= 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> form
$$
\boxed{\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),\qquad
u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r).}
$$
Equivalently, $\psi_{n\mathbf k}(\mathbf r+\mathbf R)=e^{i\mathbf k\cdot\mathbf R}\psi_{n\mathbf k}(\mathbf r)$. The index $n$ labels an <energy band>; <crystal momentum> $\hbar\mathbf k$ is defined modulo a <reciprocal lattice> vector. These are the content of <Bloch's theorem>.
Back to article page
