on the theory of measurement in quantum mechanical systems

On the Theory of Measurement in Quantum Mechanical SystemsAuthor(s): Loyal Durand IIISource: Philosophy of Science, Vol. 27, No. 2 (Apr., 1960), pp. 115-133Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/185887 .

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Philosophy of Science VOL. 27 April, I960 NO. z

ON THE THEORY OF MEASUREMENT IN QUANTUM MECHANICAL SYSTEMS*

LOYAL DURAND III** The Institute for Advanced Study

This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics.

I Introduction. The analysis of the process and meaning of measurement in quantum mechanical systems is basic to the interpretation of the quantum theory, with its many philosophic implications. While fragmentary theories of measurement have been advanced by a number of authors [1], complete expositions within the context of the quantum theory are confined to the classic work of von Neumann [2] and the more recent studies of Everett [3]. However, these theories, tenable mathematically and frequently compelling from an interpretative point of view, are in some respects in conflict with the actual physical procedures of measurements [4, 5]. On the other hand, the practical application of the quantum theory to the study of physical phenomena is well understood with relation both to the prediction of the values of microscopic quantum mechanical observables, and to the actual measurements of these observables in terms of the essentially classical phenomena of the macroscopic world. It has consequently seemed of little importance to the physicist to formulate clearly in terms of the quantum

* Received October, 1959. ** National Science Foundation Postdoctoral Fellow, now at the Physics Department, Brook-

haven National Laboratory, Upton, New York. The author would like to thank Prof J. R. Oppenheimer and the Institute for Advanced Study for the hospitality accorded him during the course of this work, and the Physics Department at Brookhaven National Laboratory for support while this paper was written.

115



116 LOYAL DURAND III

theory the general connections between the phenomena at the two extremes which make measurement possible, with the unfortunate result that statements about measurement incompatible with the procedures and analysis applied to actual experiments have gone virtually unchallenged in interpretative discussions. That spurious difficulties and conflicting views have been typical of the interpretation of quantum mechanics, particularly in its philosophic aspects, is thus not surprising [4, 5, 6, 7].

The physical and philosophical bases of the most important interpretations of the quantum theory have been analyzed recently by McKnight [5]. It was noted in particular that the interpretation most prevalent among physicists, that of the Copenhagen school as represented in the writings of Bohr [6] and Heisenberg [7], insofar as it presupposes any specific theory of measurement, is based on the physically untenable analysis given by von Neumann [2]. McKnight has furthermore illuminated a number of philosophic inadequacies in this and in the other important interpretations, and has advanced as a more adequate philosophic view on "extended latency" interpretation of quantum mechanical measurements which develops more fully ideas first advanced by Margenau [4]. The question of the detailed physical theory of measurement was left open, although a set of general criteria was proposed in terms of which the adequacy of such a theory can be judged. The present paper is in a sense complementary to those of McKnight, since its object is precisely to clarify the status of measurements in quantum mechanics by presenting a detailed quantum theory of the physical aspects of measurements. This theory is consistent with the procedures of experimental physics, and provides the complete theoretical basis which has been absent in discussions of the various interpretations of the quantum theory, or of their philosophic ramifications. Finally, since the present theory fits in well with the work of Margenau [4] and McKnight [5], the reader will be referred to their papers for a detailed exposition of its epistomological background, and of the cogent objections to the previous theories of measurement.

II. Outline of the Structure of Quantum Mechanics. The basic structure of quantum mechanics is well known. Systematizations of the theory have been given by von Neumann [2] and Margenau [8] which allow its mathematical form and its relation to the empirical world to be stated in a few axioms. The notable work of von Neumann demonstrated in particular that the non-relativistic theory is axiomatically complete and that it is capable of rigorous mathematical treatment. Since the corresponding questions of completeness and mathematical consistency have yet to be answered for the more recent relativistic quantum field theories encountered in the description of electromagnetic phenomena and of the properties of the elementary particles, we shall discuss explicitly only the non-relativistic theory. Despite some added definitional problems which are encountered, the results of this study may at once be extended to the modern field theories provided only that the Hilbert space formalism is retained.



ON THE THEORY OF MEASUREMENT IN QUANTUM MECHANICAL SYSTEMS 117

Quantum mechanics is formulated mathematically as an operator algebra in a Hilbert space, with the operator-observable identification and the probability postulate constituting the basis of the physical interpretation of the theory. The state of a physical system is completely described [4, 5] by a vector b in a Hilbert space If; to each observable A associated with the system there corresponds a Hermitian operator cA defined on this space. It is postulated that the only values of A which can appear in a measurement are the eigenvalues a of QA determined by the equation.

CA#a -aa, Ca. e H, (1)

and, furthermore, that the mean value of the observable A determined from an ensemble of measurements on identical systems in the state b is given by

<A>av KbIAkb>/KVik> (2) Here <K10> denotes the inner product in the Hilbert space, and the expectation value <KlbIc4A> is normalized by division by <010>. The state- function 0 describing the physical system is determined from the solution of the Schrodinger equation

Hf i 7iolat, (3)

where H is the operator corresponding to the classical Hamiltonian function. Much of the physical content of the theory is of course contained in the boundary conditions on 0 and in the structure of b with respect to the various constants of the motion, and is not determined by the mathematics alone.

