# Angular momentum in the fractional quantum Hall effect

## TL;DR

This paper clarifies the relationship between quantum wave functions and classical angular momentum in the lowest Landau level of the fractional quantum Hall effect, resolving apparent contradictions in azimuthal dependence and physical angular momentum.

## Contribution

It demonstrates that the quantum number m in wave functions quantifies orbit center position, not angular momentum, clarifying misconceptions in the literature.

## Key findings

- Quantum number m measures orbit center distance, not angular momentum.
- Physical angular momentum remains positive and independent of m.
- Resolves confusion caused by different wave function conventions.

## 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 $L_z$ 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 $\propto \exp(-im\phi) $ with $m\geq 0$, 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 $\propto\exp(+im\phi)$ but only on account of changing a sign (e.g., by confusing different conventions) somewhere on the way to that result.