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Finding Bloch functions

  1. Dec 26, 2011 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    Picture:

    TO5mu.png

    3. 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
     
  2. jcsd
  3. Dec 29, 2011 #2
    So no one has any idea? :\
     
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