clean mesoscopic quantum devices[@beenakker], where a high-mobility two-dimensional electron gas is trapped within a boundary of controlled shape. We attempt to begin to bring the two fields together by asking how this model system behaves from the classical dynamical point of view and what it’s quantum signatures are. We predict ranges of fields and frequencies where some novel effects may be experimentally observable. In this paper, we consider the single-electron case and leave for a future publication the many electron problem.

This paper is organized as follows: In section II we define the model with its classical and quantum-mechanical properties, elucidate the important parameters in the problem and describe the general method of solution. In section III, we investigate the properties of the classical model. Based on a combination of analytic and numerical analysis, we obtain a ‘phase diagram’ in the parameter space of the system, which separates the quasi-integrable from the chaotic regions. This phase diagram is shown in Fig.1. The vertical axis is the ratio $\epsilon=B_{ac}/B_{dc}$ of the magnitudes of the fields, and the horizontal axis is the Larmor frequency normalized to the [*a.c.*]{} drive frequency, ${\tilde\omega_c}=\omega_c/\omega_0$. This phase diagram is of paramount importance in making the connection 