What are the Bloch functions for states in valence bands?

[dingo_d]

Homework Statement

Schrödinger equation of the problem given at the picture can be written in a form:

$(E_1-E)c^a+J(1+e^{i \vec{k}\cdot\vec{a}})c^b=0$,
$J(1+e^{-i \vec{k}\cdot\vec{a}})c^a+(E_2-E)c^b=0$.

Find Bloch functions for the states from both valence bands.

Homework Equations

Picture:

The Attempt at a Solution

First I find the Bloch energies by solving the above system for E. That is, I have a non trivial solution if the determinant of the above system is zero. From that I get:

$E_{a,b}(k)=\frac{1}{2}\left [ E_1+E_2\pm\sqrt{(E_1-E_2)^2+16J^2\cos^2\left(\frac{ka}{2}\right)}\right]$

From first equation I have:

$c^a=-\frac{J(1+e^{i \vec{k}\cdot\vec{a}})}{E_1-E}c^b$

And I will use the fact that the Bloch functions should have the norm:

$|c^a|^2+|c^b|^2=1$.

From that I should get the coefficients $c^a$ and $c^b$.

But the problem is that my energy expression is too complicated. If I put it in, and try to determine the coefficients I get this giant mess :\

Is there some kind of assumption that I failed to see, that will help me simplify this problem?

Thanks

 

[dingo_d]

So no one has any idea? :\