Philosophic Foundations of Quantum Mechanics Philosophic Foundations of Quantum Mechanics By HANS REICHENBACH PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF CALIFORNIA UNIVERSITY OF CALIFORNIA PRESS B E R K E L E Y A N D L O S A N G E L E S ■ 1944 UNIVERSITY OF CALIFORNIA PRESS BERKELEYAND LOS ANGELES CALIFORNIA <> CAMBRIDGE UNIVERSITY PRESS TONDON,ENGLAND COPYRIGHT, IQ44, BY THE REGENTS Op THE UNIVERSITY OF CALIFORNIA PRINTED IN THE UNITED STATES OF AMERICA BY THE UNIVERSITY OP CALIFORNIA PRESS PREFACE Two great theoretical constructions have shaped the face of modern physics: the theory of relativity and the theory of quanta. The first has been, on the whole, the discovery of one man, since the work of Albert Einstein has remained unparalleled by the contributions of others who, like Hendrik Anton Lorentz, came very close to the foundations of special rela­ tivity, or, like Hermann Minkowski, determined the geometrical form of the theory. It is different with the theory of quanta. This theory has been de­ veloped by the collaboration of a number of men each of whom has contributed an essential part, and each of whom, in his work, has made use of the results of others. The necessity of such teamwork seems to be deeply rooted in the subject matter of quantum theory. In the first place, the development of this theory has been greatly dependent on the production of observational results and on the exactness of the numerical values obtained. Without the help of the army of experimenters who photographed spectral lines or watched the behavior of elementary particles by means of ingenious devices, it would have been im­ possible ever to carry through the theory of the quanta even after its founda­ tions had been laid. In the second place, these foundations are very different in logical form from those of the theory of relativity. They have never had the form of one unifying principle, not even after the theory has been completed. They consist of a set of principles which, despite their mathematical elegance, do not possess the suggestive character of a principle which convinces us at first sight, as does the principle of relativity. And, finally, they depart much further from the principles of classical physics than the theory of relativity ever did in its criticism of space and time; their implications include, in addition to a transition from causal laws to probability laws, a revision of philosophical ideas about the existence of unobserved objects, even of the principles of logic, and reach down to the deepest fundamentals of the theory of knowledge. In the development of the theoretical form of quantum physics, we can dis­ tinguish four phases. The first phase is associated with the names of Max Planck, Albert Einstein, and Nils Bohr. Planck's introduction of the quanta in 1900 was followed by Einstein's extension of the quantum concept toward that of a needle radiation (1905). The decisive step, however, was made in Bohr's application (1913) of the quantum idea to the analysis of the structure of the atom, which led to a new world of physical discoveries. The second phase, which began in 1925, represents the work of a younger generation which had been trained in the physics of Planck, Einstein, and Bohr, and started where the older ones had stopped. It is a most astonishing fact that this phase, which led up to quantum mechanics, began without a clear insight into what was actually being done. Louis de Broglie introduced waves as companions of particles; Erwin Schrodinger, guided by mathematical on VI PREFACE analogies with wave optics, discovered the two fundamental differential equa­ tions of quantum mechanics; Max Born, Werner Heisenberg, Pascual Jordan, and, independently of this group, Paul A. M. Dirac constructed the matrix mechanics which seemed to defy any wave interpretation. This period repre­ sents an amazing triumph of mathematical technique which, masterly applied and guided by a physical instinct more than by logical principles, determined the path to the discovery of a theory which was able to embrace all observable data. All this was done in a very short time; by 1926 the mathematical shape of the new theory had become clear. The third phase followed immediately; it consisted in the physical interpre­ tation of the results obtained. Schrodinger showed the identity of wave me­ chanics and matrix mechanics. Born recognized the probability interpretation of the waves. Heisenberg saw that the mathematical mechanism of the theory involves an unsurmountable uncertainty of predictions and a disturbance of the object by the measurement. And here once more Bohr intervened in the work of the younger generation and showed that the description of nature given by the theory was to leave open a specific ambiguity which he formu­ lated in his principle of complementarity. The fourth phase continues up to the present day; it is filled with constant extensions of the results obtained toward further and further applications, including the application to new experimental results. These extensions are combined with mathematical refinements; in particular, the adaptation of the mathematical method to the postulates of relativity is in the foreground of the investigations. We shall not speak of these problems here, since our inquiry is concerned with the logical foundations of the theory. It was with the phase of the physical interpretations that the novelty of the logical form of quantum mechanics was realized. Something had been achieved in this new theory which was contrary to traditional concepts of knowledge and reality. It was not easy, however, to say what had happened, i.e., to proceed to the philosophical interpretation of the theory. Based on the physical interpretations given, a philosophy for common use was developed by the physicists which spoke of the relation of subject and object, of pictures of reality which must remain vague and unsatisfactory, of operationalism which is satisfied when observational predictions are correctly made, and renounces interpretations as unnecessary ballast. Such concepts may appear useful for the purpose of carrying on the merely technical work of the physicist. But it seems to us that the physicist, whenever he tried to be conscious of what he did, could not help feeling a little uneasy with this philosophy. He then became aware that he was walking, so to speak, on the thin ice of a superficially frozen lake, and he realized that he might slip and break through at any moment. It was this feeling of uneasiness which led the author to attempt a philo­ sophical analysis of the foundations of quantum mechanics. Fully aware that philosophy should not try to construct physical results, nor try to prevent PREFACE vii physicists from finding such results, he nonetheless believed that a logical analysis of physics which did not use vague concepts and unfair excuses was possible. The philosophy of physics should be as neat and clear as physics itself; it should not take refuge in conceptions of speculative philosophy which must appear outmoded in the age of empiricism, nor use the operational form of empiricism as a way to evade problems of the