# Quantum Theory and Measurement

John Archibald Wheeler
Wojciech Hubert Zurek
Pages: 840
https://www.jstor.org/stable/j.ctt7ztxn5

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1. Front Matter
(pp. i-iv)
2. Bohr and Einstein in Dialogue
(pp. v-x)
(pp. xi-xiv)
4. PREFACE
(pp. xv-xx)
John Archibald Wheeler and Wojciech Hubert Zurek
(pp. xxi-xxviii)
6. ### I. QUESTIONS OF PRINCIPLE

• I.1 THE BOHR-EINSTEIN DIALOGUE
(pp. 3-8)

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 philosophical 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 to get an impression of stages in Bohr’s development of the concepts of “complementarity,” “closure,” and “phenomenon.” The book states more sharply than Bohr does in his writings the points on which Bohr disagreed with others,...

• I.1 DISCUSSION WITH EINSTEIN ON EPISTEMOLOGICAL PROBLEMS IN ATOMIC PHYSICS
(pp. 9-49)
Niels Bohr

WHEN invited by the Editor of the series, “Living Philosophers,” to write an article for this volume in which contemporary scientists are honouring the epoch-making contributions of Albert Einstein to the progress of natural philosophy 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...

• I.2 BORN’S PROBABILISTIC INTERPRETATION
(pp. 50-51)

… Schrödinger 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. Significantly, 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...

• I.2 ON THE QUANTUM MECHANICS OF COLLISIONS [Preliminary communication]
(pp. 52-55)
Max Born

Heisenberg’s quantum mechanics has so far been applied exclusively to the calculation 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...

• I.3 THE PRINCIPLE OF INDETERMINACY
(pp. 56-61)

In July [1926] I visited my parents in Munich and on this occasion I heard a lecture given by Schrödinger for the physicists in Munich about his work on wave mechanics. It was thus that I first became acquainted with the interpretation Schrödinger wanted to give his mathematical formalism of wave mechanics, and I was very disturbed about the confusion with which I believed this would burden atomic theory. Unfortunately, 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 Schrödinger’s interpretation...

• I.3 THE PHYSICAL CONTENT OF QUANTUM KINEMATICS AND MECHANICS
(pp. 62-84)
Werner Heisenberg

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 consequences 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...

• I.4 COMPLEMENTARITY
(pp. 85-86)

Complementarity is no system, no doctrine with ready-made precepts. There is no via regia to it; no formal definition of it can even be found in Bohr’s writings, and this worries many people. The French are shocked by this breach of the Cartesian rules; they blame Bohr for indulging in “clairobscur” and shrouding himself in “les brumes du Nord.” The Germans in their thoroughness have been at work distinguishing several forms of complementarity and studying, in hundreds of pages, their relations to Kant. Pragmatic Americans have dissected complementarity with the scalpel of symbolic logic and undertaken to define this gentle...

• I.4 THE QUANTUM POSTULATE AND THE RECENT DEVELOPMENT OF ATOMIC THEORY
(pp. 87-126)
Niels Bohr

Although it is with great pleasure that I follow the kind invitation of the presidency of the congress to give an account of the present state of the quantum theory in order to open a general discussion on this subject, which takes so central a position in modern physical science, it is with a certain hesitation that I enter on this task. Not only is the venerated originator of the theory present himself, but among the audience there will be several who, due to their participation in the remarkable recent development, will surely be more conversant with details of the...

• I.5 THE UNCERTAINTY PRINCIPLE
(pp. 127-128)
H. P. Robertson

The uncertainty principle is one of the most characteristic and important Consequences of the new quantum mechanics. This principle, as formulated by Heisenberg for two conjugate quantum-mechanical variables, states that the accuracy with which two such variables can be measured simultaneously is subject to the restriction that the product of the uncertainties in the two measurements is at least of orderh(Planck’s constant). Condon* has remarked that an uncertainty relation of this type can not hold in the general case where the two variables under consideration are not conjugate, and has stressed the desirability of obtaining a general formulation...

• I.6 THE WAVE MECHANICS OF α-RAY TRACKS
(pp. 129-134)
Nevill F. Mott

The present note is suggested by a recent paper by Prof. Darwin,* and is intended to show how one of the most typically particle-like properties of matter can be derived from the wave mechanics. In the theory of radioactive disintegration, as presented by Gamow, the α-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the α-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave...

