Date of Submission

Spring 2011

Academic Program

Physics; Mathematics


Christian Bracher, James Belk

Abstract/Artist's Statement

We studied the classical and quantum dynamics of electrons emitted from a point source into a two-dimensional layer subject to perpendicular homogeneous electric and magnetic fields. We solved the classical problem and found that individual trajectories feature a simple combination of drift and circular motion. However, a more complicated picture arises if one examines whole fields of trajectories emitted from a range of angles. Trajectories trace out a set of caustics that confine the classical motion as well as a series of focal points, images of the source. In general, electrons travel on several different paths from the source to a given destination. We use numerical methods to establish these paths.

In the quantum picture, the electrons propagate as waves. We construct an approximate quantum solution by means of the semiclassical method, and assign a wave to each classical trajectory. Interference of the individual waves modifies the classical distribution of the electrons and their associated electric current. We developed software that computes the semiclassical wavefunction and generates images of the current density for different strengths of the electric and magnetic fields. Using a related method, we estimate the total current emitted by the source.

A quantum mechanical calculation asserts that the total current is exponentially suppressed for an extended set of parameters. This surprising prediction is confirmed by the semiclassical study; our simulations show almost complete destructive interference of the electron waves everywhere in the plane of motion. This result may be relevant for the unusual behavior of charges observed in the Quantum Hall Effect.

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