Quantum approximation to regular and chaotic classical motion: An electron in two periodic potentials.

Abstract

An electron in a periodic crystal potential, which is subjected to an additional periodic pulsed potential, is an example of a system which can be modeled by the kicked Harper model. The relative simplicity of this model facilitates a comparison of its quantum and classical dynamics. The model has a clearly defined semiclassical limit in which wave packets follow the classical motion. A power series in \ensuremath{\Elzxh} yields systematic quantum corrections to the semiclassical limit. Numerical iterations of the quantum equations agree with the \ensuremath{\Elzxh} expansion (for a limited time), provided the corresponding classical motion is not chaotic. For chaotic motion, the agreement between classical and quantum motion disappears very quickly. The quantum solutions also show (i) precursors to classical chaos, (ii) a time-dependent analog of wave-function scarring, and (iii) quantum echoes arising from quantized phase-space orbits.