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

TL;DR

This paper studies the evolution of wave packets in a three-dimensional cubical quantum billiard, analyzing how degeneracies and symmetries influence closed orbit paths, collapse, and revival phenomena.

Contribution

It constructs and visualizes wave packet dynamics in a cubical billiard, highlighting the effects of symmetry and degeneracy on closed orbits and their shapes.

Findings

Degenerate energy levels have identical path lengths for closed orbits.

Shapes of closed orbits vary with angles despite identical path lengths.

Symmetry influences degeneracy and orbit characteristics in the billiard.

Abstract

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