How to solve 2d problems numerically.

I havn't had much classes on numerical methods in quantum mechanics and I'm wondering how one would solve a general problem involving 2d motion. With general, I mean a problem that cannot be separated. Consider for instance the hamiltonian

$\hat{H} = \frac{\widehat{p}_{x}^{2}+\widehat{p}_{y}^{2}}{2m}+x^{2}y^{2}$

How does one find the eigenvalues and eigen functions numerically?
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 Recognitions: Science Advisor 1. Use a basis of e.g. harmonic oscillator eigenfunctions and a diagonalization routine for symmetric matrices (e.g. Lapack). 2. Use a grid of points and finite difference approximation for the derivatives. Then diagonalize the matrix like in 1.