• I.7 Knowledge of the past and future in quantum mechanics
(pp. 135-136)
Albert Einstein, Richard C. Tolman and Boris Podolsky

It is well known that the principles of quantum mechanics limit the possibilities of exact prediction as to the future path of a particle. It has sometimes been supposed, nevertheless, that the quantum mechanics would permit an exact description of the past path of a particle.

The purpose of the present note is to discuss a simple ideal experiment which shows that the possibility of describing the past path of one particle would lead to predictions as to the future behaviour of a second particle of a kind not allowed in the quantum mechanics. It will hence be concluded that...

• I.8 THE EINSTEIN-PODOLSKY-ROSEN PAPER
(pp. 137-137)

[Einstein] attended [Bohr’s 1933 Solvay] lecture and followed the argument with the closest attention; he made no direct comment on it, but put at once the discussion on the general theme of the meaning of quantum theory. He had no longer any doubt about the logic of Bohr’s argumentation; but he still felt the same uneasiness as before (“Unbehagen” was his word) when confronted with the strange consequences of the theory. “What would you say of the following situation?” he asked me. “Suppose two particles are set in motion towards each other with the same, very large, momentum, and that...

• I.8 CAN QUANTUM-MECHANICAL DESCRIPTION OF PHYSICAL REALITY BE CONSIDERED COMPLETE?
(pp. 138-141)
Albert Einstein, Boris Podolsky and Nathan Rosen

ANY serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.

In attempting to judge the success of a physical theory, we may ask ourselves two questions : (1) “Is the theory correct?” and (2) “Is the description given by the theory complete?” It is only in the case in which positive answers may be given to...

(pp. 142-143)

This onslaught came down upon us as a bolt from the blue. Its effect on Bohr was remarkable. We were then in the midst of groping attempts at exploring the implications of the fluctuations of charge and current distributions, which presented us with riddles of a kind we had not met in electrodynamics. A new worry could not come at a less propitious time. Yet, as soon as Bohr had heard my report of Einstein’s argument, everything else was abandoned: we had to clear up such a misunderstanding at once. We should reply by taking up the same example and...

• I.9 QUANTUM MECHANICS AND PHYSICAL REALITY
(pp. 144-144)
Niels Bohr

In a recent article by A. Einstein, B. Podolsky and N. Rosen, which appeared in thePhysical Reviewof May 15, and was reviewed in Nature of June 22, the question of the completeness of quantum mechanical description has been discussed on the basis of a “criterion of physical reality”, which the authors formulate as follows : “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”.

Since, as the authors show, it is always possible in quantum...

• I.10 CAN QUANTUM-MECHANICAL DESCRIPTION OF PHYSICAL REALITY BE CONSIDERED COMPLETE?
(pp. 145-151)
Niels Bohr

IN a recent article¹ under the above title A. Einstein, B. Podolsky and N. Rosen have presented arguments which lead them to answer the question at issue in the negative. The trend of their argumentation, however, does not seem to me adequately to meet the actual situation with which we are faced in atomic physics. I shall therefore be glad to use this opportunity to explain in somewhat greater detail a general viewpoint, conveniently termed “complementarity,” which I have indicated on various previous occasions,² and from which quantum mechanics within its scope would appear as a completely rational description of...

• I.11 THE PRESENT SITUATION IN QUANTUM MECHANICS: A TRANSLATION OF SCHRÖDINGER’S “CAT PARADOX” PAPER
(pp. 152-167)
Erwin Schrödinger

This is a translation of Schrodinger’s three-part 1935 paper¹ inDie NaturwissenschaftenEarlier that same year there had appeared the Einstein. Podolsky, Rosen paper² (also famous in “paradoxology”) which, Schrodinger says, in a footnote, motivated his offering. Along with this article in German, Schrodinger had two closely related English-language publications.¹ But the German, aside from its one-paragraph presentation of the famous cat, covers additional territory and gives many fascinating insights into Schrodinger’s thought. The translator’s goal has been to adhere to the logical and physical content of the original, while at the same time trying to convey something of its...

• I.12 REMARKS ON THE MIND-BODY QUESTION
(pp. 168-181)
Eugene P. Wigner

F. Dyson, in a very thoughtful article,¹ points to the everbroadening scope of scientific inquiry. Whether or not the relation of mind to body will enter the realm of scientific inquiry in the near future—and the present writer is prepared to admit that this is an open question—it seems worthwhile to summarize the views to which a dispassionate contemplation of the most obvious facts leads. The present writer has no other qualification to offer his views than has any other physicist and he believes that most of his colleagues would present similar opinions on the subject, if pressed....

