# Definition Consider a periodic potential (e.g. in a crystal lattice, whether solid-state or photonic) Bloch's theorem states that the solution to the [[Schrödinger Equation]] can be broken into the form $ \psi(\pmb{r}) = e^{i \pmb{k}\cdot \pmb{r}} u(\pmb{r}) $ where $u(\pmb{r})$ is a periodic function with the same periodicity as the potential, known as the Bloch function. # Bloch Hamiltonian Consider a periodic system with Hamiltonian $\mathcal{H}$: $ \begin{align} \mathcal{H} \ket{\psi_{n\pmb{k}}} = E_{n\pmb{k}} \ket{\psi_{n\pmb{k}}} .\end{align} $ Using Bloch's Theorem we can write: $ \begin{align} \ket{\psi_{n\pmb{k}}} = e^{i \pmb{k}\cdot\pmb{r}} \ket{u_{n\pmb{k}}} .\end{align} $ which leads to the reduced [[Schrödinger Equation]] for $\ket{u_{n\pmb{k}}}$: $ \begin{align} H(\pmb{k}) \ket{u_{n\pmb{k}}} = E_{n\pmb{k}} \ket{u_{n\pmb{k}}} ,\end{align} $ where $H(\pmb{k}) = e^{-i \pmb{k}\cdot\pmb{r}} \mathcal{H} e^{+i \pmb{k}\cdot\pmb{r}}$ is the Bloch Hamiltonian.