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Phase space quantum mechanics of a free particle moving in a plane
arXiv:2504.20208v5 Announce Type: replace Abstract: The eigenstates of a free quantum particle propagating in a plane are derived in the context of phase space quantum mechanics. Two possibilities are analysed. First, the case of a particle with fixed energy and angular momentum is considered. A special choice of coordinates on the four-dimensional phase space, suitable for representing the eigenstates of the particle under consideration, is presented. A phase-space counterpart of the product of operators, known as the Moyal product, is derived in these coordinates. The eigenvalue equations are solved, and their physically acceptable solutions, called Wigner functions, are identified. Second, a particle with fixed components of the Cartesian momentum is considered. A relationship is found between the Wigner function of the particle with fixed Cartesian momentum components and some functions characterising the particle with fixed energy and angular momentum.
Abstract:The eigenstates of a free quantum particle propagating in a plane are derived in the context of phase space quantum mechanics. Two possibilities are analysed. First, the case of a particle with fixed energy and angular momentum is considered. A special choice of coordinates on the four-dimensional phase space, suitable for representing the eigenstates of the particle under consideration, is presented. A phase-space counterpart of the product of operators, known as the Moyal product, is derived in these coordinates. The eigenvalue equations are solved, and their physically acceptable solutions, called Wigner functions, are identified. Second, a particle with fixed components of the Cartesian momentum is considered. A relationship is found between the Wigner function of the particle with fixed Cartesian momentum components and some functions characterising the particle with fixed energy and angular momentum.
arXiv:2504.20208 [quant-ph]
(or arXiv:2504.20208v5 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2504.20208 Focus to learn more arXiv-issued DOI via DataCite
https://doi.org/10.1038/s41598-026-67321-3 Focus to learn more DOI(s) linking to related resources
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