are indeed no hanges. In order to have true pro esses .... one must dire t one's attention to a
part
of the system .... .
From a modern point of view, the original matrix me hani s did not
ontain the notion of a general state |Ψi for a system (not even a stati , Heisenberg-pi ture state). The only states that appeared in the theory were the stationary states |Ei i ( f. hapter 3). Even as regards stationary states, there seems to have been no notion of initial state (`it says nothing about when a given state is present'). Instead, the matri es provided a olle tive representation of all the energy eigenstates of a
losed system with Hamiltonian H . In modern notation, the matri es
onsisted of matrix elements of (Heisenberg-pi ture) observables Ω(t) = e(i/~)Ht Ω(0)e−(i/~)Ht in the energy basis: hEi | Ω(t) |Ej i = hEi | Ω(0) |Ej i e(i/~)(Ei −Ej )t .
(1)
a For Campbell's in uen e on Bohr's formulation of omplementarity, see Beller (1999, pp. 1357) and also Mehra and Re henberg (2000, pp. 18990).
206
Time, determinism, and spa etime framework
As noted in hapter 3, the formal mathemati s of matrix me hani s then seems to represent an atomi system somewhat in the manner of the Bohr-Kramers-Slater (BKS) theory ( f. hapter 9), with ea h matrix element orresponding to a virtual os illator of frequen y νij = (Ei − Ej )/h. The matrix formalism, without a notion of initial state, amounts to a stati des ription.a However, Born and Heisenberg add an intuitive physi al pi ture to the formalism, to the ee t that a subsystem of a larger ( losed) system is in fa t in one stationary state at any one time and performs random, indeterministi `quantum jumps' between su h states ( f. hapter 3). Born and Heisenberg then go on to say that `[t℄he lumsiness of the matrix theory in the des ription of pro esses developing in time
an be avoided' (p. 422) by introdu ing what we would now all the time-dependent S hrödinger equation. Here, it might appear that their view is that the mentioned `defe t of the original matrix me hani s' is removed by generalisation to a time-dependent theory. However, they add (p. 423) that: Essentially, the introdu tion of time as a numeri al variable redu es to thinking of the system under onsideration as oupled to another one and negle ting the rea tion on the latter. But this formalism is very onvenient .... .
These words give, instead, the impression that the time-dependent theory is regarded as only emergent in some approximation; the timedependent S hrödinger equation seems to have no fundamental status. Even so, this ` onvenient' formalism `leads to a further development of the statisti al view'. They in lude a time-dependent external perturbation in the (time-dependent) S hrödinger equation, and show how to al ulate the time development of any initial wave fun tion. Born and Heisenberg then argue that, following Bohr's original (1913) theory of stationary states, a system an be in only one energy eigenstate at any one time, leading to the interpretation of a superposition |Ψi = P 2 n cn (t) |Ei i as a statisti al mixture, with state probabilities |cn | . The time evolution of the wave fun tion then des ribes transition probabilities from initial to nal stationary states. This might seem lear enough, but a di ulty is then raised on erning the interpretation of a ase where the initial wave fun tion is already a superposition, resulting in `interferen e of probabilities' at later times (see se tion 6.1.2). a Even if one added a notion of initial state, be ause the only allowed states are energy eigenstates the des ription of a losed system would still be stati .
8.1 Time in quantum theory
207
Let us now onsider what S hrödinger had to say, in his report on wave me hani s, on erning time in quantum theory. (De Broglie's report does not ontain any spe ial remarks on this subje t.) In his report S hrödinger rst presents (or derives from a variational prin iple) what we would now all the time-independent S hrödinger equation for a nonrelativisti many-body system with oordinates q1 , q2 , ..., qn . After noting that the eigenfun tions ψk , with eigenvalues Ek , may be identi ed with Bohr's stationary states, S hrödinger addresses the question of time. He rst points out (p. 451) that the time-independent theory might be regarded as su ient, providing as it does a des ription of stationary states, together with expressions for jump probabilities between them: One an take the view that one should be ontent in prin iple with what has been said so far .... . The single stationary states of Bohr's theory would then
ψk , whi h do not ontain time at all. One .... an form from them .... quantities that an be aptly taken to be jump probabilities between the single stationary states. in a way be des ribed by the eigenfun tions
Here, the jump probabilities are to be obtained from matrix elements su h as (in modern notation) Z ′ hk| Qi |k i = dq qi ψk∗ ψk′ (2)
whi h an all be al ulated from the eigenfun tions ψk . S hrödinger suggests further that intera ting systems ould be treated in the same way, by regarding them as one single system. S hrödinger then goes on to dis uss this point of view and its relation with the ideas of Campbell (p. 451): On this view the
time variable would play absolutely no role in an isolated sys-
tem a possibility to whi h N. Campbell .... has re ently pointed. Limiting our attention to an isolated system, we would not per eive the passage of time in it any more than we an noti e its possible progress in spa e .... . What we would noti e would be merely a sequen e of dis ontinuous transitions, so to speak a inemati image, but without the possibility of omparing the time intervals between the transitions.
A
ording to these ideas, then, time does not exist at the level of isolated atomi systems, and our usual (ma ros opi ally-de ned) time emerges only from the statisti s of large numbers of transitions between stationary states. As S hrödinger puts it: Only se ondarily, and in fa t with in reasing pre ision the more extended the system, would a
statisti al
de nition of time result from
ounting
the
transitions taking pla e (Campbell's `radioa tive lo k'). Of ourse then one
208
Time, determinism, and spa etime framework
annot understand the jump probability in the usual way as the probability of a
single jump probability is two possibilities for jumps, the probability before the other is equal to its jump probability
transition al ulated relative to unit time. Rather, a then utterly meaningless; only with that the one may happen
divided by the sum of the two.
S hrödinger laims that this is the only onsistent view in a theory with quantum jumps, asserting that `[e℄ither all hanges in nature are dis ontinuous or not a single one'. Having sket hed a timeless view of isolated systems with dis rete quantum jumps, S hrödinger states that su h a dis rete viewpoint `still poses great di ulties', and he goes on to develop his own theory of time-dependent quantum states, in whi h ( ontinuous) time evolution does play a fundamental role even at the level of a single atomi system. Here, a general time-dependent wave fun tion ψ(q, t) a solution of the time-dependent S hrödinger equation, with arbitrary initial onditions is regarded as the des ription of the ontinuous time development of a single isolated system. From a ontemporary perspe tive, it is lear that quantum theory as we know it today is rather less radi al than some expe ted it to be in the 1920s, espe ially on erning the on epts of spa e and time. Both nonrelativisti quantum me hani s, and relativisti quantum eld theory, take pla e on a lassi al spa etime ba kground; time and spa e are ontinuous and well-de ned, even for losed systems. The evolution operator U (t, t0 ) provides a ontinuous time evolution |Ψ(t)i = U (t, t0 ) |Ψ(t0 )i for any initial quantum state, with respe t to an `external' time parameter t (even in quantum eld theory, in a given inertial frame). While S hrödinger's and de Broglie's interpretations of the wave fun tion did not gain widespread a
eptan e, their view of time in quantum theory oin ides with the one generally a
epted today.a In ontrast, the views on time expressed by Born and Heisenberg are somewhat reminis ent of views put forward by some later workers in the
ontext of anoni al quantum gravity. There, the wave fun tional Ψ[(3) G] on the spa e of 3-geometries (3) G ontains no expli it time parameter, and obeys a `timeless' S hrödinger equation HΨ = 0 (the WheelerDeWitt equation, with Hamiltonian density operator H). It is laimed a We are of ourse referring here to the standard theories as presented in textbooks. The literature ontains a number of proposals, along operational lines, alling for a `quantum spa etime' that in orporates quantum-theoreti al limits on the
onstru tion of rods and lo ks. A statisti al approa h to ausal stru ture, somewhat reminis ent of Campbell's statisti al view of time, has re ently been proposed by Hardy (2005).
8.2 Determinism and probability
209
that `time' emerges only phenomenologi ally, through the analysis of intera tion with quantum lo ks, or by the extra tion of an ee tive time variable from the 3-metri (the radius of an expanding universe being a popular hoi e) (DeWitt 1967). However, loser analysis reveals a series of di ulties with su h proposals: for example, it is di ult to ensure the emergen e of a well-behaved time parameter t su h that only one physi al state is asso iated with ea h value of t. (See, for example, Unruh and Wald (1989).) Despite some 50 years of eort, in luding the te hni al progress made in re ent years using `loop' variables to solve the equations (Rovelli 2004), the `problem of time' in anoni al quantum gravity remains unresolved.a
8.2 Determinism and probability In the published text, the rst se tion of the general dis ussion bears the title `Causality, determinism, probability'. Lorentz's opening remarks are mainly on erned with the importan e of having a lear and de nite pi ture of physi al pro esses (see se tion 8.3). He ends by addressing the question of determinism (p. 479): .... I think that this notion of probability should be pla ed at the end, and as a on lusion, of theoreti al onsiderations, and not as an
a priori
axiom,
though I may well admit that this indetermina y orresponds to experimental possibilities.
I would always be able to keep my deterministi faith for the
fundamental phenomena .... .
Lorentz seems to demand that the fundamental phenomena be deterministi , and that indeterminism should be merely emergent or ee tive. Probabilities should not be axiomati , and some theoreti al explanation is needed for the experimental limitations en ountered in pra ti e. This view would nowadays be usually asso iated with deterministi hiddenvariables theories, su h as de Broglie's pilot-wave dynami s (though it might also be asso iated with the many-worlds interpretation of Everett). De Broglie's basi equations the guidan e equation and S hrödinger equation are ertainly deterministi . De Broglie in his report, and Brillouin in the subsequent dis ussion, give examples of how these equations determine the traje tories (during interferen e and dira tion, atomi transitions, and elasti s attering). As regards probabilities, de a Barbour (1994a,b) has proposed a timeless formulation of lassi al and quantum physi s. As applied to quantum gravity, the viability of Barbour's s heme seems to depend on unproven properties of the Wheeler-DeWitt equation.
210
Time, determinism, and spa etime framework
Broglie pointed out that if an ensemble of systems with initial wave 2 fun tion Ψ0 begins with a Born-rule distribution P0 = |Ψ0 | , then as Ψ evolves, the system dynami s will maintain the distribution P = 2 |Ψ| at later times. However, nothing was said about how the initial distribution might arise in the rst pla e. Subsequent work has shown that, in de Broglie's theory, the Born-rule distribution an arise from the
omplex evolution generated by the dynami s itself, mu h as thermal distributions arise in lassi al dynami s, thereby providing an example of the kind of theoreti al explanation that Lorentz wished for (Bohm 1953; Valentini 1991a, 1992, 2001; Valentini and Westman 2005).a On this view, the initial ensemble onsidered by de Broglie orresponds to a spe ial `equilibrium' state analogous to thermal equilibrium in lassi al physi s. Lorentz goes on to say (p. 479): Could a deeper mind not be aware of the motions of these ele trons? Could one not keep determinism by making it an obje t of belief ? Must one ne essarily elevate indeterminism to a prin iple?
Here, again, we now know that de Broglie's theory provides an example of what Lorentz seems to have had in mind. For in prin iple, the theory allows the existen e of `nonequilibrium' distributions P 6= |Ψ|2 (Valentini 1991b, 1992), just as lassi al physi s allows the existen e of non-thermal distributions (not uniformly distributed on the energy surfa e in phase spa e). Su h distributions violate many of the standard quantum onstraints; in parti ular, an experimenter possessing parti les with a distribution P mu h narrower than |Ψ|2 would be able to use those parti les to perform measurements on ordinary systems more a
urate than normally allowed by the un ertainty prin iple; an experimenter possessing su h `nonequilibrium parti les' would in fa t be able to use them to observe the (normally invisible) details of the traje tories of ordinary parti les (Valentini 2002b; Pearle and Valentini 2006). From this point of view, there is indeed no need to `elevate indeterminism to a prin iple': for the urrent experimental limitations (embodied in the a Bohm (1953) onsidered the parti ular ase of an ensemble of two-level mole ules and argued that external perturbations would drive it to quantum equilibrium. A general argument for relaxation was not given, however, and soon afterwards Bohm and Vigier (1954) modi ed the dynami s by adding random u tuations that drive any system to equilibrium. This move to a sto hasti theory seems unne essary: a general H -theorem argument, analogous to the lassi al oarsegraining H -theorem, has been given (Valentini 1991a, 1992, 2001), and numeri al simulations show a very e ient relaxation with an exponential de ay of the
oarse-grained H -fun tion on the basis of the purely deterministi de BroglieBohm theory (Valentini and Westman 2005).
8.2 Determinism and probability
211
un ertainty prin iple) are not built into the laws of physi s; rather, they 2 are merely ontingent features of the equilibrium state P = |Ψ| . In the general dis ussion, as we saw in se tion 6.3, Dira expressed the view that quantum out omes o
ur when nature makes a hoi e, a view ountered by Heisenberg who laimed that the ` hoi e' is in some sense really made by the observer. As Lorentz noted at the end of this ex hange, the view that nature makes a hoi e amounts to a fundamental indeterminism, while at the same time, Dira and Heisenberg had radi ally dierent views about the meaning of this indeterminism. Dira also gave an argument for why quantum theory had to be indeterministi . In his view, the indeterminism was ne essary be ause of the inevitable disturban e involved in setting up an initial quantum state (pp. 493 f.): I should now like to express my views on determinism and the nature of the numbers appearing in the al ulations of the quantum theory ....
.
In
the lassi al theory one starts from ertain numbers des ribing ompletely the initial state of the system, and dedu es other numbers that des ribe ompletely the nal state. This deterministi theory applies only to an isolated system. But, as Professor Bohr has pointed out, an isolated system is by de nition unobservable. One an observe the system only by disturbing it and observing its rea tion to the disturban e.
Now sin e physi s is on erned only with
observable quantities the deterministi lassi al theory is untenable.
Dira 's argument seems unsatisfa tory. First of all, as a general philosophi al point, the laim that `physi s is on erned only with observable quantities' is not realisti . As is well known to philosophers of s ien e as well as to experimentalists, observation is `theory-laden': in order to arry out observations (or measurements) some body of theory is required in order to know how to arry out a orre t observation (for example, some knowledge is required of how the system being measured intera ts with the apparatus, in order to design a orre tly fun tioning apparatus). Thus, some body of theory is ne essarily on eptually prior to observation, and it is logi ally impossible to base physi al theory on `observables' only. More spe i ally, Dira laims that lassi al determinism is untenable be ause of the disturban e involved in observing a system: but there are many ases in lassi al physi s where experimenters an use their knowledge of the intera tions involved to ompensate for the disturban e aused by the measurement. Disturban e per se annot be a reason for indeterminism. It may well be that Dira had in mind the kind of `irredu ible' or `un ontrollable' disturban e that textbooks ommonly asso iate with
212
Time, determinism, and spa etime framework
the un ertainty prin iple. However, it is interesting that, in fa t, Dira goes on to say that the disturban es applied by an experimenter are under his ontrol (p. 494, Dira 's itali s): In the quantum theory one also begins with ertain numbers and dedu es others from them.
....
The disturban es that an experimenter applies to a
system to observe it are dire tly under his ontrol, and are a ts of freewill by
It is only the numbers that des ribe these a ts of freewill that an be taken as initial numbers for a al ulation in the quantum theory. Other numbers
him.
des ribing the initial state of the system are inherently unobservable, and do not appear in the quantum theoreti al treatment.
The `disturban es' refer to the experimental operations that the experimenter hooses to apply to the system, and indeed these are normally regarded as freely ontrolled by the experimenter (at least in some ee tive sense). The sense in whi h the word `disturban e' is being used here is quite dierent from the textbook sense of un ontrollable disturban e asso iated with quantum un ertainty. Dira seems to regard quantum numbers, or eigenvalues, as representing the extent to whi h an experimenter an ontrollably manipulate a system. Ma ros opi operations are under our ontrol, and through these we an prepare an initial state spe i ed by parti ular quantum numbers. For Dira , these initial numbers represent `a ts of freewill' in the form of laboratory operations. (The nal remark about `other numbers' that are unobservable, and that do not appear in quantum theory, is intriguing, and might be taken as suggesting that there are other degrees of freedom that annot be ontrolled by us.) A view quite dierent from Dira 's is expressed by Born at the very end of the general dis ussion. A
ording to Born, the onstraints on the preparation of an initial quantum state are not what distinguishes quantum from lassi al me hani s, for in lassi al physi s too (p. 520) .... the pre ision with whi h the future lo ation of a parti le an be predi ted depends on the a
ura y of the measurement of the initial lo ation.
For Born, the dieren e rather lies in the law of propagation of probability pa kets: It is then not in this that the manner of des ription of quantum me hani s, by wave pa kets, is dierent from lassi al me hani s. It is dierent be ause the laws of propagation of pa kets are slightly dierent in the two ases.
8.3 Visualisability and the spa etime framework
213
8.3 Visualisability and the spa etime framework With hindsight, from a ontemporary perspe tive, perhaps the most
hara teristi feature of quantum physi s is the apparent absen e of visualisable pro esses taking pla e within a spa etime framework. From Bell's theorem, it appears that any attempt to provide a omplete des ription of quantum systems (within a single world) will require some form of nonlo ality, leading to di ulties with relativisti spa etime. At the time of writing, we possess only one hidden-variables theory of broad s ope the pilot-wave theory of de Broglie and Bohm and in this theory there is a eld on on guration spa e (not 3-spa e) that ae ts the motion of quantum systems.a Arguably, then, pilot-wave theory does not really t into a spa etime framework: the physi s is grounded in
on guration spa e, and the intera tions en oded in the pilot wave take pla e outside of 3-spa e. Instead of a nonlo al hidden-variables theory, one might prefer to have a omplete a
ount of quantum behaviour in terms of many worlds: there too, one leaves behind ordinary spa etime as a basi framework for physi s, sin e the totality of what is real annot be mapped onto a single spa etime geometry. Generally speaking, whatever one's view of quantum theory today, the usual spa etime framework seems too restri tive, and unable to a
omodate (at least in a natural way) the phenomena asso iated with quantum superposition and entanglement.b We saw in se tion 8.1 that the quantum or matrix me hani s of Born and Heisenberg ertainly did not provide an a
ount of physi al systems in a spa etime framework. In ontrast, the initial pra ti al su
ess of S hrödinger's wave me hani s in 1926 had led some workers to think that an understanding in terms of (wave pro esses in) spa e and time might be possible after all. The resulting tension between S hrödinger on the one hand, and Bohr, Heisenberg and Pauli on the other (in the year pre eding the Solvay meeting) has been des ribed at length in se tion 4.6, where we saw that for S hrödinger the notion of `Ans hauli hkeit' in the sense of visualisability in a spa etime framework played a key role. The lash between quantum physi s and the spa etime framework was a entral theme of the fth Solvay onferen e. There was a notable a As already mentioned in se tion 5.2.1, attempts to onstru t hidden-variables theories without an ontologi al wave fun tion, along the lines pioneered by Fényes (1952) and Nelson (1966), seem to fail (Wallstrom 1994; Pearle and Valentini 2006). b A possible ex eption here is some version of quantum theory with dynami al wave fun tion ollapse.
214
Time, determinism, and spa etime framework
tension between those parti ipants who still hoped for a spa etime-based theory and those who insisted that no su h theory was possible. These dieren es are espe ially apparent in the general dis ussion where, as we have already dis ussed in se tion 7.3, di ulties were raised on erning lo ality and relativity. Lorentz, in his opening remarks at the rst session of the general dis ussion, seems to set the tone for one side of the debate, by speaking in favour of spa e and time as a basi framework for physi s (p. 477): We wish to make a representation of the phenomena, to form an image of them in our minds. Until now, we have always wanted to form these images by means of the ordinary notions of time and spa e. These notions are perhaps innate; in any ase, they have developed from our personal experien e, by our daily observations. For me, these notions are lear and I onfess that I should be unable to imagine physi s without these notions. The image that I wish to form of phenomena must be absolutely sharp and de nite, and it seems to me that we an form su h an image only in the framework of spa e and time.
Lorentz's ommittment to pro esses taking pla e in spa e and time was shared by de Broglie and S hrödinger, even though both men had found themselves unable to avoid working in terms of on guration spa e. As we saw in se tion 2.4, in his report de Broglie presented his pilot-wave dynami s, with a guiding eld in on guration spa e, only as a makeshift: he hoped that his pilot-wave dynami s would turn out to be an ee tive theory only, and that underlying it would be a theory of wave elds in 3-spa e with singularities representing parti le motion (the double-solution theory). S hrödinger, too, despite working with a many-body wave equation in on guration spa e, hoped that the physi al ontent of his theory ould be ultimately interpreted in terms of pro esses taking pla e in 3-spa e (see hapter 4). In ontrast, some of the other parti ipants, in parti ular Bohr and Pauli, wel omed indeed insisted upon the break with the spa etime framework. As Bohr put it in the general dis ussion, after Einstein's remarks on lo ality and ompleteness (p. 489): The whole foundation for [a℄ ausal spa etime des ription is taken away by quantum theory, for it is based on [the℄ assumption of observations without interferen e.
In support of Bohr's ontention, Pauli (pp. 490 f.) provided an intriguing argument to the ee t that intera tions between parti les annot be understood in a spa etime framework. Spe i ally, Pauli based his argument on the quantum-theoreti al a
ount of the long-range intera tions known as van der Waals for es.
8.3 Visualisability and the spa etime framework
215
Pauli begins his argument by saying (in agreement with Bohr) that the use of multi-dimensional on guration spa e is only a te hni al means of formulating mathemati ally the laws of mutual a tion between several parti les, a tions whi h ertainly do not allow themselves to be des ribed simply, in the ordinary way, in spa e and time.
Here, on the one hand, on guration spa e has only mathemati al signi an e. But on the other hand, as Einstein feared (see se tion 7.3), there is a
ording to Pauli no expli it a
ount of lo al a tion in spa e and time. Pauli adds that the multi-dimensional method might one day be repla ed with what we would now all quantum eld theory. He then goes on to say that, in any ase, in a
ordan e with Bohr's point of view, no matter what `te hni al means' are used to des ribe `the mutual a tions of several parti les', su h a tions ` annot be des ribed in the ordinary manner in spa e and time' (p. 491). Pauli then illustrates his point with an example. He onsiders two widely-separated hydrogen atoms, ea h in their ground state, and asks what their `energy of mutual a tion' might be. A
ording to the usual des ription in spa e and time, says Pauli, for large separations there should be no mutual a tion at all. And yet (p. 491), when one treats the same question by the multi-dimensional method, the result is quite dierent, and in a
ordan e with experiment.
What Pauli is referring to here is the problem of a
ounting for van der Waals for es between atoms and mole ules. Classi ally, mole ules with dipole moments tend to align, resulting in a mean intera tion energy ∝ 1/R6 (where R is the distan e between two mole ules). However, many mole ules exhibiting van der Waals for es have zero dipole moment; and while the lassi al orientation ee t be omes negligible at high temperatures, van der Waals for es do not. The problem of explaining van der Waals for es was nally solved by quantum theory, beginning with the work of Wang in 1927.a (Note that the ee t here is quite distin t from that of ex hange for es resulting from the Pauli ex lusion prin iple. The latter for es are important only when the relevant ele troni wave fun tions have signi ant overlap, whereas van der Waals for es o
ur between neutral atoms even when they are so far apart that their harge louds have negligible overlap.) A standard textbook al ulation of the van der Waals for e, between two hydrogen atoms (1 and 2) separated by a displa ement R, pro eeds a For a detailed review, see Margenau (1939).
216
Time, determinism, and spa etime framework
as follows.a The unperturbed energy eigenstate is the produ t Ψ = ψ01 ψ02 of the two ground states. If R = |R| is mu h larger than the Bohr radius, the lassi al ele trostati potential between the atoms is e2 (r1 · R)(r2 · R) V = 3 r1 · r2 − 3 (3) , R R2 where r1 , r2 are respe tively the positions of ele trons 1, 2 relative to their nu lei. In the state Ψ = ψ01 ψ02 the mean values of r1 , r2 both vanish, so that the expe tation value of V vanishes. However, the presen e of V perturbs the ground state of the system, to a new and entangled state Ψ′ , satisfying (H0 + V )Ψ′ = (2E0 + ∆E)Ψ′ , where H0 is the unperturbed Hamiltonian, E0 is the ground-state energy of hydrogen, and ∆E is the energy perturbation from whi h one dedu es the van der Waals for e. Standard perturbation methods show that ∆E has a leading term proportional to 1/R6 , a
ounting for the (attra tive) van der Waals for e. Pauli regarded this result whi h had been obtained by Wang from the S hrödinger equation in on guration spa e as eviden e that in quantum theory there are intera tions that annot be des ribed in terms of spa e and time. The result may be roughly understood lassi ally, as Pauli points out, by imagining that in ea h atom there is an os illating dipole moment that an indu e (and so intera t with) a dipole moment in the other atom. But su h understanding is only heuristi . In a proper treatment, using `multi-dimensional wave me hani s', the orre t result is obtained by methods that, a
ording to Pauli, annot be understood in a spa etime framework.a Pauli's point seems to have been that, be ause the perturbed wave fun tion Ψ′ is entangled, multi-dimensional on guration spa e plays a
ru ial role in bringing about the orre t result, whi h therefore annot be properly understood in terms of 3-spa e alone. If this argument seems unwarranted, it ought to be remembered that, before Wang's derivation in 1927, there had been a long history of failed attempts to explain van der Waals for es lassi ally (Margenau 1939). Disagreements over both the usefulness and the tenability of the spa etime framework are also apparent elsewhere in the general dis ussion. For example, Kramers (p. 498) asks: `What advantage do you see in a See, for example, S hi (1955, pp. 17680). a A
ording to Ehrenfest, Bohr also gave an argument (in a onversation with Einstein) against a spa etime des ription when treating many-parti le problems. See se tion 1.5.
8.3 Visualisability and the spa etime framework
217
giving a pre ise value to the velo ity v of the photons?' To this de Broglie replies (in a spirit similar to that of Lorentz above): This allows one to imagine the traje tory followed by the photons and to spe ify the meaning of these entities; one an thus onsider the photon as a material point having a position and a velo ity.
Kramers was un onvin ed: I do not very well see, for my part, the advantage that there is, for the des ription of experiments, in making a pi ture where the photons travel along well-de ned traje tories.
In this ex hange, Kramers had suggested that de Broglie's theory ould not explain radiation pressure from a single photon, thereby questioning the tenability (and not just the usefulness) of the spa etime framework for the des ription of elementary intera tions. (Cf. se tion 10.4.) Later in the general dis ussion, another argument against de Broglie's theory is provided by Pauli. As we shall dis uss at length in se tion 10.2, Pauli laims on the basis of an example that pilot-wave theory annot a
ount for the dis rete energy ex hange taking pla e in inelasti ollisions. It is interesting to note that, a
ording to Pauli, the root of the di ulty lies in the attempt to onstru t a deterministi parti le dynami s in a spa etime framework (p. 512): .... this di ulty .... is due dire tly to the ondition assumed by Mr de Broglie, that in the individual ollision pro ess the behaviour of the parti les should be ompletely determined and may at the same time be des ribed ompletely by ordinary kinemati s in spa etime.
9 Guiding elds in 3-spa e
In this hapter we address proposals (by Einstein, and by Bohr, Kramers and Slater) a
ording to whi h quantum events are in uen ed by `guiding elds' in 3-spa e. These ideas led to a predi ted violation of energymomentum onservation for single events, in ontradi tion with experiment. The ontradi tion was resolved only by the introdu tion of guiding elds in on guration spa e. All this took pla e before the fth Solvay
onferen e, but nevertheless forms an important ba kground to some of the dis ussions that took pla e there.
9.1 Einstein's early attempts to formulate a dynami al theory of light quanta Sin e the publi ation of his light-quantum hypothesis in 1905, Einstein had been engaged in a solitary struggle to onstru t a detailed theory of light quanta, and to understand the relationship between the quanta on the one hand and the ele tromagneti eld on the other.a Einstein's eorts in this dire tion were never published. We know of them indire tly: they are mentioned in letters, and they are alluded to in Einstein's 1909 le ture in Salzburg. Einstein's published papers on light quanta
ontinued for the most part in the same vein as his 1905 paper: using the theory of u tuations to make dedu tions about the nature of radiation, without giving details of a substantial theory. Einstein was essentially alone in his dualisti view of light, in whi h lo alised energy fragments a In 1900 Plan k had, of ourse, ee tively introdu ed a quantisation in the intera tion between radiation and matter; but it was Einstein in 1905 who rst proposed that radiation itself (even in free spa e) onsisted, at least in part, of spatially lo alised energy quanta. In 1918 Einstein wrote to his friend Besso: `I do not doubt anymore the reality of radiation quanta, although I still stand quite alone in this onvi tion' (original itali s, as quoted in Pais (1982, p. 411)).
218
9.1 Einstein's early attempts
219
oexisted with extended waves, until the work of de Broglie in 1923 whi h extended the dualism to all parti les, and made onsiderable progress towards a real theory (see hapter 2). A glimpse of Einstein's attempts to formulate a dynami al theory of light quanta may be obtained from a lose reading of his 1909 Salzburg le ture. There, as we saw in se tion 7.2, Einstein marshalled eviden e that light waves ontain lo alised energy fragments. He suggested that the ele tromagneti eld is asso iated with singular points at whi h the energy is lo alised, and he oered the following remarkable (if heuristi ) pi ture: I more or less imagine ea h su h singular point as being surrounded by a eld of for e whi h has essentially the hara ter of a plane wave and whose amplitude de reases with the distan e from the singular point. If many su h singularities are present at separations that are small ompared with the dimensions of the eld of for e of a singular point, then su h elds of for e will superpose, and their totality will yield an undulatory eld of for e that may dier only slightly from an undulatory eld as de ned by the urrent ele tromagneti theory of light. (Einstein 1909, English translation, p. 394)
Here, ea h light quantum is supposed to have an extended eld asso iated with it, and large numbers of quanta with their asso iated elds are supposed to yield (to a good approximation) the ele tromagneti eld of Maxwell's theory. In other words, the ele tromagneti eld as we know it emerges from the olle tive behaviour of large numbers of underlying elds asso iated with individual quanta. A similar view is expressed in a letter from Einstein to Lorentz written a few months earlier, in May 1909: I on eive of the light quantum as a point that is surrounded by a greatly extended ve tor eld, that somehow diminishes with distan e.
Whether or
not when several light quanta are present with mutually overlapping elds one must imagine a simple superposition of the ve tor elds, that I annot say. In any ase, for the determination of events, one must have equations of motion for the singular points in addition to the dierential equations for the ve tor eld. (Quoted in Howard 1990, p. 75)
From the last senten e, it is lear that Einstein's on eption was supposed to be deterministi . Einstein's view in 1909, then, is remarkably reminis ent of de Broglie's pilot-wave theory as well as of his theory of the `double solution' (from whi h de Broglie hoped pilot-wave theory would emerge, see hapter 2). Einstein seems to have thought of ea h individual light quantum as being
220
Guiding elds in 3-spa e
a
ompanied by some kind of eld in 3-spa e that ae ts the motion of the quantum. Fas inating rea tions to Einstein's ideas appear in the re orded dis ussion that took pla e after Einstein's le ture. Stark pointed out a phenomenon that seemed to speak in favour of lo alised energy quanta in free spa e: `even at great distan es, up to 10 m, ele tromagneti radiation that has left an X-ray tube for the surrounding spa e an still a hieve on entrated a tion on a single ele tron' (Einstein 1909, English translation, p. 397). Stark's point here is, again, Einstein's lo ality argument: as we noted at the end of se tion 7.2, in retrospe t it seems puzzling that this simple and ompelling argument was not widely understood mu h earlier, but here Stark learly appre iates it. However and this is probably why the argument did not gain urren y doubts were raised as to how su h a theory ould explain interferen e. Plan k spoke as follows: Stark brought up something in favor of the quantum theory, and I wish to bring up something against it; I have in mind the interferen es at the enormous phase dieren es of hundreds of thousands of wavelengths. When a quantum interferes with itself, it would have to have an extension of hundreds of thousands of wavelengths. This is also a ertain di ulty. (Einstein 1909, English translation, p. 397)
To this obje tion, Einstein gives a most interesting reply: I pi ture a quantum as a singularity surrounded by a large ve tor eld. By using a large number of quanta one an onstru t a ve tor eld that does not dier mu h from the kind of ve tor eld we assume to be involved in radiations.
I an well imagine that when rays impinge upon a boundary
surfa e, a separation of the quanta takes pla e, due to intera tion at the boundary surfa e, possibly a
ording to the phase of the resulting eld at whi h the quanta rea h the interfa e.
....
I do not see any fundamental
di ulty in the interferen e phenomena. (Einstein 1909, English translation, p. 398)
Again, the similarity to de Broglie's later ideas is striking. Einstein seems to think that the asso iated waves an ae t the motions of the quanta, in su h a way as to a
ount for interferen e. It should be noted, though, that in this ex hange it is somewhat un lear whether the subje t is interferen e for single photons or for many photons. Einstein talks about interferen e in terms of the olle tive behaviour of many quanta, rather than in terms of one quantum at a time. While single-photon interferen e with very feeble light was observed by Taylor (1909) in the same year, Stark at least seems not
9.2 The failure of energy-momentum onservation
221
to know of Taylor's results, for at this point (just before Einstein's reply) he interje ts that `the experiments to whi h Mr Plan k alluded involve very dense radiation .... . With radiation of very low density, the interferen e phenomena would most likely be dierent' (Einstein 1909, English translation, p. 397). On the other hand, Plan k may well have thought of the light-quantum hypothesis as implying that light quanta would move independently, like the mole ules of an ideal gas (in whi h ase even for intense radiation one ould onsider the motion of ea h quantum independently). Einstein ountered pre isely su h a view at the beginning of his reply to Plan k, where he states that `it must not be assumed that radiations onsist of non-intera ting quanta; this would make it impossible to explain the phenomena of interferen e'. This might be read as implying that intera tions among dierent quanta are essential, and that interferen e would not o
ur with one photon at a time. However, Einstein may simply have meant that the light quanta
annot be thought of as free parti les: they must be a
ompanied by a wave as well, in order to explain interferen e. (This last reading ts with Einstein's dis ussion, in the le ture, of thermal u tuations in radiation: these annot be obtained from a gas of free and independent parti les alone; a wave-like omponent is also needed.) While there is some un ertainty over the details of Einstein's proposal, in retrospe t, given our present understanding of how de Broglie's pilotwave theory provides a straightforward explanation of parti le interferen e (see se tion 6.1.1), Einstein's reply to Plan k seems very reasonable. However, it appears that yet another of Einstein's arguments was not appre iated by his ontemporaries. Seven years later, after having veri ed Einstein's photoele tri equation experimentally, Millikan nevertheless
ompletely reje ted Einstein's light-quantum hypothesis, whi h he alled `re kless .... be ause it ies in the fa e of the thoroughly established fa ts of interferen e' (Millikan 1916, p. 355).
9.2 The failure of energy-momentum onservation It appears that Einstein was still thinking along similar lines in the 1920s, though again without publishing any detailed theory. As we saw at the end of se tion 6.2.1, in his ollision papers of 1926 Born notes the analogy between his own work and Einstein's ideas: `I start from a remark by Einstein .... ; he said .... that the waves are there only to show the orpus ular light quanta the way, and in this sense he talked
222
Guiding elds in 3-spa e
of a ghost eld ' (Born 1926b, pp. 8034). Born gives no referen e to any published paper of Einstein's, however. How Einstein's thinking at this time ompared with that in 1909 is hard to say. Certainly, in 1909 he thought of the ele tromagneti eld as being built up from the olle tive behaviour of large numbers of ve tor elds asso iated with individual quanta. In su h a s enario, it seems plausible that in the right ir umstan es the intensity of the emergent ele tromagneti eld ould a t as, in ee t, a probability eld. (Unlike Born, Einstein would have regarded a purely probabilisti des ription as a makeshift only.) Just one year before Born's ollision papers, in 1925, Einstein gave a olloquium in Berlin where he indeed dis ussed the idea that every parti le (in luding ele trons, following de Broglie) was a
ompanied by a `Führungsfeld' or guiding eld (Pais 1982, p. 441; Howard 1990, p. 72). A
ording to Wigner, who was present at the olloquium: Yet Einstein, though in a way he was fond of it, never published it. He realized that it is in on i t with the onservation prin iples: at a ollision of a light quantum and an ele tron for instan e, both would follow a guiding eld. But these guiding elds give only the probabilities of the dire tions in whi h the two omponents, the light quantum and the ele tron, will pro eed.
Sin e
they follow their dire tions independently, .... the momentum and the energy
onservation laws would be obeyed only statisti ally .... . This Einstein ould not a
ept and hen e never took his idea of the guiding eld quite seriously. (Wigner 1980, p. 463)
In the early 1920s, then, Einstein was still thinking along lines that are reminis ent of de Broglie's work, but he never published these ideas be ause they on i ted with the onservation laws for individual events. The di ulty Einstein fa ed was over ome only by the introdu tion (through the work of de Broglie, S hrödinger, and also Born) of a guiding eld in on guration spa e a single (and generally entangled) eld that determined probabilities for all parti les olle tively. In Einstein's approa h, where ea h parti le had its own guiding eld, the possibility of entanglement was pre luded, and the orrelations were not strong enough to guarantee energy-momentum onservation for single events. As a simpli ed model of what Einstein seems to have had in mind,
onsider two parti les 1 and 2 moving towards ea h other in one dimension, with equal and opposite momenta p and −p respe tively. S hemati ally, let us represent this with an initial wave fun tion ψ(x1 , x2 , 0) = ψ1+ (x1 , 0)ψ2− (x2 , 0), where ψ1+ and ψ2− are broad pa kets (approximating plane waves) moving along +x and −x respe tively. Let the pa kets meet
9.2 The failure of energy-momentum onservation
223
in a region entred around the origin, where we imagine that an elasti
ollision takes pla e with probability 12 . At large times, we assume that ψ(x1 , x2 , t) takes the s hemati and entangled form 1 ψ(x1 , x2 , t) = √ ψ1+ (x1 , t)ψ2− (x2 , t) + ψ1− (x1 , t)ψ2+ (x2 , t) , 2
(1)
with the rst bran h orresponding to the parti les having moved freely past ea h other and the se ond orresponding to an elasti ollision reversing their motions. The initial state has zero total momentum. In the nal state, both possible out omes orrespond to zero total momentum. Thus, in quantum theory, whatever the out ome of an individual run of the experiment, momentum is always onserved. Now imagine if, instead of using a single wave fun tion ψ(x1 , x2 , t) in on guration spa e, we made use of two 3-spa e waves ψ1 (x, t), ψ2 (x, t) (one for ea h parti le). It would then seem natural that, during the ollision, the initial 3-spa e wave ψ1 (x, 0) = ψ1+ (x, 0) for parti le 1 would evolve into ψ1 (x, t) = ψ1+ (x, t) + ψ1− (x, t) ,
(2)
while the initial 3-spa e wave ψ2 (x, 0) = ψ2− (x, 0) for parti le 2 would evolve into ψ2 (x, t) = ψ2− (x, t) + ψ2+ (x, t) .
(3)
If the amplitude of ea h 3-spa e wave determined the probabilities for the respe tive parti les, there would then be four (equiprobable) possible out omes for the s attering experiment, the two stated above, together with one in whi h both parti les move to the right and one in whi h both parti les move to the left. These possibilities would orrespond, in ee t, to a nal ( on guration-spa e) wave fun tion of the separable form (4) ψ1+ + ψ1− ψ2− + ψ2+ = ψ1+ ψ2− + ψ1+ ψ2+ + ψ1− ψ2− + ψ1− ψ2+ .
The nal total momenta for the four (equiprobable) possible out omes are 0, +2p, −2p, 0. The total momentum would not be generally onserved for individual out omes, but onservation would hold on average. Note the fundamental dieren e between (1) and (4): the former is entangled, the latter is not. Einstein onsidered su h a failure of the onservation laws reason enough to reje t the idea. A theory bearing some resemblan e to this s heme was, however, proposed and published by Bohr, Kramers and Slater (1924a,b).
224
Guiding elds in 3-spa e
In the Bohr-Kramers-Slater (BKS) theory, there are no photons. Asso iated with an atom in a stationary state is a `virtual radiation eld'
ontaining all those frequen ies orresponding to transitions to and from other stationary states. The virtual eld determines the transition probabilities for the atom itself, and also ontributes to the transition probabilities for other, distant atoms. However, the resulting orrelations between widely-separated atoms are not strong enough to yield energymomentum onservation for individual events ( f. Compton's a
ount of the BKS theory in his report, p. 331).a To illustrate this, onsider again (in modern language) an atom A emitting a photon of energy Ei − Ef that is subsequently absorbed by a distant atom B. In the BKS theory, the transition at B is not dire tly
aused by the transition at A (as it would be in a simple `semi lassi al' pi ture of the emission, propagation, and subsequent absorption of a lo alised light quantum). Rather, the virtual radiation eld of A ontains terms orresponding to transitions of A, and this eld ontributes to the transition probabilities at B. Yet, if B a tually undergoes a transition
orresponding to an energy in rease Ei − Ef , atom A is not onstrained to make a transition orresponding to an energy de rease Ei − Ef . The onne tion between the atoms is merely statisti al, and energy
onservation holds only on average, not for individual pro esses.a Einstein obje ted to the BKS theory in a olloquium, and in private letters and onversations (Mehra and Re henberg 1982a, pp. 5534; Pais 1982, p. 420; Howard 1990, pp. 714) partly for the same reason he had not published his proposals: that, as he believed, energymomentum onservation would hold even in the ase of elementary intera tions between widely-separated systems.b Einstein's expe tation was subsequently on rmed by the Bothe-Geiger and Compton-Simon experiments (Bothe and Geiger 1925b, Compton and Simon 1925). In the Bothe-Geiger experiment, by means of ounter oin iden es, Compton s attering (that is, relativisti ele tron-photon s attering) was studied for individual events, to see if the outgoing s attered photon and the outgoing re oil ele tron were produ ed simultaneously. Stri t a For a detailed a
ount of the BKS theory, see Darrigol (1992, hap. 9). a Note that Slater's original theory did ontain photons, whose motions were guided (statisti ally) by the ele tromagneti eld. The photons were removed at the instigation of Bohr and Kramers (Mehra and Re henberg 1982a, pp. 5437). b In a letter to Ehrenfest dated 31 May 1924, Einstein wrote: `This idea [the BKS theory℄ is an old a quaintan e of mine, but one whom I do not regard as a respe table fellow' (Howard 1990, p. 72). Einstein listed ve riti isms, in luding the violation of the onservation laws, and a di ulty with thermodynami s (Mehra and Re henberg 1982a, p. 553).
9.2 The failure of energy-momentum onservation
225
temporal oin iden es were expe ted on the basis of the light-quantum hypothesis, but not on the basis of the BKS theory. Su h oin iden es were in fa t observed by Bothe and Geiger.a In the Compton-Simon experiment, Compton s attering was again studied, with the aim of verifying the onservation laws for individual events. Su h onservation was in fa t observed by Compton and Simon.b As a result of these experiments, it was widely on luded that the BKS theory was wrong.c It appears that, even at the time of the fth Solvay onferen e, Einstein was still thinking to some extent in terms of guiding elds in 3-spa e. Eviden e for this omes from an otherwise in omprehensible remark Einstein made during his long ontribution to the general dis ussion. There, Einstein ompared two interpretations of the wave fun tion ψ for a single ele tron. On Einstein's view I, ψ represents an ensemble or ` loud' of ele trons, while on his view II ψ is a omplete des ription of an individual ele tron. As we have dis ussed at length in se tion 7.1, Einstein argued that interpretation II is in onsistent with lo ality. Now, Einstein also made the following remark on erning the onservation of energy and momentum a
ording to interpretations I and II (p. 488): The se ond on eption goes further than the rst .... . It is only by virtue of II that the theory ontains the onsequen e that the onservation laws are valid for the elementary pro ess; it is only from II that the theory an derive the result of the experiment of Geiger and Bothe .... .
As urrently understood, of ourse, a purely `statisti al' interpretation of the wave fun tion ( f. se tion 5.2.4) would yield orre t predi tions, in agreement with the onservation laws for elementary pro esses (su h as s attering). Why, then, did Einstein assert that interpretation I would
on i t with su h elementary onservation? It must surely be that, for whatever reason, he was thinking of interpretation I as tied spe i ally to wave fun tions in 3-spa e, resulting in a failure of the onservation laws for single events as dis ussed above. In ontrast, spe i ally regarding interpretation II, Einstein expli itly asserted (p. 488) that it is `essentially tied to a multi-dimensional representation ( on guration spa e)'. The on i t between Einstein's ideas about guiding elds in 3-spa e and energy-momentum onservation was, as we have mentioned, resolved a See Compton's a
ount of `Bothe and Geiger's oin iden e experiments', pp. 350 f., and also Mehra and Re henberg (1982a, pp. 60912). b See Compton's a
ount of `Dire tional emission of s attered X-rays', pp. 351 f., and also Mehra and Re henberg (1982a, p. 612).
Cf. Bohr's remarks in the dis ussion of Compton's report, p. 360.
226
Guiding elds in 3-spa e
only by the introdu tion of a guiding eld in on guration spa e with the asso iated entanglement and nonseparability that Einstein was to nd so obje tionable. Einstein's early worries about nonseparability (long before the EPR paper) have been extensively do umented by Howard (1990). De ning `separability' as the idea that `spatiotemporally separated systems possess well-de ned real states, su h that the joint state of the omposite system is wholly determined by these two separate states' (Howard 1990, p. 64), Howard highlights the fundamental di ulty Einstein fa ed in having to hoose, in ee t, between separability and energy-momentum onservation: But as long as the `guiding' or `virtual elds' [determining the probabilities for parti le motions, or for atomi transitions℄ are assigned separately, one to ea h parti le or atom, one annot arrange of individual systems
and
both
for the merely probabilisti behavior
for orrelations between intera ting systems su -
ient to se ure stri t energy-momentum onservation in all individual events. As it turned out, it was only S hrödinger's relo ation of the wave elds from physi al spa e to on guration spa e that made possible the assignment of joint wave elds that ould give the strong orrelations needed to se ure stri t
onservation. (Howard 1990, p. 73)
Here, then, is a remarkable histori al and physi al onne tion between nonseparability or entanglement on the one hand, and energymomentum onservation on the other. The introdu tion of probability waves in on guration spa e, with generi entangled states, nally made it possible to se ure energy-momentum onservation for individual emission and s attering events, at the pri e of introdu ing a fundamental nonseparability into physi s.
10 S attering and measurement in de Broglie's pilot-wave theory
At the fth Solvay onferen e, some questions that are losely related to the quantum measurement problem (as we would now all it) were addressed in the ontext of pilot-wave theory, in both the dis ussion following de Broglie's report and in the general dis ussion. Most of these questions on erned the treatment of s attering (elasti and inelasti ); they were raised by Born and Pauli, and replies were given by Brillouin and de Broglie. Of spe ial interest is the famous and widely misunderstood obje tion by Pauli on erning inelasti s attering. Another question losely related to the measurement problem was raised by Kramers, on erning the re oil of a single photon on a mirror. In this hapter, we shall rst outline the pilot-wave theory of s attering, as urrently understood, and examine the extensive dis ussions of s attering in the ontext of de Broglie's theory that took pla e at the onferen e. We shall see that de Broglie and Brillouin orre tly answered the query raised by Born on erning elasti s attering. Further, we shall see that Pauli's obje tion on erning the inelasti ase was both more subtle and more onfused than is generally thought; in parti ular, Pauli presented his example in terms of a misleading opti al analogy (that was originally given by Fermi in a more restri ted ontext). Contrary to a widespread view, de Broglie's reply to Pauli did ontain the essential points required for a proper treatment of inelasti s attering; at the same time, Fermi's misleading analogy onfused matters, and neither de Broglie in 1927 nor Bohm in 1952 saw what the true fault with Pauli's example was. (As we shall also see, a proper pilot-wave treatment of Pauli's example was not given until 1956, by de Broglie.) We shall also outline the pilot-wave theory of quantum measurement, again as urrently understood, and we shall use this as a ontext in 227
228
S attering and measurement in de Broglie's theory
whi h to examine the question raised by Kramers, of the re oil of a single photon on a mirror, whi h de Broglie was unable to answer.
10.1 S attering in pilot-wave theory Let us rst onsider elasti s attering by a xed potential asso iated with some s attering entre or region. An in ident parti le may (for a pure quantum state) be represented by a freely-evolving wave pa ket ψinc (x, t) that is spatially nite (that is, limited both longitudinally and laterally), and that has mean momentum ℏk. During the s attering, the wave fun tion evolves into ψ = ψinc + ψsc , where ψsc is the s attered wave. At large distan es from the s attering region, and o the axis (through the s attering entre) parallel to the in ident wave ve tor k, only the s attered wave ψsc ontributes to the parti le urrent density j where j is often used in textbook derivations of the s attering ross se tion. As is well known, the mathemati s of a fully time-dependent al ulation of the s attering of a nite pa ket may be simpli ed by resorting to a time-independent treatment in whi h ψinc is taken to be an in nitelyextended plane wave eik·x . At large distan es from the s attering region the wave fun tion ψ = eik·x + ψsc (a time-independent eigenfun tion of the total Hamiltonian) has the asymptoti form ψ = eikz + f (θ, φ)
eikr r
(1)
(taking the z -axis parallel to k, using spheri al polar oordinates entred on the s attering region, and ignoring overall normalisation). The s attering amplitude f gives the dierential ross se tion dσ/dΩ = |f (θ, φ)|2 . In the standard textbook derivation of dσ/dΩ, the urrent density j is used to al ulate the rate of probability ow into an element of solid angle dΩ, where j is taken to be the urrent asso iated with ψsc only, even though ψsc overlaps with ψinc = eik·x . This is justi ed be ause the plane wave eik·x is, of ourse, merely an abstra tion used for mathemati al
onvenien e: a real in ident wave will be spatially limited, and will not overlap with the s attered wave at the lo ation of the parti le dete tor (whi h is assumed to be lo ated o the axis of in iden e, so as not to be bathed in the in ident beam). The above standard dis ussion of s attering may readily be re ast in
10.1 S attering in pilot-wave theory
229
pilot-wave terms, where v=
j 2
|ψ|
=
ℏ ∇ψ ∇S Im = m ψ m
(2)
(with ψ = |ψ| e(i/ℏ)S ) is interpreted as the a tual velo ity eld of an ensemble of parti les with positions distributed a
ording to |ψ|2 . The dierential ross se tion dσ/dΩ measures the fra tion of in ident parti les whose a tual traje tories end (asymptoti ally) in the element of solid angle dΩ. That a real in ident pa ket is always spatially nite is, of ourse, an elementary point known to every student of wave opti s. This (often impli it) assumption is essential to introdu tory textbook treatments of the s attering of light, whether by a Hertzian dipole or by a dira tion grating. If the in ident wave were a literally in nite plane wave, then the s attered wave would of ourse overlap with the in ident wave everywhere, and no matter where a dete tor was pla ed it would be ae ted by the in ident wave as well as by the s attered wave. Certainly, the parti ipants at the fth Solvay onferen e many of whom had extensive laboratory experien e were aware of this simple point. We emphasise this be ause, as we shall see, the niteness of in ident wave pa kets played a entral role in the dis ussions that took pla e regarding s attering in de Broglie's theory. Let us now onsider inelasti s attering: spe i ally, the s attering of an ele tron by a hydrogen atom initially in the ground state. The atom
an be ome ex ited by the ollision, in whi h ase the outgoing ele tron will have lost a orresponding amount of energy. Let the s attering ele tron have position xs and the atomi ele tron have position xa . In a time-dependent des ription, the total wave fun tion evolves into Ψ(xs , xa , t) = φ0 (xa )e−iE0 t/ℏ ψinc (xs , t) +
X
φn (xa )e−iEn t/ℏ ψn (xs , t) . (3)
n
Here, the rst term is an initial produ t state, where φ0 is the groundstate wave fun tion of hydrogen with ground-state energy E0 and ψinc is a ( nite) in ident pa ket. The s attering terms have omponents as shown, where the φn are the nth ex ited states of hydrogen and the ψn are outgoing wave pa kets. It may be shown by standard te hniques that, asymptoti ally, the nth outgoing pa ket ψn is entred on a radius rn = (ℏkn /m)t from the s attering region, where kn is the outgoing wave number xed by energy onservation.
230
S attering and measurement in de Broglie's theory
Be ause the outgoing (asymptoti ) pa kets ψn expand with dierent speeds, they eventually be ome widely separated in spa e. The a tual s attered ele tron with position xs (t) an o
upy only one of these nonoverlapping pa kets, say ψi , and its velo ity will then be determined by ψi alone. Further, the motion of the atomi ele tron will be determined by the orresponding φi alone, and after the s attering the atom will be (in ee t) in an energy eigenstate φi . We are using two well-known properties of pilot-wave dynami s: (a) If a wave fun tion Ψ = |Ψ| e(i/ℏ)S is a superposition of terms Ψ1 + Ψ2 + .... having no overlap in on guration spa e, then the phase gradient ∇S at the o
upied point of on guration spa e (whi h gives the velo ity of the a tual on guration) redu es to ∇Si , where Ψi = |Ψi | e(i/ℏ)Si is the o
upied pa ket. (b) If the o
upied pa ket Ψi is a produ t over ertain
on guration omponents, then the velo ities of those omponents are determined by the asso iated fa tors in the produ t. Applying (a) and (b) to the ase dis ussed here, on e the ψn have separated the total wave fun tion Ψ be omes a sum of non-overlapping pa kets, where only one pa ket Ψi = φi (xa )e−iEi t/ℏ ψi (xs , t) an ontain the a tual on guration (xa , xs ). The velo ity of the s attered ele tron is then given by x˙ s = (ℏ/m) Im(∇ψi /ψi ), while the velo ity of the atomi ele tron is given by x˙ a = (ℏ/m) Im(∇φi /φi ). Thus, there takes pla e an ee tive ` ollapse of the wave pa ket' to the state φi ψi . It is straightforward to show that, if the initial ensemble of xs , xa has distribution |Ψ|2 , the probability for ending in the ith pa ket that is, the the atom to end in the state φi will be given by R 3probability for 2 d xs |ψi (xs , t)| , in a
ordan e with the usual quantum result. Further, the ee tive ` ollapse' to the state φi ψi is for all pra ti al purposes irreversible. As argued by Bohm (1952a, p. 178), the s attered parti le will subsequently intera t with many other degrees of freedom making it very unlikely that distin t states φi ψi , φj ψj (i 6= j ) will interfere at later times, as this would require the asso iated bran hes of the total wave fun tion to overlap with respe t to every degree of freedom involved. Again, for mathemati al onvenien e, one often onsiders the limit in whi h the in ident wave ψinc is unlimited (a plane wave). This makes the al ulation of Ψ easier. In this limit, the outgoing wave pa kets ψn be ome unlimited too, and overlap with ea h other everywhere: all the terms in (3) then overlap in every region of spa e. If one naively assumed that this limit orresponded to a real situation, the outgoing ele tron would never rea h a onstant velo ity be ause it would be guided by a
10.1 S attering in pilot-wave theory
231
superposition of overlapping terms, rather than by a single term in (3). Similarly, after the s attering, the atomi ele tron would not be guided by a single eigenfun tion φi , and the atom would not nish in a de nite energy state. In any real situation, of ourse, ψinc will be limited in spa e and time, and at large times the outgoing pa kets ψn will separate: the traje tory of the s attered ele tron will be guided by only one of the ψn and (in regions outside the path of the in ident beam) will not be ae ted by ψinc . (Note that longitudinal niteness of the in ident wave ψinc leads to a separation of the outgoing waves ψn from ea h other, while lateral niteness of ψinc ensures that ψinc does not overlap with the ψn in regions o the axis of in iden e.) As we shall see, apart from the pra ti al irreversibility of the ee tive
ollapse pro ess, the above `pilot-wave theory of s attering' seems to have been more or less understood by de Broglie (and perhaps also by Brillouin) in O tober 1927. A detailed treatment of s attering was given by Bohm in his rst paper on de Broglie-Bohm theory (Bohm 1952a). In Bohm's se ond paper, however, appendix B gives a misleading a
ount of the de Broglie-Pauli en ounter at the fth Solvay onferen e (Bohm 1952b, pp. 1912), and this seems to be the sour e of the widespread misunderstandings on erning this en ounter. Citing the pro eedings of the fth Solvay onferen e, Bohm wrote the following: De Broglie's suggestions met strong obje tions on the part of Pauli, in onne tion with the problem of inelasti s attering of a parti le by a rigid rotator. Sin e this problem is on eptually equivalent to that of inelasti s attering of a parti le by a hydrogen atom, whi h we have already treated .... , we shall dis uss the obje tions raised by Pauli in terms of the latter example.
Bohm then des ribes `Pauli's argument': taking the in oming parti le to have a plane wave fun tion, all the terms in (3) overlap, so that `neither atom nor the outgoing parti le ever seem to approa h a stationary energy', ontrary to what is observed experimentally. A
ording to Bohm: Pauli therefore on luded that the interpretation proposed by de Broglie was untenable.
De Broglie seems to have agreed with the on lusion, sin e he
subsequently gave up his suggested interpretation.
Bohm then gives what he regards as his own, original answer to Pauli's obje tion: .... as is well known, the use of an in ident plane wave of in nite extent is an ex essive abstra tion, not realizable in pra ti e. A tually, both the in ident and outgoing parts of the
ψ - eld will always take the form of bounded pa kets.
232
S attering and measurement in de Broglie's theory
Moreover, .... all pa kets orresponding to dierent values of obtain lassi ally des ribable separations.
n
will ultimately
The outgoing parti le must enter
one of these pa kets, .... leaving the hydrogen atom in a de nite but orrelated stationary state.
By way of on lusion, Bohm writes: Thus, Pauli's obje tion is seen to be based on the use of the ex essively abstra t model of an in nite plane wave.
At this point, in the light of what we have said above about the use of plane waves in elementary wave opti s and in s attering theory, it is natural to ask how a physi ist of Pauli's abilities ould have made the glaring mistake that Bohm laims he made. In fa t, as we shall see in the next se tion, Pauli's obje tion was not based on a failure to appre iate the importan e of the niteness of initial wave pa kets. On the ontrary, Pauli's obje tion shows some understanding of the
ru ial role played by limited pa kets in pilot-wave theory. What really happened is that Pauli's obje tion involved a pe uliar and misleading analogy with opti s, a
ording to whi h the in ident pa ket appeared to be ne essarily unlimited, in onditions su h as to prevent the required separation into non-overlapping omponents.
10.2 Elasti and inelasti s attering: Born and Brillouin, Pauli and de Broglie In the dis ussion following de Broglie's report, Born suggests (p. 399) that de Broglie's guidan e equation will fail for an elasti ollision between an ele tron and an atom. Spe i ally, Born asks if the ele tron speed will be the same before and after the ollision, to whi h de Broglie simply replies that it will. Later in the same dis ussion, Brillouin (pp. 402 .) gives an extensive and detailed presentation, explaining how de Broglie's theory a
ounts for the elasti s attering of a photon from a mirror. Brillouin is quite expli it about the role played by the nite extension of the in ident pa ket. In his Fig. 2, Brillouin shows an in ident photon traje tory (at angle θ to the normal) guided by an in oming and laterally-limited pa ket. The pa ket is re e ted by the mirror, produ ing an outgoing pa ket that is again laterally-limited. Near the mirror there is an interferen e zone, where the in oming and outgoing pa kets overlap. As Brillouin puts it: Let us draw a diagram for the ase of a limited beam of light falling on a plane mirror; the interferen e is produ ed in the region of overlap of two beams.
10.2 Elasti and inelasti s attering
233
Brillouin sket hes the photon traje tory, whi h urves away from the mirror as it enters the interferen e zone, moves approximately parallel to the mirror while in the interferen e zone, and then moves away again, eventually settling into a re tilinear motion guided by the outgoing pa ket (Brillouin's Fig. 2). Brillouin des ribes the traje tory thus: .... at rst a re tilinear path in the in ident beam, then a bending at the edge of the interferen e zone, then a re tilinear path parallel to the mirror, with the photon travelling in a bright fringe and avoiding the dark interferen e fringes; then, when it omes out, the photon retreats following the dire tion of the re e ted light beam.
Here we have a lear des ription of an in ident photon, guided by a nite pa ket and moving uniformly towards the mirror, with the pa ket then undergoing interferen e and s attering, while the photon is eventually
arried away again with a uniform motion by a nite outgoing pa ket. (The ingoing and outgoing motions of the photon are stri tly uniform, of ourse, only in the limit where the guiding pa kets be ome in nitely broad.) Despite his des ription in terms of nite pa kets, however, in order to al ulate the pre ise motion of the photon in the interferen e zone Brillouin uses the standard devi e of treating the in oming pa ket as an in nite plane wave. In this (abstra t) approximation, he shows that the photon moves parallel to the mirror with a speed v = c sin θ. Be ause the in oming pa ket is in reality limited, a photon motion parallel to the mirror is (approximately) realised only in the interferen e zone, as Brillouin sket hes in his a
ompanying Fig. 2. While Brillouin's gure shows a laterally-limited pa ket, it is lear from subsequent dis ussion that the in ident pa ket was impli itly regarded as limited longitudinally as well (as of ourse it must be in any realisti situation). For in the general dis ussion the question of photon re e tion by a mirror was raised again, and Einstein asked (p. 499) what happens in de Broglie's theory in the ase of normal in iden e (θ = 0), for whi h the formula v = c sin θ predi ts that the photons will have zero speed. Pi
ard responded to Einstein's query, and pointed out that, indeed, the photons are stationary in the limiting ase of normal in iden e. The meaning of this ex hange between Einstein and Pi
ard is lear: if a longitudinally-limited pa ket were in ident normally on the mirror, the in ident pa ket would arry the photon towards the mirror; in the region where the in ident and re e ted pa kets overlap, the photon would be at rest; on e the pa ket has been re e ted, the
234
S attering and measurement in de Broglie's theory
photon will be arried away by the outgoing pa ket. The dis ussion of the ase of normal in iden e impli itly assumes that the in ident pa ket is longitudinally-limited. Thus, Brillouin's example of photon re e tion by a mirror illustrates the point that the use of plane waves was for al ulational onvenien e only, and that, when it ame to the dis ussion of real physi al examples, it was lear to all (and hardly worth mentioning expli itly) that in ident waves were in reality limited in extent (in all dire tions). Brillouin's mathemati al use of plane waves parallels their use in the general theory of s attering sket hed in se tion 10.1. Elasti s attering is also dis ussed in de Broglie's report, for the parti ular ase of ele trons in ident on a xed, periodi potential the potential generated by a rystal latti e. This ase is espe ially interesting in the present ontext, be ause it involves the separation of the s attered wave into non-overlapping pa kets the interferen e maxima of dierent orders, well-known from the theory of X-ray dira tion with the parti le entering just one of these pa kets (a point that is relevant to a proper understanding of the de Broglie-Pauli en ounter). Here is how de Broglie des ribes the dira tion of an ele tron wave by a rystal latti e (p. 392): .... the wave
Ψ
will propagate following the general equation, in whi h one
has to insert the potentials reated by the atoms of the rystal onsidered as
entres of for e. One does not know the exa t expression for these potentials but, be ause of the regular distribution of atoms in the rystal, one easily realises that the s attered amplitude will show maxima in the dire tions predi ted by Mr von Laue's theory. Be ause of the role of pilot wave played by the wave
Ψ,
one must then observe a sele tive s attering of the ele trons
in these dire tions.
Again, as in Brillouin's dis ussion, there is no need to mention expli itly the obvious point that the in ident wave Ψ will be spatially limited. Let us now turn to the inelasti ase. This was dis ussed by de Broglie in his report, in parti ular in the nal part, whi h in ludes a review of re ent experiments involving the inelasti s attering of ele trons by atoms of helium. De Broglie noted that, a
ording to Born's al ulations (using his statisti al interpretation of the wave fun tion), the dierential
ross se tion should show maxima as a fun tion of the s attering angle (p. 392): .... Mr Born has studied .... the ollision of a narrow beam of ele trons with an atom.
A
ording to him, the urve giving the number of ele trons that
10.2 Elasti and inelasti s attering
235
have suered an inelasti ollision as a fun tion of the s attering angle must show maxima and minima .... .
As de Broglie then dis ussed in detail, su h maxima had been observed experimentally by Dymond (though the results were only in qualitative agreement with the predi tions). Having summarised Dymond's results, de Broglie ommented (p. 393): The above results must very probably be interpreted with the aid of the new Me hani s and are to be related to Mr Born's predi tions.
We are now ready for a lose examination of Pauli's obje tion in the general dis ussion, a
ording to whi h there is a di ulty with de Broglie's theory in the ase of inelasti ollisions. That there might be su h a di ulty was in fa t already suggested by Pauli a few months earlier, in a letter to Bohr dated 6 August 1927, already quoted in hapter 2. As well as noting the ex eptional quality of de Broglie's `Stru ture' paper (de Broglie 1927b), in his letter Pauli states that he is suspi ious of de Broglie's traje tories and annot see how the theory ould a
ount for the dis rete energy ex hange seen in individual inelasti ollisions between ele trons and atoms (Pauli 1979, pp. 4045). (Su h dis rete ex hange had been observed, of ourse, in the Fran k-Hertz experiment.) This is essentially the obje tion that Pauli raises less than three months later in Brussels. In the general dis ussion (p. 511), Pauli begins by stating his belief that de Broglie's theory works for elasti ollisions: It seems to me that, on erning the statisti al results of s attering experiments, the on eption of Mr de Broglie is in full agreement with Born's theory in the ase of elasti ollisions .... .
This preliminary omment by Pauli is signi ant. For if, as Bohm asserted in 1952, Pauli's obje tion was based on `the use of the ex essively abstra t model of an in nite plane wave' (Bohm 1952b, p. 192), then Pauli would have regarded de Broglie's theory as problemati even in the elasti ase. For in an elasti ollision, if the in ident wave ψinc is in nitely extended (a plane wave), then any part of the outgoing region will be bathed in the in ident wave: the s attered parti le will inevitably be ae ted by both parts of the superposition ψinc + ψsc , and will never settle down to a onstant speed. Sin e Pauli agreed that the outgoing speed would be onstant in the elasti ase as de Broglie had asserted (in reply to Born) in the dis ussion following his report
236
S attering and measurement in de Broglie's theory
Pauli presumably understood that the nite in ident pa ket would not ae t the s attered parti le. Pauli goes on to laim that de Broglie's theory will not work for inelasti ollisions, in parti ular for the example of s attering by a rotator. To understand Pauli's point, it is important to distinguish between the real physi al situation being dis ussed, and the opti al analogy used by Pauli an analogy that had been introdu ed by Fermi as a onvenient (and as we shall see limited) means to solve the s attering problem. Pauli's obje tion is framed in terms of Fermi's analogy, and therein lies the onfusion. The real physi al set-up onsists of an ele tron moving in the (x, y)plane and olliding with a rotator. The latter is a model s attering
entre with one rotational degree of freedom represented by an angle ϕ.a Pauli took the initial wave fun tion to be ψ0 (x, y, ϕ) ∝ e(i/ℏ)(px x+py y+pϕ ϕ) ,
(4)
with pϕ restri ted to pϕ = mℏ (m = 0, 1, 2, ...). The inelasti s attering of an ele tron by a rotator had been treated by Fermi (1926) using an analogy with opti s, a
ording to whi h the (time-independent) s attering of an ele tron in two spatial dimensions by a rotator is mathemati ally equivalent to the (time-independent) s attering of a (s alar) light wave ψ(x, y, ϕ) in three spatial dimensions by an in nite dira tion grating, with ϕ interpreted as a third spatial oordinate ranging over the whole real line (−∞, +∞). The in nite `grating' onstitutes a periodi potential, arising mathemati ally from the periodi ity asso iated with the original variable ϕ. Similarly, the fun tion ψ(x, y, ϕ) is ne essarily unlimited along the ϕ-axis. By onstru tion, then, both the in ident wave and the grating are unlimited along the ϕ-axis. Fermi's analogy is useful, be ause the dierent spe tral orders for dira ted beams emerging from the grating orrespond to the possible nal (post-s attering) energy states of the rotator. However, as we shall see, Fermi's analogy has only a very limited validity. Pauli, then, presents his obje tion in terms of Fermi's opti al analogy. He says (p. 512): It is, however, an essential point that, in the ase where the rotator is in a stationary state before the ollision, the in ident wave is unlimited in the
a Classi ally, a rotator might onsist of a rigid body free to rotate about a xed axis. The quantum rotator was known to have a dis rete spe trum of quantised energy levels ( orresponding to quantised states of angular momentum), and was sometimes onsidered as a useful and simple model of a quantum system.
10.2 Elasti and inelasti s attering
237
Fig. 10.1. S attering of a laterally-limited wave by a nite dira tion grating, showing the separation of the rst-order beams from the zeroth-order beam.
dire tion of the [ϕ-℄axis. For this reason, the dierent spe tral orders of the grating will always be superposed at ea h point of on guration spa e.
If
we then al ulate, a
ording to the pre epts of Mr de Broglie, the angular velo ity of the rotator after the ollision, we must nd that this velo ity is not
onstant.
In Fermi's three-dimensional analogy, for an in ident beam unlimited along ϕ (as well as along x and y ) the s attered waves will indeed overlap everywhere. The nal on guration will then be guided by a superposition of all the nal energy states, and the nal velo ity of the on guration will not be onstant. It then appears (a
ording to Pauli) that the nal angular velo ity of the rotator will not be onstant,
ontrary to what is expe ted for a stationary state, and that there will be no de nite out ome for the s attering experiment (that is, no de nite nal energy state for the rotator). In the usual dis ussion of dira tion gratings in opti s, it is of ourse assumed that both the grating and the in ident beam are laterally limited, so that the emerging beams separate as shown in Fig. 10.1 (where only the zeroth-order and rst-order beams are drawn). But
238
S attering and measurement in de Broglie's theory
Fig. 10.2. S attering of a laterally-unlimited wave by an in nite dira tion grating.
in Fermi's analogy, there an be no su h lateral limitation and no su h separation. It might be thought that, in the ase of no lateral limitation, separation of an opti al beam would nevertheless take pla e if the in ident wave were longitudinally limited. However, as shown in Fig. 10.2, symmetry di tates that there will be no beams beyond the zeroth order.a Thus, if one a
epts Fermi's analogy with the s attering of light by an in nite grating, even if one takes a longitudinally-limited in ident light wave, there will still be no separation and the di ulty remains . The only way to obtain a separation of the s attered beams is through a lateral lo alisation along ϕ and a
ording to Pauli's argument this is impossible. A
ording to Fermi's analogy, then, it is ines apable that (for an initial stationary state) the in ident wave ψ(x, y, ϕ) is unlimited on the ϕ-axis, a Geometri ally, the presen e of diverging higher-order beams (as in Fig. 10.1) would de ne a preferred entral point on the grating. Roughly speaking, if one onsiders Fig. 10.1 in the limit of an in nite grating (and of a laterally unlimited in ident beam), the higher-order emerging beams are `pushed o to in nity', resulting in Fig. 10.2.
10.2 Elasti and inelasti s attering
239
and one annot avoid the on lusion that after the s attering the rotator need not be in a de nite energy eigenstate. There seems to be no way out of Pauli's di ulty, and de Broglie's pilot-wave theory appears to be untenable. Today, one might answer Pauli's obje tion by onsidering the measuring apparatus used to dete t the outgoing s attered parti le (or, used to measure the energy of the atom). By in luding the degrees of freedom of the apparatus in the quantum des ription, one ould obtain a separation of the total wave fun tion into non-overlapping bran hes, resulting in a de nite quantum out ome ( f. se tion 10.3). However, in 1927 the measuring apparatus was not normally onsidered to be part of the quantum system; and even today, in the pilot-wave theory of s attering (sket hed in se tion 10.1), a de nite out ome is generally guaranteed by the separation of pa kets for the s attered parti le. As we shall now show, the usual pilot-wave theory of s attering in fa t su es in Pauli's example too. To see how Pauli's obje tion would normally be met today, note rst that in Pauli's example the real physi al situation onsists of an ele tron moving in two spatial dimensions x, y and olliding with a rotator whose angular oordinate ϕ ranges over the unit ir le (from 0 to 2π ). In de ˙ y˙ Broglie's dynami s, the wave fun tion ψ(x, y, ϕ, t) yields velo ities x, for the ele tron and ϕ˙ for the rotator, where these velo ities are given by 2 the quantum urrent divided by |ψ| . As we saw in the ase of inelasti s attering by an atom, the nal wave fun tion will take the form ψ(x, y, ϕ, t) = φ0 (ϕ)e−iE0 t/ℏ ψinc (x, y, t) +
X
φn (ϕ)e−iEn t/ℏ ψn (x, y, t) , (5)
n
where now the φn are stationary states for the rotator. On e again, for nite (lo alised) ψinc , at large times the outgoing wave pa kets ψn will be entred on a radius rn = (ℏkn /m)t from the s attering region, where again kn is the outgoing wave number xed by energy
onservation. As before, be ause the ψn expand with dierent speeds they eventually be ome widely separated in spa e. The total (ele tronplus-rotator) on guration (x, y, ϕ) will then o
upy only one bran h φi (ϕ)e−iEi t/ℏ ψi (x, y, t) of the outgoing wave fun tion. The s attered ele tron will then be guided by ψi only, and the speed of the ele tron will be onstant. Further, the motion of the rotator will be determined by the stationary state φi , and will also be uniform. As long as the in ident
240
S attering and measurement in de Broglie's theory
wave ψinc is lo alised in x and y , the nal wave fun tion separates in
on guration spa e and the s attering pro ess has a de nite out ome. Pauli's obje tion therefore has a straightforward answer. On Fermi's analogy, however, Pauli's example seems to be equivalent to the s attering of a laterally-unlimited light wave in three-dimensional spa e by an in nite grating, for whi h no separation of beams an take pla e. Sin e, for the original system, we have seen that the nal state does separate, it is lear that Fermi's analogy must be mistaken in some way. To see what is wrong with Fermi's analogy, one must examine Fermi's original paper. There, Fermi introdu es the oordinates √ √ √ ξ = mx , η = my , ζ = Jϕ , (6) and writes the time-independent S hrödinger equation for the ombined rotator-plus-ele tron system of total energy E in the form ∂2ψ ∂2ψ 2 ∂ 2ψ + + + 2 (E − V )ψ = 0 , 2 2 2 ∂ξ ∂η ∂ζ ℏ
(7)
where √ the potential energy V is a periodi fun tion of ζ with period 2π J . Fermi then onsiders an opti al analogue of the wave equation (7). As Fermi puts it (Fermi 1926, p. 400): In order to see the solution of [(7)℄, we onsider the opti al analogy for the wave equation [(7)℄. In the regions far from the
ζ -axis, where V
vanishes, [(7)℄ is the
wave equation in an opti ally homogeneous medium; in the neighbourhood of the
ζ -axis,
the medium has an anomaly in the refra tive index, whi h depends
periodi ally on
√ 2π J .
ζ.
