Proof of Blochs function.

1. silverwhale

67
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:

$$\psi = \sum_k C_k e^{ikr}$$

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!!