HILBERT-SPACE FOR CHARGED-PARTICLES IN PERPENDICULAR MAGNETIC-FIELDS

We describe the quantum mechanics of two-dimensional charged particles in a perpendicular magnetic field in the planar Landau and spherical monopole configurations. These models, particularly in the work of Laughlin and Haldane, are crucial to the theoretical understanding of the quantum Hall effect. Here we present the full Hilbert space structure in each case, with special emphasis on the relationship between the two systems. The formulation in terms of stereographically projected complex coordinates makes the connection especially explicit and naturally generalizes to more complicated two-dimensional surfaces where the interaction of the particles with an external perpendicular magnetic field may be regarded as an interaction of the particles with the two-dimensional (Kähler) metric of the surface. This generalization is illustrated by the hyperbolic configuration of particles constrained to the upper sheet of a hyperboloid in the presence of a hyperbolic monopole. © 1992.