Opti ally this is nothing more than a linear grating of period
Fermi then goes on to onsider a plane wave striking the grating, and relates the outgoing beams of dierent spe tral orders to dierent types of ollisions between the ele tron and the rotator. Now, if we in lude the time dependen e ψ ∝ e−(i/ℏ)Et , we may write 2 1 ∂2ψ Eψ = − ℏ2 c2 ∂t2
(8)
p with c ≡ E/2, and (7) indeed oin ides with the wave equation of s alar opti s for the ase of a given frequen y. However and here is where Fermi's analogy breaks down be ause the `speed of light' c depends on the energy E (or frequen y E/h), the time evolution of the system in regions far from the ζ -axis is in general not equivalent to wave propagation in `an opti ally homogeneous medium'. The analogy holds
10.2 Elasti and inelasti s attering
241
only for one energy E at a time (whi h is all Fermi needed to onsider). In a realisti ase where the ( nite) in ident ele tron wave is a sum over dierent momenta and as we have said, it was understood that in any realisti ase the in ident wave would indeed be nite and therefore equal to su h a sum the problem of s attering by the rotator annot be made equivalent to the s attering of light by a grating in an otherwise opti ally homogeneous medium: on the ontrary, the `speed of light' c would have to vary as the square root of the frequen y. Pauli's presentation does not mention that Fermi's opti al analogy is valid only for a single frequen y. Sin e, as we have seen, the niteness of realisti in ident pa kets was impli it in all these dis ussions, the impression was probably given that Fermi's analogy holds generally. A nite (in x and y ) ele tron wave in ident on the rotator would then be expe ted to translate into a longitudinally nite but laterally in nite (along ζ ) light wave in ident on an in nite grating, with a resulting la k of separation in the nal state. Sin e, however, the quantity c is frequen y-dependent, in general the analogy with light is invalid and the
on lusion unwarranted. What really happens, then, in the two-dimensional s attering of an ele tron by a rotator, if one interprets the angular oordinate ϕ as a third spatial axis à la Fermi? The answer is found in a detailed dis ussion of Pauli's obje tion given by de Broglie, in a book published in 1956, whose hapter 14 bears the title `Mr Pauli's obje tion to the pilot-wave theory' (de Broglie 1956). There, de Broglie gives what is in fa t the rst proper analysis of Pauli's example in terms of pilot-wave theory (with one spatial dimension suppressed for simpli ity). The result of de Broglie's analysis an be easily seen by re onsidering the expression (5), and allowing ϕ to range over the real line, with ψ(x, y, ϕ, t) regarded as a periodi fun tion of ϕ with period 2π . Be ause ψ separates (as we have seen) into pa kets that are non-overlapping with respe t to x and y , the time evolution of ψ will be as sket hed in Fig. 10.3 (where we suppress y ) a gure that we have adapted from de Broglie's book (p. 176). The gure shows a wave moving along the x-axis towards the rotator at x = 0. The wave is longitudinally limited ( nite along x) and laterally unlimited (in nite along ϕ), and separates as shown into similar pa kets moving at dierent speeds after the s attering. (Only two of the nal pa kets are drawn.) The dierent speeds of the pa kets after the ollision orrespond, of
ourse, to the dierent possible kineti energies of the ele tron after the
ollision. Note that the ru ial separation into pa kets moving at dier-
242
S attering and measurement in de Broglie's theory
Fig. 10.3. True evolution of the ele tron-rotator wave fun tion in Pauli's example. Adapted from de Broglie (1956, p. 176).
ent speeds would be ruled out if one were to mistakenly a
ept Fermi's analogy with light s attering `in an opti ally homogeneous medium' for the outgoing pa kets would then all have to move with the same speed (that of light). A omplete reply to Pauli's original obje tion should then make two points: (1) Be ause of the frequen y-dependent `speed of light' c ≡ p E/2, Fermi's analogy with opti s is of very limited validity and annot be applied to a real ase with a nite in ident wave. (2) For a nite in ident ele tron wave, a separation of pa kets does in fa t take pla e with respe t to the spatial oordinates of the ele tron. Let us now examine how de Broglie replied to Pauli in O tober 1927, in the general dis ussion. As will be ome lear, de Broglie understood the general separation me hanism required to yield a de nite out ome, but he was misled by the (generally false) opti al analogy and phrased his answer in terms of it. De Broglie replies by rst pointing out the importan e of having a laterally-limited in ident wave, to avoid overlap among the dira t-
10.2 Elasti and inelasti s attering
243
ed beams, and to avoid overlap between these and the in ident wave (p. 513): The di ulty pointed out by Mr Pauli has an analogue in lassi al opti s. One an speak of the beam dira ted by a grating in a given dire tion only if the grating and the in ident wave are laterally limited, be ause otherwise all the dira ted beams will overlap and be bathed in the in ident wave. In Fermi's problem, one must also assume the wave
ψ
to be limited laterally in
on guration spa e.
De Broglie then notes that, if one an assume ψ to be limited laterally, then the system velo ity will be ome onstant on e the dira ted waves have separated (from ea h other and from the in ident beam): .... the velo ity of the representative point of the system will have a onstant value, and will orrespond to a stationary state of the rotator, as soon as the waves dira ted by the
ϕ-axis
will have separated from the in ident beam.
What de Broglie is des ribing here is pre isely the separation into nonoverlapping pa kets in on guration spa e, whi h the pilot wave must undergo in order for the s attering experiment to have a de nite out ome just as we have dis ussed in se tion 10.1 above. However, in Pauli's example, it appeared that the in ident ψ ould not be limited laterally. On this point, de Broglie laims that Fermi's
on guration spa e is arti ial, having been formed `by rolling out along a line the y li variable ϕ'. But as de Broglie's treatment of 1956 shows, extending the range of ϕ from the unit ir le to the real line does not really make any dieren e: it simply distributes opies of the
on guration spa e along the ϕ-axis (see Fig. 10.3). Pauli's presentation did not mention the frequen y-dependent speed p c ≡ E/2, and in his reply de Broglie did not mention it either. Indeed, it seems that neither de Broglie in 1927 nor Bohm in 1952 noti ed this misleading aspe t of Pauli's obje tion. Note that niteness of the in ident wave with respe t to x and y had not been questioned by anyone. As we have seen, the nite spatial extension of realisti wave pa kets was impli itly assumed by all. Pauli had raised the impossibility of niteness spe i ally with respe t to the ϕ-axis, and de Broglie responded on this spe i point alone. While de Broglie was misled by Fermi's analogy, even so his remarks ontain the key point, later developed in detail by Bohm that niteness of the initial pa ket will ensure a separation into non-overlapping nal pa kets in on guration spa e. As de Broglie himself put it in his book of 1956, in referen e to his 1927 reply to Pauli: `Thus I had indeed realised ....
244
S attering and measurement in de Broglie's theory
that the answer to Mr Pauli's obje tion had to rest on the fa t that the wave trains are always limited, an idea that has been taken up again by Mr Bohm in his re ent papers' (de Broglie 1956, p. 176). Finally, we point out that leaving aside the misleading nature of Fermi's opti al analogy de Broglie's audien e may well have understood his des ription of wave pa kets separating in on guration spa e, for Born had already des ribed something very similar for the Wilson
loud hamber, earlier in the general dis ussion (pp. 483 .). As we have seen in se tion 6.2, Born dis ussed the formation of a tra k in a loud
hamber in terms of a bran hing of the total wave fun tion in a multidimensional on guration spa e (formed by all the parti les involved). In parti ular, for the simple ase of an α-parti le intera ting with two atoms in one dimension, Born des ribed an initially lo alised pa ket that separates into two non-overlapping bran hes in (three-dimensional)
on guration spa e as sket hed in Born's gure. By the time de Broglie ame to reply to Pauli, then, the audien e was already familiar with the idea of a wave fun tion evolving in on guration spa e and developing non-overlapping bran hes. Thus, it seems more likely than not that this aspe t of de Broglie's reply to Pauli would have been understood.a
10.3 Quantum measurement in pilot-wave theory The general theory of quantum measurement in pilot-wave theory was rst developed by Bohm in his se ond paper on the theory (Bohm 1952b). Bohm understood that the degrees of freedom asso iated with the measurement apparatus were simply extra oordinates that should be in luded in the total system (as in, for example, Born's 1927 dis ussion of the loud hamber). Thus, denoting by x the oordinates of the `system' and y the oordinates of the `apparatus', the dynami s generates a traje tory (x(t), y(t)) for the total on guration, guided ˙ y) ˙ is by a wave fun tion Ψ(x, y, t) (where again the velo ity eld (x, given by the quantum urrent of Ψ divided by |Ψ|2 ). Bohm showed a It may also be worth noting that, in his report, when presenting the pilot-wave theory of many-body systems, de Broglie expli itly asserts (p. 387) that his probability formula in on guration spa e `fully a
ords .... with the results obtained by Mr Born for the ollision of an ele tron and an atom, and by Mr Fermi for the ollision of an ele tron and a rotator'. Be ause the printed version of de Broglie's le ture was already omplete by the time the onferen e took pla e, this remark ould not have been added after de Broglie's lash with Pauli. (De Broglie wrote to Lorentz on 11 O tober 1927 AHQP-LTZ-11, in Fren h that he had re eived the proofs of his report from Gauthier-Villars and orre ted them.)
10.3 Quantum measurement in pilot-wave theory
245
how, in the ir umstan es orresponding to a quantum measurement, Ψ separates into non-overlapping bran hes and the system oordinate x(t) is eventually guided by an ee tively `redu ed' pa ket for the system. As a simple example, let the system have an initial wave fun tion X ψ0 (x) = (9) cn φn (x) , n
where the φn are eigenfun tions of some Hermitian operator Q with eigenvalues qn . Suppose an experiment is performed that in quantum theory would be alled `a measurement of the observable Q'. This might be done by oupling the system to a `pointer' with oordinate y and initial wave fun tion g0 (y), where g0 is narrowly peaked around y = 0. An appropriate oupling might be des ribed by an intera tion Hamiltonian H = aQPy , where a is a oupling onstant and Py is the momentum operator onjugate to y . If we negle t the rest of the Hamiltonian, for an initial produ t wave fun tion Ψ0 (x, y) = ψ0 (x)g0 (y) the S hrödinger equation ∂Ψ ∂ iℏ (10) Ψ = aQ −iℏ ∂t ∂y
has the solution
Ψ(x, y, t) =
X n
cn φn (x)g0 (y − aqn t) .
(11)
Be ause g0 is lo alised, Ψ evolves into a superposition of terms that separate with respe t to y (in the sense of having negligible overlap with respe t to y ). As we saw in the ase of s attering, the nal on guration (x, y) an be in only one `bran h' of the superposition, say φi (x)g0 (y − aqi t), whi h will guide (x(t), y(t)) thereafter. And be ause the a tive bran h is a produ t in x and y , the velo ity of x will be determined by φi (x) alone. In ee t, at the end of the quantum measurement, the system is guided by a `redu ed' wave fun tion φi (x). Thus, the pointer plays the same role in the pilot-wave theory of measurement as the s attered parti le does in the pilot-wave theory of s attering. It is also readily shown that, if the initial ensemble of x, y has a 2 distribution |Ψ0 (x, y)| , then the probability of x, y ending in the bran h 2 φi (x)g0 (y − aqi t) is given by |ci | , in agreement with the standard Born rule. Finally, again as we saw in the ase of s attering, the ee tive ` ollapse' to the state φi (x)g0 (y − aqi t) is for all pra ti al purposes irreversible. As noted by Bohm (1952b, p. 182), the apparatus oordinate y will
246
S attering and measurement in de Broglie's theory
subsequently intera t with many other degrees of freedom, making it very unlikely that distin t states φi (x)g0 (y − aqi t), φj (x)g0 (y − aqj t) (i 6= j ) will interfere later on, be ause the asso iated bran hes of the total wave fun tion would have to overlap with respe t to every degree of freedom involved.
10.4 Re oil of a single photon: Kramers and de Broglie We have sket hed the pilot-wave theory of quantum measurement to emphasise that, when one leaves the limited domain of parti le s attering (by atoms, or by xed obsta les su h as mirrors or dira ting s reens), it an be ome essential to des ribe the apparatus itself in terms of pilot-wave dynami s. In some situations, the oordinates of the apparatus must be in luded in the total wave fun tion (for the `supersystem'
onsisting of system plus apparatus), to ensure that the total wave fun tion separates into non-overlapping bran hes in on guration spa e. As we shall see in hapter 11, it seems that this point was not properly appre iated by de Broglie in 1927. Indeed, most theoreti ians at the time simply applied quantum theory (in whatever form they preferred) to mi ros opi systems only. Ma ros opi apparatus was usually treated as a given lassi al ba kground. However, Born's treatment of the loud
hamber in the general dis ussion shows that the key insight was already known: the apparatus is made of atoms, and should ultimately be in luded in the wave fun tion, whi h will develop a bran hing stru ture as the measurement pro eeds. All that was needed, within pilot-wave theory, was to arry through the details properly for a general quantum measurement something that de Broglie did not see in 1927 and that Bohm did see in 1952. Now, the need to in lude ma ros opi equipment in the wave fun tion is relevant to a problem raised by Kramers in the general dis ussion,
on erning the re oil of a mirror due to the re e tion of a single photon. The dis ussion had returned to the question of how de Broglie's theory a
ounts for radiation pressure on a mirror (a subje t that had been dis ussed at the end of de Broglie's le ture). As Kramers put it (p. 498): But how is radiation pressure exerted in the ase where it is so weak that there is only one photon in the interferen e zone? .... And if there is only one photon, how an one a
ount for the sudden hange of momentum suered by the re e ting obje t?
Neither de Broglie nor Brillouin were able to give an answer. De Broglie
10.4 Re oil of a single photon
247
laimed that pilot-wave theory in its urrent form was able to give only the mean pressure exerted by an ensemble (or ` loud') of photons. What was missing from de Broglie's understanding was that, in su h a ase, the position of the mirror would have to be treated by pilot-wave dynami s and in luded in the wave fun tion. S hemati ally, let xp be the position of the photon (on an axis normal to the surfa e of the mirror) and let xm be the position of the re e ting surfa e. Let us treat the mirror as a very massive but free body, with initial wave fun tion φ(xm , 0) lo alised around xm = 0 (at t = 0). The in ident photon initially has a lo alised wave fun tion ψinc (xp , 0) dire ted towards the mirror, with mean momentum ℏk . Roughly, if the photon pa ket strikes the mirror at time t0 , then the initial total wave fun tion Ψ(xp , xm , 0) = ψinc (xp , 0)φ(xm , 0) will evolve into a wave fun tion of the s hemati form Ψ(xp , xm , t) ∼ ψref (xp , t)φ(xm − (∆p/M )(t − t0 ), 0)e(i/ℏ)(∆p)xm , (12)
where ψref is a re e ted pa ket dire ted away from the mirror, ∆p = 2ℏk is the momentum transferred to the mirror, M is the mass of the mirror, and φ(xm − (∆p/M )(t − t0 ), 0) is a pa ket (whose spreading we ignore) moving to the right with speed ∆p/M . The a tual oordinates xp , xm will be guided by Ψ in a
ordan e with de Broglie's equation, and the position xm (t) of the mirror will follow the moving pa ket. The re oil of the mirror an therefore be a
ounted for. Thus, while de Broglie had the omplete pilot-wave dynami s of a many-body system, he seems not to have understood that it is sometimes ne essary to in lude the oordinates of ma ros opi equipment in the pilot-wave des ription. Otherwise, he might have been able to answer Kramers. On the other hand, in ordinary quantum theory too, a proper explanation for the re oil of the mirror would also have to treat the mirror as part of the quantum system. If the mirror is regarded as a lassi al obje t, then quantum theory would stri tly speaking be as powerless as pilot-wave theory. Perhaps this is why Brillouin made the following remark (p. 498): No theory urrently gives the answer to Mr Kramers' question.
11 Pilot-wave theory in retrospe t
As we dis ussed in se tion 2.4, in his Solvay le ture of 1927 de Broglie presented the pilot-wave dynami s of a nonrelativisti many-body system, and outlined some simple appli ations of his `new dynami s of quanta' (to interferen e, dira tion, and atomi transitions). Further, as we saw in se tion 10.2, ontrary to a widespread misunderstanding, in the general dis ussion de Broglie's reply to Pauli's obje tion ontained the essential points needed to treat inelasti s attering (even if Fermi's misleading opti al analogy onfused matters): in parti ular, de Broglie
orre tly indi ated how de nite quantum out omes in s attering pro esses arise from a separation of wave pa kets in on guration spa e. We also saw in se tion 10.4 that de Broglie was unable to reply to a query from Kramers on erning the re oil of a single photon on a mirror: to do so, he would have had to introdu e a joint wave fun tion for the photon and the mirror. De Broglie's theory was revived by Bohm 25 years later (Bohm 1952a,b) (though with the dynami s written in terms of a law for a
eleration instead of a law for velo ity). Bohm's truly new and very important
ontribution was a pilot-wave a
ount of the general quantum theory of measurement, with ma ros opi equipment (pointers, et .) treated as part of the quantum system. In ee t, in 1952 Bohm provided a detailed derivation of quantum phenomenology from de Broglie's dynami s of 1927 (albeit with the dynami al equations written dierently). Despite this su
ess, until about the late 1990s most physi ists still believed that hidden-variables theories su h as de Broglie's ould not possibly reprodu e the predi tions of quantum theory (even for simple ases su h as the two-slit experiment, f. se tion 6.1). Or, they believed that su h theories had been disproved by experiments testing EPR-type orrelations and demonstrating violations of Bell's inequality. 248
Pilot-wave theory in retrospe t
249
As just one striking example of the latter belief, in the early 1990s James T. Cushing, then a professor of physi s and of philosophy at the University of Notre Dame, submitted a resear h proposal to the US National S ien e Foundation `for theoreti al work to be done, within the framework of Bohm's version of quantum theory, on some foundational questions in quantum me hani s' (Cushing 1996, p. 6), and re eived the following evaluation: The subje t under onsideration, the rival Copenhagen and ausal (Bohm) interpretations of the quantum theory, has been dis ussed for many years and in the opinion of several members of the Physi s Division of the NSF, the situation has been settled. The ausal interpretation is in onsistent with experiments whi h test Bell's inequalities.
Consequently ....
funding ....
a
resear h program in this area would be unwise. (Cushing 1996, p. 6)
This is ironi be ause, as even a super ial reading of Bell's original papers shows, the nonlo al theory of de Broglie and Bohm was a primary motivation for Bell's work on his famous inequalities (Bell 1964, 1966). Bell knew that pilot-wave theory was empiri ally equivalent to quantum theory, and wanted to nd out if the nonlo ality was a pe uliarity of this parti ular model, or if it was a general feature of all hidden-variables theories. Bell's on lusion was that lo al theories have to satisfy his inequality, whi h is in onsistent with EPR-type orrelations, and that therefore any viable theory must be nonlo al like pilot-wave theory. There was never any question, in experimental tests of Bell's inequality, of testing the nonlo al theory of de Broglie and Bohm; rather, it was the lass of lo al theories that was being tested.a Despite this and other misunderstandings, in re ent years the pilotwave theory of de Broglie and Bohm, with parti le traje tories guided by a physi ally-real wave fun tion, has gained wide a
eptan e as an alternative (though little used) formulation of quantum theory. While it is still o
asionally asserted that any theory with traje tories (or other hidden variables) must disagree with experiment, su h erroneous laims have be ome mu h less frequent.b This hange in attitude seems to have a See hapter 9 of Cushing (1994) for an extensive dis ussion of the generally hostile rea tions to and misrepresentations of Bohm's 1952 papers. b For example, in his book The Elegant Universe, Greene (2000) asserts that not only are quantum parti le traje tories unmeasurable (be ause of the un ertainty prin iple), their very existen e is ruled out by experiments testing the Bell inequalities: `.... theoreti al progress spearheaded by the late Irish physi ist John Bell and the experimental results of Alain Aspe t and his ollaborators have shown onvin ingly that .... [e℄le trons and everything else for that matter
annot be des ribed as simultaneously being at su h-and-su h lo ation and having su h-and-su h speed' (p. 114). This is orre ted a few years later in The Fabri of
250
Pilot-wave theory in retrospe t
been largely a result of the publi ation in 1987 of Bell's in uential book on the foundations of quantum theory (Bell 1987), several hapters of whi h onsisted of pedagogi al explanations of pilot-wave theory as an obje tive and deterministi a
ount of quantum phenomena. Leaving aside the question of whether or not pilot-wave theory (or de BroglieBohm theory) is loser to the truth about the quantum world than other formulations, it is a remarkable fa t that it took about three-quarters of a entury for the theory to be ome widely a
epted as an internally
onsistent alternative. Even today, however, there are widespread mis on eptions not only about the physi s of pilot-wave theory, but also about its history. It is generally re ognised that de Broglie worked along pilot-wave lines in the 1920s, that Bohm developed and extended the theory in 1952, and that Bell publi ised the theory in his book in 1987. But the full extent of de Broglie's ontributions in the 1920s is usually not re ognised. A areful examination of the pro eedings of the fth Solvay onferen e
hanges our per eption of pilot-wave theory, both as a physi al theory and as a part of the history of quantum physi s.
11.1 Histori al mis on eptions Many of the widespread mis on eptions about the history of pilot-wave theory are onveniently summarised in the following extra t from the book by Bohm and Hiley (1993, pp. 389): The idea of a `pilot wave' that guides the movement of the ele tron was rst suggested by de Broglie in 1927, but only in onne tion with the one-body system. De Broglie presented this idea at the 1927 Solvay Congress where it was strongly riti ised by Pauli. His most important riti ism was that, in a two-body s attering pro ess, the model ould not be applied oherently. In
onsequen e de Broglie abandoned his suggestion. The idea of a pilot wave was proposed again in 1952 by Bohm in whi h an interpretation for the many-body system was given. This latter made it possible to answer Pauli's riti ism .... a .
As we have by now repeatedly emphasised, in his 1927 Solvay report (pp. 386 f.) de Broglie did in fa t present pilot-wave theory in on guration spa e for a many-body system, not just the one-body theory the Cosmos, where, ommenting on `Bohm's approa h' Greene (2005) writes that it `does not fall afoul of Bell's results be ause .... possessing de nite properties forbidden by quantum un ertainty is not ruled out; only lo ality is ruled out .... ' (p. 206, original itali s). a Similar histori al mis on eptions appear in Cushing (1994, pp. 11821, 149).
11.1 Histori al mis on eptions
251
in 3-spa e; and further, as we saw in se tion 10.2, de Broglie's reply to Pauli's riti ism ontained the essential ideas needed for a proper rebuttal.a As we shall see below, the laim that de Broglie abandoned his theory be ause of Pauli's riti ism is also not true. Contrary to widespread belief, then, the many-body theory with a guiding wave in on guration spa e is originally due to de Broglie and not Bohm; and in 1927, de Broglie did understand the essentials of the pilot-wave theory of s attering. Thus the main ontent of Bohm's rst paper of 1952 (Bohm 1952a) whi h presents the dynami s (though in terms of a
eleration), with appli ations to s attering was already known to de Broglie in 1927. Note that the theory was regarded as only provisional, by both de Broglie in 1927 and by Bohm in 1952. In parti ular, as we saw in se tions 2.3.2 and 2.4, de Broglie regarded the introdu tion of a pilot wave in on guration spa e as a provisional measure. Bohm, on the other hand, suggested that the basi prin iples of the theory would break down at nu lear distan es of order 10−13 m (Bohm 1952a, pp. 1789). As we have said, in ontrast with de Broglie's presentation of 1927, Bohm's dynami s of 1952 was based on a
eleration, not velo ity. For de Broglie, the basi law of motion for parti les with masses mi and wave fun tion Ψ = |Ψ| e(i/~)S was the guidan e equation mi
dxi = ∇i S , dt
(1)
whereas for Bohm, the basi law of motion was the Newtonian equation d2 xi dpi = mi 2 = −∇i (V + Q) , dt dt
(2)
where Q≡−
X ℏ2 ∇2 |Ψ| i 2m |Ψ| i i
(3)
is the `quantum potential'. Taking the time derivative of (1) and using the S hrödinger equation yields pre isely (2). For Bohm, however, (1) was not a law of motion but rather a onstraint pi = ∇i S to be imposed on the initial momenta (Bohm 1952a, p. 170). This initial onstraint happens to be preserved in time by (2), whi h Bohm regarded as the true law of motion. Indeed, a To our knowledge, Bonk (1994) is the only other author to have noti ed that de Broglie's reply to Pauli was indeed along the right lines.
252
Pilot-wave theory in retrospe t
Bohm suggested that the initial onstraint pi = ∇i S ould be relaxed, leading to orre tions to quantum theory: .... this restri tion is not inherent in the on eptual stru ture .... it is quite
onsistent in our interpretation to ontemplate modi ations in the theory, whi h permit an arbitrary relation between pp. 17071)
p
and
∇S(x).
(Bohm 1952a,
For de Broglie, in ontrast, there was never any question of relaxing (1): he regarded (1) as the basi law of motion for a new form of parti le dynami s, and indeed for him (1) embodied as we saw in hapter 2 the uni ation of the prin iples of Maupertuis and Fermat, a uni ation that he regarded as the guiding prin iple of his new dynami s. (De Broglie did mention in passing, however, the alternative formulation in terms of a
eleration, both in his `Stru ture' paper ( f. se tion 2.3.1) and in his Solvay report (p. 384).) Some authors seem to believe that the `re asting' of Bohm's se ondorder dynami s into rst-order form was due to Bell (1987). But in fa t, Bell's (pedagogi al) presentation of the theory based on the guidan e equation for velo ity, and ignoring the notion of quantum potential was identi al to de Broglie's original presentation. De Broglie's rstorder dynami s of 1927 is sometimes referred to as `Bohmian me hani s'. As already noted in hapter 2, this is a misnomer: rstly be ause of de Broglie's priority, and se ondly be ause Bohm's me hani s of 1952 was a tually se ond-order in time. Another ommon histori al mis on eption on erns the re eption of de Broglie's theory at the Solvay onferen e. It is usually said that de Broglie's ideas attra ted hardly any attention. It is di ult to understand how su h an impression originated, for even a ursory perusal of the pro eedings reveals that de Broglie's theory was extensively dis ussed, both after de Broglie's le ture and during the general dis ussion. Nevertheless, in his lassi a
ount of the histori al development of quantum theory, Jammer asserts that when de Broglie presented his theory: It was immediately lear that nobody a
epted his ideas .... . In fa t, with the ex eption of some remarks by Pauli .... de Broglie's ausal interpretation was not even further dis ussed at the meeting. Only Einstein on e referred to a it en passant. (Jammer 1966, p. 357)
a Jammer adds footnoted referen es to pp. 280 and 256 respe tively of the original pro eedings, where Pauli's obje tion (involving Fermi's treatment of the rotator) appears, and where, in his main ontribution to the general dis ussion, Einstein
omments that in his opinion de Broglie `is right to sear h in this dire tion' (p. 488).
11.1 Histori al mis on eptions
253
It appears to have es aped Jammer's attention that the general dis ussion ontains extensive and varied omments on many aspe ts of de Broglie's theory (in luding a query by Einstein about the speed of photons), as does the dis ussion after de Broglie's le ture, and that support for de Broglie's ideas was expressed by Brillouin and by Einstein. In a later histori al study, again, Jammer (1974, pp. 11011) writes in referen e to the fth Solvay onferen e that de Broglie's theory `was hardly dis ussed at all', and that `the only serious rea tion ame from Pauli'. In the same study, Jammer quotes extensively from de Broglie's report and from the general dis ussion, apparently without noti ing the extensive dis ussions of de Broglie's theory that appear both after de Broglie's report and in the general dis ussion. But Jammer is by no means the only historian to have given short shrift to de Broglie's major presen e at the 1927 Solvay onferen e. In his book The Solvay Conferen es on Physi s, Mehra (1975, p. xvi) quotes de Broglie himself as saying, with referen e to his presentation of pilot-wave theory in 1927, that `it re eived hardly any attention'. But these words were written by de Broglie some 46 years later (de Broglie 1974), and de Broglie's re olle tion (or misre olle tion) after nearly half a entury is belied by the ontent of the published pro eedings.a In volume 6 of their monumental The Histori al Development of Quantum Theory, Mehra and Re henberg (2000, pp. 24650) devote several pages to the published pro eedings of the 1927 Solvay onferen e, fo ussing on the general dis ussion mainly on the omments by Einstein, Dira and Heisenberg as well as on the unpublished omments by Bohr. The rest of the general dis ussion is summarised in a single senten e (p. 250): After the Einstein-Pauli-Dira -Heisenberg ex hange, the general dis ussion turned to more te hni al problems onne ted with the des ription of photons and ele trons in quantum me hani s, as well as with the details of de Broglie's re ent ideas.
It is added that `though these points possess some intrinsi interest, they do not throw mu h light on the interpretation debate'. As we shall see in se tion 12.1, it would appear that, for Mehra and Re henberg, as indeed for most ommentators, the `interpretation debate' entred mainly around private or semi-private dis ussions between Bohr and Einstein, and that the many pages of published dis ussions a In fa t, several ommentators have drawn erroneous on lusions about de Broglie, by relying on mistaken `re olle tions' written by de Broglie himself de ades later.
254
Pilot-wave theory in retrospe t
were of omparatively little interest. Thus the remarkable downplaying of the dis ussion of de Broglie's theory, as well as of other ideas, is
oupled with a strong tenden y on the part of many authors to portray (in orre tly) the 1927 onferen e as fo ussed primarily on the
onfrontation between Bohr and Einstein. We have already noted the widespread histori al mis on eptions on erning the de Broglie-Pauli en ounter in the general dis ussion. A related mis on eption on erns de Broglie's thinking in the immediate aftermath of the Solvay onferen e. It is often asserted that, soon after the onferen e, de Broglie abandoned his theory primarily be ause of Pauli's riti ism. This is not orre t. In his book An Introdu tion to the Study of Wave Me hani s (de Broglie 1930), whi h was published just three years after the Solvay meeting, de Broglie gives three main reasons for why he onsiders his pilot-wave theory to be unsatisfa tory. First, de Broglie onsiders (p. 120) a parti le in ident on an imperfe tly re e ting mirror, and notes that if the parti le is found in the transmitted beam then the re e ted part of the wave must disappear (this being `a ne essary onsequen e of the interferen e prin iple'). De Broglie on ludes that `the wave is not a physi al phenomenon in the old sense of the word. It is of the nature of a symboli representation of a probability .... '. Here, de Broglie did not understand how pilot-wave theory a
ounts for the ee tive (and pra ti ally irreversible) ollapse of the wave pa ket, by means of a separation into non-overlapping bran hes involving many degrees of freedom ( f. se tions s at-in-pwt and 10.3). Se ond, de Broglie notes (pp. 121, 133) that a parti le in free spa e guided by a superposition of plane waves would have a rapidly-varying velo ity and energy, and he annot see how this ould be onsistent with the out omes of quantum energy measurements, whi h would oin ide stri tly with the energy eigenvalues present in the superposition. To solve this se ond problem, de Broglie would have had to apply pilot-wave dynami s to the pro ess of quantum measurement itself in luding the apparatus in the wave fun tion if ne essary as done mu h later by Bohm (see se tion10.3 ). This question of energy measurement bears some similarity to that raised by Pauli, and perhaps Pauli's query set de Broglie thinking about this problem. But even so, de Broglie in ee t gave the essen e of a orre t reply to Pauli's query, and the problem of energy measurement was posed by de Broglie himself. Third, in applying pilot-wave theory to photons, de Broglie nds (p. 132) that in some ir umstan es (spe i ally, in the interferen e zone lose to an imperfe tly re e ting mirror) the photon traje tories have superluminal
11.1 Histori al mis on eptions
255
speeds, whi h he onsiders una
eptable. De Broglie's book does not mention Pauli's riti ism. It is also often laimed that, when de Broglie abandoned his pilot-wave theory (soon after the Solvay onferen e), he qui kly adopted the views of Bohr and Heisenberg. Thus, for example, Cushing (1994, p. 121) writes: `By early 1928 he [de Broglie℄ had de ided to adopt the views of Bohr and Heisenberg'.a But de Broglie's book of 1930, in whi h the above di ulties with pilot-wave theory are des ribed, ontains a `General introdu tion' that is `the reprodu tion of a ommuni ation made by the author at the meeting of the British Asso iation for the Advan ement of S ien e held in Glasgow in September, 1928' (de Broglie 1930, p. 1). This introdu tion therefore gives an overview of de Broglie's thinking in late 1928, almost a year after the fth Solvay onferen e. While de Broglie makes it lear (p. 7) that in his view it is `not possible to regard the theory of the pilot-wave as satisfa tory', and states that the `point of view developed by Heisenberg and Bohr .... appears to
ontain a large body of truth', his on luding paragraph shows that he was still not satis ed: To sum up, the physi al interpretation of the new me hani s remains an extremely di ult question ....
the dualism of waves and parti les must be
admitted .... . Unfortunately the profound nature of the two members in this duality and the pre ise relation existing between them still remain a mystery. (de Broglie 1930, p. 10)
At that time, de Broglie seems to have a
epted the formalism of quantum theory, and the statisti al interpretation of the wave fun tion, but he still thought that an adequate physi al understanding of wave-parti le duality had yet to be rea hed. Yet another year later, in De ember 1929, doubts about the orre t interpretation of quantum theory ould still be dis erned in de Broglie's Nobel Le ture (de Broglie 1999): Is it even still possible to assume that at ea h moment the orpus le o
upies a well-de ned position in the wave and that the wave in its propagation arries the orpus le along in the same way as a wave would arry along a ork? These are di ult questions and to dis uss them would take us too far and even to the on nes of philosophy.
All that I shall say about them here is
that nowadays the tenden y in general is to assume that it is not onstantly possible to assign to the orpus le a well-de ned position in the wave.
a As eviden e for this, Cushing ites later re olle tions by de Broglie in his book Physi s and Mi rophysi s (de Broglie 1955), whi h was originally published in Fren h in 1947. Again, de Broglie's re olle tions de ades later do not seem a reliable guide to what a tually happened ir a 1927.
256
Pilot-wave theory in retrospe t
This brings us to another histori al mis on eption on erning de Broglie's work. Nowadays, de Broglie-Bohm theory is often presented as a ` ompletion' of quantum theory.a Criti s sometimes view this as an arbitrary addition to or amendment of the quantum formalism, the traje tories being viewed as an additional `baggage' being appended to an already given formalism. Regardless of the truth or otherwise of pilot-wave theory as a physi al theory, su h a view ertainly does not do justi e to the histori al fa ts. For the elements of pilot-wave theory waves guiding parti les via de Broglie's velo ity formula were already in pla e in de Broglie's thesis of 1924, before either matrix or wave me hani s existed. And it was by following the lead of de Broglie's thesis that S hrödinger developed the wave equation for de Broglie's matter waves. While S hrödinger dropped the traje tories and onsidered only the waves, nevertheless, histori ally speaking the wave fun tion ψ and the S hrödinger equation both grew out of de Broglie's phase waves.b The pilot-wave theory of 1927 was the ulmination of de Broglie's independent work from 1923, with a major input from S hrödinger in 1926. There is no sense in whi h de Broglie's traje tories were ever `added to' some pre-existing theory. And when Bohm revived the theory in 1952, while it may have seemed to Bohm's ontemporaries (and indeed to Bohm himself) that he was adding something to quantum theory, from a histori al point of view Bohm was simply reinstating what had been there from the beginning. The failure to a knowledge the priority of de Broglie's thinking from 1923 is visible even in the dis ussions of 1927. In the dis ussion following de Broglie's le ture, Pauli (p. 400) presents what he laims is the entral idea of de Broglie's theory: I should like to make a small remark on what seems to me to be the mathemati al basis of Mr de Broglie's viewpoint on erning parti les in motion on de nite traje tories. His on eption is based on the prin iple of onservation of harge ....
if in a eld theory there exists a onservation prin iple ....
it
is always formally possible to introdu e a velo ity ve tor ... and to imagine furthermore orpus les that move following the urrent lines of this ve tor.
Pauli's assertion that de Broglie's theory is based on the onservation of harge makes sense only for one parti le: for a many-body system, de Broglie's velo ity eld is asso iated with onservation of probability in on guration spa e, whereas onservation of harge is always tied a An often- ited motivation for introdu ing the traje tories is, of ourse, to solve the measurement problem. b See also the dis ussion in se tion 4.5.
11.1 Histori al mis on eptions
257
to 3-spa e. Still, the point remains that in any theory with a lo ally
onserved probability urrent, it is indeed possible to introdu e parti le traje tories following the ow lines of that urrent. This way of presenting de Broglie's theory then makes the traje tories look like an addendum to a pre-existing stru ture: given the S hrödinger equation with its lo ally onserved urrent, one an add traje tories if one wishes. But to present the theory in this way is a major distortion of the histori al fa ts and priorities. The essen e of de Broglie's dynami s
ame before S hrödinger's work, not after. Further, de Broglie obtained his velo ity law not from the S hrödinger urrent (whi h was unknown in 1923 or 1924) but from his postulated relation between the prin iples of Maupertuis and Fermat. And nally, while de Broglie's traje tory equation did not in fa t owe anything to the S hrödinger equation, again, the latter equation arose out of onsiderations (of the opti al-me hani al analogy) that had been initiated by de Broglie. It seems rather lear that the histori al priority of de Broglie's work was being downplayed by Pauli's remarks, as it has been more or less ever sin e. A related histori al mis on eption on erns the status of the S hrödinger equation in pilot-wave theory. It is sometimes argued that this equation has no natural pla e in the theory, and that therefore the theory is arti ial. For example, ommenting on what he alls `Bohm's theory', Polkinghorne (2002, pp. 55, 89) writes: There is an air of ontrivan e about it that makes it unappealing. For example, the hidden wave has to satisfy a wave equation.
ome from?
Where does this equation
The frank answer is out of the air or, more a
urately, out of
the mind of S hrödinger.
To get the right results, Bohm's wave equation
must be the S hrödinger equation, but this does not follow from any internal logi of the theory and it is simply an
ad ho
strategy designed to produ e
empiri ally a
eptable answers. .... It is on these grounds that most physi ists nd the greatest di ulty with Bohmian ideas .... the
ad ho
but ne essary
appropriation of the S hrödinger equation as the equation for the Bohmian wave has an unattra tively opportunist air to it.
Polkinghorne's omments are a fair riti ism of Bohm's 1952 reformulation of de Broglie's theory. For as we have seen, Bohm based his presentation on the Newtonian equation of motion (2) for a
eleration, with a `quantum potential' Q determined by the wave fun tion Ψ through (3): a
ording to Bohm, Ψ generates a `quantum for e' −∇i Q, whi h a
ounts for quantum ee ts. From this Newtonian standpoint, the wave equation for Ψ does indeed have nothing to do with the internal logi of the theory: it is then fair to say that, in Bohm's formulation,
258
Pilot-wave theory in retrospe t
the S hrödinger equation is `appropriated' for a purpose quite foreign to the origins of that equation. However, Polkinghorne's ritique does not apply to pilot-wave theory in its original de Broglian formulation, as a new form of dynami s in whi h parti le velo ities are determined by guiding waves (rather than parti le a
elerations being determined by Newtonian for es). For as a matter of histori al fa t, the S hrödinger equation did follow from the internal logi of de Broglie's theory. After all, S hrödinger set out in the rst pla e to nd the general wave equation for de Broglie's waves; and his derivation of that equation owed mu h to the opti al-me hani al analogy, whi h was a key omponent of de Broglie's approa h to dynami s.a It annot be said that de Broglie `appropriated' the S hrödinger equation for a purpose foreign to its origins, when the original purpose of the S hrödinger equation was in fa t to des ribe de Broglie's waves.
11.2 Why was de Broglie's theory reje ted? One might ask why de Broglie's theory did not gain widespread support soon after 1927. This question has been onsidered by Bonk (1994), who applies Bayesian reasoning to some of the dis ussions at the fth Solvay onferen e, in an attempt to understand the rapid a
eptan e of the `Copenhagen' interpretation. It has also been suggested that were it not for ertain histori al a
idents, de Broglie's theory might have triumphed in 1927 and emerged as the dominant interpretation of quantum theory (Cushing 1994, hapter 10). It is di ult to evaluate how realisti Cushing's `alternative histori al s enario' might have been. Here, we shall simply highlight two points that are usually overlooked, and whi h are relevant to any evaluation of why de Broglie's theory did not arry the day. Our rst point is that, be ause de Broglie did in fa t give a reply to Pauli's riti ism that ontained the essen e of a orre t rebuttal and be ause in ontrast de Broglie ompletely failed to reply to the di ulty raised by Kramers the question of whether or not de Broglie made a onvin ing ase at the fth Solvay onferen e (and if not, why not) should be re onsidered. Our se ond point is that there was a good te hni al reason for why a Nowadays, it is ommon in textbooks to motivate the free-parti le S hrödinger equation as the simplest equation satis ed by a plane de Broglie wave ei(k·x−ωt) with the nonrelativisti dispersion relation ℏω = (ℏk)2 /2m. This `derivation' is just as natural in pilot-wave theory as it is in standard quantum theory.
11.2 Einstein's alternative pilot-wave theory
259
`standard' quantum theory had an advantage over pilot-wave theory in 1927. By the use of a simple ` ollapse postulate' for mi rosystems, it was generally possible to a
ount for quantum measurement out omes without having to treat the measurement pro ess (in luding the apparatus) quantum-me hani ally. In ontrast, it was easy to nd examples where in pilot-wave theory it was essential to use the theory itself to analyse the measurement pro ess: as we saw in the last se tion, in his book of 1930 de Broglie ould not see how, for a parti le guided by a superposition of energy eigenfun tions, an energy measurement would give one of the results expe ted from quantum theory. Agreement with quantum theory requires an analysis of the measurement pro ess in terms of pilot-wave dynami s, with the apparatus in luded as part of the system, as shown by Bohm (1952b). In ontrast, in ordinary quantum theory it usually su es in pra ti e simply to apply a ollapse rule to the mi ros opi system alone. Su h a ollapse rule is of ourse merely pragmati , and de es pre ise formulation (there being no sharp boundary between `mi ros opi ' and `ma ros opi ', f. se tion 5.2); yet, in ordinary laboratory situations, it yields predi tions that may be ompared with experiment. These two points should be taken into a
ount in any full evaluation of why de Broglie's theory was reje ted in 1927 and shortly thereafter. Further relevant material, that seems to have never been onsidered before, onsists of omments by Heisenberg on the possibility of a deterministi pilot-wave interpretation. These omments do not appear in the pro eedings of the fth Solvay onferen e, nor are they dire ted at de Broglie's theory. Rather, they appear in a letter Heisenberg wrote to Einstein a few months earlier, on 10 June 1927, and they on ern Einstein's version of pilot-wave theory. Heisenberg's remarks ould just as well have been dire ted at de Broglie's theory, however. Both Einstein's theory and Heisenberg's omments thereon are dis ussed in the next se tion.
11.3 Einstein's alternative pilot-wave theory (May 1927) As we saw in se tion 2.3.1, de Broglie rst arrived at pilot-wave theory in a paper published in Journal de Physique in May 1927 (de Broglie 1927b). In the same month, Einstein proposed what in retrospe t appears to be an alternative version of pilot-wave theory, with parti le traje tories determined by the many-body wave fun tion but in a manner dierent from that of de Broglie's theory. This new theory
260
Pilot-wave theory in retrospe t
was des ribed in a paper entitled `Does S hrödinger's wave me hani s determine the motion of a system ompletely or only in the sense of statisti s?', whi h was presented on 5 May 1927 at a meeting of the Prussian A ademy of S ien es. On the same day Einstein wrote to Ehrenfest that `.... in a ompletely unambiguous way, one an asso iate de nite movements with the solutions [of the S hrödinger equation℄' (Howard 1990, p. 89). However, on 21 May, before the paper appeared in print, Einstein withdrew it from publi ation (Kirsten and Treder 1979, p. 135; Pais 1982, p. 444). The paper remained unpublished, but its ontents are nevertheless known from a manus ript version in the Einstein ar hive see Howard (1990, pp. 8990) and Belousek (1996).a Einstein's unpublished version of pilot-wave theory has some relevan e to his argument for the in ompleteness of quantum theory, given in the general dis ussion. As we saw in se tion 7.1, a
ording to Einstein's argument, lo ality requires `that one does not des ribe the [dira tion℄ pro ess solely by the S hrödinger wave, but that at the same time one lo alises the parti le during the propagation' (p. 488). Einstein added: `I think that Mr de Broglie is right to sear h in this dire tion' without mentioning that he himself had re ently made an attempt in the same dire tion. It quite possible that, before abandoning his version of pilot-wave theory, Einstein had onsidered presenting it at the fth Solvay onferen e. Indeed, had Einstein been happy with his new theory, there is every reason to think he would have presented it a few months later in Brussels. As dis ussed in se tion 1.3, Lorentz had asked Einstein to give a report on parti le statisti s, and while Einstein had agreed to do so, he was relu tant. Less than a month after withdrawing his pilot-wave paper, Einstein withdrew his ommittment to speak at the Solvay onferen e, writing to Lorentz on 17 June: `.... I kept hoping to be able to ontribute something of value in Brussels; I have now given up that hope. .... I did not take this lightly but tried with all my strength .... ' (quoted from Pais 1982, p. 432). It then seems indeed probable that, instead of (or in addition to) speaking about parti le statisti s, Einstein had hoped to present something like his version of pilot-wave theory. Let us now des ribe what Einstein's proposal was. (For more detailed presentations, see Belousek (1996) and Holland (2005).) Einstein's starting point is the time-independent S hrödinger equaa Ar hive referen e: AEA 2-100.00 (in German); urrently available on-line at http://www.alberteinstein.info/db/ViewDetails.do?Do umentID=34338 .
11.3 Einstein's alternative pilot-wave theory
261
tiona −
ℏ2 2 ∇ Ψ + V Ψ = EΨ 2
(4)
for a many-body system with potential energy V , total energy E , and wave fun tion Ψ on an n-dimensional on guration spa e. Einstein
onsiders (4) to de ne a kineti energy E−V =−
ℏ2 ∇2 Ψ , 2 Ψ
(5)
whi h may also be written as E−V =
1 1 (ds/dt)2 = gµν q˙µ q˙ν , 2 2
(6)
where ds2 = gµν dq µ dq ν is the line element in on guration spa e and q˙µ is the system velo ity. The theory is expressed in terms of arbitrary oordinates q µ with metri gµν . The Lapla ian is then given by ∇2 Ψ = g µν ∇µ ∇ν Ψ, where ∇µ is the ovariant derivative. Einstein writes ∇µ ∇ν Ψ as Ψµν (whi h he alls the `tensor of Ψ- urvature'), and seeks unit ve tors Aµ that extremise Ψµν Aµ Aν , leading to the eigenvalue problem (Ψµν − λgµν )Aν = 0 ,
(7)
with n real and distin t solutions λ(α) . At ea h point, the Aµ(α) de ne a lo al orthogonal oordinate system (with Eu lidean metri at that point). In this oordinate system, both Ψµν and gµν are diagonal, with ¯ αβ = λ(α) δαβ and g¯αβ = δαβ . The two expressions (5)
omponents Ψ and (6) for the kineti energy then be ome E−V =−
and
ℏ2 1 X ¯ Ψαα 2 Ψ α
E−V =
1X 2 q˙ 2 α α
(8)
(9)
a The parti le masses make no expli it appearan e, be ause they have been absorbed into the on guration-spa e metri gµν . Einstein is following S hrödinger's usage. In his se ond paper on wave me hani s, S hrödinger (1926 ) introdu ed a nonEu lidean metri determined by the kineti energy (see also S hrödinger's report, p. 450). The Lapla ian ∇2 is then understood in the Riemannian sense, and the S hrödinger equation indeed takes the form (4) with no masses appearing expli itly. Note that also in analyti al me hani s it is sometimes onvenient to write the kineti energy 12 Mµν q˙µ q˙ν as 12 (ds/dt)2 where ds2 = Mµν dq µ dq ν is a line element with metri Mµν (Goldstein 1980, pp. 36970).
262
Pilot-wave theory in retrospe t
respe tively. Einstein then introdu es the hypothesis that these two expressions mat h term by term, so that 2
q˙α = −
ℏ2 ¯ Ψαα . Ψ
(10)
¯ αα = λ(α) , and transforming ba k to the original oordinate Using Ψ P system, where q˙µ = α q˙α Aµ(α) , Einstein obtains the nal result µ
q˙ = ℏ
X α
±
r
−λ(α) µ A(α) , Ψ
(11)
whi h expresses q˙µ in terms of λ(α) and Aµ(α) , where at ea h point in
on guration spa e λ(α) and Aµ(α) are determined by the lo al values of the wave fun tion Ψ and its derivatives. (The ambiguity in sign is, a
ording to Einstein, to be expe ted for quasiperiodi motions.) Thus, a
ording to (11), the system velo ity q˙µ is lo ally determined (up to signs) by Ψ and its derivatives. This is Einstein's proposed velo ity law, to be ompared and ontrasted with de Broglie's. The manus ript ontains an additional note `added in proof', in whi h Einstein asserts that the theory he has just outlined is physi ally una
eptable, be ause it predi ts that for a system omposed of two independent subsystems, with an additive Hamiltonian and a produ t wave fun tion, the velo ities for one subsystem generally depend on the instantaneous oordinates of the other subsystem. However, Einstein adds that Grommer has pointed out that this problem ould be avoided by repla ing Ψ with ln Ψ in the onstru tion of the velo ity eld. A
ording to Einstein: `The elaboration of this idea should o
asion no di ulty ....' (Howard 1990, p. 90). Despite Einstein's apparent optimism that Grommer's modi ation would work, it has been assumed (Howard 1990, p. 90; Cushing 1994, p. 128) that Einstein withdrew the paper be ause he soon realised that Grommer's suggestion did not work. But it has been shown by Holland (2005) that the repla ement Ψ → ln Ψ does in fa t remove the di ulty raised by Einstein. Holland shows further, however, that there are other di ulties with Einstein's theory with or without Grommer's modi ation. It is not known if Einstein re ognised these other di ulties, but if he did, they would ertainly have been onvin ing grounds on whi h to abandon the s heme altogether. The di ulties raised by Holland are as follows. First, the theory applies only to a limited range of quantum states: real stationary states with E ≥ V . Se ond, the system velo ity q˙µ is de ned only in a limited
11.3 Einstein's alternative pilot-wave theory
263
domain of on guration spa e (for example, without Grommer's modi ation, λ(α) /Ψ must be negative for q˙µ to be real). Third, even where 2 q˙µ is de ned, the ontinuity equation ∇µ (|Ψ| q˙µ ) = 0 is generally not 2 satis ed (where |Ψ| is time-independent for a stationary state), so that µ the velo ity eld q˙ does not map an initial Born-rule distribution into a nal Born-rule distribution. This last feature removes any realisti hope that the theory ould reprodu e the predi tions of standard quantum theory (Holland 2005).a At the end of his manus ript (before the note added in proof), Einstein writes: `.... the assignment of ompletely determined motions to solutions of S hrödinger's dierential equation is .... just as possible as is the assignment of determined motions to solutions of the Hamilton-Ja obi dierential equation in lassi al me hani s' (Belousek 1996, pp. 441 2). Given the lose relationship between the Hamilton-Ja obi fun tion and the phase S of Ψ (the latter redu ing to the former in the shortwavelength limit), it is natural to ask why Einstein did not onsider de Broglie's velo ity eld proportional to the phase gradient ∇S whi h is a straightforward generalisation of the Hamilton-Ja obi velo ity formula. As well as being mu h simpler than Einstein's, de Broglie's velo ity eld immediately satis es Einstein's desired separability of parti le motions for produ t states.a Why did Einstein instead propose what seems a mu h more ompli ated and unwieldy s heme to generate parti le velo ities from the wave fun tion? Perhaps Einstein did not adopt de Broglie's velo ity eld simply be ause, for the real wave fun tions Einstein onsidered, the phase gradient ∇S = ℏ Im(∇Ψ/Ψ) vanishes; though it is not lear why Einstein thought one ould restri t attention to su h wave fun tions. As for the seemingly pe uliar onstru tion that Einstein did adopt, it should be noted that Einstein was (as he himself states) following S hrödinger in his use of a non-Eu lidean metri determined by the kineti energy (S hrödinger 1926 ). Further details of Einstein's onstru tion may have been related to another idea Einstein was pursuing: that quantisation onditions
ould arise from a generally- ovariant and `overdetermined' eld theory a As Holland (2005) also points out, from a modern point of view the mutual dependen e of parti le motions for produ t states, whi h Einstein found so una
eptable, need not be a real di ulty. We now know that a hidden-variables theory must be nonlo al, and there is no reason why in some theories the underlying nonlo ality ould not exist for fa torisable quantum states as well as for entangled ones. Indeed, Holland gives an example of just su h a theory. a It is not known whether or not Einstein noti ed the nonlo ality of de Broglie's theory for entangled wave fun tions.
264
Pilot-wave theory in retrospe t
that onstrains initial states, with parti les represented by singularities (Pais 1982, pp. 4648), an idea that, in retrospe t, seems somewhat reminis ent of de Broglie's double-solution theory (se tion 2.3.1). It would of ourse have been most interesting to see how physi ists would have rea ted had Einstein in fa t published his paper or presented it at the fth Solvay onferen e. It so happens that Heisenberg had heard about Einstein's theory through Born and Jordan, and on 19 May, just two days before Einstein withdrew the paper wrote to Einstein asking about it. On 10 June 1927, Heisenberg wrote to Einstein again, this time with detailed omments and arguments against what Einstein was (or had been) proposing.a At the beginning of this letter, after thanking Einstein for his `friendly letter', Heisenberg says he would like to explain why he believes indeterminism is ne essary and not just possible. He hara terises Einstein as thinking that, while all experiments will agree with the statisti al quantum theory, nevertheless it will be possible to talk about de nite parti le traje tories. Heisenberg then outlines an obje tion. He onsiders free ele trons with a onstant and very low velo ity hen e large de Broglie wavelength λ striking a grating with spa ing omparable to λ. He remarks that, in Einstein's theory, the ele trons will be s attered in dis rete spatial dire tions, and that if the initial position of a parti le were known one ould al ulate where the parti le will hit the grating. Heisenberg then asserts that one ould set up an obsta le at that point, so as to de e t the parti le in an arbitrary dire tion, independently of the rest of the grating. Heisenberg says that this ould be done, if the for es between the parti le and the obsta le a t only at short range (over distan es mu h smaller than the spa ing of the grating). Heisenberg then adds that, in a tual fa t, the ele tron will be s attered in the usual dis rete dire tions regardless of the obsta le. Heisenberg goes on to say that one ould es ape this on lusion if one `sets the motion of the parti le again in dire t relation to the behaviour of the waves'. But this means, says Heisenberg, that the size of the parti le or the range of its intera tion depends on its velo ity. Heisenberg asserts that making su h assumptions a tually amounts to giving up the word `parti le' and leads to a loss in understanding of why the simple potential energy e2 /r appears in the S hrödinger equation or in the matrix Hamiltonian fun tion. On the other hand, Heisenberg agrees that: `If you use the word parti le so liberally, I onsider it as very well a Heisenberg to Einstein, 19 May and 10 June 1927, AEA 12-173.00 and 12-174.00 (both in German).
11.4 Obje tions: in 1927 and today
265
possible that one an again also de ne parti le traje tories'. But then, adds Heisenberg, one loses the simpli ity of quantum theory, a
ording to whi h the parti le motion takes pla e lassi ally (to the extent that one an speak about motion in quantum theory). Heisenberg notes that Einstein seems willing to sa ri e this simpli ity for the sake of maintaining ausality. He remarks further that, in Einstein's on eption, many experiments are still determined only statisti ally, and `we ould only onsole ourselves with the fa t that, while for us the prin iple of
ausality is meaningless, be ause of the un ertainty relation p1 q1 ∼ h, however the dear God knows in addition the position of the parti le and thereby keeps the validity of the ausal law'. Heisenberg adds that he nds it unattra tive to try to des ribe more than just the ` onne tion between experiments' [Zusammenhang der Experimente℄. It is likely that Heisenberg had similar views of de Broglie's theory, and that if he had ommented on de Broglie's theory at the fth Solvay
onferen e he would have said things similar to the above. Heisenberg's obje tion on erning the ele tron and the grating seems to be based on inappropriate reasoning taken from lassi al physi s (a
ommon feature of obje tions to pilot-wave theory even today), and Heisenberg agrees that if the motion of the parti le is stri tly tied to that of the waves, then it should be possible to obtain onsisten y with observation (as we now know is indeed the ase for de Broglie's theory). Even so, Heisenberg seems to think that su h a highly non lassi al parti le dynami s would la k the simpli ity and intelligibility of ertain
lassi al ideas that quantum theory preserves. Finally, Heisenberg is unhappy with a theory ontaining unobservable ausal onne tions. Similar obje tions to pilot-wave theory are onsidered in the next se tion.
11.4 Obje tions: in 1927 and today It is interesting to observe that many of the obje tions to pilot-wave theory that are ommonly heard today were already voi ed in 1927. Regarding the existen e or non-existen e of de Broglie's traje tories, Brillouin in the dis ussion following de Broglie's le ture (p. 400)
ould just as well have been replying to a present-day riti of de BroglieBohm theory when he said that: Mr Born an doubt the real existen e of the traje tories al ulated by L. de Broglie, and assert that one will never be able to observe them, but he annot prove to us that these traje tories do not exist.
There is no ontradi tion
between the point of view of L. de Broglie and that of other authors .... .
266
Pilot-wave theory in retrospe t
Here we have a on i t between those who believe in hidden entities be ause of the explanatory role they play, and those who think that what annot be observed in detail should play no theoreti al role. Su h
on i ts are not un ommon in the history of s ien e: for example, a similar polarisation of views o
urred in the late nineteenth entury regarding the reality of atoms (whi h the `energeti ists' regarded as metaphysi al tions). The debate over the reality of the traje tories postulated by de Broglie has been sharpened in re ent years by the re ognition that, from the perspe tive of pilot-wave theory itself, our inability to observe those traje tories is not a fundamental onstraint built into the theory, but rather an a
ident of initial onditions with a Born-rule probability 2 distribution P = |Ψ| (for an ensemble of systems with wave fun tion Ψ). The statisti al noise asso iated with su h `quantum equilibrium' distributions sets limits to what an be measured, but for more general 2 `nonequilibrium' distributions P 6= |Ψ| , the un ertainty prin iple is violated and observation of the traje tories be omes possible (Valentini 1991b, 1992, 2002b; Pearle and Valentini 2006). Su h nonequilibrium entails, of ourse, a departure from the statisti al predi tions of quantum theory, whi h are obtained in pilot-wave theory only as a spe ial `equilibrium' ase. Arguably, the above disagreement between Born and Brillouin might nowadays turn on the question of whether one is willing to believe that quantum physi s is merely the physi s of a spe ial statisti al state. Another obje tion sometimes heard today is that a velo ity law dierent from that assumed by de Broglie is equally possible. In the dis ussion following de Broglie's le ture, S hrödinger (p. 401) raised the possibility of an alternative parti le velo ity de ned by the momentum density of a eld. Pauli pointed out that, for a relativisti eld, if the parti le velo ity were obtained by dividing the momentum density by the energy density, then the resulting traje tories would dier from those obtained by de Broglie, who assumed a velo ity de ned (in the ase of a single parti le) by the ratio of urrent density to harge density. Possibly, then as now, the existen e of alternative velo ities may have been interpreted as asting doubt on the reality of the velo ities a tually assumed by de Broglie (though the true velo ities would be ome measurable in the presen e of quantum nonequilibrium). In addition to the riti ism that the traje tories annot be observed, today it is also often obje ted that the traje tories are rather strange from a lassi al perspe tive. The pe uliar nature of de Broglie's tra-
11.4 Obje tions: in 1927 and today
267
je tories was addressed in the dis ussion following de Broglie's le ture (pp. 402 .), and again in the general dis ussion (pp. 499, 509 f.). It was, for example, pointed out that the speed of an ele tron ould be zero in a stationary state, and that for general atomi states the orbits would be very ompli ated. At the end of his dis ussion of photon re e tion by a mirror, Brillouin (p. 404) argued that de Broglie's non-re tilinear photon paths (in free spa e) were ne essary in order to avoid a paradox posed by Lewis, in whi h in the presen e of interferen e it appeared that photons would ollide with only one end of a mirror, ausing it to rotate, even though from lassi al ele trodynami s the mean radiation pressure on the mirror is expe ted to be uniform (see Brillouin's Fig. 3). This last example of Brillouin's re alls present-day debates involving
ertain kinds of quantum measurements, in whi h the traje tories predi ted by pilot-wave theory have ounter-intuitive features that some authors have labelled `surreal' (Englert et al. 1992, Aharonov and Vaidman 1996), while other authors regard these features as perfe tly understandable from within pilot-wave theory itself (Valentini 1992, p. 24, Dewdney, Hardy and Squires 1993, Dürr et al. 1993). A key question here is whether it is reasonable to expe t a theory of subquantum dynami s to onform to lassi al intuitions about measurement (given that it is the underlying dynami s that should be used to analyse the measurement pro ess). Pilot-wave theory is sometimes seen as a return to lassi al physi s (wel omed by some, riti ised by others). But in fa t, de Broglie's velo ity-based dynami s is a new form of dynami s that is simply quite distin t from lassi al theory; therefore, it is to be expe ted that the behaviour of the traje tories will not onform to lassi al expe tations. As we saw in detail in hapter 2, de Broglie did indeed originally regard his theory as a radi al departure from the prin iples of lassi al dynami s. It was Bohm's later revival of de Broglie's theory, in an unnatural pseudo-Newtonian form, that led to the widespread and mistaken per eption that de Broglie-Bohm theory onstituted a return to lassi al physi s. In more re ent years, de Broglie's original pilot-wave dynami s has again be ome re ognised as a new form of dynami s in its own right (Dürr, Goldstein and Zanghì 1992, Valentini 1992).
12 Beyond the Bohr-Einstein debate
The fth Solvay onferen e is usually remembered for the lash that took pla e between Bohr and Einstein, supposedly on erning in parti ular the possibility of breaking the un ertainty relations. It might be assumed that this lash took the form of an o ial debate that was the entrepie e of the onferen e. However, no re ord of any su h debate appears in the published pro eedings, where both Bohr and Einstein are in fa t relatively silent. The available eviden e shows that in 1927 the famous ex hanges between Bohr and Einstein a tually onsisted of informal dis ussions, whi h took pla e semi-privately (mainly over breakfast and dinner), and whi h were overheard by just a few of the parti ipants, in parti ular Heisenberg and Ehrenfest. The histori al sour es for this onsist, in fa t, entirely of a
ounts given by Bohr, Heisenberg and Ehrenfest. These a
ounts essentially ignore the extensive formal dis ussions appearing in the published pro eedings. As a result of relying on these sour es, the per eption of the onferen e by posterity has been skewed on two ounts. First, at the fth Solvay onferen e there o
urred mu h more that was memorable and important besides the Bohr-Einstein lash. Se ond, as shown in detail by Howard (1990), the real nature of Einstein's obje tions was in fa t misunderstood by Bohr, Heisenberg and Ehrenfest: for Einstein's main target was not the un ertainty relations, but what he saw as the nonseparability of quantum theory. Below we shall indi ate how these misunderstandings arose, summarise what now appear to have been Einstein's true on erns, and end by urging physi ists, philosophers and historians to re onsider what a tually took pla e in Brussels in O tober 1927, bearing in mind the deep questions that we still fa e on erning the nature of quantum physi s. 268
12.1 The standard histori al a
ount
269
12.1 The standard histori al a
ount A
ording to Heisenberg, the dis ussions that took pla e at the fth Solvay onferen e ` ontributed extraordinarily to the lari ation of the physi al foundations of the quantum theory' and indeed led to `the outward ompletion of the quantum theory, whi h now an be applied without worries as a theory losed in itself' (Heisenberg 1929, p. 495). For Heisenberg, and perhaps for others in the Copenhagen-Göttingen
amp, the 1927 onferen e seems to have played a key role in nalising the interpretation of the theory. However, the per eption that the interpretation had been nalised proved to be mistaken. As we have shown at length in hapter 5, the interpretation of quantum theory is today still an open question, and deep on erns as to its meaning have stubbornly persisted. Further, as we have seen throughout part II of this book, many of today's fundamental on erns were voi ed (often at onsiderable length) at the fth Solvay onferen e. Given that these on erns are still very mu h alive, it has evidently been a mistake to allow re olle tions of private dis ussions between Bohr and Einstein to overshadow our histori al memory of the rest of the onferen e. Note that, as we saw in hapter 1 (p. 16), while Mehra (1975, p. 152) and also Mehra and Re henberg (2000, p. 246) state that the general dis ussion was a dis ussion following `Bohr's report', in fa t Bohr did not present a report at the onferen e, nor was he invited to give one. This misunderstanding seems to have arisen be ause, at Bohr's request, a translation of his Como le ture appears in the published pro eedings, to repla e his remarks in the general dis ussion; and this has no doubt
ontributed to the ommon view that Bohr played a entral role at the onferen e, when in fa t it is lear from the pro eedings that at the o ial meetings both he and Einstein played a rather marginal role.a Further, it seems rather lear that the standard (and unbalan ed) version of events was propagated in parti ular by Bohr and Heisenberg, espe ially through their writings de ades later. There is very little independent eviden e from the time as to what was said between Bohr and Einstein. Thus, for example, Mehra and Re henberg (2000, p. 251) note the `little eviden e of the Bohr-Einstein debate in the o ial onferen e do uments', and rely on eyewitness reports by Ehrenfest, Heisenberg and Bohr to yield `a fairly onsistent a Sin e Bohr had not been invited to give a report, one might also question the propriety of his request that his remarks in the general dis ussion be repla ed by a translation of a rather lengthy paper he was already publishing elsewhere.
270
Beyond the Bohr-Einstein debate
histori al pi ture of the great epistemologi al debate between Bohr and Einstein' (p. 256). One frequently- ited pie e of ontemporary eviden e is a des ription of the onferen e written by Ehrenfest a few days later, in a letter to his students and asso iates in Leiden. This letter is ited at length by Mehra and Re henberg (2000, pp. 2513). An extra t reads: Bohr towering ompletely over everybody. .... step by step defeating everybody.
....
It was delightful for me to be present during the onversations
between Bohr and Einstein.
....
Einstein all the time with new examples.
In a ertain sense a sort of Perpetuum Mobile of the se ond kind to break the UNCERTAINTY RELATION. Bohr .... onstantly sear hing for the tools to rush one example after the other. Einstein .... jumping out fresh every morning. .... I am almost without reservation pro Bohr and ontra Einstein.
This letter has often been taken as representative of the onferen e. However, there is a marked ontrast with the published pro eedings (in whi h Bohr and Einstein are mostly silent), a ontrast whi h has not been taken into a
ount.a After examining the published pro eedings, Mehra and Re henberg (2000, pp. 25056) go on to onsider at greater length re olle tions by Heisenberg and Bohr written de ades after the onferen e
on erning the dis ussions between Bohr and Einstein. With hindsight, again given that the interpretation of quantum theory is today an open question, it would be desirable to have a more balan ed view of the
onferen e, fo ussing more on the ontent of the published pro eedings, and rather less on these later re olle tions by just two of the parti ipants. Here is an extra t from Heisenberg's re olle tion, written some 40 years later (Heisenberg 1967, p. 107): The dis ussions were soon fo ussed upon a duel between Einstein and Bohr .... . We generally met already at breakfast in the hotel, and Einstein began to des ribe an ideal experiment in whi h he thought the inner ontradi tions of the Copenhagen interpretation were espe ially learly visible. Einstein, Bohr and I walked together from the hotel to the onferen e building, and I listened to the lively dis ussion between those two people whose philosophi al attitudes were so dierent, .... at lun h time the dis ussions ontinued between Bohr and the others from Copenhagen. Bohr had usually nished the omplete analysis of the ideal experiment by late afternoon and would show it to Einstein at the supper table. Einstein had no good obje tion to this analysis, but in his heart he was not onvin ed.
a We have attempted to ompare Ehrenfest's ontemporary a
ount with that in letters written by other parti ipants soon after the onferen e, but have found nothing signi ant.
12.1 The standard histori al a
ount
271
From this a
ount, the Bohr-Einstein lash appears indeed to have been a private dis ussion, with a few of Bohr's lose asso iates in attendan e. And yet, there has been a marked tenden y to portray this dis ussion as the entrepie e of the whole onferen e. Thus, for example, in the prefa e to Mehra's book The Solvay Conferen es on Physi s, Heisenberg wrote the following about the 1927 onferen e (Mehra 1975, pp. vvi): Therefore the dis ussions at the 1927 Solvay Conferen e, from the very beginning, entred around the paradoxa of quantum theory. .... Einstein therefore suggested spe ial experimental arrangements for whi h, in his opinion, the un ertainty relations ould be evaded. But the analysis arried out by Bohr and others during the Conferen e revealed errors in Einstein's arguments. In this situation, by means of intensive dis ussions, the Conferen e ontributed dire tly to the lari ation of the quantum-theoreti al paradoxa.
Heisenberg says nothing at all about the alternative theories of de Broglie and S hrödinger, or about the views of Lorentz or Dira (for example), or about the other extensive dis ussions re orded in the pro eedings. As for the text of Mehra's book on the Solvay onferen es, the hapter devoted to the fth Solvay onferen e ontains a summary of the general dis ussion, whi h says nothing about the published dis ussions beyond providing a list of the parti ipants. Mehra's summary states that a debate took pla e between Bohr and Einstein, and that the famous Bohr-Einstein dialogue began here in 1927. Mehra then adds an appendix, reprodu ing Bohr's famous essay `Dis ussion with Einstein on epistemologi al problems in atomi physi s' (Bohr 1949), written more than 20 years after the onferen e took pla e. On e again, the published dis ussions are made to appear rather insigni ant ompared to (Bohr's re olle tion of) the informal dis ussions between Bohr and Einstein. The above essay by Bohr is in fa t the prin ipal and most detailed histori al sour e for the Bohr-Einstein debate. This essay was Bohr's
ontribution to the 1949 fests hrift for Einstein's seventieth birthday. It is reprinted as the very rst paper in Wheeler and Zurek's (1983) in uential olle tion Quantum Theory and Measurement, as well as elsewhere. It gives a detailed a
ount of Bohr's dis ussions with Einstein at the fth Solvay onferen e (as well as at the sixth Solvay onferen e of 1930). A
ording to Bohr, the dis ussions in 1927 entred around, among other things, a version of the double-slit experiment, in whi h a
ording to Einstein it was possible to observe interferen e while at the same time dedu ing whi h path the parti le had taken, a laim
on lusively refuted by Bohr. Were Bohr's re olle tions a
urate? Jammer ertainly thought so:
272
Beyond the Bohr-Einstein debate
Bohr's masterly report of his dis ussions with Einstein on this issue, though written more than 20 years after they had taken pla e, is undoubtedly a reliable sour e for the history of this episode. (Jammer 1974, p. 120)
Though as Jammer himself adds (p. 120): `It is, however, most deplorable that additional do umentary material on the Bohr-Einstein debate is extremely s anty'. We now know that, as we shall now dis uss, Bohr's re olle tion of his dis ussions with Einstein did not properly apture Einstein's true intentions, essentially be ause, at the time, no one understood what Einstein's prin ipal on ern was: the nonseparability of quantum theory.
12.2 Towards a histori al revision Separability the requirement that the joint state of a omposite of spatially separated systems should be determined by the states of the
omponent parts was a ondition basi to Einstein's eld-theoreti view of physi s (as indeed was the absen e of a tion at a distan e). As already mentioned in se tion 9.2, Howard (1990) has shown in great detail how Einstein's on erns about the failure of separability in quantum theory date ba k to long before the famous EPR paper of 1935.a There is no doubt that, by 1909, Einstein understood that if light quanta were treated like the spatially independent mole ules of an ideal gas, then the resulting u tuations were in onsistent with Plan k's formula for bla kbody radiation; and ertainly, in 1925, Einstein was on erned that Bose-Einstein statisti s entailed a mysterious interdependen e of photons. Also in 1925, as we dis ussed in se tion 9.2, Einstein's theory of guiding elds in 3-spa e in whi h spatially separated systems ea h had their own guiding wave, in a
ordan e with Einstein's separability
riterion on i ted with energy-momentum onservation for single events. Howard (pp. 8391) argues further that, in spring 1927, Einstein must have realised that S hrödinger's wave me hani s in on guration spa e violated separability, be ause a general solution to the S hrödinger equation for a omposite system ould not be written as a produ t over the omponents.b In other words, Einstein obje ted to what we would now all entanglement, and on luded that the wave fun tion in
on guration spa e ould not represent anything physi al. a See also Fine (1986, hapter 3). b Howard's argument here is somewhat ir umstantial, appealing in part to the di ulty with separability that Einstein had with his own hidden-variables amendment of wave me hani s ( f. se tion 11.3).
12.2 Towards a histori al revision
273
Einstein's on erns about separability ontinued up to and beyond the fth Solvay onferen e. While it seems that Einstein did have some early doubts about the validity of the un ertainty relations, Howard's re onstru tion shows that Einstein's main on ern lay elsewhere. The primary aim of the famous thought experiments that Einstein dis ussed with Bohr, in 1927 and subsequently, was not to defeat the un ertainty relations but to highlight the (for him disturbing) feature of quantum theory, that spatially separated systems annot be treated independently. As Howard (p. 94) puts it, regarding the 1927 Solvay onferen e: But if the un ertainty relations
really
were the main sti king point
for Einstein,
why did Einstein not say so in the published version of his remarks, or anywhere else for that matter in orresponden e or in print in the weeks and months following the Solvay meeting?
We have indeed seen in hapter 7 that Einstein's riti ism in the general dis ussion on erned lo ality and ompleteness (just like the later EPR argument), not the un ertainty relations. Bohr, in his reply, states that he does not `understand what pre isely is the point' Einstein is making. It seems rather lear that, indeed, in 1927 Bohr did not understand Einstein's point, and it is remarkable that what is most often re alled about the fth Solvay onferen e was in fa t largely a misunderstanding. A
ording to Bohr's later re olle tions (Bohr 1949), at the fth Solvay
onferen e Einstein proposed a version of the two-slit experiment in whi h measurement of the transverse re oil of a s reen with a single slit would enable one to dedu e the path of a parti le through a se ond s reen with two slits, while at the same time observing interferen e on the far side of the se ond s reen. (Consideration of this experiment must have taken pla e informally, not in the o ial dis ussions.) This experiment has been analysed in detail by Wootters and Zurek (1979), who show the ru ial role played by quantum nonseparability between the parti le and the rst s reen. While the eviden e is somewhat sket hy in this parti ular instan e, a
ording to Howard the main point that on erned Einstein in this experiment was pre isely su h nonseparability. That separability was indeed Einstein's entral on ern is learer in the later `photon-box' thought experiment he dis ussed with Bohr at the sixth Solvay onferen e of 1930, involving weighing a box from whi h a photon es apes. Again, Bohr dis usses this experiment at length in his re olle tions, where a
ording to him it was yet another of Einstein's attempts to ir umvent the un ertainty relations. Spe i ally,
274
Beyond the Bohr-Einstein debate
a
ording to Bohr, Einstein's intention was to beat the energy-time un ertainty relation, by measuring both the energy of the emitted photon (by weighing the box before and after) and its time of emission (given by a lo k ontrolling the shutter releasing the photon). On Bohr's a
ount, this attempt failed, ironi ally, be ause of the time dilation in a gravitational eld implied by Einstein's own general theory of relativity. However, it seems that in fa t, the photon-box experiment was (like Einstein's published obje tion of 1927) really a form of the later EPR argument for in ompleteness. This is shown by a letter Ehrenfest wrote to Bohr on 9 July 1931, just after Ehrenfest had visited Einstein in Berlin (Howard 1990, pp. 989). Ehrenfest reports that Einstein said he did not invent the photon-box experiment to defeat the un ertainty relations (whi h he had for a long time no longer doubted), but `for a totally dierent purpose'. Ehrenfest then explains that Einstein's real intention was to onstru t an example in whi h the hoi e of measurement at one lo ation would enable an experimenter to predi t either one or the other of two in ompatible quantities for a system that was far away at the time of the measurement. In the example at hand, if the es aped photon is re e ted ba k towards the box after having travelled a great distan e, then the time of its return may be predi ted with ertainty if the experimenter he ks the lo k reading (while the photon is still far away); alternatively, the energy (or frequen y) of the returning photon may be predi ted with ertainty if, instead, the experimenter hooses to weigh the box (again while the photon is still far away). Be ause the two possible operations take pla e while the photon is at a great distan e, the assumptions of separability and lo ality imply that both the time and energy of the returning photon are in reality determined in advan e (even if in pra ti e an experimenter annot arry out both predi tions simultaneously), leading to the on lusion that quantum theory is in omplete.a The true thrust of Einstein's argument was not appre iated at the a The reasoning here is similar to that in Einstein's own (and simpler) version of the EPR argument, whi h rst appears in a letter from Einstein to S hrödinger of 19 June 1935, one month after the EPR paper was published (Fine 1986, hap. 3). The argument whi h Einstein repeated and re ned between 1936 and 1949 runs essentially as follows. A omplete theory should asso iate one and only one theoreti al state with ea h real state of a system; in an EPR-type experiment on orrelated systems, depending on what measurement is arried out at one wing of the experiment, quantum theory asso iates dierent wave fun tions with what must (assuming lo ality) be the same real state at the other distant wing. Therefore, quantum theory is in omplete. For a detailed dis ussion, see Howard (1990).
12.2 Towards a histori al revision
275
time, perhaps be ause Bohr and his asso iates tended to identify the existen e of physi al quantities with their experimental measurability: if two quantities ould not be measured simultaneously in the same experiment, they did not exist simultaneously in the same experiment. With this attitude in mind, it would be natural to mistake Einstein's laim of simultaneous existen e for a laim of simultaneous measurability. We feel that Howard's reappraisal of the Bohr-Einstein debate, as well as being of great intrinsi interest, also provides an instru tive example of how the history of quantum physi s should be re onsidered in the light of our modern understanding of quantum theory and its open problems. There was ertainly mu h more to the fth Solvay onferen e than the Bohr-Einstein lash, and a similar reappraisal of other ru ial en ounters at that time seems overdue. If the history of quantum theory is written on the assumption that Bohr, Heisenberg and Born were right, and that de Broglie, S hrödinger and Einstein were wrong, the resulting a
ount is likely to be unsatisfa tory: opposing views will tend to be misunderstood or underestimated, supporting views over-emphasised, and valid alternative approa hes ignored. A re onsideration of the fth Solvay onferen e ertainly entails a reevaluation of de Broglie's pilot-wave theory as a oherent but (until very re ently) `forgotten' formulation of quantum theory. S hrödinger's ideas, too, seem more plausible today, in the light of modern ollapse models. One should also re onsider what Born and Heisenberg's `quantum me hani s' really was, in parti ular as on erns the role of time and the
ollapse of the wave fun tion. There is no longer a de nitive, widely-a
epted interpretation of quantum me hani s; it is no longer lear who was right and who was wrong in O tober 1927. Therefore, it seems parti ularly important at this time to return to the histori al sour es and re-evaluate them. We hope that physi ists, philosophers and historians will re onsider the signi an e of the fth Solvay onferen e, both for the history of physi s and for
urrent resear h in the foundations of quantum theory.
