# Scope of Hilbert Space of a System?

1. Dec 3, 2011

### referframe

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 Ha and system B might be described by Hilbert Space Hb.

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?).

2. Dec 3, 2011

### Fredrik

Staff Emeritus
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.

3. Dec 3, 2011