Equation cropping up in Kronig Penny model

[quantumfireball]

Homework Statement

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

Homework Equations

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

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.

 

[hulet]

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:

http://webphysics.davidson.edu/facul...nig-penney.htm

particularly Fig 8.11 Pg 297