• I.13 LAW WITHOUT LAW
(pp. 182-214)
John Archibald Wheeler

The second phase of the dialog began in Europe but continued in America from Einstein’s arrival at Princeton in October, 1933, to his death there in April, 1955. Here Einstein tried to show that quantum theory — in making what happens depend upon what the observer chooses to measure—is incompatible with any reasonable idea of reality.18Bohr’s reply19briefly summarized was this: Your concept of reality is too limited.

Of all the idealized experiments taken up by the two friends in their effort to win agreement, none is simpler than the beam splitter of fig. 4. With the final...

7. ### II. INTERPRETATIONS OF THE ACT OF MEASUREMENT

• II.1 THE THEORY OF OBSERVATION IN QUANTUM MECHANICS
(pp. 217-259)
Fritz London and Edmond Bauer

Quantum physics has brought an essential advance to science, the finding that in every experiment or measurement there inescapably enters the duality between subject and object, the action and reaction of observer and system observed, the observer and the measuring apparatus being viewable as one entity.

The classical view, disregarding the necessarily limited character of our knowledge and the retroactive effect of the measurement on the system observed, always postulated the possibility of an infinitely precise knowledge of the simultaneous values of all the parameters used for the description of the system. Heisenberg, in giving concrete significance to his principle...

• II.2 INTERPRETATION OF QUANTUM MECHANICS
(pp. 260-314)
Eugene P. Wigner

The conceptual problems generated by the generally accepted interpretation of quantum mechanics overshadow in philosophical depth those generated by the older quantum theory (see for example the collections of basic papers edited by ter Haar, 1967, and Kangro, 1972) to such an extent that one is likely to forget the latter. Nevertheless, these were very real also, though more concrete and lying more within physics proper than those generated by quantum mechanics.

The idea of the quantum emission and absorption of radiation was conceived by M. Planck (1900a,b) in order to explain the finite energy density of the black body...

• II.3 “RELATIVE STATE” FORMULATION OF QUANTUM MECHANICS
(pp. 315-323)
Hugh Everett III

THE task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity.

The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional...

• II.4 THE PROBLEM OF MEASUREMENT
(pp. 324-341)
Eugene P. Wigner

The last few years have seen a revival of interest in the conceptual foundations of quantum mechanics.¹ This revival was stimulated by the attempts to alter the probabilistic interpretation of quantum mechanics. However, even when these attempts turned out to be less fruitful than its protagonists had hoped,² the interest continued. Hence, after the subject had been dormant for more than two decades, we again hear discussions on the basic principles of quantum theory and the epistemologies that are compatible with it. As is often the case under similar circumstances, some of the early thinking had been forgotten; in fact,...

• II.5 ON THE INTERPRETATION OF MEASUREMENT IN QUANTUM THEORY
(pp. 342-350)
H. D. Zeh

The problem of measurement in quantum theory and the related problem of how to describe classical phenomena in the framework of quantum theory have received increased attention during recent years. The various contributions express very different viewpoints, and may roughly be classified as follows:

1. Those emphasizing contradictions obtained when the process of measurement is itself described in terms of quantum theory.(1)

2. Those claiming that measurement may well be explained by quantum theory in the sense that “quantum-mechanical noncausality” can be derived from statistical uncertainties inherent in the necessarily macroscopic apparatus of measurement.(2)

3. Those introducing new physical concepts like hidden variables.(3)...

8. ### III. “HIDDEN VARIABLES” VERSUS “PHENOMENON” AND COMPLEMENTARITY

• III.1 POLYELECTRONS
(pp. 353-355)
John Archibald Wheeler

Theoretical evidence for the existence of entities composed entirely of electrons and positrons is presented in the following article, together with a discussion of their properties.¹ The simplest of these entities consists of one electron and one positron, bound together in a structure similar to that of the hydrogen atom. The next higher entity is composed of two positrons and one electron or of two electrons and one positron. The bi-electron system is stable by 6.77 ev against dissociation. Against annihilation, it has a life time of 1.24 × 10−10sec., when the spins of the two particles are parallel,...

