_Bloch's theorem_ states that the [wavefunction](Wavefunction.md) subject to a [Bravais lattice](Bravais%20lattices.md) [periodic potential](Periodic%20potentials.md) is modulated by a periodic function, $u_{n\mathbf{k}}(\mathbf{r}),$ such that $\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{r}\mathbf{k}}u_{n\mathbf{k}}(\mathbf{r})$ where $e^{i\mathbf{r}\mathbf{k}}$ would be an unmodulated plane wave where $\mathbf{k}$ is the [reciprocal lattice vector of the potential](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices). This wavefunction is referred to as a _[[Bloch function]]._ %% see Kittel pg. 167 for this. %% ^0bdc3b  Equivalently, [Bloch's theorem](Bloch's%20theorem.md) states that a [wavefunction](Wavefunction.md) subject to a [Periodic potential](Periodic%20potentials.md) takes the form [$\psi_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{R}\mathbf{k}}\psi_{n\mathbf{k}}(\mathbf{r})$](Bloch%20function#^67a4eb) ^c29630 # Bloch Hamiltonians --- # Proofs and examples %%There are multiple proofs in ashcroft and Mermin. Just go there. %% ## Proof of Bloch's theorem in terms of translation operations      #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/MultiParticleQuantumSystems/SolidStatePhysics #QuantumMechanics/MathematicalFoundations