Phase-space formulation of quantum mechanics under phase-space transformations: Wigner functions of an oscillator system in crossed magnetic and electric fields

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Abstract
We give a rigorous proof for the following result that combines those discussed earlier by Krüger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.