• III.2 THE PARADOX OF EINSTEIN, ROSEN, AND PODOLSKY
(pp. 356-368)
David Bohm

15. The Paradox of Einstein, Rosen, and Podolsky. In an article in the Physical Review, † Einstein, Rosen, and Podolsky raise a serious criticism of the validity of the generally accepted interpretation of quantum theory. This objection is raised in the form of a paradox to which they are led on the basis of their analysis of a certain hypothetical experiment, which we shall discuss in detail later. Their criticism has, in fact, been shown to be unjustified, ‡ and based on assumptions concerning the nature of matter which implicitly contradict the quantum theory at the outset. Nevertheless, these implicit assumptions...

• III.3 A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF “HIDDEN” VARIABLES, I AND II
(pp. 369-396)
David Bohm

THE usual interpretation of the quantum theory is based on an assumption having very far-reaching implications,viz., that the physical state of an individual system is completely specified by a wave function that determines only the probabilities of actual results that can be obtained in a statistical ensemble of similar experiments. This assumption has been the object of severe criticisms, notably on the part of Einstein, who has always believed that, even at the quantum level, there must exist precisely definable elements or dynamical variables determining (as in classical physics) the actual behavior of each individual system, and not merely...

• III.4 ON THE PROBLEM OF HIDDEN VARIABLES IN QUANTUM MECHANICS
(pp. 397-402)
John S. Bell

To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical “dispersion free” states would be specified not only by the quantum mechanical state vector but also by additional “hidden variables”—“hidden” because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably...

• III.5 ON THE EINSTEIN PODOLSKY ROSEN PARADOX
(pp. 403-408)
John S. Bell

THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality [2]. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the...

• III.6 PROPOSED EXPERIMENT TO TEST LOCAL HIDDEN-VARIABLE THEORIES
(pp. 409-413)
John F. Clauser, Michael A. Horne, Abner Shimony and Richard A. Holt

Einstein, Podolsky, and Rosen (EPR) in a classic paper¹ presented a paradox which led them to infer that quantum mechanics is not a complete theory. They concluded that the quantum mechanical description of a physical system should be supplemented by postulating the existence of “hidden variables,” the specification of which would predetermine the result of measuring any observable of the system. They believed the predictions of quantum mechanics to be correct, but only as consequences of statistical distributions of the hidden variables. Bohr² argued in reply that no paradox can be derived from the assumption of completeness if one recognizes...

• III.7 EXPERIMENTAL TEST OF LOCAL HIDDEN-VARIABLE THEORIES
(pp. 414-417)
Stuart J. Freedman and John F. Clauser

Since quantum mechanics was first developed, there have been repeated suggestions that its statistical features possibly might be described by an underlying deterministic substructure. Such features, then, arise because a quantum state represents a statistical ensemble of “hidden-variable states.” Proofs by von Neumann and others, demonstrating the impossibility of a hid den-variable substructure consistent with quantum mechanics, rely on various assumptions concerning the character of the hidden variables.¹ Bell has argued that these assumptions are unduly restrictive. However, by considering an idealized case of two spatially separated but quantum-mechanically correlated systems, he was able to show that any hidden-variable theory...

• III.8 EXPERIMENTAL TEST OF LOCAL HIDDEN-VARIABLE THEORIES
(pp. 418-421)
Edward S. Fry and Randall C. Thompson

We have measured the linear polarization correlation of photon pairs from the 7³S1-6³P1-6¹S0cascade of Hg200. Under appropriate experimental conditions, quantum mechanics (QM) predicts there should be a very strong correlation. The essence of Bell’s theorem¹ is that any local hidden variable (LHV) theory restricts the strength of this correlation. This LHV restriction can be put in a form derived by Freedman,²$\delta =\left| R(67{{\frac{1}{2}}^{0}})/{{R}_{0}}-R(22{{\frac{1}{2}}^{0}})/{{R}_{0}} \right|-\frac{1}{4} \leqslant 0.\caption {(1)}$

Here the two photons are respectively detected on the ±Zaxes,R(φ) is the coincidence rate with an angleφbetween the transmission axes of the polarizers, andR0is the coincidence rate with polarizers removed. A...

• III.9 QUANTUM MECHANICS AND HIDDEN VARIABLES: A TEST OF BELL’S INEQUALITY BY THE MEASUREMENT OF THE SPIN CORRELATION IN LOW-ENERGY PROTON-PROTON SCATTERING
(pp. 422-434)
M. Lamehi-Rachti and W. Mittig

Since the beginning of quantum mechanics (QM) a number of physicists who contributed the most to the development of this theory had serious doubts about its logical foundations. Most of the problems were illustrated by a number of paradoxes, such as those of Einstein, Podolsky, and Rosen¹ and Schrödinger (namely, the cat paradox).² These discussions never died down and even today there is no theory of measurement which satisfies everybody.