The application of quantum theory to the problems of physics was histor- ically much aided by the prior existence of a well-developed classical physics with its accompanying epistomology and theory of measurement, to which the quantum theory could be linked by the correspondence principle of Bohr. However, the inapplicability of many of the concepts of classical physics to the discussion of quantum mechanical phenomena has often resulted in confusion in discussions of the theory and interpretation of measurement. It was early found that the predictions of the quantum theory for the results of measurements were correct only if the expectation values <0jbfeA4Ib> were interpreted in terms of ensembles of measurements, necessitating the use of statistical language in the axioms relevant to measurement [4]. The historical evolution of the statistical interpretation of the theory is well known [6], as is its consistency with the mathematical structure of the theory. One notes that any state function / can be expanded in terms of the complete orthogonal set of eigenfunctions Oa of the Hermitian operator associated with the observable A of the physical system,

- ICa #a, (4)

where the Oa satisfy Eq. (1), and the expansion coefficients Ca are given by the inner products

Ca #<akl>. (5)




Assuming that 0 and the q5a are all normalized, the condition <010> = implies that

E Ca* ca 1'. (6) a

One then obtains from Eq. (2) the expression K'kK1'kli> - - ca* ca a (7)

for the prediction of the theory for the mean value of the results of an ensemble of measurements of A on systems in the state 0. Since it is postulated in agree- ment with experiment that the only values of A which can be obtained in a measurement are the eigenvalues a, it is natural to follow the statistical view of Born and interpret the numbers Ca* Ca - Ica12 as the relative probabilities with which the various eigenvalues a will be found in the ensemble. If b is one of the eigenfunctions of the observable in question, it follows at once that

<Kale-4Aia> = a, (8) and the value a is always obtained for A in a measurement on this state. The expansion coefficients Ca are often interpreted as being transition ampli- tudes, the absolute square of which yields the probability that a system initially in the state b will be found in the state Oa, or, alternatively, the probability with which the state b is carried by the process of measurement into the state Oba. However, both statements must be treated with great care since they are fundamentally assertations about the state of a system or about the as yet unexamined process of measurement rather than about the results of a measurement of the observable [4, 5]. As will become clear later, the above statements are in fact true only in special situations. In this paper, we shall therefore restrict ourselves to the minimal postulate that the numbers Ica12 are the relative probabilities with which the various eigenvalues a will appear among the results of an ensemble of measurements of A performed on identical systems all in the state 0.

III. Comments on the General Features of Measurement. In this section we shall establish a definition of a measurement (observation) and consider in this connection several conditions of an essentially philosophic nature which any useful measuring device may reasonably be required to satisfy. It will be necessary also to discuss briefly the important distinction between a measurement and the preparation of a state, and the role of the observer in the measuring process.

a. Definition of Measurement. The definition of measurement which will be used is the following (Margenau [4, 9]):

A measurement is an operation performed on a physical system which leads to results immediately accessible to an observer. The results may consist of numbers, yes-no type discriminations, or qualitative distinctions.

Active observation of a system upon which no operation is performed directly




can also constitute a measurement, as, for example, in work on astronomy and astrophysics, in which significant scientific information may be obtained by discriminating observation of astronomical phenomena. The philosophical implications and the finer details of this definition of measurements have been discussed at length by Margenau [4, 9] and McKnight [5]. The qualifi- cation that the results of a measurement be accessible to an observer is occasionally overlooked, this oversight elevating to the status of a measurement many operations which would not ordinarily be regarded as such. The term "measurement" is ordinarily used by physicists to refer to operations the results of which enter the awareness of some observer. Thus, the act of photographing a cloud chamber track produced by an elementary particle can be regarded as that of making a measurement only because of the well known procedures which exist for extracting from the photographic image the desired information; in fact, it is unfortunately true that only with much work and a considerable amount of theoretical analysis can the physically meaningful "results of measurement" be obtained.

A class of operations closely related, but not equivalent, to measurements are those termed state-preparations. The distinction between the preparation of a state and a measurement, emphasized particularly by Margenau [4, 9] and McKnight [5] is important; failure to maintain this distinction is in fact the origin of a number of the difficulties which have been encountered in the interpretation of the quantum theory. The distinction may be illustrated by considering the use of a Nicol prism to prepare and to measure the state of polarization of a photon beam. Since the prism transmits only photons of a given, theoretically known, polarization, it can be used to prepare a photon state upon which a measurement can be made. However, to obtain a measurement of the photon polarization, it is necessary to supplement the prism by a photon detector in order to ascertain whether or not the beam is actually transmitted. A statement about undetected photons lacks the empirical reference required of the result of a measurement; e.g., "Any photons which pass through the prism are vertically polarized" is a conditional statement about a situation (transmission of photons) which has not been established to exist. While these considerations are practically obvious in the present instance, the measurement-preparation distinction can be lost in more complicated examples if care is not taken. Several more elaborate examples of this distinction are discussed in references [4], [5] and [9].

b. Role of the Observer in the Measuring Process. The term "measurement", as yet unanalyzed in the specifically quantum theoretical context, carries with it a number of connotations deriving from classical physics. Foremost among these is the concept of the observer as external to the system observed. In classical physics, it is generally assumed that the observer is separable from the system studied, and that the presence or absence of the observer is irrelevant to the behavior of that system except as the two may interact. The effects of any such classical interactions are supposed (in principle) to be




calculable in detail, so that the influence of the observer may be removed before the results of a measurement are discussed. The profound philosophic problems involved in the assumed separability of the observer from what he regards as the external world are not of present concern. Instead, it will be supposed that the relation of an observer to an essentially classical external world is adequately treated in the epistomologies of classical physics.

