Expansion method for stationary states of quantum billiards

A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensiona l infinite potential well with arbitrary shape (quantum billiard) is presen ted. The method permits the study of quantum billiards in an introductory q uantum mechanics course. According to the method, wave functions inside the : billiard are expressed in terms of an expansion of a complete set of orth onormal functions defined in a surrounding rectangle for which the Dirichle t boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementati ons of the method are described and applied to determine the energies and w ave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of ch aos in a classically chaotic generic-triangle billiard. (C) 1999 American A ssociation of Physics Teachers.