So, I was trying to see what introducing a periodic potentials would do to the single particle schrodinger wave equation and here is how I proceed
$$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi$$ since I know that $V$ is periodic, I write it as $$V=\mathop{\sum\sum}_{p,q}C_{p,q}e^{i\vec{k_{p,q}\cdot\vec{r}}}$$ Hence, my equation becomes, $$\frac{-\hbar^2}{2m}\nabla^2\psi(\vec{r})+\mathop{\sum\sum}_{p,q}C_{p,q}\psi(\vec{r})e^{i\vec{k_{p,q}\cdot\vec{r}}}=E\psi(\vec{r})$$ Taking the Fourier transform on both sides $$\frac{\hbar^2}{2m}\vec{k}\cdot\vec{k}\psi(\vec{k})+\mathop{\sum\sum}_{p,q}C_{p,q}\psi(\vec{k}-\vec{k_{p,q}})=E\psi(\vec{k})$$ $$\mathop{\sum\sum}_{p,q}C_{p,q}\psi(\vec{k}-\vec{k_{p,q}})=(E-\frac{\hbar^2}{2m}\vec{k}\cdot\vec{k})\psi(\vec{k})$$ Now, my question is that if there is anything that I can say from the last equation? Something about the nature of $\psi(\vec{k})$? To me it looks like a series in 1D, somewhat of the form $$f(x)=\sum_{k}C_kf(x-kx_0)$$ I am not able to say or deduce anything from that series, is it possible to make some mathematical conclusion about those functions $f(x)$? I would also be happy if I am directed to some literature about them. Please note, I might look really stupid, but I am not a student of mathematics or physics, I am simply an engineer.