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Prog. Theor. Phys. Vol. 82 No. 4 (1989) pp. 638-664

[ Full Text PDF : FREE ACCESS (1458K) ]

Probabilistic Interpretation and the Quantum Theory of Measurement. II

Ziro Maki

Research Institute for Fundermental Physics, Kyoto University, Kyoto 606

(Received June 2, 1989)

Abstract:

A general theory of quantum-mechanical measurement is presented. It is shown that some general conditions should be satisfied for any apparatus to carry out the ideal measurement, and the reduction of state is an inevitable consequence of the action of such an apparatus. The statistical operator (in weak sense) is defined uniquely for ensembles prepared with respect to the measured observables, without recourse to the expectation value hypothesis. The probabilistic interpretation (the Born ansatz) comes out as a natural explanation of physical contents of the statistical operator after the measurement, rather than an axiomatic presumption of quantum mechanics.


URL : http://ptp.ipap.jp/link?PTP/82/638/
DOI : 10.1143/PTP.82.638

[ Full Text PDF : FREE ACCESS (1458K) ] Citation:

References:

  1. Z. Maki, Prog. Theor. Phys. 79 (1988), 313 [PTP](to be referred to as I).
  2. Z. Maki, Soryushiron Kenkyu (Kyoto) 78 (1988), B50 (to be referred to as I′).
  3. See for example, A. Böhm, “The Rigged Hilbert Space and Quantum Mechanics” Springer Lecture Note in Physics 78 (Springer-Verlag, 1978).
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  14. D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N. Y., 1951).
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  17. This kind of randomness corresponds to the quantum fluctuations in `Class II intensive variables' in Fukuda's theory. See Ref. 11).
  18. There are few different approaches to `explain' |cλ|2 as the probability: H. Everett III, Ref. 6).
    M. Bunge, Foundation of Physics (Springer-Verlag, New York, 1967).
    M. I. Shirokov, Dubna Preprint (1981), P4-81-737.
    The last paper seems to involve a similar point to ours, but based on another axiomatic postulate of the state reduction.
  19. As for the mathematical formulation of classical variables in quantum mechanics, see for example, H. Araki in Proceedings of ISQM-Tokyo '86, ed. M. Namiki, Y. Ohnuki, Y. Murayama and S. Nomura (Phys. Soc. Japan, 1987).
  20. See for example, The Many-Worlds Interpretation of Quantum Mechanics, ed. B. S. DeWitt and N. Graham (Princeton, 1973).
  21. See, for example, A. S. Holebo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, 1982).
    G. Ludwig, Foundations of Quantum Mechanics I (Springer-Verlag, 1983).

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 84 No. 4 (1990) pp. 574-583 :
    An Algebraic Approach to the Quantum Theory of Measurements
    Ziro Maki
  2. Progress of Theoretical Physics Vol. 86 No. 4 (1991) pp. 943-958 :
    One-Particle Detectors with SU(1, 1) Coherent States in Quantum Measurement
    Kiyotaka Kakazu and Akihiko Ogawa
  3. Progress of Theoretical Physics Vol. 87 No. 1 (1992) pp. 61-76 :
    The Wigner-Araki-Yanase Theorem and Its Extension in Quantam Measurement with Generalized Coherent States
    Shoju Kudaka and Kiyotaka Kakazu

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