Quantum Potential and Symmetries in Extended Phase Space

Abstract

Abstract. The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique. This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space. It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation. The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one. The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.

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

“…Removing the quantum potential in the Husimi representation fixes the value of parameter involved and gives the well known Qfunction. In addition to the present work, in a previous paper it was shown that the quantum potential was removed for the constant, the linear and the harmonic potentials in the Wigner representation [9]. Now we are planning to look for the representations where the quantum potential would be removed in the general case.…”

“…It was shown that in the Wigner representation the quantum potential was removed from the corresponding modified Hamilton-Jacobi equation for the linear and harmonic potentials [9]. Here, we look for another possible representation in which the quantum potential could be removed.…”

“…However, unlike the q space, it does not have a simple form for a general potential V (q) in p space (see details in [5] and [9]). As an example we consider the harmonic potential and obtain the modified Hamilton-Jacobi equation and quantum potential term.…”

“…The connection of the quantum potential with quantum fluctuations and quantum geometry in terms of Weyl's curvature has been studied by F. Shojai and A. Shojai [7] and Carroll [8]. Nevertheless, unlike the external potential, the quantum potential is not a pre-assigned function of the system coordinates and can only be derived from the wave function of the system [5] or from the corresponding quantum distribution functions used to calculate the average values of the observables [9]. This representation dependent property of the quantum potential allows one to find appropriate representations where the quantum potential could be removed from the modified HamiltonJacobi equation.…”

“…Carroll [10] has shown that there are generalized quantum theories for which the quantum potential depends on the wave function. Using the extended phase space formulation of quantum mechanics [11,12,13], Nasiri [9] has shown that in the Wigner representation of phase space quantum mechanics [14] the quantum potential is removed from the dynamical equation of a particle in linear and harmonic potentials keeping the Hamilton-Jacobi equation form invariant. It seems that the Husimi representation could be another candidate to release the quantum potential.…”