
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
HamiltonJacobi 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
HamiltonJacobi equation reduces to the familiar classical form.
This happens when the parameter in the Husimi transformation
assumes a specific value corresponding to Qfunction.
Key words:
HamiltonJacobi equation; quantum potential; Husimi function; extended phase space.
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