Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 054, 12 pages      arXiv:0807.1966      doi:10.3842/SIGMA.2008.054

Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics

Dieter Schuch a and Marcos Moshinsky b
a) Institut für Theoretische Physik, Goethe-Universität Frankfurt am Main, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
b) Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México D.F., México

Received February 06, 2008, in final form June 08, 2008; Published online July 14, 2008

Abstract
For classical canonical transformations, one can, using the Wigner transformation, pass from their representation in Hilbert space to a kernel in phase space. In this paper it will be discussed how the time-dependence of the uncertainties of the corresponding time-dependent quantum problems can be incorporated into this formalism.

Key words: canonical transformations; Wigner function; time-dependent quantum mechanics; quantum uncertainties.

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References

  1. Moshinsky M., Smirnov Y.F., The harmonic oscillator in modern physics, Harwood Academic Publishers, Amsterdam, 1996.
  2. Osborne T.A., Molzahn F.H., Moyal quantum mechanics: the semiclassical Heisenberg dynamics, Ann. Physics 241 (1995), 79-127.
  3. Dias N.C., Prata J.N., Features of Moyal trajectories, J. Math. Phys. 48 (2007), 012109, 23 pages.
  4. Krivoruchenko M.I., Fuchs C., Faessler A., Semiclassical expansion of quantum characteristics for many-body potential scattering problem, Ann. Phys. (8) 16 (2007), 587-614, nucl-th/0605015.
  5. Krivoruchenko M.I., Faessler A., Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics, J. Math. Phys. 48 (2007), 052107, 22 pages, quant-ph/0604075.
  6. Schuch D., Moshinsky M., Transition from quantum to classical behavior for some simple model systems, Rev. Mex. Fis. 51 (2005), 516-524.
  7. García-Calderón G., Moshinsky M., Wigner distribution function and the representation of canonical transformations in quantum mechanics, J. Phys. A: Math. Gen. 13 (1990), L185-L188.
  8. Moshinsky M., Quesne C., Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971), 1772-1780.
  9. Mello P.A., Moshinsky M., Nonlinear canonical transformations and their representations in quantum mechanics, J. Math. Phys. 16 (1975), 2017-2028.
  10. Wigner E.P., On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749-759.
  11. Schuch D., Moshinsky M., Connection between quantum-mechanical and classical time-evolution via a dynamical invariant, Phys. Rev. A 73 (2006), 062111, 10 pages.
  12. Feynman R.P., Hibbs A.R., Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
  13. Schuch D., On the complex relations between equations describing the dynamics of wave and particle aspects, Internat. J. Quantum Chem. 42 (1992), 663-683.
  14. Ermakov V.P., Second-order differential equations, conditions of complete integrability, Univ. Izv. Kiev 20 (1880), no. 9, 1-25.
  15. Lewis H.R., Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. 18 (1967), 510-512.
  16. Schuch D., On the relation between the Wigner function and an exact dynamical invariant, Phys. Lett. A 338 (2005), 225-231.
  17. Kim Y.S., Wigner E.P., Canonical transformation in quantum mechanics, Am. J. Phys. 58 (1990), 439-448.
  18. Kim Y.S., Noz M.E., Phase space picture of quantum mechanics; group theoretical approach, Lecture Notes in Physics, Vol. 40, World Scientific, Singapore, Chapter 3.3, 1991.
  19. O'Connell R.F., The Wigner distribution function - 50th birthday, Found. Phys. 13 (1983), 83-92.
  20. Schuch D., Riccati and Ermakov equations in time-dependent and time-independent quantum systems, SIGMA 4 (2008), 043, 16 pages, arXiv:0805.1687.
  21. Dias N.C., Classicality criteria, J. Math. Phys. 43 (2002), 5882-5901, quant-ph/9912034.
  22. Dodonov V.V., Universal integrals of motion and universal invariants of quantum systems, J. Phys. A: Math. Gen. 33 (2000), 7721-7738.
  23. Atakishiyev N.M., Chumakov S.M., Rivera A.L., Wolf K.B., On the phase space description of quantum nonlinear dynamics, Phys. Lett. A 215 (1996), 128-134.
  24. Rivera A.L., Atakishiyev N.M., Chumakov S.M., Wolf K.B., Evolution under polynomial Hamiltonians in quantum and optical phase space, Phys. Rev. A 55 (1997), 876-889.
  25. Sarlet W., Class of Hamiltonians with one degree-of-freedom allowing applications of Kruskal's asymptotic theory in closed form. II, Ann. Physics 92 (1975), 248-261.

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