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Physica Scripta

On the quantum mechanical representation in phase space

Qian‐Shu Li, Xu‐Guang Hu

Published Apr 1, 1995• 21 citations
Publisher

Abstract

The theory of the quantum mechanical representation in phase space recently proposed by Torres-Vega and Frederick is rigorously verified to be extended to the bound systems whose potential energy functions may contain terms of negative powers of coordinate and the quantum Liouville equation in the theory is also generalized to the same systems. It is suggested that the kernels of projection transformation take the forms e i pq/ 2 ħ and e -(i pq/ 2 ħ ) instead of {1/√(4π ħ )}e i pq/ 2 ħ and {1/√(4π ħ )}e -(i pq/ 2 ħ ) in order to keep the consistency for projections of the wave functions in all quantum problems. The validity of quantum average values and the virial theorem in the phase space representation are discussed in detail.

DOI
10.1088/0031-8949/51/4/001

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The theory of the quantum mechanical representation in phase space recently proposed by Torres-Vega and Frederick is rigorously verified to be extended to the bound systems whose potential energy functions may contain terms of negative powers of coordinate and the quantum Liouville equation in the theory is also generalized to the same systems. It is suggested that the kernels of projection transformation take the forms e i pq/ 2 ħ and e -(i pq/ 2 ħ ) instead of {1/√(4π ħ )}e i pq/ 2 ħ and {1/√(4π ħ )}e -(i pq/ 2 ħ ) in order to keep the consistency for projections of the wave functions in all quantum problems. The validity of quantum average values and the virial theorem in the phase space representation are discussed in detail.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.
In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.