Is ∣E,l,m⟩ Always a Tensor Product in Hilbert Space?

[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?)

 

[Nate810]

The state ∣E,l,m⟩ is not necessarily the tensor product of ∣E⟩, ∣l⟩, and ∣m⟩. It depends on how they are related. In some cases, these states may be related by an operator, in which case the state would take the form of an operator acting on the product of the three states, rather than the product itself. When given a Hamiltonian, the Hilbert space you are working in is determined by the boundary conditions of the problem. For example, for the infinite well, the Hilbert space is the space of functions that vanish outside the well. Similarly, for a particle in a box, the Hilbert space is the space of functions that vanish at the walls of the box. In general, you can determine the Hilbert space by looking at the boundary conditions of the problem.As for when you can split up a Hamiltonian into two and solve them separately, this is typically done with separable Hamiltonians. This means that the Hamiltonian can be written as the sum of two separate terms that do not depend on each other, such that it can be solved independently for each term. Once the separate solutions have been found, you can take the tensor product of the solutions to get the solution to the total Hamiltonian.