Quantum Theory and Measurement [Course Book ed.] 9781400854554

Table of contents :
Bohr And Einstein In Dialogue
Contents
Preface
Acknowledgments And Copyright Information
I. Questions of Principle
II. Interpretations of The Act of Measurement
III. "Hidden Variables" Versus "Phenomenon" and Complementarity
IV. Field Measurements
V. Irreversibility And Quantum Theory
VI. Accuracy of Measurements: Quantum Limitations
Guide to some further Literature
Bibliography

Citation preview

Quantum Theory and Measurement

Princeton Series in Physics edited by Arthur S. Wightman and Philip W. Anderson Quantum Mechanics for Hamiltonians Defined as Quadratic Forms byBarry Simon Lectures on Current Algebra and Its Applications by Sam B. Treiman, Roman Jackiw, and David J. Gross Physical Cosmology by P. J. E. Peebles The Many-Worlds Interpretation of Quantum Mechanics edited by B. S. DeWitt and N. Graham The Ρ(Φ)2 Euclidean (Quantum) Field Theory by Barry Simon Homogeneous Relativistic Cosmologies by Michael P. Ryan, Jr., and Lawrence C. Shepley Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann edited by Elliott H. Lieb, B. Simon, and A. S. Wightman Convexity in the Theory of Lattice Gases by Robert B. Israel Surprises in Theoretical Physics by Rudolf Peierls The Large-Scale Structure of the Universe by P. J. E. Peebles Quantum Theory and Measurement edited by John Archibald Wheeler and Wojciech Hubert Zurek

Quantum Theory and Measurement Edited by John Archibald Wheeler and Wojciech Hubert Zurek

Princeton Series in Physics

Princeton University Press Princeton, New Jersey 1983

Copyright © 1983 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey In the United Kingdom: Princeton University Press, Guildford, Surrey All Rights Reserved Library of Congress Cataloging in Publication Data will be found on the last printed page of this book Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability Printed in the United States of America by Princeton University Press, Princeton, New Jersey ADDITIONAL COPYRIGHT INFORMATION WILL BE FOUND BEGINNING ON P. XXI

Bohr and Einstein in Dialogue

vi

BOHR-EINSTEIN DIALOGUE

According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localised in space, which move without being divided and which can be absorbed or emitted only as a whole. EINSTEIN (1905) During the elementary process of radiative loss, the molecule suffers a recoil of magnitude hv/c in a direction which is only determined by 'chance', according to the present state of the theory EINSTEIN (1916b) I am studying your great works and—when I get stuck anywhere— now have the pleasure of seeing your friendly young face before me smiling and explaining. EINSTEIN LETTER OF MAY 2, 1920, AFTER MEETING NIELS BOHR

You believe in a dice-playing God and I in perfect laws in the world of things existing as real objects, which I try to grasp in a wildly speculative way. EINSTEIN LETTER, AS QUOTED IN SCHILPP (1949), p. 176 . .. time and space are modes by which we think and not conditions in which we live. EINSTEIN AS QUOTED BY FORSEE (1963), p. 81

BOHR-EINSTEIN DIALOGUE

Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection. EINSTEIN IN SCHILPP (1949), p. 5 (1)

(2)

The dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis. The latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory. BOHR (1913a), p. 7

. . . any observation necessitates an interference with the course of the phenomena, [and requires] a final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality. 1st HALF, BOHR (1934), p. 115; 2nd HALF, BOHR (1935b), p. 697

vii

viii

BOHR-EINSTEIN DIALOGUE . . . every atomic phenomenon is closed in the sense that its observa­ tion is based on registrations obtained by means of suitable ampli­ fication devices with irreversible functioning such as, for example, permanent marks on the photographic plate, caused by the pene­ tration of electrons into the emulsion.. . . the quantum-mechani­ cal formalism permits well-defined applications referring only to such closed phenomena and must be considered a rational general­ ization of classical physics. BOHR (1958), pp. 73 and 90 . . . the concepts of space and time by their very nature acquire a meaning only because of the possibility of neglecting the inter­ action with the means of measurement. BOHR (1934), p. 99 . . . we must be prepared for the necessity of an ever extending abstraction from our customary demands for a directly visualizable description of nature. Above all, we may expect new surprises in the domain where the quantum theory meets with the theory of relativity and where unsolved difficulties still stand. BOHR (1934), p. 115 In tears, Ehrenfest said that he had to make a choice between Bohr's and Einstein's position and that he could not but agree with Bohr. [Samuel Goudsmit's account of a 1927 conversation with Ehrenfest.] PAIS (1979), p. 900 October 1927. The fifth Solvay Conference convenes [in Brussels]. All the founders of the quantum theory were there, from Planck, Einstein, and Bohr to de Broglie, Heisenberg, Schrodinger, and Dirac. During the sessions "Einstein said hardly anything beyond presenting a very simple objection to the probability interpretation. . . . Then he fell back into silence" [de Broglie, 1962, p. 150]. However, the formal meetings were not the only place of dis­ cussion. All participants were housed in the same hotel and there, in the dining room, Einstein was much livelier. Otto Stern has given this firsthand account [to Res Jost]: "Einstein came down to breakfast and expressed his misgivings about the new quantum theory, every time he had invented some beautiful experiment from which one saw that it did not work. . . . Pauli and Heisenberg who were there did not react to these matters, "ach was, das stimmt schon, das stimmt schon," ah well, it will be alright, it will be alright. Bohr on the other hand reflected on it with care and in the evening, at dinner, we were all together and he cleared up the matter in detail." PAIS (1979), p. 901

BOHR-EINSTEIN DIALOGUE At the sixth Solvay Conference, in 1930, Einstein thought he had found a counterexample to the uncertainty principle. "It was quite a shock for Bohr. . . . he did not see the solution at once. During the whole evening he was extremely unhappy, going from one to the other and trying to persuade them that it couldn't be true, that it would be the end of physics if Einstein were right; but he couldn't produce any refutation. I shall never forget the vision of the two antagonists leaving the club [of the Fondation Universitaire]: Einstein a tall majestic figure, walking quietly, with a somewhat ironical smile, and Bohr trotting near him, very excited The next morning came Bohr's triumph." ROSENFELD (1968), p. 232 The photographs on the preceding pages were taken by Paul Ehrenfest in his house in Leyden where Niels Bohr and Albert Einstein were staying as guests. The restoration of the negatives and production of the prints were done by William R. Whipple. Courtesy of the American Institute of Physics Niels Bohr Library.

IX

CONTENTS BOHR AND EINSTEIN IN DIALOGUE

ν

PREFACE

xv

ACKNOWLEDGMENTS AND COPYRIGHT INFORMATION

xxi

COMMENTARIES 1.1 TheBohr-EinsteinDialogue

50

1.3 The Principle of Indeterminacy

56

1.4 Complementarity 1.8 The Einstein-Podolsky-Rosen Paper 1.9 Bohr's Reply

I.

3

1.2 Bora's Probabilistic Interpretation

85 137 142

IV.2 TheMeasurabilityoftheElectromagneticField V.l The Decrease of Entropy by Intelligent Beings

477 537

VI.1 The Unmeasurability of the Spin of a Free Electron

699

QUESTIONS OF PRINCIPLE 1.1 BOHR (1949), Discussion with Einstein on epistemological problems in atomic physics

9

1.2 BORN (1926), On the quantum mechanics of collisions 1.3 HEISENBERG (1927), The physical content of quantum kinematics and

52

mechanics 1.4 BOHR (1928), The quantum postulate and the recent development of atomic theory

62 87

1.5 ROBERTSON (1929), The uncertainty principle

127

1.6 MOTT (1929), The wave-mechanics of alpha-ray tracks 1.7 EINSTEIN, TOLMAN, PODOLSKY (1931), Knowledge of the past and future in quantum mechanics

129 135

1.8 EINSTEIN, PODOLSKY, ROSEN (1935), Can quantum-mechanical description of physical reality be considered complete? 1.9 BOHR (1935), Quantum mechanics and physical reality

II.

138 144

1.10 BOHR (1935), Can quantum-mechanical description of physical reality be considered complete? 1.11 SCHRODINGER (1935), The present situation in quantum mechanics

145 152

1.12 WIGNER (1961), Remarks on the mind-body question 1.13 WHEELER (1979-1981), Law without law

168 182

INTERPRETATIONS OF THE ACT OF MEASUREMENT II. 1 LONDON, BAUER (1939), The theory of observation in quantum mechanics

217

II.2 WIGNER (1976), Interpretation of quantum mechanics

260

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CONTENTS

11.3 EVERETT III (1957), Relative state formulation of quantum mechanics 11.4 WIGNER (1963), The problem of measurement 11.5 ZEH (1970), On the interpretation of measurement in quantum theory

III.

315 324 342

"HIDDEN VARIABLES" VERSUS "PHENOMENON" AND COMPLEMENTARITY III.1 WHEELER (1946), Polyelectrons

111.2 BOHM (1951), The paradox of Einstein, Rosen, and Podolsky

111.3 BOHM (1952), A suggested interpretation of the quantum theory in terms of "hidden" variables, I and II 111.4 BELL (1966), On the problem of hidden variables in quantum mechanics 111.5 BELL (1964), On the Einstein Podolsky Rosen paradox 111.6 CLAUSER, HORNE, SHIMONY, HOLT (1969), Proposed experiment to test local hidden-variable theories 111.7 FREEDMAN, CLAUSER (1972), Experimental test of local hidden-variable theories 111.8 FRY, THOMPSON (1976), Experimental test of local hidden-variable theories 111.9 LAMEHI-RACHTI, MITTIG (1976), Quantum mechanics and hidden variables: A test of Bell's inequality by the measurement of the spin correlation in low-energy proton-proton scattering

353 356 369 397 403 409 414 418

422

III.10 ASPECT (1976), Proposed experiment to test the nonseparability of

quantum mechanics

435

III.11 WOOTTERS, ZUREK (1979), Complementarity in the double-slit

experiment: Quantum nonseparability and a quantitative statement of Bohr's principle III.12 BARTELL (1980), Complementarity in the double-slit experiment: On simple realizable systems for observing intermediate particle-wave behavior III.13 WICKES, ALLEY, JAKUBOWICZ (1981), A "delayed-choice" quantum mechanics experiment

IV.

443

455 457

FIELD MEASUREMENTS IV.1 LANDAU, PEIERLS (1931), Extension of the uncertainty principle to

relativistic quantum theory IV.2 BOHR, ROSENFELD (1933), On the question of the measurability of electromagnetic field quantities IV.3 BOHR, ROSENFELD (1950), Field and charge measurements in quantum electrodynamics

V.

465 479 523

IRREVERSIBILITY AND QUANTUM THEORY V.l SZILARD (1929), On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings V.2 VON NEUMANN (1932), Measurement and reversibility and The measuring process

539 549

CONTENTS

V.3 VAN HOVE (1959), The ergodic behaviour of quantum many-body systems V.4 DANERI, LOINGER, PROSPERI (1962), Quantum theory of measurement and ergodicity conditions V.5 AHARONOV, BERGMANN, LEBOWITZ (1964), Time symmetry in the quantum process of measurement V.6 MISRA, PRIGOGINE, COURBAGE (1979), Lyapounov variable: Entropyand measurement in quantum mechanics V.7 PERES (1980), Can we undo quantum measurements?

VI.

xiii

648 657 680 687 692

ACCURACY OF MEASUREMENTS: QUANTUM LIMITATIONS VI.1 VI.2 VI.3 VI.4 VI.5 VI.6 VI.7 VI.8

MOTT, MASSEY (1965), Magnetic moment of the electron ARAKI, YANASE (1960), Measurement of quantum mechanical operators YANASE (1961), Optimal measuring apparatus AHARONOV, BOHM (1961), Time in the quantum theory and the uncertainty relation for time and energy HEFFNER (1962), The fundamental noise limit of linear amplifiers HAUS, MULLEN (1962), Quantum noise in linear amplifiers PIERCE (1978), Optical channels: Practical limits with photon counting BRAGINSKY, VORONTSOV, THORNE (1980), Quantum nondemolition measurements

701 707 712 715 725 736 743 749

GUIDE TO SOME FURTHER LITERATURE

769

BIBLIOGRAPHY

787

PREFACE A textbook on quantum theory and measurement does not exist, nor is this intended to be one. This is a reference book, containing key papers on quantum theory as it relates to measurement. They are arranged in such a way, and accompanied by such supplementary references, that the collection can be used as a source book for a university course or seminar on the subject. It was so used by us in 1979-1980 and in 1980-1981 at the University of Texas. We have found that these materials are of interest to students and colleagues of all ages, not only in physics, but also in astron­ omy, philosophy, and mathematics. We suppose that the reader already has some familiarity with quantum theory, preferably at least the equiva­ lent of an undergraduate course in the subject. Quantum theory has turned out to be the overarching principle of twentieth-century physics. It would be difficult to find a single subject among the physical sciences which is not affected in its foundations or in its applications by quantum theory. We may feel lost, to be sure, in the beginning stages of the study of the subject. Our supposed knowledge of a particle with its definite track through space and time dissolves into a wave, definiteness becomes indeterminism, and predictability of place is replaced by a predictability of the properties of nuclei, atoms, molecules, solids, liquids, and gases. We soon find our­

selves armed with wonderful new tools. The more we use them, the more applications we find; and the more applications we find, the more uses of quantum theory we make. Why there is no textbook on the measurement side of quantum theory is clear to anyone who participates in a seminar on the subject, and even clearer to one who gives a course on it: puzzlement! Beyond the probability interpretation of quantum mechanics, beyond all the standard analysis of idealized experiments, beyond the prin­ ciple of indeterminacy and the limits it imposes, lie deep issues on which full agreement has not yet been reached in the physics community. They include questions like these: Does observation demand an irreversible act of amplifi­ cation such as takes place in a grain of photographic emulsion or in the elec­ tron avalanche of a Geiger counter? And if so, what does one mean by "amplification"? And by "irrevers­ ible"? Does the quantum theory of observation apply in any meaningful way to the "whole universe"? Or is it restricted, even in principle, to the light cone? And if so, whose light cone? How are the observations made by different observers to be fitted into a single consistent picture in space-time? If these are some of the issues, they lead to other still deeper questions: What is the most productive meaning to assign to the term "reality"? How are we to look at the subject, so mixed in its character, partly well-

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PREFACE

understood—and, as such, the un­ shakable foundation for all of modern physics—and partly still uncaptured frontier territory? What else is it but an unfamiliar animal, confined to an animal house? And how else can one better capture its newness than by walking around, looking at it through one window after another, seeking to combine fragmentary views into a total picture? We here take two tours around the subject. The first tour begins with Niels Bohr's account of his famous 28-year dialog with Albert Einstein. That account is followed by a great trio: Max Bora's epochal paper on the probability interpretation of quan­ tum mechanics, Werner Heisenberg's on the principle of indeterminism, and Niels Bohr's Como lecture on what quantum mechanics permits one to know. These are followed by H. P. Robertson's general formulation of indeterminism or "uncertainty," then by Nevill Mott's analysis of the mean­ ing of an α-ray track, which illustrates the principles expounded by Bohr and Born and others and shows the quan­ tum theory of observation in action. Had quantum mechanics stopped here, its deepest lesson would have escaped attention: "No elementary quantum phenomenon is a phenome­ non until it is a registered (observed) phenomenon." The door to this in­ sight—and all the questions that go with it—was opened a crack by the paper of Albert Einstein, Richard Tolman, and Boris Podolsky, and opened wider by the idealized experi­ ment proposed by Einstein, Podolsky, and Nathan Rosen. This EPR paper

reasoned that quantum mechanics is incompatible with any reasonable idea of reality. In the next two papers, Bohr replies, in effect, that the EPR concept of reality is too limited. In contrast to Bohr, Erwin Schrodinger, in his famous paper on the "cat paradox," and Eugene Wigner, in a subsequent paper, tried to connect the concept "observa­ tion" as it is employed in quantum mechanics with "consciousness." This first tour around the animal house, looking through the various windows, concludes with "law without law," an attempt to assess the situation as it stands today and to evaluate the place of quantum mechanics in the larger scheme of physics. The second tour of inspection (Sec­ tions II-VI) looks into the old windows afresh and into some new ones. It is designed for the more advanced student of the subject. It too gives a view of what is clear and well-established, but also glimpses of what is problematic and mysterious. Section II begins with the classic treatise of Fritz Wolfgang London and Edmond Bauer on the quantum theory of observation and includes notes of Wigner's 1976 lectures on interpretation of quantum mechanics, not previously published. The reader who has mastered this material or the equivalent has in hand the solid foundation of the subject. This section ends with three papers on the problem of measurement within the framework of quantum theory: Hugh Everett's "Relative State" or "Many Worlds" interpretation, and papers of Wigner and H. D. Zeh.

PREFACE

The Einstein-Podolsky-Rosen exper­ iment (Section III) deals with a system which, once united in a definite quan­ tum state, splits into two well-separated systems. It considers the correlation between the state observed for one system and the state observed for the other. It asks: Does the predicted correlation exist? And if so, how does it come about? There is an enormous literature on these questions. Out of the many possible papers we have selected thirteen for this part of our second tour around the subject. They deal with "hidden variables," John Bell's inequal­ ity designed to rule out "local hidden variables," the experiments them­ selves, and generalizations of such experiments. Up to this point, the statistical interpretation of quantum mechanics and the principle of indeterminacy have been applied to a particle or system of finitely many degrees of freedom. The next group of papers (Section IV), including one by Lev Landau and Rudolf Peierls, and two by Bohr and Leon Rosenfeld, deal with the measurement of the electro­ magnetic field, a system with infinitely many degrees of freedom. What is the connection between entropy, information, ergodicity, ir­ reversibility, thermodynamics, and quantum mechanics? The next seven selections (Section V) deal in one way or another with these topics. The second tour concludes with Section VI, eight papers on the accura­ cy achievable in measurement as it is affected by quantum limitations. The first (N. F. Mott and H. S. Massey)

xvii

deals with the impossibility of mea­ suring the spin of an electron that is free rather than bound in an atom or in some experimental combination of electric and magnetic fields. Even when a particle or an atom or a molecule admits of a measurement of its angular momentum, the mea­ suring equipment may be so light in mass that by this very reason its orientation is uncertain (papers of Huzihiro Araki and Mutsuo Yanase, and Yanase). Or the measurement of the time at which an interaction takes place may limit the accuracy with which one can determine the transfer of energy in that interaction (paper of David Bohm and Yakir Aharonov). Or a message may be sent through an amplifier or sent down a communication line and have quantum effects introduced into it along the way (papers of H. Heffner, H. A. Haus and J. A. Mullen, and John Pierce). Or in a measurement of a weak effect, like a gravitational wave from a supernova, can one circumvent quantum indeter­ minacy limits on the sensitivity of the measuring device? (paper of V. B. Braginsky, Y. I. Vorontsov and K. S. Thorne). Anyone asking for the practical bearing of the quantum theory of measurement will think of measuring devices, their sensitivity, and the im­ provements in this sensitivity that can only be achieved by exploiting modern insights to the fullest. In no way do the advances of physics spread more widely to the community than in new and improved measuring devices, whose uses range from biology to

xviii

PREFACE

medicine, from chemistry to manu­ facturing. Not otherwise can one understand why the number of kinds of devices listed in the annual "Guide to Scientific Instruments" issued by the American Association for the Ad­ vancement of Science (AAAS, 1961, 1970, 1980) has steadily increased. There is at present a heavy push to measure at the Earth the fantastically weak gravitational wave produced by a supernova in the Virgo cluster of galaxies. Central to this enterprise are considerations of quantum theory that have already led to new ideas of optimal design. Achievements on this front will surely make their way from physics to many another field and supply important areas of science and technology with measuring devices of a sensitivity previously unattainable. We regard as a welcome forerunner to the present collection an earlier collection of about half as many titles and a quarter as many pages issued for the exclusive use of members of the Physical Society of Japan. We have not entered into various schemes for the axiomatization of quantum theory and so-called "quantum logic," and therefore do not reproduce selec­ tions from such books as George W. Mackey's Mathematical Foundations of Quantum Mechanics (1963) and Josef M. Jauch's Foundations of Quantum Mechanics (1968), nor, for example, the work of Giinther Ludwig and his collaborators (1968) or P. Mittelstaedt (1978), or Beltrametti and Cassinelli (1981). Neither do we pretend to a proper historical perspective, such as surely someday will be possible thanks

to the wonderful collection of tape recordings, notebooks, and other ma­ terials inventoried in Kuhn, Heilbron, Forman, and Allen, Sources for the History of Quantum Physics (1967). The literature on quantum theory is so vast that one could fill a book this size with bibliography alone. In the annotated bibliographies for each section (located near the end of the book) we have had to content ourselves with a few especial­ ly important or representative works, seeking to provide in this way points of entry into the literature. The reader will look in vain in these selections for any detailed expo­ sition of the wealth of mechanisms that make physics such a rich subject. We leave out such topics as photo­ electric effect, scattering, chemical binding, particle theory, the LambRetherford shift, the building up of atoms, angular momentum isomers, superconductivity, and dozens of others equally important for understanding what quantum theory means for the world of today, because we assume that the reader already knows some­ thing of "quantum mechanics in action." If one already knows so much about the applications of quantum theory, what more is to be learned by the study of the quantum theory of mea­ surement? Willis Lamb (1969), some years after receiving the Nobel Prize, wrote, ". . . A discussion of the inter­ pretation of quantum mechanics on any level beyond this almost inevitably becomes rather vague. The major difficulty involves the concept of 'measurement,' which in quantum me-