One attempt to overcome these difficulties was to suppose that there are some supplementary variables outside the scope of QM (“hidden variables”) which determine the result of the individual measurement....

• III.10 PROPOSED EXPERIMENT TO TEST THE NONSEPARABILITY OF QUANTUM MECHANICS
(pp. 435-442)
Alain Aspect

The so-called nonlocality paradox of Einstein, Podolsky, and Rosen¹ has been much discussed. Bell² has shown the possibility of bringing the question into the experimental domain. Then, several experiments have been proposed and performed.3−8All these experiments are able to discriminate between quantum mechanics and “local” hidden-variable theories that fulfill Bell’s condition of locality: The setting of a measuring device does not influence the result obtained with another remote measuring device (nor does it influence the way in which particles are emitted by a distant source). Most of the experiments contradict these local hidden-variable theories,4,5,7although conflicting results exist.6,8

Although...

• III.11 COMPLEMENTARITY IN THE DOUBLE-SLIT EXPERIMENT: QUANTUM NONSEPARABILITY AND A QUANTITATIVE STATEMENT OF BOHR’S PRINCIPLE
(pp. 443-454)
William K. Wootters and Wojciech H. Zurek

In Einstein’s version of the double-slit experiment,1,2one can retain a surprisingly strong interference pattern by not insisting on a 100% reliable determination of the slit through which each photon passes. The analysis leading us to this conclusion involves the following considerations. The plate which receives the kick from each photon can either be stopped and its position measured, or released and its momentum measured. These two options give us two ways of subdividing the original ensemble of photons: (1) according to the measured position of the plate, and (2) according to the measured momentum of the plate. In case...

• III.12 COMPLEMENTARITY IN THE DOUBLE-SLIT EXPERIMENT: ON SIMPLE REALIZABLE SYSTEMS FOR OBSERVING INTERMEDIATE PARTICLE-WAVE BEHAVIOR
(pp. 455-456)
Lawrence S. Bartell

In a recent analysis of Einstein’s version of the double-slit experiment, Wootters and Zurek¹ presented a detailed treatment of the potentially observable manifestations of particle and wave characteristics of light. This approach led them to a quantitative expression of Bohr’s complementarity principle appropriate for such experiments. Einstein had originally proposed that the lateral kick imparted by a photon to an interference screen could be used to identify which slit the photon traveled through on its way to the screen. Pointing out that thermal noise in practical experiments would utterly obscure such photon-induced recoils, Wheeler² recently reformulated the Einstein experiment in...

• III.13 A “DELAYED-CHOICE” QUANTUM MECHANICS EXPERIMENT
(pp. 457-462)
William C. Wickes, Carroll O. Alley and Oleg Jakubowicz

In a traditional optical double-slit interference experiment (Fig. 1), photons from a distant point-source are incident upon a screen containing two small apertures. With a lens behind the screen, an image of the source is formed on, for example, a photographic plate situated in the focal plane of the lens. The size and shape of the image depend upon the apertures, but the image will be crossed by interference fringes of angular frequencyλ/D, whereDis the separation of the apertures andλis the wavelength of the light. Such an experiment isolates the wave character of the incident...

9. ### IV. FIELD MEASUREMENTS

• IV.1 EXTENSION OF THE UNCERTAINTY PRINCIPLE TO RELATIVISTIC QUANTUM THEORY
(pp. 465-476)
Lev Davidovich Landau and Rudolf Peierls

It is known that the application of the methods of wave mechanics to problems in which the speed of light cannot be regarded as infinite leads to absurd results. In the first place, states with negative mass appear in Dirac’s relativistic equation¹. This difficulty arises because the relation between momentum and energy in relativity theory is quadratic, so that two energy states are possible for a given momentum. In contrast to classical (h= 0) relativity theory, where the continuous change of all quantities means that transitions between the two kinds of state are impossible, such transitions cannot reasonably be...

• IV.2 THE MEASURABILITY OF THE ELECTROMAGNETIC FIELD
(pp. 477-478)

When I arrived at the Institute on the last day of February, 1931, for my annual stay, the first person I saw was Gamow. As I asked him about the news, he replied in his own picturesque way by showing me a neat pen drawing he had just made.* It represented Landau, tightly bound to a chair and gagged, while Bohr, standing before him with upraised forefinger, was saying: “Bitte, bitte, Landau, muss ichnur ein Wort sagen!” I learned that Landau and Peierls had just come a few days before with some new paper of theirs which they wanted...

