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

TL;DR

This paper demonstrates that in the Husimi representation, the quantum potential for a harmonic oscillator vanishes, simplifying the quantum dynamics to a classical Hamilton-Jacobi form at a specific transformation parameter.

Contribution

It introduces the Husimi representation as an alternative to Wigner and Schrödinger representations where the quantum potential is eliminated for the harmonic oscillator.

Findings

Quantum potential disappears in Husimi representation for harmonic oscillator.

The Hamilton-Jacobi equation reduces to classical form in this representation.

Specific parameter choice corresponds to the Q-function in Husimi transformation.

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 Schr\"odinger 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.