PREFACE

chanics means determining the value of a physical observable for a dynamic­ al system with as much precision as is possible. "I have taught graduate courses in quantum mechanics for over 20 years at Columbia, Stanford, Oxford and Yale, and for almost all of them have dealt with measurement in the follow­ ing manner. On beginning the lectures I told the students, 'You must first learn the rules of calculation in quan­ tum mechanics, and then I will tell you about the theory of measurement and discuss the meaning of the subject.' Almost invariably, the time allotted to the course ran out before I had to fulfill my promise." No one who reads among the present selections can escape some contact with the deeper meaning of the subject —and with some of the issues. Is it true that "no elementary quantum phenom­ enon is a phenomenon until it is a recorded phenomenon"? If so, what does "recording" demand? "Irrevers­ ibility"? If so, what does one mean by "irreversibility" ? If the "arrow of time" is absent from Schrodinger's equation and from quantum theory generally, what brings it into the act of measure­ ment along with all its ideological connectives, from statistical mechanics to ergodic theory, and from informa­ tion theory to thermodynamics? Is it true that the result of a measurement must be expressed in classical terms, because only in such terms can one speak in plain language to oneself— and to others? What part does com­ munication play in creating what is called "knowledge"? And from what

XlX

deeper principle arises the necessity of the quantum in a construction of the world? With a good conscience we limit ourselves here to the measurement side of quantum theory, because the formalism of the subject is so well treated in so many outstanding texts. Moreover, historic papers in the de­ velopment of that formalism have been collected in the wonderfully useful book by B. L. Van der Waerden (1967). We wish to express here our indebt­ edness to those books and above all to the book of Max Jammer (1974) which provides a wealth of historical commentary on the development of the subject to its present state. See also DeWitt and Graham (1971). We take this opportunity to thank most heartily the many colleagues who have given us their advice in the preparation of this book and to ack­ nowledge our indebtedness to the authors and publishers cited at the front for permission to reprint the selections. We could not have put the collection together without calling on the intelligence and helpfulness of Ruth Bentley, Zelda Davis, Adrienne Harding, Colleen Kieke, Jean F. Otto, Rebecca Stadtner, and Gloria TalcoveWoodward. We appreciate the care given to the project by the staff of Princeton University Press, including especially Alice Calaprice and Judith May. We thank many a favorable chance, many a kind act of hospitality, and thank, too, the University of Texas, the Center for Theoretical Physics, the Center for Statistical Mechanics and Thermodynamics, and the Nation-

XX

PREFACE

al Science Foundation (Grant PHY 7826592) for the colleagues that they have brought our way who share our concern with the measurement side of quantum theory, among them L. Bartell, J. D. Bekenstein, D. Bohm, A. Bohr, V. B. Braginsky, E. Caianiello, P. Candelas, P. C. W. Davies, B. d'Espagnat, D. Deutsch, B. S. DeWitt, R. H. Dicke, F. Dyson, H. Everett III, L. Z. Fang, R. P. Feynman, E. S. Fry, U. Gerlach, A. M. Gleason, L. P. Horwitz, A. Jaffe, F. Jenc, J. Kalckar, J. R. Klauder, D. Kondepudi, and

K. Kuchar, G. Ludwig, G. Mackey, L. Michel, C. W. Misner, P. Mittelstaedt, Y. Ne'eman, A. Peres, C. Piron, I. Prigogine, A. Qadir, L. Radicati, D. Sciama, A. Shimony, H. P. Stapp, E. C. G. Sudarshan, C. Teitelboim, K. S. Thorne, F. Tipler, W. Unruh, E. P. Wigner, W. K. Wootters, and H. D. Zeh. John Archibald Wheeler Wojciech Hubert Zurek Austin, Texas January 20, 1982

ACKNOWLEDGMENTS AND COPYRIGHT INFORMATION The editors gratefully acknowledge the permission of the Niels Bohr Library of the American Institute of Physics to reproduce here Ehrenfest's photographs of Bohr and Einstein, and the per­ missions received from authors, translators, and publishers to reprint the following items. 1.1

1.1

1.2

1.3

1.3

1.4

Commentary Based on Petersen (1968) Extracts reprinted by permission from Quantum Physics and The Philosophical Tradition by Aage Petersen. Copy­ right © 1968, Massachusetts Institute of Technology Press. Commentary of Heisenberg (1967) Extracts from "Quantum theory and its interpretations," by W. Heisenberg. Reprinted by permission from Niels Bohr: His Life and Work as Seen by His Friends and Colleagues, pp. 94-108, edited by S. Rozental. Copyright © 1967, North Holland Publishing Company. Commentary of Rosenfeld (1971a) Extracts from "Men and ideas in the history of atomic theory," by L. Rosenfeld. From Selected Papers of Leon Rosenfeld, pp. 266-296, edited by R. S. Cohen and J. J. Stachel. Copyright © 1979, D. Reidel Publishing Company, reprinted by permission of the publisher. Commentary of Heisenberg (1967) Extracts from "Quantum theory and its interpretations," by W. Heisenberg. Reprinted by permission from Niels Bohr: His Life and Work as Seen by His Friends and Colleagues, pp. 94-108, edited by S. Rozental. Copy­ right © 1967, North Holland Publishing Company. Commentary of Rosenfeld (1971a) Extracts from "Men and ideas in the history of atomic theory," by L. Rosenfeld. From Selected Papers of Leon Rosenfeld, pp. 266-296, edited by R. S. Cohen and J. J. Stachel. Copyright © 1979, D. Reidel Publishing Company, reprinted by permission of the publisher. Commentary of Rosenfeld (1963) Extracts from "Niels Bohr's contribution to epistemology," by L. Rosenfeld. From Selected Papers of Leon Rosenfeld,

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ACKNOWLEDGMENTS

1.4

1.4

1.8

1.9

IV.2

VI.1

1.1

pp. 522-535, edited by R. S. Cohen and J. J. Stachel. Copy­ right © 1979, D. Reidel Publishing Company, reprinted by permission of the publisher. Commentary of Rosenfeld (1971a) Extracts from "Men and ideas in the history of atomic theory," by L. Rosenfeld. From Selected Papers of Leon Rosenfeld, pp. 266-296, edited by R. S. Cohen and J. J. Stachel. Copyright © 1979, D. Reidel Publishing Company, reprinted by permission of the publisher. Commentary of Heisenberg (1967) Extracts from "Quantum theory and its interpretations," by W. Heisenberg. Reprinted by permission from Niels Bohr: His Life and Work as Seen by His Friends and Colleagues, pp. 94-108, edited by S. Rozental. Copyright © 1967, North Holland Publishing Company. Commentary of Rosenfeld (1967) Extracts from "Niels Bohr in the thirties: consolidation and extension of the conception of complementarity," by L. Rosenfeld. Reprinted by permission from Niels Bohr : His Life and Work as Seen by His Friends and Colleagues, pp. 114-136, edited by S. Rozental. Copyright © 1967, North Holland Publishing Company. Commentary of Rosenfeld (1967) Extracts from "Niels Bohr in the thirties: consolidation and extension of the conception of complementarity," by L. Rosenfeld. Reprinted by permission from Niels Bohr: His Life and Work as Seen by His Friends and Colleagues, pp. 114-136, edited by S. Rozental Copyright © 1967, North Holland Publishing Company. Commentary of Rosenfeld (1955) Extracts from "On quantum electrodynamics," by L. Rosenfeld. Reprinted by permission from Niels Bohr and the Development of Physics, edited by W. Pauli. Copyright 1955, Pergamon, New York. Commentary of Rosenfeld (1971b) Extracts from "Quantum theory in 1929: recollections from the first Copenhagen conference," by L. Rosenfeld. Reprinted by permission from Selected Papers of Leon Rosenfeld, pp. 305-307, edited by R. S. Cohen and J. J. Stachel. Copyright © 1979, D. Reidel Publishing Co. "Discussion with Einstein on epistemological problems in atomic physics," by Niels Bohr. Reprinted by permission

ACKNOWLEDGMENTS

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10

from Albert Einstein, Philosopher Scientist, pp. 200-241, edited by Paul Arthur Schilpp. Copyright © 1949, cur­ rently handled jointly by Open Court Publishing Company and the American Friends of The Hebrew University, Inc. "On the quantum mechanics of collisions," by M. Born. By permission from Zeitschrift fur Physik 37, pp. 863— 867. Copyright © 1926, Springer-Verlag. "The physical content of quantum kinematics and mech­ anics," by W. Heisenberg. Reprinted by permission from Zeitschrijt fur Physik 43, pp. 172-198. Copyright © 1927 by Springer-Verlag. Reprinted in Dokumente der Naturwissenschaft, Vol. 4, pp. 9-35. Copyright © 1935. "The quantum postulate and the recent development of atomic theory," by Niels Bohr. Reprinted by permission from Nature, Vol. 121, pp. 580-590. Copyright © 1928, Macmillian Journals Limited. "The uncertainty principle," by H. P. Robertson. Re­ printed by permission from Physical Review 34, pp. 163-164. Copyright © 1929, American Physical Society. "The wave mechanics of α-ray tracks," by N. F. Mott. Reprinted by permission from Proceedings A126, pp. 79-84. Copyright © 1929, The Royal Society, London. "Knowledge of past and future in quantum mechanics," by A. Einstein, R. C. Tolman, and B. Podolsky. Re­ printed by permission from Physical Reciew 37, pp. 780-781. Copyright © 1931, currently handled jointly by American Physical Society and the American Friends of the Hebrew University, Inc. "Can quantum-mechanical description of physical reality be considered complete?" by A. Einstein, B. Podolsky and N. Rosen. Reprinted by permission from Physical Review 47, pp. 777-780. Copyright © 1935, currently handled jointly by American Physical Society and the American Friends of The Hebrew University, Inc. "Quantum mechanics and physical reality," by Niels Bohr. Reprinted by permission from Nature, Vol. 136, pp. 65. Copyright © 1935, Macmillian Journals Limited. "Can quantum-mechanical description of physical reality be considered complete?" by Niels Bohr. Reprinted by permission from Physical Review 48, pp. 696-702. Copyright © 1935, American Physical Society.

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"Die gegenwartige Situation in der Quantenmechanik," by E. Schrodinger. By permission from Die Naturwissenschaften 23, pp. 807-812, 823-828, 844-849. Copyright © 1935, Springer-Verlag New York, Inc. English translation: "The present situation in quantum mechanics: a translation of Schrddinger's 'cat paradox' paper," by John D. Trimmer. Reprinted by permission from Proceedings of the American Philosophical Society, Vol. 124, pp. 323-338. Copyright © 1980, American Philosophical Society. 1.12 "Remarks on the mind-body question," pp. 171-184, from Symmetries and Reflections, science essay by E. P. Wigner. Copyright © 1967, Indiana University Press, reprinted by permission of the publisher. 1.13 "Law without law," excerpts as follows: "Frontiers of time" by John A. Wheeler. Reprinted by permission from Problems in the Foundations of Physics, Proceedings of the International School of Physics "Enrico Fermi," Course 72, pp. 395-407. Copyright © 1979, Italian Physical Society. "Beyond the Black Hole," from H. Woolf, ed., Some Strangeness in the Proportion: A Centennial Symposium to Celebrate the Achievements of Albert Einstein,

11.1

Copyright © 1980, Addison-Wesley, Reading, Mas­ sachusetts, pp. 362-363. Reprinted with permission. "The theory of observation in quantum mechanics," by F. W. London and E. Bauer. No. 775 of Actualites scientifiques et industrielles: Exposes de physique generate,

edited by Paul Langevin, Hermann et Compagnie, 1939. English translations done independently by Abner Shimony, by J. A. Wheeler and W. H. Zurek, and by James H. McGrath and Susan McLean McGrath; the editors have drawn on all three translations to produce the version here. 11.2 "Interpretation of quantum mechanics," by Eugene P. Wigner. By permission from lecture notes of Eugene P. Wigner, revised and printed here for the first time. 11.3 "Relative state formulation of quantum mechanics," by Hugh Everett III. Reprinted by permission from Reviews of Modern Physics, Vol. 29, pp. 454-462. Copyright © 1957, American Physical Society. 11.4 "The problem of measurement," by Eugene P. Wigner. Reprinted by permission from American Journal of Physics, Vol. 31, pp. 6-15, edited by John S. Rigden.

ACKNOWLEDGMENTS

Copyright © 1963, American Association of Physics Teachers. II.5

111.1

111.2

Ill.3

111.4

111.5

111.6

111.7

111.8

111.9

"On the interpretation of measurement in quantum theory," by H. D. Zeh. Reprinted by permission from Foundations of Physics, Vol. 1, pp. 69-76. Copyright © 1970, Plenum Publishing Corporation. "Polyelectrons" by John A. Wheeler. Reprinted by per­ mission from Annals of the New York Academy of Sciences, Vol. 48, pp. 219-238, edited by Bill M. Boland. Copyright © 1946, New York Academy of Science. "The paradox of Einstein, Rosen and Podolsky," by David Bohm. Reprinted by permission from Quantum Theory, Chapter 22, Sections 15-19. Copyright © 1951, PrenticeHall. "A suggested interpretation of the quantum theory in terms of 'hidden' variables, I and II," by David Bohm. Re­ printed by permission from Physical Review 85, pp. 166-179. Copyright © 1952, American Physical Society. "On the problem of hidden variables in quantum mechan­ ics," by John S. Bell. Reprinted by permission from Reviews of Modern Physics, Vol. 38, pp. 447-452. Copy­ right © 1966, American Physical Society. "On the Einstein Podolsky Rosen paradox," by John S. Bell. Reprinted by permission of J. S. Bell. Originally appeared-in Physics J, pp. 195-200. Copyright © 1964, Physics Publishing Co. "Proposed experiment to test local hidden-variable the­ ories," by John F. Clauser, Michael A. Home, Richard A. Holt, and Abner Shimony. Reprinted by permission from Physical Review Letters 23, pp. 880-884. Copyright © 1969, American Physical Society. "Experimental test of local hidden-variable theories," by Stuart J. Freedman and John F. Clauser. Reprinted by permission from Physical Review Letters 28, pp. 938-941. Copyright © 1972, American Physical Society. "Experimental test of local hidden-variable theories," by Edward S. Fry and Randall C. Thompson. Reprinted by permission from Physical Review Letters 37, pp. 465-468. Copyright © 1976, American Physical Society. "Quantum mechanics and hidden variables: A test of Bell's inequality by the measurement of the spin correlation in low-energy proton-proton scattering." by M. LamehiRachti and W. Mittig. Reprinted by permission from

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III.10

III.11

111.12

III.13

IV.1

IV.2

IV. 3

Physical Review D14, pp. 2543-2555. Copyright © 1976, American Physical Society. "Proposed experiment to test the nonseparability of quantum mechanics," by Alain Aspect. Reprinted by permission from Physical Review D14, pp. 1944-1951. Copyright © 1976, American Physical Society. "Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle," by W. K. Wootters and W. H. Zurek. Re­ printed by permission from Physical Review D19, pp. 473-484. Copyright © 1979, American Physical Society. "Complementarity in the double-slit experiment: On simple realizable systems for observing intermediate particle-wave behavior," by L. S. Bartell. Reprinted by permission from Physical Review D21, pp. 1698-1699. Copyright © 1980, American Physical Society. "A 'delayed-choice' quantum mechanics experiment," by William C. Wickes, C. O. Alley, and O. Jakubowicz. Printed here for the first time by permission of the authors. "Erweiterung des Umbestimmtheitsprinzips fur die relativistische Quantentheorie," by L. Landau and Rudolf Peierls. By permission from Zeitschrift fiir Physik 69, p. 56. Copyright © 1931 Springer-Verlag. English transla­ tion : "Extension of the uncertainty principle to relativistic quantum theory," translated and edited by Dirk ter Haar. Reprinted by permission from Collected Papers of L. D. Landau, pp. 4-51. Copyright © 1965, Gordon and Breach. "On the question of the measurability of electromagnetic field quantities," by Niels Bohr and Leon Rosenfeld. By permission from Matematisk-Fysiske Meddelelser Kongelige Danske Videnskabernes Selskab 12, No. 8. Copyright © 1933, Det Kongelige Danske Videnskabernes Selskab. English translation by Aage Petersen. Reprinted by permission from Selected Papers of Leon Rosenfeld, pp. 357412, edited by R. S. Cohen and J. J. Stachel. Copyright © 1979, D. Reidel Publishing Co. "Field and charge measurements in quantum electrody­ namics," by Niels Bohr and Leon Rosenfeld. Reprinted by permission from Physical Review 78, pp. 794-798. Copyright © 1950, American Physical Society.

ACKNOWLEDGMENTS

V.l

V.2

V.3

V.4

V.5

V.6

V.7

VI.l

"Uber die Entropieverminderung in einen thermodynamischen System bei Eingriffen intelligenter Wesen," by Leo Szilard. By permission from Zeitschrijt fiir Phvsik 53, pp. 840-856. Copyright © 1929, SpringerVerlag. English translation: "On the decrease of entropy in a thermodynamic system by the intervention of in­ telligent beings," by Anatol Rapoport and Mechthilde Knoller, revised by Carl Eckart. Reprinted by permission from Behavioral Science 9, pp. 301-310. Copyright © 1964, Society for General Systems Research. "Measurement and Reversibility" and "The Measuring Process," by John von Neumann. By permission from Mathematische Grundlagen der Quantenmechanik, Chap­ ters V and VI. Copyright © 1932, Springer-Verlag. English translation reprinted by permission from J. von Neumann, Mathematical Foundations of Quantum Mechanics, trans. Robert T. Beyer, pp. 347-445. Copy­ right © 1955, Princeton University Press. "The ergodic behaviour of quantum many-body systems," by Leon Van Hove. Reprinted by permission from Physica 25, pp. 268-276. Copyright © 1959, North Holland Publishing Company. "Quantum theory of measurement and ergodicity con­ ditions," by A. Daneri, A. Loinger, and G. M. Prosperi. Reprinted by permission from Nuclear Physics 33, pp. 297. Copyright © 1962, North Holland Publishing Company. "Time symmetry in the quantum process of measurement," by Yakir Aharonov, Peter G. Bergmann, and Joel L. Lebowitz. Reprinted by permission from Physical Review 134, pp. B1410. Copyright © 1964, American Physical Society. "Lyapounov variable: Entropy and measurement in quan­ tum mechanics," by B. Misra, I. Prigogine, and M. Courbage. Reprinted by permission from Proceedings of The National Academy of Sciences 76, pp. 4768- 4772. Copyright © 1979, The National Academy of Sciences. "Can we undo quantum measurements?" by Asher Peres. Reprinted by permission from Physical Review D22, pp. 879-883. Copyright © 1980, American Physical Society. "Magnetic moment of the electron," by N. F. Mott and

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VI.2

VI.3

VI.4

VI.5

VI.6

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Η. S. W. Massey. Reprinted by permission of Oxford Uni­ versity Press from The Theory of Atomic Collisions, 3rd ed., pp. 214-219, Section 2, Chapter IX. Copyright © 1965, Oxford University Press. "Measurement of quantum mechanical operators," by H. Araki and M. M. Yanase. Reprinted by permission from Physical Review 120, pp. 622-626. Copyright © 1960, American Physical Society. "Optimal measuring apparatus," by M. M. Yanase. Reprinted by permission from Physical Review 123, pp. 666-668. Copyright © 1961, American Physical Society. "Time in the quantum theory and the uncertainty relation for time and energy," by Y. Aharonov and D. Bohm. Reprinted by permission from Physical Review 122, pp. 1649-1658. Copyright © 1961, American Physical Society. "The fundamental noise limit of linear amplifiers," by J. Heffner. Reprinted from Proceedings of the Institute of Radio Engineers 50, pp. 1604-1608. Copyright © 1962, The Institute of Radio Engineers. "Quantum noise in linear amplifiers," by Herman A. Haus and James A. Mullen. Reprinted by permission from Physical Review 128, pp. 2407-2413. Copyright © 1962, American Physical Society. "Optical channels: Practical limits with photon counting," by J. R. Pierce. Reprinted from the Institute of Electrical and Electronics Engineers Transactions on Communications Com-26, pp. 1819-1821. Copyright © 1978, Institute of Electrical and Electronics Engineers. "Quantum nondemolition measurements," by V. B. Braginsky, Υ. I. Vorontsov, and K. S. Thorne. Reprinted from Science 209, pp. 547-557. Copyright © 1980, The American Association for the Advancement of Science.

