Main Question or Discussion Point
Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.
I've never studied physics, but I've read a bit of basic stuff on quantum mechanics and chatted to a few physicists about it and the following question has always bugged me.
Apparently in the standard formalism for quantum mechanics you represent (pure) states as elements of a Hilbert space (or rank one projections on a Hilbert space, I guess, but whatever) and observables as densely defined Hermitian operators on this space. In the finite dimensional case every Hermitian operator has a spanning eigenspace and measuring a state p with respect to some observable A just collapses that state to some eigenstate of A, where the probability of getting any given eigenstate is just the inner product of that eigenstate with p.
However, I'm very hazy as to what happens in the infinite dimensional case. I've seen various sources pretend that it's basically the same as the finite dimensional case but this can't be right as Hermitian operators on infinite dimensional Hilbert spaces need not have any eigenvectors at all. A basic example of this is the position operator acting on L^2(R), the square integrable functions on the reals. This has no eigenvectors, so how does measuring position work? Apparently in the C* formalism you represent pure states as measures and so I guess <vague bullgarbageting> you could determine whether the position of some state lies in a particular range by integrating a suitable function over that range with respect to the state </vague bullgarbageting>. However, surely you don't need anything as abstract as this to define something as simple as measuring position, and even if you do use this formalism I have no idea what happens to the state of the particle as a result of performing this measurement.
Any explanation of this would be greatly appreciated.