# Derivation of Bloch's theorem

I'm having a problem following a derivation of Bloch's theorem, looking at a one dimensional lattice with $N$ nodes and spacing a, we impose periodic boundary conditions, meaning that the wave-function of a single electron $\psi(x)$ can be decomposed in the following modes:

$$\psi(x)=\sum_{q}\phi(q)\text{e}^{iqx},$$

where

$$q = \frac{2\pi m}{N a}\; m\in \mathbb{Z}.$$

Otherwise it would be multivalued.

Schrodinger's equation gives:$$(\epsilon - \frac{\hbar^2q^2}{2m})\phi(q)=\sum_{k \in B}\tilde{V}(k)\phi(q+k)$$ where $\epsilon$ is the energy eigenvalue of $\psi$, $B$ is the reciprocal Bravais lattice ($k=\frac{2\pi n}{a}$ with $n\in \mathbb{Z}$), $\tilde{V}$ is the Fourier transform of the potential. So the $\phi$ modes separated by a vector belonging to the reciprocal Bravais lattice are coupled but I don't see how it implies that only $q$ vectors that are in this class of equivalence (whose difference belongs to $B$) should appear in the first sum (the expansion of $\psi$ in modes).

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