• IV.2 ON THE QUESTION OF THE MEASURABILITY OF ELECTROMAGNETIC FIELD QUANTITIES
(pp. 479-522)
Niels Bohr and Léon Rosenfeld

The question of limitations on the measurability of electromagnetic field quantities, rooted in the quantum of action, has acquired particular interest in the course of the discussion of the still unsolved difficulties in relativistic atomic mechanics. On the basis of exploratory considerations, Heisenberg¹ attempted to demonstrate that the connection between the limitation on the measurability of field quantities and the quantum theory of fields is similar to the relationship between the complementary limitations on the measurability of kinematical and dynamical quantities expressed in the indeterminacy principle and the non-relativistic formalism of quantum mechanics. However, in the course of a critical...

• IV.3 FIELD AND CHARGE MEASUREMENTS IN QUANTUM ELECTRODYNAMICS
(pp. 523-534)
Niels Bohr and Léon Rosenfeld

Recent important contributions¹ to quantum electrodynamics by Tomonaga Schwinger and others have shown that the problem of the interaction between charged particles and electromagnetic fields can be treated in a manner satisfying at every step the requirements of relativistic covariance. In this formulation, essential use is made of a representation of the electromagnetic field components on the one hand, and of the quantities specifying the electrified particles on the other, corresponding to a vanishing interaction between field and particles. The account of such interaction is subsequently introduced by an approximation procedure based on an expansion in powers of the nondimensional...

10. ### V. IRREVERSIBILITY AND QUANTUM THEORY

• V.1 THE DECREASE OF ENTROPY BY INTELLIGENT BEINGS
(pp. 537-538)
• V.1 ON THE DECREASE OF ENTROPY IN A THERMODYNAMIC SYSTEM BY THE INTERVENTION OF INTELLIGENT BEINGS
(pp. 539-548)
Leo Szilard

The objective of the investigation is to find the conditions which apparently allow the construction of a perpetual-motion machine of the second kind, if one permits an intelligent being to intervene in a thermodynamic system. When such beings make measurements, they make the system behave in a manner distinctly different from the way a mechanical system behaves when left to itself. We show that it is a sort of a memory faculty, manifested by a system where measurements occur, that might cause a permanent decrease of entropy and thus a violation of the Second Law of Thermodynamics, were it not...

• V.2 “MEASUREMENT AND REVERSIBILITY” AND “THE MEASURING PROCESS”
(pp. 549-647)
John von Neumann

What happens to a mixture with the statistical operator U, if a quantity ℛ with the operator R is measured in it? This operator must be thought of as measuring ℛ in each element of the ensemble and collecting the elements that have been thus treated into a new ensemble. We can answer this question -- to the extent to which it admits of an unambiguous answer.

First, let R have a pure discrete, simple spectrum, let φ1, φ2, … be the complete orthonormal set of eigenfunctions and λ1, λ2, … the corresponding eigenvalues (by assumption, all different from each...

• V.3 THE ERGODIC BEHAVIOUR OF QUANTUM MANY-BODY SYSTEMS
(pp. 648-656)
Léon van Hove

By a pertubation technique adapted to the actual properties of gases and solids (and possibly also of liquids) we have established in previous papers that under suitable conditions a quantum many-body system approaches statistical equilibrium as far as those physical quantities are concerned which are diagonal in the unperturbed representation. This result is now extended to non-diagonal quantities of a type broad enough to include all observables of actual interest. A general discussion of the resulting ergodic theorem is given, and its implications for classical statistics are briefly analyzed. The paper ends with a discussion of a recent article by...

• V.4 QUANTUM THEORY OF MEASUREMENT AND ERGODICITY CONDITIONS
(pp. 657-679)
Adriana Daneri, A. Loinger and G. M. Prosperi

As is well known, the measuring process plays a central role in quantum mechanics. Its formal mathematical theory was developed by von Neumann¹) in 1932. In recent years von Neumann’s point of view and results have been criticized under different aspects by many authors (Jordan ²), Bohm ³), Wigner, Araki and Yanase ⁴), Ludwig ⁵), Feyerabend ⁶), H. S. Green ⁷), Durand ⁷), Wakita ⁹)). Various attempts have been made (see refs.4, 5, 7—10)) to obtain a more satisfactory solution of the problem than that of von Neumann. In our opinion, the treatment given by Ludwig5, 10)...

