 Bloch's theorem

Bloch's theorem describes energy eigenstates of a Quantum Hamiltonian invariant under Bravais lattice translations. They can be chosen in the formTranslation by a lattice vector $\mathbf{R}$ therefore multiplies the state by $e^{i\mathbf{k}\cdot \mathbf{R}}$. The wavevector $\mathbf{k}$ labels crystal momentum and is equivalent modulo a reciprocal lattice vector.

This converts a problem over an entire ideal crystal into a family of cell-periodic problems, organized by the Bloch Hamiltonian. Their energy eigenvalues form energy bands over a Brillouin zone. The periodic function carries the microscopic structure missing from a bare plane wave. Boundaries, defects and disorder break exact translation invariance, so the theorem's states serve as a basis or approximation there rather than automatic exact eigenstates.