Part III The pro eedings of the 1927 Solvay
onferen e
H. A. Lorentz †
Hardly a few months have gone by sin e the meeting of the fth physi s
onferen e in Brussels, and now I must, in the name of the s ienti
ommittee, re all here all that meant to the Solvay International Institute of Physi s he who was our hairman and the moving spirit of our meetings. The illustrious tea her and physi ist, H. A. Lorentz, was taken away in February 1928 by a sudden illness, when we had just admired, on e again, his magni ent intelle tual gifts whi h age was unable to diminish in the least. Professor Lorentz, of a simple and modest demeanour, nevertheless enjoyed an ex eptional authority, thanks to the ombination of rare qualities in a harmonious whole. Theoreti ian with profound views eminent tea her in the highest forms of instru tion and tirelessly devoted to this task fervent advo ate of all international s ienti
ollaboration he found, wherever he went, a grateful ir le of pupils, dis iples and those who arried on his work. Ernest Solvay had an unfailing appre iation of this moral and intelle tual for e, and it was on this that he relied to arry through a plan that was dear to him, that of serving S ien e by organising onferen es omposed of a limited number of physi ists, gathered together to dis uss subje ts where the need for new insights is felt with parti ular intensity. Thus was born the Solvay International Institute of Physi s, of whi h Ernest Solvay followed the beginnings with a tou hing on ern and to whi h Lorentz devoted a loyal and fruitful a tivity. All those who had the honour to be his ollaborators know what he was as hairman of these onferen es and of the preparatory meetings. His thorough knowledge of physi s gave him an overall view of the problems to be examined. His lear judgement, his fair and benevolent spirit guided the s ienti ommittee in the hoi e of the assistan e it was 279
280
H. A. Lorentz †
appropriate to all upon. When we then were gathered together at a onferen e, one ould only admire without reservations the mastery with whi h he ondu ted the hairmanship. His shining intelle t dominated the dis ussion and followed it also in the details, stimulating it or preventing it from drifting, making sure that all opinions ould be usefully expressed, bringing out the nal on lusion as far as possible. His perfe t knowledge of languages allowed him to interpret, with equal fa ility, the words uttered by ea h one. Our hairman appeared to us, in fa t, gifted with an invin ible youth, in his passion for s ienti truth and in the joy he had in omparing opinions, sometimes with a shrewd smile on his fa e, and even a little mis hievousness when onfronted with an unforeseen aspe t of the question. Respe t and ae tion went to him spontaneously, reating a ordial and friendly atmosphere, whi h fa ilitated the ommon work and in reased its e ien y. True reator of the theoreti al edi e that explains opti al and ele tromagneti phenomena by the ex hange of energy between ele trons
ontained in matter and radiation viewed in a
ordan e with Maxwell's theory, Lorentz retained a devotion to this lassi al theory. All the more remarkable is the exibility of mind with whi h he followed the dis on erting evolution of the quantum theory and of the new me hani s. The impetus that he gave to the Solvay institute will be a memory and an example for the s ienti ommittee. May this volume, faithful report of the work of the re ent physi s onferen e, be a tribute to the memory of he who, for the fth and last time, honoured the onferen e by his presen e and by his guidan e.
M. Curie
Fifth physi s onferen e
The fth of the physi s onferen es, provided for by arti le 10 of the statutes of the international institute of physi s founded by Ernest Solvay, held its sessions in Brussels on the premises of the institute from 24 to 29 O tober 1927. The following took part in the onferen e:
Lorentz †, of Haarlem, Chairman. Mrs P. Curie, of Paris; Messrs N. Bohr, of Copenhagen; M. Born, of Göttingen; W. L. Bragg, of Man hester; L. Brillouin, of Paris; A. H. Compton, of Chi ago; L.-V. de Broglie, of Paris; P. Debye, of Leipzig; P. A. M. Dira , of Cambridge; P. Ehrenfest, of Leiden; A. Einstein, of Berlin; R. H. Fowler, of Cambridge; Ch.-E. Guye, of Geneva; W. Heisenberg, of Copenhagen; M. Knudsen, of Copenhagen; H. A. Kramers, of Utre ht; P. Langevin, of Paris; W. Pauli, of Hamburg; M. Plan k, of Berlin; O. W. Ri hardson, of London; [E. S hrödinger, of Zuri h;℄ C. T. R. Wilson, of Cambridge, MemMr H. A.
bers.
Vers haffelt, of Gent, ful lled the duties of Se retary. Messrs Th. De Donder, E. Henriot and Aug. Pi
ard, professors at the University of Brussels, attended the meetings of the onferen e as guests of the s ienti ommittee, Mr Ed. Herzen, professor at the
Mr J.-E.
É ole des Hautes Études de Bruxelles, as representative of the Solvay family.
281
282
Fifth physi s onferen e
Langmuir
Professor I. , of S hene tady (U. S. of Ameri a), visiting Europe, attended the meetings as a guest.
Aubel
Mr Edm. van , member of the S ienti Committee, and Mr H. , dire tor of the Meudon observatory, invited to parti ipate in the onferen e meetings, had been ex used.
Deslandres
Bragg
Sir W. H. , member of the s ienti ommittee, who had handed in his resignation before the meetings and requested to be ex used, also did not attend the sessions. The administrative ommission of the institute was omposed of:
Bordet
Messrs Jules , professor at the University of Brussels, appointed by H. M. the King of the Belgians; Armand , engineer, manager of Solvay and Co.; Mauri e , professor at the University of Brussels; Émile , professor at the University of Brussels; Ch. , engineer, appointed by the family of Mr Ernest Solvay, Administrative Se retary.
Lefébure
Henriot
Bourquin
Solvay
The s ienti ommittee was omposed of:
Lorentz Knudsen Bragg
†, professor at the University of Leiden, ChairMessrs H. A. man; M. , professor at the University of Copenhagen, Se retary; W. H. , professor at the University of London, president of the Royal Institution; Mrs Pierre , professor at the Fa ulty of S ien es of Paris; Messrs A. ,a professor, in Berlin; CharlesEug. , professor at the University of Geneva; P. , professor at the Collège de Fran e, in Paris; O. W. , professor at the University of London; Edm. van , professor at the University of Gent.
Guye
Curie Einstein Aubel
Langevin Ri hardson
Bragg
Sir W. H. , resigning member, was repla ed by Mr B. professor at the University of Madrid.
Cabrera,
To repla e its late hairman, the s ienti ommittee hose Professor .
P. Langevin
a Chosen in repla ement of Mr H. Kamerlingh Onnes, de eased.
The intensity of X-ray re e tion
a
By Mr W. L. BRAGG 1. The lassi al treatment of x-ray diffra tion phenomena The earliest experiments on the dira tion of X-rays by rystals showed that the dire tions in whi h the rays were dira ted were governed by the
lassi al laws of opti s. Laue's original paper on the dira tion of white1 radiation by a rystal, and the work whi h my father and I initiated on the re e tion of lines2 in the X-ray spe trum, were alike based on the laws of opti s whi h hold for the dira tion grating. The high a
ura y whi h has been developed by Siegbahn and others in the realm of X-ray spe tros opy is the best eviden e of the truth of these laws. Advan e in a
ura y has shown the ne essity of taking into a
ount the very small refra tion of X-rays by the rystal, but this refra tion is also determined by the lassi al laws and provides no ex eption3 to the above statement. The rst attempts at rystal analysis showed further that the strength of the dira ted beam was related to the stru ture of the rystal in a way to be expe ted by the opti al analogy. This has been the basis of most work on the analysis of rystal stru ture. When mono hromati X-rays are re e ted from a set of rystal planes, the orders of re e tion are strong, weak, or absent in a way whi h an be a
ounted for qualitatively a We follow Bragg's original English types ript, from the opy in the Ri hardson
olle tion, AHQP-RDN, do ument M-0059 (indexed as `unidenti ed author' in the mi ro lmed atalogue). Obvious typos are orre ted mostly ta itly and some of the spelling has been harmonised with that used in the rest of the volume. Dis repan ies between the original English and the published Fren h are endnoted (eds.).
283
284
W. L. Bragg
by the arrangement of atoms4 parallel to these planes. In the analysis of many stru tures, it is not ne essary to make a stri t examination of the strength of the dira ted beams. Slight displa ements of the atoms
ause the intensities of the higher orders to u tuate so rapidly, that it is possible to x the atomi positions with high a
ura y by using a rough estimate of the relative intensity of the dierent orders. When we atta k the problem of developing an a
urate quantitative theory of intensity of dira tion, many di ulties present themselves. These di ulties are so great, and the interpretation of the experimental results has often been so un ertain, that it has led5 to a natural distrust of dedu tions drawn from intensity measurements. Investigators of
rystal stru tures have relied on qualitative methods,6 sin e these were in many ases quite adequate. The development of the quantitative analysis has always interested me personally, parti ularly as a means of atta king the more ompli ated rystalline stru tures, and it would seem that at the present time the te hnique has rea hed a stage when we an rely on the results. It is my purpose in this paper to attempt a riti al survey of the present development of the subje t. It is of
onsiderable interest be ause it is our most dire t way of analysing atomi and mole ular stru ture. In any X-ray examination of a rystalline body, what we a tually measure is a series of samples7 of the oherent radiation s attered in
ertain de nite dire tions by the unit of the stru ture. This unit is, in general, the element of pattern of the rystal, while in ertain simple
ases it may be a single atom. In the examination of a small body by the mi ros ope, the obje tive re eives the radiation s attered in dierent dire tions by the body, and the information about its stru ture, whi h we get by viewing the nal image, is ontained at an earlier stage8 in these s attered beams. Though the two ases of mi ros opi and X-ray examination are so similar, there are ertain important dieren es. The s attered beams in the mi ros ope an be ombined again to form an image, and in the formation of the image the phase relationship between beams s attered in dierent dire tions plays an essential part. In the X-ray problem, sin e we an only measure the intensity of s attering in ea h dire tion, this phase relationship annot be determined experimentally, though in many ases it an be inferred.9 Further, the mi ros ope re eives the s attered beams over a ontinuous range of dire tions, whereas the geometry of the rystalline stru ture limits our examination to ertain dire tions of s attering. Thus we annot form dire tly an image of the
The intensity of X-ray re e tion
285
rystalline unit whi h is being illuminated by X-rays. We an only measure experimentally the strength of the s attered beams, and then build up an image pie e by pie e from the information we have obtained. It is important to note that in the ase of X-ray examination all work is being arried out at what is very nearly the theoreti al limit of the resolving power of our instruments. The range of wavelength whi h it is onvenient to use lies between 0.6 Å and 1.5 Å. This range is of su iently small wavelength for work with the details of rystal stru ture, whi h is always on a s ale of several Ångström units, but the wavelengths are in onveniently great for an examination into atomi stru ture. It is unfortunate from a pra ti al point of view that there is no onvenient steady sour e of radiation between the K lines of the metal palladium, and the very mu h shorter K lines of tungsten. This di ulty will no doubt be over ome, and a te hnique of `ultraviolet' X-ray mi ros opy will be developed, but at present all the a
urate work on intensity of re e tion has been done with wavelengths in the neighbourhood of 0.7 Å. We may onveniently10 divide the pro ess of analysis into three stages. a) The experimental measurement of the intensities of the dira ted beams. b) The redu tion of these observations, with the aid of theoreti al formulae, to measurements of the amplitudes of the waves s attered by a single unit of the stru ture, when a wave train of given amplitude falls on it.
) The building up of the image, or dedu tion of the form of the unit, from these measurements of s attering in dierent dire tions.
2. History of the use of quantitative methods The fundamental prin iples of a mathemati al analysis of X-ray re e tion were given in Laue's original paper [1℄, but the pre ise treatment of intensity of re e tion may be said to have been initiated by Darwin [2℄ with two papers in the Philosophi al Magazine early in 1914, in whi h he laid down the basis for a omplete theory of X-ray re e tion based on the lassi al laws of ele trodynami s.11 The very fundamental and independent treatment of the whole problem by Ewald [3℄, along
286
W. L. Bragg
quite dierent lines, has on rmed Darwin's on lusions in all essentials. These papers established the following important points. 1. Two formulae for the intensity of X-ray re e tion an be dedu ed, depending on the assumptions whi h are made. The rst of these has sin e ome to be known as the formula for the `ideally imperfe t rystal' or `mosai rystal'.a It holds for a rystal in whi h the homogeneous blo ks are so small that the redu tion in intensity of a ray passing through ea h blo k, and being partly re e ted by it, is wholly a
ounted for by the ordinary absorption oe ient. This ase is simple to treat from a mathemati al point of view, and in a tual fa t many rystals approa h this physi al ondition of a perfe t mosai . The se ond formula applies to re e tion by an ideally perfe t rystal. Here ordinary12 absorption plays no part in intensity of re e tion. This is perfe t over a nite range of glan ing angles, all radiation being re e ted within this range. The range depends on the e ien y of the atom planes in s attering. The se ond formula is entirely dierent from the rst, and leads to numeri al results of a dierent order of magnitude. 2. The a tual intensity of re e tion in the ase of ro ksalt is of the order to be expe ted from the imperfe t rystal formula. 3. The observed rapid de line in intensity of the high orders is only partly a
ounted for by the formula for re e tion, and must be due in addition to the spatial distribution of s attering matter in the atoms (ele tron distribution). 4. When a rystal is so perfe t that it is ne essary to allow for the intera tion of the separate planes, the transmitted beam is extinguished more rapidly than orresponds to the true absorption of the rystal (extin tion). 5. There exists a refra tive index for both rystalline and amorphous substan es, slightly less than unity, whi h auses small deviations from the law nλ = 2d sin θ. Another important fa tor in intensity of re e tion had been already examined theoreti ally by Debye [4℄, this being the diminution in intensia I believe we owe to Ewald the happy suggestion of the word `mosai '.
The intensity of X-ray re e tion
287
ty with rising temperature due to atomi movement. Though subsequent work has put Debye's and Darwin's formulae in modi ed and more
onvenient forms, the essential features were all ontained in these early papers. On the experimental side,the rst a
urate quantitative measurements were made by W. H. Bragg [5℄.13 The rystal was moved with onstant angular velo ity through the re e ting position, and the total amount of re e ted radiation measured. He showed that the re e tion14 from ro ksalt for a series of fa es lay on a smooth urve when plotted against the sine of the glan ing angle, emphasising that a de nite physi al onstant was being measured. This method of measurement has sin e been 15 widely used. The quantity Eω I , where E is the total energy of radiation re e ted, ω the angular velo ity of rotation, and I the total radiation falling on the rystal fa e per se ond, is independent of the experimental arrangements, and is a onstant for a given re e tion from a mosai
rystal; it is generally termed the `integrated re e tion'.16 It is related in a simple way to the energy measurements from a powdered rystal, whi h have also been employed for a
urate quantitative work. W. H. Bragg's original measurements were omparisons17 of this quantity for dierent fa es, not absolute measurements in whi h the strength of an in ident beam was onsidered. W. H. Bragg further demonstrated the existen e of the extin tion ee t predi ted by Darwin, by passing X-rays through a diamond rystal set for re e tion and obtaining an in reased absorption. He made measurements of the diminution in intensity of re e tion18 with rising temperature predi ted by Debye, and observed19 by Laue, and showed that the ee t was of the expe ted order. In the Bakerian Le ture in 1915 [6℄ he des ribed measurements in the intensity of a very perfe t
rystal, al ite, whi h seemed to show that the intensity was proportional to the s attering power of the atomi planes and not to the square of the power (this is to be expe ted from the formula for re e tion by a perfe t
rystal). In the same address he proposed the use of the Fourier method of interpreting the measurements20 whi h has been re ently used with su h su
ess by Duane, Havighurst, and Compton, and whi h is dealt with in the fourth se tion of this summary.21 At about the same time, Debye and Compton independently dis ussed the in uen e of ele troni distribution in the atom on the intensity of re e tion. The next step was made by Compton [7℄ in 1917. Darwin's formula for the mosai rystal was dedu ed by a dierent method, and was applied to the interpretation of W. H. Bragg's results with ro ksalt.
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W. L. Bragg
Compton on luded that the ele troni distribution in the atoms was of the type to be expe ted from Bohr's atomi model. Compton then published the rst measurements of the absolute intensity of ref1exion. A mono hromati beam of X-rays was obtained by re e tion from a
rystal, and this was re e ted by a se ond rotating rystal (ro ksalt and22 al ite). The absolute value of the integrated re e tion Eω I was found to be of the right order for ro ksalt when al ulated by the imperfe t rystal formula, but to be very low for al ite indi ating strong extin tion or a wrong formula, in the se ond ase. In 1921 and 1922 I published with James and Bosanquet a series of measurements on ro ksalt in whi h we tried to obtain a high a
ura y. We made absolute measurements of intensity for the strongest re e tions,a and ompared the weaker re e tions with them. Our main
ontributions in these papers were a more a
urate set of measurements of integrated re e tion for a large number of planes, and a method for estimating and orre ting for the ee t of extin tion. As Darwin showed in a paper in 1922 [9℄ on the theoreti al interpretation of our results, we only su
eeded in orre ting for extin tion of the kind he termed `se ondary' and not for `primary' extin tion.b Sin e then measurements by Havighurst [10℄, by Harris, Bates and M Innes23 [11℄ and by Bearden [12℄ have been made on the re e ting power of powdered sodium hloride when extin tion is absent. Their measurements have agreed with ours very losely indeed, on rming one's faith in intensity measurements, and showing that we were fortunate in hoosing a rystal for our examination where primary extin tion was very small. In the same papers we tried to make a areful analysis of the results in order to nd how mu h information about atomi stru ture ould be legitimately dedu ed from them, and we published urves showing the ele tron distribution in sodium and hlorine24 atoms. In this dis ussion, I have refrained from any referen e to the question of re e tion by `perfe t' rystals. The formula for re e tion by su h
rystals was rst obtained by Darwin, and has been arrived at independently by Ewald. The re e tion by su h rystals has been examined amongst others by Bergen Davis25 and Stempel [13℄, and by Mark [14℄ and predi tions of the theory have been veri ed. It is not onsidered a In our paper we failed to give due a knowledgement to Compton's absolute measurements in 1917 of whi h we were not aware at the time. b Primary extin tion is an ex essive absorption of the beam whi h is being re e ted in ea h homogeneous blo k of rystal, se ondary extin tion a statisti al ex essive absorption of the beam in the many small blo ks of a mosai rystal.
The intensity of X-ray re e tion
289
here, be ause I wish to on ne the dis ussion to those ases where a
omparison of the intensity of in ident and re e ted radiation leads to a
urate quantitative estimates of the distribution of s attering matter. This ideal an be attained with a tual rystals,26 when they are of the imperfe t or mosai type, though allowan e for extin tion is sometimes di ult in the ase of the stronger re e tions. On the other hand, it is far more di ult to know what one is measuring in the ase of rystals whi h approximate to the perfe t type. It is a fortunate ir umstan e that mathemati al formulae an be applied most easily to the type of imperfe t rystal more ommon in nature.
3. Results of quantitative analysis For the sake of on iseness, only one of the many intensity formulae will be given here, for it illustrates the essential features of them all. Let us suppose that the integrated re e tion is being measured when X-rays fall on the fa e of a rotating rystal of the mosai type. We then have ρ=
Q 1 + cos2 2θ . 2µ 2
(a) µ is the ee tive absorption oe ient, whi h may be greater than the normal oe ient, owing to the existen e of extin tion at the re e ting angle. 2
2θ is the `polarisation fa tor', whi h arises be ause (b) The fa tor 1+cos 2 the in ident rays are assumed to be unpolarised.
( ) Q=
N e2 F mc2
2
λ3 , sin 2θ
where e and m are the ele troni onstants,27 c the velo ity of light, λ the wavelength used, N the number of s attering units per unit volume, and θ the glan ing angle. (d) F is the quantity we are seeking to dedu e. It represents the s attering power of the rystal unit in the dire tion under onsideration, measured in terms of the s attering power of a single28 ele tron a
ording to the lassi al formula of J. J. Thomson. It is de ned by Ewald as the `Atomfaktor'29 when it applies to a single atom.
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W. L. Bragg
Formulae appli able to other experimental arrangements (the powder method for instan e) are very similar, and ontain the same quantity Q. Our measurements of re e tion thus lead to values of Q, and so of F , sin e all other quantities in the formulae are known. Measurements on a given rystal yield a series of values for F , and all the information that an be found out about this rystalline or atomi stru ture is represented by these values. They are the same for the same rystal whatever wavelength is employed (sin e F is a fun tion of sinλ θ ), though of ourse with shorter wavelength we have the advantage of measuring a mu h greater number of these oe ients (in reased resolving power). At this stage the ee t of the thermal agitation of the atom will be
onsidered as in uen ing the value of F . If we wish to make dedu tions about atomi stru ture, the thermal agitation must be taken into a
ount. Allowan e for it is a ompli ated matter, be ause not only do some atoms move more than others, but also they hange their relative mean positions as the temperature alters in the more omplex rystals. This will be dealt with more fully below. A series of examples will now be given to show that these quantitative formulae, when tested, lead to results whi h indi ate that the theory is on the right lines. It is perhaps more onvin ing to study the results obtained with very simple rystals, though I think that the su
ess of the theory in analysing highly omplex stru tures is also very strong eviden e, be ause we have overed su h a wide range of substan es. In the simple rystals, where the positions of the atoms are de nite, we an get the s attering power of individual atoms. The results should both indi ate the orre t number of ele trons in the atom, and should outline an atom of about the right size. When F is plotted against sinλ θ its value should tend to the number, N , of ele trons in the atom for small values of sinλ θ , and should fall away as sinλ θ in reases, at a rate whi h is reasonably explained by the spatial extension of the atom. In Fig. 1, the full lines give F urves obtained experimentally by various observers. The dotted lines are F urves al ulated for the generalised atomi model of Thomas [15℄, of appropriate atomi number. The Thomas atomi model, whi h has been shown for omparison, is most useful as it gives us the approximate ele troni distribution in an atom of any atomi number. Thomas al ulates an ideal distribution of ele trons in an atom of high atomi number. He assumes spheri al symmetry for the atom, and supposes that `ele trons are distributed uniformly in the six-dimensional phase-spa e for the motion of an ele tron, at the rate
The intensity of X-ray re e tion
291
Fig. 1. of two for ea h h3 of (six) volume'.30 He thus obtains an ideal ele tron atmosphere around the nu leus, the onstants of whi h an be simply adjusted31 so as to be suitable for any given nu lear harge. It is of ourse to be expe ted that the lower the atomi weight, the more the a tual distribution of s attering matter will depart from this arrangement, and will re e t the idiosyn rasies of the parti ular atom in question. The gure will show, however, that the a tual urves are very similar to those al ulated for Thomas' models. In parti ular, it will be lear that they tend to maximum values not far removed from the number of ele trons in the atom in ea h ase. The general agreement between the observed and al ulated F urves must mean that our measurements of F are outlining a pi ture of the atom. The agreement holds also for other atomi models than those of Thomas, whi h all lead to atoms with approximately the same spatial extension and ele troni distribution, as is well known. All these measurements of F ne essitate absolute values for the inte-
292
W. L. Bragg
grated re e tion. It is not ne essary to measure these dire tly in ea h
ase. When any one re e tion has been measured in absolute value (by
omparison of in ident and re e ted radiation), other rystals may be
ompared with it. The standard whi h has been used in every ase, as far as I am aware, is the ro ksalt rystal. Absolute measurements on this have been made by Compton [7℄, by Bragg, James and Bosanquet [8℄, and by Wasastjerna [18℄ whi h agree satisfa torily with ea h other.
4. Interpretation of measurements of F In interpreting these measurements of s attering power, we may either
al ulate the s attering of a proposed atomi model and ompare it with the observed F urve, or we may use the observations to al ulate the distribution of s attering matter dire tly. The latter method is the more attra tive, and in the hands of Duane, Havighurst, and Compton it has yielded highly interesting `images' of the atomi stru ture seen by X-rays. There is a lose analogy between the examination of a series of parallel planes by means of X-rays, and the examination of a dira tion grating, by a mi ros ope, whi h is onsidered in Abbe's theory of mi ros opi vision.a The obje tive of the mi ros ope may be
onsidered as re eiving a limited number of orders of spe tra from the grating. These spe tra in their turn build up the image viewed by the eyepie e, and the perfe tion of this image depends on the number of spe tra re eived. The strength of ea h spe tral order depends on the magnitude of the orresponding oe ient in that Fourier series whi h represents the amplitude of the light transmitted at ea h point of the grating. The extension of this well-known opti al prin iple to the X-ray eld was suggested by W. H. Bragg [6℄ in 1915. He had formed the
on lusion32 that the amplitudes of the s attered wave from ro ksalt were inversely proportional to the square of the order of re e tion, and he showed that33 `the periodi fun tion whi h represents the density of the medium must therefore be of the form34 const +
cos 4π xd cos 2nπ xd cos 2π xd + + ... + + ... ' 2 2 1 2 n2
and in this way built up a urve showing the periodi density of the ro ksalt grating. The method was not applied, however, to the mu h more a
urate measurements whi h are now available until re ently, a See for instan e the dis ussion of this theory and of A. B. Porter's experiments to illustrate it in Wood's Opti s, Chapter VIII.
The intensity of X-ray re e tion
293
when Duane and Havighurst showed how mu h ould be done with it. Duane independently arrived at a more general formula of the same type, giving the density of s attering matter at any point in the whole rystal as a triple Fourier series, whose oe ients depend on the intensity of re e tion from planes of all possible indi es. Havighurst applied this prin iple to our measurements of ro ksalt, and to measurements whi h he has made on other rystals, and obtained a pi ture of the relative density of s attering matter along ertain lines in these rystals. Compton made the further step of putting the formulae in a form whi h gives the absolute density of ele troni distribution (assuming the s attering to be by ele trons obeying the lassi al laws). Compton gives a very full dis ussion of the whole matter in his book X-rays and Ele trons.35 It is not only an extremely attra tive way of making lear just what has been a hieved by the X-ray analysis, but also the most dire t method of determining the stru ture. The formula for the distribution of s attering matter in parallel sheets, for a rystal with a entre of symmetry, is given by Compton as follows ∞
Pz =
Z 2X 2πnz + . Fn cos a a 1 a
Here z is measured perpendi ularly to the planes whi h are spa ed a distan e a apart. Pz dz is the amount Rof s attering matter between a planes at distan es z and z+dz , and Z (= 0 Pz dz ) is the total s attering matter of the rystal unit. This is a simpli ed form of Duane's formula for a Fourier series of whi h the general term is 2πn1 x 2πn2 y 2πn3 z An1 n2 n3 sin − δn1 sin − δn2 sin − δn3 , a1 a2 a3 An1 n2 n3 being proportional to the amplitude of the s attered wave from the plane (n1 n2 n3 ). Another Fourier series, due to Compton, gives the radial distribution of s attering matter, i.e. the values of Un where Un dr is the amount of s attering matter between radii r and r + dr Un =
∞ 2πnr 8πr X , nFn sin a2 1 a
where a is hosen so that values of F o
ur at onvenient intervals on the graph for F . If we know the values of F for a given atom over a su iently wide
294
W. L. Bragg
range, we an build up an image of the atom either as a `sheet distribution' parallel to a plane, or as a radial distribution of s attering matter around the nu leus. In using these methods of analysis, however, it is very ne essary to remember that we are working right at the limit of resolving power of our instruments, and in fa t are attempting a more ambitious problem than in the orresponding opti al ase. In A. B. Porter's experiments to test Abbe's theory, he viewed the image of a dira tion grating and removed any desired group of dira ted rays by
utting them o with a s reen. The rst order gives blurred lines, four or ve orders give sharper lines with a ne dark line down the entre, eight orders give two dark lines down the entre of ea h bright line and so forth. These imperfe t images are due to the absen es of the higher members in building up the Fourier series. In exa tly the same way we get false detail in our X-ray image, owing to ignoran e of the values of the higher members in the F urve. Similarly, the ne stru ture whi h a tually exists may be glossed over, sin e by using a wavelength of 0.7 Å, we annot hope to `resolve' details of atomi stru ture on a s ale of less than half this value. The ignoran e of the values of higher members of the Fourier series matters mu h less in the urve of sheet distribution than in that for radial distribution, sin e the latter onverges far more slowly. Examples of the Fourier method of analysis are given in the next paragraph. As opposed to this method of building up an image from the X-ray results,36 we may make an atomi model and test it by al ulating an F
urve for it whi h an be ompared with that obtained experimentally. This is the most satisfa tory method of testing models arrived at by other lines of resear h, for nothing has to be assumed about the values of the higher oe ients F . It is of ourse again true that our test only applies to details of the proposed model on a s ale omparable with the wavelength we are using. Sin e we an re e t X-rays right ba k from an atomi plane, we may get a resolving power for a given wavelength with the X-ray method twi e as great as the best the mi ros ope an yield. It is perhaps worth mentioning the methods I used with James and Bosanquet in our determination of the ele troni distribution in sodium and hlorine in 1922. We tried to avoid extrapolations of the F urve beyond the limit of experimental investigation. We divided the atom arbitrarily into a set of shells, with an unknown number of ele trons in ea h shell. These unknowns were evaluated by making the s attering due to them t the F urve over the observed range, this being simply done by solving simultaneous linear equations. We found we got mu h
The intensity of X-ray re e tion
295
Fig. 2. the same type of distribution however the shells were hosen, and that a limit to the ele troni distribution at a radius of about 1.1 Å in sodium and 1.8 Å in hlorine was learly indi ated. Our distribution orresponds in its general outline to that found by the mu h more dire t Fourier analysis, as the examples in paragraph 7 will show.
5. Examples of analysis
We owe to Duane [20℄ the appre iation of the very attra tive way in whi h the Fourier analysis represents the results of X-ray examinations. It has the great merit of representing, in the form of a single urve, the information yielded by all orders of re e tion from a given plane, or from the whole rystal. It is of ourse only an alternative way of
296
W. L. Bragg
interpreting the results, and the dedu tions we an make about atomi or mole ular stru tures depend in the end on the extent to whi h we an trust our experimental observations, and not on the method of analysis we use. The Fourier method is so dire t however, and its signi an e so easy to grasp, that Duane's introdu tion of it marks a great advan e in te hnique of analysis. I have reserved to paragraph 7 the more di ult problem of the arrangement of s attering matter in the atoms themselves, and the examples given here are of a simpler hara ter. They illustrate the appli ation of analysis to the general problem of the distribution of s attering matter in the whole rystal, when we are not so near the limit of resolving power. The urves in Fig. 2 represent the rst appli ation of the new method of Fourier analysis to a
urate data, arried out by Havighurst [21℄ in 1925. He used our determinations of F for sodium
hloride, and Duane's three-dimensional Fourier series, and al ulated the density of s attering matter along a ube edge through sodium and
hlorine entres, along a ube diagonal through the same atoms, and along two fa e diagonals hosen so as to pass through hlorine atoms alone or sodium atoms alone in the rystal. The atoms show as peaks in the density distribution. In the other examples, the formula for distribution in sheets has been applied to some results we have obtained in our work on rystal stru ture at Man hester. I have given them be ause I feel they are onvin ing eviden e of the power of quantitative measurements, and show that all methods of interpretations lead to the same results. Mr West and I [22℄ re ently analysed the hexagonal rystal beryl, Be3 A12 Si6 O18 ,38 whi h has a stru ture of some omplexity, depending on seven parameters. We obtained the atomi positions by the usual method of analysis, using more or less known F urves for the atoms in the rystal, and moving them about till we explained the observed F s due to the rystal unit. Fig. 3 shows the reinterpretations of this result by the Fourier method. Fig. 3a gives the ele tron density in sheets perpendi ular to the prin ipal axis of the rystal, whi h is of a very simple type. The parti ular point to be noted is the orresponden e between the position of the line B in the gure and the hump of the Fourier analysis. The line B marks the position of a group of oxygen atoms whi h lies between two other groups A and C xed by symmetry, the position of B being xed by a parameter found by familiar methods of rystal analysis. The hump represents the same group xed by the Fourier analysis, and it will be seen how losely they orrespond. In
The intensity of X-ray re e tion
297
Fig. 3a. Distribution of ele trons in sheets parallel to 0001.37
Fig. 3b. Distribution of ele trons in sheets parallel to 10¯10.39 Figs. 3b and 3 more omplex sets of planes are shown. The dotted
urve represents the interpretation of our results by Fourier analysis. The full urve is got by adding together the humps due to the separate atoms shown below, the position of these having been obtained by our X-ray analysis and their sizes by the aid of the urve in Fig. 3a in whi h the ontribution of the atoms an be separated out. The orresponden e
298
W. L. Bragg
Fig. 3 . Distribution of ele trons in sheets parallel to 11¯20.42
between the two shows that the older methods and the Fourier analysis agree. It is to be noted that the rystal had rst to be analysed by the older methods, in order that the sizes of the Fourier oe ients might be known. In Fig. 4 I have given a set of urves for the alums, re ently analysed by Professor Cork [23℄. The alums are ompli ated ubi rystals with su h formulae as KAl(SO4 )2 .12 H2 O. Wy ko40 has shown that the potassium and aluminium atoms41 o
upy the same positions in the ubi ell as the sodium atom in ro ksalt. Now we an repla e the potassium by ammonium, rubidium, aesium, or thallium, and the aluminium by
hromium, or other trivalent metals. Though the positions of the other atoms in the rystals are not yet known, they will presumably be mu h the same in all these rystals. If we represent by a Fourier series the quantitative measurements of the alums, we would expe t the density of s attering matter to vary from rystal to rystal at the points o
upied
The intensity of X-ray re e tion
299
Fig. 4. by the metal atoms, but to remain onstant elsewhere. The urves show this in the most interesting43 way. The ee t of heat motion on the movements of the atoms has already been mentioned. It was rst treated theoreti ally by Debye [4℄. Re ently Waller [24℄ has re al ulated Debye's formula, and has arrived at a modi ed form of it. Debye found that the intensities of the interferen e maxima in a simple rystal should be multiplied by a fa tor e−M , where M= x=
6h2 ϕ(x) sin2 θ , µkΘ x λ2 Θ characteristic temperature of crystal = . T absolute temperature
Without going into further detail, it is su ient to note that Waller's
300
W. L. Bragg
Fig. 5. formula diers from Debye's by making the fa tor e−2M , not e−M . James and Miss Firth [25℄ have re ently arried out a series of measurements for ro ksalt between the temperatures 86◦ abs. and 900◦ abs. They nd that Waller's formula is very losely followed up to 500◦ abs., though at higher temperatures the de line in intensity is even more rapid, as is perhaps to be expe ted owing to the rystal be oming more loosely bound. I have given the results of the measurements in Figs. 5 and 6, both as an example of the type of information whi h an be got from X-ray measurements, and be ause these a tual gures are of interest as a set of areful and a
urate measurements of s attering power. Fig. 5 shows the F urves for sodium and hlorine at dierent tem-
The intensity of X-ray re e tion
301
Fig. 6.
peratures. The rapid de line in intensity for the higher orders will be realised when it is remembered that they are proportional to F 2 . The
urve for absolute zero is an extrapolation from the others, following the Debye formula as modi ed by Waller. In Fig. 6 the same results are interpreted by the Fourier analysis. The urve at room temperatures for NaCl is pra ti ally identi al with the interpretation of our earlier gures by Compton, in his book X-rays and Ele trons,44 though the gures on whi h it is based should be more a
urate.45 The urves show the manner in whi h the sharply de ned peaks due to Cl and Na at low temperatures be ome diuse owing to heat motion at the higher temperatures. Several interesting points arise in onne tion with this analysis. In the rst pla e, James and Firth46 nd that the heat fa tor is dierent for sodium and hlorine, the sodium atoms moving with greater average amplitudes than the hlorine atoms. This has a very interesting bearing on the rystal dynami s whi h is being further investigated by Waller. To a rst approximation both atoms are ae ted equally by the elasti waves
302
W. L. Bragg
Fig. 7. travelling through the rystal, but in a further approximation it an be seen that the sodium atoms are more loosely bound than the hlorine atoms. If an atom of either kind were only xed in position by the six atoms immediately surrounding it, Waller has shown that there would be no dieren e between the motions of a sodium atom between six
hlorine atoms, or a hlorine atom between six sodium atoms. However, the hlorine is more rmly pinned in position be ause it has in addition twelve large hlorine neighbours, whereas the sodium atom is mu h less in uen ed by the twelve nearest sodium atoms. Hen e arises the dieren e in their heat motions. It is important to nd the orre t method for redu ing observations to absolute zero, and this dieren e in heat motion must be satisfa torily analysed before this is possible. In the se ond pla e, the a
ura y whi h an be attained by the experimental measurements holds out some hope that we may be able to test dire tly whether there is zero-point energy47 or not. This is being investigated by James and Waller. If a reliable atomi model is available, it would seem that the measurements an tell whether there is vibration at absolute zero or not, for the theoreti al diminution in intensity due to
The intensity of X-ray re e tion
303
the vibration is mu h larger than the experimental error in measuring F . I feel onsiderable di den e in speaking of the question of zero-point energy, and would like to have the advi e of the mathemati al physi ists present. We may al ulate, either from the measured heat fa tor or dire tly from the Fourier analyses, the average amplitude of vibration for dierent temperatures. James and Firth nd by both methods, for instan e, that at room temperature the mean amplitude of vibration for both atoms is 0.21 Å, and at 900◦ abs. it is about 0.58 Å. They examined the form whi h the Fourier urve at 0◦ abs. assumes when it is deformed by supposing all the atoms to be in vibration with the same mean amplitude. It has been already remarked that the observed F urves for atoms are very similar to those al ulated for the Thomas atomi model. The same
omparison may be made between the distributions of s attering matter. In Fig. 7 the distribution in sheets for NaCl at absolute zero is shown as a full urve. The dotted urve shows the horizontal distribution in sheets for atoms of atomi number 17 and 11. In Thomas' model the density rises towards an in nite value very lose to the nu leus, and this is represented by the very sharp peaks at the atomi entres in the dotted
urve. We would not expe t the observed distribution to orrespond to the a tual Thomas distribution at these points. Throughout the rest of the rystal the distribution is very similar. The omparison is interesting, be ause it shows how deli ate a matter it is to get the ne detail of atomi stru ture from the observations. Thomas' distribution is quite ontinuous and takes no a
ount of K, L and M sets of ele trons. The slight departures of the observed urve from the smooth Thomas
urve represent the experimental eviden e for the existen e of all the individual features of the atom.
6. The me hanism of X-ray s attering Before going on to dis uss the appli ation of the analysis to atomi stru ture, it is ne essary to onsider what is being measured when a distribution of s attering matter is dedu ed from the X-ray results. The
lassi al treatment regards the atom as ontaining a number of ele trons, ea h of whi h s atters radiation a
ording to the formula of J. J. Thomson. Sin e a vast number of atoms ontribute to the re e tion by a single rystal plane, we should obtain a pi ture of the average ele troni distribution. The quantity F should thus tend to a maximum value, at
304
W. L. Bragg
small angles of s attering, equal to the number of ele trons in the atom, and should fall away owing to their spatial distribution as sinλ θ in reases. The observed48 F urves are of this hara ter, as has been seen. When interpreted as an atomi distribution, they give atoms ontaining the
orre t number of ele trons, and this seems satisfa tory from the lassi al viewpoint. On the other hand, the eviden e of the Compton ee t would appear at rst sight to ast doubt on the whole of our analysis. What we are measuring is essentially the oherent radiation dira ted by the
rystal, whereas the Compton ee t shows that a part of the radiation whi h is s attered is of dierent wavelength. Further, this radiation of dierent wavelength is in luded with the oherent radiation, when the total amount of s attered radiation is measured, and found to agree under suitable onditions with the amount predi ted by J. J. Thomson's formula. It would therefore seem wrong to assume that we obtain a true pi ture of ele troni distribution by the aid of measurements on the
oherent radiation alone. Even before the advent of the new me hani s, Compton's original treatment of the ee t whi h he dis overed suggested a way out of this di ulty. The re oil ele tron is given an amount of energy 2 h2 ν ′ sin θ 2 , m ν λ where ν and ν ′ are the frequen ies of the modi ed and unmodi ed radiations. If the ele tron is eje ted from the atom the radiation is modi ed in wavelength, if not oherent waves are s attered. Sin e there is little modi ed s attering at small angles, the F urve will tend to a maximum equal to the number of ele trons in the atom, and any interpretation of the urve will give an atom ontaining the orre t number of ele trons. As sinλ θ in reases, more and more of the s attered radiation will be modi ed, and in al ulating the F urve this must be ′ taken into a
ount. However, if νν is not far from unity, the F urve will remain a fun tion of sinλ θ , sin e whatever riterion is applied for the s attering of modi ed or unmodi ed radiation, it will depend on the energy imparted to the s attering ele tron, whi h is itself a fun tion of sin θ λ . Our X-ray analysis would thus give us an untrue pi ture of the atom, but one whi h is onsistently the same whatever wavelength is employed. Williams [27℄ and Jaun ey [28℄ have re al ulated F urves from atomi models using this riterion, and found a better t to the experimental urves when the Compton ee t was taken into a
ount. (Examples of this loser approximation will be found in the paper by
The intensity of X-ray re e tion
305
Fig. 8. Williams [27℄ in 1926. See also a dis ussion by Kallmann and Mark [26℄).49 The point at issue is illustrated by the urves in Fig. 8. Three F
urves for hlorine are plotted in the gure. The dotted line represents the observed F urve (James and Firth). The ontinuous line is the F
urve al ulated from Hartree's [29℄ atomi model for hlorine. It shows a hump at a value of sin θ of 0.4, whi h is not present in the observed
urve. This hump arises from the fa t that the outer ele trons in the
hlorine model give negative values for F just short of this point,50 and
306
W. L. Bragg
positive values again at the point itself. All atomi models al ulated with ele troni orbits show similar irregularities whi h are not a tually observed. When, however, the Compton ee t is taken into onsideration, these outer ele trons are found to give a very small ontribution to the F urve at the large angles where the humps51 o
ur, be ause they s atter so mu h modi ed radiation. The allowan e for the Compton ee t smooths out the hump, and leads to F urves mu h more like those observed. The third urve shows the F urve due to the ontinuous Thomas distribution and is a lose t to the observed urve. I have quoted from a note by Dr Ivar Waller, in the following tentative summary of the interpretation whi h the new me hani s gives us of this phenomenon.a In a re ent letter to Nature [30℄, Waller dis usses the transition for the whole range from ordinary dispersion into Compton ee t. His note only refers to s attering by a single ele tron, but it an probably be extended to many-ele tron atoms. Waves of ontinually de reasing wavelength are supposed to fall upon the atom, and the transition is tra ed through the following stages. a) While the wavelength of the radiation remains long ompared with atomi dimensions, the dispersion formula for opti al frequen ies gradually transforms into the s attering for free ele trons given by the lassi al J. J. Thomson formula. This formula holds approximately to52 wavelengths approa hing atomi dimensions. b) At this point the s attering of oherent radiation will diminish, owing to interferen e, and be ome more on entrated in the forward dire tion of the in ident light. This is the phenomenon we are studying, with Xrays, and our F urves map out the distribution of the oherent radiation where the wavelength is of atomi dimensions.
) At the same time, the s attering of in oherent radiation will be ome appre iable, and approximate more and more losely in hange of wavelength and intensity distribution to the Compton ee t. It will have pra ti ally merged into the Compton ee t when the momentum of a quantum of the in ident light is large ompared with that orresponding to ele troni motions in the atom. d) Up to this point the Thomson formula holds for the total intensity a Spa e forbids a referen e to the many theoreti al papers whi h have ontributed towards this interpretation.
The intensity of X-ray re e tion
307
of light s attered in any dire tion, oherent and in oherent radiation being summed together. It rst eases to hold, when the frequen y displa ement due to the Compton ee t is no longer small ompared with the frequen y of the in ident light. The point of importan e for our present problem is that `the oherent part of the radiation is to be dire tly al ulated from that ontinuous distribution of ele tri ity whi h is de ned by the S hrödinger densitydistribution in the initial state of the atom'. The lassi al treatment supposes ea h point ele tron to s atter a
ording to the J. J. Thomson formula in all dire tions. In the new treatment, the ele tron is repla ed by a spatial distribution of s attering matter, and so ea h ele tron has an `F urve' of its own. It will still s atter oherent radiation in all dire tions, but its amount will fall away from that given by the lassi al formula owing to interferen e as sinλ θ in reases, and this de line will be mu h more rapid for the more diuse outer ele trons than for the
on entrated inner ele trons. The total amount of radiation T s attered in any dire tion by the ele tron is given by the Thomson formula. A fra tion f 2 T will be oherent, and will be al ulated by the laws of interferen e from the S hrödinger distribution, and the remainder, (1 − f 2 )T , will be in oherent. Thus the total oherent radiation will be F 2 T where F is al ulated from the S hrödinger distribution for the P whole atom. An amount (N − f 2 )T will be s attered with hange of wavelength. Our measurements of X-ray dira tion, if this be true,
an be trusted to measure the S hrödinger ontinuous distribution of ele tri ity in the rystal latti e. A very interesting point arises in the ase where hara teristi absorption frequen ies of the s attering atom are of shorter wavelength than the radiation whi h is being s attered. In general, this has not been so when areful intensity measurements have been made sin e atoms of low atomi weight have alone been investigated. On the lassi al analogy, we would expe t a reversal in phase of the s attered radiation, when an ele tron has a hara teristi frequen y greater than that of the in ident light. A fas inating53 experiment by Mark and Szilard [31℄ has shown that something very like this takes pla e. They investigated the (111) and (333) re e tions of RbBr, whi h are extremely weak be ause Rb and Br oppose ea h other and are nearly equal in atomi number. They found that these `forbidden' re e tions were indeed absent when the soft CuK or hard BaK radiation was used, but that SrK radiation was appre iably re e ted (SrKα λ 0.871 Å;54 absorption edges of RbK
308
W. L. Bragg
and BrK , 0.814 Å and 0.918 Å). The atoms are dierentiated be ause a reversal of phase in s attering by the K ele trons takes pla e in the one
ase and not in the other.
7. The analysis of atomi stru ture by X-ray intensity measurements It has been seen that the intensity measurements assign the orre t number of ele trons to ea h atom in a rystal, and indi ate a spatial extension of the atoms of the right order. In attempting to make the further step of dedu ing the arrangement of the ele trons in the atom, the limitations of the method begin to be very apparent. In all ases where analysis has been attempted, the atom has been treated as spheri ally symmetri al. The analysis is used to determine the amount of s attering matter Un dr between radii r and r + dr. All methods of analysis give a distribution of the same general type. I have given, for instan e, a series of analyses of sodium and hlorine in Fig. 9. In these gures, Un is plotted as ordinate against r as abs issa. The total area of the urve R ∞in ea h ase is equal to the number of ele trons in ea h atom, sin e 0 Un dr = N . The full-line urves are our original interpretations of the distribution in sodium and hlorine, based on our 1921 gures.55 The other urves are the interpretations of the same or losely similar sets of gures57 by Havighurst [32℄ and by Compton (X-rays and Ele trons) using the Fourier method of analysis. In Fig. 9a are in luded our analysis of sodium in NaCl, two analyses by Havighurst of sodium in NaCl and NaF obtained by using Duane's triple Fourier series, and an analysis of our gures58 by Compton using the Fourier formula for radial distribution. It will be seen that the general distribution of s attering matter and the limits of the atom are approximately the same in ea h ase. The same holds for the hlorine
urves in Fig. 9b. The interesting point whi h is raised is the reality of the humps whi h are shown by the Fourier analysis. We obtained similar humps in our analysis by means of shells but doubted their reality be ause we found that if we smoothed them out and re al ulated the F urve, it agreed with the observed urve within the limits of experimental error. The te hnique of measurement has greatly improved sin e then, and it would even appear from later results that we over-estimated the possible errors of our rst determinations of F . It is obvious, however, that great are must still be taken in basing on lusions on the ner details shown by any
The intensity of X-ray re e tion
309
Fig. 9.56 method of analysis. The formula whi h is used in the Fourier analysis, Un =
∞ 2πnr 4πr X 2nFn sin , a 1 a a
is one whi h onverges very slowly, sin e the su
essive oe ients Fn are multiplied by n. The observed F urve must be extrapolated to a point when F is supposed to fall to zero, and the pre ise form of the
urve rea ts very sensitively to the way in whi h this extrapolation is
arried out.
310
W. L. Bragg
Fig. 10. The urves in Fig. 10 will illustrate the extent to whi h the analysis
an be onsidered to give us information about the a tual atomi distribution. In Fig. 10a the urve shows the F values for uorine obtained by James and Randall [17℄. The ir les are points obtained59 by Havighurst from measurements on CaF, LiF, NaF;60 it will be seen that the two sets of experimental data are in very satisfa tory agreement. In Fig. 10b I have shown on the one hand Havighurst's interpretations of the F urve drawn through his points, and on the other an analysis arried out by
The intensity of X-ray re e tion
311
Fig. 11.62 Claassen [16℄ of James and Randall's using the Fourier method. The distributions are the same in their main outlines, but the peaks o
ur in quite dierent pla es. Compton (X-rays and Ele trons, p. 167) in dis ussing his diagrams of radial distribution has remarked that slight dieren es in the F urves lead to wide dieren es in details of the urves, and that too mu h
on den e should not be pla ed on these details. Havighurst [32℄ dis usses the signi an e of the analysis very fully in his paper on ele tron distribution in the atoms. Our data are not yet su iently a
urate or extensive. Nevertheless, we are so near to attaining an a
ura y of a
312
W. L. Bragg
satisfa tory order, and the results of the analysis seem to indi ate so
learly its fundamental orre tness, that it appears to be well worth while to pursue enquiry further. Work with shorter wavelengths, and at low temperatures, when heat motion is small and a large range of F values an be measured, should yield us a
urate pi tures of the atomi stru ture itself. Given a
urate data,61 the Fourier method of analysis provides a dire t way of utilising them. The radial distribution of s attering power outlined in this way is in general agreement with any reasonable atomi model. We have seen, in parti ular, that the F urves, and therefore the radial distributions, of Thomas' model63 are in approximate a
ord with those a tually observed. If it is true that the s attering of oherent radiation is to be
al ulated in all ases by the S hrödinger density distribution, we should test our model against this distribution. An interesting attempt along these lines has been re ently made by Pauling [33℄. He has used ertain simplifying assumptions to obtain an approximate S hrödinger density-distribution for many-ele tron atoms. I have shown in Fig. 11 four sets of urves. The radial ele tron distributions dedu ed by Havighurst and by Compton are shown as one urve sin e they are very similar. The gure shows also our rst analysis of ele tron distribution. Mat hed against these are plotted the generalised distribution of the Thomas model, and the S hrödinger density distribution al ulated by Pauling. We have obviously not yet rea hed a point when we an be satis ed with the agreement between theory and experiment, yet the su
ess attained so far is a distin t en ouragement to further investigation.
8. The refra tion of X-rays At Professor Lorentz's64 suggestion I have added a very brief note on the refra tion of X-rays, sin e the phenomenon is so intimately onne ted with the question of intensity of re e tion and s attering, and is another example of the su
essful appli ation of lassi al laws. The dira tion phenomenon dealt with above (intensity of re e tion) arises from the s attering of oherent radiation in all dire tions by the atoms of a rystal. The refra tive index may be onsidered as being due to the s attering in the forward dire tion of oherent radiation, whi h interferes with the primary beam. The arrangement of the s attering matter plays no part, so that the body may be rystalline or amorphous. The measurement of
The intensity of X-ray re e tion
313
the refra tive index is thus a dire t measure of the amount of oherent radiation s attered in the forward dire tion of the in ident beam. 1. Darwin [2℄ appears to have been rst in pointing out that theory assigns a refra tive index for X-rays diering from unity by about one part in a million. He predi ted that a very slight departure from the law of re e tion nλ = 2d sin θ0
would be found, the a tual angle θ being given by Darwin's formula θ − θ0 =
1−µ . sin θ cos θ
Ewald's [34℄ independent treatment of X-ray re e tion leads to an equivalent result, though the problem is approa hed along quite dierent lines. As is well known, the rst experimental eviden e of an index of refra tion was found in a departure from the re e tion laws. Stenström [35℄ A) as observed dieren es in the apparent wavelength of soft X-rays (3 ˚ measured in the dierent orders, whi h were explained by Ewald's laws of X-ray re e tion. The in reased a
ura y of X-ray spe tros opy has shown that similar deviations from the simple law of re e tion exist for harder rays, though the deviations are mu h smaller than in the ordinary X-ray region.65 Thus the deviations have been dete ted for hard rays by Duane and Patterson [36℄ and by Siegbahn and Hjalmar [37℄. It is di ult to measure the refra tive index by means of these deviations in the ordinary way, sin e they are so small, but Davis [38, 39℄ developed a very ingenious way of greatly in reasing the ee t. A rystal is ground so that the rays re e ted by the atomi planes enter or leave a fa e at a very ne glan ing angle, and thus suer a omparatively great de e tion. Compton [40℄ dis overed the total re e tion of X-rays, and measured the index of refra tion in this way. The refra tive index is slightly less than unity, hen e X-rays falling at a very ne glan ing angle on a plane surfa e of a body are totally re e ted, none of the radiation passing into the body. Compton showed that, although the refra tive index is so nearly unity, yet the riti al glan ing angle is quite appre iable. Finally, the dire t ee t of refra tion by a prism has been observed by Larsson, Siegbahn and Waller [41℄. X-rays entered one fa e of a glass prism at a very ne glan ing angle, and suered a measurable de e tion. They obtained in this way a dispersion spe trum of X-rays.
314
W. L. Bragg
2. In all ases where the frequen y of the X-radiation is great ompared with any frequen y hara teristi of the atom, the refra tive index measured by any of these methods is in lose a
ord66 with the formula ne2 , 2πmν 2 where n is the number of ele trons per unit volume in the body, e and m are the ele troni onstants, and ν the frequen y of the in ident radiation. The formula follows dire tly from the lassi al Drude-Lorentz theory of dispersion, in the limiting ase where the frequen y of the radiation is large ompared with the `free periods' of the ele trons in the atom. It an be put in the form [42℄67 1−µ=
1 − µ = 2.71 × 10−6
ρZ 2 λ , A
where λ is the wavelength in Ångström units of the in ident radiation, ρ the density of the substan e, Z and A the average atomi number and atomi weight of its onstituents (for all light atoms Z/A is very nearly 0.5).68 Expressed in this form, the order of magnitude of 1 − µ is easily grasped. The riti al glan ing angle θ for total re e tion is given by cos θ = µ ,
when e θ=
r
ne2 . πmν 2
Expressing θ in minutes of ar , and λ in Ångström units as before, r ρZ . θ = 8.0 λ A Measurements of refra tive index have been made by Compton and by Doan using the method of total re e tion, by Davis, Hatley and Nardro using re e tion in a rystal, and by Larsson, Siegbahn and Waller with a prism. A variety of substan es has been examined, and wavelengths between 0.5 and 2 Å have been used. The a
ura y of the experimental determination of 1 − µ is of the order of one to ve per ent. As long as the riti al frequen ies of the atom have not been approa hed, the results have agreed with the above formula within experimental error. Just as in the measurements of intensity of re e tion the F urves approa h a limit at small angles equal to the number of ele trons in the atom, so these measurements of refra tive index when interpreted by lassi al theory lead to a very a
urate numbering of the ele trons in the s attering units.
The intensity of X-ray re e tion
315
3. A highly interesting eld is opened up by the measurements of refra tive index for wavelengths in the neighbourhood of a riti al frequen y of the atom. It is a striking fa t that the simple dispersion formula n
µ−1=
e 2 X ns 2πm 1 νs2 − ν 2
still gives values for the refra tive index agreeing with experiment in this region, ex ept when the riti al frequen y is very losely approa hed indeed. Davis and von Nardro re e ted CuKα and CuKβ X-rays69 from iron pyrites, and found that the refra tive indi es ould be reprodu ed by substituting onstants in the formulae orresponding to two K ele trons in iron with the frequen y of the K absorption edge.70 R. L. Doan [44℄ has re ently made a series of measurements by the total re e tion method. His a
urate data support the on lusion that the Drude-Lorentz theory of dispersion represents the fa ts, `not only in regions remote from the absorption edge,71 but also in some instan es in whi h the radiation approa hes the natural frequen ies of ertain groups of ele trons'. The existen e of two K ele trons72 is very de nitely indi ated. Kallmann and Mark [43℄ have gone more deeply into the form of the dispersion urve in the neighbourhood of the riti al frequen ies. The hange in s attering power of an atom as the frequen y of the s attered radiation passes through a riti al value is of ourse another aspe t of this anomalous dispersion; the experiment of Mark and Szilard whi h showed this ee t has been des ribed above. There is ample eviden e that measurements of refra tive index will in future prove to be a most fruitful means of investigating the response of the atom to in ident radiation of frequen y very near ea h of its own hara teristi frequen ies.
316
W. L. Bragg
Referen esa [1℄ [W. Friedri h, P. Knipping and℄ M. v. Laue, Bayr. Akad. d. Wiss. Math. phys. Kl. (1912), 303. [2℄ C. G. Darwin, Phil. Mag., 27 (1914), 315, 675. [3℄ P. P. Ewald, Ann. d. Phys., 54 ([1917℄), 519. [4℄ P. Debye, Ann. d. Phys., 43 (1914), 49. [5℄ W. H. Bragg, Phil. Mag., 27 (1914), 881. [6℄ W. H. Bragg, Phil. Trans. Roy. So . [A℄, 215 (1915), 253. [7℄ A. H. Compton, Phys. Rev., 9 (1917), 29; 10 (1917), 95. [8℄ W. L. Bragg, R. W. James and C. H. Bosanquet, Phil. Mag., 41 (1921), 309; 42 (1921), 1; 44 (1922), 433. [9℄ C. G. Darwin, Phil. Mag., 43 (1922), 800. [10℄ R. J. Havighurst, Phys. Rev., 28 (1926), n. 5, 869 and 882. [11℄ L. Harris, S. J. Bates and D. A. Ma Innes,73 Phys. Rev., 28 (1926), 235. [12℄ J. A. Bearden, Phys. Rev., 27 (1926), 796; 29 (1927), 20. [13℄ Bergen Davis and W. M. Stempel, Phys. Rev., [17℄ (1921), 608. [14℄ H. Mark, Naturwiss., 13 (1925), n. 49/50, 1042. [15℄ L. H. Thomas, Pro . Camb. Phil. So ., 23 (1927), 542. [16℄ A. Claassen, Pro . Phys. So . London, 38 [pt℄ 5 (1926), 482. [17℄ R. W. James and J. T. Randall, Phil. Mag. [(7)℄, 1 (1926), 1202. [18℄ J. A. Wasastjerna, Comm. Fenn., 2 (1925), 15. [19℄ W. L. Bragg, C. G. Darwin and R. W. James, Phil. Mag. (7), 1 (1926), 897. [20℄ W. Duane, Pro . Nat. A ad. S i., 11 (1925), 489. [21℄ R. J. Havighurst, Pro . Nat. A ad. S i., 11 (1925), 502. [22℄ W. L. Bragg and J. West, Roy. So . Pro . A, 111 (1926), 691. [23℄ [J. M.℄ Cork, Phil. Mag. [(7), 4 (1927), 688℄. [24℄ I. Waller, Upsala Univ. Årsskr. 1925, [11℄; Ann. d. Phys., 83 (1927), 153. [25℄ R. W. James and E. Firth, Roy. So . Pro . [A, 117 (1927), 62℄. [26℄ [H.℄ Kallmann and H. Mark, Zeit. f. Phys., 26 (1926), [n.℄ 2, [120℄. [27℄ E. J. Williams, Phil. Mag. [(7)℄, 2 (1926), 657. [28℄ [G. E. M.℄ Jaun ey, Phys. Rev., 29 (1927), 605. [29℄ D. R. Hartree, Phil. Mag., 50 (1925), 289. [30℄ I. Waller, Nature, [120℄ (July 1927), [155℄. [31℄ H. Mark and L. Szilard, Zeit. f. Phys., 33 (1925), 688. a The style of the referen es has been modernised and uniformised (eds.).
The intensity of X-ray re e tion
317
[32℄ R. J. Havighurst, Phys. Rev., 29 (1927), 1. [33℄ L. Pauling, Roy. So . Pro . A, 114 (1927), 181. [34℄ P. P. Ewald, Phys. Zeits h., 21 (1920), 617; Zeits hr. f. Physik, 2 (1920), 332. [35℄ W. Stenström, Exper[imentelle℄ Unters[u hungen℄ d[er℄ Röntgenspektra. Dissertation, Lund (1919). [36℄ [W.℄ Duane and [R. A.℄ Patterson, Phys. Rev., 16 (1920), [526℄.74 [37℄ M. Siegbahn, Spektroskopie der Röntgenstrahlen [(Berlin: Springer, 1924)℄. [38℄ [C. C. Hatley and Bergen Davis℄, Phys. Rev., 23 (1924), 290.75 [39℄ B[ergen℄ Davis and R. von Nardro, Phys. Rev., 23 (1924), 291. [40℄ A. H. Compton, Phil. Mag., 45 (1923), 1121. [41℄ [A.℄ Larsson, [M.℄ Siegbahn and [I.℄ Waller, Naturwiss., 12 ([1924℄), 1212. [42℄ I. Waller, [Theoretis he Studien zur℄ Interferenz- und Dispersionstheorie der Röntgenstrahlen. [Dissertation, Upsala (1925)℄. [43℄ H. Kallmann and H. Mark, Ann. d. Physik, 82 (1927), 585. [44℄ R. L. Doan, Phil. Mag. [(7), 4 n.℄ 20 (1927), [100℄. A very omplete a
ount of work on intensity of re e tion is given by Compton in X-rays and Ele trons and by Ewald in volume 24 of the Handbu h der Physik by H. Geiger and K. S heel,76 Aufbau der festen Materie und seine Erfors hung dur h Röntgenstrahlen, se tion 18.
318
W. L. Bragg
Dis ussion of Mr Bragg's report
Mr Debye. To what extent an you on lude that there exists an energy at absolute zero?
Mr Bragg.
Waller and James have re ently submitted a paper to the Royal So iety in whi h they dis uss the relation between the in uen e of temperature on the intensity of re e tion (Debye ee t) and the elasti onstants of a rystal. Using the experimentally determined value of the Debye oe ient, they dedu e the s attering by an atom at rest from the s attering by the atom at the temperature of liquid air (86◦ abs.). The urve dedu ed for the s attering by a perfe tly motionless atom an of ourse take two forms, a
ording to whether or not, in interpreting the results of the experiment, one assumes the existen e of an energy at absolute zero. If one assumes the existen e of su h an energy, the urve dedu ed from the experimental results agrees with that al ulated by Hartree by applying S hrödinger's me hani s. The agreement is really very good for sodium as well as for hlorine. On the other hand, the urve that one obtains if one does not assume any energy at absolute zero deviates
onsiderably from the al ulated urve by an amount that ex eeds the possible experimental error. If these experimental resultsa are on rmed by new experiments, they provide a dire t and onvin ing proof of the existen e of an energy at absolute zero.
Mr Debye.
Would the ee t not be larger if one did the experiments with diamond?
Mr Bragg.
In the ase of diamond, it is di ult to interpret the results obtained using a single rystal, be ause the stru ture is very perfe t and the `extin tion' is strong. One would have to work with diamond powder. But I annot say if it would be easy to nd that there exists an energy at absolute zero in diamond; I should onsider it further.
Mr Fowler. Here is how Hartree al ulates the atomi elds.
Starting from Thomas' atomi eld, taken as a rst approximation, he a Note added 5 April 1928. The results to whi h allusion is made here have just been published in detail by Messrs James, Waller and Hartree in a paper entitled: `An investigation into the existen e of zero-point energy in the ro k-salt latti e by an X-ray dira tion method' (Pro . Roy. So . A, 118 (1928), 334).
Dis ussion
319
al ulates the S hrödinger fun tions for an ele tron pla ed in this eld, then the density of harge in the atom orresponding to the S hrödinger fun tions, and then the orresponding atomi eld, whi h will dier from that of Thomas. By su
essive approximations one modi es the eld until the al ulations yield the eld whi h served as a starting point. This method gives very good values for the levels orresponding to X-rays and to visible light, and leads to the atom that Mr Bragg
onsidered for omparison with experiments.
Mr Heisenberg. How an you say that Hartree's method gives
exa t results, if it has not given any for the hydrogen atom? In the
ase of hydrogen the S hrödinger fun tions must be al ulated with the aid of his dierential equation, in whi h one introdu es only the ele tri potential due to the nu leus. One would not obtain orre t results if one added to this potential the one oming from a harge distribution by whi h one had repla ed the ele tron. One may then obtain exa t results only by taking the harge density of all the ele trons, ex ept the one whose motion one wishes to al ulate. Hartree's method is
ertainly very useful and I have no obje tion to it, but it is essentially an approximation.
Mr Fowler. I may add to what I have just said that Hartree is
always areful to leave out the eld of the ele tron itself in ea h state, so that, when he onsiders an L ele tron, for example, the entral part of the eld of the whole atom is diminished by the eld of an L ele tron, as far as this may be onsidered as entral. Hartree's method would then be entirely exa t for hydrogen and in fa t he has shown that it is extremely lose to being exa t for helium. (One nds a re ent theoreti al dis ussion of Hartree's method, by Gaunt, in Pro . Cambr. Phil. So ., 24 (1928), 328.)
Mr Pauli.
In my opinion one must not perform the al ulations, 2 as in wave me hani s, by onsidering a density |ψ(x, y, z)| in threedimensional spa e,77 but must onsider a density in several dimensions |ψ(x1 , y1 , z1 , ..., xN , yN , zN )|2 ,
whi h depends on the N parti les in the atom. For su iently short waves the intensity of oherent s attered radiation is then proportional
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W. L. Bragg
to78 Z
...
Z X N 1
e
2πi λ
→−n − →, r − → 2 (n− u d k) |ψ(x1 , ..., zN )| dx1 ... dzN ,
n→ where λ is the wavelength of the in ident radiation, − u a unit ve tor in − → the dire tion of propagation, and nd the orresponding unit ve tor for the s attered radiation; the sum must be taken over all the parti les. The result that one obtains by assuming a three-dimensional density
annot be rigorously exa t; it an only be so to a ertain degree of approximation.
Mr Lorentz.
How have you al ulated the s attering of radiation by a harge distributed over a region omparable to the volume o
upied by the atom?
Mr Bragg.
To interpret the results of observation as produ ed by an average distribution of the s attering material, we applied J. J. Thomson's lassi al formula for the amplitude of the wave s attered by a single ele tron.
Mr Compton.
If we assume that there is always a onstant ratio between the harge and mass of the ele tron, the result of the
lassi al al ulation of re e tion by a rystal is exa tly the same, whether the harge and mass are assumed on entrated in parti les (ele trons) or distributed irregularly in the atom. The intensity of re e tion is determined by the average density of the ele tri harge in dierent parts of the atom. That may be represented either by the probability that a point harge o
upies this region or by the volume density of an ele tri harge distributed in a ontinuous manner through this region.
Mr Kramers. The use that one may make of the simple Thomas
model of the atom in the sear h for the laws of re e tion is extremely interesting. It would perhaps not be super uous to investigate what result would be obtained for the ele tron distribution if, instead of restri ting oneself to onsidering a single entre of attra tion, one applied Thomas' dierential equation to an in nity of entres distributed as in a
rystal grating. Has anyone already tried to solve the problem of whi h Mr Bragg has just spoken, of the al ulation of the general distribution of the ele troni density around the nu leus of a heavy atom, in the ase
Dis ussion
321
where there are many nu lei, as in a rystal?
Mr Bragg.
No, no one has yet atta ked this problem, whi h I only mentioned be ause it is interesting.
Mr Dira .
Do the s attering urves depend on the phase relations between the os illations of dierent atoms?
Mr Bragg.
No, be ause the results of our experiments give only the average s attering produ ed in ea h dire tion by a very large number of atoms.
Mr Dira .
What would happen if you had two simple os illators performing harmoni vibrations? Would they produ e a dierent s attering when in phase than when out of phase?
Mr Born.
The orre t answer to the question of s attering by an atom is ontained in the remark by Mr Pauli. Stri tly speaking there is no three-dimensional harge distribution that may des ribe exa tly how an atom behaves; one always has to onsider the total on guration of all the ele trons in the spa e of 3n dimensions. A model in three dimensions only ever gives a more or less rude approximation.
Mr Kramers
asks a question on erning the in uen e of the Compton ee t on the s attering.
Mr Bragg.
I have already said something on that subje t in my report.a Assuming a model of the atom of the old type, Jaun ey and Williams have used the riterion that the wavelength is modi ed when the re oil of the s attering ele tron is su ient to take it entirely outside the atom. Williams was the rst to apply this riterion to s attering
urves obtained with rystals. He pointed out that while the speed of the re oil ele tron depends on both the s attering angle and the wavelength, any riterion one uses is a fun tion of sinλ θ , just as the interferen e ee ts depend on sinλ θ . This implies that the existen e of the Compton ee t modi es the s attering urve su h that we an always assign the same s attering urve to no matter what type of atom, whatever the a Cf. Bragg's report, se tion 6 (eds ).
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W. L. Bragg
wavelength may be.
Mr Fowler. If I have understood properly, Mr Bragg uses theoret-
i al al ulations by Waller that have not yet been published. When light is s attered by an atom in a
ordan e with the interpretation given by Mr Waller by means of the new me hani s, the total amount of s attered light is given exa tly by J. J. Thomson's lassi al formula (ex ept for very hard γ -rays). This light is omposed of the oherent s attered radiation and of the modi ed light (Compton s attering). In the theorem of the re e tion of X-rays only the oherent s attered light must be used, and indeed it is; and this light is given exa tly by the F urves like those proposed by Hartree. These F urves for atomi s attering are obviously given simply by the lassi al s attering for ea h ele tron, diminished by interferen e.
Mr Bragg.
I should like to develop Mr Fowler's remark by re alling Waller and Wentzel's on lusions brie y sket hed in my report. The s attering by one of the ele trons in an atom partly remains the same and partly is modi ed. Within ertain limits the total amount of s attered radiation is given by J. J. Thomson's formula. A fra tion f 2 of this amount is not modi ed, f being a oe ient smaller than 1, depending on the interferen e of the spatial distribution of the harge a
ording to S hrödinger and al ulated a
ording to the lassi al laws of opti s. The remaining fra tion 1 − f 2 is modi ed.79
Mr Lorentz.
It is, without doubt, extremely noteworthy that the total s attering, omposed of two parts of quite dierent origin, agrees with Thomson's formula.
Mr Kramers makes two remarks: 1. As Mr Bragg has pointed out the importan e of there being interest in having more experimental data on erning the refrangibility of X-rays in the neighbourhood of the absorption limit, I should like to draw attention to experiments performed re ently by Mr Prins in the laboratory of Professor Coster at Groningen. By means of his apparatus (the details of the experiments and the results obtained are des ribed in a paper published re ently in Zeits hrift für Physik, 47 (1928), [479℄), Mr Prins nds in a single test the angle of total re e tion orresponding to an extended region of frequen ies. In the region of the absorption
Dis ussion
323
limit of the metal, he nds an abnormal ee t, whi h onsists mainly of a strong de rease in the angle of total re e tion on the side of the absorption limit lo ated towards the short wavelengths. This ee t is easily explained taking into a
ount the in uen e of absorption on the total re e tion, without it being ne essary to enter into the question of the hange in refrangibility of the X-rays. In fa t, the absorption may be des ribed by onsidering the refra tive index n as a omplex number, whose imaginary part is related in a simple manner to the absorption
oe ient. Introdu ing this omplex value for n in the well-known formulas of Fresnel for the intensity of re e ted rays, one nds that the sharp limit of total re e tion disappears, and that the manner in whi h the intensity of re e ted rays depends on the angle of in iden e is su h that the experiment must give an `ee tive angle of total re e tion' that is smaller than in the ase where there is no absorption and that de reases as the absorption in reases. A
ording to the atomi theory one would also expe t to nd, in the region of the absorption limit, anomalies in the real part of the refra tive index, produ ing a similar though less noti eable de rease of the ee tive angle of total re e tion on the side of the absorption edge dire ted towards the large wavelengths. Mr Prins has not yet su
eeded in showing that the experiments really demonstrate this ee t.a The theory of these anomalies in the real part of the refra tive index
onstitutes the subje t of my se ond remark. 2. Let us onsider plane and polarised ele tromagneti waves, in whi h the ele tri for e an be represented by the real part of Ee2πiνt , striking an atom whi h for further simpli ity we shall assume to be isotropi . The waves make the atom behave like an os illating dipole, giving, by expansion in a Fourier series, a term with frequen y ν . Let us represent this term by the real part of P e2πiνt , where P is a omplex ve tor having the same dire tion as the ve tor E to whi h it is, moreover, proportional. If we set P = f + ig , (1) E where f and g are real fun tions of ν , the real and imaginary parts of the refra tive index of a sample of matter are related in a simple way to the fun tions f and g of the atoms ontained in the sample. Extending the domain of values that ν may take into the negative a Continuing his resear h Mr Prins has established (February 1928) the existen e of this ee t, in agreement with the theory.
324
W. L. Bragg
region and de ning f as an even fun tion of ν , g as an odd fun tion, one easily veri es that the dispersion formulas of Lorentz's lassi al theory and also those of modern quantum me hani s are equivalent to the formula Z 1 +∞ g(ν ′ ) ′ dν , f (ν) = − (2) π −∞ ν − ν ′ R where the sign − indi ates the `prin ipal' value of the integral. This formula an easily be applied to atoms showing ontinuous absorption regions and is equivalent to the formulas proposed for these
ases by R. de Laer Kronig and by Mark and Kallmann. There is hardly any doubt that this general formula may be derived from quantum me hani s, if one duly takes into a
ount the absorption of radiation, basing oneself on Dira 's theory, for example. From a mathemati al point of view, formula (2) gives us the means to onstru t an analyti fun tion of a omplex variable ν that is holomorphi below the real axis and whose real part takes the values g(ν ′ ) on this axis. If one onsiders ν as a real variable, the integral equation (2) has the solution Z 1 +∞ f (ν ′ ) ′ g(ν) = − − (3) dν , π −∞ ν − ν ′
whi h shows that the imaginary part of the refra tive index depends on the real part in nearly the same way as the real part depends on the imaginary part. The fa t that the analyti fun tion f of the omplex variable ν , de ned by (2) for the lower half of the omplex plane, has no singularity in this half-plane, means that dispersion phenomena, when one studies them by means of waves whose amplitude grows in an exponential manner (ν omplex), an never give rise to singular behaviour for the atoms.
Mr Compton. The measurements of refra tive indi es of X-rays
made by Doan agree better with the Drude-Lorentz formula than with the expression derived by Kronig based on the quantum theory of dispersion.
Mr de Broglie.
I should like to draw attention to re ent experiments arried out by Messrs J. Thibaud and A. Soltan,a whi h tou h on the questions raised by Mr Bragg. In these experiments Messrs a C. R. A ad. S ., 185 (1927), 642.
Dis ussion
325
Thibaud and Soltan measured, by the tangent grating method, the wavelength of a ertain number of X-rays in the domain 20 to 70 Å. Some of these wavelengths had already been determined by Mr Dauvillier using dira tion by fatty-a id gratings. Now, omparing the results of Dauvillier with those of Thibaud and Soltan, one noti es that there is a systemati dis repan y between them that in reases with wavelength. Thus for the Kα line of boron, Thibaud and Soltan nd 68 Å, while Dauvillier had found 73.5 Å, that is, a dieren e of 5.5 Å. This systemati dis repan y appears to be due to the in rease of the refra tive index with wavelength. The index does not a tually play a role in the tangent grating method, while it distorts in a systemati way the results obtained by rystalline dira tion when one uses the Bragg formula. Starting from the dieren e between their results and those of Mr Dauvillier, Messrs Thibaud and Soltan have al ulated the value of the refra tive index of fatty a ids around 70 Å and found δ = 1 − µ = 10−2
thereabouts. This agrees well with a law of the form δ = Kλ2 ; sin e in the ordinary X-ray domain the wavelengths are about 100 times smaller, δ is of order 10−6 . One ould obje t that, a
ording to the Drude-Lorentz law, the presen e of K dis ontinuities of oxygen, nitrogen and arbon between 30 and 45 Å should perturb the law in λ2 . But in the X-ray domain the validity of Drude's law is doubtful, and if one uses in its pla e the formula proposed by Kallmann and Marka the agreement with the experimental results is very good. Let us note nally that the existen e of an index appre iably dierent from 1 an ontribute to explaining why large-wavelength lines, obtained with a fatty-a id grating, are broad and spread out.
Mr Lorentz makes a remark on erning the refra tive index of a
rystal for Röntgen rays and the deviations from the Bragg law. It is
lear that, a
ording to the lassi al theory, the index must be less than unity, be ause the ele trons ontained in the atoms have eigenfrequen ies smaller than the frequen y of the rays, whi h gives rise to a speed of propagation greater than c. But in order to speak of this speed, one must adopt the ma ros opi point of view, abstra ting away the mole ular dis ontinuity. Now, if one wishes to explain Laue's phenomenon in all its details, one must onsider, for example, the a tion of the vibrations a Ann. d. Phys., 82 (1927), 585.
