Chaos in the Stadium Quantum Billiards

An expansion method was used to write a MATHEMATICA program to compute the energy levels and eigenfunctions of a 2-D quantum billiard system with arbitrary shape and dirichlet boundary conditions. One integrable system, the full circle, and one non-integrable system, the stadium, were examined. Chaotic properties were sought in nearest-neighbor energy level spacing distributions (NND). It was observed that the classically non-chaotic Poisson function seemed to fit the circle’s NND better, while the classically chaotic Gaussian Orthogonal Ensemble function the stadium better. A detailed explanation of the theory and algorithm are provided, although a more rigorous energy-level analysis is desireable.