Two points are of considerable importance with respect to the observer in a quantum theory of the physical world. The externality of the observer to the system observed is implicit in the manner in which the postulates of the theory relating to measurement are stated. No reference is made in those postulates to the means by which measurements are to be made; it is assumed rather that the means and indeed, the meaning, of measurement can be specified by an observer who is not explicitly described. Secondly, as has been emphasized by Bohr [10], human observers are able to perceive and deal only with what are essentially classical objects: "however far the phenomena (of quantum physics) transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms." Both of these conditions are consistent with the structure of quantum theory. As is well known from the work of Bohr on the correspondence principle, the quantum mechanical description of a system approaches the classical description in the limit of large quantum numbers. Most macroscopic systems and the observables in terms of which they are characterized can be discussed to a very excellent degree of approximation in strictly classical terms. Pecul- iarly quantum mechanical effects are small relative to the accuracy with which (classical) observations may practically be made upon these systems. Obser- vables and systems which may be treated classically for all practical purposes will henceforth be called quasi-classical. The directly perceptible aspects of the measuring devices of physics are certainly of this class. For example, the quantum mechanical fluctuations in the (quasi-classical) position of a macroscopic pointer, the length of a yardstick, or the reading of a thermo- meter are all negligible to the accuracy with which these observables may be measured on the classical level, yet positions, lengths, and temperatures are perfectly valid observables in terms of which to describe the state of a macroscopic system. The necessary presence of a sensibly classical stage in all measurements involving human observers allows the foregoing precepts to be satisfied consistently with the specifications of quantum theory. The ultimate observer may be separated from the system on the classical level with the introduction of no practical difficulties and no more profound philosophic problems than were previously inherent in the strictly classical separation. The results of experiment may be stated in terms of (quasi-) classical observables and devices. In particular, there is no need for the quantum theory of measurement to penentrate into the domains of physiology and psychology, and the philosophic aspects of the problems of perception, awareness of a result, etc., may be discussed relative to quasi-classical phenomena within the confines of the classical epistemological schemes.




On the other hand, it is well known that the behavior of a microscopic quantum mechanical system is often inextricably connected with the means used to observe that behavior [11]. While in classical physics, one could compensate in detail for object-apparatus interactions, or could ideally make those interactions arbitrarily small, no means of detailed compensation is provided in quantum theory. Indeed, to speak of detailed classical descriptions of events is possible only under the conditions relevant to the correspondence limit. Quantum mechanical events must be considered as complete wholes, well defined (on the microscopic level) only when the conditions as well as the results of observation are specified. This limitation on the microscopic separability of the behavior of a system from the means used to observe that behavior does not, however, alter the conclusion that the ultimate (human) observer may be considered as effectively external to the system observed. That conclusion is based upon the existence of the correspondence limit, and upon its applicability to the measuring devices of physics.

c. The Measuring Device. The problems inherent in defining observables, coordinate systems, etc., have been extensively discussed [5, 12, 13]. Perhaps the main contribution of modern thought to these problems lies in the explicit recognition of the physically nonsensical nature of theoretical concepts which cannot be linked meaningfully to empirical observations. Much emphasis has therefore been given the necessity of having a number of operational definitions by which some of the fundamental observables of a theory are defined in terms of specific experiments [13]; however, a physics composed solely of operationally defined concepts devoid of logical connections would be theoretically vacuous [12]. It is necessary to have theoretical as well as operational definitions. Concepts which are not themselves operationally defined but which are nevertheless essential constituents of a logical nexus possessing many connections with experiment are acceptable components of a physical theory [14]. Elaboration of these points here is unnecessary. We shall only remark that the connection of quantum mechanics with experiment is provided by the operator-observable identification which associates with each physical observable an operator in a Hilbert space and by the probability postulate relating the expectation values of the operators to the results of measurement. Since the meaning of measurement may be understood on the quasi-classical (macroscopic) level in terms of the classical epistomological schemes, the principal problems in the development of a quantum theory of measurement concern the logical connection between the peculiarly quantum mechanical observables appropriate to the microscopic domain, and the macroscopic, quasi-classical observables which are actually measured (in the classical sense) and in terms of which the former are defined operationally. The philosophic problems associated with the existence of the peculiarly quantum mechanical observables have been discussed at length by Margenau [4] and McKnight [5], and will not be considered here. However, we may note that the physical theory of measurement to be outlined in the




next section may be interpreted very naturally in terms of the "extended latency" scheme of McKnight [5].

We will henceforth be concerned with the development of a physical theory of measurement in quantum mechanical systems, more precisely, with the enunciation of the precise logical connection between microscopic quantum mechanical observables and the macroscopic, quasi-classical quantities in terms of which they are measured. We shall assume that the quantum theoretical description of the physical world is complete, in the sense that it encompasses as a special case the description of the process of measurement connecting the quantum and the quasi-classical domains. No special physical assumptions will therefore be made about the measuring process; the attitude here assumed treats measurement as a physical process, hence one describable without reference to non-physical procedures [15]. Consequently, we will not purport to give a detailed discussion of some unique "process of measurement", but shall rather discuss the general circumstances in which a significant measurement may be said to take place, and how these circumstances are realized within the context of the quantum theory.

We turn therefore to the formulation of a set of criteria which may be used to discuss the suitability of a particular device for the measurement of a quantum mechanical observable A. The following properties would appear to be essential.

