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arXiv:quant-ph/9804015

The Particle in the box: Intermode traces in the propagator

Irene Marzoli, Iwo Białynicki‐Birula, O. M. Friesch, A. E. Kaplan, Schleich, and W. P.

Published Apr 6, 1998• 6 citations
arXivHTMLPDF

Abstract

Characteristic structures such as canals and ridges --intermode traces-- emerge in the spacetime representation of the probability distribution of a particle in a one-dimensional box. We show that the corresponding propagator already contains these structures. We relate their visibility to the factorization property of the initial wave packet.

DOI
10.48550/arxiv.quant-ph/9804015

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Characteristic structures such as canals and ridges --intermode traces-- emerge in the spacetime representation of the probability distribution of a particle in a one-dimensional box. We show that the corresponding propagator already contains these structures. We relate their visibility to the factorization property of the initial wave packet.
Generic channel and ridge structures are known to appear in the time-dependent position probability distribution of a one-dimensional quantum particle confined to a box. These structures are shown to have a detailed quantitative explanation in terms of a travelling-wave decomposition of the probability density, wherein each contributing term corresponds simultaneously to (i) a real wave propagating at a quantized velocity and (ii) to the time-averaged structure of the position distribution along a quantized direction in spacetime. The approach leads to new predictions of channel locations, widths and depths, and is able to provide more structural details than earlier approaches based on partial interference and Wigner functions. Results are also applicable to light diffracted by a periodic grating, and to the quantum rigid rotator.
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.
The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.