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Equation cropping up in Kronig Penny model

  1. Nov 25, 2008 #1
    1. The problem statement, all variables and given/known data

    how does one solve the following equation to obtain E as a function of K.

    2. Relevant equations

    cos(k*a)=P*sin(u*a)/u*a + cos(u*a),
    where u=sqrt(2*m*E)/h-bar

    3. The attempt at a solution
    theres no way this can be solved analytically,however how to go about approximating E as a function of k.
    E(k) is obviously a many to one function.
  2. jcsd
  3. Nov 2, 2009 #2
    Step1: Plot the RHS of the equation to get a sinusoidal waveform

    The LHS is just a continum of values from +1 to -1 since Cos cannot exceed these limits for all values of ka... This means the LHS gives us any number between +1 and -1

    Now imagine the graph that you plotted in step 1. Imagine two lines at +1 and -1 cutting off this graph... All points within these 2 lines represent valid energies.... all others are "forbidden energies" or band gaps

    This link might be useful to you:


    particularly Fig 8.11 Pg 297
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