Particle subject to a hopping potential between two atoms in a lattice

[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.

Homework Statement

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

Homework 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.

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!

 

[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.

 

[Trance]

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.

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

 

[TSny]

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

 

[paranormal]

Hello! It's great to see you engaging in the community and seeking help for your problem. I would suggest approaching this problem by first defining the Hamiltonian for the system. In this case, the Hamiltonian would include the kinetic energy term for the particle, as well as the hopping potential term between the two atoms in the lattice. Once you have the Hamiltonian, you can use the time-dependent Schrödinger equation to determine the evolution of the system as a function of time. This approach takes into account the quantum nature of the particle and allows for a more accurate description of its behavior.

As for the hopping potential, you are correct in thinking that it describes the likelihood of the particle hopping between the two atoms in the lattice. This potential is typically represented by a square well potential, where the particle is confined to one of the two atoms but has a probability of tunneling to the other atom.

I would also suggest looking into the concept of Bloch oscillations, which is a phenomenon that occurs in a similar system of a particle in a lattice subject to a periodic potential. This could provide some insights into the behavior of the particle in your system.

I hope this helps and good luck with your problem!