1. The device has a quasi-classical observable B, which, if not directly perceptible to an observer, can be measured using only quasi-classical devices.

2. To every discrete value of A which is to be measured, there corresponds a distinguishable value or set of values B(A) of B, while to every set of non-overlapping intervals of values in the continuum spectrum of A (if one exists), there corresponds a set of distinguishable intervals in B.

3. There must exist a time prior to the beginning of the measurement at which the values of A and B are uncorrelated.

4. At the conclusion of the measurement, the correspondence between the values of A and the values B(A) of B must be essentially one-to-one and unique for those values of A which are of interest.

The first condition is necessary if the results and description of a measurement are to be expressible in terms of the well-developed language and epistomology of classical physics. The remaining conditions essentially delineate what may be considered to be a useful measuring device and a significant measurement, and together imply the existence of what will be called a weak correlation of the observables A and B. Thus, if some value a of A is assumed to be characteristic of the object system at the beginning of the measurement, it must follow from the theory that the associated value b(a) of B will (almost always) be characteristic of the measuring device at the conclusion of the measurement, a-* b(a). Conversely, we must require that b(a) finally (almost always) implies a initially, b(a)-*a. However, the correlation must be "weak" in the sense




of (3), for if A and B are always strongly (uniquely) correlated, the state of the measuring device will determine the state of the object system, and no measurement will be possible. It should be emphasized that we have imposed no requirements on the evolution of the state of the object system during the time of the measurement, but have required only that the final values of B yield information about the initial object state. It is interesting to note that the total system of object plus measuring device is necessarily in a metastable state with respect to B, since in the presence of the object system the initial value bo of B will by hypothesis change to some different value characteristic of the initial state of the object system [16]. The initial and final values of B may also, of course, change with the time, provided the change is such that a measurement is still possible. The simplest case, however, is that in which the initial and final values of B are independent of the time (in the quasi- classical sense).

It is important to recognize that the foregoing criteria are not restrictions on the types of interactions which may be considered theoretically within the context of the quantum theory, but are rather to be used to distinguish from the various possibilities that type of interaction and measuring device which will be useful in a particular situation, that is, the type of device which will yield results of theoretical significance. The actual experiment devices and procedures of physics satisfy these conditions. Criteria of a similar nature have been used by von Neumann [2] and Everett [3] in their studies of the quantum theory of measurement, but with the additional unduly restrictive condition that the observation of a particular eigenvalue a of A require that the object system be in the eigenstate sa at the conclusion of the measurement. This condition, if regarded as a physical restriction on the admissible types of measuring devices, is neither necessary nor in accord with the types of measurement which actually occur [4, 5], nor is it necessary if it is regarded in the philosophic sense as expressing a change in the state of knowledge of the observer about the object system [4, 6]. In the present study, we will require only that the result of a measurement reflect the state of the object system at the beginning of the measurement; the final state of that system can in principle be determined by straightforward calculation.

IV. Description of Measurement in Quantized Systems. In the present section the formulation of a quantum mechanical description of measurement will be considered in detail; the approach will be based on the foregoing definitions and principles of measurement. We shall not, in particular, discuss measurements on the macroscopic, quasi-classical level, but shall assume rather that such measurements may be adequately discussed in terms of classical physics and its accompanying epistomologies. Our comments will therefore be confined to the processes by which a correlation of the type discussed in Sec. III may be established between microscopic, quantum mechanical observables and the quasi-classical observables which are directly measured.




a. Mathematical Preliminaries. The subsequent discussion will be concerned with a system composed of an apparatus subsystem denoted by (a), and an object subsystem denoted (o). We shall consider only the highly idealized case in which the statefunctions of both are known; the necessary generaliza- tion to the cases of practical interest can be given [17], but the considerations of matters of principle are unchanged. No "measuring interaction" between the subsystems of a non-physical nature (i.e., not amenable to experimental study and treatment within the context of the quantum theory) will be assumed to exist, but ordinary physical interactions will, in general, be present between the subsystems. Despite the presence of these interactions, it may be possible in certain circumstances to describe the object and apparatus subsystems as effectively independent, a condition resulting, for example, when subsystems interacting only through short range forces have a large separation in space. This effective independence of the subsystems is required at least with respect to the object and apparatus observables according to the weak correlation requirement, condition (3) above, if the interaction of the subsystems is to result in a significant measurement. Initial independence of the subsystems will henceforth be assumed. At the time at which the measurements begin, taken for convenience as t 0, the state function T(t) of the total object-apparatus system will therefore be representable, at least asymptoti- cally [18], as a product of functions 0 and 0 describing the apparatus and object systems independently.

lim T(t) - T(o) =- +(o) 0(o). (9)

T(t) is a vector in the product space If (o) 0 H (a) composed of the Hilbert spaces If (o) and H-f (a) in which the object and apparatus states are defined, and is to be a solution of the Schrodinger equation

i h aTlat - H' - [H(o) + H(a) + H(oa)]' (10)

chosen to satisfy the applicable (physical) boundary values and the asymptotic condition, Eq. (9). The total Hamiltonian for the system, H =H(o) + H(a) + H(oa), consists of parts H(o) and H(a) which determine the time evolution of the object and apparatus subsystems in the absence of interaction, and the part H(oa) which contains the interaction terms connecting the subsystems. The state function T can be represented as a linear superposition of the state functions O1,a and +: from any appropriate complete orthonormal sets describing the object and apparatus subsystems, but it is most convenient to use for the functions of the set {9,a} the eigenfunctions Oa corresponding to the object observable A which is to be studied, and for the 0,,, the eigenfunctions Ob corresponding to the correlated apparatus observable B. In particular, then, at time t = 0 (the beginning of the measurement),

