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\begin_body

\begin_layout Title
Test
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\begin_layout Theorem
For electrons in a perfect crystal
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\begin_layout Theorem
there is a basis of wave functions with the following two properties:
\end_layout

\begin_layout Standard
\begin_inset Separator plain
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\begin_layout Theorem
1) each of these wave functions is an energy eigenstate;
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\begin_layout Theorem
2) each of these wave functions is a Bloch state,
 meaning that this wave function 
\begin_inset Formula $\psi$
\end_inset

 can be written in the form 
\begin_inset Formula $\psi(r)=u(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$
\end_inset

 where 
\begin_inset Formula $u$
\end_inset

 has the same periodicity as the atomic structure of the crystal:
 
\begin_inset Formula $u_{\boldsymbol{k}}(\boldsymbol{r})=u_{\boldsymbol{k}}(\boldsymbol{r}+\boldsymbol{n}\cdot\boldsymbol{a})$
\end_inset

.
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