# Eigen functions/values for many-body Hamiltonian with creation/annihilation operators

1. ### ranytawfik

11
Problem:
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I’m trying to understand how to generally find Eigen functions/values (either analytically or numerically) for Hamiltonian with creation/annihilation operators in many-body problems.

Procedures:
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1. I setup a simple case of finite-potential well with two in-distinguishable fermions (ignore spin for the moment).
2. I got the creation/annihilation field operators.
3. I second-quantized all the elements of the Hamiltonian (kinetic energy, well/external potential, and electron-electron electrostatic interaction).

My question is how to proceed next? I have the Hamiltonian which is now function of the creation/annihilation operators. How can I solve for the many-body Eigen function/values after that?

Thanks so much.

2. ### strangerep

2,220
Re: Eigen functions/values for many-body Hamiltonian with creation/annihilation opera

ranytawfik,

It would be easier to respond if you post some of your math that you've
already obtained in the first 3 steps.

[
If you're not familiar with Latex on this forum, try this thread:
]

3. ### Bill_K

4,159
Re: Eigen functions/values for many-body Hamiltonian with creation/annihilation opera

In general, the procedure is as follows:

1) Try to find the normal modes of your system, that is, transform your creation/annihilation operators to a basis in which the Hamiltonian is diagonal: H = ∑ ak*ak.

2) Find the states for each normal mode as a quantized harmonic oscillator. The ground state is |0k> such that ak|0k> = 0. The excited states are |nk> = (ak*)n|nk>.

If you can't do step (1), you'll have to use perturbation theory.

4. ### ranytawfik

11
Re: Eigen functions/values for many-body Hamiltonian with creation/annihilation opera

Thanks strangerep and Bill. I'll try to do what you suggested, Bill, and post a follow up with the detailed Latex equations and procedures.