Quantum wave-packet revivals in circular billiards

Abstract

We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely $m=0\mathrm{}$ states (central wave packets with no momentum) the revival time is $T_{\mathrm{rev}}^{\mathrm{}(m=0\mathrm{})\mathrm{}}=8\mathrm{}\mu R^2/ħ\pi ,$ where $\mu$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that $T_{\mathrm{rev}}^{\left(m\ne 0\right)}=({\pi }^2{/2)T}_{\mathrm{rev}}^{\mathrm{}(m=0\mathrm{})\mathrm{}}\approx {5T}_{\mathrm{rev}}^{\mathrm{}(m=0\mathrm{})\mathrm{}}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ne 0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.