# Given a Hamiltonian how do you pick the most convenient Hilbert space?

golnat
For example, I have a 3D particle that experiences a harmonic oscillator potential only in the X,Y plane for all Z ie. a free particle in the Z direction. This seems like cylindrical coordinates but I'm not sure how to express the Hilbert space if I want to be able to describe states and eigenvalues.

H = (Px^2 + Py^2 + Pz^2) / (2m) + 1/2*mω^2(x^2+y^2)

I know that the energies will be characterized by the two harmonic oscillator dimension quantum numbers and also by the momentum in the z-direction, but what is the formal way to describe the Hilbert space?