# Homework Help: Particle subject to a hopping potential between two atoms in a lattice

1. Mar 13, 2013

### Trance

Hello! Long time lurker, first time poster. This is the first of a couple of questions which has totally stumped me, although I have a feeling it's easier than it first seems.

1. The problem statement, all variables and given/known data
A particle is initially located at one of two atoms. The particle is subject to Hhop ,
a discrete hopping potential Hamiltonian between the two atoms, whose strength
is constant in time. Model the particle’s location as a discrete 2-state system, and
determine the evolution as a function of time

2. Relevant equations
This is what's getting me - I'm not entirely sure what the relevant equations are! I believe a Hopping potential describes the likelyhood of the particle 'hopping' to the adjacent atom, but I'm not 100% sure about that.

3. The attempt at a solution

At first I thought to use statistical arguments, but these fail of course, because there is only one particle. The question is also for a general particle, not a specific fermion/boson, so I doubt that's the way to go. I've thought about using the Hubbard model, but from my research (We haven't explicitly studied it in my degree yet), it seems more relevant to electrons in an atom rather than particles in a lattice. I had the thought to perhaps model the situation as two Hydrogen atoms, sharing one electron, and then use the Hubbard model, but think this could be a little too specific.

Any help would be greatly appreciated!

2. Mar 13, 2013

### TSny

Hello trance.

Do you have access to Feynman's Lectures on Physics? If so, you might find it interesting to relate your problem to his treatment of the ammonia molecule in chapter 8 of volume III.

3. Mar 13, 2013

### Trance

TSny, that is a fantastic catch. Thank you so much! This describes almost exactly what I was looking for.

4. Mar 13, 2013

### TSny

Ah, Good! I was hoping (hopping) so. Feynman has a very nice discussion.