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Prog. Theor. Phys. Vol. 11 No. 4 (1954) pp. 341-373

The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space

Takehiko Takabayasi

Physical Institute, Nagoya University

(Received November 16, 1953)

Abstract:

The formulation of non-relativistic quantum mechanics in terms of ensemble in phase space is established by clarifying the subsidiary conditions for the phase space ensemble to represent a pure state, and thereby the equivalent correspondence between this formulation and the alternative formulation in terms of quantum potential previously developed is exhibited.

URL : http://ptp.ipap.jp/link?PTP/11/341/
DOI : 10.1143/PTP.11.341

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References:

1. The content of this article was reported in
   Prog. Theor. Phys. 10 (1953), 119[PTP]; ibid. 10 (1953), 121[PTP]; ibid. 10 (1953), 122, [PTP]
   in its essentials, and also in Busseiron Kenkyu 66 (1953), 48, in Japanese nearly in full.
2. P. A. M. Dirac, The Principles of Quantum Mechanics, third ed. p. 10.
3. R. P. Feynman, Rev. Mod. Phys. 20 (1948), 367[APS].
4. T. Takabayasi, Prog. Theor. Phys. 8 (1952), 143[PTP];
   also see D. Bohm, Phys. Rev. 84 (1952), 166[APS]; ibid. 84 (1952), 180[APS].
   L. de Broglie, La Physique Quantique Restra-t-elle Indéterministe? (Gauthier-Villars, Paris, 1953).
5. T. Takabayasi, Prog. Theor. Phys. 9 (1953), 187[PTP].
6. E. P. Wigner, Phys. Rev. 40 (1932), 749[APS].
7. J. E. Moyal, Proc. Cambridge Phil. Soc. 45 (1949), 99.
8. H. S. Green, J. Chem. Phys. 19 (1951), 955[CrossRef].
   J. H. Irving and R. W. Zwanzig, J. Chem. Phys. 19 (1951), 1173[CrossRef].
   H. Mori and S. Ono, Prog. Theor. Phys. 8 (1952), 327[PTP].
   H. Mori, Prog. Theor. Phys. 9 (1953), 473[PTP].
   a) J. v. Neumann, Mathematische Grundlagen der Quanteumechanik (1932).
9. H. Weyl, Gruppentheorie und Quantenmechanik (1931), p. 244.
10. Cf. P. A. M. Dirac, Rev. Mod. Phys. 17 (1945), 195[APS].
11. Cf. F. Bopp, ZS. f. Naturforsch. a 2 (1947), 202; also cf. reference 4), §1.
12. See T. Takabayasi, J. of History of Science, Japan, No. 19 (1951).
    It would be interesting to compare our viewpoint with that of
    K. Husimi, Prog. Theor. Phys. 9 (1953), 381[PTP].
13. H. A. Kramers, Physica 7 (1940), 284[CrossRef].
14. W. Weizel, ZS. f. Phys. 134 (1953), 264.
15. Compare with D. Bohm, Quantum Theory (Prentice-Hall, New York, 1951) p. 240.
16. Cf. E. Schrödinger, Naturwiss. 14 (1926), 664.
    L. Schiff, Quantum Mechanics, p. 67.

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 13 No. 1 (1955) pp. 106-108 :

   On the Structure of Dirac Wave Function

   Takehiko Takabayasi

2. Progress of Theoretical Physics Vol. 46 No. 6 (1971) pp. 1703-1712 :

   On Phase-Space Description of Quantum Mechanics

   G. J. Ruggeri

3. Progress of Theoretical Physics Vol. 110 No. 1 (2003) pp. 127-136 :

   Quantization via Star Products

   Takayuki Hori and Takao Koikawa

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