Scope of Hilbert Space of a System?

[LarryS]

In QM a system is represented by a Hilbert Space rather than a classical Phase Space. So, system A might be described by Hilbert Space H_a and system B might be described by Hilbert Space H_b.

Mathematically, Hilbert Spaces are many things, but the first thing they are, at the most fundamental level, is a set (of continuous, complex-valued, square-integrable functions, etc.).

My Question: In QM what defines the scope of the Hilbert Space for a specific system? (What elements are in its Hilbert Space set?).

As always, thanks in advance.

 

[Fredrik]

I'm not sure I understand the question. All separable infinite dimensional Hilbert spaces are isomorphic to the Hilbert space $L^2(\mathbb R^3)$ whose members are equivalence classes of square integrable functions. So I guess that's one answer.

If you meant "what are the real-world thingys that correspond to the members of that set?" you could say that each vector corresponds to an equivalence class of preparation procedures. However, if u and v are vectors, c is a complex number and u=cv, then u and v correspond to the same equivalence class of preparation procedures. So it's better to say that the 1-dimensional subspaces correspond to equivalence classes of preparation procedures. But even this is a bit unsatisfactory, since there are preparation procedures that don't correspond to state vectors.

 

[LarryS]

I'm not sure I understand the question. All separable infinite dimensional Hilbert spaces are isomorphic to the Hilbert space $L^2(\mathbb R^3)$ whose members are equivalence classes of square integrable functions. So I guess that's one answer.

If you meant "what are the real-world thingys that correspond to the members of that set?" you could say that each vector corresponds to an equivalence class of preparation procedures. However, if u and v are vectors, c is a complex number and u=cv, then u and v correspond to the same equivalence class of preparation procedures. So it's better to say that the 1-dimensional subspaces correspond to equivalence classes of preparation procedures. But even this is a bit unsatisfactory, since there are preparation procedures that don't correspond to state vectors.

Interesting reply.

I was thinking that one possible answer (for non-relativistic wave mechanics) might be "All solutions to the Schrödinger equation for that system". The set of all solutions of any differential equation is closed under scalar multiplication and vector (functions) addition.