Editing Bloch's theorem (section)

== Statement ==
{{math theorem | name = Bloch's theorem | math_statement = 
For electrons in a perfect crystal, there is a [[basis (linear algebra)|basis]] of wave functions with the following two properties:
* each of these wave functions is an energy eigenstate,
* each of these wave functions is a Bloch state, meaning that this wave function <math>\psi</math> can be written in the form <math>\;\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}),</math> where <math>u(\mathbf{r})</math> has the same periodicity as the atomic structure of the crystal, such that <math display="block">u_{\mathbf{k}}(\mathbf{x}) = u_{\mathbf{k}}(\mathbf{x} + \mathbf{n} \cdot \mathbf{a}).</math>
}}

A second and equivalent way to state the theorem is the following<ref name="ziman:1">{{cite book |last=Ziman  |first=J. M. |date=1972 |edition=2nd |title=Principles of the theory of solids |publisher=Cambridge University Press |isbn=0521297338 |pages=17–20}}</ref>

{{math theorem | name = Bloch's theorem | math_statement = 
For any wave function that satisfies the Schrödinger equation and for a translation of a lattice vector <math>\mathbf{a}</math>, there exists at least one vector <math>\mathbf{k}</math> such that: 
<math display="block">\psi_{\mathbf{k}}(\mathbf{x}+\mathbf{a}) = e^{i\mathbf{k}\cdot\mathbf{a}}\psi_{\mathbf{k}}(\mathbf{x}).</math>
}}