# Many particle physics - Hamiltonian for Fermi system

1. Oct 18, 2016

### Plaetean

1. The problem statement, all variables and given/known data
Working through problems in Mahan's 'Many Particle Physics' book, and at the end of the 1st chaper there's a question where we're asked to consider a fermion system with three energy states with eigenvalues E1, E2, E3, and matrix elements M12, M23, M13 which connect them and allow transitions between them.

The question asks us to write down a Hamiltonian for the system in terms of creation and annihilation operators, and then determine the eigenvalues for the system.

2. Relevant equations

3. The attempt at a solution
I'm really a bit lost as to where to start for this, and all I can really think of doing is writing the standard Hamiltonian for a quantum SHO as a sum over states, but I'm not confident this is remotely right.
$$H=\hbar\sum_{n=1}^{3}\omega_n(a_n^\dagger a_n + \frac{1}{2})+M_{12}+M_{23}+M_{13}$$

Is it the case that the energy eigenvalues will just be E1, E2 and E3, as the fact that transitions can occur doesn't change the actual eigenvalues of the system?

Thanks as always!

2. Oct 19, 2016

Bump!