326
W. L. Bragg
ex ited in the parti les of a rystallographi layer on a parti le of a neighbouring layer. This gives rise to series that one annot repla e by integrals. It is for this reason that I found some di ulty in the explanation of deviations from the Bragg law.80
Mr Debye. Ewald has tried to do similar al ulations. Mr Lorentz. It is very interesting to note that with Röntgen
rays one nds, in the vi inity of an absorption edge, phenomena similar to those that in lassi al opti s are produ ed lose to an absorption band. There is, however, a profound dieren e between the two ases, the absorption edge not orresponding to a frequen y that really exists in the parti les.
Notes to pp. 283298
327
Notes to the translation 1 Here and in a few other pla es, the Fren h adds (or omits) inverted
ommas. 2 [ré exion des radiations des raies℄ 3 [ne fait prévoir au un é art℄ 4 The Fren h edition adds `en ou hes' [in layers℄. 5 Types ript: `have often been .... it has led'; Fren h version: `a souvent été .... elles ont onduit'. 6 [ont eu on an e dans les méthodes quantitatives℄ 7 [portions℄ 8 [sous une forme plus primitive℄ 9 [il soit possible de les trouver℄ 10 [logiquement℄ 11 [thermodynamique℄ 12 Word omitted in the Fren h version. 13 The Fren h edition adds `Sir'. 14 [les données obtenues par ré exion℄ 15 Here following the Fren h edition; the types ript reads `total radiation'. 16 Emphasis omitted in the Fren h edition. 17 [servirent à omparer℄ 18 The Fren h omits `of re e tion'. 19 [déjà observée℄ 20 [il proposa d'employer, pour l'interprétation des mesures, la méthode de Fourier℄ 21 [rapport℄ 22 [ou℄ 23 [M Innes℄ 24 [potassium℄ 25 The types ript has a spurious omma after `Bergen'. 26 [à l'aide de ristaux℄ 27 [les deux onstantes éle troniques℄ 28 Here and in some other instan es, the Fren h renders `single' as `simple'. 29 [`fa teur atomique'℄ 30 Not printed as a quotation in the Fren h edition. 31 hoisies simplement 32 The Fren h adds `de ses expérien es'. 33 The Fren h adds `dans es onditions'. 34 This is indeed a quotation from p. 272 from the le ture by W. H. Bragg. The types ript has a omma instead of the losing quotation mark, while the Fren h edition omits the opening quotation mark. The types ript has a spurious denominator `a' instead of `d' in the se ond and third terms (but ta itly orre ts another typo in the original). 35 [dans son livre sur `les rayons X et les éle trons'℄ 36 The Fren h adds `on peut pro éder de la façon inverse, 'est-à dire'. 37 `Beryl.' omitted in Fren h edition. 38 The Fren h edition uses supers ripts throughout. 39 The Fren h edition omits the overbar in the aption. 40 [Wy kho ℄ 41 Word omitted in Fren h edition. 42 Again, the Fren h edition omits the overbar in the aption.
328
Notes to pp. 301325
43 [frappante℄ 44 [son livre sur les rayons X et les éle trons℄
45 [bien que les gures (si ) sur lesquelles la nouvelle ourbe se base soient plus exa tes℄ 46 Here and in several other pla es, the Fren h adds `Mlle'.
47 The Fren h reads `une énergie au zéro absolu (énergie de stru ture)'. 48 Word missing in the Fren h edition. 49 Bra ket printed as a footnote in the Fren h edition. 50 [tout près de e point℄ 51 [irrégularités℄ 52 [pour℄ 53 [brillante℄
˚
54 [λ SrKα = 0.871A℄ 55 [faites d'après 1921 gures℄ 56 The Fren h omits `B. J. B. gures for NaCl'. 57 [ gures℄ 58 Again, in the Fren h, the false friend ` gures'. 59 [déduits℄ 60 [CaFl, LiFl, NaFl℄ 61 [Une fois que nous disposerons de données pré ises℄ 62 The Fren h edition omits `& Compton' and has `Modèle de Pauling et S hrödinger'. 63 The Fren h translates as if the omma were after `of Thomas' model' rather than before. 64 [M. Lorentz℄ 65 The types ript reads `mu h smaller in the ordinary X-ray region', but given the ontext the text should be amended as shown (as also done in the Fren h version). 66 [parfaitement d'a
ord℄ 67 Referen e omitted in the Fren h edition. Z 68 [ A la valeur moyenne du rapport du nombre atomique au poids atomique pour ses divers onstituants (pour tous les atomes légers e rapport est à peu près égal à 0.5℄ 69 [rayons℄ 70 Types ript: `of the K adsorption edge'; Fren h version: `de la dis ontinuité K'. 71 Types ript: `adsorption edge'; Fren h: `bord d'absorption'. 72 The Fren h adds `dans la pyrite'. 73 Fren h edition: `Ma Innes'. 74 Types ript and Fren h edition both have `532'. 75 Both types ript and Fren h edition give this referen e as `B. Davis and C. C. Hatley'. The types ript has `291'. 76 Authors added in the Fren h edition. 77 Here and in the following displayed formula, the published version has square bra kets instead of absolute bars. 78 Arrow missing on
→ − rk
in the published volume.
79 The original text mistakenly states that both fra tions are `not modi ed'. 80 The mixing of rst and third person, here and in a few similar instan es throughout the dis ussions, is as in the published text.
Disagreements between experiment and the ele tromagneti theory of radiation a
By Mr Arthur H. COMPTON Introdu tion Professor W. L. Bragg has just dis ussed a whole series of radiation phenomena in whi h the ele tromagneti theory is on rmed. He has even dwelt on some of the limiting ases, su h as the re e tion of X-rays by rystals, in whi h the ele tromagneti theory of radiation gives us, at least approximately, a orre t interpretation of the fa ts, although there are reasons to doubt that its predi tions are truly exa t. I have been left the task of pleading the opposing ause to that of the ele tromagneti theory of radiation, seen from the experimental viewpoint. I have to de lare from the outset that in playing this role of the a
user I have no intention of diminishing the importan e of the ele tromagneti theory as applied to a great variety of problems.b It is, however, only by a An English version of this report (Compton 1928) was published in the Journal of the Franklin Institute. The Fren h version appears to be essentially a translation of the English paper with some additions. Whenever there are no dis repan ies, we reprodu e Compton's own English (we have orre ted some obvious typos and harmonised some of the spelling). Interesting variants are footnoted. Other dis repan ies between the two versions are reported in the endnotes (eds.). b The opening has been translated from the Fren h edition. The English version has the following dierent opening (eds.): During the last few years it has be ome in reasingly evident that the lassi al ele tromagneti theory of radiation is in apable of a
ounting for ertain large
lasses of phenomena, espe ially those on erned with the intera tion between radiation and matter. It is not that we question the wave hara ter of light the striking su
esses of this on eption in explaining polarisation and interferen e of light an leave no doubt that radiation has the hara teristi s
329
330
A. H. Compton
a quainting ourselves with the real or apparent1 failures of this powerful theory that we an hope to develop a more omplete theory of radiation whi h will des ribe the fa ts as we know them. The more serious di ulties whi h present themselves in onne tion with the theory that radiation onsists of ele tromagneti waves, propagated through spa e in a
ord with the demands of Maxwell's equations, may be lassi ed onveniently under ve heads:2 (1) Is there an ether? If there are os illations, there must be a medium in whi h these os illations are produ ed. Assuming the existen e of su h a medium, however, one en ounters great di ulties. (2) How are the waves produ ed? The lassi al ele trodynami s requires as a sour e of an ele tromagneti wave an os illator of the same frequen y as that of the waves it radiates. Our studies of spe tra,3 however, make it appear impossible that an atom should ontain os illators of the same frequen ies as the emitted rays. (3) The photoele tri ee t. This phenomenon is wholly anomalous when viewed from the standpoint of waves. (4) The s attering of X-rays, and the re oil ele trons, phenomena in whi h we nd gradually in reasing departures from the predi tions of the lassi al wave theory as the frequen y in reases. (5) Experiments on individual intera tions between quanta of radiation and ele trons. If the results of the experiments of this type are reliable, they seem to show de nitely that individual quanta of radiation, of energy hν , pro eed in de nite dire tions. The photon hypothesis.4 In order to exhibit more learly the di ulties with the lassi al theory of radiation, it will be helpful to keep in mind the suggestion that5 light onsists of orpus les. We need not think of these two views as ne essarily alternative. It may well be that the two on eptions are omplementary. Perhaps the orpus le is of waves; but it is equally true that ertain other properties of radiation are not easily interpreted in terms of waves.
The power of the ele tromagneti
theory as applied to a great variety of problems of radiation is too well known to require emphasis.
Experiment and the ele tromagneti theory
331
related to the wave in somewhat the same manner that the mole ule is related to matter in bulk; or there may be a guiding wave whi h dire ts the orpus les whi h arry the energy. In any ase, the phenomena whi h we have just mentioned suggest the hypothesis that radiation is divisible into units possessing energy hν , and whi h pro eed in de nite dire tions with momentum hν/c. This is obviously similar to Newton's old on eption of light orpus les. It was revived in its present form by Professor Einstein,6 it was defended under the name of the `Neutron Theory' by Sir William [H.℄ Bragg, and has been given new life by the re ent dis overies asso iated with the s attering of X-rays. In referring to this unit of radiation I shall use the name `photon', suggested re ently by G. N. Lewis.a This word avoids any impli ation regarding the nature of the unit, as ontained for example in the name `needle ray'. As ompared with the terms `radiation quantum' and `light quant',7 this name has the advantages of brevity and of avoiding any implied dependen e upon the mu h more general quantum me hani s or quantum theory of atomi stru ture. Virtual radiation. Another on eption of the nature of radiation whi h it will be desirable to ompare with the experiments is Bohr, Kramers and Slater's important theory of virtual radiation.b A
ording to this theory, an atom in an ex ited state is ontinually emitting virtual radiation, to whi h no energy hara teristi s are to be as ribed. The normal atoms have asso iated with them virtual os illators, of the frequen ies orresponding to jumps of the atom to all of the stationary states of higher energy. The virtual radiation may be thought of as being absorbed by these virtual os illators, and any atom whi h has a virtual os illator absorbing this virtual radiation has a ertain probability of jumping suddenly to the higher state of energy orresponding to the frequen y of the parti ular virtual os illator. On the average, if the radiation is ompletely absorbed, the number of su h jumps to levels of higher energy is equal to the number of emitting atoms whi h pass from higher to lower states. But there is no dire t onne tion between the falling of one atom from a higher to a lower state and a orresponding rise of a se ond atom from a lower to a higher state. Thus on this view the energy of the emitting atoms and of the absorbing atoms is only statisti ally onserved. a G. N. Lewis, Nature, [118℄, [874℄ (De . 18, 1926). b N. Bohr, H. A. Kramers and J. C. Slater, Phil. Mag., 47 (1924), 785; Zeits. f. Phys., 24 (1924), 69.
332
A. H. Compton
The problem of the ether
8
The onstan y of the speed of radiation of dierent wavelengths has long been onsidered as one of the most powerful arguments in favour of the wave theory of light. This onstan y suggests that a perturbation is travelling through a xed medium in spa e, the ether. If experiments like those by Mi helson and Morley's were to show the existen e of a relative motion with respe t to su h a medium, this argument would be onsiderably strengthened. For then we ould imagine light as having a speed determined with referen e to a xed axis in spa e. But, ex ept for the re ent and quite doubtful experiments by Miller,a no-one has ever dete ted su h a relative motion. We thus nd ourselves in the di ult position of having to imagine a medium in whi h perturbations travel with a de nite speed, not with referen e to a xed system of axes, but with referen e to ea h individual observer, whatever his motion. If we think of the omplex properties a medium must have in order to transmit a perturbation in this way, we nd that the medium diers so onsiderably from the simple ether from whi h we started that the analogy between a wave in su h a medium and a pertubation travelling in an elasti medium is very distant. It is true that doubts have often been expressed as to the usefulness of retaining the notion of the ether. Nevertheless, if light is truly a wave motion, in the sense of Maxwell, there must be a medium in order to transmit this motion, without whi h the notion of wave would have no meaning. This means that, instead of being a support for the wave theory, the
on ept of the ether has be ome an un omfortable burden of whi h the wave theory has been unable to rid itself. If, on the other hand, we a
ept the view suggested by the theory of relativity, in whi h for the motion of matter or energy there is a limiting speed relative to the observer, it is not surprising to nd a form of energy that moves at this limiting speed. If we abandon the idea of an ether, it is simpler to suppose that this energy moves in the form of orpus les rather than waves.
a D. C. Miller, Nat. A ad. S i. Pro ., 11 (1925), 306.
Experiment and the ele tromagneti theory
The emission of radiation
333
When we tra e a sound to its origin, we nd it oming from an os illator vibrating with the frequen y of the sound itself. The same is true of ele tri waves, su h as radio waves, where the sour e of the radiation is a stream of ele trons os illating ba k and forth in a wire. But when we tra e a light ray or an X-ray ba k to its origin, we fail to nd any os illator whi h has the same frequen y as the ray itself. The more omplete our knowledge be omes of the origin of spe trum lines, the more
learly we see that if we are to assign any frequen ies to the ele trons within the atoms, these frequen ies are not the frequen ies of the emitted rays, but are the frequen ies asso iated with the stationary states of the atom. This result annot be re on iled with the ele tromagneti theory of radiation, nor has any me hanism been suggested whereby radiation of one frequen y an be ex ited by an os illator of another frequen y. The wave theory of radiation is thus powerless to suggest how the waves originate. The origin of the radiation is onsiderably simpler when we onsider it from the photon viewpoint. We nd that an atom hanges from a stationary state of one energy to a state of less energy, and asso iated with this hange radiation is emitted. What is simpler than to suppose that the energy lost by the atom is radiated away as a single photon? It is on this view unne essary to say anything regarding the frequen y of the radiation. We are on erned only with the energy of the photon, its dire tion of emission, and its state of polarisation. The problem of the emission of radiation takes an espe ially interesting form when we onsider the produ tion of the ontinuous X-ray spe trum.a Experiment shows that both the intensity and the average frequen y of the X-rays emitted at angles less than 90 degrees with the
athode-ray stream are greater than at angles greater than 90 degrees. This is just what we should expe t due to the Doppler ee t if the X-rays are emitted by a radiator moving in the dire tion of the athode rays. In order to a
ount for the observed dissymmetry between the rays in the forward and ba kward dire tions, the parti les emitting the radiation must be moving with a speed of the order of 25 per ent that of light. This means that the emitting parti les must be free ele trons, sin e it would require an impossibly large energy to set an atom into motion with su h a speed. a The di ulty here dis ussed was rst emphasised by D. L. Webster, Phys. Rev., 13 (1919), 303.
334
A. H. Compton
But it will be re alled that the ontinuous X-ray spe trum has a sharp upper limit. Su h a sharp limit is, however, possible on the wave theory only in ase the rays ome in trains of waves of onsiderable length, so that the interferen e between the waves in dierent parts of the train
an be omplete at small glan ing angles of re e tion from the rystal. This implies that the os illator whi h emits the rays must vibrate ba k and forth with onstant frequen y a large number of times while the ray is being emitted. Su h an os illation might be imagined for an ele tron within an atom; but it is impossible for an ele tron moving through an irregular assemblage of atoms with a speed omparable with that of light. Thus the Doppler ee t in the primary X-rays demands that the rays shall be emitted by rapidly moving ele trons, while the sharp limit to the ontinuous spe trum requires that the rays be emitted by an ele tron bound within an atom. The only possible es ape from this dilemma on the wave theory is to suppose that the ele tron is itself apable of internal os illation of su h a hara ter as to emit radiation. This would, however, introdu e an undesirable omplexity into our on eption of the ele tron, and would as ribe the ontinuous X-rays to an origin entirely dierent from that of other known sour es of radiation. Here again the photon theory aords a simple solution. It is a onsequen e of Ehrenfest's adiabati prin iplea that photons emitted by a moving radiator will show the same Doppler ee t, with regard to both frequen y and intensity, as does a beam of waves.b But if we suppose that photons are radiated by the moving athode ele trons, the energy of ea h photon will be the energy lost by the ele tron, and the limit of the X-ray spe trum is ne essarily rea hed when the energy of the photon is equal to the initial energy of the ele tron, i.e., hν = eV . In this ase, if we onsider the initial state as an ele tron approa hing an atom with large kineti energy and the nal state as the ele tron leaving the atom with a smaller kineti energy, we see that the emission of the ontinuous X-ray spe trum is the same kind of event as the emission of any other type of radiation. a The adiabati prin iple onsists in the following. Sin e for a quantised quantity there should be no quantum jumps indu ed by an in nitely slowly varying external for e (in this ase, one that gently a
elerates a radiator), there is an analogy between these quantities and the lassi al adiabati invariants. Ehrenfest (1917) a
ordingly formulated a prin iple identifying the lassi al quantities to be quantised as the adiabati invariants of a system (eds.). b Cf., e.g., A. H. Compton, Phys. Rev., 21 (1923), 483.
Experiment and the ele tromagneti theory
335
Absorption of radiation. A
ording to the photon theory, absorption o
urs when a photon meets an atom and imparts its energy to the atom. The atom is thereby raised to a stationary state of higher energy pre isely the reverse of the emission pro ess. On the wave theory, absorption is ne essarily a ontinuous pro ess, if we admit the onservation of energy, sin e on no part of the wave front is there enough energy available to hange the atom suddenly from a state of low energy to a state of higher energy. What eviden e we have is, however, strongly against the atom having for any onsiderable length of time an energy intermediate between two stationary states; and if su h intermediate states annot exist, the gradual absorption of radiation is not possible. Thus the absorption of energy from waves9 is irre on ilable with the on eption of stationary states. We have seen that on the theory of virtual radiation the energy of the emitting atoms and of the absorbing atoms is only statisti ally
onserved. There is a
ording to this view therefore no di ulty with supposing that the absorbing atom suddenly jumps to a higher level of energy, even though it has not re eived from the radiation as mu h energy as is ne essary to make the jump. It is thus possible through virtual os illators and virtual radiation to re on ile the wave theory of radiation with the sudden absorption of energy, and hen e to retain the idea of stationary states.
The photoele tri effe t It is well known that the photon hypothesis was introdu ed by Einstein to a
ount for the photoele tri ee t.a The assumption that light
onsists of dis rete units whi h an be absorbed by atoms only as units, ea h giving rise to a photoele tron, a
ounted at on e for the fa t that the number of photoele trons is proportional to the intensity of the light; and the assumption that the energy of the light unit is equal to hν , where h is Plan k's onstant, made it possible to predi t the kineti energy with whi h the photoele trons should be eje ted, as expressed by Einstein's well-known photoele tri equation, 1 mc2 p − 1 = hν − wp . 1 − β2
a A. Einstein, Ann. d. Phys., 17 (1905), [132℄.10
(1)
336
A. H. Compton
Seven years elapsed before experiments by Ri hardson and Comptona and by Hughesb showed that the energy of the emitted ele trons was indeed proportional to the frequen y less a onstant,12 and that the fa tor of proportionality was lose to the value of h al ulated from Plan k's radiation formula. Millikan's more re ent pre ision photoele tri experiments with the alkali metalsc on rmed the identity of the
onstant h in the photoele tri equation with that in Plan k's radiation formula. De Broglie's beautiful experimentsd with the magneti spe trograph showed that in the region of X-ray frequen ies the same equation holds, if only we interpret the work fun tion wp as the work required to remove the ele tron from the pth energy level of the atom. Thibaud has made use of this resulte in omparing the velo ities of the photoele trons eje ted by γ -rays from dierent elements, and has thus shown that the photoele tri equation (1) holds with pre ision even for β -rays of the highest speed. Thus from light of frequen y so low that it is barely able to eje t photoele trons from metals to γ -rays that eje t photoele trons with a speed almost as great as that of light, the photon theory expresses a
urately the speed of the photoele trons. The dire tion in whi h the photoele trons are emitted is no less instru tive than is the velo ity. Experiments using the loud expansion method, performed13 by C. T. R. Wilsona and others,b have shown that the most probable dire tion in whi h the photoele tron is eje ted from an atom is nearly the dire tion of the ele tri ve tor of the in ident wave, but with an appre iable forward omponent to its motion. There is, however, a very onsiderable variation in the dire tion of emission. For example, if we plot the number of photoele trons eje ted at different angles with the primary beam we nd, a
ording to Auger, the distribution shown in Fig. 1. Ea h of these urves, taken at a dierent potential, represents the distribution of about 200 photoele tron tra ks. It will be seen that as the potential on the X-ray tube in reases, the average forward omponent of the photoele tron's motion also in reases. When polarised X-rays are used, there is a strong preponderan e of the a b
d e a b
O. W. Ri hardson and K. T. Compton, Phil. Mag., 24 (1912), 575. A. L. Hughes, Phil. Trans. A, 212 (1912), 205.11 R. A. Millikan, Phys. Rev., 7 (1916), 355. M. de Broglie, Jour. de Phys., 2 (1921), 265. J. Thibaud, C. R., 179 (1924), 165, 1053 and 1322. C. T. R. Wilson, Pro . Roy. So . A, 104 (1923), 1. A. H. Compton, Bull. Natl. Res. Coun., No. 20 (1922), 25; F. W. Bubb, Phys. Rev., 23 (1924), 137; P. Auger, C. R., 178 (1924), 1535; D. H. Loughridge, Phys. Rev., 26 (1925), 697; F. Kir hner, Zeits. f. Phys., 27 (1926), 385.
Experiment and the ele tromagneti theory
337
Fig. 1. Longitudinal distribution of photoele trons for X-rays of three dierent ee tive wavelengths, a
ording to Auger.
photoele trons in or near the plane in luding the ele tri ve tor of the in ident rays. Thus Fig. 2 shows the distribution found by Bubb of the dire tion of the photoele trons eje ted from moist air when traversed by X-rays that have been polarised by s attering at right angles from a blo k of para n. Be ause of multiple s attering in the para n, the s attered rays are not ompletely polarised, and this is probably su ient to a
ount for the fa t that some photoele trons appear to start at right angles with the ele tri ve tor. This ee t with X-rays is doubtless similar in hara ter to the sele tive photoele tri ee t dis overed many years ago by Pohl and Pringsheim, in whi h the number of ele trons eje ted by light from the liquid surfa e of sodium-potassium alloy is greater when the ele tri ve tor is in a plane perpendi ular to the surfa e than when parallel to the surfa e. Re ent experiments have shown that the dire tion in whi h the photoele trons are eje ted by X-rays is at least very nearly independent of the material from whi h the ele trons ome.a Can ele tromagneti waves produ e photoele trons? Before dis ussing the produ tion of photoele trons from the standpoint of radiation quanta, let us see what su
ess meets the attempt15 to a
ount for them on the basis of ele tromagneti waves. The fa t that they are emitted a E. A. Owen, Pro . Phys. So ., 30 (1918), 133; Auger, Kir hner, Loughridge, lo .
it.14
338
Fig. 2.
A. H. Compton
Lateral distribution of photoele trons for in ompletely polarised X-
rays, a
ording to Bubb.
approximately in the dire tion of the ele tri ve tor would suggest that the photoele trons are eje ted by the dire t a tion of the ele tri eld of the in ident rays. If this were the ase, however, we should expe t the speed of the eje ted ele trons to be greater for greater intensity of radiation, whereas experiment shows that for the same wavelength intense sunlight eje ts an ele tron no faster than does the feeble light from a star. Furthermore, the energy available from the ele tromagneti wave is wholly inadequate. Thus in a re ent experiment performed by Joe and Dobronrawov,a X-rays were produ ed by the impa t on a target of 104 to 105 ele trons per se ond. Sin e on the ele tromagneti theory an X-ray pulse is of the order of 103 waves in length or 10−16 se onds a A. Joe and N. Dobronrawov, Zeits. f. Phys., 34, 889 (1925).
Experiment and the ele tromagneti theory
339
in duration, the X-ray pulses must have followed ea h other at widely separated intervals. It was found, however, that photoele trons were o
asionally eje ted from a bismuth parti le whi h subtended a solid angle not greater than 10−5 . It is learly impossible that all the energy of an X-ray pulse whi h has spread out in a spheri al wave should spend itself on this bismuth parti le. Thus on the wave theory the eje tion of the photoele tron, whi h has almost as mu h energy as the original
athode ele tron, ould not have been a
omplished by a single16 pulse. It annot therefore be the dire t a tion of the ele tri ve tor of the wave, taken in the usual sense,17 whi h has eje ted the ele tron. We may assume, on the other hand, that the energy is gradually absorbed in the bismuth parti le of Joe's experiment until an amount hν has a
umulated, whi h is then spent in eje ting the photoele tron. We have already alled attention to the fa t that this gradual absorption hypothesis implies the existen e of stationary states in the atom having in nitesimal gradations of energy, whereas the eviden e is very strong that atoms annot endure ex ept in ertain de nitely de ned stationary states. But new di ulties also arise. Why do the photoele trons tend to start in the dire tion of the ele tri eld of the in ident wave? If we suppose that it is the gradual absorption of energy from a wave whi h liberates the ele tron, why does there exist a tenden y for the ele tron to start with a large omponent of its motion in a forward dire tion?18 The forward impulse due to the radiation pressure as19 the energy is gradually absorbed will be transferred to the atom and not left with [the℄ absorbing ele tron. The a
umulation hypothesis is thus di ult to defend. Photons and photoele trons. On the photon theory it is possible to a
ount in a simple manner for most of the properties of the photoele trons. We have seen how Einstein was able to predi t a
urately the velo ity of the photoele trons, assuming only that energy is onserved when a photon a ts on an ele tron. In order to a
ount for the dire tion of emission we must as ribe to the photon some of the properties of an ele tromagneti pulse. Bubb introdu ed the suggestiona that we as ribe to the photon a ve tor property similar to the ele tri ve tor of an ele tromagneti wave, so that when the photon traverses an atom the ele trons and the nu leus re eive impulses in opposite dire tions perpendi ular to the dire tion of propagation. Asso iated with this a F. W. Bubb, Phys. Rev., 23 (1924), 137.
340
A. H. Compton
ele tri ve tor, we should also expe t to nd a magneti ve tor. Thus if an ele tron is set in motion by the ele tri ve tor of the photon at right angles to the dire tion of propagation, the magneti ve tor of the photon will a t on the moving ele tron in the dire tion of propagation. This is stri tly analogous to the radiation pressure exerted by an ele tromagneti wave on an ele tron whi h it traverses, and means that the forward momentum of the absorbed photon is transferred to the photoele tron. In the simplest ase, where we negle t the initial momentum of the ele tron in its orbital motion in the atom, the angle between the dire tion of the in ident ray and the dire tion of eje tion is found from these assumptions to be p θ = tan−1 2/α , (2) where α = γ/λ, and γ = h/mc = 0.0242 Å. The quantity α is small
ompared with unity, ex ept for very hard X-rays and γ -rays. Thus for light, equation (2) predi ts the expulsion of photoele trons at nearly 90 degrees. This is in a
ord with the rather un ertain data whi h have been obtained with visible and ultra-violet light.a The only really signi ant test of this result is in its appli ation to X-ray photoele trons. In Fig. 1 are drawn the lines θ1 , θ2 and θ3 for the three urves, at the angles al ulated by Auger from equation (2). It will be seen that they fall very satisfa torily in the dire tion of maximum emission of the photoele trons. Similar results have been obtained by other investigators.b This may be taken as proof that a photon imparts not only its energy, but also its momentum to the photoele trons.c
a Cf. A. Parts h and W. Hallwa hs, Ann. d. Phys., 41 (1913), 247. b W. Bothe, Zeits. f. Phys., 26 (1925), 59; F. Kir hner, Zeits. f. Phys., 27 (1926), 385.20
The English version in ludes here the following footnote. Cf. also the omments by Bragg on p. 357 and the ensuing dis ussion (eds.). Sin e this was written, experiments by [D. H.℄ Loughridge (Phys.
Rev., 30
(1927), [488℄) have been published whi h show a forward omponent to the photoele tron's motion whi h seems to be greater than that predi ted by equation (2).
Williams, in experiments as yet unpublished, nds that the
forward omponent is almost twi e as great as that predi ted by this theory. These results indi ate that the me hanism of intera tion between the photon and the atom must be more omplex than here postulated. The fa t that the forward momentum of the photoele tron is found to be of the same order of magnitude as that of the in ident photon, however, suggests that the momentum of the photon is a quired by the photoele tron, while an additional forward impulse is imparted by the atom. Thus these more re ent experiments
Experiment and the ele tromagneti theory
341
Honestya obliges me to point out a di ulty that arises in this explanation of the motion of the photoele trons. It is the failure of the attempts made to a
ount properly for the fa t that the photoele trons are emitted over a wide range of angles instead of in a de nite dire tion, as would be suggested by the al ulation just outlined. The most interesting of these attempts is that of Bubb,b who takes into a
ount the momentum of the ele tron immediately before the absorption of the photon. Bubb nds a dispersion of the dire tions of emission of the photoele trons of the
orre t order of magnitude, but whi h is larger when the ele tron issues from a heavy atom than when it issues from a light one. We have seen, however, that experiment has shown this dispersion of the dire tions of emission to be notably independent of the element from whi h the photoele tron originates. Whatever may be the ause of the dispersion in the dire tions of motion of the photoele trons,21 it will readily be seen that if the time during whi h the photon exerts a for e on the ele tron is omparable with the natural period of the ele tron22 in the atom, the impulse imparted to the ele tron will be transferred in part to the positive nu leus about whi h the ele tron is moving. The fa t that the photoele trons are eje ted with a forward omponent equal, within the limits of experimental error, to the momentum of the in ident photon23 means that no appre iable part of the photon's momentum is spent on the remainder of the atom. This
an only be the ase if the time of a tion of the photon on the ele tron is short ompared with the time of revolution of the ele tron in its orbit.a also support the view that the photoele tron a quires both the energy and the momentum of the photon.
a This paragraph is present only in the Fren h edition. The orresponding one in the English edition reads: If the angular momentum of the atomi system from whi h the photoele tron is eje ted is to be onserved when a ted upon by the radiation, the ele tron
annot be eje ted exa tly in the dire tion of
θ, but must re eive an impulse in a
dire tion determined by the position of the ele tron in the atom at the instant ∗ it is traversed by the photon. Thus we should probably onsider the ele tri ve tor of the X-ray wave as de ning merely the most probable dire tion in whi h the impulse should be imparted to the ele tron. This is doubtless the
hief reason why the photoele trons are emitted over a wide range of angles instead of in a de nite dire tion, as would be suggested by the al ulation just outlined.
With the footnote: ∗ Cf. A. H. Compton, Phys. Rev., [31℄ (1928), [59℄ (eds.). b F. W. Bubb, Phil. Mag., 49 (1925), 824. a The English edition in ludes the further senten e: `Su h a short duration of
342
A. H. Compton
The photoele tri ee t and virtual radiation. It is to be noted that none of these properties of the photoele tron is in onsistent with the virtual radiation theory of Bohr, Kramers and Slater. The di ulties whi h applied to the lassi al wave theory do not apply here, sin e the energy and momentum are onserved only statisti ally. There is nothing in this theory, however, whi h would enable us to predi t anything regarding the motion of the photoele trons. The degree of su
ess that has attended the appli ation of the photon hypothesis to the motion of these ele trons has ome dire tly from the appli ation of the onservation prin iples to the individual a tion of a photon on an ele tron. The power of these prin iples as applied to this ase is surprising if the assumption is orre t that they are only statisti ally valid.
Phenomena asso iated with the s attering of X-rays As is now well known, there is a group of phenomena asso iated with the s attering of X-rays for whi h the lassi al wave theory of radiation fails to a
ount. These phenomena may be onsidered under the heads of: (1) The hange of wavelength of X-rays due to s attering, (2) the intensity of s attered X-rays, and (3) the re oil ele trons. The earliest experiments on se ondary X-rays and γ -rays24 showed a dieren e in the penetrating power of the primary and the se ondary rays. In the ase of X-rays, Barkla and his ollaboratorsa showed that the se ondary rays from the heavy elements onsisted largely of uores ent radiations hara teristi of the radiator, and that it was the presen e of these softer rays whi h was hie y responsible for the greater absorption of the se ondary rays. When later experimentsb showed a measurable dieren e in penetration even for light elements su h as arbon, from whi h no uores ent K or L radiation appears, it was natural to as ribec this dieren e to a new type of uores ent radiation, similar to the K and L types, but of shorter wavelength. Careful absorption measurementsd failed, however, to reveal any riti al absorption limit for these assumed
a b
d
intera tion is a natural onsequen e of the photon on eption of radiation, but is quite ontrary to the onsequen es of the ele tromagneti theory' (eds.). C. [G.℄ Barkla and C. A. Sadler, Phil. Mag., 16, 550 (1908).25 C. A. Sadler and P. Mesham, Phil. Mag., 24 (1912), 138; J. Laub, Ann. d. Phys., 46 (1915), 785. [C. G.℄ Barkla and [M. P.℄ White, Phil. Mag., 34 (1917), 270; J. Laub, Ann. d. Phys., 46 (1915), 785, et al. E.g., [F. K.℄ Ri htmyer and [K.℄ Grant, Phys. Rev., 15 (1920), 547.
Experiment and the ele tromagneti theory
343
`J' radiations similar to those orresponding to the K and L radiations. Moreover, dire t spe tros opi observationsa failed to reveal the existen e of any spe trum lines26 under onditions for whi h the supposed J-rays should appear. It thus be ame evident that the softening of the se ondary X-rays from the lighter elements was due to a dierent kind of pro ess than the softening of the se ondary rays from heavy elements where uores ent X-rays are present. A series of skilfully devised absorption experiments performed by J. A. Graya showed, on the other hand, that both in the ase of γ rays and in that of X-rays an in rease in wavelength a
ompanies the s attering of the rays of light elements. It was at this stage that the rst spe tros opi investigations of the se ondary X-rays from light elements were made.b A
ording to the usual ele tron theory of s attering it is obvious that the s attered rays will be of the same frequen y as the for ed os illations of the ele trons whi h emit them, and hen e will be identi al in frequen y with the primary waves whi h set the ele trons in motion. Instead of showing s attered rays of the same wavelength as the primary rays, however, these spe tra revealed lines in the se ondary rays orresponding to those in the primary beam, but with ea h line displa ed slightly toward the longer wavelengths. This result might have been predi ted from Gray's absorption measurements; but the spe trum measurements had the advantage of aording a quantitative measurement of the hange in wavelength, whi h gave a basis for its theoreti al interpretation. The spe tros opi experiments whi h have shown this hange in wavelength are too well knownc to require dis ussion. The interpretation of the wavelength hange in terms of photons being de e ted by individual27 ele trons and imparting a part of their energy to the s attering ele trons is also very familiar. For purposes of dis ussion, however, let us re all that when we onsider the intera tion of a single photon with a single ele tron the prin iples of the onservation of energy and momentum lead usd to the result that the hange in wavelength of the de e ted photon a E.g., [W.℄ Duane and [T.℄ Shimizu, Phys. Rev., 13 (1919), [289℄; ibid., 14 (1919), 389. a J. A. Gray, Phil. Mag., 26 (1913), 611; Jour. Frank. Inst., [190℄, 643 (Nov. 1920). b A. H. Compton, Bull. Natl. Res. Coun., No. 20, [18℄ ([O tober℄ 1922); Phys. Rev., 22 (1923), 409.
Cf., e.g., A. H. Compton, Phys. Rev., 22 (1923), 409; P. A. Ross, Pro . Nat. A ad., 10 (1924), 304. d A. H. Compton, Phys. Rev., [21℄ (1923), 483; P. Debye, Phys. Zeits., 24 (1923), 161.
344
A. H. Compton
is δλ =
h (1 − cos ϕ) , mc
(3)
where ϕ is the angle through whi h the photon is de e ted. The ele tron at the same time re oils from the photon at an angle of θ given by,28 1 cot θ = −(1 + α) tan ϕ ; 2
(4)
and the kineti energy of the re oiling ele tron is, Ekin = hν
2α cos2 θ . (1 + α)2 − α2 cos2 θ
(5)
The experiments show in the spe trum of the s attered rays two lines
orresponding to ea h line of the primary ray. One of these lines is of pre isely the same wavelength as the primary ray, and the se ond line, though somewhat broadened, has its entre of gravity displa ed by the amount predi ted by equation (3). A
ording to experiments by Kallman and Marka and by Sharp,b this agreement between the theoreti al29 and the observed shift is pre ise within a small fra tion of 1 per ent. The re oil ele trons. From the quantitative agreement between the theoreti al and the observed wavelengths of the s attered rays, the re oil ele trons predi ted by the photon theory of s attering were looked for with some on den e.30 When this theory was proposed, there was no dire t eviden e for the existen e of su h ele trons, though indire t eviden e suggested that the se ondary β -rays eje ted from matter by hard γ -rays are mostly of this type. Within a few months of their predi tion, however, C. T. R. Wilsonc and W. Bothed independently announ ed their dis overy. The re oil ele trons show as short tra ks, pointed in the dire tion of the primary X-ray beam, mixed among the mu h longer tra ks due to the photoele trons eje ted by the X-rays. Perhaps the most onvin ing reason for asso iating these short tra ks with the s attered X-rays omes from a study of their number. Ea h photoele tron in a loud photograph represents a quantum of truly absorbed X-ray energy. If the short tra ks are due to re oil ele trons, ea h one should represent the s attering of a photon. Thus the ratio a b
d
H. Kallman and H. Mark, Naturwiss., 13 (1925), 297. H. M. Sharp, Phys. Rev., 26 (1925), 691. C. T. R. Wilson, Pro . Roy. So . [A℄, 104 (1923), 1. W. Bothe, Zeits. f. Phys., 16 (1923), 319.
Experiment and the ele tromagneti theory
345
Nr /Np of the number of short tra ks to the number of long tra ks should be the same as the ratio σ/τ of the s attered to the truly absorbed energy31 when the X-rays pass through air. The latter ratio is known from absorption measurements, and the former ratio an be determined by ounting the tra ks on the photographs. The satisfa tory agreement between the two ratiosa for X-rays of dierent wavelengths means that on the average there is about one quantum of energy s attered for ea h short tra k that is produ ed. This result is in itself ontrary to the predi tions of the lassi al wave theory, sin e on this basis all the energy spent on a free ele tron (ex ept the insigni ant ee t of radiation pressure) should reappear as s attered X-rays. In these experiments, on the ontrary, 5 or 10 per ent as mu h energy appears in the motion of the re oil ele trons as appears in the s attered X-rays. That these short tra ks asso iated with the s attered X-rays orrespond to the re oil ele trons predi ted by the photon theory of s attering be omes lear from a study of their energies. The energy of the ele tron whi h produ es a tra k an be al ulated from the range of the tra k. The ranges of tra ks whi h start in dierent dire tions have been studieda using primary X-rays of dierent wavelengths, with the result that equation (5)33 has been satisfa torily veri ed. In view of the fa t that ele trons of this type were unknown at the time the photon theory of s attering was presented, their existen e, and the
lose agreement with the predi tions as to their number, dire tion and velo ity, supply strong eviden e in favour of the fundamental hypotheses of the theory.
Interpretation of these experiments. It is impossible to a
ount for s attered rays of altered frequen y, and for the existen e of the re oil ele trons, if we assume that X-rays onsist of ele tromagneti waves in the usual sense. Yet some progress has been made on the basis of semi- lassi al theories. It is an interesting fa t that the wavelength of the s attered ray a
ording to equation (3)34 varies with the angle just as one would expe t from a Doppler ee t if the rays are s attered from an ele tron moving in the dire tion of the primary beam. Moreover, the velo ity that must be assigned to the ele tron in order to give the proper magnitude to the hange of wavelength is that whi h the ele tron a A. H. Compton and A. W. Simon, Phys. Rev., 25 (1925), 306; J. M. Nuttall and E. J. Williams, Man hester Memoirs, 70 (1926), 1. a Compton and Simon, lo . it.32
346
A. H. Compton
would a quire by radiation pressure if it should absorb a quantum of the in ident rays. Several writersa have therefore assumed that an ele tron takes from the in ident beam a whole quantum of the in ident radiation, and then emits this energy as a spheri al wave while moving forward36 with high velo ity. This on eption that the radiation o
urs in spheri al waves, and that the s attering ele tron an nevertheless a quire suddenly the impulses from a whole quantum of in ident radiation is in onsistent with the prin iple of energy onservation. But there is the more serious experimental di ulty that this theory predi ts re oil ele trons all moving in the same dire tion and with the same velo ity. The experiments show, on the other hand, a variety of dire tions and velo ities, with the velo ity and dire tion orrelated as demanded by the photon hypothesis. Moreover, the maximum range of the re oil ele trons, though in agreement with the predi tions of the photon theory, is found to be about four times as great as that predi ted by the semi- lassi al theory. There is nothing in these experiments, as far as we have des ribed them, whi h is in onsistent with the idea of virtual os illators ontinually s attering virtual radiation. In order to a
ount for the hange of wavelength on this view, Bohr, Kramers and Slater assumed that the virtual os illators s atter as if moving in the dire tion of the primary beam, a
ounting for the hange of wavelength as a Doppler ee t. They then supposed that o
asionally an ele tron, under the stimulation of the primary virtual rays, will suddenly move forward with a momentum large ompared with the impulse re eived from the radiation pressure. Though we have seen that not all of the re oil ele trons move dire tly forward, but in a variety of dierent dire tions, the theory ould easily be extended to in lude the type of motion that is a tually observed. The only obje tion that one an raise against this virtual radiation theory in onne tion with the s attering phenomena as viewed on a large s ale, is that it is di ult to see how su h a theory ould by itself predi t the hange of wavelength and the motion of the re oil ele trons. These phenomena are dire tly predi table if the onservation of energy and momentum are assumed to apply to the individual a tions of radiation on ele trons; but this is pre isely where the virtual radiation theory denies the validity of the onservation prin iples. We may on lude that the photon theory predi ts quantitatively and a C. R. Bauer, C. R., 177 (1923), 1211; C. T. R. Wilson, Pro . Roy. So . [A℄, 104 (1923), 1; K. Fosterling, Phys. Zeits., 25 (1924), 313; O. Halpern, Zeits. f. Phys., 30 (1924), 153.35
Experiment and the ele tromagneti theory
347
in detail the hange of wavelength of the s attered X-rays and the hara teristi s of the re oil ele trons. The virtual radiation theory is probably not in onsistent with these experiments, but is in apable of predi ting the results. The lassi al theory, however, is altogether helpless to deal with these phenomena. The origin of the unmodi ed line The unmodi ed line is probably due to X-rays whi h are s attered by ele trons so rmly held within the atom that they are not eje ted by the impulse from the de e ted photons. This view is adequate to a
ount for the major hara teristi s of the unmodi ed rays, though as yet no quantitatively satisfa tory theory of their origin has been published.a It is probable that a detailed a
ount of these rays will involve de nite assumptions regarding the nature and the duration of the intera tion between a photon and an ele tron; but it is doubtful whether su h investigations will add new eviden e as to the existen e of the photons themselves. A similar situation holds regarding the intensity of the s attered Xrays. Histori ally it was the fa t that the lassi al ele tromagneti theory is unable to a
ount for the low intensity of the s attered X-rays whi h
alled attention to the importan e of the problem of s attering. But the solutions whi h have been oered by Breit,b Dira c and othersd of this intensity problem as distinguished from that of the hange of wavelength, seem to introdu e no new on epts regarding the nature of radiation or of the s attering pro ess. Let us therefore turn our attention to the experiments that have been performed on the individual pro ess of intera tion between photons and ele trons.
a Cf., however, G. E. M. Jaun ey, Phys. Rev., 25 (1925), 314 and ibid., 723; G. Wentzel, Zeits. f. Phys., 43 (1927), 14, 779; I. Waller, Nature, [120, 155℄ (July 30, 1927).37 [The footnote in the English edition ontinues with the senten e: `It is possible that the theories of the latter authors may be satisfa tory, but they have not yet been stated in a form suitable for quantitative test' (eds.).℄ b G. Breit, Phys. Rev., 27 (1926), 242.
P. A. M. Dira , Pro . Roy. So . A, [111℄ (1926), [405℄. d W. Gordon, Zeits. f. Phys., 40 (1926), 117; E. S hrödinger, Ann. d. Phys., 82 (1927), 257; O. Klein, Zeits. f. Phys., 41 (1927), 407; G. Wentzel, Zeits. f. Phys., 43 (1927), 1, 779.38
348
A. H. Compton
Intera tions between radiation and single ele trons
39
The most signi ant of the experiments whi h show departures from the predi tions of the lassi al wave theory are those that study the a tion of radiation on individual atoms or on individual ele trons. Two methods have been found suitable for performing these experiments, Geiger's point ounters, and Wilson's loud expansion photographs. (1) Test for oin iden es with uores ent X-rays. Bothe has performed an experimenta in whi h uores ent K radiation from a thin
opper foil is ex ited by a beam of in ident X-rays. The emitted rays are so feeble that only about ve quanta of energy are radiated per se ond. Two point ounters are mounted, one on either side of the opper foil in ea h of whi h an average of one photoele tron is produ ed and re orded for about twenty quanta radiated by the foil. If we assume that the uores ent radiation is emitted in quanta of energy, but pro eed[s℄ in spheri al waves in all dire tions, there should thus be about 1 han e in 20 that the re ording of a photoele tron in one hamber should be simultaneous with the re ording of a photoele tron in the other. The experiments showed no oin iden es other than those whi h were expli able by su h sour es as high-speed β -parti les whi h traverse both
ounting hambers. This result is in a
ord with the photon hypothesis,b a
ording to whi h oin iden es should not o
ur. It is, nevertheless, equally in a
ord with the virtual radiation hypothesis, if one assumes that the virtual os illators in the opper ontinuously emit virtual uores ent radiation, so that the photoele trons should be observed in the ounting
hambers at arbitrary intervals.c a W. Bothe, Zeits. f. Phys., 37 (1926), 547. b The English edition ontinues: `For if a photon of uores ent radiation produ es a β -ray in one ounting hamber it annot traverse the se ond hamber. Coin iden es should therefore not o
ur' (eds.).
At this point in the English version Compton is mu h more riti al of the BKS theory (eds.): A
ording to the virtual radiation hypothesis, however, oin iden es should have been observed. For on this view the uores ent K radiation is emitted by virtual os illators asso iated with atoms in whi h there is a va an y in the K shell. That is, the opper foil an emit uores ent K radiation only during the short interval of time following the expulsion of a photoele tron from the K shell, until the shell is again o
upied by another ele tron. This time interval −15 is so short (of the order of 10 se .) as to be sensibly instantaneous on the s ale of Bothe's experiments. Sin e on this view the virtual uores ent
Experiment and the ele tromagneti theory
349
But the experiment is important in the sense that it refutes the often suggested idea that a quantum of radiation energy is suddenly emitted in the form of a spheri al wave when an atom passes from one stationary state to another. (2) The omposite photoele tri ee t.40 Wilsona and Augerb have noti ed in their loud expansion photographs that when X-rays eje t photoele trons from heavy atoms, it often o
urs that two or more ele trons are eje ted simultaneously from the same atom. Auger has dedu ed from studying the ranges of these ele trons that, when this o
urs, the total energy of all the emitted ele trons is no larger than that of a quantum of the in ident radiation. When two ele trons are emitted simultaneously it is usually the ase that the enegy of one of them is Ekin = hν − hνK ,
whi h a
ording to the photon theory means that this ele tron is due to the absorption of an in ident photon a
ompanied by the eje tion of an ele tron from the K energy level. The se ond ele tron has in general the energy Ekin = hνK − hνL .
This ele tron an be explained as the result of the absorption by an L ele tron of the Kα-ray emitted when another L ele tron o
upies the pla e left va ant in the K orbit by the primary photoele tron. It is established that all the ele trons that are observed in the omposite photoele tri ee t have to be interpreted in the same way. Their interpretation a
ording to the photon theory thus meets with no di ulties. With regard to the virtual radiation theory, we an take two points of view: rst, under the in uen e of the ex itation produ ed by the primary virtual radiation, virtual uores ent K radiation is emitted by virtual os illators asso iated with all the atoms traversed by the primary beam. In this view, the probability that this virtual uores ent radiation will ause the eje tion of a photoele tron from the same atom as the one radiation is emitted in spheri al waves, the ounting hambers on both sides of the foil should be simultaneously ae ted, and oin ident pulses in the two
hambers should frequently o
ur.
The results of the experiment are thus
ontrary to the predi tions of the virtual radiation hypothesis.
a C. T. R. Wilson, Pro . Roy. So . A, 104 (1923), 1. b P. Auger, Journ. d. Phys., 6 (1926), 183.
350
A. H. Compton
that has emitted the primary photoele tron is so small that su h an event will almost never o
ur; se ond, we an alternatively assume that a virtual os illator emitting virtual K radiation is asso iated only with an atom in whi h there is a va ant pla e in the K shell. In this ase, sin e the virtual radiation pro eeds from the atom that has emitted the primary photoele tron, we ould expe t with extremely large probability that it should ex ite a photoele tron from the L shell of its own atom, thus a
ounting for the omposite photoele tri ee t. But in this view the virtual uores ent radiation is emitted only during a very short interval after the eje tion of the primary photoele tron, in whi h ase Bothe's uores en e experiment, des ribed above, should have shown some oin iden es. One sees thus that the virtual radiation hypothesis is irre on ilable both with the omposite photoele tri ee t and with the absen e of
oin iden es in Bothe's uores en e experiment. The photon hypothesis, instead, is in omplete a
ord with both these experimental fa ts. (3) Bothe and Geiger's oin iden e experiments.41 We have seen that a
ording to Bohr, Kramers and Slater's theory, virtual radiation42 is being ontinually s attered by matter traversed by X-rays, but only o
asionally is a re oil ele tron emitted. This is in sharp ontrast with the photon theory, a
ording to whi h a re oil ele tron appears every time a photon is s attered. A ru ial test between the two points of view is aorded by an experiment devised and brilliantly performed43 by Bothe and Geiger.a X-rays were passed through hydrogen gas, and the resulting re oil ele trons and s attered rays were dete ted by means of two dierent point ounters pla ed on opposite sides of the olumn of gas. The hamber for ounting the re oil ele trons was left open, but a sheet of thin platinum prevented the re oil ele trons from entering the
hamber for ounting the s attered rays. Of ourse not every photon entering the se ond ounter ould be noti ed, for its dete tion depends upon the produ tion of a β -ray. It was found that there were about ten re oil ele trons for every s attered photon that re orded itself. The impulses from the ounting hambers were re orded on a moving photographi lm. In observations over a total period of over ve hours, sixty-six su h oin iden es were observed. Bothe and Geiger al ulate that a
ording to the statisti s of the virtual radiation theory the han e was only 1 in 400 000 that so many oin iden es should have o
urred. a W. Bothe and H. Geiger, Zeits. f. Phys., 26 (1924), 44; 32 (1925), 639.
Experiment and the ele tromagneti theory
351
This result therefore is in a
ord with the predi tions of the photon theory, but is dire tly ontrary to the statisti al view of the s attering pro ess. (4) Dire tional emission of s attered X-rays. Additional information regarding the nature of s attered X-rays has been obtained by studying the relation between the dire tion of eje tion of the re oil ele tron and the dire tion in whi h the asso iated photon pro eeds. A
ording to the photon theory, we have a de nite relation (equation (4)) between the angle at whi h the photon is s attered and the angle at whi h the re oil ele tron is eje ted. But a
ording to any form of spreading wave theory, in luding that of Bohr, Kramers and Slater, the s attered rays may produ e ee ts in any dire tion whatever, and there should be no
orrelation between the dire tions in whi h the re oil ele trons pro eed and the dire tions in whi h the se ondary β -rays are eje ted by the s attered X-rays. A test to see whether su h a relation exists has been made,a using Wilson's loud apparatus, in the manner shown diagrammati ally in Fig. 3. Ea h re oil ele tron produ es a visible tra k, and o
asionally a se ondary tra k is produ ed by the s attered X-ray. When but one re oil ele tron appears on the same plate with the tra k due to the s attered rays, it is possible to tell at on e whether the angles satisfy equation (4). If two or three re oil tra ks appear,44 the measurements on ea h tra k
an be approximately45 weighted. Out of 850 plates taken in the nal series of readings, thirty-eight show both re oil tra ks and se ondary β -ray tra ks. On eighteen of these plates the observed angle ϕ46 is within 20 degrees of the angle
al ulated from the measured value of θ, while the other twenty tra ks are distributed at random angles. This ratio 18:20 is about that to be expe ted for the ratio of the rays s attered by the part of the air from whi h the re oil tra ks ould be measured to the stray rays from various sour es. There is only about 1 han e in 250 that so many se ondary β -rays should have appeared at the theoreti al angle. If this experiment is reliable, it means that there is s attered X-ray energy asso iated with ea h re oil ele tron su ient to produ e a β -ray, and pro eeding in a dire tion determined at the moment of eje tion of the re oil ele tron. In other words, the s attered X-rays pro eed in photons, that is47 in dire ted quanta of radiant energy. This result, like that of Bothe and Geiger, is irre on ilable with Bohr, a A. H. Compton and A. W. Simon, Phys. Rev., 26 (1925), 289.
352
A. H. Compton
Fig. 3. If the X-rays ex ite a re oil ele tron at an angle predi ts a se ondary
β -parti le
at an angle
θ,
the photon theory
ϕ.
Kramers and Slater's hypothesis of the statisti al produ tion of re oil and photoele trons. On the other hand, both of these experiments are in omplete a
ord with the predi tions of the photon theory.
Reliability of experimental eviden e While all of the experiments that we have onsidered are di ult to re on ile with the lassi al theory that radiation onsists of ele tromagneti waves, only those dealing with the individual s attering pro ess48 aord ru ial tests between the photon theory and the statisti al theory of virtual radiation. It be omes of espe ial importan e, therefore, to
onsider the errors to whi h these experiments are subje t. When two point ounters are set side by side, it is very easy to obtain
oin iden es from extraneous sour es. Thus, for example, the apparatus must be ele tri ally shielded so perfe tly that a spark on the high-tension out t that operates the X-ray tube may not produ e oin ident impulses
Experiment and the ele tromagneti theory
353
in the two ounters. Then there are high-speed α- and β -rays, due to radium emanation in the air and other radioa tive impurities, whi h may pass through both hambers and produ e spurious oin iden es. The method whi h Bothe and Geiger used to dete t the oin iden es, of49 re ording on a photographi lm the time of ea h pulse, makes it possible to estimate reliably50 the probability that the oin iden es are due to
han e. Moreover, it is possible by auxiliary tests to determine whether spurious oin iden es are o
urring for example, by operating the out t as usual, ex ept that the X-rays are absorbed by a sheet of lead. It is espe ially worthy of note that in the uores en e experiment the photon theory predi ted absen e of oin iden es, while in the s attering experiment it predi ted their presen e. It is thus di ult to see how both of these ounter experiments an have been seriously ae ted by systemati errors. In the loud expansion experiment the ee t of stray radiation is to hide the ee t sought for, rather than to introdu e a spurious ee t. It is possible that due to radioa tive ontamination and to stray s attered X-rays β -parti les may appear in dierent parts of the hamber, but it will be only a matter of han e if these β -parti les appear in the position predi ted from the dire tion of eje tion of the re oil ele trons. It was in fa t only by taking great are to redu e su h stray radiations to a minimum that the dire tional relations were learly observed in the photographs. It would seem that the only form of onsistent error that
ould vitiate the result of this experiment would be the psy hologi al51 one of misjudging the angles at whi h the β -parti les appear. It hardly seems possible, however, that errors in the measurement of these angles
ould be large enough to a
ount for the strong apparent tenden y for the angles to t with the theoreti al formula. It is perhaps worth mentioning further that the initial publi ations of the two experiments on the individual s attering pro ess were made simultaneously, whi h means that both sets of experimenters had independently rea hed a on lusion opposed to the statisti al theory of the produ tion of the β -rays. Nevertheless,52 given the di ulty of the experiments and the importan e of the on lusions to whi h they have led, it is highly desirable that both experiments should be repeated by physi ists from other laboratories.
354
A. H. Compton
Summary
The lassi al theory that radiation onsists of ele tromagneti waves propagated in all dire tions through spa e53 is intimately onne ted to the idea of the ether, whi h is di ult to on eive. It aords no adequate pi ture of the manner in whi h radiation is emitted or absorbed. It is in onsistent with the experiments on the photoele tri ee t, and is entirely helpless to a
ount for the hange of wavelength of s attered radiation or the produ tion of re oil ele trons. The theory of virtual os illators and virtual radiation whi h are asso iated statisti ally with sudden jumps of atomi energy and the emission of photoele trons and re oil ele trons, does not seem to be in onsistent with any of these phenomena as viewed on a ma ros opi s ale. This theory, however,54 retains the di ulties inherent in the on eption of the ether and seems powerless to predi t the hara teristi s of the photoele trons and the re oil ele trons. It55 is further di ult to re on ile with the
omposite photoele tri ee t and is also ontrary to Bothe's and Bothe and Geiger's oin iden e experiments and to the ray tra k experiments relating the dire tions of eje tion of a re oil ele tron and of emission of the asso iated s attered X-ray. The photon theory avoids the di ulties asso iated with the on eption of the ether.56 The produ tion and absorption of radiation is very simply onne ted with the modern idea of stationary states. It supplies a straightforward explanation of the major hara teristi s of the photoele tri ee t, and it a
ounts in the simplest possible manner for the
hange of wavelength a
ompanying s attering and the existen e of re oil ele trons. Moreover, it predi ts a
urately the results of the experiments with individual radiation quanta, where the statisti al theory fails. Unless the four57 experiments on the individual events58 are subje t to improbably large experimental errors, the on lusion is, I believe, unes apable that radiation onsists of dire ted quanta of energy, i.e., of photons, and that energy and momentum are onserved when these photons intera t with ele trons or atoms. Let me say again that this result does not mean that there is no truth in the on ept of waves of radiation. The on lusion is rather that energy is not transmitted by su h waves. The power of the wave on ept in problems of interferen e, refra tion, et ., is too well known to require emphasis. Whether the waves serve to guide the photons, or whether there is some other relation between photons and waves is another and
Experiment and the ele tromagneti theory a di ult question.
355
356
A. H. Compton
Dis ussion of Mr Compton's report
Mr Lorentz. I would like to make two omments.
First on the question of the ether. Mr Compton onsiders it an advantage of the photon theory that it allows us to do without the hypothesis of an ether whi h leads to great di ulties. I must say that these di ulties do not seem so great to me and that in my opinion the theory of relativity does not ne essarily rule out the on ept of a universal medium. Indeed, Maxwell's equations are ompatible with relativity, and one an well imagine a medium for whi h these equations hold. One an even, as Maxwell and other physi ists have done with some su
ess, onstru t a me hani al model of su h a medium. One would have to add only the hypothesis of the permeability of ponderable matter by the ether to have all that is required. Of ourse, in making these remarks, I should not wish to return in any way to these me hani al models, from whi h physi s has turned away for good reasons. One an be satis ed with the
on ept of a medium that an pass freely through matter and to whi h Maxwell's equations an be applied. In the se ond pla e: it is quite ertain that, in the phenomena of light, there must yet be something other than the photons. For instqn em in a dira tion experiment performed with very weak light, it an happen that the number of photons present at a given instant between the dira ting s reen and the plane on whi h one observes the distribution of light, is very limited. The average number an even be smaller than one, whi h means that there are instants when no photon is present in the spa e under onsideration. This learly shows that the dira tion phenomena annot be produ ed by some novel a tion among the photons. There must be something that guides them in their progress and it is natural to seek this something in the ele tromagneti eld as determined by the lassi al theory. This notion of ele tromagneti eld, with its waves and vibrations would bring us ba k, in Mr Compton's view, to the notion of ether.
Mr Compton.
It seems, indeed, di ult to avoid the idea of waves in the dis ussion of opti al phenomena. A
ording to Maxwell's theory the ele tri and magneti properties of spa e lead to the idea of waves as dire tly as did the elasti ether imagined by Fresnel. Why the spa e having su h magneti properties should bear the name of ether is perhaps simply a matter of words. The fa t that these properties of spa e immediately lead to the wave equation with velo ity c is a mu h
Dis ussion
357
more solid basis for the hypothesis of the existen e of waves than the old elasti ether. That something (E and H ) propagates like a wave with velo ity c seems evident. However, experiments of the kind we have just dis ussed show, if they are orre t, that the energy of the bundle of X-rays propagates in the form of parti les and not in the form of extended waves. So then, not even the ele tromagneti ether appears to be satisfa tory.
Mr Bragg.
In his report Mr Compton has dis ussed the average momentum omponent of the ele trons in the dire tion of motion of the photon, and he has informed us of the on lusion, at whi h several experimenters have arrived, that this forward average omponent is equal to the momentum of the light quantum whose energy has been absorbed and is found again in that of the photoele tron. I would like to report in this onne tion some results obtained by Mr Williams.a Mono hromati X-rays, with wavelength lying between 0.5 Å and 0.7 Å, enter a Wilson loud hamber ontaining oxygen or nitrogen. The traje tories of the photoele trons are observed through a stereos ope and their initial dire tions are measured. Sin e the speed of the photoele trons is exa tly known (the ionisation energy being weak by
omparison to the quantity hν ), a measurement of the initial dire tion is equivalent to a measurement of momentum in the forward dire tion. Williams nds that the average momentum omponent in this dire tion h is in all ases markedly larger than the quantity hν c or λ . These results
an be summarised by a omparison with the s heme proposed by Perrin and Auger ([P. Auger and F. Perrin℄, Journ. d. Phys. [6th series, vol. 8℄ (February 1927), [93℄). They are in perfe t agreement with the cos2 θ law, provided one assumes that the magneti impulse Tm is equal to hν 1.8 hν c and not just c as these authors assume. One should not atta h any parti ular importan e to this number 1.8, be ause the range of the examined wavelengths is too small. I mention it only to show that it is possible that the simple law proposed by Mr Compton might not be exa t. I would like to point out that this method of measuring the forward
omponent of the momentum is more pre ise than an attempt made to establish results about the most probable dire tion of emission.
Mr Wilson says that his own observations, dis ussed in his Memoir a Cf. the relevant footnote on p. 340 (eds.).
358
A. H. Compton
of 1923a (but whi h do not pretend to be very pre ise) seem to show that in fa t the forward momentum omponent of the photoele trons is, on average, mu h larger than what one would derive from the idea that the absorbed quantum yields all of its momentum to the expelled ele tron.
Mr Ri hardson. When they are expelled by ertain X-rays,
the ele trons have a momentum in the dire tion of propagation of the rays equal to 1.8 hν c . If I have understood Mr Bragg orre tly, this result is not the ee t of some spe i elementary pro ess [a tion℄, but the average result for a great number of observations in whi h the ele trons were expelled in dierent dire tions. Whether or not the laws of energy and momentum onservation apply to an elementary pro ess, it is ertain that they apply to the average result for a great number of these pro esses. Therefore, the pro ess [pro essus℄ we are talking about must be governed by the equations for momentum and energy. If for simpli ity we ignore the re nements introdu ed by relativity, these equations are hν = mv + M V c and hν =
1 1 2 mv 2 + M V , 2 2
where m and M are the masses, v and V the velo ities of the ele trons and of the positive residue; the overbars express that these are averages. The experiments show that the average value of mv is 1.8 hν c and not hν . This means that M V is not zero, so that we
annot ignore this term c in the equation. If we onsider, for instan e, the photoele tri ee t on a hydrogen atom, we have to take the ollision energy of the hydrogen nu leus into a
ount in the energy equation.
Mr Lorentz. The term
1 2 59 2 mv ?
2 1 2MV
will however be mu h smaller than
Mr Ri hardson.
It is approximately its 1850th part: that annot always be onsidered negligible.
Mr Born thinks that he is speaking also for several other members a Referen ed in footnote on p. 336 (eds.).
Dis ussion
359
in asking Mr Compton to explain why one should expe t that the momentum imparted to the ele tron be equal to hν c .
Mr Compton.
When radiation of energy hν is absorbed by an atom whi h one surely has to assume in order to a
ount for the kineti energy of the photoele tron the momentum imparted to the atom by this radiation is hν c . A
ording to the lassi al ele tron theory, when an atom omposed of a negative harge −e of mass m and a positive
harge +e of mass M absorbs energy from an ele tromagneti wave, the momenta imparted to the two elementary harges [éle trons℄ are inversely proportional to their masses. This depends on the fa t that the forward momentum is due to the magneti ve tor, whi h a ts with a for e proportional to the velo ity and onsequently more strongly on the harge having the smaller mass.60 Ee tively, the momentum is thus re eived by the harge with the smaller mass.
Mr Debye.
Is the reason why you think that the rest of the atom does not re eive any of the forward momentum purely theoreti al?
Mr Compton. The photographs of the traje tories of the photo-
ele trons show, in a
ordan e with Auger's predi tion, that the forward
omponent of the momentum of the photoele tron is, on average, the same as that of the photon. That means, learly, that on average the rest of the atom does not re eive any momentum.
Mr Dira .
I have examined the motion of an ele tron pla ed in an arbitrary for e eld a
ording to the lassi al theory, when it is subje t to in ident radiation, and I have shown in a ompletely general way that at every instant the fra tion of the rate of hange [vitesse de variation℄ of the forward momentum of the ele tron due to the in ident radiation is equal to 1c times the fra tion of the rate of hange of the energy due to the in ident radiation. The nu leus and the other ele trons of the atom produ e hanges of momentum and of energy that at ea h instant are simply added to those produ ed by the in ident radiation. Sin e the radiation must modify the ele tron's orbit, it must also hange the fra tion of the rate of hange of the momentum and of the energy that
omes from the nu leus and the other ele trons, so that it would be ne essary to integrate the motion in order to determine the total hange produ ed by the in ident radiation in the energy and the momentum.
360
A. H. Compton
Mr Born. I would like to mention here a paper by Wentzel,
a
whi h ontains a rigorous treatment of the s attering of light by atoms a
ording to quantum me hani s. In it, the author onsiders also the in uen e of the magneti for e, whi h allows him to obtain the quantum analogue of the lassi al light pressure. It is only in the limiting ase of very short wavelengths that one nds that the momentum of light hν c is
ompletely transmitted to the ele tron; in the ase of large wavelengths an in uen e of the binding for es appears.
Mr Ehrenfest.
One an show by a very simple example where the surplus of forward momentum, whi h we have just dis ussed, an have its origin. Take a box whose inner walls re e t light ompletely, but diusedly, and assume that on the bottom there is a little hole. Through the latter I shine a ray of light into the box whi h omes and goes inside the box and pushes away its lid and bottom. The lid then has a surplus of forward momentum.
Mr Bohr.
With regard to the question of waves or photons dis ussed by Mr Compton, I would like to make a few remarks, without pre-empting the general dis ussion. The radiation experiments have indeed revealed features that are not easy to re on ile within a lassi al pi ture. This di ulty arises parti ularly in the Compton ee t itself. Several aspe ts of this phenomenon an be des ribed very simply with the aid of photons, but we must not forget that the hange of frequen y that takes pla e is measured using instruments whose fun tioning is interpreted a
ording to the wave theory. There seems to be a logi al
ontradi tion here, sin e the des ription of the in ident wave as well as that of the s attered wave require that these waves be nitely extended [limitées℄ in spa e and time, while the hange in energy and in momentum of the ele tron is onsidered as an instantaneous phenomenon at a given point in spa etime. It is pre isely be ause of su h di ulties that Messrs Kramers, Slater and myself were led to think that one should
ompletely reje t the idea of the existen e of photons and assume that the laws of onservation of energy and momentum are true only in a statisti al way. The well-known experiments by Geiger and Bothe and by Compton a
a Born is presumably referring to Wentzel's se ond paper on the photoele tri ee t (Wentzel 1927). Compare Mehra and Re henberg (1987, pp. 835 .) (eds.). a This dis ussion ontribution by Bohr is reprinted and translated also in vol. 5 of Bohr's Colle ted Works (Bohr 1984, pp. 20712). (eds.).
Dis ussion
361
and Simon, however, have shown that this point of view is not admissible and that the onservation laws are valid for the individual pro esses, in a
ordan e with the on ept of photons. But the dilemma before whi h we are pla ed regarding the nature of light is only a typi al example of the di ulties that one en ounters when one wishes to interpret the atomi phenomena using lassi al on epts. The logi al di ulties with a des ription in spa e and time have sin e been removed in large part by the fa t that it has been realised that one en ounters a similar paradox with respe t to the nature of material parti les. A
ording to the fundamental ideas of Mr de Broglie, whi h have found su h perfe t
on rmation in the experiments of Davisson and Germer, the on ept of waves is as indispensable in the interpretation of the properties of material parti les as in the ase of light. We know thereby that it is equally ne essary to attribute to the wave eld a nite extension in spa e and in time, if one wishes to de ne the energy and the momentum of the ele tron, just as one has to assume a similar nite extension in the ase of the light quantum in order to be able to talk about frequen y and wavelength. Therefore, in the ase of the s attering pro ess, in order to des ribe the two hanges ae ting the ele tron and the light we must work with four wave elds (two for the ele tron, before and after the phenomenon, and two for the quantum of light, in ident and s attered), nite in extension, whi h meet in the same region of spa etime.a In su h a representation all possibility of in ompatibility with a des ription in spa e and time disappears. I hope the general dis ussion will give me the opportunity to enter more deeply into the details of this question, whi h is intimately tied to the general problem of quantum theory.
Mr Brillouin.
I have had the opportunity to dis uss Mr Compton's report with Mr Auger,b and wish to make a few omments on this topi . A purely orpus ular des ription of radiation is not su ient to understand the pe uliarities of the phenomena; to assume that energy is transported by photons hν is not enough to a
ount for all the ee ts of radiation. It is essential to omplete our information by giving the dire tion of the ele tri eld; we annot do without this eld, whose role in the wave des ription is well known. a Compare also the dis ussion ontributions below, by Pauli, S hrödinger and others (eds.). b As noted in se tion 1.4, this and other reports had been ir ulated among the parti ipants before the onferen e (eds.).
362
A. H. Compton
Fig. 1.
I shall re all in this ontext a simple argument, re ently given by Auger and F. Perrin, and whi h illustrates learly this remark. Let us onsider the emission of ele trons by an atom subje t to radiation, and let us examine the distribution of the dire tions of emission. This distribution has usually been observed in a plane ontaining the light ray and the dire tion of the ele tri eld (the in ident radiation is assumed to be polarised); let ϕ be the angle formed by the dire tion of emission of the photoele trons and the ele tri eld h; as long as the in ident radiation is not too hard, the distribution of the photoele trons is symmetri around the ele tri eld; one an then show that the probability law ne essarily takes the form A cos2 ϕ. Indeed, instead of observing the distribution in the plane of in iden e (Fig. 1), let us examine it in the plane of the wave; the same distribution law will still be valid; and it is the only one that would allow us to obtain, through the superposition of two waves polarised at right angles, an entirely symmetri distribution A cos2 ϕ + A sin2 ϕ = A .
Now, from the point of view of waves, one must ne essarily obtain this result, a beam [rayonnement℄ of natural light having no privileged dire tion in the plane of the wave. These symmetry onsiderations, whi h any theory of radiation must respe t, provide a substantial di ulty for the stru tural theories of the photon (Bubb's quantum ve tor, for instan e). Summing up, the dis ontinuity of the radiation manifests itself just in the most elementary way, through the laws of onservation of energy and momentum, but the detailed analysis of the phenomena is interpreted more naturally from the ontinuous point of view. For the problem of emission of the photoele trons, a omplete theory has been given by
Dis ussion
363
Wentzel, by means of wave me hani s.a He nds the A cos2 ϕ law of F. Perrin and Auger for radiation of low penetration; when the radiation is harder, Wentzel obtains a more omplex law, in whi h the ele trons tend to be emitted in larger numbers in the forward dire tion. His theory, however, seems in omplete with regard to this point sin e, if I am not mistaken, he has assumed the immobility of the atomi nu leus; now, nothing tells us a priori how the momentum hν c of the photon is going to be distributed between the nu leus and the emitted ele tron.
Mr Lorentz.
Allow me to point out that a
ording to the old ele tron theory, when one has a nu leus and an ele tron on whi h a beam of polarised light falls, the initial angular momentum of the system is always onserved. The angular momentum imparted to the ele tronnu leus system will be provided at the expense of the angular momentum of the radiation eld.
Mr Compton.
The on eption of the photon diers from the
lassi al theory in that, when a photoele tron is emitted, the photon is
ompletely absorbed and no radiation eld is left. The motion of the photoele tron must thus be su h that the nal angular momentum of the ele tron-nu leus system will be the same as the initial momentum of the photon-ele tron-nu leus system. This ondition restri ts the possible traje tories of the emitted photoele tron.
Mr Kramers.
In order to interpret his experiments, Mr Compton needs to know how the absorption µ is divided between a omponent τ , due to the `true' absorption, and a omponent σ due to the s attering. We do not know with ertainty that, if µ an be written in the form Cλa + D, the onstant D truly represents the s attering for large wavelengths, where Cλa is no longer small ompared to D. In general, thus, spe i measurements of σ are ne essary. Did you have su ient information regarding the values of σ and τ in your experiments?
Mr Compton. The most important ase in whi h it is ne essary
to distinguish between the true absorption τ and the absorption σ due to the s attering, is that of arbon. For this ase, Hewletta has measured a This is presumably Wentzel's treatment from his rst paper on the photoele tri ee t (Wentzel 1926). Compare again Mehra and Re henberg (1987, pp. 835 .) (eds.). a [C. W. Hewlett, Phys. Rev., 17 (1921), 284.℄
364
A. H. Compton
σ dire tly for the wavelength 0.71 Å and the total absorption µ over a large range of wavelengths. The dieren e between µ and σ for the wavelength 0.71 Å orresponds to τ for this wavelength. A
ording to Owen's formula this τ is proportional to λ3 ; we an thus al ulate τ for all wavelengths. The dieren e between this value of τ and the measured value of µ orresponds to the value of σ for the wavelengths onsidered. Sin e τ is relatively small in the ase of arbon, espe ially for small wavelengths, this pro edure yields a value for σ that annot be very impre ise.
Mr Bragg. When one onsults the original literature on this
subje t, one is stru k by how mu h the X-ray absorption measurements leave to be desired, both with regard to pre ision as well as with regard to the extent of the s ale of wavelengths for whi h they have been performed.
Mr Pauli. How large is the broadening of the modi ed rays? Mr Compton. The experiments have shown learly that the mod-
i ed ray is broader than the unmodi ed ray. In the typi al ase of the ray λ 0.7 Å s attered by arbon, the broadening is of order 0.005 angström. Unfortunately, the experiments on erning this point are far from being satisfa tory, and this number should be onsidered only as a rough approximation.
Mr Pauli.
The broadening of the modi ed ray an be interpreted theoreti ally in two ways, whi h to tell the truth redu e to the same a
ording to quantum me hani s. First, the ele tron, in a given stationary state of the atom, has a ertain velo ity distribution with regard to magnitude and dire tion. That gives rise to a broadening of the frequen y of the s attered rays through the Doppler ee t, a broadening v whose order of magnitude is ∆λ λ = c , where v denotes the average velo ity of the ele tron in the atom. In order to onvey the se ond means of explanation, I would like to sket h brie y the meaning of the Compton ee t in wave me hani s.a This meaning is based rst of all on the wave equation ! X ∂2ψ 4π 2 e2 X 2 4πi e X ∂ψ 2 2 − 2 − ϕα ϕ + m0 c ψ = 0 ∂x2α h c α ∂xα h c2 α α α a For a modern dis ussion, see Björken and Drell (1964, Chapter 9) (eds.).
Dis ussion
365
and further on the expression iSα = ψ
∂ψ 4πi e ∂ψ ∗ − ψ∗ + ϕα ψψ ∗ , ∂xα ∂xα h c
in whi h ψ is S hrödinger's fun tion, ψ ∗ the omplex onjugate value and ϕα the four-potential of the ele tromagneti eld. Given Sα , one
al ulates the radiation from lassi al ele trodynami s. If now in the wave equation one repla es ϕα by the potential of an in ident plane wave, the terms that are proportional to the amplitude of this wave an be onsidered in nitely small in the rst order, and one an apply the approximation methods of perturbation theory. This now is a point where one needs to be espe ially areful. It is all-important to know what one will take as the unperturbed eld ψ , whi h must orrespond to a solution of the wave equation for the free parti le ( orresponding to ϕα = 0).61 One nds that in order to agree with the observations, it is ne essary to take two in nitely extended mono hromati wave trains as being already present in the unperturbed solution, of whi h one orresponds to the initial state, the other to the nal state of the Compton pro ess. In my opinion this assumption, on whi h the theories of the Compton ee t by S hrödinger, Gordon and Klein are based, is unsatisfa tory and this defe t is orre ted only by Dira 's quantum ele trodynami s.a But if one makes this assumption, the urrent distribution of the unperturbed solution orresponds to that of an in nitely extended dira tion grating [un réseau in niment étendu℄ that moves with a onstant speed, and the a tion of the radiation on this grating leads to a sharp modi ed ray. If one onsiders a bound ele tron in an atom, one has to repla e one
omponent of the solution ψ in the unperturbed harge and urrent distribution by the eigenfun tion of the atom in the stationary state
onsidered, and the other omponent by a solution orresponding to the nal state of the Compton pro ess (belonging to the ontinuous spe trum of the atom), whi h at great distan e from the atom behaves more or less as a plane wave. One thus has a moving grating that rst of all depends only on the nite extension of the atom and in the se ond pla e has omponents no longer moving with the same speed at all. This gives rise to a la k of sharpness of the shifted ray of the s attered radiation. But one an show that, from the point of view of quantum me hani s, this explanation for the la k of sharpness of the shifted ray is just another form of the explanation given in the rst instan e and whi h relies on the dierent dire tions of the initial velo ities of the ele trons in the atom. a Compare below S hrödinger's ontribution and the ensuing dis ussion (eds.).
