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On the Theory of Measurement in Quantum Mechanical Systems

Published online by Cambridge University Press:  14 March 2022

Loyal Durand III*
Affiliation:
The Institute for Advanced Study

Abstract

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.

Type
Research Article
Copyright
Copyright © 1959 by Philosophy of Science Association

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Footnotes

∗∗

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.

References

1. See Ref. (4) and (5) for comparative surveys of the various theories and interpretations of measurement.Google Scholar
2. Neumann, J. von, Mathematical Foundations of Quantum Mechanics, (Princeton University Press, Princeton, 1955).Google Scholar
3. Everett, Hugh III. Rev. Mod. Phys., 29, 254 (1957).10.1103/RevModPhys.29.454CrossRefGoogle Scholar
4. Margenau, H., Phil. of Science, 4, 337 (1937); 25, 23 (1958); Physics Today, 7, 6 (1954).10.1086/286467CrossRefGoogle Scholar
5. McKnight, J. L., Phil. of Science, 24, 321 (1957); 25, 209 (1958). Also “Measurement in Quantum Mechanical Systems, an Investigation of Foundations,” doctoral dissertation, Yale University, 1957.10.1086/287553CrossRefGoogle Scholar
6. Heisenberg, W., 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.Google Scholar
7. Bohr, Niels, in Albert Einstein, Philosopher-Scientist (Tudor Publishing Company, New York, 1951); edited by P. A. Schilpp.Google Scholar
8. Margenau, H. and Murphy, G. M., 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).Google Scholar
9. See especially Margenau, H., 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.10.1086/287574CrossRefGoogle Scholar
10. Reference (7), p. 209.Google Scholar
11. This is discussed at length by Bohr, Ref. (7), pp. 201241.Google Scholar
12. Margenau, H., The Nature of Physical Reality (McGraw-Hill Book Company, New York, 1950), Chap. 12.Google Scholar
13. The strict operational viewpoint is presented by Bridgman, P. W., The Logic of Modern Physics (The Macmillan Company, New York, 1927).Google Scholar
14. Caws, P., Phil. of Science, 24, 221 (1957).10.1086/287538CrossRefGoogle Scholar
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).Google Scholar
16. Measurements of the negative kind, in which the absence of any change in B may be significant, 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.Google Scholar
17. Loyal Durand III. “On the Theory and Interpretation of Measurement in Quantum Mechanical Systems,” Institute for Advanced Study preprint, January 1958, (unpublished).Google Scholar
18. It is not always possible to require that Ψ(t) split for 0+ exactly into a product of functions ψ and ψ describing the individual subsystems, but the terms in Ψ 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 — ∞. The present analysis of measurement can be applied immediately to these theories with the understanding that the limits ± ∞ 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.Google Scholar
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.Google Scholar
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.Google Scholar
21. Green, H. S., Il Nuovo Cimento, 9, 880 (1958).10.1007/BF02903128CrossRefGoogle Scholar
22. Einstein, Podosky, and Rosen, , Phys. Rev., 47, 777 (1935). For discussions of this paradox see also N. Bohr. Phys. Rev., 48, 696 (1935), Margenau, H., Phys. Rev., 49, 240 (1936), and the articles by Bohr, Einstein and Margenau contained in (7).10.1103/PhysRev.47.777CrossRefGoogle Scholar