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

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?

DrDu
In ordinary QM (as opposed to QFT) all Hilbert spaces are equivalent, so you don't have to care.
Maybe you mean the range of definition?

I'm studying for my preliminary exam for grad school, and I have a practice problem that gives this Hamiltonian. The first part asks about the symmetries of the hamiltonian. I said energy, angular momentum about the z-axis and linear momentum in the z-direction. The next part asks, "Describe the most convenient Hilbert space to use for describing states and eigenvalues of this system, and describe why."

dextercioby