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I'm reading the book Solid State Physics by Ashcroft and Mermin. In its second proof of Bloch's Theorem on p.137, the periodic potential $U(\mathbf{r})$ and the wavefunction $\psi(\mathbf{r})$ both have been decomposed into their Fourier components $U(\mathbf{r})=\sum_{\mathbf{K}}U_{\mathbf{K}}e^{i\mathbf{K}\cdot \mathbf{r}}$ and $\psi(\mathbf{r})=\sum_{\mathbf{q}}c_{\mathbf{q}}e^{i\mathbf{q}\cdot \mathbf{r}}$. By putting those two into the Schroedinger Equation, one arrives at,

$$\sum_{\mathbf{q}}e^{i\mathbf{q}\cdot\mathbf{r}}[(\frac{\hbar^2}{2m}q^2-\varepsilon)c_{\mathbf{q}}+\sum_{\mathbf{K'}}U_{\mathbf{K'}}c_{\mathbf{q}-\mathbf{K'}}]=0,$$

where $\mathbf{K}$ is the reciprocal lattice vector and $\mathbf{K}$ has been renamed as $\mathbf{K}'$. Now, since the basis functions are orthogonal, their linear combination is zero if and only if the coefficients are zero themselves, so

$$(\frac{\hbar^2}{2m}q^2-\varepsilon)c_{\mathbf{q}}+\sum_{\mathbf{K'}}U_{\mathbf{K'}}c_{\mathbf{q}-\mathbf{K'}}=0.$$

My question is: why it is necessary to write $\mathbf{q}=\mathbf{k}-\mathbf{K}$, where $\mathbf{K}$ is a reciprocal lattice vector chosen so that $\mathbf{k}$ lies in the first Brillouin zone. Why is $\mathbf{k}$ in the first Brillouin zone rather than $\mathbf{q}$ if it is written in such way? What's the physical meaning/intuition behind such renaming process?

The book continues to state that we need to make another change of variable from $\mathbf{K'}$ to $\mathbf{K'}-\mathbf{K}$. Why is that? What's the physical meaning / intuition behind it? Sorry I'm totally confused here in this part.

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