Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals

Abstract

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by $T_{rev}=9\mu a^2/4\hslash \pi$ where $a$ and $\mu$ are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral ($60^{\circ }-60^{\circ }-60^{\circ }$) triangle, such as the half equilateral ($30^{\circ }-60^{\circ }-90^{\circ }$) triangle and other `foldings', which have related energy spectra and revival structures.

34 pages, 9 embedded .eps figures

References in corpus (7)

• The Evolution and Revival Structure of Localized Quantum Wave Packets
• Orbital Magnetism in the Ballistic Regime: Geometrical Effects
• Visualizing the collapse and revival of wavepackets in the infinite square well using expectation values
• Long-Term Evolution and Revival Structure of Rydberg Wave Packets for Hydrogen and Alkali-Metal Atoms
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