# Rayleigh Ritz with two Particles

#### wduff

Hello everyone, I have what should be a simple one to answer.

I'm solving for 2 particles in a harmonic oscillator with a gaussian bump in the middle and a delta function interaction. I'm doing all this via Rayleigh Ritz; that is, diagonalizing the Hamiltonian to find the constants in:

$$\Psi = \sum_{ij} c_{ij} \psi_{ij}$$

Where $\Psi_{ij}$ is just the standard symmetrized 2 boson wavefunction with $\psi_{i}$ and $\psi_{j}$ in a harmonic oscillator basis.

My issue is this: I end up with a rank 4 tensor: $$\langle \psi_{ij}|H|\psi_{kl}\rangle$$

I'm sure this is just inexperience, but I have no idea how to get an eigensystem out of that (I'm using mathematica). I've already done it with one particle (minus the delta function of course), which was simple enough, since the calculations produced a n x n matrix which I know how to handle. But this is a new one for me.

Any suggestions would be very appreciated. Thanks!

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#### DrClaude

Mentor
One needs to define a basis of two-particle states, which are themselves the tenor product of single-particle states.

In that two-particle basis, the Hamiltonian will be a 2x2 matrix.

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