Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: canonical transformations; Wigner function; time-dependent quantum mechanics; quantum uncertainties.
@article{SIGMA_2008_4_a53,
     author = {Dieter Schuch and Marcos Moshinsky},
     title = {Wigner {Distribution} {Functions} and the {Representation} of {Canonical} {Transformations} in {Time-Dependent} {Quantum} {Mechanics}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2008},
     volume = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a53/}
}
TY  - JOUR
AU  - Dieter Schuch
AU  - Marcos Moshinsky
TI  - Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2008
VL  - 4
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a53/
LA  - en
ID  - SIGMA_2008_4_a53
ER  - 
%0 Journal Article
%A Dieter Schuch
%A Marcos Moshinsky
%T Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics
%J Symmetry, integrability and geometry: methods and applications
%D 2008
%V 4
%U http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a53/
%G en
%F SIGMA_2008_4_a53
Dieter Schuch; Marcos Moshinsky. Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a53/

[1] Moshinsky M., Smirnov Y. F., The harmonic oscillator in modern physics, Harwood Academic Publishers, Amsterdam, 1996 | Zbl

[2] Osborne T. A., Molzahn F. H., “Moyal quantum mechanics: the semiclassical Heisenberg dynamics”, Ann. Physics, 241 (1995), 79–127 | DOI | MR

[3] Dias N. C., Prata J. N., “Features of Moyal trajectories”, J. Math. Phys., 48 (2007), 012109, 23 pp., ages | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 pp., ages ; quant-ph/0604075 | DOI | MR | Zbl

[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 | MR

[8] Moshinsky M., Quesne C., “Linear canonical transformations and their unitary representations”, J. Math. Phys., 12 (1971), 1772–1780 | DOI | MR | Zbl

[9] Mello P. A., Moshinsky M., “Nonlinear canonical transformations and their representations in quantum mechanics”, J. Math. Phys., 16 (1975), 2017–2028 | DOI | MR

[10] Wigner E. P., “On the quantum correction for thermodynamic equilibrium”, Phys. Rev., 40 (1932), 749–759 | DOI

[11] Schuch D., Moshinsky M., “Connection between quantum-mechanical and classical time-evolution via a dynamical invariant”, Phys. Rev. A, 73 (2006), 062111, 10 pp., ages | DOI

[12] Feynman R. P., Hibbs A. R., Quantum mechanics and path integrals, McGraw-Hill, New York, 1965 | Zbl

[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 | DOI

[14] Ermakov V. P., “Second-order differential equations, conditions of complete integrability”, Univ. Izv. Kiev, 20:9 (1880), 1–25

[15] Lewis H. R., “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians”, Phys. Rev. Lett., 18 (1967), 510–512 | DOI

[16] Schuch D., “On the relation between the Wigner function and an exact dynamical invariant”, Phys. Lett. A, 338 (2005), 225–231 | DOI | MR | Zbl

[17] Kim Y. S., Wigner E. P., “Canonical transformation in quantum mechanics”, Am. J. Phys., 58 (1990), 439–448 | DOI

[18] Kim Y. S., Noz M. E., Phase space picture of quantum mechanics; group theoretical approach, Chapter 3.3, Lecture Notes in Physics, 40, World Scientific, Singapore, 1991 | MR

[19] O'Connell R. F., “The Wigner distribution function – 50th birthday”, Found. Phys., 13 (1983), 83–92 | DOI

[20] Schuch D., “Riccati and Ermakov equations in time-dependent and time-independent quantum systems”, SIGMA, 4 (2008), 043, 16 pp., ages ; arXiv:0805.1687 | MR | Zbl

[21] Dias N. C., “Classicality criteria”, J. Math. Phys., 43 (2002), 5882–5901 ; quant-ph/9912034 | DOI | MR | Zbl

[22] Dodonov V. V., “Universal integrals of motion and universal invariants of quantum systems”, J. Phys. A: Math. Gen., 33 (2000), 7721–7738 | DOI | MR | Zbl

[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 | DOI

[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 | DOI | MR

[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 | DOI | MR | Zbl