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Journal of Jinggangshan University

Perturbation theory in quantum phase space based on harmonic oscillator system's

Qian‐Shu Li

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.

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The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.
Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.
The perturbation quantum system was discussed within the framework of quantum phase space representation formulated by Torres-Vega and Frederick(TF) and studied the nondegenerate perturbation theory.The energy and the wave functions of the system in quantum phase space representation has been obtained;the one dimensional anharmonic oscillator system was used for an example to analyses the behaviour of quantum system in quantum phase space representation.The actual physical system in quantum phase space representation by using the approximate methods has been discussed.
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.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick,the rigorous solutions of stationary Schro¨dinger equation for one-dimensional harmonic oscillator are solved with the methods of matrix mechanics and wave mechanics respectively.It is showed that the matrix mechanics and the wave mechanics are equivalent in phase space.The eigenfunctions in position and momentum spaces can be obtained through the Fourier-like projection transformation from the phase-space eigenfunction.