Many particle physics - Hamiltonian for Fermi system

[Plaetean]

Homework Statement

Working through problems in Mahan's 'Many Particle Physics' book, and at the end of the 1st chapter 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.

Homework Equations

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!

 

[Plaetean]

Bump!