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Journal of Beijing Union University

Theory of Quantum Phase Space and Bose-Einstein Condensate

Ping Wu

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The quantum phase-space representation established by Torres-Vega and Frederick is discussed.It is described that the wave-functions are not unique in phase space.The rigorous solutions of nonlinear Schrdinger equation,which models the Bose-Einstein condensate,are solved within the framework of the quantum phase-space representation.The eigenfunctions in position and momentum spaces can be available through the Fourier-like projection transformation from the phase-space wave-functions.

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Journal of Guangxi Normal University•Jan 1, 2007•0 citations
The quantum phase-space representation established by Torres-Vega and Frederick is discussed.It is described that the wave-functions are not unique in phase space.The rigorous solutions of nonlinear Schrdinger equation,which models the Bose-Einstein condensate,are solved within the framework of the quantum phase-space representation.The eigenfunctions in position and momentum spaces can be available through the Fourier-like projection transformation from the phase-space wave-functions.
The stationary-state nonlinear Schrödinger equation, which models the dilute-gas Bose–Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the `Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of repulsive nonlinear Schrödinger equation are solved, which models the dilute-gas Bose-Einstein condensate. The eigenfunctions in position and momentum spaces can be obtained through the “Fourier-like” projection transformation from the phase-space eigenfunctions. It shows that the wave-mechanics method in the phase-space representation could be extended to the nonlinear Schrödinger equations. The research provides the foundation for the approximate calculation in future.
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.