 Solution

For a Hamiltonian with a lattice-periodic potential $V(\mathbf{r}+\mathbf{R})=V(\mathbf{r})$, its simultaneous energy eigenstates and lattice-translation eigenstates can be chosen in the Bloch wave formEquivalently, ${\psi }_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{k}\cdot \mathbf{R}}{\psi }_{n\mathbf{k}}(\mathbf{r})$. The index $n$ labels an energy band; crystal momentum $\hslash \mathbf{k}$ is defined modulo a reciprocal lattice vector. These are the content of Bloch's theorem.