# Proof of Blochs function.

by silverwhale
Tags: blochs, function, proof
 P: 58 Hi Everybody, I am learning solid state physics and read today through Kittel. I am still stuck at the proof of Blochs theorem i.e. the proof of Blochs function. For the Schrodinger equation of a periodic potential the general ansatz: $$\psi = \sum_k C_k e^{ikr}$$ is made. Then, by looking at the Schrodinger equation in the reciprocal lattice, we see that the $C_k$ are a linear combination of $C_{k-G}$, where $G$ is the reciprocal lattice vector. Finally, the wave function is indexed by an index $k$ and rewritten as: $$\psi_k = \sum_G C_{k-G} e^{i(k-G)r},$$ where the sum goes now over G. I have two questions: 1) How can we go from the sum over k to a sum over G? 2) Is $\psi_k$ the same function as $\psi$? If so, why do we index it with k? I hope this is clear enough, and I would be glad for ANY answer!! :) I am searching for an ansatz for nearly a week now!!

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