Questions of Principle

1.1

THE BOHR-EINSTEIN DIALOGUE BOHR IN BRIEF

Complementarity: any given application of classical concepts pre­ cludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena. BOHR (1934), p. 10 The discovery of the quantum of action shows us, not only the natu­ ral limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existence of phenom­ ena independently of our observations, confronts us with a situation hitherto unknown in natural science. As we have seen, any observa­ tion necessitates an interference with the course of the phenomena, [and] of such a nature that it deprives us of the foundation under­ lying the causal mode of description. The limit, which nature herself has thus imposed upon us, of the possibility of speaking about phe­ nomena as existing objectively finds its expression, as far as we can judge, just in the formulation of quantum mechanics. BOHR (1934), p. 115 . . . atomic phenomena under different experimental conditions, must be termed complementary in the sense that each is well defined and that together they exhaust all definable knowledge about the objects concerned. The quantum-mechanical formalism . . . gives . . . an exhaustive complementary account of a very large domain of experience. BOHR (1958), p. 90 . . . one sometimes speaks of "disturbance of phenomena by ob­ servation" or "creation of physical attributes to atomic objects by measurements." Such phrases, however, are apt to cause confusion, since words like phenomena and observation, just as attributes and measurements, are here used in a way incompatible with common language and practical definition. On the lines of objective descrip­ tion, [I advocate using] the word phenomenon to refer only to ob­ servations obtained under circumstances whose description includes an account of the whole experimental arrangement. In such terminol­ ogy, the observational problem in quantum physics is deprived of any special intricacy and we are, moreover, directly reminded that every atomic phenomenon is closed in the sense that its observation is based on registrations obtained by means of suitable amplification

4

1.1 COMMENTARY

devices with irreversible functioning such as, for example, perma­ nent marks on a photographic plate, caused by the penetration of electrons into the emulsion. In this connection, it is important to realize that the quantum-mechanical formalism permits well-defined applications referring only to such closed phenomena. BOHR (1958), p. 73 . . . the finite magnitude of the quantum of action prevents alto­ gether a sharp distinction being made between a phenomenon and the agency by which it is observed .... BOHR (1934), p. 11 . . . i t . . . can make no difference, as regards observable effects ob­ tainable by a definite experimental arrangement, whether our plans for constructing or handling the instruments are fixed beforehand or whether we prefer to postpone the completion of our planning until a later moment when the particle is already on its way from one instrument to another. BOHR IN SCHILPP (1949), p. 230 . . . a subsequent measurement to a certain degree deprives the infor­ mation given by a previous measurement of its significance for pre­ dicting the future course of the phenomena. Obviously, these facts not only set a limit to the extent of the information obtainable by measurements, but they also set a limit to the meaning which we may attribute to such information. We meet here in a new light the old truth that in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience. BOHR (1934), p. 18 The experimental conditions can be varied in many ways, but the point is that in each case we must be able to communicate to others what we have done and what we have learned, and that therefore the functioning of the measuring instruments must be described within the framework of classical physical ideas. BOHR (1958), p. 89 . . . the conscious analysis of any concept stands in a relation of exclusion to its immediate application. BOHR (1934), p. 96

I am quite prepared to talk of the spiritual life of an electronic computer; to say that it is considering or that it is in a bad mood.

1.1 COMMENTARY

5

What really matters is the unambiguous description of its behaviour, which is what we observe. The question as to whether the machine really feels, or whether it merely looks as though it did, is absolutely as meaningless as to ask whether light is "in reality" waves or par­ ticles. We must never forget that "reality" too is a human word just like "wave" or "consciousness." Our task is to learn to use these words correctly—that is, unambiguously and consistently. BOHR AS QUOTED BY KALCKAR (1967), p. 234 We are suspended in language in such a way that we cannot say what is up and what is down. BOHR AS QUOTED BY PETERSEN (1968), p. 188

COMMENTARY BASED ON PETERSEN (1968) Aage Petersen was already working on his book, his Copenhagen doctoral thesis, while he was assisting Bohr (+18 November 1962) in preparing some of his last lectures. The thesis is philo­ sophical in character. It is concerned with ideas. It is not intended to be a professional history of science, nor a documentation of stages in Bohr's thinking. However, some of the sections allow one ta get an impression of stages in Bohr's development of the concepts of "complementarity," "clo­ sure," and "phenomenon." The book states more sharply than Bohr does in his writings the points on which Bohr disagreed with others, explaining, for example, why Bohr introduced the term "complementarity" when Heisenberg had already employed the term "indeterminism." Petersen (pp. 110-111 and 145) notes that Bohr insisted upon an analysis of the "possibilities of definition" over and above those "possibilities of obser­ vation" that he and Heisenberg together

had previously considered: "One of the most important issues in the measure­ ment analysis is the question of the nature and origin of the uncertainties involved in the determination of con­ jugate variables. According to Heisenberg, these uncertainties were due to discontinuous changes, imposed by the quantum on one such variable during the measurement of the other. However, as Bohr pointed out, 'a discontinuous change of energy and momentum during observation could not prevent us from ascribing accurate values to the spacetime coordinates, as well as to the momentum-energy components before and after the process.' To clarify the issue it is necessary to consider closely the possibilities of definition. ... Bohr pointed out that the conditions of description in quantum physics not only 'set a limit to the extent of the informa­ tion obtainable by measurement, but they also set a limit to the meaning which we may attribute to such informa­ tion.' More specifically, the reciprocal

6

I.l COMMENTARY

uncertainties are'. . . essentially an out­ come of the limited accuracy with which changes in energy and momentum can be defined.'" Petersen goes on to say (p. 120), "Einstein's criticism showed the need for a more rigorous formulation of the Copenhagen interpretation, and Bohr's many attempts to improve the termi­ nology illuminate the development of his attitude to the interpretation ques­ tion. [How could one] make explicit the conditions set by the formalism for applying the physical concepts unam­ biguously in the quantum domain? [For this purpose the] shift from 'intuitive understanding' to 'unambig­ uous communication' [was] an impor­ tant step." Petersen adds (pp. 120 and 122-23), "Terminologically, the principal result of Bohr's analysis of Einstein's imagi­ nary experiments was the concept of a quantum phenomenon [which first appeared in Bohr (1939)]. Bohr came to regard it as the basic element of the quantal description. ... To specify a phenomenon it is not enough to state the initial characteristics of the object, like the momentum with which it emerges from the source. The predic­ tions depend on the whole experimental arrangement and are only well defined if the whole arrangement is specified. To be able to predict the interference pattern we must be given the whole geometry of the optical bench. In other words, 'all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions, defining the initial state of the atomic system concerned and the

character of the possible predictions as regards subsequent observable proper­ ties of that system. Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is ... the combination of measurements of both kinds which constitutes a welldefined phenomenon ... .' [A phenom­ enon] is 'indivisible.' In electron inter­ ference, the physical 'process' starting at the electron's emergence from the gun and ending at its impact on the plate has no definable course. It cannot be broken up into physically welldefined steps. Unlike a classical phe­ nomenon, a quantum phenomenon is not a sequence of physical events, but a new kind of individual entity." Petersen (p. 164) recalls Bohr's state­ ment (1934, pp. 19-20): "An interesting example of ambiguity in our use of language is provided by the phrase used to express the failure of the causal mode of description, namely, that one speaks of a free choice on the part of nature. Indeed, properly speaking, such a phrase requires the idea of an external chooser, the existence of which, however, is denied already by the use of the word nature." Petersen adds (p. 172) ".. . Bohr often stressed in discussions that 'reality' is a word in our language and that this word is no different from other words in that we must learn to use it correctly ... ." Petersen continues (p. 173), "Indivisibility and closure are the two principal characteristics of a quan­ tum phenomenon. ... The phenom­ enon's 'interior' is .. . physically inscru­ table. ... In a classical physical process each infinitesimal step is 'closed', i.e. it

1.1 COMMENTARY

is a definite physical event. . .. In quan­ tum physics the object's 'behavior' is not a sequence of 'closed' steps." Here Petersen might have quoted from Bohr (1958, p. 73):"... Every atomic phenom­ enon is closed in the sense that its observation is based on registrations obtained by means of suitable ampli­ fication devices with irreversible func­ tioning such as, for example, permanent marks on a photographic plate, caused by the penetration of electrons into the emulsion." But Petersen adds (p. 177— 79),"... It is clear that Bohr considered the closure of fundamental significance not only in quantum physics but in the whole description of nature. Classical physics did not call attention to the role of this concept because classical pro­ cesses have, so to say, maximal closure. In quantum mechanics the physically describable aspects of a phenomenon are closed, but the phenomenon's physi­ cally inscrutable 'interior' is not. ... [The] question suggests itself as to whether it is possible to dispense with the classical concepts in the quantum domain or at least supplement them with new physical concepts that are less directly tied to the structure of classical theories and more adapted to the typical quantal parts of quantum mechanics. Bohr gave a negative answer to this question. He held that 'it would be a misconception to believe that the diffi­ culties of the atomic theory may be evaded by eventually replacing the concepts of classical physics by new conceptual forms.'" "Bohr was remarkably categorical about the question at issue. 'It lies in the nature of physical observation ...

7

that all experience must ultimately be expressed in terms of classical concepts .. . the unambiguous interpretation of any measurement must be essentially framed in terms of the classical physical theories, and we may say that in this sense the language of Newton and Maxwell will remain the language of physicists for all time.' 'Even when the phenomena transcend the scope of classical physical theories, the account of the experimental arrangement and the recording of observations must be given in plain language, suitably supple­ mented by technical physical terminol­ ogy. This is a clear logical demand, since the very word experiment refers to a situation where we can tell others what we have done and what we have learned.'" COMMENTARY OF HEISENBERG (1967)

The Solvay Conference in Brussels in the autumn of 1927 closed this marvel­ lous period in the history of atomic theory. Planck, Einstein, Lorentz, Bohr, de Broglie, Born, and Schrodinger, and from the younger generation Kramers, Pauli, and Dirac, were gathered here and the discussions were soon focussed to a duel between Einstein and Bohr on the question as to what extent atomic theory in its present form could be considered to be the final solution of the difficulties which had been dis­ cussed for several decades. We gener­ ally met already at breakfast in the hotel, and Einstein began to describe an ideal experiment in which he thought the inner contradictions of the Copen­ hagen interpretation were particularly

8

I.l COMMENTARY

clearly visible. Einstein, Bohr and I walked together from the hotel to the congress building, and I listened to the lively discussion between these two people whose philosophical attitudes were so different, and from time to time I added a remark on the structure of the mathematical formalism. Duringthe meeting and particularly in the pauses we younger people, mostly Pauli and I, tried to analyze Einstein's experiment, and at lunch time the discussions continued between Bohr and the others from Copenhagen. Bohr had usually finished the complete analysis of the ideal experiment late in the afternoon and showed it to Einstein at the supper table. Einstein had no good objection to this analysis, but in his heart he was not convinced. Bohr's friend Ehrenfest, who was also a close friend of Einstein, said to him, "I'm ashamed of you, Einstein. You put yourself here just in the same position as your opponents in their futile attempts to refute your relativity theory." These discussions continued even at the next Solvay meeting in 1930, and it was probably on this occasion that Einstein at break­ fast proposed the famous experiment (discussed in Bohr's article on the occasion of Einstein's 70th birthday) in which the colour of a light quantum is to be determined by weighing the source before and after the quantum's emission. As this problem involved gravity, one had to include the theory of gravity, in other words, general relativity theory in

the analysis. It was a particular triumph for Bohr that he was able to show that evening, by using just Einstein's own formulae from general relativity, that even in this experiment the uncertainty relations are valid, and that Einstein's objections were unfounded. With this the Copenhagen interpretation of quan­ tum theory seemed from now on to stand on solid ground.

COMMENTARY OF EINSTEIN (1936) There is no doubt that quantum me­ chanics has seized hold of a beautiful element of truth, and that it will be a test stone for any future theoretical basis, in that it must be deducible as a limiting case from that basis, just as electrostatics is deducible from the Maxwell equations of the electromag­ netic field or as thermodynamics is deducible from classical mechanics. However, I do not believe that quantum mechanics will be the starting point in the search for this basis, just as, vice versa, one could not go from thermo­ dynamics (resp. statistical mechanics) to the foundations of mechanics.

COMMENTARY OF EINSTEIN (BEFORE 1953) That the Lord should play with dice, all right; but that He should gamble according to definite rules, that is beyond me.

1.1 DISCUSSION WITH EINSTEIN ON EPISTEMOLOGICAL PROBLEMS IN ATOMIC PHYSICS NIELS BOHR

W

HEN invited by the Editor of the series, "Living Philos­ ophers," to write an article for this volume in which contemporary scientists are honouring the epoch-making con­ tributions of Albert Einstein to the progress of natural philos­ ophy and are acknowledging the indebtedness of our whole generation for the guidance his genius has given us, I thought much of the best way of explaining how much I owe to him for inspiration. In this connection, the many occasions through the years on which I had the privilege to discuss with Einstein epistemological problems raised by the modern development of atomic physics have come back vividly to my mind and I have felt that I could hardly attempt anything better than to give an account of these discussions which, even if no complete con­ cord has so far been obtained, have been of greatest value and stimulus to me. I hope also that the account may convey to wider circles an impression of how essential the open-minded exchange of ideas has been for the progress in a field where new experience has time after time demanded a reconsideration of our views. From the very beginning the main point under debate has been the attitude to take to the departure from customary prin­ ciples of natural philosophy characteristic of the novel develop­ ment of physics which was initiated in the first year of this cen­ tury by Planck's discovery of the universal quantum of action. This discovery, which revealed a feature of atomicity in the laws of nature going far beyond the old doctrine of the limited divis­ ibility of matter, has indeed taught us that the classical theories Originally published in Albert Einstein: Philosopher-Scientist, P. A. Schilpp, ed., pp. 200-41, The Library of Living Philosophers, Evanston (1949).

10

BOHR

of physics are idealizations which can be unambiguously applied only in the limit where all actions involved are large compared with the quantum. The question at issue has been whether the renunciation of a causal mode of description of atomic processes involved in the endeavours to cope with the situation should be regarded as a temporary departure from ideals to be ultimately revived or whether we are faced with an irrevocable step to­ wards obtaining the proper harmony between analysis and syn­ thesis of physical phenomena. To describe the background of our discussions and to bring out as clearly as possible the argu­ ments for the contrasting viewpoints, I have felt it necessary to go to a certain length in recalling some main features of the development to which Einstein himself has contributed so decisively. As is well known, it was the intimate relation, elucidated primarily by Boltzmann, between the laws of thermodynamics and the statistical regularities exhibited by mechanical systems with many degrees of freedom, which guided Planck in his in­ genious treatment of the problem of thermal radiation, leading him to his fundamental discovery. While, in his work, Planck was principally concerned with considerations of essentially statistical character and with great caution refrained from de­ finite conclusions as to the extent to which the existence of the quantum implied a departure from the foundations of mechanics and electrodynamics, Einstein's great original contribution to quantum theory (1905) was just the recognition of how physi­ cal phenomena like the photo-effect may depend directly on in­ dividual quantum effects.1 In these very same years when, in developing his theory of relativity, Einstein laid a new founda­ tion for physical science, he explored with a most daring spirit the novel features of atomicity which pointed beyond the whole framework of classical physics. With unfailing intuition Einstein thus was led step by step to the conclusion that any radiation process involves the emis­ sion or absorption of individual light quanta or "photons" with energy and momentum E — hv and P = ha (1) 1A.

Einstein, Ann. d. Phys., /7, 132, (1905).

1.1 DISCUSSIONS WITH EINSTEIN

respectively, where h is Planck's constant, while V and β are the number of vibrations per unit time and the number of waves per unit length, respectively. Notwithstanding its fertility, the idea of the photon implied a quite unforeseen dilemma, since any simple corpuscular picture of radiation would obviously be ir­ reconcilable with interference effects, which present so essential an aspect of radiative phenomena, and which can be described only in terms of a wave picture. The acuteness of the dilemma is stressed by the fact that the interference effects offer our only means of defining the concepts of frequency and wave­ length entering into the very expressions for the energy and momentum of the photon. In this situation, there could be no question of attempting a causal analysis of radiative phenomena, but only, by a combined use of the contrasting pictures, to estimate probabilities for the occurrence of the individual radiation processes. However, it is most important to realize that the recourse to probability laws under such circumstances is essentially different in aim from the familiar application of statistical considerations as practical means of accounting for the properties of mechanical systems of great structural complexity. In fact, in quantum physics we are presented not with intricacies of this kind, but with the inability of the classical frame of concepts to comprise the peculiar fea­ ture of indivisibility, or "individuality," characterizing the ele­ mentary processes. The failure of the theories of classical physics in accounting for atomic phenomena was further accentuated by the progress of our knowledge of the structure of atoms. Above all, Ruther­ ford's discovery of the atomic nucleus (1911) revealed at once the inadequacy of classical mechanical and electromagnetic con­ cepts to explain the inherent stability of the atom. Here again the quantum theory offered a clue for the elucidation of the situation and especially it was found possible to account for the atomic stability, as well as for the empirical laws governing the spectra of the elements, by assuming that any reaction of the atom resulting in a change of its energy involved a complete transition between two so-called stationary quantum states and that, in particular, the spectra were emitted by a step-like pro-

11

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ccss in which each transition is accompanied by the emission of a monochromatic light quantum of an energy just equal to that of an Einstein photon. These ideas, which were soon confirmed by the experiments of Franck and Hertz (1914) on the excitation of spectra by impact of electrons on atoms, involved a further renunciation of the causal mode of description, since evidently the interpreta­ tion of the spectral laws implies that an atom in an excited state in general will have the possibility of transitions with photon emission to one or another of its lower energy states. In fact, the very idea of stationary states is incompatible with any directive for the choice between such transitions and leaves room only for the notion of the relative probabilities of the individual transi­ tion processes. The only guide in estimating such probabilities was the so-called correspondence principle which originated in the search for the closest possible connection between the statisti­ cal account of atomic processes and the consequences to be ex­ pected from classical theory, which should be valid in the limit where the actions involved in all stages of the analysis of the phenomena are large compared with the universal quantum. At that time, no general self-consistent quantum theory was yet in sight, but the prevailing attitude may perhaps be illus­ trated by the following passage from a lecture by the writer from 1913:2 I hope that I have expressed myself sufficiently clearly so that you may appreciate the extent to which these considerations conflict with the ad­ mirably consistent scheme of conceptions which has been rightly termed the classical theory of electrodynamics. On the other hand, I have tried to convey to you the impression that—just by emphasizing so strongly this conflict—it may also be possible in course of time to establish a cer­ tain coherence in the new ideas.

Important progress in the development of quantum theory was made by Einstein himself in his famous article on radiative equilibrium in 1917,3 where he showed that Planck's law for thermal radiation could be simply deduced from assumptions 2 N. Bohr, Fysisk Tidsskrift, 12, 97, (1914). (English version in The Theory of Sfectra and Atomic Constitution, Cambridge, University Press, 1922). 3 A. Einstein, Phys. Zs., 18, 121, (1917).

1.1 DISCUSSIONS WITH EINSTEIN

conforming with the basic ideas of the quantum theory of atomic constitution. To this purpose, Einstein formulated general statistical rules regarding the occurrence of radiative transitions between stationary states, assuming not only that, when the atom is exposed to a radiation field, absorption as well as emis­ sion processes will occur with a probability per unit time pro­ portional to the intensity of the irradiation, but that even in the absence of external disturbances spontaneous emission processes will take place with a rate corresponding to a certain a priori probability. Regarding the latter point, Einstein emphasized the fundamental character of the statistical description in a most suggestive way by drawing attention to the analogy be­ tween the assumptions regarding the occurrence of the spontane­ ous radiative transitions and the well-known laws governing transformations of radioactive substances. In connection with a thorough examination of the exigencies of thermodynamics as regards radiation problems, Einstein stressed the dilemma still further by pointing out that the argu­ mentation implied that any radiation process was "unidirected" in the sense that not only is a momentum corresponding to a photon with the direction of propagation transferred to an atom in the absorption process, but that also the emitting atom will receive an equivalent impulse in the opposite direction, although there can on the wave picture be no question of a preference for a single direction in an emission process. Einstein's own attitude to such startling conclusions is expressed in a passage at the end of the article (Ioc. cit.y p. 127 f.), which may be translated as follows: These features of the elementary processes would seem to make the development of a proper quantum treatment of radiation almost unavoid­ able. The weakness of the theory lies in the fact that, on the one hand, no closer connection with the wave concepts is obtainable and that, on the other hand, it leaves to chance (Zufall) the time and the direction of the elementary processes; nevertheless, I have full confidence in the reliability of the way entered upon.

