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Schrodinger equation in the reciprocal lattice.

by silverwhale
Tags: equation, lattice, reciprocal, schrodinger
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silverwhale
#1
Oct3-13, 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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