||[Jun. 21st, 2009|09:12 pm]
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 10-20 minutes to decipher what was going on, and then the TA wanted 10-20 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 tight-binding Hamiltonian: If you read this question the obvious way -- by deleting the + symbol that doesn't mean anything, you get:
H = -t Σn=-∞∞ ( | 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.
H = Σn | 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 nearest-neighbors coupling.
Anyway. Cute problem.
Sweet problem, I'm working it now. You should post more of these, they're fun.
Also, lol electron with an "effective ass."