T (o) = (E Caba) (E db#b). (11)

The operators corresponding to the observables of the disjoint subsystems




act in the appropriate Hilbert space If(o) or if(a,), and in the direct product representation of the total Hilbert space, If = 0(o) 0 H(a), are of the form CA = 4(oQ) 0 I(a), 3 T )= 0 (o ' 3 (a), where 1(X) is the unit operator in H(j). It is of course possible to consider observables defined only with respect to the total system, with operators defined in H, but these are not of interest in connection with the process of measurement.

The evolution in time of the state function T is governed in the Schrbdinger representation of quantum mechanics by the Schrodinger equation, Eq. (10). However, the discussion of the measuring process will be somewhat simpler if we change at this point to the Heisenberg representation, in which the state function T is independent of time, while the operators are time dependent. We note first that T(t) is connected with T(o) by a unitary transform,

T(t) - U(t) TF(o), U-1(t) = U+(t), (12)

where U(t) satisfies the Schrddinger equation

H U(t) = i h 9/@t U(t) (13)

with the (asymptotic) boundary condition U(o) 1. Under this unitary transformation of the state functions, the time-independent operators 0 of the Schrodinger representation are changed into time-dependent operators 0(t),

0(t) =U-1(t) OU(t). (14)

The matrix elements of the operators are unchanged by the change of representation,

<TP'(t)JOIT(t)> = <T'(o)10(t)J T(o)>. (15)

b. Description of Measurenment. Formulation of the quantum mechanical description of the measuring process is now straightforward. If the apparatus subsystem is to constitute a device adequate for the measurement of the object observable A, it must, in accordance with the considerations of Sec. III, be characterized by a quasi-classical observable B which, while initially independent of A, is correlated with A in an essentially unique manner after a measurement. Let ? be the associated operator. We will consider first the simple case in which the initial object state is a pure state with respect to the object observable A, Ta(o) = jaO, where ka obeys Eq. (1) with eigenvalue a. The initial apparatus state A will in general not be a pure state with respect to B; indeed, since B is a quasi-classical observable, it will not be possible by observations on this level to detect small changes in 0. Thus while the initial expectation value of A will be just a,

<KT(o)feA4(o)IWy(o)> = <Kka4(0)1ha> -a, (16)

that of B will be the mean value of a distribution of eigenvalues with differing weights,

<Ta(o)1'B(o)ITa(o)> =- <|JB(o)|0> = EIdbI2b = bo (17)




where the db are the coefficients defined in Eq. (11). The spread in the values of b appearing in Eq. (17) will generally be much smaller than the accuracy with which B may be measured on the quasi-classical level. The initial independence of the object and apparatus subsystems here appears as the independence of the initial expectation values of A and B. In the absence of the interaction Hamiltonian H(oa), the subsystems will continue to evolve independently, so that at some later time t, A and B will have the expectation values

<Ka(O)IUi")(t)c.AU(o)(t)I0a(O)> and <K(o)1U_L(t)a U(a)(t)lJ(o)> = b(t).

For a significant measurement to be possible, it is in general necessary that, in the absence of object-apparatus interactions, the initial (prepared) expectation value of B remain essentially unchanged for times comparable to the duration of that measurement. [Exceptions are possible in cases in which the precise time variation of the expectation value is known.] Thus we require that (d/dt) b(t) st 0, for H(oa) -* 0, implying that [B, H(o)] - 0. That is, the apparatus system is usually prepared so that the initial state is quasi-stationary with respect to B.

On the other hand, with the inclusion in the Hamiltonian of the interaction term H(oa), it is necessary that the expectation value of B change during the course of a measurement from the value bo characteristic of the initial (metastable) apparatus state to a final value b(a,t) characteristic of the interacting systems [16]. This requires that [B, H(oa)] # 0. The operator T (t), t>O, will then operate on the components of JP(o) which specify the initial state of the object system as well as on those components specifying the initial apparatus state; this is in fact necessary if the expectation value of B for t>O is to reflect the structure of the object state. Taking Ta(0) in the form given in Eq. (11) with only one of the expansion coefficients Ca different from zero, we find for the expectation value of B assocated with a system which was initially in a pure state with respect to A [19]:

b(a ,t) < Ta(0)I(t) I T a() > -7 E, db* db<ba;bb, I1(t)I|balhb>* (18)

We will assume for simplicity that, on the quasi-classical level, the expectation value of B is for practical purposes independent of t at the conclusion of the experiment, b(a,t)-+b(a) for t>0. It is important to recognize that because of the non-vanishing of the commutator [B, H], this relation can hold only in the sense that the time-dependent fluctuations in b(a,t) are sufficiently small as to be undetectable in quasi-classical observations on the apparatus system. If that system is to constitute a device useful for the measurement of A, we must require furthermore that b(a) # b0, and that b(a) =? b(a') for a # a'. It is possible, of course, that the predicted distribution of the eigenvalues b about the mean value b(a) will have a macroscopic spread, as in the case of the finite width of the image of a point source of light formed by an optical system with a finite aperture. This causes no difficulty so long as the distributions in B corresponding to the discrete eigenvalues of A are distin-




guishable, and the mapping from the continuum eigenvalues of A to the values b(a) of B preserves the continuum property and the order of points.

