Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Quantum equations of motion and the Liouville equation

  • 129 Accesses

  • 1 Citations

Abstract

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if\(\hat F\) and Ĝ are two such functions and\(\hat F^\prime \) and Ĝ′ contain an arbitrary permutation of variables, then

$$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$

where  is a remainder commutator. In the classical limit [\(\hat F\), Ĝ] is invariant to such permutation.

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

References

  1. 1.

    J. Snygg,Am. J. Phys. 50, 906 (1982).

  2. 2.

    L. E. Ballentine,Am. J. Phys. 52, 74 (1984).

  3. 3.

    R. L. Liboff,Introduction to the Theory of Kinetic Equations (Krieger, Melbourne, Florida, 1979).

  4. 4.

    R. K. Pathria,Statistical Mechanics (Pergamon, New York, 1972).

  5. 5.

    P. A. M. Dirac,The Principles of Quantum Mechanics, 3rd edn. (Clarendon, London, 1974).

  6. 6.

    K. Gottfried,Quantum Mechanics (W. A. Benjamin, New York, 1966).

  7. 7.

    E. Merzbacher,Quantum Mechanics (Wiley, New York, 1961).

  8. 8.

    A. Messiah,Quantum Mechanics (Wiley, New York, 1961).

  9. 9.

    R. L. Liboff,Introductory Quantum Mechanics (Holden-Day, Oakland, California, 1968).

  10. 10.

    L. I. Schiff,Quantum Mechanics, 34rd edn. (McGraw Hill, New York, 1968).

  11. 11.

    D. Bohm,Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951).

  12. 12.

    R. L. Liboff,Found. Phys. 5, 271 (1975).

  13. 13.

    R. L. Liboff,Int. J. Th. Phys. 185 (1979).

  14. 14.

    J. von Neumann,Mathematical Foundations of Quantum Mechanics, (Princeton University Press, Princeton, New Jersey, 1955).

  15. 15.

    L. Reichl,A Modern Course in Statistical Physics (University of Texas Press, Austin, Texas, 1980).

  16. 16.

    P. A. M. Dirac,Proc. Cambridge Philos. Soc. 25, 62 (1929).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liboff, R.L. Quantum equations of motion and the Liouville equation. Found Phys 17, 981–991 (1987). https://doi.org/10.1007/BF00938008

Download citation

Keywords

  • Analytic Function
  • Classical Equation
  • Density Matrix
  • Closed Form
  • Related Equation