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arXiv:quant-ph/0501163 · Physical Review A

Wigner function and Schrödinger equation in phase-space representation

Dariusz Chruściński, Krzysztof Młodawski

Published May 24, 2005• 30 citations
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Abstract

We discuss a family of quasidistributions ($s$-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr\"odinger equation. It turns out that although Wigner functions satisfy the Schr\"odinger equation in phase space, they have a completely different interpretation.

DOI
10.1103/physreva.71.052104

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We discuss a family of quasidistributions (sss-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr"odinger equation. It turns out that although Wigner functions satisfy the Schr"odinger equation in phase space, they have a completely different interpretation.
This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.
In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU(2) that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO(3) case this entails a three-dimensional ``meta-phase'' space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.