Once the expectation values b(a) of B associated with object systems initially in the eigenstates Oa are known, the results of measurements on states consisting of coherent sums of the 0a may be analyzed. Let the initial state T(o) of the total system have the form given in Eq. (11). Then the initial expectation value of B is still bo, while the expectation value of A is given by

< T(o) l-4(o) I T(o) > - a 2a. (1 9)

The corresponding expectation value of B at the end of the experiment is given theoretically by

< T(o) |B (t) I T(o) > X= E,c Y ca,CadbjdbQ<0a b '/ B (t)iba0bb> (20) aabb

-

Ica J2b(a) + E X , Ca* Cadb* db <0a b I %t I0a0b> a ata' bb'

The first sum on the right hand side of Eq. (20) is just the sum of the expectation values b(a) defined in Eq. (18), each appearing with the weight Ical2 with which the corresponding eigenvalue a of A appeared in Eq. (19), and is easily interpreted if the remaining multiple sums can be neglected. Since B is assumed to be quasi-classical observable, its measurement can be discussed adequately in purely classical terms and can be effected using only well defined (and presumably understood) ffclassical" operations. We obtain, therefore, the prediction that in an ensemble of measurements of the quasi- classical observable B, assuming that the quantum mechanical spread in the values of B about the mean values b(a), Eq. (18), is undetectable classically, the distinct values b(a) will be found with the relative weights cal 2. From the unique theoretical connection of a and b(a) and an experimental observation of the relative frequency with which b(a) is observed, one can determine conversely both the relative weight ICal2 with which the eigenstate Oa of A was represented in the initial object state +(o), and the initial expectation value of A. The situation is only slightly more complicated in cases in which the spread in the values of B about the mean value b(a) is detectable on the macroscopic scale, or in which one deals with the continuum eigenvalues of a microscopic observable. In the measurement of B, the total ensemble of results must then split into sub-ensembles corresponding to the expected (distinguishable) distributions in value about the means b(a).

We turn next to the sums in Eq. (20) which represent the effects of interference phenomena involving the different (coherent) components sba of the initial object state. These sums have been neglected in the foregoing discussion, for reasons which become apparent in an analysis of the types of measurement which actually occur. It would, of course, be possible to analyze the results of an ensemble of measurements of B in terms of the initial structure of the object state if the off-diagonal matrix elements <Ma 1b B (t)l0aab> of the operator 1 (t) were all known; knowledge of the expectation values b(a) alone is insufficient for the solution of the problem. However, these




complications are absent in practice, and impose a further restriction on the measuring device. Either the interaction must be such that no matrix elements off-diagonal with respect to A occur, implying that [QA4, H] - 0 where H is the total Hamiltonian, or the measuring device must be selective, causing a physical separation of the eigenstates Xa [20]. The second alternative is that more frequently encountered. As an example of a selective measurement, one may consider observations on photon polarizations made using a Nicol prism. The state of a circularly polarized photon may be represented as a coherent superposition of two states of perpendicular linear polarizations, only one of which is transmitted by the prism. Thus, the physical interaction between the object and apparatus systems is different for the two states and results in a spacial separation of the coherent components in the final state. The measurement is completed by a photon detector which, because of its location, interacts only with the transmitted component. Thus the matrix elements of B (t) off-diagonal with respect to A in Eq. (20) will be vanishingly small; the two states do not overlap spacially near the detecting device. This example is typical of selective apparatus systems in general; the selective part of the system interacts differently with the different possible eigenstates of the object observable of interest, thereby preparing an altered state such that only the selected component can interact with the final measuring apparatus. In this case the interference terms in Eq. (20) are unimportant or disappear altogether, and the expectation value of B reduces to

< T(o)1 I (t) I (o) > Ica 12b(a). (20')

[Some of the b's may also disappear depending on the characteristics of the measuring device.] It is interesting to note that the result of Eq. (20') would also be obtained as the prediction for the results of an ensemble of measurements on an incoherent mixture of the pure states Ta(M), mixed with the weights Cal 2. Thus measurements with a selective device cannot distinguish between coherent superpositions and incoherent mixtures of pure states. Additional well-known and simply analyzed examples of selective devices are given by the atomic beams apparatus, which separates atomic eigenstates with different magnetic quantum numbers, and in the magnetic spectrometer, used to separate different eigenstates of the momentum operator. A simple example of a spin measurement with a selective device has recently been analyzed in complete detail by Green [21].

The foregoing discussion is highly idealized with respect to the treatment of the apparatus system, since it has been assumed that the initial apparatus state yb(o) can be known. Such is not the case; the initial state of the object system may indeed be known (prepared), but that of the apparatus system is at best delimited only by the results of a few quasi-classical observations. Furthermore, complete knowledge of the quantum mechanical details of the apparatus state is probably impossible even in principle. One need hardly remark that a similar problem exists relative to the complete microscopic description of a measuring device in classical physics. A statistical treatment




of the measuring process with respect to the apparatus system is therefore required; in fact, the quasi-classical observables B are often just the para- meters which reflect the "average" statistical behavior of the system, e.g., temperature, mean position or length, opacity, density, pressure, etc., and are therefore not themselves quantum mechanical observables. In this case the B's are rather related to the statistical means of the expectation values of some (quasi-classical) quantum mechanical observables B' with operators 3'. The necessary statistical treatment can be given in terms of the statistical matrix formulation of the quantum theory [17], but the conclusions of the present study are essentially unchanged.