366
A. H. Compton
For a
ording to quantum me hani s if ψ = f (x, y, z)e2πivt
is the eigenfun tion orresponding to a given stationary state of the atom, the fun tion62 Z Z Z 2πi ϕ(px , py , pz ) = f (x, y, z)e− λ (px x+py y+pz z) dxdydz ,
whi h one obtains by de omposing f in plane waves a
ording to Fourier an be interpreted in the sense that |ϕ(p)|2 dpx dpy dpz denotes the probability that in the given stationary state the omponents of the momentum of the ele tron lie between px , py , pz and px + dpx , et . Now, if through the resulting velo ity distribution of the ele trons in the atom one al ulates the broadening of the shifted line a
ording to the rst point of view, for light of su iently short wavelength with respe t to whi h the ele tron an be onsidered free in the atom (and it is only under these onditions that the pro edure is legitimate), one nds exa tly the same result as with the other method des ribed.a
Mr Compton.
Jaun ey has al ulated the broadening of the modi ed ray using essentially the method that Mr Pauli has just des ribed. Jaun ey assumed, however, that the velo ities of the ele tron are the ones given by Bohr's theory of orbital motions. The broadening thus obtained is larger than that found experimentally.
Mrs Curie.
In his very interesting report, Professor Compton has dwelt on emphasising the reasons that lead one to adopt the theory of a ollision between a quantum and a free ele tron. Along the same line of thought, I think it is useful to point out the following two views: First, the existen e of ollision ele trons seems to play a fundamental role in the biologi al ee ts produ ed on living tissues by very highfrequen y radiation, su h as the most penetrating γ -rays emitted by radioelements. If one assumes that the biologi al ee t may be attributed to the ionisation produ ed in the ells subje ted to radiation, this ee t annot depend dire tly on the γ -rays, but is due to the emission of se ondary β -rays that a
ompanies the passage of the γ -rays through matter. Before the dis overy of the ollision ele trons, only a single a Pauli was possibly the rst to introdu e the probability interpretation of the wave fun tion in momentum spa e, in a letter to Heisenberg of 19 O tober 1926 (Pauli, 1979, pp. 3478). Cf. the footnote on p. 117 (eds.).
Dis ussion
367
me hanism was known for the produ tion of these se ondary rays, that
onsisting in the total absorption of a quantum of radiation by the atom, with the emission of a photoele tron. The absorption oe ient τ relating to this pro ess varies with the wavelength λ of the primary γ -radiation, as well as with the density [ρ℄ of the absorbing matter and the atomi number N of the atoms omposing it, a
ording to the well-known relation of Bragg and Peir e τρ = ΛN 3 λ3 , where Λ is a oe ient that has a onstant value for frequen ies higher than that of the K dis ontinuity. If this relation valid in the domain of X-rays
an be applied to high-frequen y γ -rays, the resulting value of τρ for the light elements is so weak that the emission of photoele trons appears unable to explain the biologi al ee ts of radiation on the living tissues traversed.a The issue appears altogether dierent if one takes into onsideration the emission of ollision ele trons in these tissues, following Compton's theory. For a ollimated primary beam of γ -rays, the fra tion of ele tromagneti energy onverted into kineti energy of the ele trons per unit mass of the absorbing matter is given by the oe ient σa σ0 α = , 2 ρ (1 − 2α) ρ
where σρ0 is the s attering oe ient per unit mass valid for medium frequen y X-rays, a
ording to the theory of J. J. Thomson, and is lose hν to 0.2, while α is Compton's parameter α = mc 2 (h Plan k's onstant, ν primary frequen y, m rest mass of the ele tron, c speed of light). Taking α = 1.2, a value suitable for an important group of γ -rays (equivalent potential 610 kilovolts), one nds σρa = 0.02, that is, 2 per ent of the primary energy is onverted to energy of the ele tron per unit mass of absorbing matter, when e a possibility of interpreting the observed biologi al ee ts. To this dire t produ tion of ollision ele trons along the traje tory of the primary beam is added, in an extended medium, a supplementary produ tion, from the fa t that to ea h of these ele trons
orresponds a s attered quantum, with a smaller value than the primary quantum, and that this s attered quantum an in turn be subje t to the Compton ee t in the medium through whi h it propagates, with produ tion of a new ollision ele tron and of an even smaller quantum. This pro ess, inde nitely repeatable and alled the `multiple Compton ee t' seems in fa t to have been observed by ertain authors.a Not only a It is true that several authors have re ently ontested the legitima y of extending the absorption law of Bragg and Peir e to X-rays. a [B.℄ Rajewsky, Forts hritte auf dem Gebiet der Roentgenstrahlung, 35 (1926), 262.
368
A. H. Compton
is the number of ollision ele trons thereby multiplied, but, further, the primary quantum, redu ed by su
essive ollisions takes on values for whi h the absorption with emission of photoele trons be omes more and more probable. These fa ts have an important reper ussion on the te hnique of X-ray therapy. Certain authors had, in fa t, denied the usefulness of produ ing very high-voltage apparatus providing X-rays of very high frequen y and very high penetrating power, whose use is otherwise onvenient owing to the uniformity of irradiation they allow one to attain. If these rays had been devoid of e a y, one would have had to give up on their use. Su h is not the ase if one adopts the point of view of the Compton ee t, and it is then legitimate to dire t the te hnique towards the use of high voltages. Another interesting point of view to examine is that of the emission of β -rays by radioa tive bodies. Professor Compton has pointed out that among the β -rays of se ondary origin, some ould be ollision ele trons produ ed by the s attering of the primary γ -rays on the ele trons
ontained in the matter they traverse. It is in an ee t of this type that Thibaud thinks one may nd the explanation for the appearan e of the magneti spe tra of the se ondary γ -rays. These spe tra are omposed of lines that may be attributed to groups of photoele trons of the same speed, ea h of whi h is emitted by absorption in a thin metalli envelope of a group of homogeneous γ -rays emitted by a radioelement ontained in this envelope. Ea h line of photoele tri origin is a
ompanied by a band beginning at the line itself and extending towards the region of low velo ities. Thibaud thinks that this band ould be due to photoele trons expelled from the s reen by those γ -rays that, in this same s reen, had suered the Compton ee t with redu tion of frequen y. This interpretation appears plausible; however, in order to prove it, it would be ne essary to study the stru ture of the band and nd in the same spe trum the band that may be attributed to the ollision ele trons orresponding to the s attered γ -rays. An analogous problem arises regarding the emission of β -rays by radioa tive bodies with negligible thi kness, so as to eliminate, as far as possible, the se ondary ee ts due to the supports and envelopes. One then observes a magneti spe trum that may be attributed to the radioelement alone and onsisting either of a ontinuous band, or of the superposition of a ontinuous spe trum and a line spe trum. The
Dis ussion
369
latter has re eived a satisfa tory interpretation in some re ent papers (L. Meitner, Ellis, Thibaud, et .). A line is due to a group of photoele trons with the same speed expelled from the levels of the radioa tive atoms by a group of homogeneous γ -rays produ ed in their nu lei. This ee t is alled `internal onversion', sin e one assumes that the quantum emitted by an atomi nu leus is reabsorbed in the ele tron loud [enveloppe éle tronique℄ of the same atom. The great majority of observed lines nd their explanation in this hypothesis. The interpretation of the ontinuous spe trum appears to present more di ulties. Some authors attribute it only to the primary β -rays, while others onsider the possibility of a se ondary origin and invoke the Compton ee t as a possible ause of its produ tion (L. Meitner). This would be an `internal' Compton ee t, su h that a γ -ray emitted from the nu leus of an atom would experien e a ollision with one of the weakly bound ele trons at the periphery of the same atom. If that were the ase, the velo ity distribution of the emitted ollision ele trons would not be arbitrary, but would have to onform to the predi tions of Compton's theory. I have losely examined this problem, whi h has a very omplex appearan e.a Ea h group of homogeneous γ -rays is a
ompanied by s attered γ -rays, so that in the dira tion spe trum of the γ -rays, ea h line should experien e a broadening of 0.0485 Å units. The experiments on the dira tion of γ -rays are di ult and not very numerous; so far the broadening ee t has not been reported. Ea h homogeneous group of γ -rays must orrespond to a group of
ollision ele trons, whose velo ity varies ontinuously from zero to an upper limit derived from Compton's theory and whi h in the magneti spe trum orresponds to a band bounded sharply on the side of the large velo ities. The same group of γ -rays may orrespond to further groups of photoele trons expelled from the dierent levels K, L, et . of the atom through internal absorption of the s attered γ -rays. For ea h group of photoele trons, the velo ity of emission lies between two well-de ned limits. The upper limit orresponds to the surplus energy of the primary γ -rays with respe t to the extra tion work W hara teristi of the given level; the lower limit orresponds to the surplus energy, with respe t to the same work, of the γ -rays s attered in the dire tion opposite to that of the primary rays, and having experien ed be ause of a [M.℄ Curie, Le Journal de Physique et le Radium, 7 (1926), 97.
370
A. H. Compton
that the highest loss of frequen y. In the magneti spe trum, ea h group of photoele trons will be represented by a band equally well bounded on the side of the large and of the small velo ities, with the same dieren e between the extreme energies for ea h band. It is easy to see that in the same magneti spe trum the dierent bands
orresponding to the same group of γ -rays may partially overlap, making it di ult to analyse the spe trum omparing the distribution of β -rays with that predi ted by theory. For substan es emitting several groups of γ -rays, the di ulty must be ome onsiderable, unless there are large dieren es in their relative ee tiveness in produ ing the desired ee t. Let us also point out that the ontinuous spe trum due to the Compton ee t may be superposed with a ontinuous spe trum independent of this ee t (that may be attributed for instan e to the primary β -rays). Examination of the experimental data available so far does not yet allow one to draw on lusions onvin ingly. Most of the spe tra are very
omplex, and their pre ise study with respe t to the energy distributions of the β -rays will require very detailed work. In ertain simple spe tra su h as that of the β -rays of RaD, one observes lines of photoele tri origin that may be attributed to a single group of mono hromati γ rays. These lines form the upper edge of bands extending towards low velo ities and probably arising from photoele trons produ ed by the s attered γ -rays. In ertain magneti spe tra obtained from the β -rays of mesothorium 2 in the region of low velo ities, one noti es in the ontinuous spe trum a gap that might orrespond, for the group of primary γ -rays with 58 kilovolts, to the separation between the band due to the ollision ele trons and that due to the photoele trons of the s attered γ -rays.a
Mr S hrödinger
, at the invitation of Mr Ehrenfest, draws on the bla kboard in oloured halk the system of four wave trains by whi h he has tried to represent the Compton ee t in an ans hauli h way [d'une façon intuitive℄b (Ann. d. Phys. 4th series, vol. 82 (1927), 257).c
Mr Bohr. The simultaneous onsideration of two systems of waves
has not the aim of giving a ausal theory in the lassi al sense, but one a D. K. Yovanovit h and A. Prola, Comptes Rendus, 183 (1926), 878. b For dis ussions of the notion of Ans hauli hkeit, see se tions 3.4.7, 4.6 and 8.3 (eds.).
S hrödinger (1928, p. x) later remarked on a mistake pointed out to him by Ehrenfest in the gure as published in the original paper (eds.).
Dis ussion
371
an show that it leads to a symboli analogy. This has been studied in parti ular by Klein. Furthermore, it has been possible to treat the problem in more depth through the way Dira has formulated S hrödinger's theory. We nd here an even more advan ed renun iation of Ans hauli hkeit [intuitivité℄, a fa t very hara teristi of the symboli methods in quantum theory.
Mr Lorentz.
Mr S hrödinger has shown how one an explain the Compton ee t in wave me hani s. In this explanation one onsiders the waves asso iated with the ele tron (e) and the photon (ph), before (1) and after (2) the en ounter. It is natural to think that, of these four systems of waves e1 , ph1 , e2 and ph2 , the latter two are produ ed by the en ounter. But they are not determined by e1 and ph1 , be ause one an for example hoose arbitrarily the dire tion of e2 . Thus, for the problem to be well-de ned, it is not su ient to know e1 and ph1 ; another pie e of data is ne essary, just as in the ase of the ollision of two elasti balls one must know not only their initial velo ities but also a parameter that determines the greater or lesser e
entri ity of the
ollision, for instan e the angle between the relative velo ity and the
ommon normal at the moment of the en ounter. Perhaps one ould introdu e into the explanation given by Mr S hrödinger something that would play the role of this a
essory parameter.
Mr Born. I think it is easy to understand why three of the four
waves have to be given in order for the pro ess to be determined; it su es to onsider analogous ir umstan es in the lassi al theory. If the motions of the two parti les approa hing ea h other are given, the ee t of the ollision is not yet determined; it an be made determinate by giving the position of losest approa h or an equivalent pie e of data. But in wave me hani s su h mi ros opi data are not available. That is why it is ne essary to pres ribe the motion of one of the parti les after the ollision, if one wants the motion of the se ond parti le after the ollision to be determined. But there is nothing surprising in this, everything being exa tly as in lassi al me hani s. The only dieren e is that in the old theory one introdu es mi ros opi quantities, su h as the radii of the atoms that ollide, whi h are eliminated from subsequent
al ulations, while in the new theory one avoids the introdu tion of these quantities.
372
Notes to pp. 329348
Notes to the translation 1 The words `réels ou apparents' are present only in the Fren h version. 2 The English version has only four headings (starting with `(1) How are the waves produ ed?'), and a
ordingly omits the next se tion, on `The problem of the ether', and later referen es to the ether. 3 [d'après les résultats de l'étude des spe tres℄ 4 The English edition distinguishes se tions and subse tions more systemati ally than the Fren h edition, and in this and other small details of layout we shall mostly follow the former. 5 [rappeler qu'il existe une théorie dans laquelle℄ 6 The words `le professeur' are present only in the Fren h edition. 7 [`élément de radiation' ou `quantum de lumière'℄ 8 This se tion is present only in the Fren h version. 9 [énergie ondulatoire℄ 10 The original footnote gives page `145'.
213'.
11 The English edition has ` 12 [à part une onstante℄ 13 [perfe tionnée℄
14 The se ond part of the footnote is printed only in the English edition. 15 The Fren h edition here in ludes the lause `qui a été faite'. 16 Here and in several pla es in the following, the Fren h edition has `simple' where the English one has `single'. 17 [l'a tion dire te du ve teur éle trique de l'onde, prise dans le sens ordinaire℄ 18 [dans la dire tion de propagation de l'onde℄ 19 [puisque℄ 20 This footnote is only present in the English edition. 21 The pre eding lause is only present in the Fren h edition. 22 [la période de l'éle tron dans son mouvement orbital℄ 23 In the English edition this reads: `The fa t that the photoele trons re eive the momentum of the in ident photon'. 24 [sur les rayons X se ondaires et les rayons
γ℄
25 This footnote appears only in the Fren h edition. 26 [ne fournirent au une preuve de l'existen e d'un spe tre de raies℄ 27 This word is missing in the Fren h edition. 28 The English edition reads `(1
+ x)'.
29 [prédit℄ 30 [on eut quelque on an e dans les éle trons de re ul℄ 31 The Fren h edition uses dis ussion (where
ρ
ρ
instead of
τ
in the text, but uses
τ
in the
is used for matter density). The English edition uses
t. 32 Footnote mark missing in the Fren h edition. 33 The Fren h edition gives (4). 34 The Fren h edition gives (2). 35 This footnote is present only in the English edition. 36 This word is missing in the Fren h edition. 37 The Fren h edition reads `J. Waller'. 38 The page numbers for Wentzel appear only in the Fren h edition. 39 The English edition des ribes only three experiments, omitting the
Notes to pp. 349366
373
se tion on the omposite photoele tri ee t as well as referen es to it later. 40 This se tion is present only in the Fren h edition. 41 This and the next se tion are of ourse numbered (2) and (3) in the English edition. 42 [rayonnement de uores en e℄ 43 [une expérien e ru iale entre les deux points de vue a été imaginée et brillamment réalisée℄ 44 The Fren h edition in ludes also `en même temps'. 45 [d'une façon appropriée℄ 46 The Fren h edition has `θ '. 47 The words `photons, 'est-à-dire' are present only in the Fren h edition. 48 [au phénomène de la diusion par les éle trons individuels℄ 49 [ou℄ 50 [ave ertitude℄ 51 [physiologique℄ 52 This senten e is only printed in the Fren h edition. 53 The English edition omits referen e to the ether and ontinues dire tly with `aords no adequate pi ture'. 54 The English edition ontinues dire tly with: `seems powerless'. 55 The English edition ontinues: `is also ontrary to'. 56 In the English edition this reads simply: `A
ording to the photon theory, the produ tion .... '. 57 In the English edition: `three'. 58 [pro essus℄ 59 Overbars have been added. 60 The Fren h text reads `ave moins d'intensité sur la harge ayant la plus petite masse'. This is evidently an error: the (transverse) velo ity of the
harges stems from the ele tri eld, whi h imparts the larger velo ity to the harge with the smaller mass, whi h therefore experien es the larger magneti for e. 61 The printed text reads `ϕα ω '. 62 Bra kets in the exponent added.
The new dynami s of quanta
a
By Mr Louis de BROGLIE I. Prin ipal points of view
b
1. First works of Mr Louis de Broglie [1℄. In his rst works on the Dynami s of Quanta, the author of the present report started with the following idea: taking the existen e of elementary orpus les of matter and radiation as an experimental fa t, these orpus les are supposed to be endowed with a periodi ity. In this way of seeing things, one no longer on eives of the `material point' as a stati entity pertaining to only a tiny region of spa e, but as the entre of a periodi phenomenon spread all around it. Let us onsider, then, a ompletely isolated material point and, in a system of referen e atta hed to this point, let us attribute to the postulated periodi phenomenon the appearan e of a stationary wave de ned by the fun tion u(x0 , y0 , z0 , t0 ) = f (x0 , y0 , z0 ) cos 2πν0 t0 .
In another Galilean system x, y , z , t, the material point will have a re tilinear and uniform motion with velo ity v = βc. Simple appli ation a Our translation of the title (`La nouvelle dynamique des quanta') re e ts de Broglie's frequent use of the word `quantum' to refer to a (pointlike) parti le, an asso iation that would be lost if the title were translated as, for example, `The new quantum dynami s' (eds.). b On beginning this exposition, it seems right to underline that Mr Mar el Brillouin was the true pre ursor of wave Me hani s, as one may realise by referring to the following works: C. R. 168 (1919), 1318; 169 (1919), 48; 171 (1920), 1000. Journ. Physique 3 (1922), 65.
374
The new dynami s of quanta
375
of the Lorentz transformation shows that, as far as the phase is on erned, in the new system the periodi phenomenon has the appearan e of a plane wave propagating in the dire tion of motion whose frequen y and phase velo ity are ν0 ν=p , 1 − β2
V =
c2 c = . v β
The appearan e of this phase propagation with a speed superior to c, as an immediate onsequen e of the theory of Relativity, is quite striking. There exists a noteworthy relation between v and V . The formulas giving ν and V allow us in fa t to de ne a refra tive index of the va uum, for the waves of the material point of proper frequen y ν0 , by the dispersion law r c ν2 n= = 1 − 02 . V ν One then easily shows that 1 ∂(nν) 1 = , v c ∂ν
that is, that the velo ity v of the material point is equal to the group velo ity orresponding to the dispersion law. With the free material point being thus de ned by wave quantities, the dynami al quantities must be related ba k to these. Now, sin e the frequen y ν transforms like an energy, the obvious thing to do is to assume the quantum relation W = hν ,
a relation that is valid in all systems, and from whi h one derives the undulatory de nition of the proper mass m0 m0 c2 = hν0 .
Let us write the fun tion representing the wave in the system x, y, z, t in the form u(x, y, z, t) = f (x, y, z, t) cos
2π ϕ(x, y, z, t) . h
Denoting by W and p the energy and momentum, one easily shows that
376
L. de Broglie
one hasa,b W =
∂ϕ , ∂t
−−−−→ − → p = −grad ϕ .
The fun tion ϕ is then none other than the Ja obi fun tion.a One dedu es from this that, in the ase of uniform re tilinear motion, the prin iples of least a tion and of Fermat are identi al. To look for a generalisation of these results, let us now assume that the material point moving in a eld derived from a potential fun tion F (x, y, z, t) is represented by the fun tion u(x, y, z, t) = f (x, y, z, t) cos
2π ϕ(x, y, z, t) , h
where ϕ is the Ja obi fun tion of the old Dynami s. This assimilation of the phase into the Ja obi fun tion then leads us to assume the following two relations, whi h establish a general link between me hani al quantities and wave quantities: W = hν =
∂ϕ , ∂t
−−−−→ hν − → p = = −grad ϕ . V
One then dedu es that, for the waves of the new Me hani s, the spa e o
upied by the eld has a refra tive index s 2 ν2 F − 02 . 1− n= hν ν Hamilton's equations show in addition that, here again, the velo ity of the moving body is equal to the group velo ity.b These on eptions lead to an interpretation of the stability onditions introdu ed by quantum theory. If, indeed, one onsiders a losed traje tory, the phase must be a single-valued fun tion along this urve, and as a result one is led to write the Plan k ondition1 I (p · dl) = k · h (k integer) . The Sommerfeld onditions for quasi-periodi motions may also be derived. The phenomena of quantum stability thus appear to be analogous to phenomena of resonan e, and the appearan e of whole numbers a These are the relativisti guidan e equations of de Broglie's early pilot-wave theory of 192324, for the spe ial ase of a free parti le (eds.). −−−−→ b The ve tor `grad ϕ' is the ve tor whose omponents are ∂ϕ/∂x, ∂ϕ/∂y , ∂ϕ/∂z . a Usually alled the Hamilton-Ja obi fun tion (eds.). b In the ase of motion of a point harge in a magneti eld, spa e behaves like an anisotropi medium (see Thesis, p. 39).
The new dynami s of quanta
377
here be omes as natural as in the theory of vibrating strings or plates. Nevertheless, as we shall see, the interpretation that has just been re alled still onstitutes only a rst approximation. The appli ation of the new on eptions to orpus les of light leads to di ulties if one onsiders their proper mass to be nite. One avoids these di ulties by assuming that the properties of the orpus les of light are dedu ed from those of ordinary material points by letting the proper mass tend to zero. The two speeds v and V then both tend to c, and in the limit one obtains the two fundamental relations of the theory of light quanta hν = W ,
hν =p, c
with the aid of whi h one an a
ount for Doppler ee ts, radiation pressure, the photoele tri ee t and the Compton ee t. The new wave on eption of Me hani s leads to a new statisti al Me hani s, whi h allows us to unify the kineti theory of gases and the theory of bla kbody radiation into a single do trine. This statisti s
oin ides with that proposed independently by Mr Bose [2℄; Mr Einstein [3℄ has shown its s ope and lari ed its signi an e. Sin e then, numerous papers [4℄ have developed it in various dire tions. Let us add a few remarks. First, the author of this report has always assumed that the material point o
upies a well-de ned position in spa e. As a result, the amplitude f should ontain a singularity or at the very least have abnormally high values in a very small region. But, in fa t, the form of the amplitude plays no role in the results reviewed above. Only the phase intervenes: hen e the name phase waves originally given to the waves of the new Me hani s. On the other hand, the author, after having redu ed the old forms of Dynami s to geometri al Opti s, realised learly that this was only a rst stage. The existen e of dira tion phenomena appeared to him to require the onstru tion of a new Me hani s `whi h would be to the old Me hani s (in luding that of Einstein) what wave Opti s is to geometri al Opti s'.a It is Mr S hrödinger who has had the merit of de nitively onstru ting the new do trine. 2. The work of Mr E. S hrödinger [5℄. Mr S hrödinger's fundamental idea seems to have been the following: the new Me hani s must begin from wave equations, these equations being onstru ted in su h a way a Revue Générale des S ien es, 30 November 1924, p. 633.
378
L. de Broglie
that in ea h ase the phase of their sinusoidal solutions should be a solution of the Ja obi equation in the approximation of geometri al Opti s. Instead of onsidering waves whose amplitude ontains a singularity, Mr S hrödinger systemati ally looks at waves of lassi al type, that is to say, waves whose amplitude is a ontinuous fun tion. For him, the waves of the new Me hani s are therefore represented by fun tions Ψ that one an always write in the anoni al form 2π ϕ, h a being a ontinuous fun tion and ϕ being in the rst approximation a solution of the Ja obi equation. We may understand the words `in the rst approximation' in two dierent ways: rst, if the onditions that legitimate the use of geometri al Opti s are realised, the phase ϕ will obey the equation alled the equation of geometri al Opti s, and this equation will have to be identi al to that of Ja obi; se ond, one must equally re over the Ja obi equation if one makes Plan k's onstant tend to zero, be ause we know in advan e that the old Dynami s must then be ome valid. Let us rst onsider the ase of the motion of a single material point in a stati eld derived from the potential fun tion F (x, y, z). In his rst Memoir S hrödinger shows that the wave equation, at least in the approximation of Newton's Me hani s, is in this ase Ψ = a cos
8π 2 m0 (E − F )Ψ = 0 . h2 It is also just this equation that one arrives at beginning from the dispersion law noted in the rst se tion. Having obtained this equation, Mr S hrödinger used it to study the quantisation of motion at the atomi s ale (hydrogen atom, Plan k os illator, et .). He made the following fundamental observation: in the problems onsidered in mi rome hani s, the approximations of geometri al Opti s are no longer valid at all. As a result, the interpretation of the quantum onditions proposed by L. de Broglie shows only that the Bohr-Sommerfeld formulas orrespond to the approximation of the old Dynami s. To resolve the problem of quantisation rigorously, one must therefore onsider the atom as the seat of stationary waves satisfying ertain onditions. S hrödinger assumed, as is very natural, that the wave fun tions must be nite, single-valued and ontinuous over all spa e. These onditions de ne a set of fundamental fun tions (Eigenfunktionen) △Ψ +
The new dynami s of quanta
379
for the amplitude, whi h represent the various stable states of the atomi system being onsidered. The results obtained have proven that this new quantisation method, to whi h Messrs Léon Brillouin, G. Wentzel and Kramers [6℄ have made important ontributions, is the orre t one. For Mr S hrödinger, one must look at ontinuous waves, that is to say, waves whose amplitude does not have any singularities. How an one then represent the `material point' ? Relying on the equality of the velo ity of the moving body and the group velo ity, S hrödinger sees the material point as a group of waves (Wellenpaket2 ) of losely neighbouring frequen ies propagating in dire tions ontained within the interior of a very narrow one. The material point would then not be really pointlike; it would o
upy a region of spa e that would be at least of the order of magnitude of its wavelength. Sin e, in intra-atomi phenomena, the domain where motion takes pla e has dimensions of the order of the wavelengths, there the material point would no longer be de ned at all; for Mr S hrödinger, the ele tron in the atom is in some sense `smeared out' [`fondu'℄, and one an no longer speak of its position or velo ity. This manner of on eiving of material points seems to us to raise many di ulties; if, for example, the quantum of ultraviolet light o
upies a volume whose dimensions are of the order of its wavelength, it is quite di ult to on eive that this quantum ould be absorbed by an atom of dimensions a thousand times smaller. Having established the wave equation for a material point in a stati eld, Mr S hrödinger then turned to the Dynami s of many-body systems [la Dynamique des systèmes℄. Still limiting himself to the Newtoniana approximation, and inspired by Hamilton's ideas, he arrived at the following statement: Given an isolated system whose potential energy is F (q1 , q2 , ..., qn ), the kineti energy is a homogeneous quadrati form in the momenta pk and one may write X 2T = mkl pk pl , kl
the m being fun tions of the q . If m denotes the determinant mkl and if E is the onstant of energy in the lassi al sense, then a
ording to S hrödinger one must begin with the wave equation X ∂ 1 ∂Ψ 8π 2 1 m+ 2 m− 2 mkl l + 2 (E − F )Ψ = 0 , ∂qk ∂q h kl
kl
whi h des ribes the propagation of a wave in the on guration spa e a That is, nonrelativisti (eds.).
380
L. de Broglie
onstru ted by means of the variables q . Setting
Ψ = a cos
2π ϕ, h
and letting h tend to zero, in the limit one indeed re overs the Ja obi equation 1 X kl ∂ϕ ∂ϕ +F =E . m 2 ∂q k ∂q l kl
To quantise an atomi system, one will here again determine the fundamental fun tions of the orresponding wave equation. We annot re all here the su
esses obtained by this method (papers by Messrs S hrödinger, Fues,3 Manneba k [7℄, et .), but we must insist on the di ulties of a on eptual type that it raises. Indeed let us
onsider, for simpli ity, a system of N material points ea h possessing three degrees of freedom. The on guration spa e is in an essential way formed by means of the oordinates of the points and yet Mr S hrödinger assumes that in atomi systems material points no longer have a learly de ned position. It seems a little paradoxi al to onstru t a on guration spa e with the oordinates of points that do not exist. Furthermore, if the propagation of a wave in spa e has a lear physi al meaning, it is not the same as the propagation of a wave in the abstra t
on guration spa e, for whi h the number of dimensions is determined by the number of degrees of freedom of the system. We shall therefore have to return later to the exa t meaning of the S hrödinger equation for many-body systems. By a transformation of admirable ingenuity, Mr S hrödinger has shown that the quantum Me hani s invented by Mr Heisenberg and developed by Messrs Born, Jordan, Pauli, et ., an be translated into the language of wave Me hani s. By omparison with Heisenberg's matrix elements, he was able to derive the expression for the mean harge density of the atom from the fun tions Ψ, an expression to whi h we shall return later. The S hrödinger equations are not relativisti . For the ase of a single material point, various authors [8℄ have given a more general wave equation that is in a
ord with the prin iple of Relativity. Let e be the − → ele tri harge of the point, V and A the two ele tromagneti potentials.
The new dynami s of quanta
381
The equation that the wave Ψ, written in omplex form, must satisfy isa " # 4π 2 e2 2 ∂Ψ 4πi e V ∂Ψ X 2 2 2 − 2 m0 c − 2 (V − A ) Ψ = 0 . + Ax Ψ + h c c ∂t ∂x h c xyz
As Mr O. Klein [9℄ and then the author [10℄ have shown, the theory of the Universe with ve dimensions allows one to give the wave equation a more elegant form in whi h the imaginary terms, whose presen e is somewhat sho king for the physi ist, have disappeared. We must also make a spe ial mention of the beautiful Memoirs in whi h Mr De Donder [11℄ has onne ted the formulas of wave Me hani s to his general theory of Einsteinian Gravity. 3. The ideas of Mr Born [12℄. Mr Born was stru k by the fa t that the ontinuous wave fun tions Ψ do not allow us to say where the parti le whose motion one is studying is and, reje ting the on ept of the Wellenpaket, he onsiders the waves Ψ as giving only a statisti al representation of the phenomena. Mr Born seems even to abandon the idea of the determinism of individual physi al phenomena: the Quantum Dynami s, he wrote in his letter to Nature, `would then be a singular fusion of me hani s and statisti s .... . A knowledge of Ψ enables us to follow the ourse of a physi al pro ess in so far as it is quantum me hani ally determinate: not in a ausal sense, but in a statisti al one'.4 These on eptions were developed in a mathemati al form by their author, in Memoirs of fundamental interest. Here, by way of example, is how he treats the ollision of an ele tron and an atom. He writes the S hrödinger equation for the ele tron-atom system, and he remarks that before the ollision, the wave Ψ must be expressed by the produ t of the fundamental fun tiona representing the initial state of the atom and the plane wave fun tion orresponding to the uniform re tilinear motion of the ele tron. During the ollision, there is an intera tion between the ele tron and the atom, an intera tion that appears in the wave equation as the mutual potential energy term. Starting from the initial form of Ψ, Mr Born derives by methods of su
essive approximation its nal form after the ollision, in the ase of an elasti ollision, whi h does not modify the internal state of the atom, as well as in the ase of an inelasti
ollision, where the atom passes from one stable state to another taking a This is the omplex, time-dependent Klein-Gordon equation in an external ele tromagneti eld (eds.). a That is, eigenfun tion (eds.).
382
L. de Broglie
energy from or yielding it to the ele tron. A
ording to Mr Born, the nal form of Ψ determines the probability that the ollision may produ e this or that result. The ideas of Mr Born seem to us to ontain a great deal of truth, and the onsiderations that shall now be developed show a great analogy with them.
II. Probable meaning of the ontinuous waves Ψ [13℄ 4. Case of a single material point in a stati eld. The body of experimental dis overies made over forty years seems to require the idea that matter and radiation possess an atomi stru ture. Nevertheless,
lassi al opti s has with immense su
ess des ribed the propagation of light by means of the on ept of ontinuous waves and, sin e the work of Mr S hrödinger, also in wave Me hani s one always onsiders ontinuous waves whi h, not showing any singularities, do not allow us to de ne the material point. If one does not wish to adopt the hypothesis of the `Wellenpaket', whose development seems to raise di ulties, how an one re on ile the existen e of pointlike elements of energy with the su
ess of theories that onsider the waves Ψ? What link must one establish between the orpus les and the waves? These are the hief questions that arise in the present state of wave Me hani s. To try to answer this, let us begin by onsidering the ase of a single
orpus le arrying a harge e and moving in an ele tromagneti elda − → de ned by the potentials V and A . Let us suppose rst that the motion is one for whi h the old Me hani s (in relativisti form) is su ient. If we write the wave Ψ in the anoni al form Ψ = a cos
2π ϕ, h
the fun tion ϕ is then, as we have seen, the Ja obi fun tion, and the velo ity of the orpus le is de ned by the formula of Einsteinian Dynami s −−−−→ e − → − → 2 grad ϕ + c A . v = −c ∂ϕ ∂t − eV
(I)
We propose to assume by indu tion that this formula is still valid when the old Me hani s is no longer su ient, that is to say when ϕ a Here we leave aside the ase where there also exists a gravitational eld. Besides, the onsiderations that follow extend without di ulty to that ase.
The new dynami s of quanta
383
is no longer a solution of the Ja obi equation.a If one a
epts this hypothesis, whi h appears justi ed by its5 onsequen es, the formula (I) ompletely determines the motion of the orpus le as soon as one is given its position at an initial instant. In other words, the fun tion ϕ, just like the Ja obi fun tion of whi h it is the generalisation, determines a whole lass of motions, and to know whi h of these motions is a tually des ribed it su es to know the initial position. Let us now onsider a whole loud of orpus les, identi al and without intera tion, whose motions, determined by (I), orrespond to the same fun tion ϕ but dier in the initial positions. Simple reasoning shows that if the density of the loud at the initial moment is equal to ∂ϕ − eV , Ka2 ∂t where K is a onstant, it will subsequently remain onstantly given by this expression. We an state this result in another form. Let us suppose there be only a single orpus le whose initial position we ignore; from the pre eding, the probability for its presen e [sa probabilité de présen e℄ at a given instant in a volume dτ of spa e will be ∂ϕ 2 π dτ = Ka (II) − eV dτ . ∂t In brief, in our hypotheses, ea h wave Ψ determines a ` lass of motions', and ea h one of these motions is governed by equation (I) when one knows the initial position of the orpus le. If one ignores this initial position, the formula (II) gives the probability for the presen e of the
orpus le in the element of volume dτ at the instant t. The wave Ψ then appears as both a pilot wave (Führungsfeld of Mr Born) and a probability wave. Sin e the motion of the orpus le seems to us to be stri tly determined by equation (I), it does not seem to us that there is any reason to renoun e believing in the determinism of individual physi al phenomena,a and it is in this that our on eptions, whi h are very similar in other respe ts to those of Mr Born, appear nevertheless to dier from them markedly. Let us remark that, if one limits oneself to the Newtonian approxima2 tion, in (I) and (II) one an repla e: ∂ϕ ∂t − eV by m0 c , and one obtains a Mr De Donder assumes equation (I) as we do, but denoting by ϕ not the phase of the wave, but the lassi al Ja obi fun tion. As a result his theory and ours diverge as soon as one leaves the domain where the old relativisti Me hani s is su ient. a Here, that is, of the motion of individual orpus les.
384
L. de Broglie
the simpli ed forms 1 −−−−→ e − → − → v =− grad ϕ + A , m0 c
(I′ )
π = const · a2 .
(II′ )
There is one ase where the appli ation of the pre eding ideas is done in a remarkably lear form: when the initial motion of the orpus les is uniform and re tilinear in a region free of all elds. In this region, the
loud of orpus les we have just imagined may be represented by the homogeneous plane wave6 vx 2π W t− 2 ; Ψ = a cos h c here a is a onstant, and this means that a orpus le has the same probability to be at any point of the loud. The question of knowing how this homogeneous plane wave will behave when penetrating a region where a eld is present is analogous to that of determining the form of an initially plane light wave that penetrates a refra ting medium. In his Memoir `Quantenme hanik der Stossvorgänge', Mr Born has given a general method of su
essive approximation to solve this problem, and Mr Wentzel [14℄ has shown that one an thus re over the Rutherford formula for the de e tion of β -rays by a harged entre. We shall present yet another observation on the Dynami s of the material point su h as results from equation (I): for the material point one an always write the equations of the Dynami s of Relativity even when the approximation of the old me hani s is not valid, on ondition that one attributes to the body a variable proper mass M0 given by the formula r h2 a . M0 = m20 − 2 2 4π c a 5. The interpretation of interferen e. The new Dynami s allows us to interpret the phenomena of wave Opti s in exa tly the way that was foreseen, a long time ago now, by Mr Einstein.a In the ase of light, the wave Ψ is indeed the light wave of the lassi al theories.b,c a Cf. hapter 9 (eds.). b We then onsider Ψ as the `light variable' without at all spe ifying the physi al meaning of this quantity.
By ` lassi al theories' de Broglie seems to mean s alar wave opti s. In the general dis ussion (p. 509), de Broglie states that the physi al nature of Ψ for photons is unknown (eds.).
The new dynami s of quanta
385
If we onsider the propagation of light in a region strewn with xed obsta les, the propagation of the wave Ψ will depend on the nature and arrangement of these obsta les, but the frequen y h1 ∂ϕ ∂t will not vary (no Doppler ee t). The formulas (I) and (II) will then take the form 2
c −−−−→ − → v = − grad ϕ ; hν
π = const·a2 .
The se ond of these formulas shows immediately that the bright and dark fringes predi ted by the new theory will oin ide with those predi ted by the old. To re ord the fringes, for example by photography, one an do an experiment of short duration with intense irradiation, or an experiment of long duration with feeble irradiation (Taylor's experiment); in the rst ase one takes a mean in spa e, in the se ond ase a mean in time, but if the light quanta do not a t on ea h other the statisti al result must evidently be the same. Mr Bothe [15℄ believed he ould dedu e, from ertain experiments on the Compton ee t in a eld of interferen e, the inexa titude of the rst formula written above, the one giving the velo ity of the quantum, but in our opinion this on lusion an be ontested. 6. The energy-momentum tensor of the waves Ψ. In one of his Memoirs [16℄, Mr S hrödinger gave the expression for the energy-momentum tensor in the interior of a wave Ψ.a Following the ideas expounded here, the wave Ψ represents the motion of a loud of orpus les; examining the expression given by S hrödinger and taking into a
ount the relations (I) and (II), one then per eives that it de omposes into one part giving the energy and momentum of the parti les, and another that an be interpreted as representing a state of stress existing in the wave around the parti les. These stresses are zero in the states of motion onsistent with the old Dynami s; they hara terise the new states predi ted by wave Me hani s, whi h thus appear as ` onstrained states' of the material point and are intimately related to the variability of the proper mass M0 . Mr De Donder has also drawn attention to this fa t, and he was led to denote the amplitude of the waves that he onsidered by the name of `internal stress potential'. The existen e of these stresses allows one to explain how a mirror re e ting a beam of light suers a radiation pressure, even though a a Cf. S hrödinger's report, se tion II (eds.).
386
L. de Broglie
ording to equation (I), be ause of interferen e, the orpus les of light do not `strike' its surfa e.a 7. The dynami s of many-body systems. We must now examine how these on eptions may serve to interpret the wave equation proposed by S hrödinger for the Dynami s of many-body systems. We have pointed out above the two di ulties that this equation raises. The rst, relating to the meaning of the variables that serve to onstru t the on guration spa e, disappears if one assumes that the material points always have a quite de nite position. The se ond di ulty remains. It appears to us ertain that if one wants to physi ally represent the evolution of a system of N orpus les, one must onsider the propagation of N waves in spa e, ea h of the N propagations being determined by the a tion of the N − 1 orpus les onne ted to the other waves.a Nevertheless, if one fo usses one's attention only on the orpus les, one an represent their states by a point in on guration spa e, and one an try to relate the motion of this representative point to the propagation of a titious wave Ψ in on guration spa e. It appears to us very probable that the waveb 2π Ψ = a(q1 , q2 , ..., qn ) cos ϕ(t, q1 , ..., qn ) , h a solution of the S hrödinger equation, is only a titious wave whi h, in the Newtonian approximation, plays for the representative point of the system in on guration spa e the same role of pilot wave and of probability wave that the wave Ψ plays in ordinary spa e in the ase of a single material point. Let us suppose the system to be formed of N points having for re tangular oordinates N N x11 , x12 , x13 , ..., xN 1 , x2 , x3 .
In the on guration spa e formed by means of these oordinates, the representative point of the system has for [velo ity℄ omponents along the axis xki 1 ∂ϕ , vxki = − mk ∂xki a Cf. Brillouin's example in the dis ussion at the end of de Broglie's le ture (eds.). a Cf. se tion 2.4 (eds.).
b The amplitude a is time-independent be ause de Broglie is assuming the timeindependent S hrödinger equation. Later in his report, de Broglie applies his dynami s to a non-stationary wave fun tion as well, for the ase of an atomi transition (eds.).
The new dynami s of quanta
387
mk being the mass of the k th orpus le. This is the relation that repla es (I′ ) for many-body systems. From this, one dedu es that the probability for the presen e of the representative point in the element of volume dτ of on guration spa e is π dτ = const·a2 dτ .
This new relation repla es relation (II′ ) for many-body systems. It fully a
ords, it seems to us, with the results obtained by Mr Born for the ollision of an ele tron and an atom, and by Mr Fermi [17℄ for the
ollision of an ele tron and a rotator.a Contrary to what happens for a single material point, it does not appear easy to nd a wave Ψ that would de ne the motion of the system taking Relativity into a
ount. 8. The waves Ψ in mi rome hani s. Many authors think it is illusory to wonder what the position or the velo ity of an ele tron in the atom is at a given instant. We are, on the ontrary, in lined to believe that it is possible to attribute to the orpus les a position and a velo ity even in atomi systems, in a way that gives a pre ise meaning to the variables of on guration spa e. This leads to on lusions that deserve to be emphasised. Let us
onsider a hydrogen atom in one of its stable states. A
ording to S hrödinger, in spheri al oordinatesb the orresponding fun tion Ψn is of the form Ψn = F (r, θ)[A cos mα + B sin mα]
sin 2π Wn t cos h
(m integer)
with
2π 2 m0 e4 . n2 h2 If we then apply our formula (I′ ), we on lude that the ele tron is motionless in the atom, a on lusion whi h would evidently be inadmissible in the old Me hani s. However, the examination of various questions and notably of the Zeeman ee t has led us to believe that, in its stable states, the H atom must rather be represented by the fun tion 2π mh Ψn = F (r, θ) cos Wn t − α , h 2π Wn = m0 c2 −
a Cf. the remarks by Born and Brillouin in the dis ussion at the end of de Broglie's le ture, and the de Broglie-Pauli en ounter in the general dis ussion at the end of the onferen e (pp. 511 .) (eds.). b r , radius ve tor; θ, latitude; α, longitude.
388
L. de Broglie
whi h, being a linear ombination of expressions of the type written above, is equally a
eptable.a If this is true the ele tron will have, from (I′ ), a uniform ir ular motion of speed v=
1 mh . m0 r 2π
It will then be motionless only in states where m = 0. Generally speaking, the states of the atom at a given instant an always be represented by a fun tion X Ψ= cn Ψ n , n
the Ψn being S hrödinger's Eigenfunktionen. In parti ular, the state of transition i → j during whi h the atom emits the frequen y νij would be given by (this appears to be in keeping with S hrödinger's ideas) Ψ = ci Ψ i + cj Ψ j , ci and cj being two fun tions of time that hange very slowly ompared with the trigonometri fa tors of the Ψn , the rst varying from 1 to 0 and the se ond from 0 to 1 during the transition. Writing the fun tion ′ Ψ in the anoni al form a cos 2π h ϕ, whi h is always possible, formula (I ) will give the velo ity of the ele tron during the transition, if one assumes the initial position to be given. So it does not seem to be impossible to arrive in this way at a visual representation of the transition.a Let us now onsider an ensemble of hydrogen atoms that are all in the same state represented by the same fun tion X Ψ= cn Ψ n . n
The position of the ele tron in ea h atom is unknown to us, but if, in our imagination, we superpose all these atoms, we obtain a mean atom where the probability for the presen e of one of the ele trons in an element of volume dτ will be given by the formula (II),b K being determined by
a In his memoir, `Les moments de rotation et le magnétisme dans la mé anique ondulatoire' (Journal de Physique 8 (1927), 74), Mr Léon Brillouin has impli itly assumed the hypothesis that we formulate in the text. a In this example, de Broglie is applying his dynami s to a ase where the wave fun tion Ψ has a time-dependent amplitude a (eds.). b In this se tion on atomi physi s (`mi rome hani s') de Broglie onsiders the non-relativisti approximation, using the limiting formula (I′ ) ex ept in this paragraph where he reverts to the relativisti formulas (I) and (II), for the purpose of omparison with the relativisti formulas for harge and urrent density obtained by other authors (eds.).
The new dynami s of quanta
389
the fa t that the total probability for all the possible positions must be − → → equal to unity. The harge density ρ and the urrent density J = ρ− v in the mean atom are then, from (I) and (II), ∂ϕ 2 − eV , ρ = Kea ∂t −−−−→ e − − → → J = −Kec2 a2 grad ϕ + A c
and these formulas oin ide, apart from notation, with those of Messrs Gordon, S hrödinger and O. Klein [18℄. Limiting ourselves to the Newtonian approximation, and for a moment ¯ the onjugate denoting by Ψ the wave written in omplex form, and by Ψ fun tion, it follows that ¯ . ρ = const·a2 = const·ΨΨ
This is the formula to whi h Mr S hrödinger was led in reformulating the matrix theory; it shows that the ele tri dipole moment of the mean atom during the transition i → j ontains a term of frequen y νij , and thus allows us to interpret Bohr's frequen y relation. Today it appears ertain that one an predi t the mean energy radiated by an atom by using the Maxwell-Lorentz equations, on ondition → that one introdu es in these equations the mean quantities ρ and ρ− v a whi h have just been de ned. One an thus give the orresponden e prin iple an entirely pre ise meaning, as Mr Debye [19℄ has in fa t shown in the parti ular ase of motion with one degree of freedom. It seems indeed that lassi al ele tromagnetism an from now on retain only a statisti al value; this is an important fa t, whose meaning one will have to try to explore more deeply. To study the intera tion of radiation with an ensemble of atoms, it is rather natural to onsider a `mean atom', immersed in a `mean light' whi h one de nes by a homogeneous plane wave of the ve tor potential. The density ρ of the mean atom is perturbed by the a tion of the light and one dedu es from this the s attered radiation. This method, whi h gives good mean predi tions, is related more or less dire tly to the theories of s attering by Messrs S hrödinger and Klein [20℄, to the theory of the Compton ee t by Messrs Gordon and S hrödinger [21℄, and to the Memoirs of Mr Wentzel [22℄ on the photoele tri ee t and the Compton ee t, et . The s ope of this report does not permit us to dwell any further on this interesting work. a Cf. S hrödinger's report, p. 455, and the ensuing dis ussion, and se tion 4.5 (eds.).
390
L. de Broglie
9. Con lusions and remarks. So far we have onsidered the orpus les as `exterior' to the wave Ψ, their motion being only determined by the propagation of the wave. This is, no doubt, only a provisional point of view: a true theory of the atomi stru ture of matter and radiation should, it seems to us, in orporate the orpus les in the wave phenomenon by onsidering singular solutions of the wave equations. One should then show that there exists a orresponden e between the singular waves and the waves Ψ, su h that the motion of the singularities is onne ted to the propagation of the waves Ψ by the relation (I).a In the ase of no [external℄ eld, this orresponden e is easily established, but it is not so in the general ase. → We have seen that the quantities ρ and ρ− v appearing in the MaxwellLorentz equations must be al ulated in terms of the fun tions Ψ, but that does not su e to establish a deep link between the ele tromagneti quantities and those of wave Me hani s. To establish this link,b one should probably begin with singular waves, for Mr S hrödinger has very rightly remarked that the potentials appearing in the wave equations are those that result from the dis ontinuous stru ture of ele tri ity and not those that ould be dedu ed from the fun tions Ψ. Finally, we point out that Messrs Uhlenbe k and Goudsmit's hypothesis of the magneti ele tron, so ne essary to explain a great number of phenomena, has not yet found its pla e in the s ope of wave Me hani s.
III. Experiments showing preliminary dire t eviden e for the new Dynami s of the ele tron 10. Phenomena whose existen e is suggested by the new on eptions. The ideas that have just been presented lead one to onsider the motion of an ele tron as guided by the propagation of a ertain wave. In many usual ases, the old Me hani s remains entirely adequate as a rst approximation; but our new point of view, as Elsasser7 [23℄ pointed out already in 1925, ne essarily raises the following question: `Could one not observe ele tron motions that the old Me hani s would be in apable of predi ting, and whi h would therefore be hara teristi of wave Mea Cf. se tion 2.3.1 (eds.). b The few attempts made till now in this dire tion, notably by Mr Bateman (Nature 118 (1926), 839) and by the author (Ondes et mouvements, Chap. VIII, and C. R. 184 (1927), 81) an hardly be regarded as satisfa tory.
The new dynami s of quanta
391
hani s? In other words, for ele trons, ould one not nd the analogue of the phenomena of dira tion and interferen e?'a These new phenomena, if they exist, must depend on the wavelength of the wave asso iated with the ele tron motion. For an ele tron of speed v , the fundamental formula hν p= V gives p h h 1 − β2 V = = . λ= ν p m0 v If β is not too lose to 1, it su es to write λ=
h . m0 v
Let V be the potential dieren e, expressed in volts, that is apable of imparting the speed v to the ele tron; numeri ally, for the wavelength in entimetres, one will havea λ=
7.28 12.25 = √ × 10−8 . v V
To do pre ise experiments, it is ne essary to use ele trons of at least a few volts: from whi h one has an upper limit for λ of a few angstroms. One then sees that, even for slow ele trons, the phenomena being sought are analogous to those shown by X-rays and not to those of ordinary light. As a result, it will be di ult to observe the dira tion of a beam of ele trons by a small opening, and if one wishes to have some
han e of obtaining dira tion by a grating, one must either onsider those natural three-dimensional gratings, the rystals, or use ordinary gratings under a very grazing in iden e, as has been done re ently for X-rays. On making slow ele trons pass through a rystalline powder or an amorphous substan e, one ould also hope to noti e the appearan e of rings analogous to those that have been obtained and interpreted in the a This is not a quotation: these words do not appear in the ited 1925 paper by Elsasser. Further, it was de Broglie who rst suggested ele tron dira tion, in a paper of 1923 (see se tion 2.2.1) (eds.). a Here we have adopted the following values: h = 6.55 × 10−27 erg-se onds , m0 = 9 × 10−28 gr , e = 4.77 × 10−10 e.s.u. .
392
L. de Broglie
X-ray domain by Messrs Hull, Debye and S herrer, Debierne, Keesom and De Smedt, et . The exa t theoreti al predi tion of the phenomena to be observed along these lines is still not very advan ed. Let us onsider the dira tion of a beam of ele trons with the same velo ity by a rystal; the wave Ψ will propagate following the general equation, in whi h one has to insert the potentials reated by the atoms of the rystal onsidered as entres of for e. One does not know the exa t expression for these potentials but, be ause of the regular distribution of atoms in the rystal, one easily realises that the s attered amplitude will show maxima in the dire tions predi ted by Mr von Laue's theory. Be ause of the role of pilot wave played by the wave Ψ, one must then observe a sele tive s attering of the ele trons in these dire tions. Using his methods, Mr Born has studied another problem: that of the
ollision of a narrow beam of ele trons with an atom. A
ording to him, the urve giving the number of ele trons that have suered an inelasti
ollision as a fun tion of the s attering angle must show maxima and minima; in other words, these ele trons will display rings on a s reen pla ed normally to the ontinuation of the in ident beam. It would still be premature to speak of agreement between theory and experiment; nevertheless, we shall present experiments that have revealed phenomena showing at least broadly the predi ted hara ter. 11. Experiments by Mr E.G. Dymond [24℄. Without feeling obliged to follow the hronologi al order, we shall rst present Mr Dymond's experiments: A ask of puri ed helium ontained an `ele tron gun', whi h onsisted of a brass tube ontaining an in andes ent lament of tungsten and in whose end a slit was ut. This gun dis harged a well- ollimated beam of ele trons into the gas, with a speed determined by the potential dieren e (50 to 400 volts) established between the lament and the wall of the tube. The wall of the ask had a slit through whi h the ele trons
ould enter a hamber where the pressure was kept low by pumping and where, by urving their traje tories, a magneti eld brought them onto a Faraday ylinder. Mr Dymond rst kept the orientation of the gun xed and measured the speed of the ele trons thus s attered by a given angle. He noti ed that most of the s attered ele trons have the same energy as the primary ele trons; they have therefore suered an elasti ollision. Quite a large number of ele trons have a lower speed orresponding to an energy loss
The new dynami s of quanta
393
from about 20 to 55 volts: this shows that they made the He atom pass from the normal state 11 S to the ex ited state 21 S . One also observes a lower proportion of other values for the energy of the s attered ele trons; we shall not dis uss the interpretation that Mr Dymond has given them, be ause what interests us most here is the variation of the number of s attered parti les with the s attering angle θ. To determine this number, Mr Dymond varied the orientation of the gun inside the ask, and for dierent s attering angles olle ted the ele trons that suered an energy loss equivalent to 20 to 55 volts; he onstru ted a series of
urves of the angular distribution of these ele trons for dierent values of the tension V applied to the ele tron gun. The angular distribution
urve shows a very pronoun ed maximum for a low value of θ, and this maximum appears to approa h θ = 0 for in reasing values of V . Another, less important, maximum appears towards θ = 50◦ for a primary energy of about a hundred volts, and then moves for in reasing values of V towards in reasing θ. Finally, a very sharp maximum appears for a primary energy of about 200 volts at θ = 30◦ , and then seems independent of V . These fa ts are summarised in the following table given by Dymond:8 V (volts) 48.9 72.3 97.5 195 294 400
.... .... .... .... .... ....
Positions of the maxima (◦ ) 24 8 5 50 < 2.5 30 59 < 2.5 30 69 < 2.5 30 70
The above results must very probably be interpreted with the aid of the new Me hani s and are to be related to Mr Born's predi tions. Nevertheless, as Mr Dymond very rightly says, `the theoreti al side of the problem is however not yet su iently advan ed to give detailed information on the phenomena to be expe ted, so that the results above reported annot be said to substantiate the wave me hani s ex ept in the most general way'.9 12. Experiments by Messrs C. Davisson and L. H. Germer. In 1923, Messrs Davisson and Kunsman [25℄ published pe uliar results on the s attering of ele trons at low speed. They dire ted a beam of ele trons, a
elerated by a potential dieren e of less than 1000 volts, onto a blo k of platinum at an in iden e of 45◦ and determined the distribution
394
L. de Broglie
of s attered ele trons by olle ting them in a Faraday ylinder. For potentials above 200 volts, one observed a steady de rease in s attering for in reasing values of the deviation angle, but for smaller voltages the urve of angular variation showed two maxima. By overing the platinum with a deposit of magnesium, one obtained a single small maximum for ele trons of less than 150 volts. Messrs Davisson and Kunsman attributed the observed phenomena to the a tion of various layers of intra-atomi ele trons on the in ident ele trons, but it seems rather, a
ording to Elsasser's opinion, that the interpretation of these phenomena is a matter for the new Me hani s. Resuming analogous experiments with Mr Germer [26℄, Mr Davisson obtained very important results this year, whi h appear to on rm the general predi tions and even the formulas of Wave Me hani s. The two Ameri an physi ists sent homogeneous beams of ele trons onto a rystal of ni kel, ut following one of the 111 fa es of the regular o tahedron (ni kel is a ubi rystal). The in iden e being normal, the phenomenon ne essarily had to show the ternary symmetry around the dire tion of the in ident beam. In a ubi rystal ut in this manner, the fa e of entry is ut obliquely by three series of 111 planes, three series of 100 planes, and six series of 110 planes. If one takes as positive orientation of the normals to these series of planes the one forming an a ute angle with the fa e of entry, then these normals, together with the dire tion of in iden e, determine distinguished azimuths, whi h Messrs Davisson and Germer all azimuths (111), (100), (110), and for whi h they studied the s attering; be ause of the ternary symmetry, it evidently su es to explore a single azimuth of ea h type. Let us pla e ourselves at one of the distinguished azimuths and let us
onsider only the distribution of Ni atoms on the fa e of entry of the
rystal, whi h we assume to be perfe t. These atoms form lines perpendi ular to the azimuth being onsidered and whose equidistan e d is known from rystallographi data. The dierent dire tions of s attering being identi ed in the azimuthal plane by the angle θ of o-latitude, the waves s attered by the atoms in the fa e of entry must be in phase in dire tions su h that one has nλ n 12.25 √ · 10−8 (n integer) . θ = arcsin = arcsin d d V
One must then expe t to observe maxima in these dire tions, for the s attering of the ele trons by the rystal. Now here is what Messrs Davisson and Germer observed. By gradually
The new dynami s of quanta
395
varying the voltage V that a
elerates the ele trons one observes, in the neighbourhood of ertain values of V , very distin t s attering maxima in dire tions whose o-latitude is a
urately given by the above formula (provided one sets in general n = 1, and sometimes n = 2). There is dire t numeri al on rmation of the formulas of the new Dynami s; this is evidently a result of the highest importan e. However, the explanation of the phenomenon is not omplete: one must explain why the s attering maxima are observed only in the neighbourhood of ertain parti ular values of V , and not for all values of V . One interpretation naturally omes to mind: we assumed above that only the fa e of entry of the rystal played a role, but one an assume that the ele tron wave penetrates somewhat into the rystal and, further, in reality the fa e of entry will never be perfe t and will be formed by several parallel 111 planes forming steps. In these onditions, it is not su ient to onsider the interferen e of the waves s attered by a single reti ular plane at the surfa e, one must take into a
ount the interferen e of the waves s attered by several parallel reti ular planes. In order for there to be a strong s attering in a dire tion θ, θ and V must then satisfy not only the relation written above, but also another relation whi h is easy to nd; the s attering must then be sele tive, that is to say, o
ur with [signi ant℄ intensity only for ertain values of V , as experiment shows. Of ourse, the theory that has just been outlined is a spe ial
ase of Laue's general theory. Unfortunately, as Messrs Davisson and Germer have themselves remarked, in order to obtain an exa t predi tion of the fa ts in this way, it is ne essary to attribute to the separation of the 111 planes next to the fa e of entry a smaller value (of about 30%) than that provided by Crystallography and by dire t measurements by means of X-rays. It is moreover not unreasonable to assume that the very super ial reti ular planes have a spa ing dierent from those of the deeper planes, and one
an even try to onne t this idea to our urrent on eptions on erning the equilibrium of rystalline gratings. If one a
epts the pre eding hypothesis, the s attering must be produ ed by a very small number of reti ular planes in the entirely super ial layer of the rystal; the on entration of ele trons in preferred dire tions must then be mu h less pronoun ed than in the ase of s attering by a whole unlimited spatial grating. Is it nevertheless su ient in order to explain the `peaks' observed by Davisson and Germer? To this question, Mr Patterson has re ently provided an a rmative answer, by showing that the involvement of just two super ial reti ular planes
396
L. de Broglie
already su es to predi t exa tly the variations of the sele tive re e tion of ele trons observed in the neighbourhood of θ = 50◦ ,
V = 54 volts .
To on lude, we an do no better than quote the on lusion of Mr Patterson [27℄: `The agreement of these results with al ulation seems to indi ate that the phenomenon an be explained as a dira tion of waves in the outermost layers of the rystal surfa e. It also appears [....℄ that a omplete analysis of the results of su h experiments will give valuable information as to the onditions prevailing in the a tual surfa e, and that a new method has been made available for the investigation of the stru ture of rystals in a region whi h has up to the present almost
ompletely es aped observation'.10 13. Experiments by Messrs G. P. Thomson and A. Reid [28℄. Very re ently, Messrs Thomson and Reid have made the following results known: if a narrow pen il of homogeneous athode rays passes normally through a elluloid lm, and is then re eived on a photographi plate pla ed parallel to the lm at 10 m behind it, one observes rings around the entral spot. With rays of 13 000 volts, a photometri examination has revealed the existen e of three rings. By gradually in reasing the energy of the ele trons, one sees the rings appear around 2500 volts, and they have been observed up to 16 500 volts. The radii of the rings de rease when the energy in reases and, it seems, approximately in inverse proportion to the speed, that is, to our wavelength λ. These observations are very interesting, and again on rm the new
on eptions in broad outline. Is it a question here of an atomi phenomenon analogous to those observed by Dymond, or else of a phenomenon of mutual interferen e falling into one of the ategories studied by Debye and S herrer, Hull, Debierne, Keesom and De Smedt? We are unable to say, and we limit ourselves to remarking that here the ele trons used are relatively fast; this is interesting from the experimental point of view, be ause it is mu h easier to study ele trons of a few thousand volts than ele trons of about a hundred volts.
The new dynami s of quanta
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L. de Broglie
[13℄ L. de Broglie, C. R., 183 (1926), 447, and 184 (1927), 273. Nature, 118 (1926), 441. J. de Phys. (6), VIII (1927), 225. C. R., 185 (1927), 380. (See also E. Madelung, Zts. f. Phys., 40 (1926), 322.) [14℄ G. Wentzel, Zts. f. Phys., 40 (1926), 590. [15℄ W. Bothe, Zts. f. Phys., 41 (1927), 332. [16℄ E. S hrödinger, Ann. der Phys., 82 (1927), 265. [17℄ E. Fermi, Zts. f. Phys., 40 (1926), 399. [18℄ W. Gordon and E. S hrödinger, lo . it. in [8℄ and [16℄. O. Klein, Zts. f. Phys., 41 (1927), 407. [19℄ P. Debye, Physik. Zts., 27 (1927), 170. [20℄ E. S hrödinger and O. Klein, lo . it. in [5℄ and [18℄. [21℄ W. Gordon, lo . it. in [8℄. E. S hrödinger, Ann. der Phys., 82 (1927), 257. [22℄ G. Wentzel, Zts. f. Phys., 43 (1927), 1; 41 (1927), 828. (See also G. Be k, Zts. f. Phys., 41 (1927), 443; J. R. Oppenheimer, Zts. f. Phys., 41 (1927), 268.) [23℄ [W.℄ Elsasser, Naturwissens h., 13 (1925), 711. [24℄ E. G. Dymond, Nature, 118 (1926), 336. Phys. Rev., 29 (1927), 433. [25℄ C. Davisson and C. H. Kunsman, Phys. Rev., 22 (1923), 242. [26℄ C. Davisson and L. H. Germer, Nature, 119 (1927), 558. [27℄ A. L. Patterson, Nature, 120 (1927), 46. [28℄ C. P. Thomson and A. Reid, Nature, 119 (1927), 890.
Dis ussion
399
Dis ussion of Mr de Broglie's report
Mr Lorentz. I should like to see learly how, in the rst form of
your theory, you re overed Sommerfeld's quantisation onditions. You obtained a single ondition, appli able only to the ase where the orbit is losed: the wave must, after travelling along the orbit, nish in phase when it omes ba k to the initial point. But in most ases the traje tory is not losed; this happens, for example, for the hydrogen atom when one takes relativity into a
ount; the traje tory is then a rosette, and never omes ba k to its initial point. How did you nd the quantisation onditions appli able to these multiperiodi problems?
Mr de Broglie.
The di ulty is resolved by onsidering pseudoperiods, as I pointed out in my Thesis ( hap. III, p. 41). When a system is multiperiodi , with partial periods τ1 , τ2 , ..., τn , one an prove that one an nd quasi-periods τ that are nearly exa tly whole multiples of the partial periods: τ = m1 τ1 + ε1 = m2 τ2 + ε2 ... = mn τn + εn ,
the m1 , m2 , ..., mn being integers and the ε1 , ε2 , ..., εn as small as one likes. The traje tory then never omes ba k to its initial point, but at the end of a quasi-period τ it omes ba k as losely as one likes to the initial position. One will then be led to write that, at the end of a quasi-period, the wave nishes in phase; now, there is an in nite number of quasi-periods, orresponding to all kinds of systems of values of the integers m1 , m2 , ..., mn . In order that the wave nishes in phase after any one of these quasi-periods, it is ne essary that one have11 Z Z Z pn dqn = nn h , p2 dq2 = n2 h , ... , p1 dq1 = n1 h , τ1
τ2
τn
whi h gives exa tly Sommerfeld's onditions.a
Mr Born.
The de nition of the traje tory of a parti le that Mr de Broglie has given seems to me to present di ulties in the ase of a
ollision between an ele tron and an atom. In an elasti ollision, the speed of the parti le must be the same after the ollision as before. I a Darrigol (1993, pp. 3423, 3645) shows that this derivation is faulty: the ondition that the wave should nish in phase after any quasi-period does not imply the n separate onditions listed above.
400
L. de Broglie
should like to ask Mr de Broglie if that follows from his formula.
Mr de Broglie. That follows from it, indeed. Mr Brillouin. It seems to me that no serious obje tion an be
made to the point of view of L. de Broglie. Mr Born an doubt the real existen e of the traje tories al ulated by L. de Broglie, and assert that one will never be able to observe them, but he annot prove to us that these traje tories do not exist. There is no ontradi tion between the point of view of L. de Broglie and that of other authors, sin e, in his report (8, p. 38912 ) L. de Broglie shows us that his formulasa are in exa t agreement with those of Gordon, at present a
epted by all physi ists.
Mr Pauli. I should like to make a small remark on what seems to
me to be the mathemati al basis of Mr de Broglie's viewpoint on erning parti les in motion on de nite traje tories. His on eption is based on the prin iple of onservation of harge: ∂s2 ∂s3 ∂ρ ∂s1 + + + =0 ∂t ∂x ∂y ∂z
or
4 X ∂sk =0, ∂xk
(a)
k=1
whi h is a onsequen e of the wave equation, when one sets isk = ψ
∂ψ 4πi e ∂ψ ∗ − ψ∗ + Φk ψψ ∗ . ∂xk ∂xk h c
Mr de Broglie introdu es, in pla e of the omplex fun tion ψ , the two real fun tions a and ϕ de ned by ψ = ae
2πi h ϕ
,
ψ ∗ = ae−
2πi h ϕ
.
Substituting these expressions into the expression for sk yields: 4π 2 ∂ϕ e sk = + Φk . a h ∂xk c
From this follow the expressions given by Mr de Broglie for the velo ity ve tor, de ned by s1 s2 s3 v1 = (b) , v2 = , v3 = . ρ ρ ρ Now if in a eld theory there exists a onservation prin iple of the a That is, de Broglie's equations for the mean harge and urrent density to be used in semi lassi al radiation theory (eds.).
Dis ussion
401
form (a), it is always formally possible to introdu e a velo ity ve tor (b), depending on spa e and time, and to imagine furthermore orpus les that move following the urrent lines of this ve tor. Something similar was already proposed in opti s by Slater; a
ording to him, light quanta should always move following the lines of the Poynting ve tor. Mr de Broglie now introdu es an analogous representation for material parti les. In any ase, I do not believe that this representation may be developed in a satisfa tory manner; I intend to return to this during the general dis ussion.a
Mr S hrödinger.
If I have properly understood Mr de Broglie, the velo ity of the parti les must have its analogue in a ve tor eld
omposed of the three spatial omponents of the urrent in a fourdimensional spa e, after division of these by the omponent with respe t to time (that is, the harge density). I should like simply to re all now that there exist still other ve tor quantities of a eld, whi h an be made to orrespond with the velo ity of the parti les, su h as the omponents of the momentum density (see Ann. d. Phys.13 82, 265). Whi h of the two analogies is the more onvin ing?
Mr Kramers.
The fa t that with independent parti les in motion one annot onstru t an energy-momentum tensor having the properties required by Maxwell's theory onstitutes nevertheless a di ulty.
Mr Pauli.
The quotient of the momentum by the energy density whi h Mr S hrödinger onsiders would in fa t lead in a relativisti
al ulation to other parti le traje tories than would the quotient of the densities of urrent and of harge.
Mr Lorentz. In using his formulas for the velo ity of the ele tron,
has Mr de Broglie not al ulated this velo ity in parti ular ases, for example for the hydrogen atom?
Mr de Broglie.
When one applies the formula for the velo ity to a wave fun tion representing a stable state of the hydrogen atom a
ording to Mr S hrödinger, one nds ir ular orbits. One does not a See pp. 511 . (eds.).
402
L. de Broglie
re over the ellipti al orbits of the old theory (see my report, 8).
Mr Ehrenfest. Can the speed of an ele tron in a stationary orbit
be zero?
Mr de Broglie. Yes, the speed of the ele tron an be zero. Mr S hrödinger. Mr de Broglie says that in the ase of the
hydrogen atom his hypothesis leads to ir ular orbits. That is true for the parti ular solutions of the wave equation that one obtains when one separates the problem in polar oordinates in spa e; perhaps it is still true for the solutions that one obtains by making use of paraboli or ellipti al oordinates. But in the ase of a degenera y (as he onsiders it here) it is, in reality, not at all the parti ular solutions whi h have a signi an e, but only an arbitrary linear ombination, with onstant
oe ients, of all the parti ular solutions belonging to the same eigenvalue, be ause there is no means of distinguishing between them, all linear
ombinations being equally justi ed in prin iple. In these onditions, mu h more ompli ated types of orbit will ertainly appear. But I do not believe that in the atomi domain one may still speak of `orbits'.
Mr Lorentz. Does one know of su h more ompli ated orbits? Mr S hrödinger. No, one does not know of them; but I simply
wanted to say that if one nds ir ular orbits, that is due to a fortuitous
hoi e of parti ular solutions that one onsiders, and this hoi e annot be motivated in a way that has no arbitrariness.
Mr Brillouin.
Perhaps it is not super uous to give some examples that illustrate well the meaning of Mr L. de Broglie's formulas, and that allow one to follow the motion of the parti les guided by the phase wave. If the wave is plane and propagates freely, the traje tories of the parti les are the rays normal to the wave surfa e. Let us suppose that the wave is re e ted by a plane mirror, and let θ be the angle of in iden e; the wave motion in front of the mirror is given by a superposition of the in ident wave
ψ1 = a1 cos 2π
x sin θ − z cos θ t − T λ
Dis ussion
403
Fig. 1. and the re e ted wave ψ2 = a1 cos 2π
whi h gives
,
x sin θ t − T λ
t x sin θ + z cos θ − T λ
2πz cos θ ψ = 2a1 cos cos 2π λ
This wave is put in L. de Broglie's anoni al form ψ = a cos
.
2π ϕ h
with a = 2a1 cos
2πz cos θ λ
and
ϕ=h
x sin θ t − T λ
.
Let us then apply L. de Broglie's formulas, in the simpli ed form given on page 385 (5);14 and let us suppose that it is a light wave guiding the photons; the velo ity of these is 2
c −−−−→ − → v = − grad ϕ . hν We see that the proje tiles move parallel to the mirror, with a speed vx = c sin θ, less than c. Their energy remains equal to hν , be ause their mass has undergone a variation, a
ording to the following formula (report by L. de Broglie, p. 384):15 r r h2 a h hν a 2 M 0 = m0 − 2 2 = = 2 cos θ . − 4π c a 2πc a c
404
L. de Broglie
Fig. 2. The mass of the photons, whi h is zero in the ase where the wave propagates freely, is then assumed to take a non-zero value in the whole region where there is interferen e or deviation of the wave. Let us draw a diagram for the ase of a limited beam of light falling on a plane mirror; the interferen e is produ ed in the region of overlap of two beams. The traje tory of a photon will be as follows: at rst a re tilinear path in the in ident beam, then a bending at the edge of the interferen e zone, then a re tilinear path parallel to the mirror, with the photon travelling in a bright fringe and avoiding the dark interferen e fringes; then, when it omes out, the photon retreats following the dire tion of the re e ted light beam. No photon a tually strikes the mirror, nevertheless the mirror suers
lassi al radiation pressure; it is in order to explain this fa t that L. de Broglie assumes the existen e of spe ial stresses in the interferen e zone; these stresses, when added to the tensor of momentum ux transported by the photons, reprodu e the lassi al Maxwell tensor; there is then no dieren e in the me hani al ee ts produ ed by the wave during its re e tion by the mirror. These remarks show how L. de Broglie's system of hypotheses preserves the lassi al formulas, and avoids a ertain number of awkward paradoxes. One thus obtains, for example, the solution to a urious problem posed by G. N. Lewis (Pro . Nat. A ad. 12 (1926), 22 and 439), whi h was the subje t of dis ussions between this author and R. C. Tolman and S. Smith (Pro . Nat. A ad. 12 (1926), 343 and 508). Lewis assumed that the photons always follow the path of a light
Dis ussion
405
Fig. 3.
ray of geometri al opti s, but that they hoose, among the dierent rays, only those that lead from the luminous sour e to a bright fringe situated on an absorbing body. He then onsidered a sour e S whose light is re e ted by two mirrors AA and BB; the light beams overlap, produ ing interferen e [zones℄ in whi h one pla es a s reen CD; the dimensions are assumed to be su h that there is a bright fringe on one of the edges D of the s reen and a dark fringe on the other edge C. Following the hypothesis of Lewis, the photons would follow only the paths SBD and SAD, whi h end at the bright fringe D; no photon will take the path SAC or SBC. All the photons ome to strike the mirror AA on the edge A, so one ould predi t that this mirror would suer a torque; if one made it movable around an axis O, it would tend to turn in the dire tion of the arrow. This paradoxi al on lusion is entirely avoided by L. de Broglie, sin e his system of hypotheses preserves the values of radiation pressure. This example shows learly that there is a ontradi tion between the hypothesis of re tilinear paths for the photons (following the light rays) and the ne essity of nding photons only where a bright interferen e fringe is produ ed, no photon going through the regions of dark fringes.
Mr Lorentz
draws attention to a ase where the lassi al theory and the photon hypothesis lead to dierent results on erning the ponderomotive for es produ ed by light. Let us onsider re e tion by the hypotenuse fa e of a glass prism, the angle of in iden e being larger than the angle of total re e tion. Let us pla e a16 se ond prism behind the rst, at a distan e of the order of magnitude of the wavelength, or only
406
L. de Broglie
a fra tion of this length. Then, the re e tion will no longer be total. The light waves that penetrate the layer of air rea h the se ond prism before their intensity is too mu h weakened, and there give rise to a beam transmitted in the dire tion of the in ident rays. If, now, one al ulates the Maxwell stresses on a plane situated in the layer of air and parallel to its surfa es, one nds that, if the angle of in iden e ex eeds a ertain value (60◦ for example), there will be an attra tion between the two prisms. Su h an ee t an never be produ ed by the motion of orpus les, this motion always giving rise to a [positive℄ pressure as in the kineti theory of gases. What is more, in the lassi al theory one easily sees the origin of the `negative pressure'. One an distinguish two ases, that where the ele tri os illations are in the plane of in iden e and that where this is so for the magneti os illations. If the in iden e is very oblique, the os illations of the in ident beam that I have just mentioned are only slightly in lined with respe t to the normal to the hypotenuse fa e, and the same is true for the orresponding os illations in the layer of air. One then has approximately, in the rst ase an ele tri eld su h as one nds between the ele trodes of a apa itor, and in the se ond ase a magneti eld su h as exists between two opposite magneti poles. The ee t would still remain if the se ond prism were repla ed by a glass plate, but it must be very di ult to demonstrate this experimentally.
