Band structure and quantum Poincaré sections of a classically chaotic quantum rippled channel

We obtain the energy band spectra, eigenfunctions, and quantum Poincaré sections of a free particle moving in a two-dimensional channel bounded by a periodically varying (ripple) wall and a flat wall. Classical Poincaré sections show a generic transition from regular to chaotic motion as the size of the ripple is increased. The energy band structure is obtained for two representative geometries corresponding to a wide and a narrow channel. The comparison of numerical results with the level-splitting predictions of low-order quantum degenerate perturbation theory elucidate some aspects of the classical-quantum correspondence. For larger ripple amplitudes the conduction bands for narrow channels become flat and nearly equidistant at low energies. Quantum-classical correspondence is discussed with the aid of quantum Poincaré (Husimi) plots.