Wigner's representation of quantum mechanics in integral form and its applications

Abstract

We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\mathbf{J}$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their potential fulfill Liouville's theorem of volume preservation in quantum mechanics. Contrary to a recent suggestion, our proof shows that the non-Liouvillian character of quantum phase-space dynamics cannot be transformed away.

arXiv comment4 pages, 1 figures

References in corpus (2)

• Sub-Planck spots of Schroedinger cats and quantum decoherence
• Wigner flow reveals topological order in quantum phase space dynamics

Cited by in corpus (13)

• Wigner`s quantum phase-space current in weakly-anharmonic weakly-excited two-state systems
• Anharmonic quantum mechanical systems do not feature phase space trajectories
• Non-classicality from the phase-space flow analysis of the Weyl-Wigner quantum mechanics
• Testing non-classicality with exact Wigner currents for an anharmonic quantum system
• Quantum Kerr oscillators' evolution in phase space: Wigner current, symmetries, shear suppression and special states
• Dynamic shear suppression in quantum phase space
• Phase-space quantum profile of Pöschl-Teller two-level systems
• Classical and statistical limits of the quantum singular oscillator
• Quasiprobability currents on the sphere
• State-space distribution and dynamical flow for closed and open quantum systems
• Phase-space elementary information content of confined Dirac spinors
• Equivalence of a harmonic oscillator to a free particle and Eisenhart lift
• Isospectral mapping for quantum systems with energy point spectra to polynomial quantum harmonic oscillators