When I had the great experience of meeting Einstein for the first time during a visit to Berlin in 1920, these fundamental

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questions formed the theme of our conversations. The discus­ sions, to which I have often reverted in my thoughts, added to all my admiration for Einstein a deep impression of his detached attitude. Certainly, his favoured use of such picturesque phrases as "ghost waves (Gespensterfelder) guiding the photons" im­ plied no tendency to mysticism, but illuminated rather a pro­ found humour behind his piercing remarks. Yet, a certain dif­ ference in attitude and outlook remained, since, with his mastery for co-ordinating apparently contrasting experience without abandoning continuity and causality, Einstein was perhaps more reluctant to renounce such ideals than someone for whom re­ nunciation in this respect appeared to be the only way open to proceed with the immediate task of co-ordinating the multifari­ ous evidence regarding atomic phenomena, which accumulated from day to day in the exploration of this new field of knowl­ edge. In the following years, during which the atomic problems at­ tracted the attention of rapidly increasing circles of physicists, the apparent contradictions inherent in quantum theory were felt ever more acutely. Illustrative of this situation is the dis­ cussion raised by the discovery of the Stern-Gerlach effect in 1922. On the one hand, this effect gave striking support to the idea of stationary states and in particular to the quantum theory of the Zeeman effect developed by Sommerfeld 3 on the other hand, as exposed so clearly by Einstein and Ehrenfest,4 it pre­ sented with unsurmountable difficulties any attempt at forming a picture of the behaviour of atoms in a magnetic field. Similar paradoxes were raised by the discovery by Compton (1924) of the change in wave-length accompanying the scattering of X-rays by electrons. This phenomenon afforded, as is well known, a most direct proof of the adequacy of Einstein's view regarding the transfer of energy and momentum in radiative processes; at the same time, it was equally clear that no simple picture of a corpuscular collision could offer an exhaustive description of the phenomenon. Under the impact of such difficulties, doubts 'A. Einstein and P. Ehrenfest, Zs. f. Phys., n, 31, (1922).

15

1.1 DISCUSSIONS WITH EINSTEIN

were for a time entertained even regarding the conservation of energy and momentum in the individual radiation processes;5 a view, however, which very soon had to be abandoned in face of more refined experiments bringing out the correlation be­ tween the deflection of the photon and the corresponding elec­ tron recoil. The way to the clarification of the situation was, indeed, first to be paved by the development of a more comprehensive quantum theory. A first step towards this goal was the recogni­ tion by de Broglie in 1925 that the wave-corpuscle duality was not confined to the properties of radiation, but was equally unavoidable in accounting for the behaviour of material par­ ticles. This idea, which was soon convincingly confirmed by ex­ periments on electron interference phenomena, was at once greeted by Einstein, who had already envisaged the deep-going analogy between the properties of thermal radiation and of gases in the so-called degenerate state.® The new line was pur­ sued with the greatest success by Schrodinger (1926) who, in particular, showed how the stationary states of atomic systems could be represented by the proper solutions of a wave-equation to the establishment of which he was led by the formal analogy, originally traced by Hamilton, between mechanical and optical problems. Still, the paradoxical aspects of quantum theory were in no way ameliorated, but even emphasized, by the apparent contradiction between the exigencies of the general superposi­ tion principle of the wave description and the feature of in­ dividuality of the elementary atomic processes. At the same time, Heisenberg (1925) had laid the foundation of a rational quantum mechanics, which was rapidly developed through important contributions by Born and Jordan as well as by Dirac. In this theory, a formalism is introduced, in which the kinematical and dynamical variables of classical mechanics are replaced by symbols subjected to a non-commutative algebra. Notwithstanding the renunciation of orbital pictures, Hamilton's canonical equations of mechanics are kept unaltered and 5N. Bohr, H. A. Kramers and J. C. Slater, Phil. Mag., 47, ' A . E i n s t e i n , Berl. Ber., ( 1 9 2 4 ) , 2 6 1 , and ( 1 9 2 5 ) , 3 and

78J, (1924). 18.

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Planck's constant enters only in the rules of commutation __ h q p — p q = V — I— (2) 2π holding for any set of conjugate variables q and p. Through a representation of the symbols by matrices with elements re­ ferring to transitions between stationary states, a quantitative formulation of the correspondence principle became for the first time possible. It may here be recalled that an important pre­ liminary step towards this goal was reached through the estab­ lishment, especially by contributions of Kramers, of a quantum theory of dispersion making basic use of Einstein's general rules for the probability of the occurrence of absorption and emission processes. This formalism of quantum mechanics was soon proved by Schrodinger to give results identical with those obtainable by the mathematically often more convenient methods of wave theory, and in the following years general methods were gradually established for an essentially statistical description of atomic processes combining the features of individuality and the requirements of the superposition principle, equally characteristic of quantum theory. Among the many advances in this period, it may especially be mentioned that the formalism proved capable of incorporating the exclusion principle which governs the states of systems with several electrons, and which already before the advent of quantum mechanics had been derived by Pauli from an analysis of atomic spectra. The quantitative comprehension of a vast amount of empirical evidence could leave no doubt as to the fertility and adequacy of the quantum-mechanical formal­ ism, but its abstract character gave rise to a widespread feeling of uneasiness. An elucidation of the situation should, indeed, de­ mand a thorough examination of the very observational prob­ lem in atomic physics. This phase of the development was, as is well known, initiated in 1927 by Heisenberg,7 who pointed out that the knowledge obtainable of the state of an atomic system will al­ ways involve a peculiar "indeterminacy." Thus, any measure­ ment of the position of an electron by means of some device, 7 W.

Heisenberg, Zs.

f.

Phys.,

43, 172,

(1927).

17

1.1 DISCUSSIONS WITH EINSTEIN

like a microscope, making use of high frequency radiation, will, according to the fundamental relations (i), be connected with a momentum exchange between the electron and the measuring agency, which is the greater the more accurate a position meas­ urement is attempted. In comparing such considerations with the exigencies of the quantum-mechanical formalism, Heisenberg called attention to the fact that the commutation rule (2) imposes a reciprocal limitation on the fixation of two conjugate variables, q and φ, expressed by the relation Aq ' Ap ~ h,

(3)

where Aq and Ap are suitably defined latitudes in the deter­ mination of these variables. In pointing to the intimate con­ nection between the statistical description in quantum mechanics and the actual possibilities of measurement, this so-called in­ determinacy relation is, as Heisenberg showed, most important for the elucidation of the paradoxes involved in the attempts of analyzing quantum effects with reference to customary physi­ cal pictures. The new progress in atomic physics was commented upon from various sides at the International Physical Congress held in September 1927, at Como in commemoration of Volta. In a lecture on that occasion,8 I advocated a point of view con­ veniently termed "complementarity," suited to embrace the characteristic features of individuality of quantum phenomena, and at the same time to clarify the peculiar aspects of the ob­ servational problem in this field of experience. For this purpose, it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the ac­ count of all evidence must be expressed in classical terms. The argument is simply that by the word "experiment" we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable ap­ plication of the terminology of classical physics. This crucial point, which was to become a main theme of the 8 Atti del Congresso Internazionale dei Fisici, Como, Settembre 1927 (reprinted in Nature, 121, 78 and 580, 1928).

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discussions reported in the following, implies the impossibility of any sharp separation between the behaviour of atomic ob­ jects and, the interaction with the measuring instruments which serve to define the conditions tmder which the phenomena ap­ pear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of in­ teraction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be compre­ hended within a single picture, but must be regarded as com­ plementary in the sense that only the totality of the phenomena exhausts the possible information about the objects. Under these circumstances an essential element of ambiguity is involved in ascribing conventional physical attributes to atomic objects, as is at once evident in the dilemma regarding the corpuscular and wave properties of electrons and photons, where we have to do with contrasting pictures, each referring to an essential aspect of empirical evidence. An illustrative ex­ ample, of how the apparent paradoxes are removed by an ex­ amination of the experimental conditions under which the com­ plementary phenomena appear, is also given by the Compton effect, the consistent description of which at first had presented us with such acute difficulties. Thus, any arrangement suited to study the exchange of energy and momentum between the electron and the photon must involve a latitude in the spacetime description of the interaction sufficient for the definition of wave-number and frequency which enter into the relation (ι). Conversely, any attempt of locating the collision between the photon and the electron more accurately would, on account of the unavoidable interaction with the fixed scales and clocks de­ fining the space-time reference frame, exclude all closer account as regards the balance of momentum and energy. As stressed in the lecture, an adequate tool for a complement­ ary way of description is offered precisely by the quantummechanical formalism which represents a purely symbolic scheme permitting only predictions, on lines of the correspond­ ence principle, as to results obtainable under conditions specified

1.1 DISCUSSIONS WITH EINSTEIN

by means of classical concepts. It must here be remembered that even in the indeterminacy relation (3) we are dealing with an implication of the formalism which defies unambiguous expres­ sion in words suited to describe classical physical pictures. Thus, a sentence like "we cannot know both the momentum and the position of an atomic object" raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the conditions for the un­ ambiguous use of space-time concepts, on the one hand, and dynamical conservation laws, on the other hand. While the com­ bination of these concepts into a single picture of a causal chain of events is the essence of classical mechanics, room for regulari­ ties beyond the grasp of such a description is just afforded by the circumstance that the study of the complementary phenome­ na demands mutually exclusive experimental arrangements. The necessity, in atomic physics, of a renewed examination of the foundation for the unambiguous use of elementary physi­ cal ideas recalls in some way the situation that led Einstein to his original revision on the basis of all application of space-time concepts which, by its emphasis on the primordial importance of the observational problem, has lent such unity to our world picture. Notwithstanding all novelty of approach, causal de­ scription is upheld in relativity theory within any given frame of reference, but in quantum theory the uncontrollable inter­ action between the objects and the measuring instruments forces us to a renunciation even in such respect. This recognition, how­ ever, in no way points to any limitation of the scope of the quantum-mechanical description, and the trend of the whole argumentation presented in the Como lecture was to show that the viewpoint of complementarity may be regarded as a ra­ tional generalization of the very ideal of causality.

At the general discussion in Como, we all missed the pre­ sence of Einstein, but soon after, in October 1927, I had the opportunity to meet him in Brussels at the Fifth Physical Con­ ference of the Solvay Institute, which was devoted to the theme "Electrons and Photons." At the Solvay meetings, Einstein had from their beginning been a most prominent figure, and several

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of us came to the conference with great anticipations to learn his reaction to the latest stage of the development which, to our view, went far in clarifying the problems which he had himself from the outset elicited so ingeniously. During the discussions, where the whole subject was reviewed by contributions from many sides and where also the arguments mentioned in the preceding pages were again presented, Einstein expressed, how­ ever, a deep concern over the extent to which causal account in space and time was abandoned in quantum mechanics. To illustrate his attitude, Einstein referred at one of the ses­ sions® to the simple example, illustrated by Fig. i, of a particle (electron or photon) penetrating through a hole or a narrow slit in a diaphragm placed at some distance before a photo­ graphic plate. On account of the diffraction of the wave con-

Fic. ι

nected with the motion of the particle and indicated in the figure by the thin lines, it is under such conditions not possible to predict with certainty at what point the electron will arrive at the photographic plate, but only to calculate the probability that, in an experiment, the electron will be found within any given region of the plate. The apparent difficulty, in this de­ scription, which Einstein felt so acutely, is the fact that, if in the experiment the electron is recorded at one point A of the plate, ' Institut International de Physique Solvay, Raffort et discussions du J e Conseil, Paris 1928, 253ff.

1.1 DISCUSSIONS WITH EINSTEIN

then it is out of the question of ever observing an effect of this electron at another point (B), although the laws of ordinary wave propagation offer no room for a correlation between two such events. Einstein's attitude gave rise to ardent discussions within a small circle, in which Ehrenfest, who through the years had been a close friend of us both, took part in a most active and helpful way. Surely, we all recognized that, in the above ex­ ample, the situation presents no analogue to the application of statistics in dealing with complicated mechanical systems, but rather recalled the background for Einstein's own early con­ clusions about the unidirection of individual radiation effects which contrasts so strongly with a simple wave picture (cf. p. 205). The discussions, however, centered on the question of whether the quantum-mechanical description exhausted the pos­ sibilities of accounting for observable phenomena or, as Einstein maintained, the analysis could be carried further and, especially, of whether a fuller description of the phenomena could be ob­ tained by bringing into consideration the detailed balance of energy and momentum in individual processes. To explain the trend of Einstein's arguments, it may be il­ lustrative here to consider some simple features of the mo­ mentum and energy balance in connection with the location of a particle in space and time. For this purpose, we shall examine the simple case of a particle penetrating through a hole in a diaphragm without or with a shutter to open and close the hole, as indicated in Figs. 2a and 2b, respectively. The equidistant parallel lines to the left in the figures indicate the train of plane waves corresponding to the state of motion of a particle which, before reaching the diaphragm, has a momentum P related to the wave-number σ by the second of equations (1). In accord­ ance with the diffraction of the waves when passing through the hole, the state of motion of the particle to the right of the diaphragm is represented by a spherical wave train with a suit­ ably defined angular aperture θ and, in case of Fig. 2b, also with a limited radial extension. Consequently, the description of this state involves a certain latitude A-p in the momentum component of the particle parallel to the diaphragm and, in the case of a

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diaphragm with a shutter, an additional latitude AE of the kinetic energy. Since a measure for the latitude Aq in location of the par­ ticle in the plane of the diaphragm is given by the radius a of the hole, and since θ~ (i/oa), we get, using (i), just Af ~ $P ~ (h/Aq), in accordance with the indeterminacy rela­ tion (3). This result could, of course, also be obtained directly by noticing that, due to the limited extension of the wave-field at the place of the slit, the component of the wave-number parallel to the plane of the diaphragm will involve a latitude Δσ ~ (1 /a) ~ {1/Aq). Similarly, the spread of the frequencies

Fic.

FIG.

2a

2B

of the harmonic components in the limited wave-train in Fig. 2b is evidently Av ~ (ι/At), where At is the time interval during which the shutter leaves the hole open and, thus, represents the latitude in time of the passage of the particle through the dia­ phragm. From (1), we therefore get AE'At ~ h}

(4)

again in accordance with the relation (3) for the two conjugated variables E a n d t . From the point of view of the laws of conservation, the origin of such latitudes entering into the description of the state of the particle after passing through the hole may be traced to the pos­ sibilities of momentum and energy exchange with the diaphragm

1.1 DISCUSSIONS WITH EINSTEIN

or the shutter. In the reference system considered in Figs. 2a and 2b, the velocity of the diaphragm may be disregarded and only a change of momentum Δρ between the particle and the dia­ phragm needs to be taken into consideration. The shutter, how­ ever, which leaves the hole opened during the time Δ/, moves with a considerable velocity ν ~ («/Δ/), and a momentum transfer Δρ involves therefore an energy exchange with the par­ ticle, amounting to νΔρ ~ (ι/Δί) Δς Δρ ~ (Α/Δ/), being just of the same order of magnitude as the latitude ΔΕ given by (4) and, thus, allowing for momentum and energy balance. The problem raised by Einstein was now to what extent a control of the momentum and energy transfer, involved in a location of the particle in space and time, can be used for a further specification of the state of the particle after passing through the hole. Here, it must be taken into consideration that the position and the motion of the diaphragm and the shutter have so far been assumed to be accurately co-ordinated with the space-time reference frame. This assumption implies, in the description of the state of these bodies, an essential latitude as to their momentum and energy which need not, of course, noticeably affect the velocities, if the diaphragm and the shutter are sufficiently heavy. However, as soon as we want to know the momentum and energy of these parts of the measuring ar­ rangement with an accuracy sufficient to control the momentum and energy exchange with the particle under investigation, we shall, in accordance with the general indeterminacy relations, lose the possibility of their accurate location in space and time. We have, therefore, to examine how far this circumstance will affect the intended use of the whole arrangement and, as we shall see, this crucial point clearly brings out the complementary character of the phenomena. Returning for a moment to the case of the simple arrange­ ment indicated in Fig. 1, it has so far not been specified to what use it is intended. In fact, it is only on the assumption that the diaphragm and the plate have well-defined positions in space that it is impossible, within the frame of the quantum-mechani­ cal formalism, to make more detailed predictions as to the point

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of the photographic plate where the particle will be recorded. If, however, we admit a sufficiently large latitude in the knowl­ edge of the position of the diaphragm it should, in principle, be possible to control the momentum transfer to the diaphragm and, thus, to make more detailed predictions as to the direction of the electron path from the hole to the recording poirt. As regards the quantum-mechanical description, we have to deal here with a two-body system consisting of the diaphragm as well as of the particle, and it is just with an explicit application of conservation laws to such a system that we are concerned in the Compton effect where, for instance, the observation of the recoil of the electron by means of a cloud chamber allows us to predict in what direction the scattered photon will eventually be observed. The importance of considerations of this kind was, in the course of the discussions, most interestingly illuminated by the examination of an arrangement where between the diaphragm with the slit and the photographic plate is inserted another

& " I

FIG. 3

diaphragm with two parallel slits, as is shown in Fig. 3. If a parallel beam of electrons (or photons) falls from the left on the first diaphragm, we shall, under usual conditions, observe on the plate an interference pattern indicated by the shading of the photographic plate shown in front view to the right of the figure. With intense beams, this pattern is built up by the ac­ cumulation of a large number of individual processes, each giving rise to a small spot on the photographic plate, and the distribution of these spots follows a simple law derivable from

1.1 DISCUSSIONS WITH EINSTEIN

the wave analysis. The same distribution should also be found in the statistical account of many experiments performed with beams so faint that in a single exposure only one electron (or photon) will arrive at the photographic plate at some spot shown in the figure as a small star. Since, now, as indicated by the broken arrows, the momentum transferred to the first diaphragm ought to be different if the electron was assumed to pass through the upper or the lower slit in the second dia­ phragm, Einstein suggested that a control of the momentum transfer would permit a closer analysis of the phenomenon and, in particular, to decide through which of the two slits the elec­ tron had passed before arriving at the plate. A closer examination showed, however, that the suggested control of the momentum transfer would involve a latitude in the knowledge of the position of the diaphragm which would exclude the appearance of the interference phenomena in ques­ tion. In fact, if ω is the small angle between the conjectured paths of a particle passing through the upper or the lower slit, the difference of momentum transfer in these two cases will, ac­ cording to (i), be equal to Λσω and any control of the mo­ mentum of the diaphragm with an accuracy sufficient to measure this difference will, due to the indeterminacy relation, involve a minimum latitude of the position of the diaphragm, comparable with ι/σω. If, as in the figure, the diaphragm with the two slits is placed in the middle between the first diaphragm and the photographic plate, it will be seen that the number of fringes per unit length will be just equal to σω and, since an uncertainty in the position of the first diaphragm of the amount of ι/σω will cause an equal uncertainty in the positions of the fringes, it follows that no interference effect can appear. The same re­ sult is easily shown to hold for any other placing of the second diaphragm between the first diaphragm and the plate, and would also be obtained if, instead of the first diaphragm, an­ other of these three bodies were used for the control, for the purpose suggested, of the momentum transfer. This point is of great logical consequence, since it is only the circumstance that we are presented with a choice of either trac­ ing the path of a particle or observing interference effects, which

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allows us to escape from the paradoxical necessity of concluding that the behaviour of an electron or a photon should depend on the presence of a slit in the diaphragm through which it could be proved not to pass. We have here to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements (cf. p. 210) and are just faced with the impossibility, in the analysis of quantum effects, of drawing any sharp separation between an independent behaviour of atomic objects and their interaction with the meas­ uring instruments which serve to define the conditions under which the phenomena occur. Our talks about the attitude to be taken in face of a novel situation as regards analysis and synthesis of experience touched naturally on many aspects of philosophical thinking, but, in spite of all divergencies of approach and opinion, a most humor­ ous spirit animated the discussions. On his side, Einstein mock­ ingly asked us whether we could really believe that the pro­ vidential authorities took recourse to dice-playing (". . . ob der liebe Gott wiirfelt"), to which I replied by pointing at the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in every-day language. I remember also how at the peak of the discussion Ehrenfest, in his affectionate manner of teasing his friends, jokingly hinted at the apparent similarity between Einstein's attitude and that of the opponents of relativity theory; but instantly Ehrenfest added that he would not be able to find relief in his own mind before concord with Einstein was reached. Einstein's concern and criticism provided a most valuable in­ centive for us all to reexamine the various aspects of the situa­ tion as regards the description of atomic phenomena. To me it was a welcome stimulus to clarify still further the role played by the measuring instruments and, in order to bring into strong relief the mutually exclusive character of the experimental con­ ditions under which the complementary phenomena appear, I tried in those days to sketch various apparatus in a pseudorealistic style of which the following figures are examples. Thus, for the study of an interference phenomenon of the type

