- #1
Sheng
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Homework Statement
I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear,
Homework Equations
w refers to wannier function, which relates to the Bloch function
##\mathbf{R}## is this case should be zero.
The Bloch function
$$\psi_{n\mathbf{k}}=e^{i\mathbf{k \cdot r}}u_{n\mathbf{k}}$$, where ##u_{n\mathbf{k}}## is the cell periodic part.
The Attempt at a Solution
Using the given relation ##\mathbf{r\psi_{k}}##, I manage to get the following the equation
$$\langle w \vert \mathbf{r} \vert w \rangle = \left( \frac{\Omega}{8\pi^3} \right)^2 \int_{BZ} d\mathbf{k} d\mathbf{k}' i e^{i(\mathbf{k-k'}) \cdot r} \langle u_{\mathbf{k}} \vert \nabla u_{\mathbf{k'}} \rangle $$,
but I cannot find a way to factorize the exponential term out or to reduce the order of magnitude the prefactor.
Any help is appreciated.
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