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Journal of Physics A Mathematical and General

Spacetime structures in simple quantum systems

Frank Großmann, Jan Michael Rost, Wolfgang P. Schleich

Published May 7, 1997• 73 citations
Publisher

Abstract

Recently W Kinzel [1995 Phys. Bl. 51 1190] has argued that even simple quantum systems can exhibit surprising phenomena. As an example he presented the formation of canals and ridges in the time-dependent probability density of a particle caught in a square well with infinitely high walls. We show how these structures emerge from the wavefunction and present a simple derivation of their location in the spacetime continuum.

DOI
10.1088/0305-4470/30/9/004

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Recently W Kinzel [1995 Phys. Bl. 51 1190] has argued that even simple quantum systems can exhibit surprising phenomena. As an example he presented the formation of canals and ridges in the time-dependent probability density of a particle caught in a square well with infinitely high walls. We show how these structures emerge from the wavefunction and present a simple derivation of their location in the spacetime continuum.
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.
Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.