1.1 DISCUSSIONS WITH EINSTEIN

indicated in Fig. 3, it suggests itself to use an experimental ar­ rangement like that shown in Fig. 4, where the solid parts of the apparatus, serving as diaphragms and plate-holder, are

FIG. 4

firmly bolted to a common support. In such an arrangement, where the knowledge of the relative positions of the diaphragms and the photographic plate is secured by a rigid connection, it is obviously impossible to control the momentum exchanged be­ tween the particle and the separate parts of the apparatus. The only way in which, in such an arrangement, we could insure that the particle passed through one of the slits in the second diaphragm is to cover the other slit by a lid, as indicated in the figure; but if the slit is covered, there is of course no question of any interference phenomenon, and on the plate we shall simply observe a continuous distribution as in the case of the single fixed diaphragm in Fig. 1. In the study of phenomena in the account of which we are dealing with detailed momentum balance, certain parts of the whole device must naturally be given the freedom to move independently of others. Such an apparatus is sketched in Fig. 5, where a diaphragm with a slit is suspended by weak springs from a solid yoke bolted to the support on which also other immobile parts of the arrangement are to be fastened. The scale on the diaphragm together with the pointer on the bearings of

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the yoke refer to such study of the motion of the diaphragm, as may be required for an estimate of the momentum transferred to it, permitting one to draw conclusions as to the deflection suffered by the particle in passing through the slit. Since, how­ ever, any reading of the scale, in whatever way performed, will

FIG. S

involve an uncontrollable change in the momentum of the diaphragm, there will always be, in conformity with the in­ determinacy principle, a reciprocal relationship between our knowledge of the position of the slit and the accuracy of the momentum control. In the same semi-serious style, Fig. 6 represents a part of an arrangement suited for the study of phenomena which, in con­ trast to those just discussed, involve time co-ordination ex­ plicitly. It consists of a shutter rigidly connected with a robust clock resting on the support which carries a diaphragm and on which further parts of similar character, regulated by the same clock-work or by other clocks standardized relatively to it, are also to be fixed. The special aim of the figure is to underline that a clock is a piece of machinery, the working of which can completely be accounted for by ordinary mechanics and will be

1.1 DISCUSSIONS WITH EINSTEIN

affected neither by reading of the position of its hands nor by the interaction between its accessories and an atomic particle. In securing the opening of the hole at a definite moment, an ap­ paratus of this type might, for instance, be used for an accurate measurement of the time an electron or a photon takes to come from the diaphragm to some other place, but evidently, it would leave no possibility of controlling the energy transfer to

FIG. 6

the shutter with the aim of drawing conclusions as to the energy of the particle which has passed through the diaphragm. If we are interested in such conclusions we must, of course, use an arrangement where the shutter devices can no longer serve as accurate clocks, but where the knowledge of the moment when the hole in the diaphragm is open involves a latitude connected with the accuracy of the energy measurement by the general relation (4). The contemplation of such more or less practical arrange­ ments and their more or less fictitious use proved most instruc­ tive in directing attention to essential features of the problems. The main point here is the distinction between the objects under investigation and the measuring instruments which serve to de­ fine, in classical terms, the conditions under which the

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phenomena appear. Incidentally, we may remark that, for the illustration of the preceding considerations, it is not relevant that experiments involving an accurate control of the mo­ mentum or energy transfer from atomic particles to heavy bodies like diaphragms and shutters would be very difficult to perform, if practicable at all. It is only decisive that, in contrast to the proper measuring instruments, these bodies together with the particles would in such a case constitute the system to which the quantum-mechanical formalism has to be applied. As re­ gards the specification of the conditions for any well-defined application of the formalism, it is moreover essential that the whole experimental arrangement be taken into account. In fact, the introduction of any further piece of apparatus, like a mirror, in the way of a particle might imply new interference effects essentially influencing the predictions as regards the results to be eventually recorded. The extent to which renunciation of the visualization of atomic phenomena is imposed upon us by the impossibility of their subdivision is strikingly illustrated by the following ex­ ample to which Einstein very early called attention and often has reverted. If a semi-reflecting mirror is placed in the way of a photon, leaving two possibilities for its direction of propaga­ tion, the photon may either be recorded on one, and only one, of two photographic plates situated at great distances in the two directions in question, or else we may, by replacing the plates by mirrors, observe effects exhibiting an interference between the two reflected wave-trains. In any attempt of a pictorial representation of the behaviour of the photon we would, thus, meet with the difficulty: to be obliged to say, on the one hand, that the photon always chooses one of the two ways and, on the other hand, that it behaves as if it had passed both ways. It is just arguments of this kind which recall the impossibility of subdividing quantum phenomena and reveal the ambiguity in ascribing customary physical attributes to atomic objects. In particular, it must be realized that—besides in the account of the placing and timing of the instruments forming the experi­ mental arrangement—all unambiguous use of space-time con­ cepts in the description of atomic phenomena is confined to the

1.1 DISCUSSIONS WITH EINSTEIN

recording of observations which refer to marks on a photo­ graphic plate or to similar practically irreversible amplification effects like the building of a water drop around an ion in a cloud-chamber. Although, of course, the existence of the quantum of action is ultimately responsible for the properties of the materials of which the measuring instruments are built and on which the functioning of the recording devices depends, this circumstance is not relevant for the problems of the ade­ quacy and completeness of the quantum-mechanical description in its aspects here discussed. These problems were instructively commented upon from different sides at the Solvay meeting,10 in the same session where Einstein raised his general objections. On that occasion an interesting discussion arose also about how to speak of the appearance of phenomena for which only predictions of statisti­ cal character can be made. The question was whether, as to the occurrence of individual effects, we should adopt a terminology proposed by Dirac, that we were concerned with a choice on the part of "nature" or, as suggested by Heisenberg, we should say that we have to do with a choice on the part of the "ob­ server" constructing the measuring instruments and reading their recording. Any such terminology would, however, appear dubious since, on the one hand, it is hardly reasonable to endow nature with volition in the ordinary sense, while, on the other hand, it is certainly not possible for the observer to influence the events which may appear under the conditions he has arranged. To my mind, there is no other alternative than to admit that, in this field of experience, we are dealing with individual phe­ nomena and that our possibilities of handling the measuring in­ struments allow us only to make a choice between the different complementary types of phenomena we want to study. The epistemological problems touched upon here were more explicitly dealt with in my contribution to the issue of Naturwissenschaften in celebration of Planck's 70th birthday in 1929· In this article, a comparison was also made between the lesson derived from the discovery of the universal quantum of action w Ibid.,

248S.

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and the development which has followed the discovery of the finite velocity of light and which, through Einstein's pioneer work, has so greatly clarified basic principles of natural philos­ ophy. In relativity theory, the emphasis on the dependence of all phenomena on the reference frame opened quite new ways of tracing general physical laws of unparalleled scope. In quan­ tum theory, it was argued, the logical comprehension of hitherto unsuspected fundamental regularities governing atomic phe­ nomena has demanded the recognition that no sharp separation can be made between an independent behaviour of the objects and their interaction with the measuring instruments which de­ fine the reference frame. In this respect, quantum theory presents us with a novel situation in physical science, but attention was called to the very close analogy with the situation as regards analysis and syn­ thesis of experience, which we meet in many other fields of human knowledge and interest. As is well known, many of the difficulties in psychology originate in the different placing of the separation lines between object and subject in the analysis of various aspects of psychical experience. Actually, words like "thoughts" and "sentiments," equally indispensable to illus­ trate the variety and scope of conscious life, are used in a simi­ lar complementary way as are space-time co-ordination and dynamical conservation laws in atomic physics. A precise for­ mulation of such analogies involves, of course, intricacies of terminology, and the writer's position is perhaps best indicated in a passage in the article, hinting at the mutually exclusive relationship which will always exist between the practical use of any word and attempts at its strict definition. The principal aim, however, of these considerations, which were not least in­ spired by the hope of influencing Einstein's attitude, was to point to perspectives of bringing general epistemological prob­ lems into relief by means of a lesson derived from the study of new, but fundamentally simple physical experience. At the next meeting with Einstein at the Solvay Conference in 1930, our discussions took quite a dramatic turn. As an ob­ jection to the view that a control of the interchange of momen-

33

1.1 DISCUSSIONS WITH EINSTEIN

turn and energy between the objects and the measuring in­ struments was excluded if these instruments should serve their purpose of defining the space-time frame of the phenomena, Einstein brought forward the argument that such control should be possible when the exigencies of relativity theory were taken into consideration. In particular, the general relationship be­ tween energy and mass, expressed in Einstein's famous formula

E ----- mc2

(5)

should allow, by means of simple weighing, to measure the total energy of any system and, thus, in principle to control the energy transferred to it when it interacts with an atomic object. As an arrangement suited for such purpose, Einstein pro­ posed the device indicated in Fig. 7, consisting of a box with

FIG. 7

a hole in its side, which could be opened or closed by a shutter moved by means of a clock-work within the box. If, in the be­ ginning, the box contained a certain amount of radiation and the clock was set to open the shutter for a very short interval at a chosen time, it could be achieved that a single photon was released through the hole at a moment known with as great accuracy as desired. Moreover, it would apparently also be possible, by weighing the whole box before and after this event, to measure the energy of the photon with any accuracy wanted,

34

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in definite contradiction to the reciprocal indeterminacy of time and energy quantities in quantum mechanics. This argument amounted to a serious challenge and gave rise to a thorough examination of the whole problem. At the out­ come of the discussion, to which Einstein himself contributed effectively, it became clear, however, that this argument could not be upheld. In fact, in the consideration of the problem, it was found necessary to look closer into the consequences of the identification of inertial and gravitational mass implied in the application of relation (5). Especially, it was essential to take into account the relationship between the rate of a clock and its position in a gravitational field—well known from the red-shift of the lines in the sun's spectrum—following from Einstein's principle of equivalence between gravity effects and the phenomena observed in accelerated reference frames. Our discussion concentrated on the possible application of an apparatus incorporating Einstein's device and drawn in Fig. 8 in the same pseudo-realistic style as some of the preceding figures. The box, of which a section is shown in order to ex­ hibit its interior, is suspended in a spring-balance and is fur­ nished with a pointer to read its position on a scale fixed to the balance support. The weighing of the box may thus be per­ formed with any given accuracy Am by adjusting the balance to its zero position by means of suitable loads. The essential point is now that any determination of this position with a given accuracy Aq will involve a minimum latitude Ap in the control of the momentum of the box connected with Aq by the rela­ tion (3). This latitude must obviously again be smaller than the total impulse which, during the whole interval T of the balancing procedure, can be given by the gravitational field to a body with a mass Am, or Ap ~

h

c2hs.m

Together with the formula (5), this relation again leads to ΔΓ·Δ£ > h, in accordance with the indeterminacy principle. Consequently, a use of the apparatus as a means of accurately measuring the energy of the photon will prevent us from controlling the moment of its escape. The discussion, so illustrative of the power and consistency of relativistic arguments, thus emphasized once more the neces­ sity of distinguishing, in the study of atomic phenomena, be­ tween the proper measuring instruments which serve to define the reference frame and those parts which are to be regarded as objects under investigation and in the account of which quantum effects cannot be disregarded. Notwithstanding the most sug­ gestive confirmation of the soundness and wide scope of the quantum-mechanical way of description, Einstein nevertheless, in a following conversation with me, expressed a feeling of dis­ quietude as regards the apparent lack of firmly laid down prin­ ciples for the explanation of nature, in which all could agree. From my viewpoint, however, I could only answer that, in dealing with the task of bringing order into an entirely new field of experience, we could hardly trust in any accustomed principles, however broad, apart from the demand of avoiding logical inconsistencies and, in this respect, the mathematical formalism of quantum mechanics should surely meet all re­ quirements. The Solvay meeting in 1930 was the last occasion where, in common discussions with Einstein, we could benefit from the stimulating and mediating influence of Ehrenfest, but shortly before his deeply deplored death in 1933 he told me that Einstein was far from satisfied and with his usual acuteness had discerned new aspects of the situation which strengthened his critical attitude. In fact, by further examining the possibili­ ties for the application of a balance arrangement, Einstein had perceived alternative procedures which, even if they did not allow the use he originally intended, might seem to enhance

1.1 DISCUSSIONS WITH EINSTEIN

the paradoxes beyond the possibilities of logical solution. Thus, Einstein had pointed out that, after a preliminary weighing of the box with the clock and the subsequent escape of the photon, one was still left with the choice of either repeating the weigh­ ing or opening the box and comparing the reading of the clock with the standard time scale. Consequently, we are at this stage still free to choose whether we want to draw conclusions either about the energy of the photon or about the moment when it left the box. Without in any way interfering with the photon between its escape and its later interaction with other suitable measuring instruments, we are, thus, able to make accurate pre­ dictions pertaining either to the moment of its arrival or to the amount of energy liberated by its absorption. Since, however, according to the quantum-mechanical formalism, the specifica­ tion of the state of an isolated particle cannot involve both a well-defined connection with the time scale and an accurate fixation of the energy, it might thus appear as if this formalism did not offer the means of an adequate description. Once more Einstein's searching spirit had elicited a peculiar aspect of the situation in quantum theory, which in a most strik­ ing manner illustrated how far we have here transcended cus­ tomary explanation of natural phenomena. Still, I could not agree with the trend of his remarks as reported by Ehrenfest. In my opinion, there could be no other way to deem a logically consistent mathematical formalism as inadequate than by dem­ onstrating the departure of its consequences from experience or by proving that its predictions did not exhaust the possibilities of observation, and Einstein's argumentation could be directed to neither of these ends. In fact, we must realize that in the problem in question we are not dealing with a single specified experimental arrangement, but are referring to two different, mutually exclusive arrangements. In the one, the balance to­ gether with another piece of apparatus like a spectrometer is used for the study of the energy transfer by a photon; in the other, a shutter regulated by a standardized clock together with another apparatus of similar kind, accurately timed relatively to the clock, is used for the study of the time of propagation of a photon over a given distance. In both these cases, as also as-

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sumed by Einstein, the observable effects are expected to be in complete conformity with the predictions of the theory. The problem again emphasizes the necessity of considering the whole experimental arrangement, the specification of which is imperative for any well-defined application of the quantummechanical formalism. Incidentally, it may be added that para­ doxes of the kind contemplated by Einstein are encountered also in such simple arrangements as sketched in Fig. 5. In fact, after a preliminary measurement of the momentum of the dia­ phragm, we are in principle offered the choice, when an elec­ tron or photon has passed through the slit, either to repeat the momentum measurement or to control the position of the dia­ phragm and, thus, to make predictions pertaining to alternative subsequent observations. It may also be added that it obviously can make no difference as regards observable effects obtainable by a definite experimental arrangement, whether our plans of constructing or handling the instruments are fixed beforehand or whether we prefer to postpone the completion of our plan­ ning until a later moment when the particle is already on its way from one instrument to another. In the quantum-mechanical description our freedom of con­ structing and handling the experimental arrangement finds its proper expression in the possibility of choosing the classically defined parameters entering in any proper application of the formalism. Indeed, in all such respects quantum mechanics ex­ hibits a correspondence with the state of affairs familiar from classical physics, which is as close as possible when considering the individuality inherent in the quantum phenomena. Just in helping to bring out this point so clearly, Einstein's concern had therefore again been a most welcome incitement to explore the essential aspects of the situation. The next Solvay meeting in 1933 was devoted to the prob­ lems of the structure and properties of atomic nuclei, in which field such great advances were made just in that period due to the experimental discoveries as well as to new fruitful applica­ tions of quantum mechanics. It need in this connection hardly be recalled that just the evidence obtained by the study of arti-

1.1 DISCUSSIONS WITH EINSTEIN

ficial nuclear transformations gave a most direct test of Ein­ stein's fundamental law regarding the equivalence of mass and energy, which was to prove an evermore important guide for re­ searches in nuclear physics. It may also be mentioned how Ein­ stein's intuitive recognition of the intimate relationship between the law of radioactive transformations and the probability rules governing individual radiation effects (cf. p. 205) was confirmed by the quantum-mechanical explanation of spontaneous nuclear disintegrations. In fact, we are here dealing with a typical ex­ ample of the statistical mode of description, and the comple­ mentary relationship between energy-momentum conservation and time-space co-ordination is most strikingly exhibited in the well-known paradox of particle penetration through potential barriers. Einstein himself did not attend this meeting, which took place at a time darkened by the tragic developments in the political world which were to influence his fate so deeply and add so greatly to his burdens in the service of humanity. A few months earlier, on a visit to Princeton where Einstein was then guest of the newly founded Institute for Advanced Study to which he soon after became permanently attached, I had, how­ ever, opportunity to talk with him again about the epistemological aspects of atomic physics, but the difference between our ways of approach and expression still presented obstacles to mutual understanding. While, so far, relatively few persons had taken part in the discussions reported in this article, Ein­ stein's critical attitude towards the views on quantum theory adhered to by many physicists was soon after brought to public attention through a paper11 with the title "Can Quantum-Me­ chanical Description of Physical Reality Be Considered Com­ plete?," published in 1935 by Einstein, Podolsky and Rosen. The argumentation in this paper is based on a criterion which the authors express in the following sentence: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quan­ tity, then there exists an element of physical reality correspond" A. Einstein, B. Podolsky and N. Rosen, Phys. Reu., 47, 777, (1935).

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ing to this physical quantity." By an elegant exposition of the consequences of the quantum-mechanical formalism as regards the representation of a state of a system, consisting of two parts which have been in interaction for a limited time interval, it is next shown that different quantities, the fixation of which can­ not be combined in the representation of one of the partial sys­ tems, can nevertheless be predicted by measurements pertaining to the other partial system. According to their criterion, the authors therefore conclude that quantum mechanics does not "provide a complete description of the physical reality," and they express their belief that it should be possible to develop a more adequate account of the phenomena. Due to the lucidity and apparently incontestable character of the argument, the paper of Einstein, Podolsky and Rosen cre­ ated a stir among physicists and has played a large role in gen­ eral philosophical discussion. Certainly the issue is of a very subtle character and suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization. It will be seen, however, that we are here dealing with problems of just the same kind as those raised by Einstein in previous discussions, and, in an article which appeared a few months later,12 I tried to show that from the point of view of comple­ mentarity the apparent inconsistencies were completely re­ moved. The trend of the argumentation was in substance the same as that exposed in the .foregoing pages, but the aim of re­ calling the way in which the situation was discussed at that time may be an apology for citing certain passages from my article. Thus, after referring to the conclusions derived by Einstein, Podolsky and Rosen on the basis of their criterion, I wrote: Such an argumentation, however, would hardly seem suited to affect the soundness of quantum-mechanical description, which is based on a co­ herent mathematical formalism covering automatically any procedure of measurement like that indicated. The apparent contradiction in fact discloses only an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics. Indeed the finite interaction between object and measuring agencies conditioned 12 N.

Bohr, Phys. Rev., 48, 696, (1935).

1.1 DISCUSSIONS WITH EINSTEIN

by the very existence of the quantum of action entails—because of the impossibility of controlling the reaction of the object on the measuring instruments, if these are to serve their purpose—the necessity of a final re­ nunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality. In fact, as we shall see, a criterion of reality like that proposed by the named authors contains— however cautious its formulation may appear—an essential ambiguity when it is applied to the actual problems with which we are here con­ cerned.

As regards the special problem treated by Einstein, Podolsky and Rosen, it was next shown that the consequences of the for­ malism as regards the representation of the state of a system consisting of two interacting atomic objects correspond to the simple arguments mentioned in the preceding in connection with the discussion of the experimental arrangements suited for the study of complementary phenomena. In fact, although any pair q and p, of conjugate space and momentum variables obeys the rule of non-commutative multiplication expressed by (2), and can thus only be fixed with reciprocal latitudes given by (3), the difference qx —• q2 between two space-co-ordinates re­ ferring to the constituents of the system will commute with the sum ρλ -j- p2 of the corresponding momentum components, as follows directly from the commutability of q± with pi and q-i with pi. Both qx — q2 and px -(- p2 can, therefore, be accurately fixed in a state of the complex system and, consequently, we can predict the values of either qx or p\ if either q% or />2, respec­ tively, are determined by direct measurements. If, for the two parts of the system, we take a particle and a diaphragm, like that sketched in Fig. 5, we see that the possibilities of specifying the state of the particle by measurements on the diaphragm just correspond to the situation described on p. 220 and further discussed on p. 230, where it was mentioned that, after the particle has passed through the diaphragm, we have in princi­ ple the choice of measuring either the position of the diaphragm or its momentum and, in each case, to make predictions as to subsequent observations pertaining to the particle. As repeatedly stressed, the principal point is here that such measurements de­ mand mutually exclusive experimental arrangements.

