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American Journal of Physics

Exploring the propagator of a particle in a box

S. A. Fulling, K. S. Güntürk

Published Dec 20, 2002• 14 citations
Publisher

Abstract

The propagator of a particle in a one-dimensional box is quite different from the propagator for infinite space. It has a complicated but regular structure of everywhere dense singularities. We investigate the case of periodic boundary conditions both numerically and theoretically, leaving the slightly more complicated case of reflecting boundary conditions to the suggested problems. The singularity structure is shown to match up with the fractional revivals observed by many authors in wave packet studies.

DOI
10.1119/1.1509415

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The propagator of a particle in a one-dimensional box is quite different from the propagator for infinite space. It has a complicated but regular structure of everywhere dense singularities. We investigate the case of periodic boundary conditions both numerically and theoretically, leaving the slightly more complicated case of reflecting boundary conditions to the suggested problems. The singularity structure is shown to match up with the fractional revivals observed by many authors in wave packet studies.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.
Fractional revivals of wave packets in an infinite square well are scrutinized with a viewpoint rooted in Fourier analysis, and a compact relation, expressing the wavefunction Ψ( x , t ) for certain values of t in terms of spatially displaced copies of Ψ( x ,0), is derived without appealing to a classical analogy ; conditions for the appearance of an interference pattern in a plot of the position probability density of a wave packet are deduced, along with related results obtained previously from different considerations, with minimal effort. The specific case of a packet of Gaussian shape is analysed in greater detail to provide concrete illustrations of fractional revivals with or without interference between overlapping copies of wave packets, depending on whether a finite value of the so-called classical time can or cannot be assigned to the wave packet.