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

SIGMA 4 (2008), 014, 7 pages      arXiv:0802.0482      https://doi.org/10.3842/SIGMA.2008.014
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

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

Samira Bahrami a and Sadolah Nasiri b
a) Department of Physics, Zanjan University, Zanjan, Iran
b) Institute for Advanced Studies in Basic Sciences, Iran

Received October 08, 2007, in final form January 23, 2008; Published online February 04, 2008

Abstract
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.

Key words: Hamilton-Jacobi equation; quantum potential; Husimi function; extended phase space.

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