Angular momentum in the fractional quantum Hall effect

Abstract

Suppose a classical electron is confined to move in the xy plane under the influence of a constant magnetic field in the positive z direction. It then traverses a circular orbit with a fixed positive angular momentum Lz with respect to the center of its orbit. It is an underappreciated fact that the quantum wave functions of electrons in the ground state (the so-called lowest Landau level) have an azimuthal dependence ∝ exp (−imϕ) with m≥0, seemingly in contradiction with the classical electron having positive angular momentum. We show here that the gauge-independent meaning of that quantum number m is not angular momentum, but that it quantizes the distance of the center of the electron's orbit from the origin, and that the physical angular momentum of the electron is positive and independent of m in the lowest Landau levels. We note that some textbooks and some of the original literature on the fractional quantum Hall effect do find wave functions that have the seemingly correct azimuthal form ∝ exp (+imϕ) but only on account of changing a sign (e.g., by confusing different conventions) somewhere on the way to that result.