Nonperturbative Solution for Bloch Electrons in Constant Magnetic Fields

A general theoretical approach for the nonperturbative Bloch solution of Schrödinger’s equation in the presence of a constant magnetic field is presented. Using a singular gauge transformation based on a lattice of magnetic flux lines, an equivalent quantum system with a periodic vector potential is obtained. For rational magnetic fields this system forms a magnetic superlattice for which Bloch’s theorem then applies. Extensions of the approach to particles with spin and many-body systems and connections to the theory of magnetic translation groups are discussed.