Wigner functions for the diamagnetic hydrogen atom: quantum fringes

Phase-space distributions (Wigner functions) for eigenstates of the diamagnetic hydrogen atom at fixed scaled energy have been obtained for the intermediate regime in between regularity and full chaos. States localized on the three major fixed points of the classical Poincare surface of section have been considered. While previous numerical work was primarily concerned with phase-space localization on period orbits, here the oscillatory structure (quantum fringes) forms the main subject. The fringes are delocalized relative to the dominant positive structures. Although the latter are strongly localized on fixed points and their associated invariant manifolds, the fringes permeate the whole energy surface. Fringes originating in the ergodic part of phase-space experience an abrupt change in character, in addition to the expected fall in amplitude, within stable islands. As a result, a single quantum state can provide a striking map of the global structure of phase-space. Phase-space is partitioned into relatively strong irregular scar fringes in the ergodic region and weaker regular Airy type fringes inside stable islands. Quantum probability distributions near bifurcations and confluences have also been examined. A comparison with Husimi distributions has been carried out and it is shown that the latter have lost some dynamically interesting structures.