Molecular Electronics 
[Jun. 21st, 200909:12 pm]
drostie

I only really aced one out of the four big questions on my molecular electronics exam. Oddly enough, it also had a serious typo in the Hamiltonian, and didn't really define its own language. It took me 1020 minutes to decipher what was going on, and then the TA wanted 1020 minutes to verify that my version was indeed correct. Here's the problem:Consider the motion of an electron on a linear chain of hopping sites separated by a distance a. The motion is described by the following tightbinding Hamiltonian:H = t Σ_{n=∞}^{∞} (  n > < n+1+  n+1 >  < n  ) a) What is the meaning of t? What is the unit of t? b) Show that plane waves are eigenfunctions of this Hamiltonian. c) Find the expression for the dispersion relation for the electrons and plot it. How large is the band width? d) Show that at low energy the dispersion relation is that of a free electron with an effective ass. Find the expression of the effective mass as a function of the band width. If you read this question the obvious way  by deleting the + symbol that doesn't mean anything, you get:H = Σ_{n}  n > < n+1  n+1 > < n  BUT  and this is a big but  since wavefunctions are always normalized, < a  a > is just trivially 1. That makes this the sum of all  n > < n  states, but if they're exhaustive and orthogonal, then that's just the identity matrix. So the Hamiltonian is trivially "H = t." No good.
The key here is that there's an extra  symbol which is in between a ket and a bra on the rightmost part  and it doesn't mean anything there. It was misplaced, and the interaction is supposed to read: n > < n+1  +  n+1 > < n  Which is indeed a nearestneighbors coupling.
Anyway. Cute problem. 

