Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples

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Abstract

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 δ-function-like spikes at distance values (L) corresponding to the lengths of periodic orbits or closed trajectories. We show how these Fourier transforms can be rather 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 variations) in two dimensions. Such calculations provide a novel, well-motivated, and relatively straightforward example of numerical Fourier transform techniques and make interesting connections between quantum energy levels and classical trajectories in a way which is seldom stressed in the undergraduate curriculum.

Original languageEnglish (US)
Pages (from-to)1167-1175
Number of pages9
JournalAmerican Journal of Physics
Volume65
Issue number12
DOIs
StatePublished - Jan 1 1997

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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