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dingo_d
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Homework Statement
Schrodinger equation of the problem given at the picture can be written in a form:
[itex](E_1-E)c^a+J(1+e^{i \vec{k}\cdot\vec{a}})c^b=0[/itex],
[itex]J(1+e^{-i \vec{k}\cdot\vec{a}})c^a+(E_2-E)c^b=0[/itex].
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:
[itex]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][/itex]
From first equation I have:
[itex]c^a=-\frac{J(1+e^{i \vec{k}\cdot\vec{a}})}{E_1-E}c^b[/itex]
And I will use the fact that the Bloch functions should have the norm:
[itex]|c^a|^2+|c^b|^2=1[/itex].
From that I should get the coefficients [itex]c^a[/itex] and [itex]c^b[/itex].
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