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Chaos Solitons & Fractals

Quantum infinite square well with an oscillating wall

M. L. Glasser, J. Mateo, J. Negro, Luis Miguel Nieto

Published Oct 17, 2008• 32 citations
Publisher
DOI
10.1016/j.chaos.2008.07.055

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The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.
We revisit the analysis of sharp infinite potentials within the path integral formalism using the image method [1]. We show that the use of a complete set of energy eigenstates that satisfy the boundary conditions of an infinite wall precisely generates the propagator proposed in Ref. [1]. We then show the validity of the image method by using supersymmetric quantum mechanics to relate a potential without a sharp boundary to the infinite square well and derive its propagator with an infinite number of image charges. Finally, we show that the image method readily generates the propagator for the half-harmonic oscillator, a potential that has a sharp infinite boundary at the origin and a quadratic potential in the allowed region, and leads to the well known eigenvalues and eigenfunctions.