Read the sections in Kittel about the Kronig-Penny model, then run the Kronig-Penny applet at http://fermi.la.asu.edu/ccli/applets/kp/kp.html. The applet is a bit buggy on Windows and takes a long time to load, but runs fine on Macintosh. The File, Options and Help menus are especially problematic and I suggest you NOT use them (they do nothing useful anyways). The color coding of the Bloch function u and wave function psi is: red=real, blue=imaginary, orange=magnitude.
1. Set the potential V=0 and observe the free electron dispersion relation and wave functions as you adjust the wave function energy E. Note how the wave function and Bloch function change as the energy passes through the band crossings at E close to 10 and 40. Explain this behavior.
2. Choose potential strength V and width W so that band gaps appear. Compare the wave function magnitudes at energies slightly below and slightly above the band gap. How do the positions of maximum probability relate to peaks of the potential? Why does the wave function magnitude match the Bloch function magnitude?