Notes to pp. 376405
407
Notes to the translation 1 The integral sign is printed as `
R
' in the original. 0 2 The Fren h uses `Wellenpa ket' throughout. 3 Mis-spelt as `Fuess'.
4 We follow the original English, whi h is a translation by Oppenheimer, from Born (1927, p. 355). De Broglie translates 'me hani s' as `dynamique' and in ludes the words `La Dynamique des Quanta' in the quotation, where Born has `it' (referring to `quantum me hani s'). 5 The Fren h reads ` es' [these℄ rather than `ses' [its℄. 6 `v ' is misprinted as `V '.
7 Consistently mis-spelt throughout the text as `Elsaesser'. 8 For larity, the presentation of the table has been slightly altered. 9 We have used the original English (Dymond 1927, p. 441). 10 We use here the original English text (Patterson 1927, p. 47). De Broglie
hanges `of these results' to `des résultats expérimentaux', omits the itali s and translates `valuable' by R `exa ts'.
11 The last equality is misprinted as ` p dq = nn hn '. τn n 12 The original reads `p. 18', whi h is presumably in referen e to de Broglie's Gauthier-Villars `preprint', in all likelihood ir ulated before the
onferen e (preprints of other le tures were ir ulated as mimeographs). Cf. hapter 1, p. 19
13 `Ann.
de Phys.'
in the original.
14 This is `p. 117' in the original. 15 Again, `p. 117' in the original. 16 Misprinted as `au' instead of `un'.
Quantum me hani s
a
by Messrs Max BORN and Werner HEISENBERG Introdu tion Quantum me hani s is based on the intuition that the essential dieren e between atomi physi s and lassi al physi s is the o
urren e of dis ontinuities (see in parti ular [1,4,5863℄).b Quantum me hani s should thus be onsidered a dire t ontinuation of the quantum theory founded by Plan k, Einstein and Bohr. Bohr in parti ular stressed repeatedly, already before the birth of quantum me hani s, that the dis ontinuities must lead to the introdu tion of new kinemati al and me hani al
on epts, so that indeed lassi al me hani s and its orresponding on eptual s heme should be abandoned [1,4℄. Quantum me hani s tries to introdu e the new on epts through a pre ise analysis of what is `observable in prin iple'. In fa t, this does not mean setting up the prin iple that a sharp division between `observable' and `unobservable' quantities is possible and ne essary. As soon as a on eptual s heme is given, one an infer from the observations to other fa ts that are a tually not observable dire tly, and the boundary between `observable' and `unobservable'1 quantities be omes altogether indeterminate. But if the on eptual s heme itself is still unknown, it will be expedient to enquire only about the observations themselves, without drawing
on lusions from them, be ause otherwise wrong on epts and prejudi es a Our translation follows the German types ript in AHQP-RDN, do ument M-0309. Dis repan ies between the types ript and the published version are reported in the endnotes. The published version is reprinted in Heisenberg (1984, ser. B, vol. 2, pp. 5899) (eds.). b Numbers in square bra kets refer to the bibliography at the end.
408
Quantum me hani s
409
taken over from before will blo k the way to re ognising the physi al relationships [Zusammenhänge℄. At the same time the new on eptual s heme provides the ans hauli h ontent of the new theory.a From a theory that is ans hauli h in this sense, one an thus demand only that it is onsistent in itself and that it allows one to predi t unambiguously the results for all experiments on eivable in its domain. Quantum me hani s is meant as a theory that is in this sense ans hauli h and
omplete for the mi rome hani al pro esses [46℄.2 Two kinds of dis ontinuities are hara teristi of atomi physi s: the existen e of orpus les (ele trons, light quanta) on the one hand, and the o
urren e of dis rete stationary states (dis rete3 energy values, momentum values et .) on the other. Both kinds of dis ontinuities an be introdu ed in the lassi al theory only through arti ial auxiliary assumptions. For quantum me hani s, the existen e of dis rete stationary states and energy values is just as natural as the existen e of dis rete eigenos illations in a lassi al os illation problem [4℄. The existen e of orpus les will perhaps later turn out to be redu ible just as easily to dis rete stationary states of the wave pro esses (quantisation of the ele tromagneti waves on the one hand, and of the de Broglie waves on the other) [4℄, [54℄. The dis ontinuities, as the notion of `transition probabilities' already shows, introdu e a statisti al element into atomi physi s. This statisti al element forms an essential part of the foundations of quantum me hani s (see in parti ular [4,30,38,39,46,60,61,62℄);4 a
ording to the latter, for instan e, in many ases the ourse of an experiment is determinable from the initial onditions only statisti ally, at least if in xing the initial onditions one takes into a
ount only the experiments on eivable in prin iple up to now. This onsequen e of quantum me hani s is empiri ally testable. Despite its statisti al hara ter, the theory nevertheless a
ounts for the apparently fully ausal determination of ma ros opi 5 pro esses. In parti ular, the prin iples of onservation of energy and momentum hold exa tly also in quantum me hani s. There seems thus to be no empiri al argument against a
epting fundamental indeterminism for the mi ro osm.
a For the notion of Ans hauli hkeit, see the omments in se tions 3.4.7, 4.6 and 8.3 (eds.).
410
M. Born and W. Heisenberg
I. The mathemati al methods of quantum me hani s
a
The phenomenon for whose study the mathemati al formalism of quantum me hani s was rst developed is the spontaneous radiation of an ex ited atom. After innumerable attempts to explain the stru ture of the line spe tra with lassi al me hani al models had proved inadequate, one returned to the dire t des ription of the phenomenon on the basis of its simplest empiri al laws (Heisenberg [1℄). First among these is Ritz's
ombination prin iple, a
ording to whi h the frequen y of ea h spe tral line of an atom appears as the dieren e of two terms νik = Ti − Tk ; thus the set of all lines of the atom will be best des ribed by spe ifying a quadrati array [S hema℄, and sin e ea h line possesses besides its frequen y also an intensity and a phase, one will write in ea h position of the array an elementary os illation fun tion with omplex amplitude: q11 e2πiν11 t q21 e2πiν21 t .........
q12 e2πiν12 t q22 e2πiν22 t .........
... ... . ...
(1)
This array is understood as representing a oordinate q as a fun tion of time in a similar way as the totality of terms of the Fourier series X q(t) = qn e2πiνn t , νn = nν0 n
in the lassi al theory; ex ept that now be ause of the two indi es the sum no longer makes sense. The question arises of whi h expressions
orrespond to fun tions of the lassi al oordinate, for instan e to the square q 2 . Now, su h arrays ordered by two indi es o
ur as matri es in mathemati s in the theory of quadrati forms and of linear transformations; the omposition of two linear transformations, X X blj zj , xk = akl yl , yl = j
l
to form a new one,
xk =
X
ckj zj ,
j
then orresponds to the omposition or multipli ation of the matri es X akl blj = ckj . ab = c, that is, (2) l
a Se tion 3.3 ontains additional material on the less familiar aspe ts of the formalisms presented here (eds.).
Quantum me hani s
411
This multipli ation in general is not ommutative. It is natural to apply this re ipe to the array of the atomi os illations (Born and Jordan [2℄, Dira [3℄); it is immediately evident that be ause of Ritz's formula νik = Ti − Tk no new frequen ies appear, just as in the lassi al theory in the multipli ation of two Fourier series, and herein lies the rst justi ation for the pro edure. By repeated appli ation of additions and multipli ations one an de ne arbitrary matrix fun tions. The analogy with the lassi al theory leads further to allowing as representatives of real quantities only those matri es that are `Hermitian', that is, whose elements go over to the omplex onjugate numbers under permutation of the indi es. The dis ontinuous nature of the atomi pro esses here is put into the theory from the start as empiri ally established. However, this does not establish yet the onne tion with quantum theory and its hara teristi onstant h. This is also a hieved, by arrying over the ontent6 of the Bohr-Sommerfeld quantum onditions in a form given by Kuhn and Thomas, in whi h they are written as relations between the Fourier oe ients of the oordinates q and momenta p. In this way one obtains the matrix equation h ·1 , (3) 2πi where 1 means the unit matrix. The matrix p thus does not ` ommute' with q . For several degrees of freedom the ommutation relation (3) holds for every pair of onjugate quantities, while the qk ommute with ea h other, the pk with ea h other, and also the pk with the non orresponding qk . In order to onstru t the new me hani s (Born, Heisenberg and Jordan [4℄), one arries over as far as possible the notions of the lassi al theory. It is possible to de ne the dierentiation of a matrix with respe t to time and that of a matrix fun tion with respe t to an argument matrix. One an thus arry over to the matrix theory the anoni al equations pq − qp =
∂H dq = , dt ∂p
dp ∂H =− , dt ∂q
where one should understand H(p, q) as the same fun tion of the matri es p, q that o
urs in the lassi al theory as a fun tion of the numbers p, q . (To be sure,7 ambiguity an o
ur be ause of the non ommutativity of the multipli ation; for example, p2 q is dierent from pqp.) This pro edure was tested in simple examples (harmoni and anharmoni os illator). Further, one an prove the theorem of onservation of energy, whi h for non-degenerate systems (all terms Tk dierent from ea h other,
412
M. Born and W. Heisenberg
or: all frequen ies νik dierent from zero) here takes the form: for the solutions p, q of the anoni al equations the Hamiltonian fun tion H(p, q) be omes a diagonal matrix W . It follows immediately that the elements of this diagonal matrix represent the terms Tn of Ritz's formula multiplied by h (Bohr's frequen y ondition). It is parti ularly important to realise that onversely the requirement H(p, q) = W
(diagonal matrix)
is a omplete substitute for the anoni al equations of motion, and leads to unambiguously determined solutions even if one allows for degenera ies (equality of terms, vanishing frequen ies). By a matrix with elements that are harmoni fun tions of time, one
an of ourse represent only quantities ( oordinates) that orrespond to time-periodi quantities of the lassi al theory. Therefore y li oordinates (angles), whi h in rease proportionally to time, annot be treated at present.a Nevertheless, one easily manages to subje t rotating systems to the matrix method by representing the Cartesian omponents of the angular momentum with matri es [4℄.8 One obtains thereby expressions for the energyb that dier hara teristi ally from the orresponding lassi al ones; for instan e the modulus9 of the total p angular h h j(j + 1), momentum is not equal to 2π j (j = 0, 1, 2, . . .), but to 2π in a
ordan e with empiri al rules that Landé and others had derived from the term splitting in the Zeeman ee t.c Further, one obtains for the hanges in the angular quantum numbers [Rotationsquantenzahlen℄ the orre t sele tion rules and intensity formulas, as had already been arrived at earlier by orresponden e arguments and on rmed by the Utre ht observations.d Pauli [6℄, avoiding angular variables, even managed to work out the hydrogen atom with matrix me hani s, at least with regard to the energy values and some aspe ts of the intensities. Asking for the most general oordinates for whi h the quantum me hani al laws are valid leads to the generalisation of the notions of anoni al variables and anoni al transformations known from the lassi al theory. Dira [3℄ has noted that the ontent of the expressions su h as a This point is taken up again shortly after eq. (10). (eds.). b Angular momentum is of ourse responsible for a hara teristi splitting of the energy terms (eds.).
For Landé's work on the anomalous Zeeman ee t, see Mehra and Re henberg (1982a, se . IV.4, esp. pp. 46776 and 4825) (eds.). d For the `Utre ht observations' see Mehra and Re henberg (1982a, se . VI.6, pp. 6478) and Mehra and Re henberg (1982b, se . III.4, esp. pp. 15461) (eds.).
Quantum me hani s
413
− ql pk ) − δkl , whi h appear in the ommutation relations of the type (3) orresponds10 to that of the Poisson bra kets, whose vanishing in lassi al me hani s hara terises a system of variables as anoni al. Therefore also in quantum me hani s one will denote as anoni al every system of matrix pairs p, q that satisfy the ommutation relations, and as a anoni al transformation every transformation that leaves these relations invariant. One an write these with the help of an arbitrary matrix S in the forma 2πi h (pk ql
P = S −1 pS,
Q = S −1 qS ,
(4)
and in a ertain sense this is the most general anoni al transformation. Then for an arbitrary fun tion one has f (P, Q) = S −1 f (p, q)S .
Now one an also arry over the main idea of the Hamilton-Ja obi theory [4℄. Indeed, if the Hamiltonian fun tion H is given as a fun tion of any known anoni al matri es p0 , q0 , then the solution of the me hani al problem de ned by H redu es to nding a matrix S that satis es the equation S −1 H(p0 , q0 )S = W .
(5)
This is an analogue of the Hamilton-Ja obi dierential equation of lassi al me hani s. Exa tly as there, also here perturbation theory an be treated most
learly with the help of equation (5). If H is given as a power series in some small parameter H = H0 + λH1 + λ2 H2 + . . .
and the me hani al problem is solved for λ = 0, that is, H0 = W0 is known as a diagonal matrix, then the solution to (5) an be obtained easily as a power series S = 1 + λS1 + λ2 S2 + . . .
by su
essive approximations. Among the numerous appli ations of this pro edure, only the derivation of Kramers' dispersion formula shall be mentioned here, whi h results if one assumes that the light-emitting and the s attering systems are weakly oupled and if one al ulates the perturbation on the latter ignoring the ba krea tion [4℄.a a Cf. p. 99 above (eds.). a In other words, one onsiders just the s attering system under an external
414
M. Born and W. Heisenberg
The theory of the anoni al transformations leads to a deeper on eption, whi h later be ame essential in understanding the physi al meaning of the formalism. To ea h matrix a = (anm ) one an asso iate a quadrati (more pre isely: Hermitian) forma X anm ϕn ϕ¯m nm
of a sequen e of variables ϕ1 , ϕ2 . . ., or also a linear transformation of the sequen e of variables ϕ1 , ϕ2 . . ., into another one ψ1 , ψ2 . . .11 X (6) ψn = anm ϕm , m
where provisionally the meaning of the variables ϕn and ψn shall be left unspe i ed; we shall return to this. A transformation (6) is alled `orthogonal' if it maps the identity form into itself X X (7) ϕn ϕ¯n = ψn ψ¯n . n
n
Now these orthogonal transformations of the auxiliary variables ϕn immediately turn out to be essentially identi al to the anoni al transformations of the q and p matri es; the Hermitian hara ter and the
ommutation relations are preserved. Further, one an repla e the matrix equation (5) by the equivalent requirement [4℄: the form X Hnm (q0 , p0 )ϕn ϕ¯m nm
is to be transformed orthogonally into a sum of squares X wn ψn ψ¯n .
(8)
n
The fundamental problem of me hani s is thus none other than the prin ipal axes problem for surfa es of se ond order in in nite-dimensional spa e, o
urring everywhere in pure and applied mathemati s and variously studied. As is well known, this is equivalent to asking for the values of the parameter W for whi h the linear equations X W ϕn = (9) Hnm ϕm m
perturbation (Born, Heisenberg and Jotrdan [4℄, se tion 2.4, in parti ular eq. (32)). See also Mehra and Re henberg (1982 , h. III, esp. pp. 934 and 1039) (eds.). a ϕ¯ denotes the omplex number onjugate to ϕ.
Quantum me hani s
415
have a non-identi ally vanishing solution. The values W = W1 , W2 , . . . are alled eigenvalues of the form H ; they are the energy values (terms) of the me hani al system. To ea h eigenvalue Wn orresponds an eigensolution ϕk = ϕkn .a The set of these eigensolutions evidently again forms a matrix and it is easy to see that this is identi al with the transformation matrix S appearing in (5).b The eigenvalues, as is well known, are invariant under orthogonal transformations of the ϕk ,12 and sin e these orrespond to the anoni al substitutions of the p and q matri es, one re ognises immediately the
anoni al invarian e of the energy values Wn . While the quantum theoreti al matri es do not belong to the lass of matri es ( nite and bounded in nite13 ) investigated by the mathemati ians (espe ially by Hilbert and his s hool), one an nevertheless arry over the main aspe ts of the known theory to the more general ase. The pre ise formulation of these theorems14 has been re ently given by J. von Neumann [42℄ in a paper to whi h we shall have to return.15 The most important result that is a hieved in this way is the theorem that a form annot always be de omposed into a sum of squares (8), but that there also o
ur invariant integral omponents Z (10) W ψ(W )ψ(W )dW ,
where the sequen e of variables ψ1 , ψ2 , . . . has to be omplemented by the ontinuous distribution ψ(W ). In this way the ontinuous spe tra appear in the theory in the most natural way. But this implies by no means that in this domain the
lassi al theory omes again into its own. Also here the hara teristi dis ontinuities of quantum theory remain; also in the ontinuous spe trum a (spontaneous) state transition onsists of a `jump' of the system from a point W ′ to another one W ′′ with emission of a wave q(W, W ′ )e2πiνt with the frequen y ν = h1 (W ′ − W ′′ ). The main defe t of matrix me hani s onsists in its lumsiness, even helplessness, in the treatment of non-periodi quantities, su h as angular variables or oordinates that attain in nitely large values (e.g. hyperboli traje tories). To over ome this di ulty two essentially dierent routes have been taken, the operator al ulus of Born and Wiener [21℄, and the so- alled16 q-number theory of Dira [7℄. a This is the notation used by Born and Heisenberg: the nth eigensolution is represented by an in nite ve tor with omponents labelled by k (eds.). b This point is made more expli it after eq. (17). See also the relevant ontributions by Dira and by Kramers in the general dis ussion, pp. 492 and 496 (eds.).
416
M. Born and W. Heisenberg
The latter starts from the idea that a great part of the matrix relations
an be obtained without an expli it representation of the matri es, simply on the basis of the rules for operating with the matrix symbols. These depart from the rules for numbers only in that the multipli ation is generally not ommutative. Dira therefore onsiders abstra t quantities, whi h he alls q-numbers (as opposed to the ordinary -numbers) and with whi h he operates a
ording to the rules of the non ommutative algebra. It is therefore a kind of hyper omplex number system. The
ommutation relations are of ourse preserved. The theory a quires an extraordinary resemblan e to the lassi al one; for instan e, one an introdu e angle and a tion variables w, J and expand any q-number into a Fourier series with respe t to the w; the oe ients are fun tions of the J and turn out to be identi al to the matrix elements if one repla es the J by integer multiples of h. By his method Dira has a hieved important results, for instan e worked out the hydrogen atom independently of Pauli [7℄ and determined the intensity of radiation in the Compton ee t [12℄. A drawba k of this formalism apart from the quite tiresome dealing with the non ommutative algebra is the ne essity to repla e at a ertain point of the al ulation ertain q-numbers with ordinary numbers (e.g. J = hn), in order to obtain results omparable with experiment. Spe ial `quantum onditions' whi h had disappeared from matrix me hani s are thus needed again. The operator al ulus diers from the q-number method in that it does not introdu e abstra t hyper omplex numbers, but on rete, onstru tible mathemati al obje ts that obey the same laws, namely operators or fun tions in the spa e of in nitely many variables. The method is by E kart [22℄ and was then developed further by many others following on from S hrödinger's wave me hani s, espe ially by Dira [38℄ and Jordan [39℄ and in an impe
able mathemati al form by J. von Neumann [42℄; it rests roughly on the following idea. A sequen e of variables ϕ1 , ϕ2 , . . . an be interpreted as a point in an P in nite-dimensional spa e. If the sum of squares n |ϕn |2 onverges, then it represents a measure of distan e, a Eu lidean metri [Massbestimmung℄, in this spa e; this metri spa e of in nitely many dimensions is alled for short a Hilbert spa e. The anoni al transformations of matrix me hani s orrespond thus to the rotations of the Hilbert spa e. Now, however, one an also x a point in this spa e other than by the spe i ation of dis rete oordinates ϕ1 , ϕ2 , . . . . Take for instan e a
omplete, normalised orthogonal system of fun tions f1 (q), f2 (q), . . . ,
Quantum me hani s that is one for whi h17 Z 1 for n = m fn (q)fm (q)dq = δnm = 0 for n 6= m ;
417
(11)
the variable q an range here over an arbitrary, also multi-dimensional domain. If one then sets (Lan zos [23℄)18 ( P ϕ(q) = n ϕn fn (q) (12) P ′ ′′ ′ ′′ H(q , q ) = nm Hnm fn (q )fm (q ) , the linear equations (9)19 turn into the integral equation20 Z W ϕ(q ′ ) = H(q ′ , q ′′ )ϕ(q ′′ )dq ′′ .
(13)
This relation established through (12) means thus nothing but a hange of the oordinate system in the Hilbert spa e, given by the orthogonal transformation matrix fn (q) with one dis rete and one ontinuous index. One sees thus that the preferen e for `dis rete' oordinate systems in the original version of the matrix theory is by no means something essential. One an just as well use ` ontinuous matri es' su h as H(q ′ , q ′′ ). Indeed, the spe i representation of a point in the Hilbert spa e by proje tion onto ertain orthogonal oordinate axes does not matter at all; rather, one an summarise equations (9) and (13) in the more general equation W ϕ = Hϕ ,
(14)
where H denotes a linear operator whi h transforms the point ϕ of the Hilbert spa e into another. The equation requires to nd those points ϕ whi h under the operation H only suer a displa ement along the line joining them to the origin.21 The points satisfying this ondition determine an orthogonal system of axes, the prin ipal axes frame of the operator H ; the number of axes is nite or in nite, in the latter
ase distributed dis retely or ontinuously, and the eigenvalues W are the lengths of the prin ipal axes. The linear operators in the Hilbert spa e are thus the general on ept that an serve to represent a physi al quantity mathemati ally. The al ulus with operators pro eeds obviously a
ording to the same rules as the one with Dira 's q-numbers; they22
onstitute a realisation of this abstra t notion. So far we have analysed the situation with the example of the Hamiltonian fun tion, but the same holds for any quantum me hani al quantity. Any oordinate q an be written, instead of as a matrix with dis rete indi es qnm , also as a
418
M. Born and W. Heisenberg
fun tion of two ontinuous variables q(q ′ , q ′′ ) by proje tion onto an orthogonal system of fun tions, or, more generally, an also be onsidered as a linear operator in the Hilbert spa e; then it has eigenvalues that are invariant, and eigensolutions with respe t to ea h orthogonal oordinate system. The same holds for a momentum p and every fun tion of q and p, indeed for every quantum me hani al `quantity'. While in the
lassi al theory physi al quantities are represented by variables that an take numeri al values from an arbitrary value range, a physi al quantity in quantum theory is represented by a linear operator and the sto k of values that it an take by the eigenvalues of the orresponding prin ipal axes problem in the Hilbert spa e. In this view, S hrödinger's wave me hani s [24℄ appears formally as a spe ial ase. The simplest operator whose hara teristi values are all the real numbers, is in fa t the multipli ation of a fun tion F (q) by the real number q ; one writes it simply q . Then, however, the eigenfun tions are `improper' fun tions; for a
ording to (14) they must have the property of being everywhere zero ex ept if W = q . Dira [38℄ has introdu ed for the representation of su h improper fun tions whi h should always be zero when s 6= 0, but the `unit fun tion' δ(s) R ,+∞ for whi h nonetheless −∞ δ(s)ds = 1 should hold. Then one an write down the (normalised) eigenfun tions ϕ(q, W ) = qδ(W − q)
(15)
belonging to the operator q . The onjugate to the operator q is the dierential operator p=
h ∂ ; 2πi ∂q
(16)
indeed, the ommutation relation (3) holds, whi h means just the trivial identity h h d dF (pq − qp)F (q) = = (qF ) − q F (q) . 2πi dq dq 2πi
If one now onstru ts a Hamiltonian fun tion out of p, q (or out of several su h onjugate pairs), then equation (14) be omes a dierential equation for the quantity ϕ(q): H(q,
h ∂ )ϕ(q) = W ϕ(q) . 2πi ∂q
(17)
This is S hrödinger's wave equation, whi h appears here as a spe ial ase of the operator theory. The most important point about this formulation
Quantum me hani s
419
of the quantum laws (apart from the great advantage of onne ting to known mathemati al methods) is the repla ement of all `quantum
onditions', su h as were still ne essary in Dira 's theory of q-numbers, by the simple requirement that the eigenfun tion ϕ(q) = ϕ(q, W ) should be everywhere nite in the domain of de nition of the variables q ; from this, in the event, a dis ontinuous spe trum of eigenvalues Wn (along with ontinuous ones) arises automati ally. But S hrödinger's eigenfun tion ϕ(q, W ) is a tually nothing but the transformation matrix S of equation (5), whi h one an indeed also write in the form HS = SW ,
analogous to (17). Dira [38℄ has made this state of aairs even learer by writing the operators q and p and thereby also H as integral operators, as in (13); then one has to set Z ′ qF (q ) = q ′′ δ(q ′ − q ′′ )F (q ′′ )dq ′′ = q ′ F (q ′ ) , Z (18) h dF h ′ ′ , δ (q − q ′′ )F (q ′′ )dq ′′ = pF (q ′ ) = 2πi 2πi dq ′
where, however, the o
urren e of the derivative of the singular fun tion δ has to be taken into the bargain. Then S hrödinger's equation (17) takes the form (13). The dire t passage to the matrix representation in the stri t sense takes pla e by inverting the formulas (12), in whi h one identi es the orthogonal system fn (q) with the eigenfun tions ϕ(q, Wn ) belonging to the dis rete spe trum. If T is an arbitrary operator ( onstru ted from q h ∂ and p = 2πi ∂q ), de ne the orresponding matrix Tnm by the oe ients of the expansion X T ϕn (q) = (19) Tnm ϕm (q) m
or
Tnm =
Z
ϕm (q)T ϕn (q)dq ;
(19a)
then one easily sees that equation (17) is equivalent to (9). The further development of the formal theory has taken pla e in lose
onne tion with its physi al interpretation, to whi h we therefore turn rst.
420
M. Born and W. Heisenberg
II. Physi al interpretation
The most noti eable defe t of the original matrix me hani s onsists in the fa t that at rst it appears to give information not about a tual phenomena, but rather only about possible states and pro esses. It allows one to al ulate the possible stationary states of a system; further it makes a statement about the nature of the harmoni os illation that
an manifest itself as a light wave in a quantum jump. But it says nothing about when a given state is present, or when a hange is to be expe ted. The reason for this is lear: matrix me hani s deals only with
losed periodi systems, and in these there are indeed no hanges. In order to have true pro esses,23 as long as one remains in the domain of matrix me hani s, one must dire t one's attention to a part of the system; this is no longer losed and enters into intera tion with the rest of the system. The question is what matrix me hani s an tell us about this. Imagine, for instan e, two systems 1 and 2 weakly oupled to ea h other (Heisenberg [35℄, Jordan [36℄).a For the total system onservation of energy then holds; that is, H is a diagonal matrix. But for a subsystem, for instan e 1, H (1) is not onstant, the matrix has elements o the diagonal.24 The energy ex hange an now be interpreted in two ways: for one, the periodi elements of the matrix of H (1) (or of H (2) ) represent a slow beating, a ontinuous os illation of the energy to and fro; but at the same time, one an also des ribe the pro ess with the
on epts of the dis ontinuum theory and say that system 1 performs quantum jumps and arries over the energy that is thereby freed to system 2 as quanta, and vi e versa. But one an now show that these two apparently very dierent views do not ontradi t ea h other at all. This rests on a mathemati al theorem that states the following: (1) (1) Let f (Wn ) be any fun tion of the energy values Wn of the isolated subsystem 1; if one forms the same fun tion of the matrix H (1) that represents the energy of system 1 in the presen e of the oupling to system 2, then f (H (1) ) is a matrix that does not onsist only of diagonal elements f (H (1) )nn . But these represent the time-averaged value of the quantity f (H (1) ). The ee t of the oupling is thus measured by the dieren e25 δfn = f (H (1) )nn − f (Wn(1) ) . a The form of the result as given here is similar to that in Heisenberg [35℄. For further details, see the dis ussion in se tion 3.4.4 (eds.).
Quantum me hani s
421
The rst part of the said theorem now states that δfn an be brought into the form26 X δfn = (20) {f (Wm ) − f (Wn )}Φnm . m
This an be interpreted thus: the time average of the hange in f due to the oupling is the arithmeti mean, with ertain weightings Φnm , of all possible jumps of f for the isolated system. These Φnm will have to be alled `transition probabilities'. The se ond part of the theorem determines the Φnm through the features of the
oupling. Namely, if p01 , q10 , p02 , q20 are oordinates satisfying the evolution equations of the un oupled systems, for whi h therefore H (1) and H (2) on their own are diagonal matri es, one an then think of the energy, in luding the intera tion, as expressed as a fun tion of these quantities. Then the solution of the me hani al problem a
ording to (5)27 redu es to onstru ting a matrix S that satis es the equation S −1 H(p01 , q10 , p02 , q20 )S = W .
Denoting the states of system 1 by n1 , those of system 2 by n2 , a state of the total system is given by n1 n2 ,28 and to ea h transition n1 n2 → m1 m2
orresponds an element of S , Sn1 n2 ,m1 m2 . Then the result is:29 X |Sn1 n2 ,m1 m2 |2 . Φn1 m1 = (21) n2 m2
The squares of the elements of the S -matrix thus determine the transition probabilities. The individual sum term |Sn1 n2 ,m1 m2 |2 in (21) obviously means that omponent of the transition probability for the jump n1 → m1 of system 1 that is indu ed by the jump n2 → m2 of system 2. By means of these results the ontradi tion between the two views from whi h we started is removed. Indeed, for the mean values, whi h alone may be observed, the on eption of ontinuous beating always leads to the same result as the on eption of quantum jumps. If one asks the question when a quantum jump o
urs, the theory provides no answer. At rst it seemed as if there were a gap here whi h might be lled with further probing. But soon it be ame apparent that this is not so, rather, that it is a failure of prin iple, whi h is deeply an hored in the nature of the possibility of physi al knowledge [physikalis hes Erkenntnisvermögen℄. One sees that quantum me hani s yields mean values orre tly, but
422
M. Born and W. Heisenberg
annot predi t the o
urren e of an individual event. Thus determinism, held so far to be the foundation of the exa t natural s ien es, appears here to go no longer un hallenged. Ea h further advan e in the interpretation of the formulas has shown that the system of quantum me hani al formulas an be interpreted onsistently only from the point of view of a fundamental indeterminism, but also, at the same time, that the totality of empiri ally as ertainable fa ts an be reprodu ed by the system of the theory. In fa t, almost all observations in the eld of atomi physi s have a statisti al hara ter; they are ountings, for instan e of atoms in a ertain state. While the determinateness of an individual pro ess is assumed by lassi al physi s, in fa t it plays pra ti ally no role, be ause the mi ro oordinates that exa tly determine an atomi pro ess an never all be given; therefore by averaging they are eliminated from the formulas, whi h thereby be ome statisti al statements. It has be ome apparent that quantum me hani s represents a merging of me hani s and statisti s, in whi h the unobservable mi ro oordinates are eliminated. The lumsiness of the matrix theory in the des ription of pro esses developing in time an be avoided by making use of the more general formalisms30 we have des ribed above. In the general equation (14) one an easily introdu e time expli itly by invoking the theorem of
lassi al me hani s that energy W and time t behave as anoni ally
onjugate quantities; in quantum me hani s it orresponds to having a
ommutation relation W t − tW =
Thus for W one an posit the operator
h . 2πi h ∂ 2πi ∂t .
Equation (14) then reads
h ∂ϕ = Hϕ , 2πi ∂t
(22)
and here one an onsider H as depending expli itly on time. A spe ial
ase of this is the equation h ∂ h ∂ ϕ(q) = 0 , )− H(q, 2πi ∂q 2πi ∂t
(22a)
given by S hrödinger [24℄,31 whi h stands to (17) in the same relation
Quantum me hani s as (22) to (14), as well as the form: Z h ∂ϕ(q ′ ) = H(q ′ , q ′′ )ϕ(q ′′ )dq ′′ , 2πi ∂t
423
(22b)
mu h used by Dira , whi h relates to the integral formula (13). Essentially, the introdu tion of time as a numeri al variable redu es to thinking of the system under onsideration as oupled to another one and negle ting the rea tion on the latter. But this formalism is very
onvenient and leads to a further development of the statisti al view,a namely, if one onsiders the ase where an expli itly time-dependent perturbation V (t) is added to a time-independent energy fun tion H 0 , so that one has the equation h ∂ϕ 0 = H + V (t) ϕ 2πi ∂t
(23)
(Dira [37℄, Born [34℄).32 Now if ϕ0n are the eigenfun tions of the operator H 0 , whi h for the sake of simpli ity we assume to be dis rete, the desired quantity ϕ an be expanded in terms of these: X ϕ(t) = (24) cn (t)ϕ0n . n
The cn (t) are then the oordinates of ϕ in the Hilbert spa e with respe t to the orthogonal system ϕ0n ; they an be al ulated from the dierential equation (23), if their initial values cn (0) are given. The result an be expressed as: X cn (t) = (25) Snm (t)cm (0) , m
where Snm (t) is an orthogonal matrix depending33 on t and determined by V (t). The temporal pro ess is thus represented by a rotation of the Hilbert spa e or by a anoni al transformation (4) with the time-dependent matrix S . Now how is one to interpret this? From the point of view of Bohr's theory a system an always be in only one quantum state. To ea h of these belongs an eigensolution ϕ0n of the unperturbed system. If now one wishes to al ulate what happens to a system that is initially in a ertain state, say the k th, one has to hoose ϕ = ϕ0k as the initial ondition for equation (23), i.e. cn (0) = 0 for n 6= k , and ck (0) = 1. But then, after the perturbation is a See the dis ussion in se tion 3.4 (eds.).
424
M. Born and W. Heisenberg
over, cn (t) will have be ome equal to Snk (t), and the solution onsists of a superposition of eigensolutions. A
ording to Bohr's prin iples it makes no sense to say a system is simultaneously in several states. The only possible interpretation seems to be statisti al: the superposition of several eigensolutions expresses that through the perturbation the initial state an go over to any other quantum state, and it is lear that as measure for the transition probability one has to take the quantity Φnk = |Snk (t)|2 ;
be ause then one obtains again equation (20) for the average hange of any state fun tion. This interpretation is supported by the fa t that one establishes the validity of Ehrenfest's adiabati theorem (Born [34℄); one an show that under an in nitely slow a tion, one has Φnn → 1,
Φnk → 0
(n 6= k) ,
that is, the probability of a jump tends to zero. But this assumption also leads immediately to an interpretation of the cn (t) themselves: the |cn (t)|2 must be the state probabilities [Zustandswahrs heinli hkeiten℄. Here, however, one runs into a di ulty of prin iple that is of great importan e, as soon as one starts from an initial state for whi h not all the cn (0) ex ept one vanish. Physi ally, this ase o
urs if a system is given for whi h one does not know exa tly the quantum state in whi h it is, but knows only the probability |cn (0)|2 for ea h quantum state. As a matter of fa t, the phases [Ar us℄ of the omplex quantities cn (0) still remain inde nite; if one sets cn (0) = |cn (0)|eiγn , then the γn denote some phases whose meaning needs to be established. The probability distribution at the end of the perturbation a
ording to (25) is then X 2 |cn (t)|2 = (26) Snm (t)cm (0) m
and not
X m
|Snm (t)|2 |cm (0)|2 ,
(27)
as one might suppose from the usual probability al ulus. Formula (26), following Pauli, an be alled the theorem of the interferen e of probabilities; its deeper meaning has be ome lear only through the wave me hani s of de Broglie and S hrödinger, whi h we shall presently dis uss. Before this, however, it should be noted that
Quantum me hani s
425
this `interferen e' does not represent a ontradi tion with the rules of the probability al ulus, that is, with the assumption that the |Snk |2 are quite usual probabilities.a In fa t, the omposition rule (27) follows from the on ept of probability for the problem treated here when and only when the relative number, that is, the probability |cn |2 of the atoms in the state n, has been established beforehand experimentally.34 In this ase the phases γn are unknown in prin iple,35 so that (26) then naturally goes over to (27) [46℄. It should be noted further that the formula (26) goes over to the expression (27) if the perturbation fun tion pro eeds totally irregularly as a fun tion of time. That is for instan e the ase when the perturbation is produ ed by `white light'.a Then, on average, the surplus terms in (26) drop out and one obtains (27). In this way it is easy to derive the Einstein oe ient Bnm for the probability per unit radiation of the quantum jumps indu ed by light absorption (Dira [37℄, Born [30℄). But, in general, a
ording to (26) the knowledge of the probabilities |cn (0)|2 is by no means su ient to al ulate the ourse of the perturbation, rather one has to know also the phases γn . This ir umstan e re alls vividly the behaviour of light in interferen e phenomena. The intensity of illumination on a s reen is by no means always equal to the sum of the light intensities of the individual beams of rays that impinge on the s reen, or, as one an well say, it is by no means equal to the sum of the light quanta that move in the individual beams; instead it depends essentially on the phases of the waves. Thus at this point an analogy between the quantum me hani s of orpus les and the wave theory of light be omes apparent. As a matter of fa t this onne tion was found by de Broglie in a quite dierent way. It is not our purpose to dis uss this. It is enough to formulate the result of de Broglie's onsiderations, and their further development by S hrödinger, and to put it in relation to quantum me hani s. The dual nature of light waves, light quanta orresponds to the analogous dual nature of material parti les; these also behave in a ertain respe t like waves. S hrödinger has set up the laws of propagation of these waves [24℄ and has arrived at equation [(17)℄,36 here derived in a dierent way. His view, however, that these waves exhaust the essen e of matter and that parti les are nothing but wave pa kets, not a The notation |Snm |2 would probably be learer, at least a
ording to the reading of this passage proposed in se tion 3.4.6 (eds.). a Compare also Born's dis ussion in Born (1926 [34℄) (eds.).
426
M. Born and W. Heisenberg
only stands in ontradi tion with the prin iples of Bohr's empiri ally very well-founded theory, but also leads to impossible on lusions; here therefore it shall be left to one side. Instead we attribute a dual nature to matter also: its des ription requires both orpus les (dis ontinuities) and waves ( ontinuous pro esses). From the viewpoint of the statisti al approa h to quantum me hani s it is now lear why these an be re on iled: the waves are probability waves. Indeed, it is not the probabilities themselves, rather ertain `probability amplitudes' that propagate
ontinuously and obey dierential or integral equations, as in lassi al
ontinuum physi s; but additionally there are dis ontinuities, orpus les whose frequen y is governed by the square of these amplitudes. The most de nite support for this on eption is given by ollision phenomena for material parti les (Born [30℄). Already Einstein [16℄, when he dedu ed from de Broglie's daring theory the possibility of `dira tion' of material parti les,a ta itly assumed that it is the parti le number that is determined by the intensity of the waves. The same o
urs in the interpretation given by Elsasser [17℄ of the experiments by Davisson and Kunsman [18,19℄ on the re e tion of ele trons by rystals; also here one assumes dire tly that the number of ele trons is a maximum in the dira tion maxima. The same holds for Dymond's [20℄ experiments on the dira tion of ele trons by helium atoms. The appli ation of wave me hani s to the al ulation of ollision pro esses takes a form quite analogous to the theory of dira tion of light by small parti les. One has to nd the solution to S hrödinger's wave equation (17) that goes over at in nity to a given in ident plane wave; this solution behaves everywhere at in nity like an outgoing37 spheri al wave. The intensity of this spheri al wave in any dire tion ompared to the intensity of the in oming wave determines the relative number of parti les de e ted in this dire tion from a parallel ray. As a measure of the intensity one has to take a ` urrent ve tor'38 whi h an be onstru ted from the solution ϕ(q, W ), and whi h is formed quite analogously to the Poynting ve tor of the ele tromagneti theory of light, and whi h measures the number of parti les rossing a unit surfa e in unit time. In this way Wentzel [31℄ and Oppenheimer [32℄ have derived wave me hani ally the famous Rutherford law for the s attering of α-parti les by heavy nu lei.b a Note that the rst predi tion of su h dira tion appears in fa t to have been made by de Broglie in 1923; f. se tion 2.2.1 (eds.). b Cf. Born (1969, Appendix XX). The urrent ve tor, as de ned there, is the usual h 1 j = 2πi (ψ∗ ∇ψ − ψ∇ψ) (eds.). 2m
Quantum me hani s
427
If one wishes to al ulate the probabilities of ex itation and ionisation of atoms [30℄, then one must introdu e the oordinates of the atomi ele trons as variables on an equal footing with those of the olliding ele tron. The waves then propagate no longer in three-dimensional spa e but in multi-dimensional on guration spa e. From this one sees that the quantum me hani al waves are indeed something quite dierent from the light waves of the lassi al theory. If one onstru ts the urrent ve tor just de ned for a solution of the generalised S hrödinger equation (22), whi h des ribes time evolution, one sees that the time derivative of the integral Z |ϕ|2 dq ′ , ranging over an arbitrary domain of the independent numeri al variables q ′ , an be transformed into the surfa e integral of the urrent ve tor over the boundary of that domain. From this it emerges that |ϕ|2
has to be interpreted as parti le density or, better, as probability density. The solution ϕ itself is alled `probability amplitude'. The amplitude ϕ(q ′ , W ′ ) belonging to a stationary state thus yields via |ϕ(q ′ , W ′ )|2 the probability that for given energy W ′ the oordinate q ′ is in some given element dq ′ .39 But this an be generalised immediately. In fa t, ϕ(q ′ , W ′ ) is the proje tion of the prin ipal axis W ′ of the operator H onto the prin ipal axis q ′ of the operator q . One an therefore say in general (Jordan [39℄): if two physi al quantities are given by the operators q and Q and if one knows the prin ipal axes of the former, for instan e, a
ording to magnitude and dire tion,40 then from the equation Qϕ(q ′ , Q′ ) = Q′ ϕ(q ′ , Q′ )
one an determine the prin ipal axes Q′ of Q41 and their proje tions ϕ(q ′ , Q′ ) on the axes of q . Then |ϕ(q ′ , Q′ )|2 dq ′ is the probability that for given Q′ the value of q ′ falls in a given interval dq ′ . If onversely one imagines the prin ipal axes of Q as given, then those of q are obtained42 through the inverse rotation; from this one easily re ognises that ϕ(Q′ , q ′ ) is the orresponding amplitude,43 so that |ϕ(Q′ , q ′ )|2 dQ′ means the probability, given q ′ , to nd the value of Q′ in h ∂ dQ′ . If for instan e one takes for Q the operator p = 2πi ∂q , then one
428
M. Born and W. Heisenberg
has the equation h ∂ϕ = p′ ϕ , 2πi ∂q ′
thus ϕ = Ce
2πi ′ ′ h q p
.
(28)
This is therefore the probability amplitude for a pair of onjugate quantities. For the probability density one obtains |ϕ|2 = C , that is, for given q ′ every value p′ is equally probable. This is an important result, sin e it allows one to retain the on ept of ` onjugate quantity' even in the ase where the dierential de nition fails, namely when the quantity q has only a dis rete spe trum or even when it is only apable of taking nitely many values. The latter for instan e is the ase for angles with quantised dire tion [ri htungsgequantelte Winkel℄,a say for the magneti ele tron, or in the Stern-Gerla h experiments. One an then, as Jordan does, all by de nition a quantity p onjugate to q , if the orresponding probability amplitude has the expression (28). As the amplitudes are the elements of the rotation matrix of one orthogonal system into another, they are omposed a
ording to the matrix rule: Z ϕ(q ′ , Q′ ) =
ψ(q ′ , β ′ )χ(β ′ , Q′ )dβ ′ ;
(29)
in the ase of dis rete spe tra, instead of the integral one has nite or in nite sums. This is the general formulation of the theorem of the interferen e of probabilities. As an appli ation, let us look again at formula (24). Here cn (t) was the amplitude for the probability that the system at time t has energy Wn ; ϕ0n (q ′ ) is the amplitude for the probability that for given energy Wn the oordinate q ′ has a given value. Thus X ϕ(q ′ , t) = cn (t)ϕ0n (q ′ ) n
expresses the amplitude for the probability that q ′ at time t has a given value. Alongside the on ept of the relative state probability |ϕ(q ′ , Q′ )|2 , a This was a standard term referring to the fa t that in the presen e of an external magneti eld, the proje tion of the angular momentum in the dire tion of the eld has to be quantised (quantum number m). Therefore, the dire tion of the angular momentum with respe t to the magneti eld an be said to be quantised. Cf. Born (1969, p. 121) (eds.).
Quantum me hani s
429
there also o
urs the on ept of the transition probability,44 namely, every time one onsiders a system as depending on an external parameter, be it time or any property of a weakly oupled external system. Then the system of prin ipal axes of any quantity be omes dependent on this parameter; it experien es a rotation, represented by an orthogonal transformation S(q ′ , q ′′ ), in whi h the parameter enters (as in formula (25)). The quantities |S(q ′ , q ′′ )|2 are the `transition probabilities';45 in general, however, they are not independent, instead the `transition amplitudes' are omposed a
ording to the interferen e rule.
III. Formulation of the prin iples and delimitation of their s ope After the general on epts of the theory have been developed through analysis of empiri al ndings, the dual task arises, rst of giving a system of prin iples as simple as possible and onne ted dire tly to the observations, from whi h the entire theory an be dedu ed as from a mathemati al system of axioms, and se ond of riti ally s rutinising experien e to assure oneself that no observation on eivable by today's means stands in ontradi tion to the prin iples. Jordan [39℄ has formulated su h a `system of axioms', whi h takes the following statements as fundamental:46 1) One requires for ea h pair of quantum me hani al quantities q, Q the existen e of a probability amplitude ϕ(q ′ , Q′ ), su h that |ϕ|2 gives the probability47 that for given Q′ the value of q ′ falls in a given in nitesimal interval. 2) Upon permutation of q and Q, the orresponding amplitude should be ϕ(Q′ , q ′ ). 3) The theorem (29) of the omposition of probability amplitudes. 4) To ea h quantity q there should belong a anoni ally onjugate one p, de ned by the amplitude (28). This is the only pla e where the quantum
onstant h appears.48 Finally one also takes as obvious that, if the quantities q and Q are identi al, the amplitude ϕ(q ′ , q ′′ ) be omes equal to the `unit matrix' δ(q ′ − q ′′ ), that is, always to zero, ex ept when q ′ = q ′′ . This assumption
430
M. Born and W. Heisenberg
and the multipli ation theorem 3) together hara terise the amplitudes thus de ned as the oe ients of an orthogonal transformation; one obtains the orthogonality onditions simply by stating that the omposition of the amplitude belonging to q, Q with that belonging to Q, q must yield the identity. One an then redu e all given quantities, in luding the operators, to amplitudes by writing them as integral operators as in formula (13). The non ommutative operator multipli ation is then a onsequen e of the axioms and loses all the strangeness atta hed to it in the original matrix theory. Dira 's method [38℄ is ompletely equivalent to Jordan's formulation, ex ept in that he does not arrange the prin iples in axiomati form.a This theory now indeed summarises all of quantum me hani s in a system in whi h the simple on ept of the al ulable probability [bere henbare Wahrs heinli hkeit℄49 for a given event plays the main role.50 It also has some short omings, however. One formal short oming is the o
urren e of improper fun tions, like the Dira δ , whi h one needs for the representation of the unit matrix for ontinuous ranges of variables. More serious is the ir umstan e that the amplitudes are not dire tly measurable quantities, rather, only the squares of their moduli; the phase fa tors are indeed essential for how dierent phenomena are onne ted [für den Zusammenhang der vers hiedenen Ers heinungen wesentli h℄, but are only indire tly determinable, exa tly as phases in opti s are dedu ed indire tly by ombining measurements of intensity. It is, however, a tried and proven prin iple, parti ularly in quantum me hani s, that one should introdu e as far as possible only dire tly observable quantities as fundamental on epts of a theory. This defe t51 is related mathemati ally to the fa t that the de nition of probability in terms of the amplitudes does not express the invarian e under orthogonal transformations of the Hilbert spa e ( anoni al transformations). These gaps in the theory have been lled by von Neumann [41,42℄. There is52 an invariant de nition of the eigenvalue spe trum for arbitrary operators, and of the relative probabilities, without presupposing the existen e of eigenfun tions or indeed using improper fun tions. Even though this theory has not yet been elaborated in all dire tions, one
an however say with ertainty that a mathemati ally irreproa hable grounding of quantum me hani s is possible. Now the se ond question has to be answered: is this theory in a a There are nevertheless some dieren es between the approa hes of Dira and Jordan. Cf. Darrigol (1992, pp. 3434) (eds.).
Quantum me hani s
431
ord with the totality of our experien e? In parti ular, given that the individual pro ess is only statisti ally determined, how an the usual deterministi order be preserved in the omposite ma ros opi phenomena?53 The most important step in testing the new on eptual system in this dire tion onsists in the determination of the boundaries within whi h the appli ation of the old ( lassi al) words and on epts is allowed, su h as `position, velo ity, momentum, energy of a parti le (ele tron)' (Heisenberg [46℄). It now turns out that all these quantities an be individually exa tly measured and de ned, as in the lassi al theory, but that for simultaneous measurements of anoni ally onjugate quantities (more generally: quantities whose operators do not ommute) one annot get below a hara teristi limit of indetermina y [Unbestimmtheit℄.a To determine this, a
ording to Bohr [47℄54 one an start quite generally from the empiri ally given dualism between waves and orpus les. One has essentially the same phenomenon already in every dira tion of light by a slit. If a wave impinges perpendi ularly on an (in nitely long) slit of width q1 , then the light distribution as a fun tion of the deviation angle ϕ is given a
ording to Kir hho by the square of the modulus of the quantity πq 1 Z + q21 sin sin ϕ 2πi λ , a e λ sin ϕq dq = 2a πq1 q1 − 2 sin ϕ λ and thus ranges over a domain whose order of magnitude is given by55 sin ϕ1 = qλ1 and gets ever larger with de reasing slit width q1 . If one
onsiders this pro ess from the point of view of the orpus ular theory, and if the asso iation given by de Broglie of frequen y and wavelength with energy and momentum of the light quantum is valid, h =P , λ then the momentum omponent perpendi ular to the dire tion of the slit is h p = P sin ϕ = sin ϕ . λ hν = W,
One sees thus that after the passage through the slit the light quanta a Here and in the following, the hoi e of translation re e ts the hara teristi terminology of the original. Born and Heisenberg use the terms `Unbestimmtheit' (indetermina y) and `Ungenauigkeit' (impre ision), while the standard German terms today are `Unbestimmtheit' and `Uns härfe' (unsharpness) (eds.).
432
M. Born and W. Heisenberg
have a distribution whose amplitude is given by e
2πi λ
sin ϕ·q
=e
2πi h p·q
,
pre isely as quantum me hani s requires for two anoni ally onjugate variables; further, the width of the domain of the variable p that ontains the greatest number of light quanta is p1 = P sin ϕ1 =
Pλ h = . q1 q1
By general onsiderations of this kind one arrives at the insight that the impre isions (average errors) of two anoni ally onjugate variables p and q always stand in the relation p1 q1 ≥ h .
(30)
The narrowing of the range of one variable, whi h forms the essen e of a measurement, widens unfailingly the range of the other. The same follows immediately from the mathemati al formalism of quantum me hani s on the basis of formula (28). The a tual meaning of Plan k's
onstant h is thus that it is the universal measure of the indetermina y that enters the laws of nature through the dualism of waves and
orpus les. That quantum me hani s is a mixture of stri tly me hani al and statisti al prin iples an be onsidered a onsequen e of this indetermina y. Indeed, in the lassi al theory one may x the state of a me hani al system by, for instan e, measuring the initial values of p and q at a
ertain instant. In quantum me hani s su h a measurement of the initial state is possible only with the a
ura y (30). Thus the values of p and q are known also at later times only statisti ally. The relation between the old and the new theory an therefore be des ribed thus: In lassi al me hani s one assumes the possibility of determining exa tly the initial state; the further development is then determined by the laws themselves. In quantum me hani s, be ause of the impre ision relation, the result of ea h measurement an be expressed by the hoi e of appropriate initial values for probability fun tions; the quantum me hani al laws determine the hange (wave-like propagation) of these probability fun tions. The result of future experiments however remains in general indeterminate and only the expe tation56 of the result is statisti ally onstrained. Ea h new experiment repla es the probability fun tions valid until now with
Quantum me hani s
433
new ones, whi h orrespond to the result of the observation; it separates the physi al quantities into known and unknown (more pre isely and less pre isely known) quantities in a way hara teristi of the experiment. That in this view ertain laws, like the prin iples of onservation of energy and momentum, are stri tly valid, follows from the fa t that they are relations between ommuting quantities (all quantities of the kind q or all quantities of the kind p).a The transition from mi ro- to ma rome hani s results naturally from the impre ision relation be ause of the smallness of Plan k's onstant h. The fa t of the propagation,57 the `melting away' of a `wave or probability pa ket' is ru ial to this. For some simple me hani al systems (free ele tron in a magneti or ele tri eld (Kennard [50℄), harmoni os illator (S hrödinger [25℄)), the quantum me hani al propagation of the wave pa ket agrees with the propagation of the system traje tories that would o
ur in the lassi al theory if the initial onditions were known only with the pre ision restri tion (30). Here the purely lassi al treatment of α and β parti les, for instan e in the dis ussion of Wilson's photographs, immediately nds its justi ation. But in general the statisti al laws of the propagation of a `pa ket' for the lassi al and the quantum theory are dierent; one has parti ularly extreme examples of this in the ases of `dira tion' or `interferen e' of material rays, as in the already mentioned experiments of Davisson, Kunsman and Germer [18,19℄ on the re e tion of ele trons by metalli rystals. That the totality of experien e an be tted into the system of this theory an of ourse be established only by al ulation and dis ussion of all the experimentally a
essible ases. Individual experimental setups,58 in whi h the suspi ion of a ontradi tion with the pre ision limit (30) might arise, have been dis ussed [46,47℄; every time the reason for the impossibility of xing exa tly all determining data ould be exhibited intuitively [ans hauli h aufgewiesen℄. There remains only to survey the most important onsequen es of the theory and their experimental veri ation.
a A similar but more expli it phrasing is used by Born (1926e, le ture 15): assuming that H(p, q) = H(p) + H(q), the time derivatives not only of H but also of all
omponents of momentum and of angular momentum have the form f (q) + g(p) with suitable fun tions f and g . Born states that sin e all q ommute with one another and all p ommute with one another, the expressions f (q) + g(p) will vanish under the same ir umstan es as in lassi al me hani s (eds.).
434
M. Born and W. Heisenberg
IV. Appli ations of quantum me hani s
In this se tion we shall brie y dis uss those appli ations of quantum me hani s that stand in lose relation to questions of prin iple. Here the Uhlenbe k-Goudsmit theory of the magneti ele tron shall be mentioned rst. Its formulation and the treatment of the anomalous Zeeman ee ts with the matrix al ulus raise no di ulties [11℄; the treatment with the method of eigenos illations su
eeds only with the help of the general Dira -Jordan theory (Pauli [45℄). Here, two three-dimensional wave fun tions are asso iated with ea h ele tron. It be omes natural thereby to look for an analogy between matter waves and polarised light waves, whi h in fa t an be arried through to a ertain extent (Darwin [49℄, Jordan [53℄). What is ommon to both phenomena is that the number of terms is nite, so the representative matrix is also nite (two arrangements [Einstellungen℄ for the ele tron, two dire tions of polarisation for light). Here the de nition of the onjugate quantity by means of dierentiation thus fails; one must resort to Jordan's de nition by means of the probability amplitudes (formula (28)). From among the other appli ations, the quantum me hani s of manybody problems shall be mentioned [28,29,40℄. In a system that ontains a number of similar parti les [glei her Partikel℄,59 there o
urs between them a kind of `resonan e' and from that results a de omposition of the system of terms into subsystems that do not ombine (Heisenberg, Dira [28,37℄). Wigner has systemati ally investigated this phenomenon by resorting to group theoreti methods, and has set up the totality of the non- ombining systems of terms [40℄; Hund has managed to derive the majority of these results by omparatively elementary means [48℄. A spe ial role is played by the `symmetri ' and `antisymmetri ' subsystems of terms; in the former every eigenfun tion remains un hanged under permutation of arbitrary similar parti les, in the latter it hanges sign under permutation of any two parti les. In applying this theory to the spe tra of atoms with several ele trons it turns out that the Pauli equivalen e rulea allows only the antisymmetri subsystem.60 On the basis of this insight one an establish quantum me hani ally the systemati s of the line spe tra and of the ele tron grouping throughout the whole periodi system of elements. If one has a large number of similar parti les, whi h are to be given a statisti al treatment (gas theory), one obtains dierent statisti s a That is, the Pauli prin iple applied to ele trons that are `equivalent' in the sense of having the same quantum numbers n and l; f. Born (1969, p. 178). (eds.).
Quantum me hani s
435
depending on whether one hooses the orresponding wave fun tion a
ording to the one or the other subsystem. The symmetri system is
hara terised by the fa t that no new state arises under permutation of the parti les from61 a state des ribed by a symmetri eigenfun tion; thus all permutations that belong to the same set of quantum numbers (lie in the same ` ell') together always have the weight 1. This orresponds to the Bose-Einstein statisti s [56,16℄. In the antisymmetri term system two quantum numbers may never be ome equal, be ause otherwise the eigenfun tion vanishes; a set of quantum numbers orresponds therefore either to no proper fun tion at all or at most to one, thus the weight of a state is 0 or 1. This is the Fermi-Dira statisti s [57,37℄. Bose-Einstein statisti s holds for light quanta, as emerges from the validity of Plan k's radiation formula. Fermi-Dira statisti s ertainly holds for (negative) ele trons, as emerges from the above-mentioned systemati s of the spe tra on the basis of Pauli's equivalen e rule, and with great likelihood also for the positive elementary parti les (protons); one an infer this from observations of band spe tra [28,43℄ and in parti ular from the spe i heat of hydrogen at low temperatures [55℄.62 The assumption of Fermi-Dira statisti s for the positive and negative elementary parti les of matter has the onsequen e that Bose-Einstein statisti s holds for all neutral stru tures, e.g. mole ules (symmetry of the eigenfun tions under permutation of an even number63 of parti les of matter). Within quantum me hani s, in whi h a many-body problem is treated in on guration spa e, the new statisti s of Bose-Einstein and Fermi-Dira has a perfe tly legitimate pla e, unlike in the lassi al theory, where an arbitrary modi ation of the usual statisti s is impossible; nevertheless the restri tions made on the form of the eigenfun tions appear as an arbitrary additional assumption. In parti ular, the example of light quanta indi ates that the new statisti s is related in an essential way to the wave-like properties of matter and light. If one de omposes the ele tromagneti os illations of a avity into spatial harmoni omponents, ea h of these behaves like a harmoni os illator as regards time evolution; it now turns out that under quantisation of this system of os illators a solution results that behaves exa tly like a system of light quanta obeying Bose-Einstein statisti s [4℄. Dira has used this fa t for a onsistent treatment of ele trodynami al problems [51,52℄, to whi h we shall return brie y. The orpus ular stru ture of light thus appears here as quantisation of light waves, su h as vi e versa the wave nature of matter manifests itself in the `quantisation' of the orpus ular motion. Jordan has shown
436
M. Born and W. Heisenberg
[54℄ that one an pro eed analogously with ele trons; one has then to de ompose the S hrödinger fun tion of a avity into fundamental and harmoni s and to quantise ea h of these as a harmoni os illator, in su h a way in fa t that Fermi-Dira statisti s is obtained. The new quantum numbers, whi h express the `weights' in the usual many-body theory, have thus only the values 0 and 1. Therefore one has again here a ase of nite matri es, whi h an be treated only with Jordan's general theory. The existen e of ele trons thus plays the same role in the formal elaboration of the theory as that of light quanta; both are dis ontinuities no dierent in kind from the stationary states of a quantised system. However, if the material parti les stand in intera tion with ea h other, the development of this idea might run into di ulties of a deep nature. The results of Dira 's investigations [51,52℄ of quantum ele trodynami s onsist above all in a rigorous derivation of Einstein's transition probabilities for spontaneous emission.a Here the ele tromagneti eld (resolved into quantised harmoni os illations) and the atom are onsidered as a oupled system and quantum me hani s is applied in the form of the integral equation (13). The intera tion energy appearing therein is obtained by arrying over lassi al formulas. In this onne tion, the nature of absorption and s attering of light by atoms is lari ed. Finally, Dira [52℄ has managed to derive a dispersion formula with damping term; this in ludes also the quantum me hani al interpretation of Wien's experiments on the de ay in lumines en e of anal rays.b His method
onsists in onsidering the pro ess of the s attering of light by atoms as a
ollision of light quanta. However, sin e one an indeed attribute energy and momentum to the light quantum but not easily a spatial position, there is a failure of the wave me hani al ollision theory (Born [30℄), in whi h one presupposes knowledge of the intera tion between the ollision partners as a fun tion of the relative position. It is thus ne essary to use the momenta as independent variables, and an operator equation of matrix hara ter instead of S hrödinger's wave equation. Here one has a ase where the use of the general points of view whi h we have emphasised in this report annot be avoided. At the same time, the theory of Dira reveals anew the deep analogy between ele trons and light quanta.
a As opposed to the indu ed emission dis ussed on p. 425 (eds.). b See above, p. 143 (eds.).
Quantum me hani s
Con lusion
437
By way of summary, we wish to emphasise that while we onsider the last-mentioned enquiries, whi h relate to a quantum me hani al treatment of the ele tromagneti eld, as not yet ompleted [unabges hlossen℄, we onsider quantum me hani s to be a losed theory [ges hlossene Theorie℄, whose fundamental physi al and mathemati al assumptions are no longer sus eptible of any modi ation. Assumptions about the physi al meaning of quantum me hani al quantities that ontradi t Jordan's or equivalent postulates will in our opinion also ontradi t experien e. (Su h ontradi tions an arise for example if the square of the modulus of the eigenfun tion is interpreted as harge density.a ) On the question of the `validity of the law of ausality' we have this opinion: as long as one takes into a
ount only experiments that lie in the domain of our urrently a quired physi al and quantum me hani al experien e, the assumption of indeterminism in prin iple, here taken as fundamental, agrees with experien e. The further development of the theory of radiation will hange nothing in this state of aairs, be ause the dualism between orpus les and waves, whi h in quantum me hani s appears as part of a ontradi tion-free, losed theory [abges hlossene Theorie℄, holds in quite a similar way for radiation. The relation between light quanta and ele tromagneti waves must be just as statisti al as that between de Broglie waves and ele trons. The di ulties still standing at present in the way of a omplete theory of radiation thus do not lie in the dualism between light quanta and waves whi h is entirely intelligible instead they appear only when one attempts to arrive at a relativisti ally invariant, losed formulation of the ele tromagneti laws; all questions for whi h su h a formulation is unne essary an be treated by Dira 's method [51,52℄. However, the rst steps also towards over oming these relativisti di ulties have already been made.
a See S hrödinger's report, espe ially his se tion I, and se tion 4.4 above (eds.).
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M. Born and W. Heisenberg
Bibliographya [1℄ W. Heisenberg, Über quanten[theoretis he℄ Umdeutung kinematis her und me hanis her Beziehungen, Z. f. Phys., 33 (1925), 879. [2℄ M. Born and P. Jordan, Zur Quantenme hanik, I, Z. f. Phys., 34 (1925), 858. [3℄ P. Dira , The fundamental equations of quantum me hani s, Pro . Roy. So . A, 109 (1925), 642. [4℄ M. Born, W. Heisenberg and P. Jordan, Zur Quantenme hanik, II, Z. f. Phys., 35 (1926), 557. [5℄ N. Bohr, Atomtheorie und Me hanik, Naturwiss., 14 (1926), 1. [6℄ W. Pauli, Über das Wasserstospektrum vom Standpunkt der neuen Quantenme hanik, Z. f. Phys., 36 (1926), 336. [7℄ P. Dira , Quantum me hani s and a preliminary investigation of the hydrogen atom, Pro . Roy. So . A, 110 (1926), 561. [8℄ G. E. Uhlenbe k and S. Goudsmit, Ersetzung der Hypothese vom unme hanis hen Zwang dur h eine Forderung bezügli h des inneren Verhaltens jedes einzelnen Elektrons, Naturw., 13 (1925), 953.64 [9℄ L. H. Thomas, The motion of the spinning ele tron, Nature, 117 (1926), 514. [10℄ J. Frenkel, Die Elektrodynamik des rotierenden Elektrons, Z. f. Phys., 37 (1926), 243. [11℄ W. Heisenberg and P. Jordan, Anwendung der Quantenme hanik auf das Problem der anomalen Zeemaneekte, Z. f. Phys., 37 (1926), 263. [12℄ P. Dira , Relativity quantum me hani s with an appli ation to Compton s attering, Pro . Roy. So . [A℄, 111 (1926), 405. [13℄ P. Dira , The elimination of the nodes in quantum me hani s, Pro . Roy. So . [A℄, 111 (1926), 281. [14℄ P. Dira , On quantum algebra, Pro . Cambridge Phil. So ., 23 (1926), 412. [15℄ P. Dira , The Compton ee t in wave me hani s, Pro . Cambridge Phil. So ., 23 (1926), 500. a The style of the bibliography has been both modernised and uniformised. Amendments and fuller details (when missing) are given in square bra kets, mostly without ommentary. Amendments in the Fren h edition of mistakes in the types ript (wrong initials, spelling of names et .) are taken over also mostly without ommentary. Mistakes o
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Quantum me hani s
439
[16℄ A. Einstein, Quantentheorie des einatomigen idealen Gases, II, Berl. Ber. (1925), [3℄.65 [17℄ W. Elsasser, Bemerkungen zur Quanten[me hanik℄ freier Elektronen, Naturw., 13 (1925), 711. [18℄ C. Davisson and [C.℄ Kunsman, [The s℄ attering of low speed ele trons by [Platinum℄ and [Magnesium℄, Phys. Rev., 22 (1923), [242℄.66 [19℄ C. Davisson and L. Germer, The s attering of ele trons by a single
rystal of Ni kel, Nature, 119 (1927), 558. [20℄ E. G. Dymond, On ele tron s attering in Helium, Phys. Rev., 29 (1927), 433. [21℄ M. Born and N. Wiener, Eine neue Formulierung der Quantengesetze für periodis he und [ni htperiodis he℄ Vorgänge, Z. f. Phys., 36 (1926), 174. [22℄ C. E kart, Operator al ulus and the solution of the equation[s℄ of quantum dynami s, Phys. Rev., 28 (1926), 711. [23℄ K. Lan zos, Über eine feldmässige Darstellung der neuen Quantenme hanik, Z. f. Phys., 35 (1926), 812. [24℄ E. S hrödinger, Quantisierung als Eigenwertproblem, I to IV, Ann. d. Phys., 79 (1926), 361; ibid., 489; ibid., 80 (1926), 437; ibid., 81 (1926), 109. [25℄ E. S hrödinger, Der stetige Übergang von der Mikro- zur Makrome hanik, Naturw., 14 (1926), 664. [26℄ E. S hrödinger, Über das Verhältnis der Heisenberg-Born-Jordans hen Quantenme hanik zu der meinen, Ann. d. Phys., 79 (1926), 734. [27℄ P. Jordan, Bemerkung über einen Zusammenhang zwis hen Duanes Quantentheorie der Interferenz und den de Broglies hen Wellen, Z. f. Phys., 37 (1926), 376. [28℄ W. Heisenberg, Mehrkörperproblem und Resonanz in der Quantenme hanik, I and II, Z. f. Phys., 38 (1926), 411; Ibid. 41 (1927), 239. [29℄ W. Heisenberg, Über die Spektren von Atomsystemen mit zwei Elektronen, Z. f. Phys., 39 (1926), 499.67 [30℄ M. Born, Zur Quantenme hanik der Stossvorgänge, Z. f. Phys., 37 (1926), 863; [Quantenme hanik der Stossvorgänge℄, ibid., 38 (1926), 803. [31℄ G. Wentzel, Zwei Bemerkungen über die Zerstreuung korpuskularer Strahlen als Beugungsers heinung, Z. f. Phys., 40 (1926), 590. [32℄ J. R. Oppenheimer, Bemerkung zur Zerstreuung der α-Teil hen, Z. f. Phys., 43 (1927), 413.
440
M. Born and W. Heisenberg
[33℄ W. Elsasser, Diss. Göttingen, [Zur Theorie der Stossprozesse bei Wassersto℄, Z. f. Phys., [45 (1927), 522℄.68 [34℄ M. Born, Das Adiabatenprinzip in der Quantenme hanik, Z. f. Phys., 40 ([1926℄), 167.69 [35℄ W. Heisenberg, S hwankungsers heinungen und Quantenme hanik, Z. f. Phys., 40 (1926), 501. [36℄ P. Jordan, Über quantenme hanis he Darstellung von Quantensprüngen, Z. f. Phys., 40 ([1927℄), 661. [37℄ P. Dira , On the theory of quantum me hani s, Pro . Roy. So . A, 112 (1926), 661. [38℄ P. Dira , The physi al interpretation of quantum dynami s, Pro . Roy. So . A, 113 ([1927℄), 621. [39℄ P. Jordan, Über eine neue Begründung der Quantenme hanik, Z. f. Phys., 40 ([1927℄), [809℄; Se ond Part, ibid., 44 (1927), 1. [40℄ E. Wigner, Über ni htkombinierende Terme in der neueren Quantentheorie, First Part, Z. f. Phys., 40 (1926), 492; Se ond Part, ibid., 40 (1927), 883. [41℄ D. Hilbert, [J.℄. v. Neumann and L. Nordheim, [Über die Grundlagen der Quantenme hanik,℄ Mathem. Ann., 98 [1928℄, 1. [42℄ [J.℄ v. Neumann, [Mathematis he Begründung der Quantenme hanik,℄ Gött. Na hr., 20 May 1927, [1℄. [43℄ F. Hund, Zur Deutung der Molekelspektr[en℄, Z. f. Phys., 40 ([1927℄), 742; ibid., 42 (1927), 93; ibid., 43 (1927), 805. [44℄ W. Pauli, Über Gasentartung und Paramagnetismus, Z. f. Phys., 41 (1927), 81. [45℄ W. Pauli, Zur Quantenme hanik des magnetis hen Elektrons, Z. f. Phys., 43 (1927), 601. [46℄ W. Heisenberg, Über den ans hauli hen Inhalt der quantentheoretis hen Kinematik und Me hanik, Z. f. Phys., 43 (1927), 172.70 [47℄ N. Bohr, Über den begri i hen Aufbau der Quantentheorie, forth oming [im Ers heinen℄. [48℄ F. Hund, Symmetrie haraktere von Termen bei Systemen mit glei hen Partikeln in der Quantenme hanik, Z. f. Phys., 43 (1927), 788. [49℄ [C. G.℄ Darwin, The ele tron as a ve tor wave, Nature, 119 (1927), 282. [50℄ E. Kennard, Zur Quantenme hanik einfa her Bewegungstypen, Z. f. Phys., 44 (1927), 326. [51℄ P. Dira , The quantum theory of emission and absorption of radiation, Pro . Roy. So . A, 114 (1927), 243.
Quantum me hani s
441
[52℄ P. Dira , The quantum theory of dispersion, Pro . Roy. So . A,
114 (1927), 710.
[53℄ P. Jordan, Über die Polarisation der Li htquanten, Z. f. Phys., 44 (1927), 292. [54℄ P. Jordan, Zur Quantenme hanik der Gasentartung, Z. f. Phys., forth oming [im Ers heinen℄ [44 (1927), 473℄. [55℄ D. Dennison, A note on the spe i heat of [the℄ Hydrogen [mole ule℄, Pro . Roy. So . A, [115℄ (1927), 483. On statisti s also: [56℄ N. S. Bose, Plan ks Gesetz und Li htquantenhypothese, Z. f. Phys., 26 (1924), 178. [57℄ E. Fermi, [Sulla quantizzazione del gas perfetto monatomi o℄, Lin ei Rend., 3 (1926), 145. General surveys: [58℄ M. Born (Theorie des Atombaus?) [Probleme der Atomdynamik ℄, Le tures given at the Massa husetts Institute of Te hnology (Springer, [1926℄).71 [59℄ W. Heisenberg, [Über q℄uantentheoretis he Kinematik und Me hanik, Mathem. Ann., 95 (1926), 683. [60℄ W. Heisenberg, Quantenme hanik, Naturw., 14 (1926), [989℄. [61℄ M. Born, Quantenme hanik und Statistik, Naturw., 15 (1927), 238.72 [62℄ P. Jordan, Kausalität und Statistik in der modernen Physik, Naturw., 15 (1927), 105. [63℄ P. Jordan, Die Entwi klung der neuen Quantenme hanik, Naturw., 15 (1927), 614, 636.73
442
M. Born and W. Heisenberg
Dis ussion of Messrs Born and Heisenberg's reporta
Mr Dira . I should like to point out the exa t nature of the orre-
sponden e between the matrix me hani s and the lassi al me hani s. In
lassi al me hani s one an work out a problem by two methods: (1) by taking all the variables to be numbers and working out the motion, e.g. by Newton's laws, whi h means one is al ulating the motion resulting from one parti ular set of numeri al values for the initial oordinates and momenta, and (2) by onsidering the variables to be fun tions of the J 's (a tion variables)74 and using the general transformation theory of dynami s and thus determining simultaneously the motion resulting from all possible initial onditions.75 The matrix theory orresponds to this se ond lassi al method. It gives information about all the states of the system simultaneously. A dieren e between the matrix method and the se ond lassi al method arises sin e in the latter one requires to treat simultaneously only states having nearly the same J 's (one uses, ∂ ), while in the matrix theory one must for instan e, the operators ∂J treat simultaneously states whose J 's dier by nite amounts. To get results omparable with experiment when one uses the se ond
lassi al method,76 one must substitute numeri al values for the J 's in the fun tions of the J 's obtained from the general treatment. One has to do the same in the matrix theory. This gives rise to a di ulty sin e the results of the general treatment are now matrix elements, ea h referring in general to two dierent sets of J 's. It is only the diagonal elements, for whi h these two sets of J 's oin ide, that have a dire t physi al interpretation.
Mr Lorentz. I was very surprised to see that the matri es satisfy
equations of motion. In theory that is very beautiful, but to me it is a great mystery, whi h, I hope, will be lari ed. I am told that by all these
onsiderations one has ome to onstru t matri es that represent what one an observe in the atom, for instan e the frequen ies of the emitted radiation. Nevertheless, the fa t that the oordinates, the potential energy, and so on, are now represented by matri es indi ates that the quantities have lost their original meaning and that one has made a huge step in the dire tion of abstra tion. Allow me to draw attention to another point that has stru k me. Let
a The two dis ussion ontributions by Dira follow his manus ript in AHQP, mi ro lm 36, se tion 10. Deviations in the Fren h edition (whi h may or may not be due to Dira ) are reported in endnote, as well as interesting variants or
an ellations in the manus ript, and pun tuation has been slightly altered (eds.).
Dis ussion
443
us onsider the elements of the matri es representing the oordinates of a parti le in an atom, a hydrogen atom for instan e, and satisfying the equations of motion. One an then hange the phase of ea h element of the matri es without these easing to satisfy the equations of motion; one an, for instan e, hange the time. But one an go even further and
hange the phases, not arbitrarily, but by multiplying ea h element by a fa tor of the form ei(δm −δn ) , and this is quite dierent from a hange of time origin.a Now these matrix elements ought to represent emitted radiation. If the emitted radiation were what is at the basis of all this, one ould expe t to be able to hange all phases in an arbitrary way. The abovementioned fa t then leads us very naturally to the idea that it is not the radiation that is the fundamental thing: it leads us to think that behind the emitted os illations are hidden some true os illations, of whi h the emitted os illations are dieren e os illations. In this way then, in the end there would be os illations of whi h the emitted os illations are dieren es, as in S hrödinger's theory,a and it seems to me that this is ontained in the matri es. This ir umstan e indi ates the existen e of a simpler wave substrate.
Mr Born.
Mr Lorentz is surprised that the matri es satisfy the equations of motion; with regard to this I would like to note the analogy with omplex numbers. Also here we have a ase where in an extension of the number system the formal laws are preserved almost ompletely. Matri es are some kind of hyper omplex numbers, whi h are distinguished from the ordinary numbers by the fa t that the law of ommutativity no longer holds.
Mr Dira .
The arbitrary phases o
urring in the matrix method
orrespond exa tly77 to the arbitrary phases in the se ond lassi al method, where the variables are fun tions of the J 's and w's (a tion and angle variables). There are arbitrary78 phases in the w's, whi h may have dierent values for ea h dierent set of values for the J 's. This is
ompletely analogous79 to the matrix theory, in whi h ea h arbitrary phase is asso iated with a row and olumn, and therefore with a set of a This orresponds of ourse to the hoi e of a phase fa tor eiδn for ea h stationary state. This point (among others) had been raised in the orresponden e between Lorentz and Ehrenfest in the months pre eding the onferen e. See Lorentz to Ehrenfest, 4 July 1927, AHQP-EHR-23 (in Dut h) (eds.). a See se tion 4.4 (eds.).
444
M. Born and W. Heisenberg
values for the J 's.
Mr Born. The phases α
whi h Mr Lorentz has just mentioned are asso iated with the dierent energy levels, quite like in lassi al me hani s. I do not think there is anything mysterious hiding behind this. n
Mr Bohr.
