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?

Thanks in advance.

 

[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?

 

[golnat]

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]

Technically the existing symmetries (such as space-time ones) would help you get a suitable Hilbert space starting from L^(R^n), n = dof. For the uniparticle Hamiltonian in post #1, the Hilbert space would be L^2((0,r),dr) X {e^(im\phi) | m in Z} X L^2 (R,dz).