41

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The argumentation of the article was summarized in the following passage: From our point of view we now see that the wording of the abovementioned criterion of physical reality proposed by Einstein, Podolsky, and Rosen contains an ambiguity as regards the meaning of the expres­ sion 'without in any way disturbing a system.' Of course there is in a case like that just considered no question of a mechanical disturbance of the system under investigation during the last critical stage of the measur­ ing procedure. But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behaviour of the system. Since these conditions constitute an inherent element of the description of any phe­ nomenon to which the term "physical reality" can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantum-mechanical description is essentially incom­ plete. On the contrary, this description, as appears from the preceding discussion, may be characterized as a rational utilization of all pos­ sibilities of unambiguous interpretation of measurements, compatible with the finite and uncontrollable interaction between the objects and the measuring instruments in the field of quantum theory. In fact, it is only the mutual exclusion of any two experimental procedures, permitting the unambiguous definition of complementary physical quantities, which provides room for new physical laws, the coexistence of which might at first sight appear irreconcilable with the basic principles of science. It is just this entirely new situation as regards the description of physical phenomena that the notion of complementarity aims at characterizing.

Rereading these passages, I am deeply aware of the ineffi­ ciency of expression which must have made it very difficult to appreciate the trend of the argumentation aiming to bring out the essential ambiguity involved in a reference to physical at­ tributes of objects when dealing with phenomena where no sharp distinction can be made between the behaviour of the objects themselves and their interaction with the measuring in­ struments. I hope, however, that the present account of the dis­ cussions with Einstein in the foregoing years, which contributed so greatly to make us familiar with the situation in quantum physics, may give a clearer impression of the necessity of a radical revision of basic principles for physical explanation in order to restore logical order in this field of experience.

1.1 DISCUSSIONS WITH EINSTEIN

Einstein's own views at that time are presented in an article "Physics and Reality," published in 1936 in the Journal of the Franklin Institute.™." Starting from a most illuminating exposi­ tion of the gradual development of the fundamental principles in the theories of classical physics and their relation to the problem of physical reality, Einstein here argues that the quan­ tum-mechanical description is to be considered merely as a means of accounting for the average behaviour of a large num­ ber of atomic systems and his attitude to the belief that it should offer an exhaustive description of the individual phe­ nomena is expressed in the following words: "To believe this is logically possible without contradiction; but it is so very con­ trary to my scientific instinct that I cannot forego the search for a more complete conception." Even if such an attitude might seem well-balanced in itself, it nevertheless implies a rejection of the whole argumentation exposed in the preceding, aiming to show that, in quantum me­ chanics, we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recog­ nition that such an analysis is in principle excluded. The peculiar individuality of the quantum effects presents us, as regards the comprehension of well-defined evidence, with a novel situation unforeseen in classical physics and irreconcilable with conven­ tional ideas suited for our orientation and adjustment to or­ dinary experience. It is in this respect that quantum theory has called for a renewed revision of the foundation for the unam­ biguous use of elementary concepts, as a further step in the de­ velopment which, since the advent of relativity theory, has been so characteristic of modern science. In the following years, the more philosophical aspects of the situation in atomic physics aroused the interest of ever larger circles and were, in particular, discussed at the Second Interna­ tional Congress for the Unity of Science in Copenhagen in July 1936. In a lecture on this occasion,14 I tried especially to "A. Einstein, Journ. Frankl. Imt., 22/, 349, (1936). 14 N. Bohr, Erkenntnu t 6, 293, (1937), and Philosofhy of Science, 4, 289,

(1937)·

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stress the analogy in epistemological respects between the limi­ tation imposed on the causal description in atomic physics and situations met with in other fields of knowledge. A principal purpose of such parallels was to call attention to the necessity in many domains of general human interest to face problems of a similar kind as those which had arisen in quantum theory and thereby to give a more familiar background for the apparently extravagant way of expression which physicists have developed to cope with their acute difficulties. Besides the complementary features conspicuous in psychol­ ogy and already touched upon (cf. p. 224), examples of such relationships can also be traced in biology, especially as regards the comparison between mechanistic and vitalistic viewpoints. Just with respect to the observational problem, this last question had previously been the subject of an address to the Interna­ tional Congress on Light Therapy held in Copenhagen in 1932,15 where it was incidentally pointed out that even the psycho-physical parallelism as envisaged by Leibniz and Spin­ oza has obtained a wider scope through the development of atomic physics, which forces us to an attitude towards the prob­ lem of explanation recalling ancient wisdom, that when search­ ing for harmony in life one must never forget that in the drama of existence we are ourselves both actors and spectators. Utterances of this kind would naturally in many minds evoke the" impression of an underlying mysticism foreign to the spirit of science J at the above mentioned Congress in 1936 I there­ fore tried to clear up such misunderstandings and to explain that the only question was an endeavour to clarify the condi­ tions, in each field of knowledge, for the analysis and synthesis of experience.14 Yet, I am afraid that I had in this respect only little success in convincing my listeners, for whom the dissent among the physicists themselves was naturally a cause of scepti­ cism as to the necessity of going so far in renouncing customary demands as regards the explanation of natural phenomena. Not least through a new discussion with Einstein in Princeton in 1937, where we did not get beyond a humourous contest con18 II e Congres international de la Lumi^re, Copenhague 1932 (reprinted in Nature i /3/, 421 and 457, 1933).

1.1 DISCUSSIONS WITH EINSTEIN

cerning which side Spinoza would have taken if he had lived to see the development of our days, I was strongly reminded of the importance of utmost caution in all questions of terminology and dialectics. These aspects of the situation were especially discussed at a meeting in Warsaw in 1938, arranged by the International In­ stitute of Intellectual Co-operation of the League of Nations.16 The preceding years had seen great progress in quantum phy­ sics due to a number of fundamental discoveries regarding the constitution and properties of atomic nuclei as well as due to important developments of the mathematical formalism taking the requirements of relativity theory into account. In the last respect, Dirac's ingenious quantum theory of the electron of­ fered a most striking illustration of the power and fertility of the general quantum-mechanical way of description. In the phe­ nomena of creation and annihilation of electron pairs we have in fact to do with new fundamental features of atomicity, which are intimately connected with the non-classical aspects of quan­ tum statistics expressed in the exclusion principle, and which have demanded a still more far-reaching renunciation of ex­ planation in terms of a pictorial representation. Meanwhile, the discussion of the epistemological problems in atomic physics attracted as much attention as ever and, in commenting on Einstein's views as regards the incompleteness of the quantum-mechanical mode of description, I entered more directly on questions of terminology. In this connection I warned especially against phrases, often found in the physical literature, such as "disturbing of phenomena by observation" or "creating physical attributes to atomic objects by measure­ ments." Such phrases, which may serve to remind of the ap­ parent paradoxes in quantum theory, are at the same time apt to cause confusion, since words like "phenomena" and "obser­ vations," just as "attributes" and "measurements," are used in a way hardly compatible with common language and practical definition. As a more appropriate way of expression I advocated the aple

New Theories in Physics (Paris 1938), 11.

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plication of the word 'phenomenon exclusively to refer to the observations obtained under specified circumstances, including an account of the whole experimental arrangement. In such terminology, the observational problem is free of any special intricacy since, in actual experiments, all observations are ex­ pressed by unambiguous statements referring, for instance, to the registration of the point at which an electron arrives at a photographic plate. Moreover, speaking in such a way is just suited to emphasize that the appropriate physical interpretation of the symbolic quantum-mechanical formalism amounts only to predictions, of determinate or statistical character, pertain­ ing to individual phenomena appearing under conditions de­ fined by classical physical concepts. Notwithstanding all differences between the physical prob­ lems which have given rise to the development of relativity theory and quantum theory, respectively, a comparison of purely logical aspects of relativistic and complementary argu­ mentation reveals striking similarities as regards the renuncia­ tion of the absolute significance of conventional physical attri­ butes of objects. Also, the neglect of the atomic constitution of the measuring instruments themselves, in the account of actual experience, is equally characteristic of the applications of rela­ tivity and quantum theory. Thus, the smallness of the quantum of action compared with the actions involved in usual experi­ ence, including the arranging and handling of physical ap­ paratus, is as essential in atomic physics as is the enormous number of atoms composing the world in the general theory of relativity which, as often pointed out, demands that dimen­ sions of apparatus for measuring angles can be made small com­ pared with the radius of curvature of space. In the Warsaw lecture, I commented upon the use of not directly visualizable symbolism in relativity and quantum theory in the following way: Even the formalisms, which in both theories within their scope offer ade­ quate means of comprehending all conceivable experience, exhibit deepgoing analogies. In fact, the astounding simplicity of the generalization of classical physical theories, which are obtained by the use of multidimen­ sional geometry and non-commutative algebra, respectively, rests in both

1.1 DISCUSSIONS WITH EINSTEIN

cases essentially on the introduction of the conventional symbol V — i. The abstract character of the formalisms concerned is indeed, on closer examination, as typical of relativity theory as it is of quantum mechanics, and it is in this respect purely a matter of tradition if the former theory is considered as a completion of classical physics rather than as a first fundamental step in the thoroughgoing revision of our conceptual means of comparing observations, which the modern development of physics has forced upon us.

It is, of course, true that in atomic physics we are confronted with a number of unsolved fundamental problems, especially as regards the intimate relationship between the elementary unit of electric charge and the universal quantum of action; but these problems are no more connected with the epistemological points here discussed than is the adequacy of relativistic argumentation with the issue of thus far unsolved problems of cosmology. Both in relativity and in quantum theory we are concerned with new aspects of scientific analysis and synthesis and, in this connection, it is interesting to note that, even in the great epoch of critical philosophy in the former century, there was only question to what extent a friori arguments could be given for the adequacy of space-time co-ordination and causal connection of experience, but never question of rational generalizations or inherent limitations of such categories of human thinking. Although in more recent years I have had several occasions of meeting Einstein, the continued discussions, from which I always have received new impulses, have so far not led to a common view about the epistemological problems in atomic physics, and our opposing views are perhaps most clearly stated in a recent issue of Dialectical bringing a general discussion of these problems. Realizing, however, the many obstacles for mutual understanding as regards a matter where approach and background must influence everyone's attitude, I have wel­ comed this opportunity of a broader exposition of the develop­ ment by which, to my mind, a veritable crisis in physical science has been overcome. The lesson we have hereby received would seem to have brought us a decisive step further in the never"N. Bohr, Oialectlcai i, 312 (1948).

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ending struggle for harmony between content and form, and taught us once again that no content can be grasped without a formal frame and that any form, however useful it has hitherto proved, may be found to be too narrow to comprehend new ex­ perience. Surely, in a situation like this, where it has been difficult to reach mutual understanding not only between philosophers and physicists but even between physicists of different schools, the difficulties have their root not seldom in the preference for a certain use of language suggesting itself- from the different lines of approach. In the Institute in Copenhagen, where through those years a number of young physicists from various countries came together for discussions, we used, when in trouble, often to comfort ourselves with jokes, among them the old saying of the two kinds of truth. To the one kind belong statements so sim­ ple and clear that the opposite assertion obviously could not be defended. The other kind, the so-called "deep truths," are statements in which the opposite also contains deep truth. Now, the development in a new field will usually pass through stages in which chaos becomes gradually replaced by order; but it is not least in the intermediate stage where deep truth prevails that the work is really exciting and inspires the imagination to search for a firmer hold. For such endeavours of seeking the proper balance between seriousness and humour, Einstein's own personality stands as a great example and, when expressing my belief that through a singularly fruitful co-operation of a whole generation of physicists we are nearing the goal where logical order to a large extent allows us to avoid deep truth, I hope that it will be taken in his spirit and may serve as an apology for several utterances in the preceding pages. The discussions with Einstein which have formed the theme of this article have extended over many years which have wit­ nessed great progress in the field of atomic physics. Whether our actual meetings have been of short or long duration, they have always left a deep and lasting impression on my mind, and when writing this report I have, so-to-say, been arguing with Einstein all the time even when entering on topics ap-

1.1 DISCUSSIONS WITH EINSTEIN

parently far removed from the special problems under debate at our meetings. As regards the account of the conversations I am, of course, aware that I am relying only on my own memory, just as I am prepared for the possibility that many features of the development of quantum theory, in which Einstein has played so large a part, may appear to himself in a different light. I trust, however, that I have not failed in conveying a proper impression of how much it has meant to me to be able to benefit from the inspiration which we all derive from every contact with Einstein. NIELS BOHR UNIVERSITETETS INSTITUT FOR TEORETISK FYSIK COPENHAGEN, DENMARK

49

1.2

BORN'S PROBABILISTIC INTERPRETATION

COMMENTARY OF ROSENFELD (1971A)

. . . Schrodinger made no secret of his intention to substitute simple classical pictures for the strange conceptions of quantum mechanics, for whose abstract character he expressed deep "aversion"; he was conscious that this last sentiment was shared by all the older generation of physicists, who had not accepted the necessity of giving up their habitual ways of thinking when dealing with phenomena on the atomic scale. Signif­ icantly, he turned towards the chevroned peers of the classical era— Lorentz, Planck, Einstein—who did not grudge him praise and encouragement, and shunned the founders of quantum mechanics. The latter, however, who had taken no notice of de Broglie, looked into Schrodinger's ideas most critically; he had indeed enforced the claim to be taken seriously by solving the wave equation for the hydrogen atom and obtaining Rydberg's formula by a calculation much simpler and more elegant than Pauli's quantum algebra. This could hardly be a fortuitous coin­ cidence, and indeed both Pauli and Schrodinger soon discovered that the two theories so different in conception, were mathematically equivalent: for instance, the quantum amplitude gov­ erning the transition between two sta­ tionary states could be readily computed with the help of the wave amplitudes corresponding to these states. Would this formal equivalence clinch the issue in favor of Schrodinger's contention

that the proper quantal concepts can be altogether dispensed with? Far from it, Heisenberg had at once seen that this contention was untenable: Schrodin­ ger's way of treating the charge density as a classical source of radiation would even prevent him from obtaining Planck's law for the distribution of thermal radiation. How could such an elementary, but fatal objection have escaped, not only Schrodinger himself, but above all the creators of the quan­ tum theory of radiation, who gave him such uncritical support? Almost regret­ ting to have ushered in ideas whose revolutionary consequences they had not foreseen, these great masters desper­ ately clung to the "sound philosophy" of classical physics, without realizing its limitations; an emotional resistance dimmed their judgment. The pioneers of quantum mechanics, on their side, were now confronted with a new chal­ lenge: Schrodinger's wave theory of matter clearly provided another formal approach to the consistent description of atomic phenomena they were striving for, but its physical content still eluded them. It did not last long, however, before they realized that wave mechanics was just the appropriate technique they wanted for dealing with the aperiodic phenomena intractable by the quantum algebra. This was explicitly demon­ strated by Born, who showed how to treat atomic collisions by a transposi­ tion of the mathematical methods ap­ plied to the analogous classical problem

1.2 COMMENTARY

of the scattering of light waves by a polarizable medium. Born's argument, moreover, embodied an essentially new feature, of decisive importance, with regard to the physical interpretation of the intensity of the matter waves. The optical analogy suggested a comparison of this intensity with that of a classical light wave, which the quantum theory of radiation interprets as the density of the statistical distribution of the asso­ ciated photons. Born pointed out that the usual particle description of the atomic collision process could be main­ tained if one adopted a similar statis­ tical relationship between the matter

51

waves and the associated atomic par­ ticles. He accordingly proposed to interpret the wave intensity, not as the density of an actual distribution of matter, as Schrodinger imagined, but as a density of probability for the presence of a particle. Thus, the formal equiva­ lence of wave mechanics and Heisenberg's quantum mechanics became physically meaningful: it established a complete harmony between the statis­ tical meaning of the wave intensity and the statistical character of the rules of quantum algebra for the calculation of transition probabilities.

1.2 ON THE QUANTUM MECHANICS OF COLLISIONS [Preliminary communication]r

MAX BORN Through the investigation of collisions it is argued that quantum mechanics in the Schrddinger form allows one to describe not only stationary states but also quantum jumps. Heisenberg's quantum mechanics has so far been applied exclusively to the cal­ culation of stationary states and vibration amplitudes associated with transitions (I purposely avoid the word "transition probabilities"). In this connection the formalism, further developed in the meantime, seems to be well validated. However, questions of this kind deal with only one aspect of quantum theory. Beside them there shows up as equally important the question of the nature of the "transitions" themselves. On this point opinions seem to be divided. Many assume that the problem of transitions is not encompassed by quantum mechanics in its present form, but that here new conceptual developments will be necessary. I myself, impressed with the closed character of the logical nature of quantum mechanics, came to the presumption that this theory is complete and that the problem of transitions must be contained in it. I believe that I have now succeeded in proving this. Bohr has already directed attention to the fact that all difficulties of principle associated with the quantum approach which meet us in the emission and absorp­ tion of light by atoms also occur in the interaction of atoms at short distances and consequently in collision processes. In collisions one deals not with mysterious wave fields, but exclusively with systems of material particles, subject to the for­ malism of quantum mechanics. I therefore attack the problem of investigating more closely the interaction of the free particle (α-ray or electron) and an arbitrary atom and of determining whether a description of a collision is not possible within the framework of existing theory. Of the different forms of the theory only Schrodinger's has proved suitable for this process, and exactly for this reason I might regard it as the deepest formula­ tion of the quantum laws. The course of my reasoning is the following. If one wishes to calculate quantum mechanically the interaction of two systems, f This report was originally intended for die Naturwissenschaften, but could not be accepted there for lack of space. I hope that its publication in this journal [Zeitschnft fiir Physik] does not seem out of place [M.B.].

Originally published under the title, "Zur Quantenmechanik der Stossvorgange," Zeitschrift fiir Physik, 37, 863-67 (1926); reprinted in Dokumente der Naturwissensehaft, 1, 48-52 (1962) and in M. Born (1963); translation into English by J A.W. and W.H.Z., 1981.

1.2 PROBABILITY INTERPRETATION

53

then, as is well known, one cannot, as in classical mechanics, pick out a state of the one system and determine how this is influenced by a state of the other system, since all states of both systems are coupled in a complicated way. This is true also in an aperiodic process, such as a collision, where a particle, let us say an electron, comes in from infinity and then goes off to infinity. There is no escape from the conclusion that, as well before as after collision, when the electron is far enough away and the coupling is small enough, a definite state must be specifiable for the atom and likewise a definite rectilinear motion for the electron. The problem is to formulate mathematically this asymptotic behavior of the coupled particles. I did not succeed in doing this with the matrix form of quantum mechanics, but did with the Schrodinger formulation. According to Schrodinger, the atom in its nth quantum state is a vibration of a state function of fixed frequency W°/h spread over all of space. In particular, an electron moving in a straight line is such a vibratory phenomenon which corre­ sponds to a plane wave. When two such waves interact, a complicated vibration arises. However, one sees immediately that one can determine it through its asymptotic behavior at infinity. Indeed one has nothing more than a "diffraction problem" in which an incoming plane wave is refracted or scattered at an atom. In place of the boundary conditions which one uses in optics for the description of the diffraction diaphragm, one has here the potential energy of interaction be­ tween the atom and the electron. The task is clear. We have to solve the Schrodinger wave equation for the system atom-plus-electron subject to the boundary condition that the solution in a preselected direction of electron space goes over asymptotically into a plane wave with exactly this direction of propagation (the arriving electron). In a thus selected solution we are further interested principally in a behavior of the "scattered" wave at infinity, for it describes the behavior of the system after the collision. We spell this out a little further. Let ιt/*¾¾)* · · · be the eigenfunctions of the unperturbed atom (we assume that there is only a discrete spectrum). The unperturbed electron, in straight-line motion, corresponds to eigenfunctions sin (2π/λ)(αχ + βγ + yz + δ), a continuous manifold of plane waves. Their wave­ length, according to de Broglie, is connected with the energy of translation τ by the relation τ = Ρ/(2μλ2). The eigenfunction of the unperturbed state in which the electron arrives from the + ζ direction, is thus ^ n A q k , Z) = Ά°( + oo into the above function. The question is now how this solution behaves "after the collision."

54

BORN

The calculation gives this result: The scattered wave created by this perturba­ tion has asymptotically at infinity the form: β, 7) sin k n Jax + βγ + yz + δ ) φ ^ ) .

This means that the perturbation, analyzed at infinity, can be regarded as a super­ position of solutions of the unperturbed problem. If one calculates the energy belonging to the wavelength A„im according to the de Broglie formula, one finds \Y n rf

T

m

= /ιν^ -4- T nm ' n v

where the v°m are the frequencies of the unperturbed atom. If one translates this result into terms of particles, only one interpretation is possible. Φ„τ,„(χ β, y) gives the probability* for the electron, arriving from the

z-

direction, to be thrown out into the direction designated by the angles α, β, y, with the phase change δ. Here its energy τ has increased by one quantum hv°m at the cost of the energy of the atom (collision of the first kind for W° < W„, hv°nm < 0; collision of the second kind

> W0m, hv°nm > 0).