c. Measurements on Correlated Quantities. The possible structure of the initial state function qb(o) of the object system, Eq. (11), is of considerable interest. In the foregoing discussion, we have assumed that i(o) can be expanded in a series of the eigenfunctions 6a of the operator Qe4 without reference to any other object observables. However, the expansion coefficients Ca will depend on the structure of i(o) with respect to any remaining observables. The object-apparatus interactions may also depend on these quantities which are not of interest. Nevertheless, the result of a significant measurement of A should by definition depend only on the state of the object system with respect to A, and not on unobserved quantities. The apparent difficulty disappears when the measurement of correlated observables is examined in detail. If only a specific state of some portion of the "unobserved" part of the object system is consistent with a particular assumed state of the "observed" part [that is, the part which is of interest in the measurement considered], then the "measured" and the "unmeasured" observables will be correlated. A measurement on the observable A of interest is then tantamount to a measurement of its correlates as well, and the entire set of correlated observables belongs in fact to the observed part of the system. The single index a then specifies in the foregoing sections the state of the object system relative to this total set of observables. Spurious difficulties have at times been introduced into the interpretation of quantum mechanics by the neglect of the essential point that correlated observables can be considered only in these sets: to consider one of the observables to have values independent of the values of the others is meaningless. For example, the final momenta in the elastic scattering of two particles are not independent; from the knowledge of the initial momenta Pl, P2 and an observation of the final momentum p', one can infer using the conservation of momentum that the momentum of the second particle was p' = Pi + P2 - p'. One often speaks, however, as if the values of p' and p' were independently indeterminate before a measurement, while afterwards the measured value p' determined the value of its correlate. The measuring process is therefore viewed as introducung a correlation where none was previously present. This misconception lies at the root of the "'paradox" of Einstein, Podosky, and Rosen [22]. Actually, the possible relative values of the correlated observables are determined

2




by the theory; the only uncertainty before a measurement relates to the particular set of values which will be found.

It is possible also that the observed and the unobserved parts of the object system are correlated, but that the state with respect to the unmeasured quantities does not influence appreciably the object-apparatus interaction. In this case all reference to the unobserved parts of the object system can be suppressed in discussing the process of measurement. Thus the spin state of the nucleus is irrelevant in discussions of the gross fine structure [but not the hyperfine structure] of atomic energy levels. A significant measurement will also be possible if the observed and the unobserved parts of the object system are uncorrelated. The state of the unobserved part may then be subjected to a specific preparation without influencing the quantities of interest; different total object states will then interact differently with the apparatus solely because of differences in the state with respect to the observable of interest. For example, the structure of an atomic state with respect to the z-component of the angular momentum is independent of the value of J2 except for the restriction I jz <J. Observations to determine that structure in different circumstances may always be made starting with the same eigenstate of J 2; the outcome of the measurements then reflects only differences in the structure with respect to Jz and not with respect to the extraneous observable J2. [It should be noted in this connection that the initial apparatus state is assumed always to be prepared in the same way before a measurement.] The case in which the observable of interest is not correlated with other, unmeasured object observables corresponds quite generally to situations in which the total state is specified in terms of a set of constants of the motion with respect to Ho, only one of which is observed.

Successive measurements on a single system also find a natural explanation within the present theory of measurement. At a time t at the completion of a single observation of A (or more precisely, of B(A)), the state of the object system is given according to Eqs. (12) by T(t) U(t) TF(o), where U(t) depends explicitly on the nature of the interaction between the object and apparatus subsystems which resulted in the first measurement. One may now consider the measurement of a second observable A' in terms of a quasi- classical observable B', starting with T(t) as the initial state and following the procedure which has been discussed. In particular, even if the value b(a) of B corresponding to the eigenvalue a of A is found in the first observation, T(t) is not in general the associated eigenstate Ta(t). While this is in direct contradiction to the theories of von Neumann [2] and Everett [3], and dis- agrees also with some versions of the Copenhagen interpretation of measurement [4, 5, 6, 7], it is certainly in accord with the way in which experiments are actually performed and analyzed. The results of a set of successive measurements consist of a collection of sequences b1(al), b2(a2).... of values corresponding to the individual steps in each succession of observations. These results are again to be interpreted statistically, and care must be taken with respect to correlations between the results of successive observations similar to that




discussed above. The detailed analysis will, however, be left to the reader as an example which displays the principles of measurement which have been proposed herein.

V. Concluding Remarks. This paper has been concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It was noted that the quantum theory permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr [7] of those macroscopic phenomena in terms of which the observer forms his perceptions. The essential remark at this point was that the process of measurement in quantum mechanics could be understood at this level by transcribing from the strictly classical observables of Newtonian physics to the corresponding quasi-classical observables the rules for the measurement of the former. There is in fact no practically observable distinction between observables of the two kinds insofar as they are supposed to enter the description of actual physical systems. It was therefore assumed that the measurement of quasi-classical observables could be adequately discussed in the language of classical physics and its accompanying epistomologies. The remaining (physical) problem was delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A to a classically measurable observable B could be said to result in a measurement of A. Suitable restrictions on the types of apparatus which could lead to significant measurements were therefore proposed and analyzed in terms of the mathematical structure of the theory. Together, the statistical interpretation of quantum mechanics and the existence of appropriate correlations between object and apparatus observables permit the translation of theoretical statements about the expectation values of abstract operators in a Hilbert space, evaluated in the initial object state, into statistical assertions about the results which will be found in an ensemble of quasi-classical observations. Conversely, from the results of such an ensemble in conjunction with the theory, significant information can be obtained about the initial object system. It is in this sense that the expectation value of a quantum mechanical observable is to be regarded as a prediction for the result of a set of measurements, or in which a set of measurements on the classical level is to be regarded as yielding information about a microscopic quantum mechanical system.

