From localized to extended states in a time-dependent quantum model

Abstract

The problem of a particle inside a rigid box with one of the walls oscillating periodically in time is studied quantum mechanically. In the classical limit, this model was introduced by Fermi in the context of cosmic ray physics. The classical solutions can go from being quasiperiodic to chaotic, as a function of the amplitude of the wall oscillation. In the quantum case, we calculate the spectral properties of the corresponding evolution operator, i.e.: the quasi-energy eigenvalues and eigenvectors. The specific form of the wall oscillation, e.g. $\ell (t) = \sqrt {1 + 2\delta \left| t \right|}$, with |t| ≤ 1/2, and ℓ(t + 1) = ℓ(t) , is essential to the solutions presented here. It is found that as ℏ increases with δfixed, the nearest neighbor separation between quasi-energy eigenvalues changes from showing no energy level repulsion to energy level repulsion. This transition, from Poisson-like statistics to Gaussian-Orthogonal-Ensemble-like statistics is tested by looking at the distribution of quasi-energy level nearest neighboor separations and the Δ₃(L) statistics. These results are also correlated to a transition between localized to extended states in energy space. The possible relevance of the results presented here to experiments in quasi-one-dimensional atoms is also discussed.

Talk presented at the Second International Conference on Quantum Chaos held in Cuernavaca, Mor., Mex. January 1986