The vector $\mathbf{k}$ is the [wave vector](matter%20waves.md) referred to as the [k-vector](k-vector.md) in [crystal lattices](Crystal%20lattices.md) and the index $n$ is an index accounting for the fact that we may have more than one [wavefunction](Wavefunction.md) with a given wave vector in a given [periodic potential.](Periodic%20potentials.md) This index is referred to as the [band](Bands.md) index. ^c01e0d If we examine the geometry of [periodic potentials](Periodic%20potentials.md) we find that lattices can be characterized with a [lattice vector](Periodic%20potentials.md#Lattice%20vectors%20and%20reciprocal%20lattices) $\mathbf{R}$ such that $u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=u_{n\mathbf{k}}(\mathbf{r})$ meaning that for a [Bloch function,](Bloch%20function.md) [$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{r}\mathbf{k}}u_{n\mathbf{k}}(\mathbf{r}),$](Bloch's%20theorem#^0bdc3b) $\psi_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{R}\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}u_{n\mathbf{k}}(\mathbf{r})$and thus we may define a Bloch function as a [Wavefunction](Wavefunction.md) such that $\psi_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{R}\mathbf{k}}\psi_{n\mathbf{k}}(\mathbf{r})$ ^67a4eb Note that here, with vectors $\mathbf{r}$ and $\mathbf{k}$ we model Bloch functions in $2$ or more dimensions, while [below](Bloch%20function.md#One-dimensional%20Bloch%20functions) we consider Bloch functions for one-dimensional models. # One-dimensional Bloch functions In many models we consider, we may only need to write the Bloch function in one dimension. #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/MultiParticleQuantumSystems/SolidStatePhysics