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

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 ̄ h or
vanishing anharmonicity, does not pointwise converge to classical
dynamics.