Particles in Periodic Potentials

Non-interacting electrons in a periodic potential are an important example of a physical system with a space group symmetry. Due to the lack of interactions among the electrons, one only needs to solve the eigenvalue problem of a single electron in that potential. The energy levels are subsequently filled following the Pauli exclusion principle, and ignoring the spin-orbit interaction also allows us to disregard the electron’s spin. Sect. 14.1 focuses on the Schrödinger equation of the system and derives the well-known Bloch theorem with group-theoretical arguments. We explain in Sect. 14.2 why determining the eigenstates in a certain part of the Brillouin zone is sufficient, since all other eigenstates can be derived from these.