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Applied Mechanics and Materials

One-Dimensional Harmonic Oscillator in Quantum Phase Space

Jun Lu, Xue Mei Wang, Ping Wu

Published Oct 24, 2011• 0 citations
Publisher

Abstract

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

DOI
10.4028/www.scientific.net/amm.110-116.3750

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Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.
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.
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.