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 $T_{rev} = 9 mu a^2/4hbar pi$ 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.
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机译：利用等边三角形无限大阱（或台球）的能量本征态为闭合形式这一事实，我们使用周期轨道理论和短周期理论，研究了能量本征值谱与该几何中经典闭合路径之间的联系。波浪包的术语半经典行为。我们还讨论了波包的复兴，并表明对于所有波包，在$ T_ {rev} = 9 mu a ^ 2 / 4hbar pi $给出的时间确实存在复兴，其中$ a $和$ mu $是点粒子的一侧和质量。对于最初位于台球内部特殊对称点处的零动量波包，我们发现了具有更短复兴时间的精确复兴的其他情况。最后，我们讨论等边（$ 60 ^ {circ} -60 ^ {circ} -60 ^ {circ} $）三角形的简单变化，例如半等边（$ 30 ^ {circ} -60 ^ {circ} -90 ^ {circ} $）三角形和其他“褶皱”，它们具有相关的能谱和复兴结构。
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