Electron wave-functions in a magnetic field

TL;DR

This paper critically examines the standard Landau quantization for an electron in a magnetic field, revealing gauge symmetry issues and deriving a more consistent wave function with only one quantum number.

Contribution

It introduces a gauge-symmetry respecting wave function formulation, correcting Landau's approach and clarifying the physical quantum numbers involved.

Findings

Landau's wave functions break translation gauge symmetry

A new physical wave function with a single quantum number is derived

Landau's wave functions are a limiting case of the new solutions

Abstract

The problem of a single electron in a magnetic field is revisited from first principles. It is shown that the standard quantization, used by Landau, is inconsistent for this problem, whence Landau's wave functions spontaneously break the gauge symmetry of translations in the plane. Because of this Landau's (and Fock's) wave functions have a spurious second quantum number. The one-body wave function of the physical orbit, with only one quantum number, is derived, and expressed as a superposition of Landau's wave functions. Conversely, it is shown that Landau's wave functions are a limiting case of physical solutions of a different problem, where two quantum numbers naturally appear. When the translation gauge symmetry is respected, the degeneracy related to the choice of orbit center does not appear in the one-body problem.