- #1
baranas
- 14
- 0
How to understand that Bloch wave solutions can be completely characterized
by their behaviour in a single Brillouin zone? Given Bloch wave:
\begin{equation*}
\psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) \exp (i\mathbf{k}\mathbf{r})
\end{equation*}
I can write wavefunction for momentum ##\mathbf{k}' = \mathbf{k} + \mathbf{G}##.
\begin{equation*}
\psi_{\mathbf{k}+\mathbf{G}}(\mathbf{r}) = u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G}\mathbf{r})\exp (i\mathbf{k}\mathbf{r})
\end{equation*}
As I understand ##u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G}\mathbf{r}) \neq u_{\mathbf{k}}(\mathbf{r})##, so:
\begin{equation*}
\psi_{\mathbf{k}+\mathbf{G}}(\mathbf{r}) \neq \psi_{\mathbf{k}}(\mathbf{r}).
\end{equation*}
Or I am wrong? I know that another way to analyze this problem is to notice that
##u_k## and ##u_k'## are periodic and do the Fourier expansion:
\begin{equation*}
u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G})=\sum_{\mathbf{G}_{i}}C_{\mathbf{k}',\mathbf{G}_{i}}\exp\left(i\mathbf{G}\mathbf{r}\right)\exp\left(i\mathbf{G}_{i}\mathbf{r}\right)
\end{equation*}
However I still don't see why [tex] C_{\mathbf{k}',\mathbf{G}_{i}}\exp\left(i\mathbf{G}\mathbf{r}\right) = C_{\mathbf{k},\mathbf{G}_{i}}[/tex]
by their behaviour in a single Brillouin zone? Given Bloch wave:
\begin{equation*}
\psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) \exp (i\mathbf{k}\mathbf{r})
\end{equation*}
I can write wavefunction for momentum ##\mathbf{k}' = \mathbf{k} + \mathbf{G}##.
\begin{equation*}
\psi_{\mathbf{k}+\mathbf{G}}(\mathbf{r}) = u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G}\mathbf{r})\exp (i\mathbf{k}\mathbf{r})
\end{equation*}
As I understand ##u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G}\mathbf{r}) \neq u_{\mathbf{k}}(\mathbf{r})##, so:
\begin{equation*}
\psi_{\mathbf{k}+\mathbf{G}}(\mathbf{r}) \neq \psi_{\mathbf{k}}(\mathbf{r}).
\end{equation*}
Or I am wrong? I know that another way to analyze this problem is to notice that
##u_k## and ##u_k'## are periodic and do the Fourier expansion:
\begin{equation*}
u_{\mathbf{k}'}(\mathbf{r})\exp(i\mathbf{G})=\sum_{\mathbf{G}_{i}}C_{\mathbf{k}',\mathbf{G}_{i}}\exp\left(i\mathbf{G}\mathbf{r}\right)\exp\left(i\mathbf{G}_{i}\mathbf{r}\right)
\end{equation*}
However I still don't see why [tex] C_{\mathbf{k}',\mathbf{G}_{i}}\exp\left(i\mathbf{G}\mathbf{r}\right) = C_{\mathbf{k},\mathbf{G}_{i}}[/tex]