Schrodinger equation in the reciprocal lattice.

[silverwhale]

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! :)

 

[BigJDubs]

</code>The Schrodinger equation in momentum space is a representation of the original Schrodinger equation in the reciprocal lattice space. The solution of this equation for each value of k gives you a wavefunction psi_k and an energy eigenvalue E_k. The wavefunction psi_k is given by:psi_k = C_k * exp(i*k*r), where C_k is the coefficient obtained from the solution of the equation and r is the position vector.The energy eigenvalue E_k is obtained by solving the equation:E_k = (hbar^2*k^2)/(2m) - V, where hbar is Planck's constant, m is the mass of the particle, and V is the potential energy. Hope this helps!