Wave packet construction in three-dimensional quantum billiards: Visualizing the closed orbit, collapse and revival of wave packets in the cubical billiard

摘要

We examine the dynamical evolution of wave packets in a cubical billiardwhere three quantum numbers ($n_x,n_y,n_z$) determine its energy spectrum andconsequently its dynamical behavior. We have constructed the wave packet in thecubical billiard and have observed its time evolution for various closedorbits. The closed orbits are possible for certain specific values of quantumnumbers ($n_x,n_y,n_z$) and initial momenta ($k_x,k_y,k_z$). We observe that acubical billiard exhibits degenerate energy levels and the path lengths of theclosed orbits for these degenerate energy levels are identical. In spite of theidentical path lengths, the shapes of the closed orbits for degenerate levelsare different and depend upon angles $\theta$ and $\phi$ which we term as thesweep and the elevation angle respectively. These degenerate levels owe theirorigin to the symmetries prevailing in the cubical billiard and degeneratelevels disappear completely or partially for a parallelepiped billiard as thesymmetry breaks due to commensurate or incommensurate ratio of sides.