• V.5 TIME SYMMETRY IN THE QUANTUM PROCESS OF MEASUREMENT
(pp. 680-686)
Yakir Aharonov, Peter G. Bergmann and Joel L. Lebowitz

ONE of the perennially challenging problems of theoretical physics is that of the “arrow of time.” Everyday experience teaches us that the future is qualitatively different from the past, that our practical powers of prediction differ vastly from those of memory, and that complex physical systems tend to develop in the course of time in patterns distinct from those of their antecedents. On the other hand, all the “microscopic” laws of physics ever seriously propounded and widely accepted are entirely symmetric with respect to the direction of time; they are form-invariant with respect to time reversal.1, 2

Thede facto...

• V.6 LYAPOUNOV VARIABLE: ENTROPY AND MEASUREMENT IN QUANTUM MECHANICS
(pp. 687-691)
Baidyanath Misra, Ilya Prigogine and Maurice Courbage

No other question in theoretical physics seems to have caused as much controversial discussions over as long a period of time as the question of the dynamical meaning of irreversibility expressed in the second law of thermodynamics With the advent of quantum mechanics and the discovery of the apparently irreversible exponential decay of unstable particles, this question has gained added theoretical importance. Irreversibility is now an essential feature of gross macroscopic phenomena such as the familiar transport processes and it also seems to be intrinsic in such basic processes as the “wave packet reduction” caused by measurement and the decay...

• V.7 CAN WE UNDO QUANTUM MEASUREMENTS?
(pp. 692-696)
Asher Peres

The measurement process in quantum physics was analyzed long ago by von Neumann¹ who showed that it could formally be described as the transformation of a pure state$\Psi =\Sigma {{c}_{n}}{{\phi }_{n}}$into a mixture$\rho =\Sigma {{|{{c}_{n}}|}^{2}}{{P}_{n}}$. Here, the φnare eigenstates of the dynamical variable being measured, and thePnare the corresponding projection operators.

This irreversible transformation, commonly called the “collapse of the wave packet,” cannot follow from the Schrödinger equation, since the latter generates a unitary mapping of the Hilbert space of states. In fact, the coupling of the eigenstates of the measured system to those of the measuring apparatus...

11. ### VI. ACCURACY OF MEASUREMENTS:: QUANTUM LIMITATIONS

• VI.1 THE UNMEASURABILITY OF THE SPIN OF A FREE ELECTRON
(pp. 699-700)

Bohr gave much thought to the spin of the electron and to Dirac’s theory. I never felt quite at ease about his argument that the spin cannot be observed by classical means although he always succeeded in showing the fallacies in any proposed experimental setup.

Bohr as a lecturer is a different matter. It is much glossed, but very little written about. Perhaps the only one who has put his view of it in print so far is Larmor; in a speech (later published)* at the Maxwell celebrations in Cambridge in 1931, he commented upon Maxwell’s reputation of being ‘a...

• VI.1 MAGNETIC MOMENT OF THE ELECTRON
(pp. 701-706)
Nevill F. Mott and Harrie S.W. Massey

We have discussed so far only the magnetic moment of the atom. We shall not review here the evidence, derived from the anomalous Zeeman effect, from the gyromagnetic effect, etc., that theelectronhas a fourth degree of freedom, a magnetic moment —eħ/2mc, and a mechanical moment ½ħ. We shall content ourselves with remarking that according to the Schrödinger theory the ground state of the hydrogen atom is not degenerate, and therefore, in order to account for the splitting in a magnetic field revealed by the Stern–Gerlach experiment, it is necessary to assume that the electron has a fourth...

• VI.2 MEASUREMENT OF QUANTUM MECHANICAL OPERATORS
(pp. 707-711)
Huzihiro Araki and Michael Yanase

IT was pointed out by Wigner¹ that the presence of a conservation law puts a limitation on the measurement of an operator which does not commute with the conserved quantity. The limitation is such that the measurement of such an operator is only approximately possible. An approximate measurement can be done by a measuring apparatus which is large enough in the sense that the apparatus should be a superposition of sufficiently many states with different quantum numbers of the conserved quantity. He has proved these statements for a simple case where the x component of the spin of a spin...