The issue of the meaning of the arbitrary phases, raised by Mr Lorentz, is of very great importan e, I think, in the dis ussion of the onsisten y of the methods of quantum theory. Although the
on ept of phase is indispensable in the al ulations, it hardly enters the interpretation of the observations.
Notes to pp. 408425
445
Notes to the translation 1 Here and in a number of pla es in the following, the Fren h edition omits quotation marks present in the German types ript. They are ta itly restored in this edition. 2 The Fren h edition gives `[47℄'. 3 [diskrete℄ [déterminées℄ 4 The Fren h edition omits `[60℄'. 5 [makroskopis he℄ [mi ros opiques℄ 6 [dur h sinngemässe Übertragung℄ [par une extension logique℄ 7 In the Fren h edition, the parentheti al remark is given as a footnote. 8 The types ript does not give the referen e number, only the bra kets. The Fren h edition omits the referen e entirely. The mentioned results are to be found in se tion 4.1 of Born, Heisenberg and Jordan (1926 [4℄). 9 Word omitted in the Fren h edition. 10 [sind sinngemässe Übertragungen℄ [sont des extensions logiques℄ 11 Misprint in the Fren h edition: summation index `n' in the equation. 12 [orthogonale Transformationen der
ϕk ℄
[transformations orthogonales
ϕk ℄ 13 [bes hränkte unendli he℄ [partiellement in nies℄ 14 [Sätze℄ [prin ipes℄ 15 [no h zurü kzukommen haben℄ [n'avons pas à revenir℄ 16 Word omitted in the Fren h edition. 17 The overbar is missing in the original types ript (only here), but is in luded in the Fren h edition. 18 The types ript reads: `Lan zos [ ℄', the referen e number is added in the Fren h edition. 19 The types ript onsistently gives this referen e as `(q)', the Fren h edition as `(9)'. 20 Equation number missing in the Fren h edition. 21 [eine Vers hiebung längs ihrer Verbindungslinie mit dem Nullpunkt℄ [un dépla ement de leur droite de jon tion ave l'origine℄ 22 [sie℄ [ es règles℄ 23 [Vorgänge℄ [phénomènes℄ 24 Both the manus ript and the Fren h edition read `H1 ' and `H2 ' in this (1) paragraph and two paragraphs later, and `H ' in the intervening paragraph. We have uniformised the notation. 25 The Fren h edition onsistently reads `δfn '.
P
m {f (Wn ) − f (Wm )}Φnm ' in both the types ript and the Fren h edition, but it should be as shown
26 The right-hand side of this equation reads ` (see above, p. 115). 27 The Fren h edition gives `(2)'.
28 Both the types ript and the Fren h edition read (only here) `n1 , n2 '. 29 Both the types ript and the Fren h edition read `Φnm '. 30 Singular in the Fren h edition. 31 The types ript in ludes the square bra kets but no referen e number. The Fren h edition omits the referen e entirely. 32 Only bra kets in the types ript, referen es omitted in the Fren h edition. 33 [abhängige℄ [indépendante℄ 2 2 34 The Fren h edition reads `(cnk ) ' instead of `|cn | ' and `nk ' instead of `n'.
35 The Fren h edition reads `pnk ' instead of `γn '.
446
Notes to pp. 426434
36 Both the types ript and the Fren h edition give `(11)', but this should evidently be either `(17)' or `(22a)'. 37 The adje tive is omitted in the Fren h edition. 38 [`Strahlvektor'℄ [`ve teur radiant'℄ 39 The absolute square is missing in the German types ript, but is added in the Fren h edition. 40 [des einen, etwa, na h Grösse und Ri htung℄ [de l'un, par exemple en grandeur et en dire tion℄ 41 [von
Q℄
[et
Q℄
42 The Fren h edition has a prime on `q '. 43 The overbar is missing in the German types ript, but is added in the Fren h edition. 44 Throughout this paragraph, the Fren h edition translates `Übergang' as `transformation' instead of `transition'. 45 `S ' missing in the Fren h edition. 46 [das folgende Sätze zugrunde legt℄ [qui est à la base des théorèmes suivants℄. Note that `Satz' an indeed mean both `statement' and `theorem'. 47 The Fren h edition omits absolute bars. 48 The `h' is present in the Fren h edition but not in the types ript. 49 In the types ript, this is typed over an (illegible) previous alternative. Jordan in his habilitation le ture (1927f [62℄) uses the term `angebbare Wahrs heinli hkeit' (`assignable probability' in Oppenheimer's translation (Jordan 1927g)). 50 [in dem der einfa he Begri der bere henbaren Wahrs heinli hkeit für ein bestimmtes Ereignis die Hauptrolle spielt℄ [dans lequel la simple notion de la probabilité al ulable joue le r le prin ipale pour un événement déterminé℄ 51 [Überstand℄ [défaut℄. The word `Überstand' may be hara terising the phases as some kind of surplus stru ture, but it is quite likely a mistyping of `Übelstand', whi h an indeed be translated as `defe t', as in the Fren h version. 52 [Es gibt℄ [Cet auteur donne℄ 53 [Wie kann insbesondere bei der nur statistis hen Bestimmtheit des Einzelvorgangs in den zusammengesetzten makroskopis hen Ers heinungen die gewohnte deterministis he Ordnung aufre ht erhalten werden?℄ [En parti ulier omment, vu la détermination uniquement statistique des pro essus individuels dans les phénomènes ma ros opiques
ompliqués, l'ordre déterministe auquel nous sommes a
outumés peut-il être onservé?℄ 54 This referen e is to a supposedly forth oming `Über den begri i hen Aufbau der Quantentheorie'. Yet, no su h published or unpublished work by Bohr is extant. Some pages titled `Zur Frage des begri i hen Aufbaus der Quantentheorie' are ontained in the folder `Como le ture II' in the Niels Bohr ar hive, mi ro lmed in AHQP-BMSS-11, se tion 4. See also Bohr (1985, p. 478). We wish to thank Feli ity Pors, of the Niels Bohr ar hive, for orresponden e on this point. 55 The Fren h edition in orre tly reads `sin ϕ1 56 [Erwartung℄ [attente℄
=
q1 '. λ
57 [Ausbreitung℄ [extension℄ 58 [einzelne Versu hsanordnungen℄ [des essais isolés℄
Notes to pp. 434443
447
59 Again, the terminology has hanged both in German and in English. The term `similar parti les' for `identi al parti les' is used for instan e by Dira (1927a [37℄). 60 [nur das antisymmetris he Teilsystem zulässt℄ [ne permet pas le système antisymétrique℄ 61 [aus℄ [dans℄ 62 The Fren h edition gives `[56℄'. 63 Both the German version followed here and the Fren h version (`a whole number of parti les') seem rather infeli itous. 64 Both the types ript and the Fren h edition add `(Magnetelektron)'. The Fren h edition reads `Nature'.
65 Both the types ript and the Fren h edition read `p. 5'. 66 Types ript and published volume read `Pt' and `Mg', as well as `243'. 67 In the Fren h edition: `409'. 68 This is indeed the (abridged) published version of Elsasser's Göttingen dissertation. 69 In the Fren h edition: `Das Adiabatenprinzip in den Quanten', as well as `1927' (the latter as in the types ript). 70 In both the types ript and the Fren h edition the title of the paper is given as `Über den ans hauli hen Inhalt der Quantenme hanik'. 71 Date given as `1927' in types ript and volume. 72 In the Fren h edition: `288'. 73 The Fren h edition omits `614'. 74 The manus ript in ludes also `and
w's'
and `and angle', both an elled.
75 The Fren h edition breaks up and rearranges this senten e. 76 The Fren h edition omits the temporal lause. 77 The Fren h edition reads `trouvent une analogie'. 78 In the manus ript this repla es the an elled word `unknown'. 79 In the manus ript this repla es ` orresponds exa tly'.
Wave me hani s
a
By Mr E. SCHRÖDINGER Introdu tion Under this name at present two theories are being arried on, whi h are indeed losely related but not identi al. The rst, whi h follows on dire tly from the famous do toral thesis by L. de Broglie, on erns waves in three-dimensional spa e. Be ause of the stri tly relativisti treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave me hani s. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like pro ess in the spa e of position oordinates (q -spa e) of an arbitrary me hani al system.1 We shall therefore all it the multi-dimensional wave me hani s. Of ourse this use of the q -spa e is to be seen only as a mathemati al tool, as it is often applied also in the old me hani s; ultimately, in this version also, the pro ess to be des ribed is one in spa e and time. In truth, however, a omplete uni ation of the two on eptions has not yet been a hieved. Anything over and above the motion of a single ele tron ould be treated so far only in the multi-dimensional version; also, this is the one that provides the mathemati al solution to the problems posed by the Heisenberg-Born matrix me hani s. For these reasons I shall pla e it rst, hoping in this way also to illustrate better the a Our translation follows S hrödinger's German types ript in AHQP-RDN, do ument M-1354. Dis repan ies between the types ript and the Fren h edition are endnoted. Interspersed in the German text, S hrödinger provided his own summary of the paper (in Fren h). We translate this in the footnotes. The Fren h version of this report is also reprinted in S hrödinger (1984, vol. 3, pp. 30223) (eds.).
448
Wave me hani s
449
hara teristi di ulties of the as su h more beautiful four-dimensional version.a
I. Multi-dimensional theory
Given a system whose on guration is des ribed by the generalised position oordinates q1 , q2 , . . . , qn , lassi al me hani s onsiders its task as being that of determining the qk as fun tions of time, that is, of exhibiting all systems of fun tions q1 (t), q2 (t), . . . , qn (t) that orrespond to a dynami ally possible motion of the system. Instead, a
ording to wave me hani s the solution to the problem of motion is not given by a system of n fun tions of the single variable t, but by a single fun tion ψ of the n variables q1 , q2 , . . . , qn and perhaps of time (see below). This is determined by a partial dierential equation with q1 , q2 , . . . , qn (and perhaps t) as independent variables. This hange of role of the qk , whi h from dependent be ome independent variables, appears to be the
ru ial point. More later on the meaning of the fun tion ψ , whi h is still ontroversial. We rst des ribe how it is determined, thus what
orresponds to the equations of motion of the old me hani s. First let the system be a onservative one. We start from its Hamiltonian fun tion H =T +V ,
that is, from the total energy expressed as a fun tion of the qk and the
anoni ally- onjugate momenta pk . We take H to be a homogeneous quadrati fun tion of the qk and of unity and repla e in it ea h pk by ∂ψ h ∂ψ 2π ∂qk and unity by ψ . We all the fun tion of the qk , ∂qk and ψ thus obtained L (be ause in wave me hani s it plays the role of a Lagrange fun tion). Thus h ∂ψ + V ψ2 . L = T qk , (1) 2π ∂qk Now we determine ψ(q1 , q2 , . . . , qn ) by the requirement that under variation of ψ , Z Z δ Ldτ = 0 with (2) ψ 2 dτ = 1 . a Summary of the introdu tion: Currently there are in fa t two [theories of℄ wave me hani s, very losely related to ea h other but not identi al, that is, the relativisti or four-dimensional theory, whi h on erns waves in ordinary spa e, and the multi-dimensional theory, whi h originally on erns waves in the
on guration spa e of an arbitrary system. The former, until now, is able to deal only with the ase of a single ele tron, while the latter, whi h provides the solution to the matrix problems of Heisenberg-Born, omes up against the di ulty of being put in relativisti form. We start with the latter.
450
E. S hrödinger
The integration is to be performed over the whole of q -spa e (on whose perhaps in nitely distant boundary, ∂ψ must disappear). However, dτ is not simply the produ t of the dqk , rather the `rationally measured' volume element in q -spa e: − 12 ∂2T dτ = dq1 dq2 . . . dqn ± (3) ∂p1 . . . ∂pk (it is the volume element of a Riemannian q -spa e, whose metri , as for instan e also in Hertz's me hani s,a is determined by the kineti energy). Performing the variation, taking the normalisation onstraint with the multiplier [Fa tor℄ −E , yields the Euler equation ∆ψ +
8π 2 (E − V )ψ = 0 h2
(4)
(∆ stands for the analogue of the Lapla e operator in the generalised Riemannian sense). As is well known, Z Ldτ = E
for a fun tion that satis es the Euler equation (4) and the onstraint in (2). Now, it turns out that equation (4) in general does not have, for every E -value, a solution ψ that is single-valued and always nite and
ontinuous together with its rst and se ond derivatives; instead, in all spe ial ases examined so far, this is the ase pre isely for the E -values that Bohr's theory would des ribe as stationary energy levels of the system (in the ase of dis repan ies, the re al ulated values explain the fa ts of experien e better than the old ones). The word `stationary' used by Bohr is thus given a very pregnant meaning by the variation problem (4). We shall refer to these values as eigenvalues, Ek , and to the orresponding solutions ψk as eigenfun tions.b We shall number the eigenvalues always in in reasing order and shall number repeatedly those with multiple eigensolutions. The ψk form a normalised omplete orthogonal system in the q -spa e, with respe t to whi h every well-behaved fun tion of the qk an be expanded in a series. Of ourse this does not mean that every well-behaved fun tion solves the homogeneous equation (4) and a For S hrödinger's interest in Hertz's work on me hani s, see Mehra and Re henberg (1987, pp. 52232) (eds.). b As a rule, in ertain domains of the energy axis2 the eigenvalue spe trum is
ontinuous, so that the index k is repla ed by a ontinuous parameter. In the notation we shall generally not take this into a
ount.
Wave me hani s
451
thus the variation problem, be ause (4) is indeed an equation system, ea h single eigensolution ψk satisfying a dierent element of the system, namely the one with E = Ek .a One an take the view that one should be ontent in prin iple with what has been said so far and its very diverse spe ial appli ations. The single stationary states of Bohr's theory would then in a way be des ribed by the eigenfun tions ψk , whi h do not ontain time at all .a One would nd that one an derive mu h more from them that is worth knowing, in parti ular, one an form from them, by xed general rules, quantities that an be aptly taken to be jump probabilities between the single stationary states. Indeed, it an be shown for instan e that the integral Z (5) qi ψk ψk′ dτ ,
extended to the whole of q -spa e, yields pre isely the matrix element bearing the indi es k and k ′ of the `matrix q ' in the Heisenberg-Born theory; similarly, the elements of all matri es o
urring there an be
al ulated from the wave me hani al eigenfun tions. The theory as it stands, restri ted to onservative systems, ould treat already even the intera tion between two or more systems, by onsidering these as one single system, with the addition of a suitable term in the potential energy depending on the oordinates of all subsystems. Even the intera tion of a material system with the radiation eld is not out of rea h, if one imagines the system together with ertain ether os illators (eigenos illations of a avity) as a single onservative system, positing suitable intera tion terms. On this view the time variable would play absolutely no role in an isolated system a possibility to whi h N. Campbell (Phil. Mag., [1℄ (1926), [1106℄) has re ently pointed. Limiting our attention to an isolated system, we would not per eive the passage of time in it any more a Summary of the above: Wave me hani s demands that events in a me hani al system that is in motion be des ribed not by giving n generalised oordinates q1 , q2 . . . qn as fun tions of the time t, but by giving a single fun tion [ψ℄ of the n variables q1 , q2 . . . qn and maybe of the time t. The system of equations of motion of lassi al me hani s orresponds in wave me hani s to a single partial dierential equation, eq. (4), whi h an be obtained by a ertain variational pro edure. E is a Lagrange multiplier, V is the potential energy, a fun tion of the oordinates; h is Plan k's onstant, ∆ denotes the Lapla ian in q -spa e, generalised in the sense of Riemann. One nds in spe i ases that nite and ontinuous solutions, `eigenfun tions' ψk of eq. (4), exist only for ertain `eigenvalues' Ek of E . The set of these fun tions forms a omplete orthogonal system in the oordinate spa e. The eigenvalues are pre isely the `stationary energy levels' of Bohr's theory. a Cf. se tion 8.1 (eds.).
452
E. S hrödinger
than we an noti e its possible progress in spa e, an assimilation of time to the spatial oordinates that is very mu h in the spirit of relativity. What we would noti e would be merely a sequen e of dis ontinuous transitions, so to speak a inemati image, but without the possibility of
omparing the time intervals between the transitions. Only se ondarily, and in fa t with in reasing pre ision the more extended the system, would a statisti al de nition of time result from ounting the transitions taking pla e (Campbell's `radioa tive lo k'). Of ourse then one annot understand the jump probability in the usual way as the probability of a transition al ulated relative to unit time. Rather, a single jump probability is then utterly meaningless; only with two possibilities for jumps, the probability that the one may happen before the other is equal to its jump probability divided by the sum of the two. I onsider this view the only one that would make it possible to hold on to `quantum jumps' in a oherent way. Either all hanges in nature are dis ontinuous or not a single one. The rst view may have many attra tions; for the time being however, it still poses great di ulties. If one does not wish to be so radi al and give up in prin iple the use of the time variable also for the single atomisti system, then it is very natural to assume that it is ontained hidden also in equation (4). One will onje ture that equation system (4) is the amplitude equation of an os illation equation, from whi h time has been eliminated by settinga (6)
ψ ∼ e2πiνt .
E must then be proportional to a power of ν , and it is natural to set E = hν . Then the following is the os illation equation that leads to (4) with the ansatz (6):b ∆ψ −
8π 2 4πi ∂ψ Vψ− =0. h2 h ∂t
(7)
Now this is satis ed not just by a single3 ψk e2πiνk t
(νk =
Ek ), h
a S hrödinger introdu es the time-dependent equation in his fourth paper on quatisation (1926g). There (p. 112), S hrödinger leaves the sign of time undetermined, settling on the same onvention as in (6) the opposite of today's
onvention on pp. 114-15. As late as S hrödinger (1926h, p. 1065), one reads that `the most general solution of the wave-problem will be (the real part of) [eq. (27) of that paper℄'. Instead the wave fun tion is hara terised as `essentially
omplex' in S hrödinger (1927 , fn. 3 on p. 957) (eds.). b Re all that S hrödinger does not in fa t set m = 1, but absorbs the mass in the de nition of ∆ (eds.).
Wave me hani s
453
but by an arbitrary linear ombination ψ=
∞ X
ck ψk e2πiνk t
(8)
k=1
with arbitrary (even omplex) onstants ck . If one onsiders this ψ as the des ription4 of a ertain sequen e of phenomena in the system, then this is now given by a ( omplex) fun tion of the q1 , q2 , . . . , qn and of time, a fun tion whi h an even be given arbitrarily at t = 0 (be ause of the ompleteness5 and orthogonality of the ψk ); the os illation equation (7), or its solution (8) with suitably hosen ck , then governs the temporal development. Bohr's stationary states orrespond to the eigenos illations of the stru ture (one ck = 1, all others = 0). There now seems to be no obsta le to assuming that equation (7) is valid immediately also for non- onservative systems (that is, V may
ontain time expli itly). Then, however,6 the solution no longer has the simple form (8). A parti ularly interesting appli ation hereof is the perturbation of an atomi system by an ele tri alternating eld. This leads to a theory of dispersion, but we must forgo here a more detailed des ription of the same. From (7) there always follows Z d (9) dτ ψψ ∗ = 0 . dt (An asterisk shall always denote the omplex onjugate.7 ) Instead of the earlier normalisation ondition (2), one an thus require Z dτ ψψ ∗ = 1 , (10)
whi h in the onservative ase, equation (8), means ∞ X
ck c∗k = 1 .a
(11)
k=1
a Summary of the above: Even limiting oneself to what has been said up to now, it would be possible to derive mu h of interest from these results, for instan e the transition probabilities, formula (5) yielding pre isely the matrix element qi (k, k ′ ) for the same me hani al problem formulated a
ording to the Heisenberg-Born theory. Although we have restri ted ourselves so far to onservative systems, it would be possible to treat in this way also the mutual a tion between several systems and even between a material system and the radiation eld; one would only have to add all relevant systems to the system under onsideration. Time does not appear at all in our onsiderations and one ould imagine that the only events that o
ur are sudden transitions from one quantum state of the total system to another quantum state, as Mr N. Campbell has re ently thought. Time would be de ned only statisti ally by ounting the quantum jumps (Mr Campbell's
454
E. S hrödinger
What does the ψ -fun tion mean now, that is, how does the system des ribed by it really look like in three dimensions? Many physi ists today are of the opinion that it does not des ribe8 the o
urren es in an individual system,9 but only the pro esses in an ensemble of very many like onstituted systems that do not sensibly in uen e one another10 and are all under the very same onditions. I shall skip this point of view, sin e others are presenting it.a I myself have so far found useful the following perhaps somewhat naive but quite on rete idea [dafür re ht greifbare Vorstellung℄. The lassi al system of material points does not really exist, instead there exists something that ontinuously lls the entire spa e and of whi h one would obtain a `snapshot' if one dragged the lassi al system, with the amera shutter open, through all its on gurations, the representative point in q -spa e spending in ea h volume element dτ a time that is proportional to the instantaneous value of ψψ ∗ . (The value of ψψ ∗ for only one value of the argument t is thus in question.) Otherwise stated: the real system is a superposition of the
lassi al one in all its possible states, using ψψ ∗ as `weight fun tion'. The systems to whi h the theory is applied onsist lassi ally of several12
harged point masses. In the interpretation just dis ussed13 the harge of every single one of these is distributed ontinuously a ross spa e, the individual point mass with harge e yielding to the harge in the three-dimensional volume element dx dy dz the ontribution14 e
Z
′
ψψ ∗ dτ .
(12)
The prime on the integration sign means: one has to integrate only over the part of the q -spa e orresponding to a position of the distinguished point mass within dxdydz . Sin e ψψ ∗ in general depends on time, these harges u tuate; only in the spe ial ase of a onservative system `radioa tive lo k'). Another, less radi al, point of view is to assume that time is hidden already in the family of equations (4) parametrised by E , this family being the amplitude equation of an os illation equation, from whi h time has been eliminated by the ansatz (6). Assuming hν = E one arrives at eq. (7), whi h, be ause it no longer ontains the frequen y ν , is solved by the series (8), where the ck are arbitrary, generally omplex, onstants. Now ψ is a fun tion of the q1 , q2 . . . qn as well as of time t and, by a suitable hoi e of the ck , it an be adjusted to an arbitrary initial state. Nothing prevents us now from making the time appear also in the fun tion V this is the theory of non- onservative systems, one of whose most important appli ations is the theory of dispersion. The important relation (9), whi h follows from eq. (7), allows one in all ases to normalise ψ a
ording to eq. (10). a See the report by Messrs Born and Heisenberg.11
Wave me hani s
455
os illating with a single eigenos illation are they distributed permanently, so to speak stati ally. It must now be emphasised that by the laim that there are15 these
harge densities (and the urrent densities arising from their u tuation), we an mean at best half of what lassi al ele trodynami s would mean by that. Classi ally, harge and urrent densities are (1) appli ation points, (2) sour e points of the ele tromagneti eld. As appli ation points they are ompletely out of the question here; the assumption that these harges and urrents a t, say, a
ording to Coulomb's or Biot-Savart's law dire tly on one another, or are dire tly ae ted in su h a way by external elds, this assumption is either super uous or wrong (N.B. de fa to wrong), be ause the hanges in the ψ fun tion and thereby in the harges are indeed to be determined through the os illation equation (7) thus we must not think of them as determined also in another way, by for es a ting on them. An external ele tri eld is to be taken a
ount of in (7) in the potential fun tion V , an external magneti eld in a similar way to be dis ussed below, this is the way their appli ation to the harge distribution is expressed in the present theory. Instead, our spatially distributed harges prove themselves ex ellently as sour e points of the eld, at least for the external a tion of the system, in parti ular with respe t to its radiation. Considered as sour e points in the sense of the usual ele trodynami s, they yield largely16
orre t information about its frequen y, intensity and polarisation.a In most ases, the harge is in pra ti e on ned to a region that is small
ompared to the wavelengths of the emitted light. The radiation is then determined by the resulting dipole moment [elektris hes Moment℄ of the
harge distribution. A
ording to the prin iples determined above, this is al ulated from the lassi al dipole moment of an arbitrary on guration by performing an average using ψψ ∗ Z (13) Mqu = M l ψψ ∗ dτ . A glan e at (8) shows that in Mqu the dieren es of the νk will appear as emission frequen ies; sin e the νk are the spe tros opi term values, our pi ture provides an understanding17 of Bohr's frequen y ondition. The integrals that appear as amplitudes of the dierent partial os illations of the dipole moment represent a
ording to the remarks on (5) the elements of Born and Heisenberg's `dipole moment matrix'. By evaluata See the dis ussion after the report, as well as se tion 4.4 (eds.).
456
E. S hrödinger
ing these integrals one obtained the orre t polarisations and intensities of the emitted light in many spe ial ases, in parti ular intensity zero in all ases where a line allowed by the frequen y ondition is missing a
ording to experien e (understanding 18 of the sele tion prin iple). Even though all these results, if one so wishes, an be deta hed from the pi ture of the u tuating harges and be represented in a more abstra t form, yet they put quite beyond doubt that the pi ture is tremendously useful for one who has the need for Ans hauli hkeit!19,a In no way should one laim that the provisional attempt of a lassi alele trodynami oupling of the eld to the harges generating the eld is already the last word on this issue. There are internal20 reasons for doubting this. First, there is a serious di ulty in the question of the rea tion of the emitted radiation on the emitting system, whi h is not yet expressed by the wave equation (7), a
ording to whi h also su h wave forms of the system that ontinuously emit radiation ould and would in fa t always persist unabated. Further, one should onsider the following. We always observe the radiation emitted by an atom only through its a tion on another atom or mole ule. Now, from the wave me hani al standpoint we an onsider two harged point masses that belong to the same atom, neither as a ting dire tly on ea h other in their pointlike form (standpoint of lassi al me hani s), nor are we allowed to think this of their `smeared out' wave me hani al harge distributions (the wrong move taunted above). Rather, we have to take a
ount of their lassi al potential energy, onsidered as a fun tion in q -spa e, in the oe ient V of the wave equation (7). But then, when we have two dierent atoms, it will surely not be orre t in prin iple to insert the elds generated by the spread-out harges of the rst at the position a Summary of the above: The physi al meaning of the fun tion ψ appears to be that the system of harged point parti les imagined by lassi al me hani s does not in fa t exist, but that there is a ontinuous distribution of ele tri harge, whose density an be al ulated at ea h point of ordinary spa e using ψ or rather ψψ∗ , the square of the absolute value of ψ. A
ording to this idea, the quantum (or: real) system is a superposition of all the possible on gurations of the lassi al system, the real fun tion ψψ∗ in q -spa e o
urring as `weighting fun tion'. Sin e ψψ∗ in general ontains time, u tuations of harge must o
ur. What we mean by the existen e of these ontinuous and u tuating harges is not at all that they should a t on ea h other a
ording to Coulomb's or Biot-Savart's law the motion of these harges is already ompletely governed by eq. (7). But what we mean is that they are the sour es of the ele tri elds and magneti elds pro eeding from the atom, above all the sour es of the observed radiation. In many a ase one has obtained wonderful agreement with experiment by al ulating the radiation of these u tuating harges using lassi al ele trodynami s. In parti ular, they yield a omplete and general explanation of Bohr's `frequen y ondition' and of the spe tral `polarisation and sele tion rules'.
Wave me hani s
457
of the se ond in the wave equation for the latter. And yet we do this when we al ulate the radiation of an atom in the way des ribed above and now treat wave me hani ally the behaviour of another atom in this radiation eld. I say this way of al ulating the intera tion between the
harges of dierent atoms an be at most approximate, but not orre t in prin iple. For within one system it is ertainly wrong. But if we bring the two atoms loser together, then the distin tion between the harges of one and those of the other gradually disappears, it is a tually never a distin tion of prin iple.21 The oherent wave me hani al route would surely be to ombine both the emitting and the re eiving system into a single one and to des ribe them through a single wave equation with appropriate oupling terms, however large the distan e between emitter and re eiver may be. Then one ould be ompletely silent about the pro esses in the radiation eld. But what would be the orre t oupling terms? Of ourse not the usual Coulomb potentials, as soon as the distan e is equal to several wavelengths!22 (One realises from here that without important amendments the entire theory in reality an only be applied to very small systems.) Perhaps one should use the retarded potentials. But these are not fun tions in the ( ommon) q -spa e, instead they are something mu h more ompli ated. Evidently we en ounter here the provisional limits of the theory and must be happy to possess in the pro edure depi ted above an approximate treatment that appears to be very useful.a
a Summary of the above: However, there are reasons to believe that our u tuating and purely lassi ally radiating harges do not provide the last word on this question. Sin e we observe the radiation of an atom only by its ee t on another atom or mole ule (whi h we shall thus also treat quite naturally by the methods of wave me hani s), our pro edure redu es to substituting into the wave equation of one system the potentials that would be produ ed a
ording to the lassi al laws by the extended harges of another system. This way of a
ounting for the mutual a tion of the harges belonging to two dierent systems annot be absolutely
orre t, sin e for the harges belonging to the same system it is not. The orre t method of al ulating the in uen e of a radiating atom on another atom would be perhaps to treat them as one total system a
ording to the methods of wave me hani s. But that does not seem at all possible, sin e the retarded potentials, whi h should no doubt o
ur, are not simply fun tions of the on guration of the systems, but something mu h more ompli ated. Evidently, at present these are the limits of the method!
458
E. S hrödinger
II. Four-dimensional theory
If one applies the multi-dimensional version of wave me hani s to a single ele tron of mass m and harge e moving in a spa e with the ele trostati potential ϕ and to be des ribed by the three re tangular oordinates x, y, z , then the wave equation (7) be omes 8π 2 4πi ∂ψ 1 ∂2ψ ∂2ψ ∂2ψ (14) − 2 eϕψ − + + =0. 2 2 2 m ∂x ∂y ∂z h h ∂t 1 (N.B. The fa tor m derives from the fa t that, given the way of deter√ √ mining the metri of the q -spa e through the kineti energy, x m, y m, √ z m should be used as oordinates rather than x, y, z .23 ) It now turns out that the present equation is nothing else but the ordinary threedimensional wave equation for de Broglie's `phase waves' of the ele tron, ex ept that the equation in the above form is shortened or trun ated in a way that one an all `negle ting the in uen e of relativity'. In fa t, in the ele trostati eld de Broglie gives the following expressiona for the wave velo ity u of his phase waves, depending on the potential ϕ (i.e. on position) and on the frequen y ν :24 hν mc2 . u = cp ν = (15) 0 h (hν − eϕ)2 − h2 ν02
If one inserts this into the ordinary three-dimensional wave equation ∂ 2ψ 1 ∂2ψ ∂2ψ ∂2ψ + + − 2 2 =0, 2 2 2 ∂x ∂y ∂z u ∂t
and uses (6) to eliminate the frequen y ν from the equation, one has25 ∂2 ∂2 ∂2 (∆ = ∂x 2 + ∂y 2 + ∂z 2 ) 1 ∂ 2 ψ 4πieϕ ∂ψ 4π 2 e2 ϕ2 2 − ν0 ψ = 0 . + 2 ∆ψ − 2 2 + (16) c ∂t hc2 ∂t c h2 Now if one onsiders that in the ase of `slow ele tron motion' (a) the o
urring frequen ies are always very nearly equal to the rest frequen y ν0 , so that in order of magnitude the derivative with respe t to time in 26 (16) is equal to a multipli ation by 2πiν0 , and that (b) eϕ h in this ase is always small with respe t to ν0 ; and if one then sets in equation (16) ψ = e2πiν0 t ψ˜ ,
(17)
and disregarding squares of small quantities, one obtains for ψ˜ exa tly a Cf. the formula for the refra tive index on p. 376 of de Broglie's report (eds.).
Wave me hani s
459
equation (14) derived from the multi-dimensional version of wave me hani s. As laimed, this is thus indeed the ` lassi al approximation' of the wave equation holding for de Broglie's phase waves.a The transformation (17) here shows us that, onsidered from de Broglie's point of view, the multi-dimensional theory is ommitted to a so to speak trun ated view of the frequen y, in that it subtra ts on e and for all from all frequen ies the rest frequen y ν0 (N.B. In al ulating the harge density from ψψ ∗ ,27 the additional fa tor is of ourse irrelevant sin e it has modulus28 1.29 )b Let us now keep to the form (16) of the wave equation. It still requires an important generalisation. In order to be truly relativisti it must be invariant with respe t to Lorentz transformations. But if we perform su h a transformation on our ele tri eld, hitherto assumed to be stati , then it loses this feature and a magneti eld appears by itself next to it. In this way one derives almost unavoidably the form of the wave equation in an arbitrary ele tromagneti eld. The result an be put in the following transparent form, whi h makes the
omplete equivalen e [Glei hbere htigung℄ of time and the three spatial
oordinates fully expli it : " 2 2 2πie 2πie ∂ ∂ + ax + + ay ∂x hc ∂y hc # (18) 2 2 2πie 2πie 1 ∂ 4π 2 ν02 ∂ ψ = 0 . + az + + iϕ − + ∂z hc ic ∂t hc c2 (N.B. a is the ve tor potential.30 In evaluating the squares one has to take a
ount of the order of the fa tors, sin e one is dealing with operators, and further of Maxwell's relation: ∂ax ∂ay ∂az 1 ∂(iϕ) + + + = 0 .) ∂x ∂y ∂z ic ∂t
(19)
This wave equation is of very manifold interest. First, as shown by a That is, the nonrelativisti approximation (eds.). b Summary of the above: The three-dimensional wave equation, eq. (14), obtained by applying the multi-dimensional theory to a single ele tron in an ele trostati potential eld ϕ, is none other than the nonrelativisti approximation of the wave equation that results from Mr L. [d℄e Broglie's ideas for his `phase waves'. The latter, eq. (16), is obtained by substituting into the ordinary wave equation expression (15), whi h Mr [d℄e Broglie has derived for the phase velo ity u as a fun tion of the frequen y ν and of the potential ϕ (that is, of the oordinates x, y, z , on whi h ϕ will depend) and by eliminating from the resulting formula the frequen y ν by means of (6).
460
E. S hrödinger
Gordon,a it an be derived in a way very similar to what we have seen above for the amplitude equation of onservative systems, from a variational prin iple, whi h now obtains in four dimensions, and where time plays a perfe tly symmetri al role with respe t to the three spatial
oordinates. Further: if one adds to the Lagrange fun tion of Gordon's variational prin iple the well-known Lagrange fun tion of the Maxwell eld in va uo (that is, the half-dieren e of the squares of the magneti and the ele tri eld strenghts) and varies in the spa etime integral of the new Lagrange fun tion thus obtained not only ψ , but also the potential
omponents ϕ, ax , ay , az , one obtains as the ve Euler equations along with the wave equation (18) also the four retarded potential equations for ϕ, ax , ay , az .b (One ould also say: Maxwell's se ond quadruple of equations, while the rst, as is well-known, holds identi ally in the potentials.32 ) It ontains as harge and urrent density quadrati forms in ψ and its rst derivatives33 that agree ompletely with the rule whi h we had given in the multi-dimensional theory for al ulating the true
harge distribution from the ψ -fun tion. Se ond, one an further de nec a stress-energy-momentum tensor of the harges, whose ten omponents are also quadrati forms of ψ and its rst derivatives, and whi h together with the well-known Maxwell tensor obeys the laws of onservation of energy and of momentum (that is, the sum of the two tensors has a vanishing divergen e).d But I shall not bother you here with the rather omplex mathemati al development of these issues, sin e the view still ontains a serious in onsisten y. Indeed, a
ording to it, it would be the same potential
omponents ϕ, ax , ay , az whi h on the one hand a t to modify the wave equation (18) (one ould say: they a t on the harges as movers 35 ) and whi h on the other hand are determined in turn, via the retarded a b
d
W. Gordon, Zeits hr. f. Phys., 40 (1926), 117. E. S hrödinger, Ann. d. Phys., 82 (1927), [265℄.31 E. S hrödinger, lo . it.34 Summary of the above: In order to generalise equation (16) so that it may apply to an arbitrary ele tromagneti eld, one subje ts it to a Lorentz transformation, whi h automati ally makes a magneti eld appear. One arrives at eq. (18), in whi h time enters in a perfe tly symmetri al way with the spatial oordinates. Gordon has shown that this equation derives from a four-dimensional variational prin iple. By adding to Gordon's Lagrangian the well-known Lagrangian of the free eld and by varying along with ψ also the four omponents of the potential, one derives from a single variational prin iple besides eq. (18) also the laws of ele tromagnetism with ertain homogeneous quadrati fun tions of ψ and its rst derivatives as harge and urrent densities. These agree well with what was said in the previous hapter regarding the al ulation of the u tuating harges using the ψ fun tion. One nds a de nition of the stress-energy-momentum tensor, whi h, added to Maxwell's tensor, satis es the onservation laws.
Wave me hani s
461
potential equations, by these same harges, whi h o
ur as sour es in the latter equations. (That is: the wave equation (18) determines the ψ fun tion, from the latter one derives the harge and urrent densities, whi h as sour es determine the potential omponents.) In reality, however, one operates otherwise in the appli ation of the wave equation (18) to the hydrogen ele tron, and one must operate otherwise to obtain the orre t result: one substitutes in the wave equation (18) the already given potentials of the nu leus and of possible external elds (Stark and Zeeman ee t). From the solution for ψ thus obtained one derives the u tuating harge densities dis ussed above, whi h one in fa t36 has to use for the determination from sour es of the emitted radiation; but one must not add a posteriori to the eld of the nu leus and the possible external elds also the elds produ ed by these harges at the position of the atom itself in equation (18) 37 something totally wrong would result. Clearly this is a painful la una. The pure eld theory is not enough, it has to be supplemented by performing a kind of individualisation of the harge densities oming from the single point harges of the
lassi al model, where however ea h single `individual' may be spread over the whole of spa e, so that they overlap. In the wave equation for the single individual one would have to take into a
ount only the elds produ ed by the other individuals but not its self- eld. These remarks, however, are only meant to hara terise the general nature of the required supplement, not to onstitute a programme to be taken
ompletely literally.a We wish to present also the remarkable spe ial result yielded by the relativisti form (18) of the wave equation for the hydrogen atom. One would at rst expe t and hope to nd the well-known Sommerfeld formula for the ne stru ture of terms. Indeed one does obtain a ne stru ture and one does obtain Sommerfeld's formula, however the result
ontradi ts experien e, be ause it is exa tly what one would nd in the Bohr-Sommerfeld theory, if one were to posit the radial as well as the azimuthal quantum number as half-integers [halbzahlig℄, that is, half a Summary of the above: However, these last developments run into a great di ulty. From their dire t appli ation would follow the logi al ne essity of taking into a
ount in the wave equation, for instan e in the ase of the hydrogen atom, not only the potential arising from the nu leus, but also the potentials arising from the u tuating harges; whi h, apart from the enormous mathemati al
ompli ations that would arise, would give ompletely wrong results. The eld theory (`Feldtheorie') appears thus inadequate; it should be supplemented by a kind of individualisation of the ele trons, despite these being extended over the whole of spa e.
462
E. S hrödinger
of an odd integer. Today this result is not as disquieting as when it was rst en ountered.a In fa t, it is well-known that the extension of Bohr's theory through the Uhlenbe k-Goudsmit ele tron spin [Elektronendrall℄, required by many other fa ts of experien e, has to be supplemented in turn by the move to se ondary quantum `half'-numbers [`halbe' Nebenquantenzahlen℄ in order to obtain good results. How the spin is represented in wave me hani s is still un ertain. Very promising suggestionsb point in the dire tion that instead of the s alar ψ a ve tor should be introdu ed. We annot dis uss here this latest turn in the theory.c
III. The many-ele tron problem The attemptsa to derive numeri al results by means of approximation methods for the atom with several ele trons, whose amplitude equation (4) or wave equation (7) annot be solved dire tly, have led to the remarkable result that a tually, despite the multi-dimensionality of the original equation, in this pro edure one always needs to al ulate only with the well-known three-dimensional eigenfun tions of hydrogen; indeed one has to al ulate ertain three-dimensional harge distributions that result from the hydrogen eigenfun tions a
ording to the prin iples presented above, and one has to al ulate a
ording the prin iples of
lassi al ele trostati s the self-potentials and intera tion potentials of these harge distributions; these onstants then enter as oe ients in a system of equations that in a simple way determines in prin iple the behaviour of the many-ele tron atom. Herein, I think, lies a hint that with the furthering of our understanding `in the end everything will indeed be ome intelligible in three dimensions again'.38 For this reason I want to elaborate a little on what has just been said. a E. S hrödinger, Ann. d. Phys., 79 (1926), [361℄, p. 372. b C. G. Darwin, Nature, 119 (1927), 282, Pro . Roy. So . A, 116 (1927), 227.
Summary of the above: For the hydrogen atom the relativisti equation (18) yields a result that, although disagreeing with experien e, is rather remarkable, that is: one obtains the same ne stru ture as the one that would result from the Bohr-Sommerfeld theory by assuming the radial and azimuthal quantum numbers to be `integral and a half', that is, half an odd integer. The theory has evidently to be ompleted by taking into a
ount what in Bohr's theory is alled the spin of the ele tron. In wave me hani s this is perhaps expressed (C. G. Darwin) by a polarisation of the ψ waves, this quantity having to be modi ed from a s alar to a ve tor. a See in parti ular A. Unsold, Ann. d. Phys., 82 (1927), 355.
Wave me hani s
463
Let ψk (x, y, z) and Ek ;
(k = 1, 2, 3, . . .)
be the normalised eigenfun tions (for simpli ity assumed as real) and
orresponding eigenvalues of the one-ele tron atom with Z -fold positive nu leus, whi h for brevity we shall all the hydrogen problem. They satisfy the three-dimensional amplitude equation ( ompare equation (4)): 1 ∂2ψ 8π 2 Ze2 ∂2ψ ∂2ψ ψ + E + + + m ∂x2 ∂y 2 ∂z 2 h2 r p (r = x2 + y 2 + z 2 ) .
= 0
(20)
If only one eigenos illation is present, one has the stati harge distribution39 ρkk = −eψk2 .
(21)
If one imagines two being ex ited with maximal strength, one adds to ρkk + ρll a harge distribution os illating with frequen y |Ek − El |/h, whose amplitude distribution is given by 2ρlk = −2eψk ψl .
(22)
The spatial integral of ρkl vanishes when k 6= l (be ause of the orthogonality of the ψk ) and it is −e for k = l. The harge distribution resulting from the presen e of two eigenos illations together has thus at every instant the sum zero. One an now form the ele trostati potential energies Z Z ρkl (x, y, z)ρk′ l′ (x′ , y ′ , z ′ ) pk,l;k′ ,l′ = . . . dx dy dz dx′ dy ′ dz ′ , (23) r′ p where r′ = (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2 and the indi es k, l, k ′ , l′ may exhibit arbitrary degenera ies (to be sure, in the ase k = k ′ , l = l′ , p is twi e the potential self-energy of the harge distribution ρkl ; but that is of no importan e). It is the onstants p that ontrol also the many-ele tron atom. Let us sket h this. Let the lassi al model now onsist of n ele trons and a Z -fold positively harged nu leus at the origin. We shall use the wave equation in the form (7). It be omes 3n-dimensional,40 say thus 1 4πi ∂ψ 8π 2 (∆1 + ∆2 + . . . + ∆n )ψ − 2 (Vn + Ve )ψ − =0. m h h ∂t
(24)
464
E. S hrödinger
Here ∆σ =
∂2 ∂2 ∂2 + + ; ∂x2σ ∂yσ2 ∂zσ2
σ = 1, 2, 3, . . . , n .
(25)
We have onsidered the potential energy fun tion as de omposed in two parts, Vn + Ve ; Vn should orrespond to the intera tion of all n ele trons with the nu leus, Ve to their intera tion with one another, thereforea Vn = −Ze2 Ve = +e2
n X 1 , r σ=1 σ
(26)
X′ 1 rστ
(27)
(σ,τ )
h i p p rσ = x2σ + yσ2 + zσ2 , rστ = (xσ − xτ )2 + (yσ − yτ )2 + (zσ − zτ )2 .
As the starting point for an approximation pro edure we hoose now the eigensolutions of equation (24) with Ve = 0, that is with the intera tion between the ele trons disregarded. The eigenfun tions are then produ ts of hydrogen eigenfun tions, and the eigenvalues are sums of the orresponding eigenvalues of hydrogen. As a matter of fa t, one easily shows that41 ψk1 ...kn = ψk1 (x1 y1 z1 ) . . . ψkn (xn yn zn )e
2πit h (Ek1 +...+Ekn )
(28)
always satis es equation (24) (with Ve = 0). And if one takes all possible sequen es of numbers [Zahlenkombinationen℄ for the k1 , k2 , . . . , kn , then these produ ts of ψk form a omplete orthogonal system in the 3ndimensional q -spa e one has thus integrated the approximate equation
ompletely. One now aims to solve the full42 equation (24) (with Ve 6= 0) by expansion with respe t to this omplete orthogonal system, that is one makes this ansatz: ψ=
∞ X
k1 =1
...
∞ X
ak1 ...kn ψk1 ...kn .
(29)
kn =1
But of ourse the oe ients a annot be onstants, otherwise the above sum would again be only a solution of the trun ated equation with Ve = 0. It turns out, however, that it is enough to onsider the a as fun tions a Analogously to eq. (12), the prime on the summation sign should be interpreted as meaning that the sum is to be taken over all pairs with σ 6= τ (eds.).
Wave me hani s
465
of time alone (`method of the variation of onstants').a Substituting (29) into (24) one nds that the following onditions on the time dependen e of the a must hold:43,44 ∞ ∞ X 2πi X 2πit dak1 ...kn = vk1 ...kn ,l1 ...ln al1 ...ln e h (El1 ...ln −Ek1 ...kn ) ... dt h l1 =1
ln =1
[k1 . . . kn = 1, 2, 3 . . . ]. (30)
Here we have set for brevity Ek1 + . . . + Ekn = Ek1 ...kn .
(31)
The v are onstants, indeed they are prima fa ie 3n-tuple integrals ranging over the whole of q -spa e (Additional explanation:45 Where do these 3n-tuple integrals ome from? They derive from the fa t that after substituting (29) into (24) one repla es the latter equation by the mathemati ally equivalent ondition that its left-hand side shall be orthogonal to all fun tions of the omplete orthogonal system in R3n . The system (30) expresses this ondition.) Writing this out one has vk1 ...kn ,l1 ...ln = Z 3n-fold Z . . . dx1 . . . dzn Ve ψk1 (x1 , y1 , z1 ) . . . ψkn (xn , yn , zn ) (32) ψl1 (x1 , y1 , z1 ) . . . ψln (xn , yn , zn ) .
If one now onsiders the simple stru ture of Ve given in (27), one re ognises that the v an be redu ed to sextuple integrals, in fa t ea h of them is a nite sum of some of the Coulomb potential energies de ned in (23). Indeed, if in the nite sum representing Ve , we fo us on an individual term, for example e2 /rστ , this ontains only the six variables xσ , . . . , zτ . One an thus immediately perform in (32) pre isely 3n − 6 integrations on this term, yielding (be ause of the orthogonality and normalisation of the ψk ) the fa tor 1, if kρ = lρ for all indi es ρ that oin ide neither with σ nor with τ , and yielding instead the fa tor 0 if even just for a single ρ dierent from σ and τ one has: kρ 6= lρ . (One sees thus that very many terms disappear.) For the non-vanishing terms, it is easy to see that they oin ide with one of the p de ned in (23). QEDa a P. A. M. Dira , Pro . Roy. So . A, 112 (1926), [661℄ p. 674. a Summary of the above: Calling ψk and Ek the eigenfun tions and eigenvalues of the problem for one ele tron, harge −e, in the eld of a nu leus +Ze (hydrogen problem), let us form the harge distributions (21) and (22), the former orresponding to the existen e of a single normal mode, the latter to the
ooperation of two of them. Taken as harge densities in ordinary ele trostati s,
466
E. S hrödinger
Let us now have a somewhat loser look at the equation system (30), whose oe ients, as we have just seen, have su h a relatively simple stru ture, and whi h determines the varying amplitudes of our ansatz46 (29) as fun tions of time. We an allow ourselves to introdu e a somewhat simpler symboli notation, by letting the string of indi es k1 , k2 , . . . , kn be represented by the single index k , and similarly l1 , l2 , . . . , ln by l. One then has ∞
2πi X 2πit dak = vkl al e h (El −Ek ) . dt h
(33)
l=1
(One must not onfuse, however, El , Ek with the single47 eigenvalues of the hydrogen problem, whi h were earlier denoted in the same way.48 ) This is now a system of in nitely many dierential equations, whi h we annot solve dire tly: so, pra ti ally nothing seems to have been gained. In turn, however, we have as yet also negle ted nothing: with exa t solutions ak of (33), (29) would be an exa t solution of (24). This is pre isely where I want to pla e the main emphasis, greater than on the pra ti al implementation of the approximation pro edure, whi h shall be sket hed below only for the sake of ompleteness. In prin iple the equations (33) determine the solution of the many-ele tron problem exa tly;49 and they no longer ontain anything multi-dimensional; their oe ients are simple Coulomb energies of harge distributions that already o
ur in the hydrogen problem. Further, the equations (33) determine the solutions of the many-ele tron problem a
ording to (29) as a ombination of produ ts of the hydrogen eigenfun tions. While these produ ts (denoted above by ψk1 k2 ...kn )50 are still fun tions on the 3n-dimensional q -spa e, any two of them yield in the al ulation of the three-dimensional harge distributions in the many-ele tron atom, as is easily seen, a harge distribution whi h if it is not identi ally zero ea h of these would have a ertain potential energy and there would even be a
ertain mutual potential energy between two of them, assumed to oexist. These are the onstants pkl;k′ l′ in (23). With these givens, let us atta k the problem of the n-ele tron atom. Dividing the potential energy in the wave equation (24) for this problem into two terms and negle ting at rst the term Ve , due to the mutual a tion between the ele trons, the eigensolutions would be given by (28), that is, by the produ ts of n hydrogen fun tions. From these produ ts, taken in all
ombinations, form the series (29), whi h will yield the exa t solution of equation (24), provided that the oe ients ak1 k2 ...kn are fun tions of time satisfying the equations (30); (see the abbreviation (31)). The oe ients v in (30) are onstants, de ned originally by the 3n-tuple integrals (32), whi h however, thanks to the simple form of Ve (see (27)), redu e to sextuple integrals, namely pre isely to the
onstants pk,l;k′ ,l′ (see (23)).
Wave me hani s
467
orresponds again to a hydrogen distribution (denoted above by ρkk or ρkl ). These onsiderations are the analogue of the onstru tion of the higher atoms from hydrogen traje tories in Bohr's theory. They reinfor e the hope that by delving more deeply one will be able to interpret and understand the results of the multi-dimensional theory in three dimensions.a Now, as far as the approximation method is on erned, it onsists in fa t of onsidering the ontribution Ve made to the potential energy fun tion V by the intera tion of the ele trons with one another, to be as far as possible small as ompared to the a tion of the nu leus. The vkl are then onsidered small ompared to the eigenvalue dieren es El −Ek , ex ept if El = Ek . The al will then vary slowly by omparison to the powers of e appearing on the right-hand side of equation (33), as long as the latter are not equal to 1, and all those terms on the right-hand side for whi h this is not the ase will yield only small u tuations of short period of the ak and an be negle ted in the approximation.51 Thereby, rst, the sums on the right be ome nite, be ause in fa t always only a nite number of eigenvalues oin ide. Se ond, the in nitely many equations separate into groups; ea h group ontains only a nite number of al and
an be integrated very easily.52 This is the rst step of the approximation pro edure, whi h in theory an be ontinued inde nitely, but be omes more and more umbersome. We shall not enter into details. One an also transform the untrun ated system of dierential equations (33) at a single stroke into a system of ordinary linear equations (with in nitely many unknowns!) by setting a l = cl e
2πit h (E−El )
,
(34)
where the quantity E and the quantities cl are unknown onstants. Substituting into (33) one nds (E − Ek )ck =
∞ X
vkl cl ;
(k = 1, 2, 3, . . .) .
(35)
l=1
a Summary of the above: Although the system of eqs. (30) (abbreviated to (33)) does not admit a dire t solution, the number of equations as well as the number of unknown fun tions being in nite, it seems to me very interesting that the solution to the multi-dimensional problem is provided in prin iple by a system of equations whose oe ients have su h simple meanings in three dimensions. Further, one realises that the harge distribution that orresponds to the solution (29) of the n-ele tron problem turns out to be the superposition of the distributions ρkk and ρkl that o
ur already in the hydrogen problem. The hope of interpreting and of understanding the multi-dimensional theory in three dimensions is thus strengthened.
468
E. S hrödinger
This equation system oin ides with the Heisenberg-Born `prin ipal axes problem'. If the vkl are very small quantities, then, if not all cl are to be very small, E must be lose to one of the El , let us say to Ek . In the rst approximation then only ck , and all those cl for whi h El = Ek , are dierent from zero. The problem thus separates in the rst approximation into a denumerable set of nite prin ipal axes problems.a
a Summary of the above: One an embark on the solution of the system of equations (33) by an approximation method. Positing (34), the onstants E and cl have to satisfy the system (35) of ordinary linear homogeneous equations, whose number as well as that of the unknown onstants, however, is in nite. It is only by assuming all oe ients vkl to be small that one an on lude that E has to be very lose to one of the values El , for instan e Ek , and that [cl ℄ approximately vanishes, unless El is equal to Ek . Sin e there is only a nite number of El that oin ide with Ek , the problem redu es in the rst approximation to a problem of a nite number of `prin ipal axes', or rather to an in nity of su h nite problems. As a matter of fa t, the equations (35) oin ide with the problem of an in nite number of prin ipal axes, whi h the Heisenberg-Born me hani s redu es to.
Dis ussion
469
Dis ussion of Mr S hrödinger's report
Mr S hrödinger.
It would seem that my des ription in terms of a snapshot was not very fortunate, sin e it has been misunderstood. Perhaps the following explanation is learer. The interpretation of Born is well-known, who takes ψψ ∗ dτ to be the probability for the system being in the volume element dτ of the on guration spa e. Distribute a very large number N of systems in the on guration spa e, taking the above probability as `frequen y fun tion'. Imagine these systems as superposed in real spa e, dividing however by N the harge of ea h point mass in ea h system. In this way, in the limiting ase where N = ∞ one obtains the wave me hani al pi ture of the system.
Mr Bohr. You have said that from the harge distribution ψψ dτ ∗
and the lassi al laws you obtain the frequen y and intensity of light, but do the remarks about di ulties you made later indi ate that what you had obtained was not orre t?
Mr S hrödinger.
The di ulty I mentioned is the following. If one expands the general solution as a series with respe t to the eigenfun tions X ψ= ck ψk k
and if one al ulates the intensity of the radiation resulting from ψk and ψl together, one nds that it be omes proportional to c2k c2l . However, a
ording to the old theory, only the square of the amplitude orresponding to the `initial level' should appear here; that of the ` nal level' should be repla ed by 1.
Mr Bohr. Has Dira not found the solution to the di ulty? Mr S hrödinger. Dira 's results are ertainly very interesting
and point the way toward a solution, if they do not ontain it already. Only, we should rst ome to an understanding in physi al terms [nous devrions d'abord nous entendre en langage physique℄. I nd it still impossible, for the time being, to see an answer to a physi al question in the assertion that ertain quantities obey a non ommutative algebra, espe ially when these quantities are meant to represent numbers of atoms. The relation between the ontinuous spatial densities, des ribed earlier, and the observed intensities and polarisations of the spe tral
470
E. S hrödinger
rays is [too natural℄a for me to deny all meaning to these densities only be ause some di ulties appear that are not yet resolved.
Mr Born. It seems to me that interpreting the quantity ψψ
as a harge density leads to di ulties in the ase of quadrupole moments. The latter in fa t need to be taken into a
ount in order to obtain the radiation, not only for theoreti al reasons, but also for experimental reasons. For brevity let us set ∗
e2 ψψ ∗ = e2 |ψ|2 = Ψ
and let us onsider, for example, the ase of two parti les; Ψ be omes a fun tion of x1 and x2 , where for brevity x1 stands for all the oordinates of the rst parti le; x2 has a similar meaning. The ele tri density is then, a
ording to S hrödinger, Z Z ρ(x) = Ψ(x, x2 )dx2 + Ψ(x1 , x)dx1 . In wave me hani s the quadrupole moment Z Z x1 x2 Ψ(x1 , x2 )dx1 dx2
annot, as far as I an tell, be expressed using the fun tion ρ(x). I would like to know how one an, in this ase, redu e the radiation of the quadrupole to the motion of a harge distribution ρ(x) in the usual three-dimensional spa e.
Mr S hrödinger.
I an assure you that the al ulation of the dipole moments is perfe ly orre t and rigorous and that this obje tion by Mr Born is unfounded. Does the agreement between wave me hani s and matrix me hani s extend to the possible radiation of a quadrupole? That is a question I have not examined. Besides, we do not possess observations on this point that ould allow us to use a possible disagreement between the two approa hes to de ide between them.
Mr Fowler asks for explanations regarding the method for solving
the equations in the ase of the many-ele tron problem.
Mr De Donder. Equation (24) of Mr S hrödinger's report an be
a The Fren h here reads `trop peu naturelle', whi h has the exa t opposite meaning. The ontext would seem, however, to justify the amendment (eds.).
Dis ussion
471
extended to the ase in whi h the n harged parti les are dierent and where the external a tions as well as the intera tions an be des ribed, in spa etime, by a gravitational eld [ hamp gravi que℄.a The quantum equation thus obtained is the sum of the quantum equations for the n parti les taken separately, ea h of the equations being divided by the (rest) mass of the orresponding parti le. Thus, for instan e, the quantum equation for the nu leus will not enter if one assumes, as a rst approximation, that the mass of the nu leus is in nitely large with respe t to that of an ele tron. When there is intera tion, the problem is mu h more omplex. One
an, as Mr S hrödinger indi ates, onsider the a tion of the nu leus as an external a tion a ting on the ele trons of the loud [ ouronne℄, and the (ele trostati ) a tions between the ele trons in this loud as a perturbation; but that is only a rst approximation. In order to a
ount for relativisti and ele tromagneti ee ts I have assumed that the mole ular systems have an additive hara ter.a One an thus re over, as a spe ial
ase, the above-mentioned method of quantisation by S hrödinger.
Mr Born.
In Göttingen we have embarked on a systemati al ulation of the matrix elements that appear in perturbation theory, with the aim of olle ting them in tables up to the prin ipal quantum number 10. Part of these al ulations, whi h are very extended, has already been done. My oworker Mr Biemüller has used them to al ulate the lower terms of the helium atom a
ording to the usual perturbation method up to perturbations of the se ond order. The agreement of the ground term with the empiri al value, despite the defe ts of the pro edure, is hardly worse than in the re ently published paper by Kellner [Zeits hr. f. Phys., 44 (1927), 91℄, who has applied a more pre ise method (Ritz's pro edure).
Mr Lorentz. Do you see the out ome of this long labour as
satisfa tory?
Mr Born. The al ulation has not attained yet the pre ision of
the measurements. The al ulations we have done applying the ordinary a Th. De Donder, L'équation fondamentale de la Chimie quantique, Comptes Rendus A ad. S i. Paris, session of 10 O tober 1927, pp. 698700. See esp. eq. (10). a For more details, one an onsult our note: `L'équation de quanti ation des molé ules omprenant n parti ules éle trisées', published after this meeting, in the Bull. A . R. Belg., Cl. des S ien es, session of 5 November 1927.
472
E. S hrödinger
perturbation method [méthode des perturbations ordinaires℄ onsist of a series expansion with respe t to the inverse of the nu lear harge Z , of the form b c E = Z a + + 2 + ... . Z Z The three terms shown have been al ulated. Nevertheless, in the ase of helium (Z = 2) the pre ision is not yet as good as in the al ulations done by Kellner using Ritz's approximation method.
Mr Lorentz. But you hope however to improve your results. Mr Born. Yes, only the onvergen e of the series is very slow. Mr Heisenberg. On the subje t of this approximation method,
Mr S hrödinger says at the end of his report that the dis ussion he has given reinfor es the hope that when our knowledge will be deeper it will be possible to explain and to understand in three dimensions the results provided by the multi-dimensional theory. I see nothing in Mr S hrödinger's al ulations that would justify this hope. What Mr S hrödinger does in his very beautiful approximation method, is to repla e the n-dimensional dierential equations by an in nity of linear equations. That redu es the problem, as Mr S hrödinger himself states, to a problem with ordinary matri es, in whi h the oe ients an be interpreted in three-dimensional spa e. The equations are thus `threedimensional' exa tly in the same sense as in the usual matrix theory. It thus seems to me that, in the lassi al sense, we are just as far from understanding the theory in three dimensions as we are in the matrix theory.
Mr S hrödinger.
I would not know how to express more pre isely my hope of a possible formulation in a three-dimensional spa e. Besides, I do not believe that one would obtain simpler al ulational methods in this way, and it is probable that one will always do al ulations using the multi-dimensional wave equation. But then one will be able to grasp its physi al meaning better. I am not pre isely sear hing for a threedimensional partial dierential equation. Su h a simple formulation is surely impossible. If I am not satis ed with the urrent state of the problem, it is be ause I do not understand yet the physi al meaning of its solution.
Dis ussion
473
What Mr Heisenberg has said is mathemati ally unex eptionable, but the point in question is that of the physi al interpretation. This is indispensable for the further development of the theory. Now, this development is ne essary. For one must agree that all urrent ways of formulating the results of the new quantum me hani s only orrespond to the lassi al me hani s of a tions at a distan e. As soon as light
rossing times be ome relevant in the system, the new me hani s fails, be ause the lassi al potential energy fun tion no longer exists. Allow me, to show that my hope of a hieving a three-dimensional
on eption is not quite utopian, to re all what Mr Fowler has told us on the topi of Mr Hartree's approximation method.a It is true that this method abstra ts from what one alls the `ex hange terms' (whi h orrespond, for instan e, to the distan e between the ortho and para terms of neutral helium). But, abstra ting from that, it already a hieves the three-dimensional aim I tend to. Should one de lare a priori impossible that Hartree's method might be modi ed or developed in su h a way as to take into a
ount the ex hange terms while working with a satisfa tory three-dimensional model?
Mr Born.
Regarding the question of knowing whether it is possible to des ribe a many-ele tron problem by a eld equation in three dimensions, I would like to point out the following. The number of quantum numbers of an atom rises by three with ea h additional ele tron; it is thus equal to 3n for n ele trons. It seems doubtful that there should be an ordinary, three-dimensional eigenvalue problem, whose eigenvalues have a range of size ∞3n [dont la valeur ara téristique ait une multitude de ∞3n dimensions℄.b Instead, it follows from re ent papers by Dira and Jordanc that one an build on a three-dimensional os illation equation if one onsiders the eigenfun tion itself not as an ordinary number, but as one of Dira 's q-numbers, that is, if one quantises again its amplitude as a fun tion of time. An n-quanta os illation with this amplitude then yields together with the three spatial quantum numbers the ne essary range [multitude℄ of quantum numbers. From this point of view the number of ele trons in a system appears itself as a quantum number, that is, the ele trons themselves appear as dis ontinuities of the same a See the dis ussion after Bragg's report, p. 318 (eds.). b The Fren h text here appears to make little sense, but Born is possibly referring to the dimension of the spa e of solutions (eds.).
Cf. se tion IV of Born and Heisenberg's report (eds.).
474
E. S hrödinger
nature as the stationary states.
Mr S hrödinger. Pre isely the stru ture of the periodi system
is already ontained in the physi s [mé anique℄ of the three-dimensional hydrogen problem. The degrees of degenera y 1, 4, 9, 16, et ., multiplied by 2, yield pre isely the periodi numbers [nombres de périodes℄. The fa tor 2 that I have just mentioned derives from the spin [giration (spin)℄. From the point of view of wave me hani s, the apparently mysterious `Pauli a tion' of the rst two ele trons on the third (whi h they prevent from also following an orbit with quantum number 1) means stri tly speaking nothing other than the non-existen e of a third eigenfun tion with prin ipal quantum number 1. This non-existen e is pre isely a property of the three-dimensional model, or of the three-dimensional equation. The multi-dimensional equation has too many eigenfun tions; it is this [elle℄ that makes the `Pauli ex lusion' (Pauliverbot) ne essary to eliminate this defe t.a
a The Fren h text refers to the four-dimensional equation (`l'équation à quatre dimensions') as having too many solutions. This reading ould be orre t, in the sense that the ex lusion prin iple was rst introdu ed in the ontext of the relativisti (four-dimensional) Bohr-Sommerfeld theory of the atom, but the above reading seems mu h more natural in ontext. Note that S hrödinger throughout his report uses `vierdimensional' and `vieldimensional', whi h ould be easily
onfused, for `four-dimensional' and `many-dimensional', respe tively (eds.).
Notes to pp. 448464
475
Notes to the translation 1 Here and in the following, the Fren h edition omits some itali s, whi h are quite hara teristi of S hrödinger's writing style and whi h we ta itly restore. 2 [Energiegerade℄ [série des énergies℄ 3 Bra ket added in the Fren h edition. 4 [als Bes hreibung℄ [ omme la dé nition℄ 5 [Vollständigkeit℄ [perfe tion℄ 6 [freili h℄ [évidemment℄ 7 Printed as a footnote in the Fren h edition. 8 [sie ni ht .... bes hreibe℄ [qu'ils ne dé rivent pas℄ 9 [Einzelsystem℄ [système déterminé℄ 10 This lause is omitted in the Fren h edition. 11 [von anderer Seite vertreten℄ [défendue par d'autres℄. Footnote only in the Fren h edition. 12 [aus einer Anzahl℄ [d'un grand nombre℄ 13 [Dur h die eben bespro hene Deutung℄ [Ainsi que nous venons de le voir℄ 14 The equation number is missing in the Fren h edition, and the following senten e is printed as a footnote. 15 [es
gebe℄
[sont données℄
16 [weitgehend℄ [tout à fait℄
17 [Verständnis℄
[interprétation℄
18 [Verständnis℄ [signi ation℄
19 No ex lamation mark in the Fren h edition. 20 [innere℄ [intimes℄ 21 [eine prinzipielle℄ [essentielle℄ 22 No ex lamation mark in the Fren h edition. 23 Printed as a footnote in the Fren h edition. 24 In the Fren h edition this equation number is given to the following equation (unnumbered in the types ript). 25 Bra ket printed as a footnote in the Fren h edition, with the addition: `∆ stands for the Lapla ian'. eϕ eϕ 26 [ h ℄ [e h ℄ ∗ 27 [die Ladungsdi hte aus ψψ ℄ [la densité de harge
ψψ ∗ ℄
28 [Betrag℄ [valeur℄ 29 Bra ket printed as a footnote in the Fren h edition. 30 The rest of the bra ket is printed as a footnote in the Fren h edition. 31 Both types ript and Fren h edition give `365' as page number. 32 The Fren h edition adds this to the footnote. 33 [treten darin in
ψ
und seinen ersten Ableitungen quadratis he Formen
auf ℄ [y gurent dans
ψ
et ses premières dérivées des formes
quadratiques℄ 34 Footnote only in the Fren h edition. 35 [bewegend℄ [par
le mouvement℄
36 [allerdings℄ [ ertainement℄ 37 [(18)℄ [(8)℄ 38 The Fren h edition omits the inverted ommas. 39 The equation number is missing in the printed volume. 40 [3n-dimensional℄ [tridimensionelle℄
476
Notes to pp. 464467
41 Misprint in the Fren h edition: the
Ek
are not in the exponent.
42 [komplet[t℄e℄ [ omplexe℄ 43 [dass für die Abhängigkeit der bestehen℄ [que pour que
a
a
von der Zeit folgende Forderungen
dépende du temps les onditions suivantes
doivent être satisfaites℄ 44 Two misprints in the Fren h edition: the the
ki
run to
Ek
are not in the exponent, and
n.
45 Printed as a footnote in the Fren h edition. 46 [unseres Ansatzes℄ [de notre expression fondamentale℄ 47 [den
einzelnen℄
[les
diverses℄
48 Bra ket printed as footnote in the Fren h edition. 49 [bestimmen die Lösung exa t℄ [déterminent la solution℄ 50 Misprint in the Fren h edition: `ψ(k1 , k2 , . . . , kn )'. 51 Both the types ript and the Fren h edition read `cl ' and `ck ' instead of `al ' and `ak '. 52 Again, both the types ript and the Fren h edition read `ck ' instead of `ak '.
General dis ussion of the new ideas presented a
Causality, determinism, probability Mr Lorentz. I should like to draw attention to the di ulties one
en ounters in the old theories. We wish to make a representation of the phenomena, to form an image of them in our minds. Until now, we have always wanted to form these images by means of the ordinary notions of time and spa e. These notions are perhaps innate; in any ase, they have developed from our personal experien e, by our daily observations. For me, these notions are
lear and I onfess that I should be unable to imagine physi s without these notions. The image that I wish to form of phenomena must be absolutely sharp and de nite, and it seems to me that we an form su h an image only in the framework of spa e and time. For me, an ele tron is a orpus le that, at a given instant, is present at a de nite point in spa e, and if I had the idea that at a following moment the orpus le is present somewhere else, I must think of its traje tory, whi h is a line in spa e. And if the ele tron en ounters an atom and penetrates it, and after several in idents leaves the atom, I make up a theory in whi h the ele tron preserves its individuality; that is to say, I imagine a line following whi h the ele tron passes through the atom. a As mentioned in se tion 1.6, the Bohr ar hives ontain a opy of the galley proofs of the general dis ussion, dated 1 June 1928.1 A few of the ontributions in these proofs seem to have been still largely unedited: they ontain some gaps and in omplete senten es, some more olloquial formulations, and in at least one
ase a senten e that was dropped from the published volume. We reprodu e in endnotes the most substantial examples of these alternative versions. For most of the dis ussion ontributions by Dira , we have followed his manus ript version.2 For Bohr's dis ussion ontributions, we have used material from Bohr (1985) and from notes taken by Ri hardson3 (also mentioned in se tion 1.6). See our notes for further details (eds.).
477
478
General dis ussion
Obviously, su h a theory may be very di ult to develop, but a priori it does not seem to me impossible. I imagine that, in the new theory, one still has ele trons. It is of
ourse possible that in the new theory, on e it is well-developed, one will have to suppose that the ele trons undergo transformations. I happily
on ede that the ele tron may dissolve into a loud. But then I would try to dis over on whi h o
asion this transformation o
urs. If one wished to forbid me su h an enquiry by invoking a prin iple, that would trouble me very mu h. It seems to me that one may always hope one will do later that whi h we annot yet do at the moment. Even if one abandons the old ideas, one may always preserve the old lassi ations [dénominations℄. I should like to preserve this ideal of the past, to des ribe everything that happens in the world with distin t images. I am ready to a
ept other theories, on ondition that one is able to re-express them in terms of lear and distin t images. For my part, despite not having yet be ome familiar with the new ideas that I now hear expressed,a I ould visualise these ideas thus. Let us take the ase of an ele tron that en ounters an atom; let us suppose that the ele tron leaves the atom and that at the same time there is emission of a light quantum. One must onsider, in the rst pla e, the systems of waves that orrespond to the ele tron and to the atom before the ollision. After the ollision, we will have new systems of waves. These systems of waves an be des ribed by a fun tion ψ de ned in a spa e with a large number of dimensions and satisfying a dierential equation. The new wave me hani s will work with this equation and will determine the fun tion ψ before and after the ollision. Now, there are phenomena that tea h us that there is something else in addition to the waves, namely orpus les; one an, for example, perform an experiment with a Faraday ylinder; one must then take into a
ount the individuality of the ele trons and also of the photons. I think I would nd that, to explain the phenomena, it su es to assume that the expression ψψ ∗ gives the probability that the ele trons and the photons exist in a given volume; that would su e to explain the experiments. But the examples given by Mr Heisenberg tea h me that I will have thus attained everything that experiment allows me to attain. However, I think that this notion of probability should be pla ed at a In fa t, Lorentz had followed the re ent developments rather losely. In parti ular, he had orresponded extensively with Ehrenfest and with S hrödinger, and had even delivered seminars and le tures on wave me hani s and on matrix me hani s at Leiden, Cornell and Calte h. See se tion 1.3 (eds.).
Causality, determinism, probability
479
the end, and as a on lusion, of theoreti al onsiderations, and not as an a priori axiom, though I may well admit that this indetermina y
orresponds to experimental possibilities. I would always be able to keep my deterministi faith for the fundamental phenomena, of whi h I have not spoken. Could a deeper mind not be aware of the motions of these ele trons? Could one not keep determinism by making it an obje t of belief? Must one ne essarily elevate indeterminism to a prin iple?
Mr Bohr
expounds his point of view with respe t to the problems of quantum theory. The original published pro eedings add `(see the pre eding arti le)'.
In the
pro eedings, the arti le pre eding the general dis ussion is a Fren h translation of the German version of Bohr's Como le ture (Bohr 1928) (published in
Naturwissens haften).
As des ribed in se tion 1.6, this arti le was in luded
at Bohr's request, to repla e his remarks made at this point in the general dis ussion.
(In our translation of the pro eedings, we have omitted this
well-known arti le.) The extant notes relating to Bohr's remarks at this point are parti ularly fragmentary.
Kal kar's introdu tion to volume 6 of Bohr's
Colle ted Works
(Bohr 1985) des ribes the orresponding part of notes (taken by Kramers and by Vers haelt) in the Bohr ar hives as too in omplete to warrant reprodu tion in that volume, but provides the following summary and omparison with the printed versions of the Como le ture:
`The notes over the wave- orpus le
aspe ts of light and matter ( orresponding to the rst se tions of the printed le ture). The
γ -ray
mi ros ope is analysed, although the notes are somewhat
in omplete here (as in many other pla es), and the r le of the nite wave trains is dis ussed in onne tion with the momentum measurement through the Doppler ee t (as in the printed versions). After some questions .... Bohr
ontinues by dis ussing the signi an e of the phase and omments on the Stern-Gerla h experiment and the inobservability of the phase in a stationary state .... ' (Bohr 1985, p. 37). Further details of what Bohr said at this point may be obtained from notes 4 Below, we reprodu e the
on the general dis ussion taken by Ri hardson.
relevant parts of these notes, and omment on their relation to Bohr's paper translated in the pro eedings. The rst part of Ri hardson's notes relating to Bohr reads as follows:
E = hν
ei2π(τx x+τy y+τz z−νt)
p = hτ
Int[er℄f[eren℄ e. ∆x∆τx ∆t∆ν
?h → ∞ [?℄ ∼ 1 ∼ 1
∆x∆px ∆t∆E
∼ h ∼ h
480
General dis ussion
Fig. A.
This orresponds to part of se tion 2 of Bohr's paper translated in the pro eedings. There Bohr introdu es the on epts of energy and momentum for plane waves, and the idea that waves of limited extent in spa etime are obtained through the `interferen e' (that is, superposition) of dierent plane waves, the resulting waves satisfying (at best) the given relations. (As a onsequen e, a group of waves has no well-de ned phase, a point Bohr takes up again below.) This is used to justify Bohr's idea of omplementarity between a ausal pi ture (in the sense of energy-momentum onservation for elementary pro esses) and a spa etime pi ture. Ri hardson's notes then ontinue as shown in Fig. A. The
γ -ray mi ros ope
is dis ussed in se tion 3 of Bohr (1928) (the se tion on measurement, whi h also dis usses momentum measurements based on the Doppler ee t). Bohr appears to have inserted a dis ussion of these experiments as an illustration of the un ertainty-type relations above. The next part of Ri hardson's notes returns to se tion 2 of the paper, and is reprodu ed in Fig. B. This orresponds in fa t to the subsequent paragraphs of se tion 2, in whi h Bohr applies the notion of omplementarity to resolve the per eived paradoxes related to the s attering of radiation by free ele trons (note the extended as opposed to pointlike region of s attering in the diagram, and see Bohr's ontribution to the dis ussion of Compton's report, p. 60) as well as the per eived paradoxes related to ollisions ( f. se tion 3.4.2).
Causality, determinism, probability
481
Fig. B. Possibly,
harge.
I
stands for `Impuls' (that is, momentum),
R
for radiation,
q
for
The next part of Ri hardson's notes, shown in Fig. C, instead relates to part of se tion 6 of Bohr's paper (se tions 4 and 5 of the paper are, respe tively, a review of the orresponden e prin iple and of matrix me hani s, and a dis ussion and ritique of wave me hani s).
In se tion 6 of the published
paper, Bohr raises the following puzzle. A
ording to Bohr, in any observation that distinguishes between dierent stationary states one has to disregard the past history of the atom, but, paradoxi ally, the theory assigns a phase to a stationary state.
However, sin e the system will not be stri tly isolated,
one will work with a group of waves, whi h (as mentioned in se tion 2) has no well-de ned phase. Bohr then illustrates this with the Stern-Gerla h experiment.
The ondition for distinguishability of the eigenstates of the
hydrogen atom is that the angular spreading of the beam should be greater than that given by dira tion at the slit (ε
> α),
whi h translates into the
time-energy un ertainty relation. As Bohr mentions, Heisenberg (1927) uses this as an illustration of the un ertainty relation, while Bohr uses it as an illustration of how knowledge of the phase is lost. (This se tion also dis usses the limit of high quantum numbers.) The nal se tion 7 of the paper (`The problem of elementary parti les') has no parallel in Ri hardson's notes.
