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American Journal of Physics

On the Interpretation of Measurement within the Quantum Theory

Leon N. Cooper, Deborah van Vechten

Published Dec 1, 1969• 34 citations
Publisher

Abstract

An interpretation of the process of measurement is proposed which can be placed wholly within the quantum theory. The entire system including the apparatus and even the mind of the observer can be considered to develop according to the Schrödinger equation. No separation, in principle, of the observer and the observed is necessary; nor is it necessary to introduce either the type I process of von Neumann or wave function reduction.

DOI
10.1119/1.1975279

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A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II

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In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of "hidden" variables, developed in a previous paper. We find that in principle, these "hidden" variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.Finally, we show that von Neumann's proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann's assumptions.In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.

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A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I

David Böhm

The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

Physical Review•Jan 15, 1952•5,507 citations
An interpretation of the process of measurement is proposed which can be placed wholly within the quantum theory. The entire system including the apparatus and even the mind of the observer can be considered to develop according to the Schrödinger equation. No separation, in principle, of the observer and the observed is necessary; nor is it necessary to introduce either the type I process of von Neumann or wave function reduction.
It is well-known that developments of quantum mechanical states according to the Schrödinger equation during a measurement seem to prevent measurements from having definite results. For, the usually assumed (idealized) Schrödinger development of the measured object and the measuring apparatus during a measurement typically results in a state of the entire system which is a superposition of the eigenstates of the measured observable and measuring observable. And the most common interpretations of quantum mechanics state that an observable does not have a definite value if the quantum mechanical state is a superposition of the eigenstates of the observable. Many ways out of this problem have been suggested. I want to discuss an interpretation suggested by S. Kochen (Kochen 1985), and suggest similar interpretations, each of which accept the universal validity of the Schrödinger equation, and yet yield definite results for measurements.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.
In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of "hidden" variables, developed in a previous paper. We find that in principle, these "hidden" variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.Finally, we show that von Neumann's proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann's assumptions.In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.