Wigner`s quantum phase-space current in weakly-anharmonic weakly-excited two-state systems

Abstract

There are no phase-space trajectories for anharmonic quantum systems, but Wigner's phase-space representation of quantum mechanics features Wigner current~$\mathbf{J}$. This current reveals fine details of quantum dynamics -- finer than is ordinarily thought accessible according to quantum folklore invoking Heisenberg's uncertainty principle. Here, we focus on the simplest, most intuitive, and analytically accessible aspects of~$\mathbf{J}$. We investigate features of~$\mathbf{J}$ for bound states of time-reversible, weakly-anharmonic one-dimensional quantum-mechanical systems which are weakly-excited. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd potentials. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline patterns can be characterised by~$\mathbf{J}$'s discrete stagnation points, how these arise and how a quantum system's dynamics is constrained by the stagnation points' topological charge conservation. We additionally show that quantum dynamics in phase space, in the case of vanishing Planck constant $\hslash$ or vanishing anharmonicity, does not pointwise converge to classical dynamics.

11 pages, 10 figures

References in corpus (4)

• Wigner flow reveals topological order in quantum phase space dynamics
• Efficient method to generate time evolution of the Wigner function for open quantum systems
• Wigner's representation of quantum mechanics in integral form and its applications
• Shape invariance in phase space

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