I don’t get how to start this question? what do they mean with \textbf{q}=(\pi/2,0)?
This is a k vector: \vec{k} of the form \vec{q}=(k_x, k_y). You should use these k values to construct a Hamiltonian H in this, and it’s ‘energy equivalent’ point.
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Do you maybe also know how to <\frac{\pi}{a}|V(x)|\frac{\pi}{a}> is supposed to be done?
First of all, you should integrate over an eigenstate with \vec{k} the vector \vec{q} and it’s equivalent vector \vec{k'} corresponding to \vec{q'}=(-\pi/a, 0), or more general you should integrate as follows: W = \langle \vec{k}|V|\vec{k'}\rangle, where \vec{k}'=\vec{k}+\vec{G} (where \vec{G} links equivalent lattice points). In 2D this becomes a 2D Fourier transform of the form:
W = \langle \vec{k}|V(\vec{r})|\vec{k'}\rangle=\frac{1}{a^2} \int_0^a\int_0^a d\vec{r} \times e^{i(\vec{k}-\vec{k'}) \cdot\vec{r}} \times V{(\vec{r}})
Where the pre factor becomes 1/a^2 and the boundaries [0, a] because of the fact that it’s a ‘square lattice’.
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