Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane

TL;DR

This paper explores various approaches to the Landau problem, including canonical transformations and linear potentials, providing algebraic solutions and extending the analysis to half-plane geometries relevant for quantum Hall effects.

Contribution

It introduces a complete algebraic solution to the Landau problem with a linear potential and extends it to half-plane geometries, linking to quantum Hall physics.

Findings

Canonical transformations relate to gauge choices.

Algebraic solution for linear potential case.

Extension to half-plane geometry for quantum Hall relevance.

Abstract

The ordinary Landau problem of a charged particle in a plane subjected to a perpendicular homogeneous and static magnetic field is reconsidered from different points of view. The role of phase space canonical transformations and their relation to a choice of gauge in the solution of the problem is addressed. The Landau problem is then extended to different contexts, in particular the singular situation of a purely linear potential term being added as an interaction, for which a complete purely algebraic solution is presented. This solution is then exploited to solve this same singular Landau problem in the half-plane, with as motivation the potential relevance of such a geometry for quantum Hall measurements in the presence of an electric field or a gravitational quantum well.