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 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 J. We investigate features of 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 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 ℏ or vanishing anharmonicity, does not pointwise converge to classical dynamics.