One can always choose Eigenstates \(\psi\) of \(H\) so, that for each \(\psi\) we find a wave vector \(\vec{k}\) with
| \begin{equation*} \psi(\vec{r}+\vec{R}) = e^{i \vec{k}\vec{R}} \psi(\vec{r}) \qquad , \label{bloch4} \end{equation*} | (4.5) |
This may be rewritten as
| \begin{equation*} \psi(\vec{r}+\vec{R}) = e^{i \vec{k}(\vec{r}+\vec{R})} e^{-i \vec{k}\vec{r}} \psi(\vec{r}) = e^{i \vec{k}(\vec{r}+\vec{R})} u(\vec{r}) \qquad , \label{bloch5} \end{equation*} | (4.6) |
with
| \begin{equation*} u(\vec{r}) = e^{-i \vec{k}\vec{r}} \psi(\vec{r}) \end{equation*} | (4.7) |
being a (lattice) periodic function. Thus Eq. (4.2) and (4.5) are equivalent.
© J. Carstensen (Quantum Mech.)