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Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

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Abstract

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary “*” operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.

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REFERENCES

  1. N. Bohr, Phil. Mag. (Series 6) 26, 857 (1913).

    Google Scholar 

  2. W. Wilson, Phil. Mag. 29, 795 (1915).

    Google Scholar 

  3. A. Sommerfeld, Ann. Phys. (Leipzig) 51, 1 (1916).

    Google Scholar 

  4. G. Wentzel, Z. Phys. 38, 518 (1926).

    Google Scholar 

  5. H. Kramers, Z. Phys. 39, 828 (1926).

    Google Scholar 

  6. L. Brillouin, C. R. Acad. Sc. (Paris) 183, 24 (1926).

    Google Scholar 

  7. E. Madelung, Z. Phys. 40, 322 (1926).

    Google Scholar 

  8. V. P. Maslov, Theéorie des Perturbations et Méthodes Asymptotiques (Dunod, Paris, 1972).

    Google Scholar 

  9. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), p. 439.

    Google Scholar 

  10. R. G. Littlejohn, J. Stat. Phys. 68, 7 (1992).

    Google Scholar 

  11. J. B. Keller, Ann. Phys. 4, 180 (1958).

    Google Scholar 

  12. E. J. Heller, J. Chem. Phys. 62, 1544 (1975).

    Google Scholar 

  13. R. G. Littlejohn, Phys. Rev. Lett. 54, 1742 (1985).

    Google Scholar 

  14. R. G. Littlejohn, Phys. Lett. 138, 193 (1986).

    Google Scholar 

  15. H. Weyl, Z. Phys. 46, 1 (1928).

    Google Scholar 

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

    Google Scholar 

  17. J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).

    Google Scholar 

  18. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Lett. 106, 121 (1984).

    Google Scholar 

  19. B. Lesche and T. H. Seligman, J. Phys. A: Math. Gen. 19, 91 (1986).

    Google Scholar 

  20. A. Ram, Geometric Analysis and Lie Theory in Mathematics and Physics, A. L. Carey and M. K. Murray, eds. (Cambridge University Press, Cambridge, 1998), Chap. 2.

    Google Scholar 

  21. J. C. Light and J. M. Yuan, J. Chem. Phys. 58, 660 (1973).

    Google Scholar 

  22. J. C. Light, Reduced Density Operators with Applications to Physical and Chemical Systems-II, R. Erdahl, ed. (Queen's University, Kingston, Ontario, Canada, 1974), p. 171, Queen's Papers on Pure and Applied Mathematics, No. 40.

    Google Scholar 

  23. R. M. Stratt and W. H. Miller, J. Chem. Phys. 67, 5894 (1977).

    Google Scholar 

  24. N. Snider, J. Chem. Phys. 69, 3545 (1978).

    Google Scholar 

  25. B. Grammaticos and A. Voros, Ann. Phys. (N.Y.) 123, 359 (1979).

    Google Scholar 

  26. L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1927).

    Google Scholar 

  27. E. Fermi, Z. Phys. 48, 73 (1928).

    Google Scholar 

  28. W. Lenz, Z. Phys. 77, 713 (1932).

    Google Scholar 

  29. N. H. March, Adv. Phys. 6, 1 (1957).

    Google Scholar 

  30. E. H. Lieb and B. Simon, Phys. Rev. Lett. 31, 681 (1973).

    Google Scholar 

  31. E. H. Lieb, Rev. Mod. Phys. 53, 603 (1981).

    Google Scholar 

  32. L. Spruch, Rev. Mod. Phys. 63, 151 (1991).

    Google Scholar 

  33. D. A. Kirzhnits, Sov. Phys. JETP 5, 64 (1957).

    Google Scholar 

  34. S. Golden, Rev. Mod. Phys. 32, 322 (1960).

    Google Scholar 

  35. B. K. Jennings, R. K. Bhaduri, and M. Brack, Nucl. Phys. A 253, 29 (1975).

    Google Scholar 

  36. R. G. Parr, K. Rupnik, and S. K. Ghosh, Phys. Rev. Lett. 56, 1555 (1986).

    Google Scholar 

  37. S. Golden, Phys. Rev. 107, 1283 (1957).

    Google Scholar 

  38. S. Nordholm, J. Chem. Phys. 86, 363 (1987).

    Google Scholar 

  39. G. Landi, An Introduction to Noncommutative Spaces and Their Geometries (Springer, Berlin/Heidelberg, 1997), pp. 9-10.

    Google Scholar 

  40. M. Head-Gordon and J. C. Tully, J. Chem. Phys. 103, 10137 (1995).

    Google Scholar 

  41. B. Poirier and J. C. Light, J. Chem. Phys. 113, 211 (2000).

    Google Scholar 

  42. C. F. von Weizsäcker, Z. Phys. 96, 431 (1935).

    Google Scholar 

  43. B. Poirier and J. C. Light, J. Chem. Phys. 111, 4869 (1999).

    Google Scholar 

  44. B._G. Englert and J. Schwinger, Phys. Rev. A 29, 2339 (1984).

    Google Scholar 

  45. B. Farb and R. K. Dennis, Noncommutative Algebra (Springer, New York, 1991), pp. 10-12.

    Google Scholar 

  46. B. Poirier, unpublished.

  47. B. Poirier, Phys. Rev. A 56, 120 (1997).

    Google Scholar 

  48. J. J. Morehead, J. Math. Phys. 36, 5431 (1995).

    Google Scholar 

  49. J. B. Keller and S. I. Rubinow, Ann. Phys. 9, 24 (1960).

  50. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985).

    Google Scholar 

  51. R. E. Langer, Phys. Rev. 51, 669 (1937).

    Google Scholar 

  52. B. Poirier, J. Math. Phys. 40, 6302 (1999).

    Google Scholar 

  53. M. V. Berry and A. M. Ozorio de Almeida, J. Phys. 6, 1451 (1973).

    Google Scholar 

  54. J. D. Jackson, Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975), Chap. 2.1.

    Google Scholar 

  55. W. E. Milne, Phys. Rev. 35, 863 (1930).

    Google Scholar 

  56. L. A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974).

    Google Scholar 

  57. S. W. McDonald and A. N. Kaufmann, Phys. Rev. Lett. 42, 1189 (1979).

    Google Scholar 

  58. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 409.

    Google Scholar 

  59. C. A. Hodges, Can. J. Phys. 51, 1428 (1973).

    Google Scholar 

  60. D. R. Murphy, Phys. Rev. A 24, 1682 (1981).

    Google Scholar 

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  1. Département de Chimie, Université de Montréal, C.P. 6128, Succursale centre-ville, Montréal, Québec, H3C 3J7, Canada

    Bill Poirier

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Poirier, B. Algebraically Self-Consistent Quasiclassical Approximation on Phase Space. Foundations of Physics 30, 1191–1226 (2000). https://doi.org/10.1023/A:1003632404712

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