Advertisement

Time-Dependent Variational Principle for the Expectation Value of an Observable: Mean-Field Applications

  • Marcel Vénéroni
Part of the NATO ASI Series book series (NSSB, volume 130)

Abstract

In non equilibrium statistical mechanics one is often interested in making some prediction at a time t1 from the knowledge of some properties at an earlier time t0. In quantum statistical mechanics all the knowledge about the state (whether pure or not) of the system of interest is included, at a given time, in the density operator. Let us assume that this system has been prepared in such a way that the density operator D(t0), characterizing the initial state, is known. At some later time t1 one intends to perform a measurement of some observable A. The theoretical problem is the prediction of the average value of A at the time t1. In principle the calculation of this expectation value
$$A({{t}_{1}};D,{{t}_{0}})\equiv TrAD({{t}_{1}},{{t}_{0}})$$
(1.1)
requires only the knowledge of the Hamiltonian H of the system, since the exact evolution of the density operator D(t, t0) is given by the Liouville-von Neumann equation (2.4).

Keywords

Variational Principle Density Operator Slater Determinant Heisenberg Equation Heisenberg Picture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Balian and M. Vénéroni, to be published in Annals of Physics; a first account is given in Phys. Rev. Lett. 47(1981), 1353, 1765 (E).Google Scholar
  2. [2]
    K. Gottfried, “Quantum mechanics”, p. 240, Benjamin, New York, 1966.Google Scholar
  3. [3]
    See Balian’s lectures in the present volume.Google Scholar
  4. [4]
    B.A. Lippmann and J. Schwinger, Phys. Rev. 79 (1950), 469.CrossRefGoogle Scholar
  5. [5]
    J.P. Blaizot and G. Ripka, Phys. Lett. 105B (1981), 1.CrossRefGoogle Scholar
  6. [6]
    N.G. Van Kampen, Physica Norvegica 5(1971), 279Google Scholar
  7. [7]
    M. Gaudin, Nucl. Phys. 15 (1960), 89.CrossRefGoogle Scholar
  8. [8]
    R. Balian and M. Vénéroni, Phys. Lett. 136B(1984), 301, and in preparation;Google Scholar
  9. R. Balian, P. Bonche, H. Flocard and M. Vénéroni, Nucl. Phys. A 428 (1984), 79c – 94c.CrossRefGoogle Scholar
  10. [9]
    R. Balian and E. Brézin, Nuovo Cimento B64 (1969), 37.CrossRefGoogle Scholar
  11. [10]
    S. Levit, Phys. Rev. C21 (1980), 1594.Google Scholar
  12. Y. Alhassid and S.E. Koonin, Phys. Rev. C23 (1981), 1590.Google Scholar
  13. See also references in H. Reinhardt, Nucl. Phys. A390 (1982), 70.CrossRefGoogle Scholar
  14. [11]
    J.W. Negele, Rev. Mod. Phys. 54 (1982), 913.CrossRefGoogle Scholar
  15. K.T.R. Davies, K.R. Sandhya Devi, S.E. Koonin and M.R. Strayer, in “Heavy ion science” (D.A. Bromley, Ed.), Plenum, New York, 1983.Google Scholar
  16. [12]
    See for instance D. Brink, Nucl. Phys. A409 (1983) p. 220c;Google Scholar
  17. S. Yamaji and M. Tohyama, Phys. Lett. 147B (1984), 399.CrossRefGoogle Scholar
  18. [13]
    P. Bonche and H. Flocard, to be published in Nucl. Phys.A.Google Scholar
  19. [14]
    H.J. Lipkin, N. Meshkov and A.J. Glick, Nucl. Phys. 62 (1965), 188, 199, 211.Google Scholar
  20. [15]
    P. Bonche, S.E. Koonin and J.W. Negele, Phys. Rev. C13 (1976) 1226.CrossRefGoogle Scholar
  21. [16]
    J.B. Marston, Senior Thesis, Kellog Radiation Laboratory, Caltech (1984), unpublished.Google Scholar
  22. [17]
    T. Troudet and D. Vautherin, Phys. Rev. C31 (1985), 278.Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Marcel Vénéroni
    • 1
  1. 1.Division de Physique ThéoriqueInstitut de Physique NucléaireOrsayFrance

Personalised recommendations