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

TL;DR

This paper analyzes the Wigner phase-space current in weakly-anharmonic, weakly-excited two-state quantum systems, revealing detailed quantum dynamics and classifying potential types based on current patterns and stagnation points.

Contribution

It introduces a classification of weakly-anharmonic potentials into three classes and characterizes their Wigner current patterns and topological features.

Findings

Wigner current patterns differ across potential classes

Stagnation points' topology constrains quantum dynamics

Quantum phase-space dynamics do not converge pointwise to classical in certain limits

Abstract

There are no phase-space trajectories for anharmonic quantum systems, but Wigner's phase-space representation of quantum mechanics features Wigner current~$\bf J$. 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~$\bf J$. We investigate features of~$\bf J$ 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~$\bf J$'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 $\hbar$ or vanishing anharmonicity, does not pointwise converge to classical dynamics.