  Lowest Landau level wavefunctions are eigenstates of the Hamiltonian of a
charged particle in two dimensions under a uniform magnetic field. They are
known to be holomorphic both in real and momentum spaces, and also exhibit
uniform, translationally invariant, geometrical properties in momentum space.
In this paper, using the Stone-von Neumann theorem, we show that lowest Landau
level wavefunctions are indeed the only possible states with unit Chern number
satisfying these conditions. We also prove the uniqueness of their direct
analogs with higher Chern numbers and provide their expressions.
