For each vector \(\vec{R}\) of the Bravais lattice we define a translation operator \(T_R\) by
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| \begin{equation*} T_R f(\vec{r}) = f(\vec{r}+\vec{R}) \end{equation*} | (4.8) |
for an arbitrary function \(f\).
Since the Hamiltonian
is periodic, we find
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| \begin{equation*} \label{tr1} T_R H \psi(\vec{r}) = H(\vec{r}+\vec{R}) \psi(\vec{r}+\vec{R}) = H(\vec{r}) \psi(\vec{r}+\vec{R}) = H T_R \psi(\vec{r}) \end{equation*} | (4.9) |
Eq. (4.9) holds for all state functions \(\psi\); so we can write
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| \begin{equation*} T_R H = H T_R \qquad \mbox{or} \qquad [T_R, H] = 0 \qquad . \label{tr2} \end{equation*} | (4.10) |
Successively applying the translation leads to
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| \begin{equation*} T_R T_{R'} \psi(\vec{r}) = T_{R'} T_R \psi(\vec{r}) = \psi\left(\vec{r}+\vec{R}+\vec{R}'\right) \qquad , \label{tr3} \end{equation*} | (4.11) |
i.e.
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| \begin{equation*} T_R T_{R'} = T_{R'} T_R = T_{R+R'} \qquad . \label{tr4} \end{equation*} | (4.12) |
Following Eq. (4.10) we can choose the Eigenvectorsystem of \(H\) to be simultaneously a Eigenvectorsystem of \(T_R\):
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| \begin{equation*} \begin{split} H \psi & = E \psi\\ T_R \psi & = c(\vec{R}) \psi \end{split} \end{equation*} | (4.13) |
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| \begin{equation*} T_R T_{R'} \psi(\vec{r}) = c(\vec{R}') T_R \psi(\vec{r}) = c(\vec{R}')c(\vec{R}) \psi(\vec{r}) \qquad . \end{equation*} | (4.14) |
Thus for the Eigenvalues we find
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| \begin{equation*} c(\vec{R}+\vec{R}') = c(\vec{R})c(\vec{R}') \qquad . \label{tr6} \end{equation*} | (4.15) |
Let’s have a closer look on the primitive translation vectors of the Bravais lattice \(\vec{a}_i\). For adequate variables \(x_i\) we can write
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| \begin{equation*} c(\vec{a}_i) = e^{i 2 \pi x_i} \qquad . \label{tr7} \end{equation*} | (4.16) |
Thus for a general translation vector
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| \begin{equation*} \vec{R} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 \end{equation*} | (4.17) |
we find by successively applying Eq. (4.15)
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| \begin{equation*} c(\vec{R}) = c(\vec{a}_1)^{n_1} * c(\vec{a}_2)^{n_2} * c(\vec{a}_3)^{n_3} \end{equation*} | (4.18) |
which may be rewritten as
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| \begin{equation*} c(\vec{R}) = e^{i \vec{k} \vec{R}} \end{equation*} | (4.19) |
using Eq. (4.16) and the following definitions:
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| \begin{equation*} \vec{k} = x_1 \vec{b}_1 + x_2 \vec{b}_2 + x_3 \vec{b}_3 \end{equation*} | (4.20) |
and
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| \begin{equation*} \vec{a}_i \vec{b}_j = 2 \pi \delta_{i,j} \quad . \end{equation*} | (4.21) |
The above defined vectors \(\vec{b}\) are the basic vectors of the reciprocal
lattice.
Summing up, we have shown that for every vector of the real space the Eigenvectors \(\psi\) of \(H\) can be chosen to fulfill the following relation:
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| \begin{equation*} T_R \psi(\vec{r}) = \psi(\vec{r} + \vec{R}) = c(\vec{R}) \psi(\vec{r}) = e^{i \vec{k} \vec{R}} \psi(\vec{r}) \qquad . \end{equation*} | (4.22) |
This is the definition of the Bloch theorem according to Eq. (4.5). Additional illustrative information you can find in the MaWi II script.
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© J. Carstensen (Quantum Mech.)