It is hoped that the present work will elucidate the physical aspects of the quantum theory of measurement, permitting a clearer discussion of the interpretative problems associated with quantum mechanics, and of their philosophic implications, unburdened by spurious difficulties originating in inadequate examinations of the actual process and theory of measurement.




REFERENCES

1. See Ref. (4) and (5) for comparative surveys of the various theories and interpretations of measurement.

2. J. VON NEUMANN, Mathematical Foundations of Quantum Mechanics, (Princeton University Press, Princeton, 1955).

3. HUGH EVERETT, III. Rev. Mod. Phys., 29, 254 (1957). 4. H. MARGENAU, Phil. of Science, 4, 337 (1937); 25, 23 (1958); Physics Today, 7, 6 (1954).

5. J. L. MCKNIGHT, Phil. of Science, 24, 321 (1957); 25, 209 (1958). Also "Measurement in Quantum Mechanical Systems, an Investigation of Foundations," doctoral dissertation, Yale University, 1957.

6. W. HEISENBERG, in Niels Bohr and the Development of Physics (Pergamon Press, London, 1955), edited by W. Pauli; Niels Bohr, in Albert Einstein, Philosopher-Scientist (Tudor Publishing Company, New York, 1951), edited by P. A. Schilpp.

7. NIELS BOHR, in Albert Einstein, Philosopher-Scientist (Tudor Publishing Company, New York, 1951); edited by P. A. Schilpp.

8. H. MARGENAU and G. M. MURPHY, The Mathematics of Physics and Chemistry, (D. van Nostrand Company, Inc. New York, 1951), Chap. 11. R. B. Lindsay and H. Margenau, Foundation of Physics (John Wiley and Sons, New York, 1936).

9. See especially H. MARGENAU, Phil. of Science, 25, 23 (1958), for a detailed analysis of the meaning of measurement in quantum physics and of the distinction between measurement and the preparation of a state.

10. Reference (7), p. 209. 11. This is discussed at length by Bohr, Ref. (7), pp. 201-241. 12. H. MARGENAU, The Nature of Physical Reality (McGraw-Hill Book Company, New York,

1950), Chap. 12. 13. The strict operational viewpoint is presented by P. W. Bridgman, The Logic of Modern

Physics (The Macmillan Company, New York, 1927). 14. P. CAWS, Phil. of Science, 24, 221 (1957). 15. A different attitude is characteristic of several other discussions of measurement. See in

particular the comments on the subject of McKnight (5) and Margenau (9). 16. Measurements of the negative kind, in which the absence of any change in B may be signif-

icant, can be completed only through the observation of a change in some other quasi- classical observable C which verifies at least that the apparatus is functioning, and generally also that the object system was present. Thus, in the example given in Sec. IIIa to illustrate the distinction between a measurement and the preparation of a state, the failure to observe any photons which have passed through the Nicol prism becomes a significant measurement with respect to the photon polarization only after it has been verified that a photon beam is indeed incident on the prism. Frequent use is also made of this type of measurement in the study of the new particles, for which the absence of a particular decay mode may be of great theoretical significance. However, for reasons of simplicity, we shall not consider explicitly the theory of measurements of this type, since no new matters of principle are involved.

17. LOYAL DURAND III. "On the Theory and Interpretation of Measurement in Quantum Mechanical Systems," Institute for Advanced Study preprint, January 1958, (unpublished).

18. It is not always possible to require that 'F(t) split for t - 0+ exactly into a product of functions b and b describing the individual subsystems, but the terms in 'I expressing the correlation may usually be made so small as to be negligible practically. A similar asymptotic conditions plays an important role in the modern quantum field theories, with the time t = 0 replaced by t --* - . The present analysis of measurement can be

applied immediately to these theories with the understanding that the limits t-- ? which appear there are to be interpreted as meaning at times remote in the future or past compared to the times characteristic of the interaction of the subsystems.




19. It is important to recognize that we do not require a unique correlation between the eigenvalues of A and the eigenvalues of B, but only between the eigenvalues of A and the expectation values of B. This is a much weaker condition that the former, which has been used by von Neumann (2) and Everett (3), and is in fact the most that can be required if the observable B is to be quasi-classical in nature. See also the remarks on statistical observables at the end of Sec. IVb.

20. It may, of course, also be possible to infer the value of A from measurements on some correlated quantum mechanical observable A' for which the interference effects are absent. This will be discussed in Sec IVc.

21. H. S. GREEN, II Nuovo Cimento, 9, 880 (1958). 22. EINSTEIN, PODOSKY, and ROSEN, Phys. Rev., 47, 777 (1935). For discussions of this paradox

see also N. BOHR. Phys. Rev., 48, 696 (1935), H. MARGENAU, Phys. Rev., 49, 240 (1936), and the articles by Bohr, Einstein and Margenau contained in (7).



on the theory of measurement in quantum mechanical systems

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