The Eigenstate \(\psi\) of a one electron Hamiltonian
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| \begin{equation*} H = - \frac{\hbar^2\Delta^2}{2m}+U(\vec{r}) \end{equation*} | (4.1) |
with \(U(\vec{r} + \vec{R}) = U(\vec{r})\) for all \(\vec{R}\) in the Bravais lattice can be chosen as
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| \begin{equation*} \psi_{nk}(\vec{r}) = e^{i \vec{k}\vec{r}} u_{nk}(\vec{r}) \qquad , \label{bloch1} \end{equation*} | (4.2) |
with \(u_{n,k}\) being a function with the periodicity of the lattice:
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| \begin{equation*} u(\vec{r} + \vec{R} )= u(\vec{r}) \label{bloch2} \end{equation*} | (4.3) |
From Eq. (4.2) and (4.3) follows
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| \begin{equation*} \psi_{nk}(\vec{r}+\vec{R}) = e^{i \vec{k}\vec{R}} \psi_{nk}(\vec{r}) \qquad , \label{bloch3} \end{equation*} | (4.4) |
which allows to define the Bloch-Theorem in an alternative way:
One
can always choose Eigenstates \(\psi\) of \(H\) so, that for each \(\psi\) we find a
wave vector \(\vec{k}\) with
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| \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
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| \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
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| \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.
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© J. Carstensen (Quantum Mech.)