For electrons in a perfect crystal there is a basis of wave functions with the following two properties:

1) each of these wave functions is an energy eigenstate; 2) each of these wave functions is a Bloch state, meaning that this wave function \psi ψ can be written in the form \psi(r)=u(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}} ψ ( r ) = u ( r ) e i k⋅ r where u u has the same periodicity as the atomic structure of the crystal: u_{\boldsymbol{k}}(\boldsymbol{r})=u_{\boldsymbol{k}}(\boldsymbol{r}+\boldsymbol{n}\cdot\boldsymbol{a}) u k ( r ) = u k ( r + n⋅ a ) .