How to solve 2d problems numerically.

[ehj]

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?

 

[DrDu]

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.