• VI.3 OPTIMAL MEASURING APPARATUS
(pp. 712-714)
Michael Yanase

IT was shown recently that a quantum mechanical operator which does not commute with the operator of a conserved quantity can be measured only approximately. There is a finite probability that the measurement is unsuccessful, but this probability can be very small if the measuring apparatus contains a large amount of the conserved quantity.¹ It was shown, in particular, that if the product of the probability of an unsuccessful measurement and of the maximum value of the conserved quantity which is present in the measuring apparatus exceeds a certain value, no contradiction with the conservation law occurs. The objective of...

• VI.4 TIME IN THE QUANTUM THEORY AND THE UNCERTAINTY RELATION FOR TIME AND ENERGY
(pp. 715-724)
Yakir Aharonov and David Bohm

AS is well known, the uncertainty relations in quantum mechanics can be regarded in two closely related ways. First of all, they are a direct mathematical consequence of the replacement of classical numbers by operators, and of adding the basic principle that the statistical distributions of the corresponding observables can be obtained by means of the usual formulas from the wave function and its probability interpretation.¹ Secondly, however, it can be shown by analyses such as that of the Heisenberg microscope experiment that they are also limitations on the possible accuracy of measurements.²

These considerations apply to observables such as...

• VI.5 THE FUNDAMENTAL NOISE LIMIT OF LINEAR AMPLIFIERS
(pp. 725-735)
H. Heffner

SINCE THE advent of the maser, there have been a number of treatments of the noise figure or noise temperature of this and other potentially low noise devices such as the parametric amplifier.1–4Most of these have treated each specific device as a quantum system and have determined a limiting noise temperature arising because of amplified spontaneous emission. Although the details of the calculations differ, investigations of the minimum noise temperature due to this effect yield values of the order ofhv/kfor both the maser and the parametric amplifier.

The maser and the parametric amplifiers are phase preserving...

• VI.6 QUANTUM NOISE IN LINEAR AMPLIFIERS
(pp. 736-742)
H. A. Haus and James A. Mullen

THE availability of coherent signals at optical frequencies has stimulated research in their use for communication purposes. Ways of processing optical frequencies are considered that are similar to those of the low end of the coherent frequency spectrum. With the use of classical communication techniques, classical performance criteria will be applied. One purpose of this paper is to extend classical noise performance criteria to linear quantum amplifiers in whichthe predominant noise is quantum mechanicalin nature. These criteria will be applied to a wide class of linear quantum mechanical amplifiers.

The purpose of a sensitive linear amplifier is to...

• VI.7 OPTICAL CHANNELS: PRACTICAL LIMITS WITH PHOTON COUNTING
(pp. 743-748)
John R. Pierce

Consider a free-space path such as we might have between space vehicles. For a wavelength λ, a distanceLand transmitting antennas of effective areasATandARthe ratio of received powerPRto transmitted powerPTis¹

PR/PT=ATAR/λ²L².(1)

This suggests the use of a short wavelength.

Going to optical wavelengths requires very smooth antenna surfaces and very precise pointing. Further, at optical frequencies we encounter quantum effects. Here we disregard antenna and pointing problems and consider how quantum effects will limit a communication system.

In receiving a signal mixed with Johnson noise, we have the...

• VI.8 QUANTUM NONDEMOLITION MEASUREMENTS
(pp. 749-768)
Vladimir B. Braginsky, Yuri I. Vorontsov and Kip S. Thorne

Scientists have understood since the 1920’s that the physical laws which govern atoms, molecules, and elementary particles are very different from the laws of everyday experience. The special laws of the atomic and molecular “microworld” are called quantum mechanics; those of everyday experience are classical mechanics. The laws of quantum mechanics were forced on physicists and chemists in the 1920’s as the only possible way to understand the spectral properties of the light emitted by atoms and molecules.

Quantum mechanics tells us that, whenever a person measures some property of an electron (or of any other object in the microworld),...

12. GUIDE TO SOME FURTHER LITERATURE
(pp. 769-786)

The following papers are cited, not with any thought of providing a comprehensive bibliography, but rather with the aim of providing a few points of entry to an enormous literature. The traditional detective novel is loaded with clues, most of them irrelevant, distractive or downright deceptive. The few truly decisive items have no star to distinguish them. That is how the author conceals his plot. If we have distinguished no items with a star in our bibliography, it is because we do notknowthe plot!

Consciousness:Wigner reasons that an observation is only then an observation when it becomes...

13. BIBLIOGRAPHY
(pp. 787-811)
14. Back Matter
(pp. 812-812)