# Schrodinger equation in the reciprocal lattice.

by silverwhale
Tags: equation, lattice, reciprocal, schrodinger
 P: 64 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: $$\left( \frac{\hbar^2 k^2}{2m} - E \right) C_\vec{k} + \sum_\vec{G} V_\vec{G} C_{\vec{k}-\vec{G}} = 0.$$ 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 $\vec{k}$ there is one equation. Next, they say that for each $\vec{k}$ there is a solution $\psi_{\vec{k}}$ with a corresponding energy eigenvalue $E_\vec{k}$. I do not understand what $\psi_{\vec{k}}$ is. From each equation I get only one value for $C_\vec{k}$. And by looking at all the different values for $\vec{k}$ I get all the different algebraic equations from which I can extract the $C_\vec{k}$ with which I can construct $\psi$, the original wave function. That's what I thought.. Where does $\psi_{\vec{k}}$ come from? And related to this where does $E_k$ come from? Where is the eigenvalue equation that gives the k indexed wavefunction and eigenvalue? Or is $\psi_{\vec{k}}$ simply given by: $$\psi_{\vec{k}} = C_{\vec{k}} e^{i \vec{k} \cdot \vec{r}}?$$ I do not understand. I searched many books but didn't find any answer.. Thanks for your help in advance! :)

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