The part of the notes relating to Bohr's
remarks at this point on ludes instead with the following (expli itly labelled `Bohr'):
482
General dis ussion
Fig. C. 1. [blank℄ 2. Stationary states, past lost ∵ [be ause℄ phase indetermination Stern & Gerla h's Exp[erimen℄t. 3. S hroedinger's ψ , prob[abilit℄y of ele tron at a given pla e at a given time [?℄, un ertainty ∆ν∆t ∼ 1 γ×
v c
Mr Brillouin
. Mr Bohr insists on the un ertainty of simultaneous measurements of position and momentum; his point of view is losely
onne ted to the notion of ells in phase spa e introdu ed by Plan k a very long time ago. Plan k assumed that if the representative point of a system is in a ell (of size ∆p∆q = h) one annot distinguish it from another point in the same ell. The examples brought by Mr Bohr aptly make pre ise the physi al meaning of this quite abstra t notion.
Mr De Donder
. The onsiderations that Mr Bohr has just developed are, I think, in lose relation with the following fa t: in the
Causality, determinism, probability
483
Einsteinian Gravitationa of a ontinuous system or of a pointlike system, there appear not the masses and harges of the parti les, but entities τ (m) and τ (e) in four dimensions; note that these generalised masses and
harges, lo alised in spa etime, are onserved along their worldlines.
Mr Born. Mr Einstein has onsidered the following problem: A
radioa tive sample emits α-parti les in all dire tions; these are made visible by the method of the Wilson loud [ hamber℄. Now, if one asso iates a spheri al wave with ea h emission pro ess, how an one understand that the tra k of ea h α parti le appears as a (very nearly) straight line? In other words: how an the orpus ular hara ter of the phenomenon be re on iled here with the representation by waves? To do this, one must appeal to the notion of `redu tion of the probability pa ket' developed by Heisenberg.a The des ription of the emission by a spheri al wave is valid only for as long as one does not observe ionisation; as soon as su h ionisation is shown by the appearan e of
loud droplets, in order to des ribe what happens afterwards one must `redu e' the wave pa ket in the immediate vi inity of the drops. One thus obtains a wave pa ket in the form of a ray, whi h orresponds to the orpus ular hara ter of the phenomenon. Mr Paulib has asked me if it is not possible to des ribe the pro ess without the redu tion of wave pa kets, by resorting to a multidimensional spa e, whose number of dimensions is three times the number of all the parti les present (α-parti les and atoms hit by the radiation). This is in fa t possible and an even be represented in a very ans hauli h manner [d'une manière fort intuitive℄ by means of an appropriate simpli ation, but this does not lead us further as regards the fundamental questions. Nevertheless, I should like to present this ase here as an example of the multi-dimensional treatment of su h problems. I assume, for simpli ity, that there are only two atoms that may be hit. One then has to distinguish two ases: either the two atoms 1 and 2 lie on the same ray starting from the origin (the pla e where the preparation a Th. De Donder, Théorie des hamps gravi ques (Mémorial des s ien es mathématiques, part 14, Paris, 1926). See esp. equations (184), (184) and (188), (188). One an also onsult our le tures: The Mathemati al Theory of Relativity (Massa husetts Institute of Te hnology), Cambridge, Mass., 1927. See esp. equations (23), (24) and (28), (29). a Born is referring here in parti ular to Heisenberg's un ertainty paper (Heisenberg 1927) (eds.). b Cf. Pauli's letter to Bohr, 17 O tober 1927, dis ussed in se tion 6.2.1 (eds.).
484
General dis ussion
Fig. 1. is), or they do not lie on the same ray. If we represent by ε the probability that an atom will be hit, we have the following probability diagram:a I. The points 1 and 2 are lo ated on the same ray starting from the origin. Number of parti les hit
Probability
0 1 2
1−ε 0 ε
a In the following tables, the probability for the number of parti les hit to equal 1 should be read as the probability for ea h ase in whi h the number of parti les hit equals 1 (eds.).
Causality, determinism, probability
485
II. The points 1 and 2 are not on the same ray. Number of parti les hit
Probability
0 1 2
1 − 2ε ε 0
This is how one should express the probability of events in the ase of re tilinear propagation. To make possible a graphi al representation of the phenomenon, we will simplify it further by assuming that all the motions take pla e following only a single straight line, the axis x. We must then distinguish the two ases where the atoms lie on the same side and on either side of the origin. The orresponding probabilities are the following: I. The points 1 and 2 are lo ated on the same side. Number of parti les hit
Probability
0 1 2
1 2
0 1 2
II. The points 1 and 2 are lo ated on dierent sides. Number of parti les hit
Probability
0 1 2
0 1 2
0
Now, these relations an be represented by the motion of a wave pa ket in a spa e with three dimensions x0 , x1 , x2 . To the initial state there
orresponds: In ase I, the point x0 = 0, x1 = a x2 = b In ase II, the point x0 = 0, x1 = a x2 = −b
where a and b are positive numbers. The wave pa ket at rst lls the spa e surrounding these points and subsequently moves parallel to the axis x0 , dividing itself into two pa kets of the same size going in opposite dire tions. Collisions are produ ed when x0 = x1 or x0 = x2 , that is to say, on two planes of whi h one, P1 , is parallel to the axis x2 and uts the plane x0 x1 following the bise tor of the positive quadrant, while the se ond, P2 , is parallel to the axis x1 and uts the plane x0 x2 following the bise tor of the positive quadrant. As soon as the wave pa ket strikes
486
General dis ussion
the plane P1 , its traje tory re eives a small kink in the dire tion x1 ; as soon as it strikes P2 the traje tory re eives a kink in the dire tion x2 (Fig. 1). Now, one immediately sees in the gure that the upper part of the wave pa ket, whi h orresponds to ase I, strikes the planes P1 , P2 on the same side of the plane x1 x2 , while the lower part strikes them5 on dierent sides. The gure then gives an ans hauli h representation of the ases indi ated in the above diagram. It allows us to re ognise immediately whether, for a given size of wave pa ket, a given state, that is to say a given point x0 , x1 , x2 , an be hit or not. To the `redu tion' of the wave pa ket orresponds the hoi e of one of the two dire tions of propagation +x0 , −x0 , whi h one must take as soon as it is established that one of the two points 1 and 2 is hit, that is to say, that the traje tory of the pa ket has re eived a kink. This example serves only to make lear that a omplete des ription of the pro esses taking pla e in a system omposed of several mole ules is possible only in a spa e of several dimensions.
Mr Einstein.
Despite being ons ious of the fa t that I have not entered deeply enough into the essen e of quantum me hani s, nevertheless I want to present here some general remarks.b One an take two positions towards the theory with respe t to its postulated domain of validity, whi h I wish to hara terise with the aid of a simple example. Let S be a s reen provided with a small opening O (Fig. 2), and P a hemispheri al photographi lm of large radius. Ele trons impinge on S in the dire tion of the arrows. Some of these go through O, and be ause of the smallness of O and the speed of the parti les, are dispersed uniformly over the dire tions of the hemisphere, and a t on the lm. Both ways of on eiving the theory now have the following in ommon. There are de Broglie waves, whi h impinge approximately normally on S and are dira ted at O. Behind S there are spheri al waves, whi h rea h the s reen P and whose intensity at P is responsible [massgebend℄ for what happens at P.c a
a The extant manus ript in the Einstein ar hives6 onsists of the rst four paragraphs only, whi h we have translated here (footnoting signi ant dieren es from the published Fren h) (eds.). b The published Fren h has: `I must apologise for not having gone deeply into quantum me hani s. I should nevertheless want to make some general remarks' (eds.).
In the published Fren h, the German expression `ist massgebend' is misrendered
Causality, determinism, probability
487
Fig. 2. We an now hara terise the two points of view as follows. 1. Con eption I. The de Broglie-S hrödinger waves do not orrespond to a single ele tron, but to a loud of ele trons extended in spa e. The theory gives no information about individual pro esses, but only about the ensemble of an in nity of elementary pro esses. 2. Con eption II. The theory laims to be a omplete theory of individual pro esses. Ea h parti le dire ted towards the s reen, as far as an be determined by its position and speed, is des ribed by a pa ket of de Broglie-S hrödinger waves of short wavelength and small angular width. This wave pa ket is dira ted and, after dira tion, partly rea hes the lm P in a state of resolution [un état de résolution℄. 2
A
ording to the rst, purely statisti al, point of view |ψ| expresses the probability that there exists at the point onsidered a parti ular parti le of the loud, for example at a given point on the s reen. A
ording to the se ond, |ψ|2 expresses the probability that at a given instant the same parti le is present at a given point (for example on the s reen). Here, the theory refers to an individual pro ess and laims to des ribe everything that is governed by laws. as `donne la mesure' [gives the measure℄ instead of as `is responsible'. This is of some signi an e for the interpretation of Einstein's remarks as a form of the later EPR argument; see se tion 7.1 (eds.).
488
General dis ussion
The se ond on eption goes further than the rst, in the sense that all the information resulting from I results also from the theory by virtue of II, but the onverse is not true.a It is only by virtue of II that the theory ontains the onsequen e that the onservation laws are valid for the elementary pro ess; it is only from II that the theory an derive the result of the experiment of Geiger and Bothe, and an explain the fa t that in the Wilson [ loud℄ hamber the droplets stemming from an α-parti le are situated very nearly on ontinuous lines. But on the other hand, I have obje tions to make to on eption II. The s attered wave dire ted towards P does not show any privileged dire tion. If |ψ|2 were simply regarded as the probability that at a
ertain point a given parti le is found at a given time, it ould happen that the same elementary pro ess produ es an a tion in two or several pla es on the s reen. But the interpretation, a
ording to whi h |ψ|2 expresses the probability that this parti le is found at a given point, assumes an entirely pe uliar me hanism of a tion at a distan e, whi h prevents the wave ontinuously distributed in spa e from produ ing an a tion in two pla es on the s reen. In my opinion, one an remove this obje tion only in the following way, that one does not des ribe the pro ess solely by the S hrödinger wave, but that at the same time one lo alises the parti le during the propagation. I think that Mr de Broglie is right to sear h in this dire tion. If one works solely with the S hrödinger waves, interpretation 2 II of |ψ| implies to my mind a ontradi tion with the postulate of relativity. I should also like to point out brie y two arguments whi h seem to me to speak against the point of view II. This [view℄ is essentially tied to a multi-dimensional representation ( on guration spa e), sin e only this 2 mode of representation makes possible the interpretation of |ψ| pe uliar to on eption II. Now, it seems to me that obje tions of prin iple are opposed to this multi-dimensional representation. In this representation, indeed, two on gurations of a system that are distinguished only by the permutation of two parti les of the same spe ies are represented by two dierent points (in on guration spa e), whi h is not in a
ord with the new results in statisti s. Furthermore, the feature of for es of a ting only at small spatial distan es nds a less natural expression in on guration spa e than in the spa e of three or four dimensions. a The Fren h has `I' and `II' ex hanged in this senten e, whi h is illogi al (eds.).
Mr Bohr.
Causality, determinism, probability
489
I feel myself in a very di ult position be ause I don't understand what pre isely is the point whi h Einstein wants to [make℄. No doubt it is my fault. .... As regards general problem I feel its di ulties. I would put problem in other way. I do not know what quantum me hani s is. I think we are dealing with some mathemati al methods whi h are adequate for des ription of our experiments. Using a rigorous wave theory we are laiming something whi h the theory annot possibly give. [We must realise℄ that we are away from that state where we ould hope of des ribing things on lassi al theories. Understand same view is held by Born and Heisenberg. I think that we a tually just try to meet, as in all other theories, some requirements of nature, but di ulty is that we must use words whi h remind of older theories. The whole foundation for
ausal spa etime des ription is taken away by quantum theory, for it is based on assumption of observations without interferen e. .... ex luding interferen e means ex lusion of experiment and the whole meaning of spa e and time observation .... be ause we [have℄ intera tion [between obje t and measuring instrument℄ and thereby we put us on a quite dierent standpoint than we thought we ould take in lassi al theories. If we speak of observations we play with a statisti al problem. There are ertain features omplementary to the wave pi tures (existen e of individuals). .... .... The saying that spa etime is an abstra tion might seem a philosophi al triviality but nature reminds us that we are dealing with something of pra ti al interest. Depends on how I onsider theory. I may not have understood, but I think the whole thing lies [therein that the℄ theory is nothing else [but℄ a tool for meeting our requirements and I think it does. a
Mr Lorentz
. To represent the motion of a system of n material points, one an of ourse make use of a spa e of 3 dimensions with n points or of a spa e of 3n dimensions where the systems will be represented by a single point. This must amount to exa tly the same a These remarks by Bohr do not appear in the published Fren h. We have reprodu ed them from Bohr's Colle ted Works, vol. 6 (Bohr 1985, p. 103), whi h
ontains a re onstru tion of Bohr's remarks from notes by Vers haelt (held in the Bohr ar hive). The tentative interpolations in square bra kets are by the editor of Bohr (1985), J. Kal kar (eds.).
490
General dis ussion
thing; there an be no fundamental dieren e. It is merely a question of knowing whi h of the two representations is the most suitable, whi h is the most onvenient. But I understand that there are ases where the matter is di ult. If one has a representation in a spa e of 3n dimensions, one will be able to return to a spa e of 3 dimensions only if one an reasonably separate the 3n oordinates into n groups of 3, ea h orresponding to a point, and I
ould imagine that there may be ases where that is neither natural nor simple. But, after all, it ertainly seems to me that all this on erns the form rather than the substan e of the theory.
Mr Pauli
. I am wholly of the same opinion as Mr Bohr, when he says that the introdu tion of a spa e with several dimensions is only a te hni al means of formulating mathemati ally the laws of mutual a tion between several parti les, a tions whi h ertainly do not allow themselves to be des ribed simply, in the ordinary way, in spa e and time. It may perfe tly well be that this te hni al means may one day be repla ed by another, in the following fashion. By Dira 's method one
an, for example, quantise the hara teristi vibrations of a avity lled with bla kbody radiation, and introdu e a fun tion ψ depending on the amplitudes of these hara teristi vibrations of unlimited number. One
an similarly use, as do Jordan and Klein, the amplitudes of ordinary four-dimensional material waves as arguments of a multi-dimensional fun tion φ. This gives, in the language of the orpus ular pi ture, the probability that at a given instant the numbers of parti les of ea h spe ies present, whi h have ertain kinemati al properties (given position or momentum), take ertain values. This pro edure also has the advantage that the defe t of the ordinary multi-dimensional method, of whi h Mr Einstein has spoken and whi h appears when one permutes two parti les of the same spe ies, no longer exists. As Jordan and Klein have shown, making suitable assumptions on erning the equations that this fun tion φ of the amplitudes of material waves in ordinary spa e must satisfy,7 one arrives exa tly at the same results as by basing oneself on S hrödinger's multi-dimensional theory. To sum up, I wish then to say that Bohr's point of view, a
ording to whi h the properties of physi al obje ts of being de ned and of being des ribable in spa e and time are omplementary, seems to be more general than a spe ial te hni al means. But, independently of su h a means, one an, a
ording to this idea, de lare in any ase that the
Causality, determinism, probability
491
mutual a tions of several parti les ertainly annot be des ribed in the ordinary manner in spa e and time. To make lear the state of things of whi h I have just spoken, allow me to give a spe ial example. Imagine two hydrogen atoms in their ground state at a great distan e from ea h other, and suppose one asks for their energy of mutual a tion. Ea h of the two atoms has a perfe tly isotropi distribution of harge, is neutral as a whole, and does not yet emit radiation. A
ording to the ordinary des ription of the mutual a tion of the atoms in spa e and time, one should then expe t that su h a mutual a tion does not exist when the distan e between the two neutral spheres is so great that no notable interpenetration takes pla e between their harge louds. But when one treats the same question by the multi-dimensional method, the result is quite dierent, and in a
ordan e with experiment. The lassi al analogy to this last result would be the following: Imagine inside ea h atom a lassi al os illator whose moment p varies periodi ally. This moment produ es a eld at the lo ation of the other atom whose periodi ally variable intensity is of order E ∼ rp3 , where r is the distan e between the two atoms. When two of these os illators a t on ea h other, a polarisation o
urs with the following potential energy,
orresponding to an attra tive for e between the atoms, 1 1 1 − αE 2 ∼ αp2 6 , 2 2 r
where α represents the polarisability of the atom. In speaking of these os illators, I only wanted to point out a lassi al analogy with the ee t that one obtains as a result of multi-dimensional wave me hani s. I had found this result by means of matri es, but Wang has derived it dire tly from the wave equation in several dimensions. In a paper by Heitler and London, whi h is likewise on erned with this problem, the authors have lost sight of the fa t that, pre isely for a large distan e between the atoms, the ontribution of polarisation ee ts to the energy of mutual a tion, a ontribution whi h they have negle ted, outweighs in order of magnitude the ee ts they have al ulated.
Mr Dira .
a
I should like to express my ideas on a few questions.
a Here we mostly follow the English version from Dira 's manus ript.8 (The Fren h translation may have been done from a types ript or fairer opy.) We generally follow the Fren h paragraphing, and we uniformise Dira 's notation. Interesting variants, an ellations and additions will be noted, as will signi ant deviations from the published Fren h (eds.).
492
General dis ussion
The rst is the one that has just been dis ussed and I have not mu h to add to this dis ussion. I shall just mention the explanation that the quantum theory would give of Bothe's experiment.9 The di ulty arises from10 the inadequa y of the 3-dimensional wave pi ture. This pi ture
annot distinguish between the ase when there is a probability p of a light-quant being in a ertain small volume, and the ase when there is a probability 12 p of two light-quanta being in the volume, and no probability for only one. But the wave fun tion in many-dimensional spa e does distinguish between these ases. The theory of Bothe's experiment in many-dimensional spa e would show that, while there is a ertain probability for a light-quantum appearing in one or the other of the ounting hambers, there is no probability of two appearing simultaneously. At present the general theory of the wave fun tion in many-dimensional spa e ne essarily involves the abandonment of relativity.11 One might, perhaps, be able to bring relativity into the general quantum theory in the way Pauli has mentioned of quantising 3-dimensional waves, but this would not lead to greater Ans hauli hkeit12 in the explanation of results su h as Bothe's. I shall now show how S hrödinger's expression for the ele tri density appears naturally in the matrix theory. This will show the exa t signi ation of this density and the limitations whi h must be imposed on its use. Consider an ele tron moving in an arbitrary eld, su h as that of an H atom. Its oordinates x, y, z will be matri es. Divide the spa e up into a number of ells, and form that fun tion of x, y, z that is equal to 1 when the ele tron is in a given ell and 0 otherwise. This fun tion of the matri es x, y, z will also be a matrix.a There is one su h matrix for ea h ell whose matrix elements will be fun tions of the oordinates a, b, c of the ell, so that it an be written A(a, b, c). Ea h of these matri es represents a quantity that if measured experimentally must have either the value 0 or 1. Hen e ea h of these matri es has the hara teristi values 0 and 1 and no others. If one takes the two matri es A(a, b, c) and A(a′ , b′ , c′ ), one sees that they must ommute,13 sin e one an give a numeri al value to both simultaneously; for example, if the ele tron is known to be in the ell a, b, c, it will ertainly not be in a The published version has: `Divide the spa e up into a large number of small ells, and onsider the fun tion of three variables ξ , η, ζ that is equal to 1 when the point ξ , η, ζ is in a given ell and equal to 0 when the point is elsewhere. This fun tion, applied to the matri es x, y , z , gives another matrix' (eds.).
Causality, determinism, probability
493
the ell a′ , b′ , c′ , so that if one gives the numeri al value 1 to A(a, b, c), one must at the same time give the numeri al value 0 to A(a′ , b′ , c′ ). We an transform ea h of the matri es A into a diagonal matrix A∗ by a transformation14 of the type A∗ = BAB −1 .
Sin e all the matri es A(a, b, c) ommute,15 they an be transformed simultaneously into diagonal matri es by a transformation of this type. The diagonal elements of ea h matrix A∗ (a, b, c) are its hara teristi values, whi h are the same as the hara teristi values of A(a, b, c), that is, 0 and 1. Further, no two A∗ matri es, su h as A∗ (a, b, c) [and℄ A∗ (a′ , b′ , c′ ),
an both have 1 for the same diagonal element, as a simple argument shows that A∗ (a, b, c)+A∗ (a′ , b′ , c′ ) must also have only the hara teristi values 0 and 1. We an without loss of generality assume that ea h A∗ has just one diagonal element equal to 1 and all the others zero. By transforming ba k, by means of the formula A(a, b, c) = B −1 A∗ (a, b, c)B ,
we now nd that the matrix elements of A(a, b, c) are of the form −1 A(a, b, c)mn = Bm Bn ,
i.e. a fun tion of the row multiplied by a fun tion of the olumn. It should be observed that the proof of this result is quite independent of equations of motion and quantum onditions. If we take these −1 and Bn are apart from onstants just into a
ount, we nd that Bm S hrödinger's eigenfun tions ψ¯m and ψn at the point a, b, c. Thus S hrödinger's density fun tion ψ¯m (x, y, z)ψm (x, y, z) is a16 diagonal element of the matrix A referring to a ell about the point x, y, z . The true quantum expression for the density is the whole matrix. Its diagonal elements give only the average density, and must not be used when the density is to be multiplied by a dynami al variable represented by a matrix. I should now like to express my views on determinism and the nature of the numbers appearing in the al ulations of the quantum theory, as they appear to me after thinking over Mr Bohr's remarks of yesterday.17 In the lassi al theory one starts from ertain numbers des ribing ompletely the initial state of the system, and dedu es other numbers that des ribe ompletely the nal state. This deterministi theory applies only to an isolated system.
494
General dis ussion
But, as Professor Bohr has pointed out, an isolated system is by de nition unobservable. One an observe the system only by disturbing it and observing its rea tion to the disturban e. Now sin e physi s is on erned only with observable quantities the deterministi lassi al theory is untenable.18 In the quantum theory one also begins with ertain numbers and dedu es others from them. Let us inquire into the distinguishing hara teristi s19 of these two sets of numbers. The disturban es that an experimenter applies to a system to observe it are dire tly under his
ontrol, and are a ts of freewill by him. It is only the numbers that des ribe these a ts of freewill that an be taken as initial numbers for a
al ulation in the quantum theory. Other numbers des ribing the initial state of the system are inherently unobservable, and do20 not appear in the quantum theoreti al treatment. Let us now onsider the nal numbers obtained as the result of an experiment. It is essential that the result of an experiment shall be a permanent re ord. The numbers that des ribe su h a result must help to not only des ribe the state of the world at the instant the experiment is ended, but also help to des ribe the state of the world at any subsequent time. These numbers des ribe what is ommon to all the events in a
ertain hain of ausally onne ted events, extending inde nitely into the future. Take as an example a Wilson loud expansion experiment. The ausal
hain here onsists of the formation of drops of water round ions, the s attering of light by these drops of water, and the a tion of this light on a photographi plate, where it leaves a permanent re ord. The numbers that form the result of the experiment des ribe all of the events in this
hain equally well and help to des ribe the state of the world at any time after the hain began. One ould perhaps extend the hain further into the past.21 In the example one ould, perhaps, as ribe the formation of the ions to a β -parti le, so that the result of the experiment would be numbers des ribing the tra k of a β -parti le. In general one tries with the help of theoreti al onsiderations to extend the hain as far ba k into the past as possible, in order that the numbers obtained as the result of the experiment may apply as dire tly as possible to the pro ess under investigation.22 This view of the nature of the results of experiments ts in very well with the new quantum me hani s. A
ording to quantum me hani s the state of the world at any time is des ribable by a wave fun tion ψ ,
Causality, determinism, probability
495
whi h normally varies a
ording to a ausal law, so that its initial value determines its value at any later time. It may however happen that at a ertain time t1 , ψ an be expanded in the form ψ=
X
cn ψn ,
n
where the ψn 's are wave fun tions of su h a nature that they annot interfere with one another at any time subsequent to t1 . If su h is the
ase, then the world at times later than t1 will be des ribed not by ψ but by one of the ψn 's. The parti ular ψn that it shall be must be regarded as hosen by nature.23 One may say that nature hooses whi h ψn it is to be, as the only information given by the theory is that the probability of any ψn being hosen is |cn |2 .24 The value of the su x n that labels the parti ular ψn hosen may be the result of an experiment, and the result of an experiment must always be su h a number. It is a number des ribing an irrevo able hoi e of nature, whi h must ae t the whole of the future ourse of events.a As an example take the ase of a simple ollision problem. The wave pa ket representing the in ident ele tron gets s attered in all dire tions. One must take for the wave fun tion after the pro ess not the whole s attered wave, but on e again a wave pa ket moving in a de nite dire tion. From the results of an experiment, by tra ing ba k a hain of ausally onne ted events one ould determine in whi h dire tion the ele tron was s attered and one would thus infer that nature had hosen this dire tion. If, now, one arranged a mirror to re e t the ele tron wave s attered in one dire tion d1 so as to make it interfere with the ele tron wave s attered in another dire tion d2 , one would not be able to distinguish between the ase when the ele tron is s attered in the dire tion d2 and when it is s attered in the dire tion d1 and re e ted ba k into d2 . One would then not be able to tra e ba k the hain of ausal events so far, and one would not be able to say that nature had hosen a dire tion as soon as the ollision o
urred, but only [that℄ at a later a The last two senten es appear dierently in the published version: `The hoi e, on e made, is irrevo able and will ae t the whole future state of the world. The value of n hosen by nature an be determined by experiment and the results of all experiments are numbers des ribing su h hoi es of nature'. Dira 's notes ontain a similar variant written in the margin: `The value of n hosen by nature may be determined by experiment. The result of every experiment onsists of numbers determining one of these hoi es of nature, and is permanent sin e su h a hoi e is irrevo able and ae ts the whole future state of the world' (eds.).
496
General dis ussion
time nature hose where the ele tron should appear. The25 interferen e between the ψn 's ompels nature to postpone her hoi e.
Mr Bohr.
Quite see that one must go into details of pi tures, if one wants to ontrol or illustrate general statements. I think still that you may simpler put it in my way. Just this distin tion between observation and de nition allows to let the quantum me hani s appear as generalisation. What does mean: get re ords whi h do not allow to work ba kwards. Even if we took all mole ules in photographi plate one would have losed system. If we tell of a re ord we give up de nition of plate. Whole point lies in that by observation we introdu e something whi h does not allow to go on. a
....
Mr Born
. I should like to point out, with regard to the onsiderations of Mr Dira , that they seem losely related to the ideas expressed in a paper by my ollaborator J.26 von Neumann, whi h will appear shortly. The author of this paper shows that quantum me hani s an be built up using the ordinary probability al ulus, starting from a small number of formal hypotheses; the probability amplitudes and the law of their omposition do not really play a role there.
Mr Kramers
. I think the most elegant way to arrive at the results of Mr Dira 's onsiderations is given to us by the methods he presented in his memoir in the Pro . Roy. So ., ser. A, vol. 113, p. 621. Let us onsider a fun tion of the oordinates q1 , q2 , q3 of an ele tron, that is equal to 1 when the point onsidered is situated in the interior of a
ertain volume V of spa e and equal to zero for every exterior point, and let us represent by ψ(q, α) and ψ(α, q) the transformation fun tions that allow us to transform a physi al quantity F , whose form is known as a matrix (q ′ , q ′′ ), into a matrix (α′ , α′′ ), α1 , α2 , α3 being the rst integrals of the equation of motion. The fun tion f , written as a matrix (q ′ , q ′′ ), will then take the form f (q ′ )δ(q ′ − q ′′ ), where δ(q ′ − q ′′ ) represents
a Again, these remarks do not appear in the published Fren h and we have reprodu ed them from Bohr's Colle ted Works (Bohr 1985, p. 105) (eds.).
Causality, determinism, probability
497
Dira 's unit matrix. As a matrix (α′ , α′′ ), f will then take the form Z ′ ′′ ¯ ′ , q ′ )dq ′ f (q ′ )δ(q ′ − q ′′ )dq ′′ ψ(q ′′ , α′′ ) f (α , α ) = ψ(α Z ¯ ′ , q ′ )dq ′ ψ(q ′ , α′′ ) , ψ(α = V
the integral having to be extended over the whole of the onsidered volume. The diagonal terms of f (α′ , α′′ ), whi h may be written in the form Z ¯ , f (α) = ψ ψdq will dire tly represent, in a
ordan e with Dira 's interpretation of the matri es, the probability that, for a state of the system hara terised by given values of α, the oordinates of the ele tron are those of a point situated in the interior of V . As ψ is nothing other than the solution of S hrödinger's wave equation, we arrive at on e at the interpretation of the expression ψ ψ¯ under dis ussion.
Mr Heisenberg. I do not agree with Mr Dira when he says
that, in the des ribed experiment, nature makes a hoi e. Even if you pla e yourself very far away from your s attering material, and if you measure after a very long time, you are able to obtain interferen e by taking two mirrors. If nature had made a hoi e, it would be di ult to imagine how the interferen e is produ ed. Evidently, we say that this hoi e of nature an never be known before the de isive experiment has been done; for this reason, we an make no real obje tion to this
hoi e, be ause the expression `nature makes a hoi e' then implies no physi al observation. I should rather say, as I did in my last paper, that the observer himself makes the hoi e,a be ause it is only at the moment where the observation is made that the ` hoi e' has be ome a physi al reality and that the phase relationship in the waves, the power of interferen e, is destroyed.
Mr Lorentz. There is then, it seems to me, a fundamental a From Heisenberg's publi ation re ord, it is lear that he is here referring to his un ertainty paper, whi h had appeared in May 1927. There we nd the statement that `all per eiving is a hoi e from a plenitude of possibilities' (Heisenberg 1927, p. 197). When Heisenberg says, in his above omment on Dira , that the observer `makes' the hoi e, he seems to mean this in the sense of the observer bringing about the hoi e (eds.).
498
General dis ussion
dieren e of opinion on the subje t of the meaning of these hoi es made by nature. To admit the possibility that nature makes a hoi e means, I think, that it is impossible for us to know in advan e how phenomena will take pla e in the future. It is then indeterminism that you wish to ere t as a prin iple. A
ording to you there are events that we annot predi t, whereas until now we have always assumed the possibility of these predi tions.
Mr Kramers
Photons
. During the dis ussion of Mr de Broglie's report, Mr Brillouin explained to us how radiation pressure is exerted in the ase of interferen e and that one must assume an auxiliary stress. But how is radiation pressure exerted in the ase where it is so weak that there is only one photon in the interferen e zone? And how does one obtain the auxiliary tensor in this ase?
Mr de Broglie
. The proof of the existen e of these stresses an be made only if one onsiders a loud of photons.
Mr Kramers
. And if there is only one photon, how an one a
ount for the sudden hange of momentum suered by the re e ting obje t?
Mr Brillouin. No theory urrently gives the answer to Mr Kramers'
question.
Mr Kramers
. No doubt one would have to imagine a ompli ated me hanism, that annot be derived from the ele tromagneti theory of waves?
Mr de Broglie
. The dualist representation by orpus les and asso iated waves does not onstitute a de nitive pi ture of the phenomena. It does not allow one to predi t the pressures exerted on the dierent points of a mirror during the re e tion of a single photon. It gives only the mean value of the pressure during the re e tion of a loud of photons.
Mr Kramers
. What advantage do you see in giving a pre ise value to the velo ity v of the photons?
Photons
499
Mr de Broglie
. This allows one to imagine the traje tory followed by the photons and to spe ify the meaning of these entities; one an thus
onsider the photon as a material point having a position and a velo ity.
Mr Kramers. I do not very well see, for my part, the advantage
that there is, for the des ription of experiments, in making a pi ture where the photons travel along well-de ned traje tories.
Mr Einstein. During re e tion on a mirror, Mr L. de Broglie
assumes that the photons move parallel to the mirror with a speed c sin θ; but what happens if the in iden e is normal? Do the photons then have zero speed, as required by the formula (θ = 0)?
Mr Pi
ard. Yes. In the ase of re e tion, one must assume that
the omponent of the velo ity of the photons parallel to the mirror is
onstant. In the interferen e zone, the omponent normal to the mirror disappears. The more the in iden e in reases, the more the photons are slowed down. One thus indeed arrives at stationary photons in the limiting ase of normal in iden e.a
Mr Langevin
. In this way then, in the interferen e zone, the photons no longer have the speed of light; they do not then always have the speed c?
Mr de Broglie
. No, in my theory the speed of photons is equal to c only outside any interferen e zone, when the radiation propagates freely in the va uum. As soon as there are interferen e phenomena, the speed of the photons be omes smaller than c.
Mr De Donder
. I should like to show how the resear h of Mr L. de Broglie is related to mine on some points. By identifying the ten equations of the gravitational eld and the four equations of the ele tromagneti eld with the fourteen equations
a Note that here the wave train is ta itly assumed to be limited longitudinally. Cf. our dis ussion of the de Broglie-Pauli en ounter, se tion 10.2 (eds.).
500
General dis ussion
of the wave me hani s of L. Rosenfeld, I have obtaineda a prin iple of
orresponden e that lari es and generalises that of O. Klein.b In my prin iple of orresponden e, there appear the quantum urrent and the quantum tensor. I will give the formulas for them later on; let it su e to remark now that the example of orresponden e that Mr de Broglie has expounded is in harmony with my prin iple. Mr L. Rosenfelda has given another example. Here, the mass is
onserved and, moreover, one resorts to the quantum urrent. We add that this model of quantisation is also in luded, as a parti ular ase, in our prin iple of orresponden e. Mr Lorentz has remarked, with some surprise, that the ontinuity equation for harge is preserved in Mr de Broglie's example. Thanks to our prin iple of orresponden e, and to Rosenfeld's ompatibility27 theorem, one an show that it will always be so for the total urrent (in luding the quantum urrent) and for the theorem of energy and momentum. The four equations that express this last theorem are satis ed by virtue of the two generalised quantum equations of de Broglie-S hrödinger. One further small remark, to end with. Mr de Broglie said that relativisti systems do not exist yet. I have given the theory of ontinuous or holonomi systems.b But Mr de Broglie gives another meaning to the word system; he has in mind intera ting systems, su h as the Bohr atom, the system of three bodies, et . I have remarked re entlyc that the quantisation of these systems should be done by means of a (ds)2 taken in a on guration spa e with 4n dimensions, n denoting the number of parti les. In a paper not yet published, I have studied parti ular systems
alled additive.
Mr Lorentz
. The stresses of whi h you speak and whi h you all quantum, are they those of Maxwell?
Mr De Donder
. Our quantum stresses must ontain the Maxwell stresses as a parti ular ase; this results from the fa t that our prin iple a Bull. A . Roy. de Belgique, Cl. des S . (5) XIII, ns. 89, session of 2 August 1927, 5049. See esp. equations (5) and (8). b Zeits hr. f. Phys. 41, n. 617 (1927). See esp. equations (18), p. 414. a L. Rosenfeld, `L'univers à inq dimensions et la mé anique ondulatoire (quatrième
ommuni ation)', Bull. A . Roy. Belg., Cl. des S ., O tober 1927. See esp. paragraphs 4 and 5. b C. R. A ad. S . Paris, 21 February 1927, and Bull. A . Roy. Belgique, Cl. des S ., 7 Mar h 1927.
Bull. A . Roy. Belgique, Cl. des S ., 2 August 1927. See esp. form. (22).
Photons and ele trons
501
of orresponden e is derived (in part, at least) from Maxwell's equations, and from the fa t that these quantum stresses here formally play the same role as the stresses of ele trostri tiona in Einsteinian Gravity. Let us re all, on this subje t, that our prin iple of orresponden e is also derived from the fundamental equations of Einsteinian Gravity. Mr de Broglie has, by means of his al ulations, thus re overed the stresses of radiation.
Mr Langevin
Photons and ele trons
makes a omparison between the old and modern statisti s. Formerly, one de omposed the phase spa e into ells, and one evaluated the number of representative points attributing an individuality to ea h onstituent of the system. It seems today that one must modify this method by suppressing the individuality of the onstituents of the system, and substituting instead the individuality of the states of motion. By assuming that any number of onstituents of the system an have the same state of motion, one obtains the statisti s of Bose-Einstein. One obtains a third statisti s, that of Pauli-Fermi-Dira ,28 by assuming that there an be only a single representative point in ea h ell of phase spa e. The new type of representation seems more appropriate to the on eption of photons and parti les: sin e one attributes a omplete identity of nature to them, it appears appropriate to not insist on their individuality, but to attribute an individuality to the states of motion. In the report of Messrs Born and Heisenberg, I see that it results from quantum me hani s that the statisti s of Bose-Einstein is suitable for mole ules, that of Pauli-Dira for ele trons and protons. This means that for photons29 and mole ules there is superposition, while for protons and ele trons there is impenetrability. Material parti les are then distinguished from photons30 by their impenetrability.31
Mr Heisenberg
. There is no reason, in quantum me hani s, to prefer one statisti s to another. One may always use dierent statisti s, whi h an be onsidered as omplete solutions of the problem of quantum a For more details, see our Note: `L'éle trostri tion déduite de la gravi que einsteinienne', Bull. A . Roy. Belgique, Cl. des S ., session of 9 O tober 1926, 6738.
502
General dis ussion
me hani s. In the urrent state of the theory, the question of intera tion has nothing to do with the question of statisti s. We feel nevertheless that Einstein-Bose statisti s ould be the more suitable for light quanta, Fermi-Dira statisti s for positive and negative ele trons.a The statisti s ould be onne ted with the dieren e between radiation and matter, as Mr Bohr has pointed out. But it is di ult to establish a link between this question and the problem of intera tion. I shall simply mention the di ulty reated by ele tron spin.
Mr Kramers
reminds us of Dira 's resear h on statisti s, whi h has shown that Bose-Einstein statisti s an be expressed in an entirely dierent manner. The statisti s of photons, for example, is obtained by
onsidering a avity lled with bla kbody radiation as a system having an in nity of degrees of freedom. If one quantises this system a
ording to the rules of quantum me hani s and applies Boltzmann statisti s, one arrives at Plan k's formula, whi h is equivalent to Bose-Einstein statisti s applied to photons. Jordan has shown that a formal modi ation of Dira 's method allows one to arrive equally at a statisti al distribution that is equivalent to Fermi statisti s. This method is suggested by Pauli's ex lusion prin iple.
Mr Dira
points out that this modi ation, onsidered from a general point of view, is quite arti ial. Fermi statisti s is not established on exa tly the same basis as Einstein-Bose statisti s, sin e the natural method of quantisation for waves leads pre isely to the latter statisti s for the parti les asso iated with the waves. To obtain Fermi statisti s, Jordan had to use an unusual method of quantisation for waves, hosen spe ially so as to give the desired result. There are mathemati al errors in the work of Jordan that have not yet been redressed. b
Mr Kramers
. I willingly grant that Jordan's treatment does not seem as natural as the manner by whi h Mr Dira quantises the solution of the S hrödinger equation. However, we do not yet understand why nature requires this quantisation, and we an hope that one day we will nd the deeper reason for why it is ne essary to quantise in one way in one ase and in another way in the other.
Mr Born. An essential dieren e between Debye's old theory, in
a That is, for protons and ele trons (eds.). b On this riti ism by Dira , f. Kragh (1990, pp. 12830) (eds.).
Photons and ele trons
503
whi h the hara teristi vibrations of the bla kbody avity are treated like Plan k os illators, and the new theory is this, that both yield quite exa tly Plan k's radiation formula (for the mean density of radiation), but that the old theory leads to inexa t values for the lo al u tuations of radiation, while the new theory gives these values exa tly.
Mr Heisenberg
. A
ording to the experiments, protons and ele trons both have an angular momentum and obey the laws of the statisti s of Fermi-Dira ; these two points seem to be related. If one takes two parti les together, if one asks, for example, whi h statisti s one must apply to a gas made up of atoms of hydrogen, one nds that the statisti s of Bose-Einstein is the right one, be ause by permuting two H atoms, we permute one positive ele tron and one negative ele tron,a so that we hange the sign of the S hrödinger fun tion twi e. In other words, Bose-Einstein statisti s is valid for all gases made up of neutral mole ules, or more generally, omposed of systems whose harge is an even multiple of e. If the harge of the system is an odd multiple of e, the statisti s of Fermi-Dira applies to a olle tion of these systems. The He nu leus does not rotate and a olle tion of He nu lei obeys the laws of Bose-Einstein statisti s.
Mr Fowler asks if the ne details of the stru ture of the bands of
helium agree better with the idea that we have only symmetri states of rotation of the nu lei of helium than with the idea that we have only antisymmetri states.
Mr Heisenberg
. In the bands of helium, the fa t that ea h se ond line disappears tea hes us that the He nu leus is not endowed with a spinning motion. But it is not yet possible to de ide experimentally, on the basis of these bands, if the statisti s of Bose-Einstein or that of Fermi-Dira must be applied to the nu leus of He.
Mr S hrödinger. You have spoken of experimental eviden e in
favour of the hypothesis that the proton is endowed with a spinning motion just like the ele tron, and that protons obey the statisti al law of Fermi-Dira . What eviden e are you alluding to?
Mr Heisenberg. The experimental eviden e is provided by the a That is, we permute the two protons, and also the two ele trons (eds.).
504
General dis ussion
work of Dennisona on the spe i heat of the hydrogen mole ule, work whi h is based on Hund's resear h on erning the band spe tra of hydrogen. Hund found good agreement between his theoreti al s heme and the experimental work of Dilke, Hop eld and Ri hardson, by means of the hypotheses mentioned by Mr S hrödinger. But for the spe i heat, he found a urve very dierent from the experimental urve. The experimental urve of the spe i heat seemed rather to speak in favour of Bose-Einstein statisti s. But the di ulty was elu idated in the paper by Dennison, who showed that the systems of `symmetri ' and `antisymmetri ' terms (with regard to protons) do not ombine in the time ne essary to arry out the experiment. At low temperature, a transition takes pla e about every three months. The ratio of statisti al weights of the systems of symmetri and antisymmetri terms is 1 : 3, as in the helium atom. But at low temperatures the spe i heat must be al ulated as if one had a mixture of two gases, an `ortho' gas and a `para' gas. If one wished to perform experiments on the spe i heat with a gas of hydrogen, kept at low temperature for several months, the result would be totally dierent from the ordinary result.
Mr Ehrenfest
wishes to formulate a question that has some relation to the re ent experiments by Mr Langmuir on the disordered motion of ele trons in the ow of ele tri ity through a gas. In the well-known Pauli ex lusion (Pauliverbot), one introdu es (at least in the language of the old quantum theory) a parti ular in ompatibility relation between the quantum motions of the dierent parti les of a single system, without speaking expli itly of the role possibly played by the for es a ting between these parti les. Now, suppose that through a small opening one allows parti les that, so to speak, do not exert for es on ea h other, to pass from a large spa e into a small box bounded by quite rigid walls with a ompli ated shape, so that the parti les en ounter the opening and leave the box only at the end of a su iently long time. Before entering the box, if the parti les have almost no motion relative to one another, the Pauli ex lusion intervenes. After their exit, will they have very dierent energies, independently of the weakness of the mutual a tion between the parti les? Or else what role do these for es play in the produ tion of Pauli's in ompatibility ( hoi e of antisymmetri solutions of the wave equation)? a Pro . Roy. So . A 114 (1927), 483.
Photons and ele trons
505
Mr Heisenberg
. The di ulty with Mr Ehrenfest's experiment is the following: the two ele trons must have dierent energies. If the energy of intera tion of the two ele trons is very small, the time τ1 required for the ele trons to ex hange an appre iable amount of energy is very long. But to nd experimentally whi h state, symmetri or antisymmetri , the system of the two ele trons in the box is in, we need a ertain time τ2 whi h is at least ∼ 1/ν , if hν is the [energy℄ dieren e between the symmetri and antisymmetri states. Consequently, τ1 ∼ τ2 and the di ulty disappears.
Mr Ri hardson
. The eviden e for a nu lear spin is mu h more
omplete than Mr Heisenberg has just said. I have re ently had o
asion to lassify a large number of lines in the visible bands of the spe trum of the H2 mole ule. One of the hara teristi features of this spe trum is a rather pronoun ed alternation in the intensity of the su
essive lines. The intensities of the lines of this spe trum were re ently measured by Ma Lennan, Grayson-Smith and Collins. Unfortunately, a large number of these lines overlap with ea h other, so that the intensity measurements must be a
epted only with reservations. But nevertheless, I think one an say, without fear of being mistaken, that all the bands that are su iently well-formed and su iently free of in uen es of the lines on ea h other (so that one an have on den e in the intensity measurements) have lines, generally numbered 1, 3, 5, ..., that are intrinsi ally three times more intense than the intermediate lines, generally numbered 2, 4, 6, ... . By intrinsi intensity, I mean that whi h one obtains after having taken into a
ount the ee ts on the intensity of temperature and quantum number (and also, of ourse, the ee ts of overlap with other lines, where it is possible to take this into a
ount). In other words, I wish to say that the onstant c of the intensity formula − m+ 1 2 h2 ( 2) 1 J =c m+ , e 8πKkT 2 where m is the number of the line and K the moment of inertia of the mole ule, is three times bigger for the odd-numbered lines than for the even-numbered ones. This means that the ratio 3 : 1 applies, with an a
ura y of about 5%, for at least ve dierent vibration states of a three-ele tron state of ex itation. It also applies to another state, whi h is probably 31 P if the others are 33 P . It is also shown, but in a less
506
General dis ussion
pre ise way, that it applies to two dierent vibration states of a state of ex itation with four ele trons. At present, then, there is a great deal of experimental eviden e that this nu lear spin persists through the dierent states of ex itation of the hydrogen mole ule.
Mr Langmuir. The question has often been raised of a similarity
in the relation between light waves and photons on the one hand, and de Broglie waves and ele trons on the other. How far an this analogy be developed? There are many remarkable parallels, but also I should like to see examined if there are no fundamental dieren es between these relations. Thus, for example, an ele tron is hara terised by a onstant
harge. Is there a onstant property of the photon that may be ompared with the harge of the ele tron? The speed of the ele tron is variable; is that of the photon also? The ele tromagneti theory of light has suggested a multitude of experiments, whi h have added onsiderably to our knowledge. The wave theory of the ele tron explains the beautiful results of Davisson and Germer. Can one hope that this theory will be as fertile in experimental suggestions as the wave theory of light has been?32
Mr Ehrenfest
. When one examines a system of plane waves of ellipti ally polarised light, pla ing oneself in dierently moving oordinate systems, these waves show the same degree of ellipti ity whatever system one pla es oneself in. Passing from the language of waves to that of photons, I should like to ask if one must attribute an ellipti al polarisation (linear or ir ular in the limiting ases) to ea h photon? If the reply is a rmative, in view of the invarian e of the degree of ellipti ity in relativity, one must distinguish as many spe ies of photons as there are degrees of ellipti ity. That would yield, it seems to me, a new dieren e between the photon and the spinning ele tron. If, on the other hand, one wishes above all to retain the analogy with the ele tron, as far as I an see one omes up against two di ulties: 1. How then must one des ribe linearly polarised light in the language of photons? (It is instru tive, in this respe t, to onsider the way in whi h the two linearly polarised omponents, emitted perpendi ularly to the magneti eld by a ame showing the Zeeman ee t, are absorbed by a se ond ame pla ed in a magneti eld with antiparallel orientation.) Mr Zeeman, to whom I posed the question, was kind enough to
Photons and ele trons
507
perform the experiment about a year ago, and he was able to noti e that the absorption is the same in parallel and antiparallel elds, as one
ould have predi ted, in fa t, by onsiderations of ontinuity. 2. For ele trons, whi h move always with a speed less than that of light, the universality of the spin may be expressed as follows, that one transforms the orresponding antisymmetri tensor into a system of oordinates arried with the ele tron in its translational motion (`at rest'). But photons always move with the speed of light!
Mr Compton. Can light be ellipti ally polarised when the photon
has an angular momentum?
Mr Ehrenfest
. Be ause the photons move with the speed of light, I do not really understand what it means when one says that ea h photon has a universal angular momentum just like an ele tron. Allow me to remind you of yet another property of photons. When two photons move in dire tions that are not exa tly the same, one an say quite arbitrarily that one of the photons is a radio-photon and the other a γ -ray photon, or inversely. That depends quite simply on the moving system of oordinates to whi h one refers the pair of photons.
Mr Lorentz. Can you make them identi al by su h a transfor-
mation?
Mr Ehrenfest
. Perfe tly. If they move in dierent dire tions. One an then give them the same olour by adopting a suitable frame of referen e. It is only in the ase where their worldlines are exa tly parallel that the ratio of their frequen ies remains invariant.
Mr Pauli
. The fa t that the spinning ele tron an take two orientations in the eld allowed by the quanta seems to invite us at rst to
ompare it to the fa t that there are, for a given dire tion of propagation of the light quanta, two hara teristi vibrations of bla kbody radiation, distinguished by their polarisation. Nevertheless there remain essential dieren es between the two ases. While in relativity one des ribes waves by a (real) sextuple ve tor Fix = −Fxi , for the spinning ele tron one has proposed the following two modes of des ription for the asso iated de Broglie waves: 1. One des ribes these waves by two omplex fun tions ψα , ψβ (and so by four real fun tions); but these fun tions transform
508
General dis ussion
in a way that is hardly intuitive during the hange from one system of oordinates to another. That is the route I followed myself. Or else: 2. Following the example of Darwin, one introdu es a quadruple ve tor with generally omplex omponents (and so eight real fun tions in total). But this pro edure has the in onvenien e that the ve tor involves a redundan y [indétermination℄, be ause all the veri able results depend on only two omplex fun tions. These two modes of des ription are mathemati ally equivalent, but independently of whether one de ides in favour of one or the other, it seems to me that one annot speak of a simple analogy between the polarisation of light waves and the polarisation of de Broglie waves asso iated with the spinning ele tron. Another essential dieren e between ele trons and light quanta is this, that between light quanta there does not exist dire t (immediate) mutual a tion, whereas ele trons, as a result of their arrying an ele tri harge, exert dire t mutual a tions on ea h other.
Mr Dira .
I should like to point out an important failure in the analogy between the spin of ele trons and the polarisation of photons. In the present theory of the spinning ele tron one assumes that one
an spe ify the dire tion of the spin axis of an ele tron at the same time as its position, or at the same time as its momentum. Thus the spin variable of an ele tron ommutes33 with its oordinate and with its momentum variables. The ase is dierent for photons. One
an spe ify a dire tion of polarisation for plane mono hromati light waves, representing photons of given momentum, so that the polarisation variable ommutes with the momentum variables. On the other hand, if the position of a photon is spe i ed, it means one has an ele tromagneti disturban e on ned to a very small volume,34 and one annot give a de nite polarisation, i.e. a de nite dire tion for the ele tri ve tor, to this disturban e. Thus the polarisation variable of a photon does not
ommute with its oordinates. a
Mr Lorentz
. In these dierent theories, one deals with the probability ψψ ∗ . I should like to see quite learly how this probability an exist when parti les move in a well-de ned manner following ertain laws. In the ase of ele trons, this leads to the question of motions in the eld ψ (de Broglie). But the same question arises for light quanta. a Again, here we follow Dira 's original English (eds.).
Photons and ele trons
509
Do photons allow us to re over all the lassi al properties of waves? Can one represent the energy, momentum and Poynting ve tor by photons? One sees immediately that, when one has an energy density and energy ow, if one wishes to explain this by photons then the number of photons per unit volume gives the density, and the number of photons per se ond that move a ross a unit surfa e gives the Poynting ve tor. The photons will then have to move with a speed dierent from that of light. If one wished to assign always the same speed c to the photons, in some ases one would have to assume a superposition of several photon
urrents. Or else one would have to assume that the photons annot be used to represent all the omponents of the energy-momentum tensor. Some of the terms must be ontinuous in the eld. Or else the photons are smeared out [fondus℄. A related question is to know whether the photons an have a speed dierent from that of light and whether they an even be at rest. That would altogether displease me. Could we speak of these photons and of their motion in a eld of radiation?
Mr de Broglie. When I tried to relate the motion of the photons
to the propagation of the waves ψ of the new me hani s, I did not worry about putting this point of view in a
ord with the ele tromagneti on eption of light waves, and I onsidered only waves ψ of s alar hara ter, whi h one has normally used until now.
Mr Lorentz
. One will need these waves for photons also. Are they of a dierent nature than light waves? It would please me less to have to introdu e two types of waves.
Mr de Broglie
. At present one does not know at all the physi al nature of the ψ -wave of the photons. Can one try to identify it with the ele tromagneti wave? That is a question that remains open. In any ase, one an provisionally try to develop a theory of photons by asso iating them with waves ψ .
Mr Lorentz. Is the speed of the wave equal to that of light? Mr de Broglie. In my theory, the speed of photons is equal to
c, ex ept in interfering elds. In general, I nd that one must assign to
510
General dis ussion
a moving orpus le a proper mass M0 given by the formula r h2 a M0 = m20 − 2 2 , 4π c a the fun tion a a being al ulated at the point where the moving body is lo ated at the given moment (a is the amplitude of the wave ψ ). For photons, one has m0 = 0 .
Thus, when a photon moves freely, that is to say, is asso iated with an ordinary plane wave, M0 is zero and, to have a nite energy, the photon must have speed c. But, when there is interferen e, a a be omes dierent from zero, M0 is no longer zero and the photon, to maintain the same energy, must have a speed less than c, a speed that an even be zero.
Mr Lorentz. The term
would be ome imaginary.
a a
must be negative, otherwise the mass
Mr de Broglie
. In the orpus ular on eption of light, the existen e of dira tion phenomena o
uring at the edge of a s reen requires us to assume that, in this ase, the traje tory of the photons is urved. The supporters of the emission theory said that the edge of the s reen exerts a for e on the orpus le. Now, if in the new me hani s as I develop it, one writes the Lagrange equations for the photon, one sees appear on the right-hand side of these equations a term proportional to the gradient of M0 . This term represents a sort of for e of a new kind, whi h exists only when the proper mass varies, that is to say, where there is interferen e. It is this for e that will urve the traje tory of the photon when its wave ψ is dira ted by the edge of a s reen. Furthermore, for a loud of photons the same Lagrange equations lead one to re over the internal stresses pointed out by Messrs S hrödinger and De Donder.a One nds, indeed, the relations ∂ ik T + Πik = 0 , k ∂x
where the tensor T ik is the energy-momentum tensor of the orpus les T ik = ρ0 ui uk . a Cf. S hrödinger (1927b) and De Donder's omments above (eds.).
Photons and ele trons
511
The tensor Πik , whi h depends on derivatives of the amplitude of the wave ψ and is zero when this amplitude is onstant, represents stresses existing in the loud of orpus les, and these stresses allow us to re over the value of the radiation pressure in the ase of re e tion of light by a mirror. The tensor T ik + Πik is ertainly related to the Maxwell tensor but, to see learly how, one would have to be able to larify the relationship existing between the wave ψ of the photons and the ele tromagneti light wave.
Mr Pauli.
It seems to me that, on erning the statisti al results of s attering experiments, the on eption of Mr de Broglie is in full agreement with Born's theory in the ase of elasti ollisions, but that it is no longer so when one also onsiders inelasti ollisions. I should like to illustrate this by the example of the rotator, whi h was already mentioned by Mr de Broglie himself. As Fermib has shown, the treatment by wave me hani s of the problem of the ollision of a parti le that moves in the (x, y) plane and of a rotator situated in the same plane, may be made lear in the following manner.c One introdu es a on guration spa e of three dimensions, of whi h two oordinates orrespond to the x and y of the olliding parti le, while as third oordinate one hooses the angle ϕ of the rotator. In the ase where there is no mutual a tion between the rotator and the parti le, the fun tion ψ of the total system is given by35 a
ψ(x, y, ϕ) = Ae2πi[ h (px x+py y+pϕ ϕ)−νt] , 1
where one has put h (m = 0, 1, 2, ...) . 2π In parti ular, the sinusoidal os illation of the oordinate ϕ orresponds to a stationary state of the rotator. A
ording to Born, the superposition of several partial waves of this type, orresponding to dierent values of m and by onsequen e of pϕ ,36 means that there is a probability dierent from zero for several stationary states of the rotator, while a
ording to the point of view of Mr de Broglie, in this ase the rotator no longer has a onstant angular velo ity and an also exe ute os illations in ertain
ir umstan es. pϕ = m
a Cf. se tion 10.2 (eds.). b Zeits hr. f. Phys. 40 (1926), 399.
See se tion 10.2 for a dis ussion of Fermi's argument (eds.).
512
General dis ussion
Now, in the ase of a nite energy of intera tion between the olliding parti le and the rotator, if we study the phenomenon of the ollision by means of the wave equation in the spa e (x, y, ϕ), a
ording to Fermi the result an be interpreted very simply. Indeed, sin e the energy of intera tion depends on the angle ϕ in a periodi manner and vanishes at large distan es from the rotator, that is to say from the axis ϕ, in the spa e (x, y, ϕ) we are dealing simply with a wave that falls on a grating and, in parti ular, on a grating that is unlimited in the dire tion of the axis ϕ. At large distan es from the grating, waves ome out only in xed dire tions in on guration spa e, hara terised by integral values of the dieren e m′ − m′′ . Fermi has shown that the dierent spe tral orders orrespond simply to the dierent possible ways of transferring the energy of the olliding parti le to the rotator, or onversely. Thus to ea h spe tral order of the grating orresponds a given stationary state of the rotator after the ollision. It is, however, an essential point that, in the ase where the rotator is in a stationary state before the ollision, the in ident wave is unlimited in the dire tion of the axis. For this reason, the dierent spe tral orders of the grating will always be superposed at ea h point of on guration spa e. If we then al ulate, a
ording to the pre epts of Mr de Broglie, the angular velo ity of the rotator after the ollision, we must nd that this velo ity is not onstant. If one had assumed that the in ident wave is limiteda in the dire tion of the axis ϕ, it would have been the same before the ollision. Mr de Broglie's point of view does not then seem to me ompatible with the requirement of the postulate of the quantum theory, that the rotator is in a stationary state both before and after the
ollision. To me this di ulty does not appear at all fortuitous or inherent in the parti ular example of the rotator; in my opinion, it is due dire tly to the
ondition assumed by Mr de Broglie, that in the individual ollision pro ess the behaviour of the parti les should be ompletely determined and may at the same time be des ribed ompletely by ordinary kinemati s in spa etime. In Born's theory, agreement with the quantum postulate is realised thus, that the dierent partial waves in on guration spa e, of whi h the general solution of the wave equation after the ollision is omposed, are appli able [indiquées℄ separately in a statisti al way. a The Fren h reads `illimitée' [unlimited℄, whi h we interpret as a misprint. Pauli seems to be saying that if, on the other hand, the in ident wave had been taken as limited, then before the ollision the rotator ould not have been in a stationary state and its angular velo ity ould not have been onstant (eds.).
Photons and ele trons
513
But this is no longer possible in a theory that, in prin iple, onsiders it possible to avoid the appli ation of notions of probability to individual
ollision pro esses.
Mr de Broglie
. Fermi's problem is not of the same type as that whi h I treated earlier; indeed, he makes on guration spa e play a part, and not ordinary spa e. The di ulty pointed out by Mr Pauli has an analogue in lassi al opti s. One an speak of the beam dira ted by a grating in a given dire tion only if the grating and the in ident wave are laterally limited, be ause otherwise all the dira ted beams will overlap and be bathed in the in ident wave. In Fermi's problem, one must also assume the wave ψ to be limited laterally in on guration spa e.
Mr Lorentz. The question is to know what a parti le should do
when it is immersed in two waves at the same time.
Mr de Broglie. The whole question is to know if one has the right
to assume the wave ψ to be limited laterally in on guration spa e. If one has this right, the velo ity of the representative point of the system will have a onstant value, and will orrespond to a stationary state of the rotator, as soon as the waves dira ted by the ϕ-axis will have separated from the in ident beam.
One an say that it is not possible to assume the in ident beam to be limited laterally, be ause Fermi's on guration spa e is formed by the superposition of identi al layers of height 2π in the dire tion of the ϕ-axis; in other words, two points of on guration spa e lying on the same parallel to the ϕ-axis and separated by a whole multiple of 2π represent the same state of the system. In my opinion, this proves above all the arti ial hara ter of on guration spa es, and in parti ular of that whi h one obtains here by rolling out along a line the y li variable ϕ.
Mr De Donder
. In the ourse of the dis ussion of Mr L. de Broglie's report, we explained how we obtained our Prin iple of Corre-
514
General dis ussion
sponden e; thanks to this prin iple, one will havea
√ c X −h an ¯ .n − 2 e Φa ψ ψ¯ , ψ ψ¯.n − ψψ g −gK 2 e n 2iπ c √ X X aα bβ = −g ψ.α ψ¯.β + ψ¯.α ψ.β − γ ab L γ γ
ρ(e) ua + Λa = ρ(m) ua ub + Πab
α
β
(a, b, n = 1, ..., 4; α, β = 0, 1, ..., 4) .
The rst relation represents the total urrent (≡ ele troni urrent + quantum urrent) as a fun tion of ψ and of the potentials g an , Φa . Re all that one has set
L≡
XX α
β
1 ψψ , γ αβ ψ.α ψ.β + k 2 µ2 − 2χ
γ ab ≡ g ab , γ 0a ≡ −αΦa , γ 00 ≡ α2 Φa Φa − ξα2 ≡ 2χ , χ ≡
8πG , c2
1 , ξ
G ≡ 6.7 × 10−8 c.g.s.
We have already mentioned the examples (or models) of orresponden e found respe tively by L. de Broglie and L. Rosenfeld. To be able to show learly a new solution to the problem relating to photons that Mr L. de Broglie has just posed, I am going to display the formulas
on erning the two above-mentioned models.a
a I adopt here L. Rosenfeld's notation, so as to fa ilitate the omparison with his formulas, given later. a L. Rosenfeld, `L'univers à inq dimensions et la mé anique ondulatoire', Bull. A . Roy. Belgique, Cl. des S ., O tober 1927. See respe tively the formulas (*38), (*31), (*27), (21), (1), (8), (35), (28), (29), (*35).
Photons and ele trons Model of L. de Broglie. Quantum urrent Λa ≡ 0.
515
Model of L. Rosenfeld. Quantum urrent Λa = 2K 2 A′2 C.a , where A′ is the modulus of ψ and where the potential C ≡ S ′ − S . The fun tion S satis es the lassi al Ja obi equation; the fun tion S ′ satis es the modi ed Ja obi equation; one then has 1 , 2χ 1 A′ = µ2 − + 2 ′ . 2χ K A
γ αβ S.α S.β = µ2 − ′ ′ γ αβ S.α S.β
The quantum potential C produ es the dieren e between physi al quantisation and geometri al quantisation. 2 Re all that µ ≡ m0ec , where m0 and e are respe tively the mass (at rest) and harge of the parti le under
onsideration. We have also put k ≡ iK ≡ i
2π e . h c
Charge density ρ(e) = 2K 2 A′2 µ′ , Charge density ρ(e) = 2K 2 A′2 µ. where we have put Here then one retains, at the same ′ time, the mass m0 and the harge e. A µ′2 = µ2 + 2 ′ , K A whi h, retaining the harge e, redu es to substituting for the mass m0 the modi ed mass of L. de Broglie: r h2 A′ M0 ≡ m20 + 2 2 ′ . 4π c A Let us respe tively apply these formulas to the problem of the photon pointed out by Mr L. de Broglie. The proper mass m0 of the photon is zero; in the model of Mr L. de Broglie, this mass must be repla ed by the modi ed mass M0 ; on the ontrary, in the model of Mr L. Rosenfeld, one
516
General dis ussion
uses only the proper mass m0 ≡ 0. In the two models, the harge density ρ(e) is zero. Finally, in the rst model, the speed of the photon must vary; in ontrast, in the se ond model, one an assume that this speed is always that of light. These on lusions obviously speak in favour of the model of L. Rosenfeld, and, in onsequen e, also in favour of the physi al existen e of our quantum urrent Λa (a = 1, 2, 3, 4). This urrent will probably play a dominant role in still unexplained opti al phenomena.a
Mr Lorentz
. Let us take an atom of hydrogen and let us form the S hrödinger fun tion ψ .37 We onsider ψψ ∗ as the probability for the presen e of the ele tron in a volume element. Mr Born has mentioned all the traje tories in the lassi al theory: let us take them with all possible phases,a but let us now take the ψ orresponding to a single value Wn of energy and then let us form ψψ ∗ . Can one say that this produ t ψn ψn∗ represents the probability that the ele trons move with the given energy Wn ? We think that the ele tron annot es ape from a
ertain sphere. The atom is limited, whereas ψ extends to in nity. That is disagreeable.38
Mr Born. The idea that ψψ
represents a probability density has great importan e in appli ations. If, for example, in the lassi al theory an ele tron had two equilibrium positions separated by a onsiderable potential energy, then lassi ally, for a su iently weak total energy only one os illation ould ever take pla e, around one of the two equilibrium positions. But a
ording to quantum me hani s, ea h eigenfun tion extends from one domain into the other; for this reason there always exists a probability that a parti le, whi h at rst vibrates in the neighbourhood of one of the equilibrium positions, jumps to the other. Hund has made important appli ations of this to mole ular stru ture. This phenomenon probably also plays a role in the explanation of metalli ondu tion. ∗
Mr de Broglie. In the old theory of the motion of an ele tron in a On this subje t, Mr L. Brillouin has kindly drawn my attention to the experiments by Mr F. Wolfers: `Sur un nouveau phénomène en optique: interféren es par diusion' (Le Journal de Physique et le Radium (VI) 6, n. 11, November 1925, 35468). a The `phases' of lassi al traje tories seems to be meant in the sense of a tion-angle variables (eds.).
Photons and ele trons
517
the hydrogen atom, an ele tron of total energy e2 m 0 c2 − W = p r 1 − β2
annot es ape from a sphere of radius R=−
e2 W − m 0 c2
be ause the value of the term √m0 c
2
1−β 2
has m0 c2 as a lower limit.
In my on eption one must take
M 0 c2 e2 − W = p , r 1 − β2
as the expression for the energy, where M0 is the variable proper mass whi h I have already de ned. Cal ulation shows that the proper mass M0 diminishes when r in reases, in su h a way that an ele tron of energy W is no longer at all onstrained to be in the interior of a sphere of radius R.
Mr Born
. Contrary to Mr S hrödinger's opinion, that it is nonsense to speak of the lo ation and motion of an ele tron in the atom, Mr Bohr and I are of the opinion that this manner of speaking always has a meaning when one an spe ify an experiment allowing us to measure the oordinates and the velo ities with a ertain approximation. Again in Ri hardson's notes on the general dis ussion ( f. p. 479), the following text together with Fig. D (both labelled `Bohr'), and a similar gure with the shaded region labelled `B', appear immediately after notes on De Donder's lengthy exposition just above, and learly refer to remarks Bohr made on the topi being addressed here:
B[ohr℄ says it has no point to worry about the paradox that the ele tron in the atom is in a xed path (ellipse or ir le) and the probability that it should be found in a given pla e is given by the produ t ψ ψ¯ whi h is a ontinuous fun tion of spa e extending from zero to ∞. He says if we take a region su h as B a long way from the atom in order to nd if the ele tron is there we must illuminate it with long light waves and the frequen y of these is so low that the ele tron is out of the region by reason of its motion in the stationary state before it has been illuminated long enough for the photoele tri a t to o
ur. I am really not sure if this is right. But, anyway, it is no obje tion to pulling it out with an
518
General dis ussion
Fig. D. intense stati ele tri eld & this appears to be what is happening in the W experiments.
Mr Pauli
. One an indeed determine the lo ation of the ele tron outside the sphere, but without modifying its energy to the point where an ionisation of the atom o
urs.
Mr Lorentz
. I should like to make a remark on the subje t of wave pa kets.a When Mr S hrödinger drew attention to the analogy between me hani s and opti s, he suggested the idea of passing from orpus ular me hani s to wave me hani s by making a modi ation analogous to that whi h is made in the passage from geometri al opti s to wave opti s.39 The wave pa ket gave a quite striking pi ture of the ele tron, but in the atom the ele tron had to be ompletely smeared out [fondu℄, the pa ket having the dimensions of the atom. When the dimensions of the wave pa ket be ome omparable to those of the traje tories of the
lassi al theory, the material point would start to spread; having passed this stage, the ele tron will be ompletely smeared out. The mathemati al di ulty of onstru ting wave pa kets in the atom is due to the fa t that we do not have at our disposal wavelengths su iently small or su iently lose together. The frequen ies of stable a Cf. also the dis ussion of the Lorentz-S hrödinger orresponden e in se tion 4.3 (eds.).
Photons and ele trons
519
waves in the atom (eigenvalues) are more or less separated from ea h other; one annot have frequen ies very lose together orresponding to states diering by very little, be ause the onditions at in nity would not be satis ed. To onstru t a pa ket, one must superpose waves of slightly dierent wavelengths; now, one an use only eigenfun tions ψn , whi h are sharply dierent from ea h other. In atoms, then, one annot have wave pa kets. But there is a di ulty also for free ele trons, be ause in reality a wave pa ket does not, in general, retain its shape in a lasting manner. Lo alised [limités℄ wave pa kets do not seem able to maintain themselves; spreading takes pla e. The pi ture of the ele tron given by a wave pa ket is therefore not satisfying, ex ept perhaps during a short enough time. What Mr Bohr does is this: after an observation he again lo alises [limite℄ the wave pa ket so as to make it represent what this observation has told us about the position and motion of the ele tron; a new period then starts during whi h the pa ket spreads again, until the moment when a new observation allows us to arry out the redu tion again. But I should like to have a pi ture of all that during an unlimited time.40
Mr S hrödinger. I see no di ulty at all in the fa t that on orbits
of small quantum number one ertainly annot onstru t wave pa kets that move in the manner of the point ele trons of the old me hani s. The fa t that this is impossible is pre isely the salient point of the wave me hani al view, the basis of the absolute powerlessness of the old me hani s in the domain of atomi dimensions. The original pi ture was this, that what moves is in reality not a point but a domain of ex itation of nite dimensions, in parti ular at least of the order of magnitude of a few wavelengths. When su h a domain of ex itation propagates along a traje tory whose dimensions and radii of urvature are large ompared with the dimensions of the domain itself, one an abstra t away the details of its stru ture and onsider only its progress along the traje tory. This progress takes pla e following exa tly the laws of the old me hani s. But if the traje tory shrinks until it be omes of the order of magnitude of a few wavelengths, as is the ase for orbits of small quantum number, all its points will be ontinually inside the domain of ex itation and one
an no longer reasonably speak of the propagation of an ex itation along a traje tory, whi h implies that the old me hani s loses all meaning. That is the original idea. One has sin e found that the naive identi ation of an ele tron, moving on a ma ros opi orbit, with a wave pa ket en ounters di ulties and so annot be a
epted to the letter.
520
General dis ussion
The main di ulty is this, that with ertainty the wave pa ket spreads in all dire tions when it strikes an obsta le, an atom for example. We know today, from the interferen e experiments with athode rays by Davisson and Germer, that this is part of the truth, while on the other hand the Wilson loud hamber experiments have shown that there must be something that ontinues to des ribe a well-de ned traje tory after the ollision with the obsta le. I regard the ompromise proposed from dierent sides, whi h onsists of assuming a ombination of waves and point ele trons, as simply a provisional manner of resolving the di ulty.
Mr Born. Also in the lassi al theory, the pre ision with whi h the
future lo ation of a parti le an be predi ted depends on the a
ura y of the measurement of the initial lo ation. It is then not in this that the manner of des ription of quantum me hani s, by wave pa kets, is dierent from lassi al me hani s. It is dierent be ause the laws of propagation of pa kets are slightly dierent in the two ases.
Notes to pp. 477501
521
Notes to the translation 1 Mi ro lmed in AHQP-BMSS-11, se tion 5. 2 AHQP-36, se tion 10. 3 These notes are to be found in the Ri hardson olle tion in Houston, in luded with the opy of Born and Heisenberg's report (mi ro lmed in AHQP-RDN, do ument M-0309). 4 In luded in AHQP-RDN, do ument M-0309. 5 Fren h edition: `les' is misprinted as `le'. 6 AEA 16-617.00 (in German, with trans ription and ar hival omments). 7 The Fren h text has `ψ ' instead of `φ', and `doit satisfaire dans l'espa e ordinaire' instead of the other way round. Note that
φ
is a fun tional of
`material' waves whi h themselves propagate in ordinary spa e. 8 AHQP-36, se tion 10. 9 The Fren h adds: `dé rite par M. Compton'. 10 The Fren h reads: `provient uniquement de'. 11 Dira 's manus ript omits `At present'. 12 The Fren h reads `intuitivité'. 13 Instead of ` ommute' the Fren h has `permuter leurs valeurs'. 14 The Fren h reads: `transformation anonique'. 15 Instead of ` ommute' the Fren h has ` hangent de valeur'. 16 Dira 's manus ript reads `the'. 17 In Dira 's manus ript, the words `determinism and' are an elled and possibly reinstated. They appear in the Fren h, whi h also omits `of yesterday'. 18
unsatisfa tory>
,
the latter seems reinstated.
The Fren h has `indéfendable'. 19 Instead of `the distinguishing hara teristi s' the Fren h has `l'essen e physique'. 20
<do>,
{would} appears above the line, { an} below. The Fren h reads
`ne gurent pas'. 21 This senten e does not appear in the Fren h. 22 In the Fren h, this senten e appears at the beginning of the paragraph. 23 This senten e does not appear in the Fren h. 2 24 Dira 's manus ript has `cn '. 25
possibility> {the existen e} of> .
26 The Fren h has `F.'. 27 Misprinted as ` omptabilité', despite having been orre ted in the galley proofs. 28 In the printed text, the word `Dira ' is mispla ed to later in the paragraph. 29 Misprinted as `protons'. 30 Again misprinted as `protons'. 31 The version of this ontribution in the galley proofs reads as follows:
Mr Langevin makes a omparison between the old and modern statisti s.
Formerly, one de omposed the phase spa e, into ells and one evaluated the representative points.
522
Notes to pp. 506519
It seems that one must modify this method by suppressing the individuality of the representative points and [blank℄ Third method: that of Pauli. This type of representation seems more appropriate to the on eption of photons and parti les [blank℄ attribute identity of nature, attribute at the same time individuality representing a state. In the report of Messrs Born and Heisenberg, I see that it results from quantum me hani s that the statisti s of Bose-Einstein is suitable for mole ules, that of Pauli-Dira , instead, is suitable for ele trons. This means that for [blank℄ there is superposition, while for photons and ele trons there is impenetrability. 32 The galley proofs ontain the following version of this ontribution:
Mr Langmuir would like to see established learly a parallel between
ele trons and photons. What hara terises an ele tron? A well-de ned
harge. What hara terises the photon? Its velo ity, perhaps? What is the analogy, what are the dieren es? Ele tron: de Broglie waves; photon: ele tromagneti waves. For ertain respe ts, this parallelism is
lear, but perhaps it an be pursued to the end? What are the suggestions in the way of experiments?
33 The Fren h renders ` ommute' throughout with ` hanger'. 34 The Fren h adds: `à un instant donné'. 35 `h' misprinted as `λ'. 36 `pϕ ' misprinted as `ϕ'. 37 `ψ ' missing in the original, with a spa e instead. 38 Here the galley proofs in lude an additional senten e: If one took the integral extended over the whole of this spa e, the exterior part would be omparable. 39 The original mistakenly reads `geometri al me hani s' and ` orpus ular opti s'. 40 The version in the galley proofs reads as follows. (Note that in the ase of this and the pre eding ontribution by Lorentz in the galley proofs, the published version was learly not edited by him, sin e he had died at the beginning of February.)
Mr Lorentz. I should like to make a remark on the subje t of wave
pa kets.
When Mr S hrödinger drew attention to the analogy between me hani s and opti s, he suggested the idea of passing from geometri al me hani s to wave me hani s by making a modi ation analogous to that whi h is made in the passage from orpus ular opti s to wave opti s. The wave pa ket was a quite striking pi ture, but in the atom the ele tron is
ompletely smeared out, the pa ket being of the dimensions of the
Notes to pp. 519519
523
atom [blank℄, material point that would start to spread [blank℄, passed, these ele trons are ompletely smeared out. Mathemati al di ulty, wave pa kets in the atom, more or less distinguished frequen ies (eigenvalues), but you ould not have frequen ies very lose together by states diering by mu h or little [par des états tant soit peu diérants℄, be ause one would not have the
onditions at in nity. To onstru t a pa ket, one must superpose waves of slightly dierent wavelengths; now, one an use only eigenfun tions ψn , whi h are sharply dierent from ea h other. Thus one does not have the waves with whi h one ould build a pa ket. In atoms, then, one annot have the wave pa kets; it is the same for free ele trons. All these wave pa kets will end up dissolving. In reality a wave pa ket does not last; wave pa kets that would remain lo alised [limités℄ do not seem to maintain themselves; spreading takes pla e; the pi ture is therefore not satisfying [blank℄, short enough time perhaps [blank℄. What Mr Bohr does is this [small blank℄ after an observation we have again lo alised [limité℄ [blank℄; a new period starts [blank℄. But I should like to have a pi ture of all that during an inde nite time.
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