VISUALIZING CLASSICAL PERIODIC-ORBITS FROM THE QUANTUM ENERGY-SPECTRUM VIA THE FOURIER-TRANSFORM - SIMPLE INFINITE WELL EXAMPLES

The Fourier transform of the density of quantized energy levels for a quantum mechanical particle in a two-dimensional (2-D) infinite well ( or billiard geometry) is known to exhibit delta-function-like spikes a t distance values (L) corresponding to the lengths of periodic orbits or closed trajectories. We show how these Fourier transforms can be ra ther easily calculated numerically for simple infinite well geometries including the square and rectangular well in 2 D, the cubical well in three dimensions, as well as the circular infinite well (and variatio ns) in two dimensions. Such calculations provide a novel, well-motivat ed, and relatively straightforward example of numerical Fourier transf orm techniques and make interesting connections between quantum energy levels and classical trajectories in a way which is seldom stressed i n the undergraduate curriculum. (C) 1997 American Association of Physi cs Teachers.