Bandgapalicious

So, one thing I realized about my solid state physics class is that I really have managed to avoid learning the essential material for most of the semester. I've been reading ahead in my quantum book, and finally got to the stuff that makes the really really basic solid state stuff (that we did back in friggin' August) make some sense: for example, where do bandstructures come from? Turns out they arise (at least, in the theory) from a very technical point in Bloch's theorem (which was not at all well explained by our textbook). But at least it makes sense now. To wit:

If V(x+a)=V(x) (periodicity), then the V(x) that satisfy the TISE H*psi(x)=E_n*psi(x) follow the eigenfunction condition psi(x+a)=exp(i*K*a)*psi(x). If you use V(x) as a series of evenly spaced delta functions and use this to find the eigenfunctions, you end up with an expression for k (and therefore E), where your result is:

cos(K*a)=cos(k*a)+(m*alpha)/[(hbar^2*k)*sin(k*a)]

Since |cos(K*a)|<=1, you get a set of allowed k values: whenever the RHS of the above equation > 1, you get a 'forbidden zone' for k, which tells you what energies aren't allowed. This is where bandgaps come from.

Finishing up my solid state project this weekend. I think I mostly get what's going on...hope so, anyway. I wonder if we're going to even have a final in this class.