Schrodinger's quantum mechanics therefore gives quite a definite answer to the question of the effect of the collision; but there is no question of any causal description. One gets no answer to the question, "what is the state after the colli­ sion," but only to the question, "how probable is a specified outcome of the collision" (where naturally the quantum mechanical energy relation must be fulfilled). Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which condition a definite outcome for the collision. Ought we to hope later to discover such prop­ erties (like phases or the internal atomic motions) and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment—as to the impossibility of prescribing conditions for a causal evolution—is a preestablished harmony founded on the nonexistence of such conditions? I myself am inclined to give up determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive. In practical terms indeterminism is present for experimental as well as for theoretical physicists. The "yield function" Φ so much investigated by experimen­ talists is now also sharply defined theoretically. One can determine it from the potential energy of interaction, V(x, y, z; qk). However, the calculations required * Addition in proof. More careful consideration shows that the probability is proportional to the square of the quantity Φ„ τ „.

1.2 PROBABILITY INTERPRETATION

55

for this purpose are too complicated to communicate here. I will only clarify briefly the meaning of the function Φη πι . If, for example, the atom before the collision is in the normal state η = 1, then it follows from the equation τ + K m = τ - ZtvO1

=

> 0>

that, for an electron with less energy than the lowest excitation energy of the atom, the final state is also necessarily m = 1, or that W ui must be equal to τ. Then we have "elastic reflection" of the electron with the yield function O 1A . If τ increases beyond the first excitation level, then there occurs, besides reflection, also excita­ tion with the yield Φ 1τ2 , etc. If the target atom is in the excited state η = 2 and τ < /iν2then there occur reflection with yield Φ 2τ2 and collisions of the second kind with the yield Φ 2τ1 · If the kinetic energy τ > Z jvj 1 , then further excitation is also possible. The formulas thus reproduce completely the qualitative character of collisions. The quantitative predictions of the formulas for particular cases require extensive investigation. I do not exclude the possibility that the strict connection of mechanics and statistics as it comes to light here will demand a revision of basic ideas of thermo­ dynamics and statistical mechanics. I also believe that the problem of radiation of light—and irradiation—has to be handled in a way entirely analogous to the "boundary value problem" of the wave equation, and will lead to a rational theory of radiation damping and linebreadths in agreement with the theory of light quanta. An extended treatment will appear shortly in this journal.

1.3

THE PRINCIPLE OF INDETERMINACY

COMMENTARY OF HEISENBERG (1967)

In July [1926] I visited my parents in Munich and on this occasion I heard a lecture given by Schrodinger for the physicists in Munich about his work on wave mechanics. It was thus that I first became acquainted with the interpre­ tation Schrodinger wanted to give his mathematical formalism of wave me­ chanics, and I was very disturbed about the confusion with which I believed this would burden atomic theory. Un­ fortunately, nothing came of my attempt during the discussion to put things in order. My argument that one could not even understand Planck's radiation law on the basis of Schrodinger's interpre­ tation convinced no one. Wilhelm Wien, who held the chair of experimental physics at the University of Munich, answered rather sharply that one must really put an end to quantum jumps and the whole atomic mysticism, and the difficulties I had mentioned would cer­ tainly soon be solved by Schrodinger. I no longer remember whether or not I wrote to Bohr of this encounter in Munich. Be that as it may, Bohr shortly afterwards invited Schrodinger to Copenhagen and asked him not only to lecture on his wave mechanics, but also to stay in Copenhagen so long that there would be adequate time to discuss the interpretation of quantum theory. As far as I remember these discus­ sions took place in Copenhagen around September 1926 and in particular they left me with a very strong impression of

Bohr's personality. For though Bohr was an unusually considerate and oblig­ ing person, he was able in such a discus­ sion, which concerned epistemological problems which he considered to be of vital importance, to insist fanatically and with almost terrifying relentlessness on complete clarity in all arguments. He would not give up, even after hours of struggling, before Schrodinger had ad­ mitted that this interpretation was in­ sufficient, and could not even explain Planck's law. Every attempt from Schrodinger's side to get round this bitter result was slowly refuted point by point in infinitely laborious discussions. It was perhaps from over-exertion that after a few days Schrodinger became ill and had to lie abed as a guest in Bohr's home. Even here it was hard to get Bohr away from Schrodinger's bed and the phrase, "But, Schrodinger, you must at least admit that..." could be heard again and again. Once Schrodinger burst out almost desperately, "If one has to go on with these damned quan­ tum jumps, then I'm sorry that I ever started to work on atomic theory." To which Bohr answered, "But the rest of us are so grateful that you did, for you have thus brought atomic physics a decisive step forward." Schrodinger fi­ nally left Copenhagen rather discour­ aged, while we at Bohr's Institute felt that at least Schrodinger's interpreta­ tion of quantum theory, an interpreta­ tion rather too hastily arrived at using the classical wave-theories as models, was now disposed of, but that we still

1.3 COMMENTARY

lacked some important ideas before we could really reach a full understanding of quantum mechanics. From now on the discussions be­ tween Bohr and his co-workers in Copenhagen became more and more concentrated on the central problem in quantum theory: how the mathematical formalism was to be applied to each individual problem, and thus how the frequently discussed paradoxes, such as e.g. the apparent contradiction between the wave and particle models, could be cleared up. Ever new imaginary experi­ ments were thought up, each displaying the paradoxes in a more clear-cut way than its predecessors, and we tried to guess what answer nature would prob­ ably give to each experiment.

Left alone in Copenhagen [when Bohr went on vacation at the end of February 1927] I too was able to give my thoughts freer play, and I decided to make the above uncertainty the central point in the interpretation. Remem­ bering a discussion I had had long be­ fore with a fellow student in Gottingen, I got the idea of investigating the possibility of determining the position of a particle with the aid of a gamma-ray microscope, and in this way soon ar­ rived at an interpretation which I be­ lieved to be coherent and free of contradictions. I then wrote a long letter to Pauli, more or less the draft of a paper, and Pauli's answer was decidedly positive and encouraging. When Bohr returned from Norway, I was already able to present him with the first version of a paper along with the letter from

57

Pauli. At first Bohr was rather dis­ satisfied. He pointed out to me that certain statements in this first version were still incorrectly founded, and as he always insisted on relentless clarity in every detail, these points offended him deeply. Further, he had probably al­ ready grown familiar, while he was in Norway, with the concept of comple­ mentarity which would make it possible to take the dualism between the wave and particle picture as a suitable starting point for an interpretation. This concept of complementarity fitted well the fun­ damental philosophical attitude which he had always had, and in which the limitations of our means of expressing ourselves entered as a central philo­ sophical problem. He therefore took objection to the fact that I had not started from the dualism between par­ ticles and waves. After several weeks of discussion, which were not devoid of stress, we soon concluded, not least thanks to Oskar Klein's participation, that we really meant the same, and that the uncertainty relations were just a special case of the more general com­ plementarity principle. Thus, I sent my improved paper to the printer and Bohr prepared a detailed publication on complementarity. COMMENTARY OF ROSENFELD (1971A)

Bohr was now impatient to come to grips with the outstanding epistemological issue. He was convinced that in the quantum theory of matter just as in that of radiation, one faced an irreduc­ ible particle-wave dualism, and it seemed to him that the way was clear to the

58

1.3 COMMENTARY

elucidation of its epistemological sig­ nificance. The emphasis must be laid here on the role of a scientific theory as a means of unambiguous communica­ tion of experience. In atomic theory it is found convenient to use for this purpose, according to the circum­ stances, either the language appropriate to the description of wave phenomena, or the language of particle mechanics, both modes of description being subject to essential limitations: it is a necessary task of the theory to formulate these limitations so as to fix the conditions of validity of each type of idealized de­ scription. Night after night in the first months of 1927, Bohr and Heisenberg pondered together over these questions; again, they were at loggerheads over the strategy: Bohr expected the answer would follow from a direct analysis of the definition of the idealized concepts, Heisenberg argued that the answer was hidden in the formal structure of the theory and that a closer scrutiny of this structure should bring it to light. Towards the end of February, Vhey parted, after a last fruitless discussion, Bohr having decided to seek a much needed recreation in the snowy hills of Norway. On the same night, however, Heisenberg had an inspiration that put him on the right track. He vividly remembered a discussion he had with Einstein a year before about the new quantum mechanics; his attempt at arguing that a good theory ought only to operate with observable quantities had elicited from Einstein the pointed retort, which had made a strong impres­ sion on him: "Only the theory itself can decide what is and is not observable."

This remark, he thought, was the key to the whole problem: all one had to do was to investigate, for each given phe­ nomenon, which conclusions one could draw from its theoretical analysis, ac­ cording to the principles of quantum mechanics, about possible limitations to its observability.

Heisenberg based his argumenta­ tion on very general "indeterminacy relations" which he had shown to be deeply rooted in the formal structure of quantum mechanics. The mechanical variables are always associated in "con­ jugate" pairs . . . . H eisenberg naturally wanted to clinch the argument by analyzing particular examples of idealized experimental sit­ uations which could illustrate the appli­ cation of the indeterminacy relations. He was anxious, in particular, to dem­ onstrate the impossibility of assigning a trajectory to a moving electron: let us imagine we should attempt to localize the electron by means of a microscope·, in order to achieve sufficient resolution, we should have to illuminate it with radiation of very high frequency, in the range of the gamma-rays emitted by radioactive substances; the scattering of such photons by the electron, according to the theory of the Compton effect, would throw the electron completely out of the path prescribed by its previ­ ous velocity. Bohr's reaction to these considerations was very characteristic: he at once realized that Heisenberg again had forged the weapon that was needed to master the problem, but was soon dissatisfied with the smart way in

1.3 COMMENTARY

59

which he was wielding it. The idea of cepts describing the various aspects of bringing out the origin of the limitations experience; the indeterminacy relations of classical description by analyzing were part of this correspondence, their imaginary processes of observation was physical content could be exhibited by of the type that would appeal to Bohr, the discussion of appropriate experi­ and he took it up eagerly, but in his own ments. Bohr wanted to pursue the episinfinitely patient manner, examining temological analysis one step further, the argument from all sides, leaving no and in particular to understand the point in the shadow. He was not long logical nature of the mutual exclusion to discover that Heisenberg's discussion of the two aspects opposed in the of the gamma-ray microscope touched particle-wave dualism. From this point only one aspect of the question—the of view the indeterminacy relations role of the frequency of the radiation, appear in a new light. The conjugate but did not really go to the heart of the variables for which they hold refer, matter: the Compton recoil of the elec­ respectively, to the two mutually exclu­ tron could only be a source of indeter­ sive classical pictures centered on the minacy for its momentum to the extent idealized concepts of particle and wave: to which the scattering process itself is to the former are attached the momen­ not completely determined. Now, he tum and energy variables, to the latter pointed out, the formation of an image the coordinates of space and time. The in a microscope requires a bundle of particle concept is used to describe the rays of finite aperture, and it is this exchanges of momentum and energy latitude in the direction of the photons between atomic systems, and between impinging on the electron that implies such systems and radiation: it is the a corresponding latitude in the electron physical support, so to speak, of the recoil. On the other hand a large aper­ conservation laws governing such ex­ ture increases the accuracy of the elec­ changes, irrespectively of any spacetron's localization; and a simple com­ time localization. The wave concept, or putation shows that the two contrary rather the derived density function, effects of the aperture on the indeter- describes the localization of atomic minacies of position and momentum systems and the distribution of radia­ are such as to impose on their product tion in space and time; it excludes the the lower limit given by the relation (2), specification of momentum and energy, independently of the aperture and of since only a wave of infinite spatial and the frequency of the radiation. temporal extension can have a sharply . . . B o h r l o o k e d u p o n s u c h i m a g i n a r y defined periodicity, necessary for a experiments in a rather different spirit sharp definition of momentum and en­ from Heisenberg. The latter was satis­ ergy. The odd situation we meet in fied with the knowledge that he had now atomic theory is the necessity of making established a complete correspondence use of these two conflicting pictures in between the mathematical structure of order to deal with all the conditions the theory and the usual physical con­ under which we can observe the atomic

60

1.3 COMMENTARY

phenomena. The indeterminacy rela­ tions are therefore essential to ensure the consistency of the theory, by as­ signing the limits within which the use of classical concepts belonging to the two extreme pictures may be applied without contradiction. For this novel logical relationship, which called in Bohr's mind echoes of his philosophical meditations over the duality of our mental activity, he proposed the name "complementarity," conscious that he was here breaking new ground in epistemology. The solemnity of the occasion did not dawn immediately upon Heisenberg. He was not well-disposed toward the idea of a particle-wave dualism, and Bohr was himself still struggling to find the right expression for the new ideas that were taking shape in his mind. However eager he was to let Heisenberg share his elation at the prospects they disclosed, his eagerness was not suffi­ ciently matched by clarity to make impression on a reluctant interlocutor. The memory of the fruitless discussions with Schrodinger still lingered in Heisenberg's mind, and he regarded with deep suspicion any attempt to make more than formal use of the con­ cept of matter wave. He stressed, quite rightly, that according to the corre­ spondence principle, it is the particle concept that has a direct physical meaning with respect to atomic con­ stituents, whereas matter waves are merely mathematical auxiliaries. To this Bohr would oppose the case of radiation, where the correspondence principle, on the contrary, points to the electromagnetic waves as the funda­

mental classical concept, and leaves only a symbolic part to the photons. Moreover, Oskar Klein, who was then Bohr's assistant, had carefully discussed the relation of matter waves to the cor­ respondence principle, and shown how the latter gave as reliable a guidance for the physical interpretation of wave me­ chanics as for that of the quantum algebra: but Klein's paper had not found grace with Heisenberg. He stuck obstinately to the view that his own scheme of quantum mechanics formed a sufficient basis for a complete descrip­ tion of the phenomena. Matters did not improve when Bohr was shown the hastily written paper in which Heisenberg had developed his arguments. Bohr started handling the manuscript as he would have done one of his own, namely as a rough draft which could eventually lead to an acceptable text; and with his usual optimism, he expected Heisenberg to welcome this scrutiny: the latter was simply annoyed to find Bohr raising so many objections to the inaccuracies and careless statements of the paper, and proposing new formulations with which he was in no mood to agree. There is no telling how things would have ended if Pauli's arrival on the scene had not relieved the tension. Heisenberg had informed Pauli of his ideas and received from him approval and encouragement ; now, Pauli was in a position both to lend Bohr a dispassionate ear and to gain a hearing from Heisenberg. He was able to explain to the latter that there was no disagreement between his and Bohr's ways of analyzing the physical content of atomic theory, but that Bohr

1.3 COMMENTARY

had gone further and deeper in the analysis of its logical structure. Even so, all Heisenberg could be persuaded to do about his paper was to append to it a "remark added on the proof," in which he declared in substance that he had missed essential points, whose clarifi­ cation would be found in a forthcoming paper by Bohr. This addendum must

61

have puzzled many readers: it is not often that the announcement of a deci­ sive progress in our insight into the workings of nature is qualified by such a warning. As to Bohr's "forthcoming" publica­ tion, more than a year elapsed before it appeared in print....

1.3 THE PHYSICAL CONTENT OF QUANTUM KINEMATICS AND MECHANICS WERNER HEISENBERG First we define the terms velocity, energy, etc. (for example, for an electron) which remain valid in quantum mechanics. It is shown that canonically conjugate quantities can be determined simulta­ neously only with a characteristic indeterminacy (§1). This indeter­ minacy is the real basis for the occurrence of statistical relations in quantum mechanics. Its mathematical formulation is given by the Dirac-Jordan theory (§2). Starting from the basic principles thus obtained, we show how microscopic processes can be understood by way of quantum mechanics (§3). To illustrate the theory, a few special gedankenexperiments are discussed (§4). We believe we understand the physical content of a theory when we can see its qualitative experimental consequences in all simple cases and when at the same time we have checked that the application of the theory never contains inner contradictions. For example, we believe that we understand the physical content of Einstein's concept of a closed 3-dimensional space because we can visualize consistently the experimental consequences of this concept. Of course these con­ sequences contradict our everyday physical concepts of space and time. However, we can convince ourselves that the possibility of employing usual space-time concepts at cosmological distances can be justified neither by logic nor by ob­ servation. The physical interpretation of quantum mechanics is still full of internal discrepancies, which show themselves in arguments about continuity versus dis­ continuity and particle versus wave. Already from this circumstance one might conclude that no interpretation of quantum mechanics is possible which uses ordinary kinematical and mechanical concepts. Of course, quantum mechanics arose exactly out of the attempt to break with all ordinary kinematic concepts and to put in their place relations between concrete and experimentally determinable numbers. Moreover, as this enterprise seems to have succeeded, the mathematical scheme of quantum mechanics needs no revision. Equally unnecessary is a revi­ sion of space-time geometry at small distances, as we can make the quantummechanical laws approximate the classical ones arbitrarily closely by choosing sufficiently great masses, even when arbitrarily small distances and times come into question. But that a revision of kinematical and mechanical concepts is necessary Originally published under the title, "LJber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," ZeitschriJt fur Physik, 43, 172-98 (1927); reprinted in Dokumente der Naturwissenschaft, 4, 9-35 (1963); translation into English by J.A.W. and W.H.Z., 1981.

1.3 PRINCIPLE OF INDETERMINISM

63

seems to follow directly from the basic equations of quantum mechanics. When a definite mass m is given, in our everyday physics it is perfectly understandable to speak of the position and the velocity of the center of gravity of this mass. In quantum mechanics, however, the relation pq — qp = —ih between mass, posi­ tion, and velocity is believed to hold. Therefore we have good reason to become suspicious every time uncritical use is made of the words "position" and "velocity." When one admits that discontinuities are somehow typical of processes that take place in small regions and in short times, then a contradiction between the con­ cepts of "position" and "velocity" is quite plausible. If one considers, for example, the motion of a particle in one dimension, then in continuum theory one will be able to draw (Fig. 1) a worldline x(t) for the track of the particle (more precisely, its center of gravity), the tangent of which gives the velocity at every instant. In contrast, in a theory based on discontinuity there might be in place of this curve a series of points at finite separation (Fig. 2). In this case it is clearly meaningless to speak about one velocity at one position (1) because one velocity can only be defined by two positions and (2), conversely, because any one point is associated with two velocities.

FIGURE 1

The question therefore arises whether, through a more precise analysis of these kinematic and mechanical concepts, it might be possible to clear up the contradic­ tions evident up to now in the physical interpretations of quantum mechanics and to arrive at a physical understanding of the quantum-mechanical formulas.* * The present work has arisen from efforts and desires to which other investigators have already given clear expression, before the development of quantum mechanics. I call attention here especially to Bohr's papers on the basic postulates of quantum theory (for example, Zeits. f. Physik, 13, 117 [1923]) and Einstein's discussions on the connection between wave field and light quanta. The problems dealt with here are discussed most clearly in recent times, and the problems arising are partly answered, by W. Pauli ("Quantentheorie," Handbuch der Physik, Vol. XXIII, cited hereafter as I.e.); quantum mechanics has changed only slightly the formulation of these problems as given by Pauli. It is also a special pleasure to thank here Herrn Pauli for the repeated stimulus I have received from our oral and written discussions, which have contributed decisively to the present work.

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§1. CONCEPTS: POSITION, PATH, VELOCITY, ENERGY In order to be able to follow the quantum-mechanical behavior of any object one has to know the mass of this object and its interactions with any fields and other objects. Only then can the Hamiltonian function be written down for the quantummechanical system. (The following considerations ordinarily refer to nonrelativistic quantum mechanics, as the laws of quantum electrodynamics are still very incom­ pletely known.)* About the "Gestalt" (construction) of the object any further assumption is unnecessary; one most usefully employs the word "Gestalt" to designate the totality of these interactions. When one wants to be clear about what is to be understood by the words "position of the object," for example of the electron (relative to a given frame of reference), then one must specify definite experiments with whose help one plans to measure the "position of the electron"; otherwise this word has no meaning. There is no shortage of such experiments, which in principle even allow one to determine the "position of the electron" with arbitrary accuracy. For example, let one illuminate the electron and observe it under a microscope. Then the highest attainable accuracy in the measurement of position is governed by the wavelength of the light. However, in principle one can build, say, a 7-ray microscope and with it carry out the determination of position with as much accuracy as one wants. In this measurement there is an important feature, the Compton effect. Every obser­ vation of scattered light coming from the electron presupposes a photoelectric effect (in the eye, on the photographic plate, in the photocell) and can therefore also be so interpreted that a light quantum hits the electron, is reflected or scattered, and then, once again bent by the lens of the microscope, produces the photoeffect. At the instant when position is determined—therefore, at the moment when the photon is scattered by the electron—the electron undergoes a discontinuous change in momentum. This change is the greater the smaller the wavelength of the light employed—that is, the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum therefore can be known up to magnitudes which correspond to that discontinuous change. Thus, the more precisely the position is determined, the less precisely the momen­ tum is known, and conversely. In this circumstance we see a direct physical interpretation of the equation pq — qp = — ih. Let q t be the precision with which the value q is known Ull is, say, the mean error of q), therefore here the wavelength of the light. Let P1 be the precision with which the value ρ is determinable; that is, here, the discontinuous change of ρ in the Compton effect. Then, according to the elementary laws of the Compton effect P1 and qx stand in the relation

* Quite recently, however, great advances in this domain have been made in the papers of P. Dirac [Proc. Roy. Soc. Al 14, 243 (1927) and papers to appear subsequently].

