Skip to main content
Log in

Quantum mechanics without wave functions

  • Part IV. Invited Papers Dedicated To David Bohm
  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

The phase space formulation of quantum mechanics is based on the use of quasidistribution functions. This technique was pioneered by Wigner, whose distribution function is perhaps the most commonly used of the large variety that we find discussed in the literature. Here we address the question of how one can obtain distribution functions and hence do quantum mechanics without the use of wave functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. H. Weyl,Z. Phys. 46, 1 (1927); H. Weyl,Gruppentheorie und Quantenmechanik (Hirzel, Leipzig, 1928), Chap. 4.

    Google Scholar 

  2. E. Wigner,Phys. Rev. 40, 749 (1932).

    Article  Google Scholar 

  3. For a recent review, see, e.g., M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,Phys. Rep. 106, 121 (1984).

    Article  Google Scholar 

  4. B. K. Jennings, R. K. Bhaduri, and M. Brack,Phys. Rev. Lett. 34, 228 (1975);Nucl. Phys. A 253, 29 (1975).

    Article  Google Scholar 

  5. J. P. Dahl and M. Springborg,Mol. Phys. 47, 1001 (1982).

    Google Scholar 

  6. M. Springborg and J. P. Dahl, inLocal Density Approximation in Quantum Chemistry and Solid State Physics, J. P. Dahl and J. Avery, eds. (Plenum, New York, 1984), p. 381.

    Google Scholar 

  7. J. T. Devreese, F. Brosens, and L. F. Lemmens,Phys. Rev. B 21, 1349 (1980);21, 1363 (1980); L. Wang and R. F. O'Connell,Phys. Rev. 34B, 5160 (1986).

    Article  Google Scholar 

  8. R. Dickman and R. F. O'Connell,Phys. Rev. Lett. 55, 1703 (1985);Phys. Rev. B 32, 471 (1985);Phys. Rev. B 34, 5678 (1986).

    Article  Google Scholar 

  9. P. Carruthers and F. Zachariasen,Rev. Mod. Phys. 55, 249 (1983).

    Article  Google Scholar 

  10. R. J. Gluaber,Phys. Rev. Lett. 10, 84 (1963);Phys. Rev. 131, 2766 (1963); E. C. G. Sudarshan,Phys. Rev. Lett. 10, 277 (1963); M. O. Scully and W. E. Lamb, Jr.,Phys. Rev. 159, 208 (1967).

    Article  Google Scholar 

  11. J. G. Kirkwood,Phys. Rev. 44, 31 (1933).

    Article  Google Scholar 

  12. See, e.g., D. Kastler,Commun. Math. Phys. 1, 14 (1965); G. Loupias and Miracle-Solé,Commun. Math. Phys. 2, 31 (1966); K. E. Cahill and R. J. Glauber,Phys. Rev. 177, 1857, 1883 (1969); G. S. Agarwal and E. Wolf,Phys. Rev. D 2, 2161 (1970).

    Article  Google Scholar 

  13. R. F. O'Connell and E. P. Wigner,Phys. Lett. 83A, 145 (1981);85A, 121 (1981); R. F. O'Connell, L. Wang, and H. A. Williams,Phys. Rev. A 30, 2187 (1984); R. F. O'Connell and L. Wang,Phys. Lett. A 107, 9 (1985);Phys. Rev. A 31, 1707 (1985).

    Google Scholar 

  14. J. E. Moyal,Proc. Camb. Phil. Soc. 45, 99 (1949).

    Google Scholar 

  15. L. Cohen,J. Math. Phys. 7, 781 (1966);17, 1863 (1976);ibid. inFrontiers of Nonequilibrium Satistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986), pp. 97–117.

    Article  Google Scholar 

  16. F. Bayen, M. Flato, C. Fronsdal, A. Lichnérowicz, and D. Sternheimer,Ann. Phys. (N.Y.) 111, 61, 111 (1978).

    Article  Google Scholar 

  17. J. M. Gracia-Bondia,Phys. Rev. A 30, 691 (1984).

    Article  Google Scholar 

  18. R. Dickman and R. F. O'Connell,Microstructures and Superlattices 2, 57 (1986).

    Article  Google Scholar 

  19. H. J. Groenewold,Physica 12, 405 (1946).

    Article  Google Scholar 

  20. F. Bopp,Werner Heisenberg und die Physik unserer Zeit (Vieweg, Braunschweig, 1961), p. 128.

    Google Scholar 

  21. In related work, Bayenet al. in Ref. 16, and later Gracia-Bondia in Ref. 17, considered the so-called evolution operator exp(−itĤ), which becomes the unnormalized density operator for a canonical ensemble at temperatureT=1/kβ, if one makes the substitutionit → β.

  22. F. Bloch,Z. Phys. 74, 295 (1932).

    Article  Google Scholar 

  23. I. Oppenheim and J. Ross,Phys. Rev. 107, 28 (1957).

    Article  Google Scholar 

  24. A. Alastuey and B. Jancovici,Physica 102A, 327 (1980).

    Google Scholar 

  25. K. Imre, E. Ozizmir, M. Rosenbaum, and P. Zweifel,J. Math. Phys. 8, 1097 (1967).

    Article  Google Scholar 

  26. D. Bohm and B. J. Hiley,Found. Phys. 11, 179 (1981).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L., O'Connell, R.F. Quantum mechanics without wave functions. Found Phys 18, 1023–1033 (1988). https://doi.org/10.1007/BF01909937

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01909937

Keywords

Navigation