We may define a [translation operator](Translation%20operator%20in%20quantum%20mechanics.md) $\hat{T}_\mathbf{R}$ on that applies to any function $f(\mathbf{r})$ for a [lattice vector,](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) $\mathbf{R}$ such that $\hat{T}_\mathbf{R}f(\mathbf{r}) = f(\mathbf{r}+\mathbf{R})$ ^ac7cb4 We must then show that this [translation operator](Proof%20of%20Bloch's%20theorem%20in%20terms%20of%20translation%20operations#^ac7cb4) gives rise to [wavefunctions,](Wavefunction.md#Position%20space%20wavefunctions) $\psi(\mathbf{r})$ that are also [Bloch functions](Bloch%20function.md) provided that their [Hamiltonian](Bloch's%20theorem.md#Bloch%20Hamiltonians) describes a [periodic potential](Periodic%20potentials.md) that is also a [Bravais lattice.](Bravais%20lattices.md) Since such a Hamiltonian must also be periodic it follows that $\hat{T}_\mathbf{R}\hat{H}(\mathbf{r})\psi(\mathbf{r}) = \hat{H}(\mathbf{r}+\mathbf{R})\psi(\mathbf{r}+\mathbf{R})=\hat{H}(\mathbf{r})\psi(\mathbf{r}+\mathbf{R})=\hat{H}(\mathbf{r})\hat{T}_\mathbf{R}\psi(\mathbf{r}).$ Therefore, $\hat{T}_\mathbf{R}\hat{H}=\hat{H}\hat{T}_{\mathbf{R}}$ meaning that Bloch Hamiltonians [commute](Commutators%20in%20quantum%20mechanics.md#Commutation%20relations%20with%20both%20observables%20and%20time%20evolution%20operators) with [lattice vector](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) translation operators. ^aa86b7 If we then define a second [translation operator,](Proof%20of%20Bloch's%20theorem%20in%20terms%20of%20translation%20operations#^ac7cb4) $\hat{T}_\mathbf{R'}$ to represent a second successive translation on the [Bravais lattice](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) we find that when we apply it to [$\psi(\mathbf{r})$,](Wavefunction.md#Position%20space%20wavefunctions) $\hat{T}_\mathbf{R}\hat{T}_\mathbf{R'}\psi(\mathbf{r})=\psi(\mathbf{r}+\mathbf{R}'+\mathbf{R})=\hat{T}_\mathbf{R'}\hat{T}_\mathbf{R}\psi(\mathbf{r}),$ meaning that translations [commute](Commutators%20in%20quantum%20mechanics.md) and we may define $\hat{T}_\mathbf{R}\hat{T}_\mathbf{R'}=\hat{T}_{\mathbf{R}+\mathbf{R'}}.$ ^7df1fa %%Note if it's the case that translation operators always commute then this step could be simplified.%% It follows from the fact that $\hat{T}_\mathbf{R}$ and $\hat{H}$ [commute](Proof%20of%20Bloch's%20theorem%20in%20terms%20of%20translation%20operations#^aa86b7) that $\hat{T}_\mathbf{R}$ and $\hat{H}$ [share](Commuting%20operators.md) at least one non-trivial [eigenstate](Wavefunction.md) meaning that we can always pick an eigenstate $\psi$ such that [$\hat{H}\psi = E\psi$](Schrödinger%20equation%20(time%20independent).md) and $\hat{T}_R\psi=c(\mathbf{R})\psi$ ^ec95ad It follows from the fact that [$\hat{T}_\mathbf{R'}\hat{T}_\mathbf{R}=\hat{T}_\mathbf{R}\hat{T}_\mathbf{R'}=\hat{T}_{\mathbf{R}+\mathbf{R'}}$](Proof%20of%20Bloch's%20theorem%20in%20terms%20of%20translation%20operations#^7df1fa) that $\hat{T}_\mathbf{R'}\hat{T}_\mathbf{R}\psi=\hat{T}_\mathbf{R'}c(\mathbf{R})\psi=c(\mathbf{R}')c(\mathbf{R})\psi=c(\mathbf{R}+\mathbf{R}')\psi,$ This means that $c(\mathbf{R}')c(\mathbf{R})\psi=c(\mathbf{R}+\mathbf{R}')$ so that if we write $c(\mathbf{a}_i)=e^{2\pi i x_i},$ then the above algebraic conditions are satisfied. Here $\mathbf{a}_i$ is a [primitive lattice vector](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) and for a [Bravais lattice](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) where [$\mathbf{R}=n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3$](Periodic%20potentials#^13f06b) and thus $c(\mathbf{R})=c(\mathbf{a}_1)^{n_1}c(\mathbf{a}_2)^{n_2}c(\mathbf{a}_3)^{n_3} = e^{2\pi i x_1 \mathbf{a}_1 n_1}e^{2\pi i x_2 \mathbf{a}_2 n_2}e^{2\pi i x_3 \mathbf{a}_3 n_3} = e^{i\mathbf{k}\mathbf{R}}$ where $\mathbf{k}$ is the [reciprocal lattice vector](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) following from the fact that [$\mathbf{b}_i\mathbf{a_j} = 2\pi\delta_{ij}$.](periodic%20potentials#^1fe7d8) Thus substituting into our [eigenfunction equation](Proof%20of%20Bloch's%20theorem%20in%20terms%20of%20translation%20operations#^ec95ad) $T_\mathbf{R}\psi(\mathbf{r})=e^{i\mathbf{k}\mathbf{R}}\psi(\mathbf{r})$ thus we obtain a [Bloch function](Bloch%20function#^67a4eb) $\psi(\mathbf{r}+\mathbf{R})=e^{i\mathbf{k}\mathbf{R}}\psi(\mathbf{r}),$ which satisfies our [second definition of Bloch's theorem](Bloch's%20theorem#^c29630) $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\blacksquare$ ^7b39f7 #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/MultiParticleQuantumSystems/SolidStatePhysics #QuantumMechanics/MathematicalFoundations #Proofs