logic of interpretations. Di­ rected by this principle the author has tried in the present book to develop a philosophical interpretation of quantum physics which is free from meta­ physics, and yet allows us to consider quantum mechanical results as state­ ments about an atomic world as real as the ordinary physical world. It scarcely will appear necessary to emphasize that this philosophical analysis is carried through in deepest admiration of the work of the physicists, and that it does not pretend to interfere with the method of physical inquiry. All that is intended in this book is clarification of concepts; nowhere in this presenta­ tion, therefore, is any contribution toward the solution of physical problems to be expected. Whereas physics consists in the analysis of the physical world, philosophy consists in the analysis of our knowledge of the physical world. The present book is meant to be philosophical in this sense. The division of the book is so planned that the first part presents the general ideas on which quantum mechanics is based; this part, therefore, outlines our philosophical interpretation and summarizes its results. The presentation is such that it does not presuppose mathematical knowledge, nor an acquaintance with the methods of quantum physics. In the second part we present the out­ lines of the mathematical methods of quantum mechanics; this is so written that a knowledge of the calculus should enable the reader to understand the exposition. Since we possess today a number of excellent textbooks on quantum mechanics, such an exposition may appear unnecessary; we give it, however, in order to open a short cut toward the mathematical foundations of quantum mechanics for all those who do not have the time for thorough studies of the subject, or who would like to see in a short review the methods which they have applied in many individual problems. Our presentation, of course, makes no claim to be complete. The third part deals with the various interpretations of quantum mechanics; it is here that we make use of both the philosophical ideas of the first part and the mathematical formulations of the second. The properties of the different interpretations are discussed, and an interpretation in terms of a three-valued logic is constructed which appears as a satisfactory logical form of quantum mechanics. I am greatly indebted to Dr. Valentin Bargmann of the Institute of Ad­ vanced Studies in Princeton for his advice in mathematical and physical questions; numerous improvements in the presentation, in Part II in particular, are due to his suggestions. I wish to thank Dr. Norman C. Dalkey of the Uni­ versity of California, Los Angeles, and Dr. Ernest H. Hutten, formerly at Los Angeles, now in the University of Chicago, for the opportunity of discussing via PREFACE with them questions of a logical nature, and for their assistance in matters of style and terminology. Finally I wish to thank the staff of the University of California Press for the care and consideration with which they have edited my book and for their liberality in following my wishes concerning some deviations from established usage in punctuation. A presentation of the views developed in this book, including an exposition of the system of three-valued logic introduced in § 32, was given by the author at the Unity of Science Meeting in the University of Chicago on September 5,1941. Hans R eichenbach Department of Philosophy, University of California, Los Angeles June, 19J.2 C O N T E N T S Part I: General Considerations PAGE § 1. Causal laws and probability laws........................................................ 1 § 2. The probability distributions.............................................................. 5 § 3. The principle of indeterminacy............................................................ 9 § 4. The disturbance of the object by the observation........................... . 14 § 5. The determination of unobserved objects.......................................... . 17 § 6. Waves and corpuscles............................................................................ . 20 § 7. Analysis of an interference experiment.............................................. . 24 § 8. Exhaustive and restrictive interpretations........................................ . 32 Part II: Outlines op the M athematics op Quantum M echanics § 9. Expansion of a function in terms of an orthogonal set................... . 45 § 10. Geometrical interpretation in the function space............................. . 52 §11. Reversion and iteration of transformations....................................... . 58 § 12. Functions of several variables and the configuration space............ . 64 § 13. Derivation of Schrodinger's equation from de Broglie’s principle... . 66 § 14. Operators, eigen-functions, and eigen-values of physical entities.. . 72 §15. The commutation rule.......................................................................... . 76 § 16. Operator matrices.................................................................................. . 78 §17. Determination of the probability distributions................................ . 81 § 18. Time dependence of the ^-function.................................................... . 85 § 19. Transformation to other state functions............................................ . 90 § 20. Observational determination of the ^-function................................ . 91 § 21. Mathematical theory of measurement............................................... . 95 § 22. The rules of probability and the disturbance by the measurement. 100 § 23. The nature of probabilities and of statistical assemblages in quantum mechanics.......................................................................... . 105 X CONTENTS Part III: Interpretations PAGE §24. Comparison of classical and quantum mechanical statistics.......... in §25. The corpuscle interpretation................................................................. 118 §26. The impossibility of a chain structure................................................. 122 §27. The wave interpretation........................................................................ 129 §28. Observational language and quantum mechanical language............ 136 §29. Interpretation by a restricted meaning............................................... 139 §30. Interpretation through a three-valued logic........................................ 144 §31. The rules of two-valued logic................................................................ 148 §32. The rules of three-valued logic.............................................................. 150 §33. Suppression of causal anomalies through a three-valued logic........ 160 §34. Indeterminacy in the object language.................................................. 166 §35. The limitation of measurability............................................................ 169 §36. Correlated systems.................................................................................. 170 §37. Conclusion................................................................................................ 176 179