Journal of Statistical Physics

, Volume 52, Issue 5–6, pp 1203–1220 | Cite as

The statistical wave function

  • R. D. Levine


Statistical considerations are applied to quantum mechanical amplitudes. The physical motivation is the progress in the spectroscopy of highly excited states, The corresponding wave functions are “strongly mixed.” In terms of a basis set of eigenfunctions of a zeroth-order Hamiltonian with good quantum numbers, such wave functions have contributions from many basis states. The vector x is considered whose components are the expansion coefficients in that basis. Any amplitude can be written as a·x. It is argued that the components of x and hence other amplitudes can be regarded as random variables. The maximum entropy formalism is applied to determine the corresponding distribution function. Two amplitudes a·x and b·x are independently distributed if b·a=0. It is suggested that the theory of quantal measurements implies that, in general, one can one determine the distribution of amplitudes and not the amplitudes themselves.

Key words

Fluctuations spectra intensities statistical theories mixing chaos maximum entropy 


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  1. 1.
    H. Reiss, A. D. Hammerich, and E. W. Montroll,J. Stat. Phys. 42:647 (1986).Google Scholar
  2. 2.
    U. Dinur and R. D. Levine,Chem. Phys. 5:17 (1975).Google Scholar
  3. 3.
    J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1955).Google Scholar
  4. 4.
    E. W. Montroll,Proc. Natl. Acad. Sci. USA 78:7839 (1981).Google Scholar
  5. 5.
    J. Schwinger,Proc. Natl. Acad. Sci. USA 45:1552 (1959).Google Scholar
  6. 6.
    R. C. Tolman,The Principles of Statistical Mechanics (Clarendon Press, Oxford, 1938).Google Scholar
  7. 7.
    E. J. Heller,J. Chem. Phys. 72:1337 (1980); E. B. Stechel and E. J. Heller,Annu. Rev. Phys. Chem. 35:563 (1984).Google Scholar
  8. 8.
    Y. Alhassid and R. D. Levine,Phys. Rev. Lett. 57:2879 (1986).Google Scholar
  9. 9.
    R. D. Levine,Ado. Chem. Phys. 70:53 (1988).Google Scholar
  10. 10.
    R. B. Gerber, V. Buch, and M. A. Ratner,Chem. Phys. Lett. 89:171 (1982).Google Scholar
  11. 11.
    M. V. Berry,J. Phys. A 17:2083 (1977).Google Scholar
  12. 12.
    R. B. Bernstein, A. Dalgarno, H. S. W. Massey, and I. C. Percival,Proc. Soc. A 274:427 (1963).Google Scholar
  13. 13.
    R. D. Levine,Quantum Mechanics of Molecular Rate Processes (Clarendon Press, Oxford, 1969).Google Scholar
  14. 14.
    R. D. Levine and R. B. Bernstein,J. Chem. Phys. 53:686 (1970).Google Scholar
  15. 15.
    H. Feshbach,Ann. Phys. 19:287 (1962).Google Scholar
  16. 16.
    H. S. Taylor,Int. J. Quant. Chem. 31:747 (1987).Google Scholar
  17. 17.
    C. E. Porter,Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965).Google Scholar
  18. 18.
    T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong,Rev. Mod. Phys. 53:585 (1981).Google Scholar
  19. 19.
    J. Brickmann, Y. M. Engel, and R. D. Levine,Chem. Phys. Lett. 137:441 (1987).Google Scholar
  20. 20.
    P. A. P. Moran,An Introduction to Probability Theory (Clarendon Press, Oxford, 1968).Google Scholar
  21. 21.
    E. T. Jaynes,Phys. Rev. 108:171 (1957).Google Scholar
  22. 22.
    I. C. Percival,Adv. Chem. Phys. 36:1 (1977).Google Scholar
  23. 23.
    R. D. Levine, inLarge Finite Systems, J. Jortner, A. Pullman, and B. Pullman, eds. (Reidel, Dordrecht, 1987).Google Scholar
  24. 24.
    R. D. Levine,J. Chem. Phys. 84:910 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • R. D. Levine
    • 1
  1. 1.Fritz Haber Research Center for Molecular DynamicsHebrew UniversityJerusalemIsrael

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