Solution
= Solution
<Bloch theorem> follows because lattice translations commute with $H$ and with each other: simultaneous eigenstates obey $T_R\psi=e^{ik\cdot R}\psi$, so $\psi_k(r)=e^{ik\cdot r}u_k(r)$ with $u_k$ lattice-periodic. Conversely this form has the required translation character.