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There are no phase-space trajectories for anharmonic quantum systems, butWigner's phase-space representation of quantum mechanics features Wignercurrent~$\bf J$. This current reveals fine details of quantum dynamics -- finerthan is ordinarily thought accessible according to quantum folklore invokingHeisenberg's uncertainty principle. Here, we focus on the simplest, mostintuitive, and analytically accessible aspects of~$\bf J$. We investigatefeatures of~$\bf J$ for bound states of time-reversible, weakly-anharmonicone-dimensional quantum-mechanical systems which are weakly-excited. Weestablish that weakly-anharmonic potentials can be grouped into three distinctclasses: hard, soft, and odd potentials. We stress connections between eachother and the harmonic case. We show that their Wigner current fieldlinepatterns can be characterised by~$\bf J$'s discrete stagnation points, howthese arise and how a quantum system's dynamics is constrained by thestagnation points' topological charge conservation. We additionally show thatquantum dynamics in phase space, in the case of vanishing Planck constant$\hbar$ or vanishing anharmonicity, does not pointwise converge to classicaldynamics.

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