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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:

[tex] \psi = \sum_k C_k e^{ikr} [/tex]

is made.

Then, by looking at the Schrodinger equation in the reciprocal lattice, we see that the [itex] C_k [/itex] are a linear combination of [itex] C_{k-G} [/itex], where [itex]G[/itex] is the reciprocal lattice vector.

Finally, the wave function is indexed by an index [itex] k [/itex] and rewritten as:

[tex] \psi_k = \sum_G C_{k-G} e^{i(k-G)r}, [/tex]

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 [itex] \psi_k [/itex] the same function as [itex] \psi [/itex]? 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!!