Hilbert Space and Tensor Products

1. Feb 28, 2012

lackrange

I never thought about this stuff much before, but I am getting confused by a couple of things.
For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a Hilbert Space, but now I am confused by the lingo. If it is a tensor product, then is there a difference between their Hilbert spaces? Given a Hamiltonian, when can you split it up into two and solve them separately and then take the tensor product of the solutions to call it the solution to the total Hamiltonian?

Also, when given a Hamiltonian, how do you know what Hilbert space you are working in? I know wave functions are square integrable, but for instance in the infinite well problem, the eigenfunctions vanish outside the well, so when we say eigenfunctions of an observable are complete, with respect to which Hilbert space? (I know in the infinite well problem it is the space of functions that vanish outside the well, but how do you know in general what your eigenfunctions span?)