Projection operator proof of Bloch's theorem

Abstract

Bloch's theorem has been a cornerstone of solid-state phenomena. It is a property of the regular arrangement of atoms in solids. It is a cornerstone in the band theory of solids. However, in traditional proofs of Bloch's theorem this explicit relation to the translational symmetry group is not very directly apparent. We present here a proof of Bloch's theorem which is totally based on group theoretical considerations, using projection operators to irreducible representations of the lattice translation group, so that the connection between Bloch's theorem and lattice translation symmetry is totally apparent.