We prove that a spin-½ charged particle moving in a plane under the influence of a perpendicular magnetic field has ($N-1\mathrm{}$) zero-energy states, where $N$ is the closest integer to the total flux in units of the flux quantum. The ($N-1\mathrm{}$) independent wave functions are calculated explicitly. The result, which is extremely simple to prove, is an example of the Atiyah-Singer index theorem when applied to the Euclidean two-dimensional Dirac equation.

• Received 19 June 1978

DOI:https://doi.org/10.1103/PhysRevA.19.2461