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Schrodinger equation in the reciprocal lattice. 
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#1
Oct313, 03:00 AM

P: 62

Hi Everybody,
I am learning solid state physics using a German book called "Festkorperphysik" written by Gross and Marx. Now, in page 336 the Schrodinger equation in momentum space is introduced: [tex] \left( \frac{\hbar^2 k^2}{2m}  E \right) C_\vec{k} + \sum_\vec{G} V_\vec{G} C_{\vec{k}\vec{G}} = 0. [/tex] Then the authors go on and say that this set of algebraic equations is a representation of the Schrodinger equation in the reciprocal space (reciprocal lattice). I guess they mean set because for each value of [itex] \vec{k}[/itex] there is one equation. Next, they say that for each [itex] \vec{k} [/itex] there is a solution [itex] \psi_{\vec{k}} [/itex] with a corresponding energy eigenvalue [itex] E_\vec{k} [/itex]. I do not understand what [itex] \psi_{\vec{k}} [/itex] is. From each equation I get only one value for [itex] C_\vec{k} [/itex]. And by looking at all the different values for [itex] \vec{k} [/itex] I get all the different algebraic equations from which I can extract the [itex] C_\vec{k} [/itex] with which I can construct [itex] \psi[/itex], the original wave function. That's what I thought.. Where does [itex] \psi_{\vec{k}} [/itex] come from? And related to this where does [itex] E_k [/itex] come from? Where is the eigenvalue equation that gives the k indexed wavefunction and eigenvalue? Or is [itex] \psi_{\vec{k}} [/itex] simply given by: [tex] \psi_{\vec{k}} = C_{\vec{k}} e^{i \vec{k} \cdot \vec{r}}?[/tex] I do not understand. I searched many books but didn't find any answer.. Thanks for your help in advance! :) 


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