Abstract

We study the quantum mechanics of a two-dimensional version of the Fermi accelerator. The model consists of a free particle constrained to move inside a disk with radius varying periodically in time. A complete quantum mechanical solution of the problem is possible for a specific choice of the time-periodic oscillation radius. The quasi-energy spectral (QES) properties of the model are obtained from direct evaluation of finite-dimensional approximations to the time evolution operator. As the effective ħ is changed from large to small the statistics of the QES eigenvalues change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the QES eigenfunctions is studied using the X2 test with v degrees of freedom. The Porter-Thomas distribution is shown to apply in the COE regime, while the Poisson regime does not fit the X2 with v = 0. We find that the Poisson regime is associated with exponentially localized QES eigenfunctions whereas they are extended in the COE regime.

Introduction

Although a complete understanding of the quantum manifestations of classical chaos (QMCC) is not yet in sight, significant progress has been made in obtaining partial answers to this question. This progress has primarily been achieved from studies of lower-dimensional models: two-dimensional for energy conserving models and one-dimensional when energy is not conserved. In particular, very few studies have been carried out in two-dimensional time-dependent problems, for even in the classical limit the theoretical analysis is nontrivial.