TY - JOUR

T1 - Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform

T2 - Simple infinite well examples

AU - Robinett, Richard Wallace

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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.

AB - 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.

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U2 - 10.1119/1.18750

DO - 10.1119/1.18750

M3 - Article

AN - SCOPUS:0031488203

SN - 0002-9505

VL - 65

SP - 1167

EP - 1175

JO - American Journal of Physics

JF - American Journal of Physics

IS - 12

ER -