Symmetry Transformation in Extended Phase Space: the Harmonic Oscillator in the Husimi Representation

Samira Bahrami ; Sadolah Nasiri

Symmetry, integrability and geometry: methods and applications, Tome 4 (2008)

Résumé

In a previous work the concept of quantum potential is generalized into extended phase space (EPS) for a particle in linear and harmonic potentials. It was shown there that in contrast to the Schrödinger quantum mechanics by an appropriate extended canonical transformation one can obtain the Wigner representation of phase space quantum mechanics in which the quantum potential is removed from dynamical equation. In other words, one still has the form invariance of the ordinary Hamilton–Jacobi equation in this representation. The situation, mathematically, is similar to the disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates. Here we show that the Husimi representation is another possible representation where the quantum potential for the harmonic potential disappears and the modified Hamilton–Jacobi equation reduces to the familiar classical form. This happens when the parameter in the Husimi transformation assumes a specific value corresponding to $Q$-function.

Détail

Keywords: Hamilton–Jacobi equation; quantum potential; Husimi function; extended phase space.

Bibliographie

[1] Bohm D., Hiley B. J., “Unbroken quantum realism, from microscopic to macroscopic levels”, Phys. Rev. Lett., 55 (1985), 2511–2514 | DOI | MR

[2] Holland P. R., The quantum theory of motion, Cambridge University Press, 1993, 68–69 | MR

[3] Takabayashi T., “The formulation of quantum mechanics in terms of ensemble in phase space”, Progr. Theoret. Phys., 11 (1954), 341–373 | DOI | MR

[4] Muga J. G., Sala R., Snider R. F., “Comparison of classical and quantum evolution of phase space distribution functions”, Phys. Scripta, 47 (1993), 732–739 | DOI

[5] Brown M. R., The quantum potential: the breakdown of classical symplectic symmetry and the energy of localization and dispersion,, quant-ph/9703007

[6] Holland P. R., “Quantum back-reaction and the particle law of motion”, J. Phys. A: Math. Gen., 39 (2006), 559–564 | DOI | MR | Zbl

[7] Shojai F., Shojai A., “Constraints algebra and equation of motion in Bohmian interpretation of quantum gravity”, Classical Quantum Gravity, 21 (2004), 1–9 ; gr-qc/0409035 | DOI | MR | Zbl

[8] Carroll R., Fluctuations, gravity, and the quantum potential, gr-qc/0501045

[9] Nasiri S., “Quantum potential and symmetries in extended phase space”, SIGMA, 2 (2006), 062, 12 pp., ages ; quant-ph/0511125 | MR | Zbl

[10] Carroll R., Some fundamental aspects of a quantum potential, quant-ph/0506075

[11] Sobouti Y., Nasiri S., “A phase space formulation of quantum state functions”, Internat. J. Modern Phys. B, 7 (1993), 3255–3272 | DOI | MR

[12] Nasiri S., Sobouti Y., Taati F., “Phase space quantum mechanics – direct”, J. Math. Phys., 47 (2006), 092106, 15 pp., ages ; quant-ph/0605129 | DOI | MR | Zbl

[13] Nasiri S., Khademi S., Bahrami S., Taati F., “Generalized distribution functions in extended phase space”, Proceedings QST4, Vol. 2, ed. V. K. obrev, Heron Press Sofia, 2006, 820–826

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

[15] Lee H. W., “Theory and application of the quantum phase space distribution functions”, Phys. Rep., 259 (1995), 147–211 | DOI | MR

[16] de Gosson M., “Symplectically covariant Schrödinger equation in phase space”, J. Phys. A: Math. Gen., 38 (2005), 9263–9287 ; math-ph/0505073 | DOI | MR | Zbl

[17] Jannussis A., Patargias N., Leodaris A., Phillippakis T., Streclas A., Papatheos V., Some remarks on the nonnegative quantum mechanical distribution functions, Preprint, Department of Theoretical Physics, University of Patras, 1982 | MR

[18] Husimi K., “Some formal properties of the density matrix”, Proc. Phys.-Math. Soc. Japan, 22 (1940), 264–314