Ground state of a spin-½ charged particle in a two-dimensional magnetic field

We prove that a spin-\textonehalf{} charged particle moving in a plane under the influence of a perpendicular magnetic field has ($N\ensuremath{-}1$) zero-energy states, where $N$ is the closest integer to the total flux in units of the flux quantum. The ($N\ensuremath{-}1$) 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.