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Proof of Blochs function. 
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#1
Oct413, 05:08 AM

P: 64

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: [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_{kG} [/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_{kG} e^{i(kG)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!! 


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