1.3 PRINCIPLE OF INDETERMINISM

65

P i h ~ h.

(1)

That this relation (1) is a straightforward mathematical consequence of the rule pq - qp = —iti will be shown below. Here we can note that equation (1) is a

precise expression for the facts which one earlier sought to describe by the division of phase space into cells of magnitude h. For the determination of the position of the electron one can also do other experiments—for example, collision experi­ ments. A precise measurement of the position demands collisions with very fast particles, because for slow electrons the diffraction phenomena—which, according to Einstein, are consequences of de Broglie waves (as, for example, in the Ramsauer effect)—prevent a sharp specification of location. In a precise measurement of position the momentum of the electron again changes discontinuously. An ele­ mentary estimate of the precision using the formulas for de Broglie waves leads once more to relation (1). Throughout this discussion the concept of "position of the electron" seems well enough defined, and only a word need be added about the "size" of the electron. When two very fast particles hit the electron one after the other within a very short time interval Δί, then the positions of the electron defined by the two particles lie very close together at a distance Δ/. From the regularities which are observed for α-particles we conclude that Al can be pushed down to a magnitude of the order of 10"12 cm if only At is sufficiently small and particles are selected with sufficiently great velocity. This is what we mean when we say that the electron is a corpuscle whose radius is not greater than IO"12 cm. We turn now to the concept of "path of the electron." By path we understand a series of points in space (in a given reference system) which the electron takes as "positions" one after the other. As we already know what is to be understood by "position at a definite time," no new difficulties occur here. Nevertheless, it is easy to recognize that, for example, the often used expression, the "Is orbit of the elec­ tron in the hydrogen atom," from our point of view has no sense. In order to mea­ sure this Is "path" we have to illuminate the atom with light whose wavelength is considerably shorter than 10~8 cm. However, a single photon of such light is enough to eject the electron completely from its "path" (so that only a single point of such a path can be defined). Therefore here the word "path" has no definable meaning. This conclusion can already be deduced, without knowledge of the recent theories, simply from the experimental possibilities. In contrast, the contemplated measurement of position can be carried out on many atoms in a Is state. (In principle, atoms in a given "stationary" state can be selected, for example, by the Stern-Gerlach experiment.) There must therefore exist for a definite state—for example, the Is state—of the atom a probability function for the location of the electron which corresponds to the mean value for the classical orbit, averaged over all phases, and which can be determined through

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the measurement with an arbitrary precision. According to Born,* this function is given by φ l s {q) where φ ls (q) designates the Schrodinger wave function belonging to the Is state. With a view to later generalizations I should like to say— with Dirac and Jordan—that the probability is given by S( 1.s, q)S(ls, q), where S(ls, q) designates that column of the matrix S(E, q) of transformation from E to q that belongs to the energy E = E l s . In the fact that in quantum theory only the probability distribution of the position of the electrons can be given for a definite state, such as Is, one can recog­ nize, with Born and Jordan, a characteristically statistical feature of quantum theory as contrasted to classical theory. However, one can say, if one will, with Dirac, that the statistics are brought in by our experiments. For plainly even in classical theory only the probability of a definite position for the electron can be given as long as we do not know the phase of [the motion of the electron in] the atom. The distinction between classical and quantum mechanics consists rather in this: classically we can always think of the phase as determined through suitable experiments. In reality, however, this is impossible, because every experiment for the determination of phase perturbs or changes the atom. In a definite stationary "state" of the atom, the phases are in principle indeterminate, as one can see as a direct consequence of the familiar equations Et — tE = — ih

or

Jw — wJ = — ih,

where J is the action variable and w is the angle variable. The word "velocity" can easily be defined for an object by measurements when the motion is free of force. For example, one can illuminate the object with red light and by way of the Doppler effect in the scattered light determine the velocity of the particle. The determination of the velocity is the more exact the longer the wavelength of the light that is used, as then the change in velocity of the particle, per light quantum, by way of the Compton effect is so much less. The determina­ tion of position becomes correspondingly inexact, in agreement with equation (1). If one wants to measure the velocity of the electron in the atom at a definite instant, then, for example, one will let the nuclear charge and the forces arising * The statistical interpretation of de Broglie waves was first formulated by A. Einstein (Sitzungsber. d. preussische Akad. d Wiss., p. 3 [1925]). This statistical feature of quantum mechanics then played an essential role in M. Born, W. Heisenberg, and P. Jordan, Quantenmechanik II (Zeits. /. Physik, 35, 557 [1926]), especially chapter 4, §3, and P. Jordan (Zeits. f. Physik, 37, 376 [1926]). It was analyzed mathematically in a seminal paper of M. Born (Zeits. f. Physik, 38, 803 [1926]) and used for the inter­ pretation of collision phenomena One finds how to base the probability picture on the theory of the transformation of matrices in the following papers: W. Heisenberg (Zeits. f. Physik, 40, 501 [1926]), P. Jordan (Zeits. f. Physik, 40, 661 [1926]), W. Pauli (remark in Zeits. J. Physik, 41, 81 [1927]), P. Dirac (Proc. Roy. Soc. Al 13, 621 [1926]), and P. Jordan (Zeits. f. Physik, 40, 809 [1926]). The statistical side of quantum mechanics is discussed more generally in P. Jordan (Naturwiss., 15, 105 [1927]) and M Born (Naturwiss., 15, 238 [1927]).

1.3 PRINCIPLE OF INDETERMINISM

67

from the other electrons suddenly be taken away, so that the motion from then on is force-free, and one will then carry out the measurement described above. As above, one can again easily convince oneself that a [momentum] function p(t) cannot be defined for a given state—such as the ls-state—of an atom. On the contrary, there is again a probability function for ρ in this state which according to Dirac and Jordan has the value S( 1 s, p)S(ls, p). Here S(ls, p) again designates that column of the matrix S(E, p)—that transforms from E to ρ—which belongs to E = Els. Finally we come to experiments which allow one to measure the energy or the value of the action variable J. Such experiments are especially important because only with their help can we define what we mean when we speak of the discon­ tinuous change of the energy and of J. The Franck-Hertz collision experiments allow one to base the measurement of the energy of the atom on the measurement of the energy of electrons in rectilinear motion, because of the validity of the law of conservation of energy in quantum theory. This measurement in principle can be carried out with arbitrary accuracy if only one forgoes the simultaneous deter­ mination of the position of the electron or its phase (see the determination of p, above), corresponding to the relation Et — tE = — ih. The Stern-Gerlach experi­ ment allows one to determine the magnetic or an average electric moment of the atom, and therefore to measure quantities which depend only on the action variable J. The phases remain undetermined in principle. It makes as little sense to speak of the frequency of the light wave at a definite instant as of the energy of an atom at a definite moment. Correspondingly, in the Stern-Gerlach experiment the accuracy of the energy measurement decreases as we shorten the time during which the atom is under the influence of the deflecting field.* Specifically, an upper bound is given for the deviating force through the circumstance that the potential energy of that deflecting force can at most vary inside the beam by an amount which is considerably smaller than the differences in energy of the stationary states. Only then will a determination of the energy of the stationary states be at all possible. Let E 1 be an amount of energy which satisfies this condition (E i also fixes the precision of the energy measurement). Then E l Id specifies the highest allowable value for the deflecting force, if d is the breadth of the beam (measurable through the spacing of the slits employed). The angular deviation of the atomic beam is then E^tjdp, where we designate by the time during which the atoms are under the influence of the deflecting field, and by ρ the momentum of the atoms in the direction of the beam. This deflection must be of at least the same order of magnitude as the natural broadening of the beam brought about by the diffraction by the slits, if any measurement is to be possible. The diffraction angle is roughly λ/d if λ denotes the de Broglie wavelength; thus, * In this connection see W. Pauh, i.e., p. 61

68

HEISENBERG Xjd ~ E^Jdp,

or, as λ = h/p, E 1 I 1 ~ h.

(2)

This equation corresponds to equation (1) and shows how a precise determina­ tion of energy can only be obtained at the cost of a corresponding uncertainty in the time.

§2. THE DIRAC-JORDAN THEORY We might summarize and generalize the results of the preceding section in this statement: All concepts which can be used in classical theory for the description of a mechanical system can also be defined exactly for atomic processes in analogy to the classical concepts. The experiments which provide such a definition themselves suffer an indeterminacy introduced purely by the observational procedures we use when we ask of them the simultaneous determination of two canonically con­ jugate quantities. The magnitude of this indeterminacy is given by relation (1) (generalized to any canonically conjugate quantities whatsoever). It is natural in this respect to compare quantum theory with special relativity. According to relativity, the word "simultaneous" cannot be defined except through experiments in which the velocity of light enters in an essential way. If there existed a "sharper" definition of simultaneity, as, for example, signals which propagate infinitely fast, then relativity theory would be impossible. However, because there are no such signals, or, rather, because already in the definition of simultaneity the velocity of light appears, there is room left for the postulate of the constancy of the speed of light so that this postulate does not contradict any meaningful use of the words "position, velocity, time." We find a similar situation with the definition of the concepts of "position of an electron" and "velocity" in quantum theory. All ex­ periments which we can use for the definition of these terms necessarily contain the uncertainty implied by equation (1), even though they permit one to define exactly the concepts ρ and q taken in isolation. If there existed experiments which allowed simultaneously a "sharper" determination of ρ and q than equation (1) permits, then quantum mechanics would be impossible. Thus only the uncertainty which is specified by equation (1) creates room for the validity of the relations which find their most pregnant expression in the quantum-mechanical commuta­ tion relations, pq - qp = -ih. That uncertainty makes possible this equation without requiring that the physical meaning of the quantities ρ and q be changed. For those physical phenomena whose quantum-mechanical formulation is still

1.3 PRINCIPLE OF INDETERMINISM

69

unknown (for example, electrodynamics), equation (1) makes a demand which may be useful for the discovery of the new laws. For quantum mechanics equation (1) can be derived from the Dirac-Jordan formulation by a slight generalization. If, for any definite state variable t] we determine the position q of the electron as q' with an uncertainty then we can express this fact by a probability amplitude which differs appreciably from zero only in a region of spread For example, one can write proportional to exp

near

(3a)

with therefore proportional to exp

(3b)

Then for the probability amplitude for any given value of p we have (4)

For S(q, p), according to Jordan, we can write (5) Then, according to (4), differs appreciably from zero only for values of p for which is not significantly greater than 1. Specifically, employing (3), we find i is proportional to

that is, proportional to

and thus SS is proportional to exp where (6) The assumption (3) for corresponds therefore to the experimental fact that the value p' is measured for p and the value q' for q [with the limit (6) on the precision]. From the purely mathematical point of view it is characteristic of the Dirac-

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Jordan formulation of quantum mechanics that the relations between p, q, E, etc. can be described as equations connecting very general matrices in such a way that any predetermined quantum-theoretic quantity appears as a diagonal matrix. The possibility of writing things in such a way is evident when one pictures the matrices as tensors (for example, moment-of-inertia tensors) in a multidimensional space between which there are mathematical connections. One can always pick the axis of the coordinate system in which one expresses these relations along the principal axes of one of these tensors. Finally, one also can always characterize the mathematical relation between two tensors A and B through the transformation equations which take a coordinate system oriented along the principal axes of A over into another oriented along the principal axes of B. This latter formulation corresponds to the Schrodinger theory. In contrast, one will view Dirac's g-number formulation as the formulation of quantum mechanics that is really "invariant" and independent of all coordinate systems. When we want to derive physical results from that mathematical framework, then we have to associate numbers with the quantum-theoretical magnitudes—that is, with the matrices (or "tensors" in multidimensional space). This task is to be understood in these terms: In that multidimensional space a definite direction is arbitrarily prescribed (by the nature of the experimental setup); and it is asked what is the "value" of the matrix (for example, in that picture, what is the value of the moment of inertia) in this given direction. This question only has a well-defined meaning when the given direction coincides with the direction of one of the principal axes of that matrix. In this case there is an exact answer for the question. But also when the prescribed direction differs only little from one of the principal axes of the matrix one can still speak of the "value" of the matrix in the prescribed direction up to a definite uncertainty determined by the angle between the two directions. One can therefore say that associated with every quantum-theoretical quantity or matrix is a number which gives its "value" within a certain definite statistical error. The statistical error depends on the coordinate system. For every quantum-theoretical quantity there exists a coordinate system in which the statistical error for this quantity is zero. Therefore a definite experiment can never give exact information on all quantumtheoretical quantities. Rather, it divides physical quantities into "known" and "unknown" (or more and less accurately known quantities) in a way characteristic of the experiment in question. The results of two experiments can be derived exactly one from the other only then when the two experiments divide the physical quantities in the same way into "known" and "unknown" (that is, when the tensors in that multidimensional space frequently invoked—for ease of visualization—are "looked at" in both experiments from the same direction). When two experiments use different divisions into "known" and "unknown," then their results can be related only statistically. For a more detailed discussion of this statistical connection let a gedankenexperiment be considered. Let a Stern-Gerlach atomic beam be sent first through

1.3 PRINCIPLE OF INDETERMINISM

3

a field F t which is so strongly inhomogeneous in the direction of the beam that it induces many transitions by sudden reversal in the force on the spin. Then let the atomic beam run free up to a definite distance from F v But there let a second field F 2 begin, as inhomogeneous as Between and let it be possible to measure the number of atoms in the different stationary states through an optionally applied magnetic field. Let all radiation by the atoms be neglected. If we know that an atom was in a state of energy before it passed F 1 , then we can express this fact by ascribing to the atom a wave function—for example, in pspace—with the definite energy E„ and the undetermined phase

After passage through the field F u this function is changed into* (7)

Here we can make some arbitrary determination of the so that the are uniquely determined by F j . The matrix transforms the energy values before the transition through F x to the values after the transition. If after F t we carry out a determination of the stationary state, say, by use of an inhomogeneous magnetic field, then we will find that the atom has jumped from the wth state to the mth state with a probability When we find experimentally that an atom has indeed jumped to the mth state, then we have to ascribe to it in all calculations thereafter, not the function but simply the function with an undetermined phase. Through the experimental determination, "mth state," we select out of the multitude of different possibilities a definite one, m. However, at the same time we disturb everything that was still contained in the phase relations between the quantities as detailed below. In the transition of the atomic beam through F 2 , what happened at F j repeats itself. Let be the coefficients of the transformation matrix which transform the energies before F 2 to the energies after If no determination of the state is carried out between F 1 and F 2 , then the eigenfunction is transformed according to the following scheme, (8)

Let

be called

If the stationary state of the atom is determined

beyond F 2 , then one will find the state / with the probability In contrast, if between and one determines the state—and finds for it the value then * See P. D i r a c (Proc.

Roy

Soc. A112,

661 [ 1 9 2 6 ] ) a n d M . B o r n

Physik,

40, 167 [ 1 9 2 6 ] ) .

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the probability for "/" beyond F2 is given by d m l d m l . In many repetitions of the entire experiment (in which each time the state is determined between F 1 and F 2 ) one will therefore observe the state /, beyond F 2 , with the relative frequency Z n i = Σ nmCnmd id i· This expression does not agree at all with enlenl. For this reason c

m

m

m

Jordan (I.e.) has spoken of an "interference of probabilities." I cannot agree. The two kinds of experiments which lead respectively to enlenl and Znl are physically distinct. In one case the atom experiences no disturbance between F1 and F2. In the other case it is perturbed by the apparatus which determines its stationary state. This apparatus has as a consequence that the "phase" of the atom changes by an amount that is in principle uncheckable, as the momentum of an electron likewise changes with a determination of its position (see §1). The magnetic field for the determination of the state between F1 and F2 will separate the eigenvalues E. In the observation of the path of the atomic beam the atoms are slowed down by statistically different and uncheckable amounts (I think here, say, of Wilson cloudchamber pictures). This has as a consequence that the final transformation matrix (from the energy value before entry into F1 to the energy after exit from F2) is no longer given by £ cnmdml, but every term in this sum has additionally an unknown m

phase factor. No expectation is therefore open to us, except that the mean value of enlenl averaged over all these expected phase alterations is equal to Znl. A simple calculation confirms that this is the case. We can therefore deduce from one experiment the possible results of another by definite statistical rules. The other experiment itself selects out of the plenitude of all possibilities a quite definite one, and thereby limits the possibilities for all later experiments. Such an interpretation of the equation for the transformation matrix S or the Schrodinger wave equation is only possible because the sum of solutions is again a solution. In this circumstance we see the deep significance of the linearity of Schrddinger wave equations. On that account they can be understood only as equations for waves in phase space; and on that account we may regard as hopeless every attempt to replace these equations by nonlinear equations, for example in the relativistic case (for more than one electron).

§3. THE TRANSITION FROM MICRO- TO MACROMECHANICS

It seems to me that the concepts of kinematics and mechanics in quantum theory are sufficiently clarified by the analysis of the words "position of an electron," "velocity," "energy," etc., in the preceding sections that physical understanding of macroscopic processes from the standpoint of quantum mechanics must also be possible. The transition from micro- to macromechanics has already been treated by Schrodinger,* but I do not believe that Schrodinger's considerations * E. Schrodinger, Naturwiss., 14, 664 (1926).

1.3 PRINCIPLE OF INDETERMINISM

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get to the heart of the problem, and this is why: According to Schrodinger, in a high state of excitation a sum of eigenfunctions ought to be able to give a wave packet of limited extent which—through periodic changes in its form—will carry out the periodic motions of the classical "electron." There is an argument against this outlook: If the wave packet had such properties as ascribed to it by this view, then the radiation sent out by the atom would be representable as a Fourier series in which the frequencies of the higher vibrations were integer multiples of the basic frequency. The frequencies of the spectral lines sent out by the atom are, however, according to quantum mechanics, never integer multiples of the basic frequency—except in the special case of the harmonic oscillator. Thus Schrodinger's reasoning is only viable for the case of the harmonic oscillator treated by him; in all other cases a wave packet spreads out in the course of time over the whole immediate neighborhood of the atom. The higher the state of excitation of the atom, the slower is that spreading of the wave packet. However, if one waits long enough it happens. The reasoning developed above about the radiation sent out by the atom might at first sight be used against all experiments which look for a direct transition from quantum to classical mechanics at high quantum numbers. For that reason the attempt was made earlier to circumvent such reasoning by referring to the natural radiation broadening of stationary states—certainly wrongly, first of all because this way out is blocked already in the case of the hydrogen atom on account of the weakness of the radiation for high states, and secondly, because the transition from quantum to classical mechanics ought to be understandable without calling on electrodynamics. Bohr* has already referred many times to these well-known difficulties which stand in the way of a direct connection between quantum and classical theory. We have spelled them out again here so explicitly only because in recent times they seem to be forgotten. I believe that one can fruitfully formulate the origin of the classical "orbit" in this way: the "orbit" comes into being only when we observe it. For example, let an atom be given in a state of excitation η = 1000. The dimensions of the orbit in this case are already relatively large so that, in accordance with §1, it is enough to use light of relatively low wavelength to determine the position of the electron. If the position determination is not to be too fuzzy then the Compton recoil will put the atom in some state of excitation between, say, 950 and 1050. Simultaneously, the momentum of the electron can be determined from the Doppler effect with a precision given by (1). One can characterize the experimental finding by a wavepacket, or, better, a probability-amplitude packet, in q-space of a spread given by the wavelength of the light used, and built up primarily out of eigenfunctions between the 950th and 1050th eigenfunction—and by a corresponding packet in p-space. Let a new determination of position be made after some time with the same precision. Its result, according to §2, can be predicted only statistically. All * N Bohr, "Grundpostulate der Quantentheorie," I.e.

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positions count as likely (with calculable probability) which lie within the bounds of the now broadened wavepacket. The situation would be no different in classical theory, for there too the result of the second position measurement would be predictable only statistically because of the uncertainty in the first measurement. Also, the orbits of classical theory would spread out like the wavepacket. However, statistical laws themselves are different in quantum mechanics and in classical theory. The second determination of the position selects a definite "q" from the totality of possibilities and limits the options for all subsequent measurements. After the second position determination the results of later measurements can only be calculated when one again ascribes to the electron a "smaller" wavepacket of extension λ (wavelength of the light used in the observation). Thus every position determination reduces the wavepacket back to its original extension λ. The "values" of the quantities p, q are known throughout all the experiments with a certain precision. The values of ρ and q stay within the precision limits fixed by the classical equations of motion. This can be seen directly from the quantum-mechanical equations, ρ = -