I'm taking a course on quantum mechanics and I'm getting to the part where some of the mathematical foundations are being formulated more rigorously. However when it comes to Hilbert spaces, I'm somewhat confused.
The way I understand it is as follows:
The Hilbert space describing for example a particle moving in the $x$-direction contains all the possible state vectors, which are abstract mathematical objects describing the state of the particle. We can take the components of these abstract state vectors along a certain base to make things more tangible. For example:
- The basis consisting of state vectors having a precisely defined position. In which case the components of the state vector will be the regular configuration space wave function
- The basis consisting of state vectors having a precisely defined momentum. In which case the components of the state vector will be the regular momentum space wave function
- The basis consisting of state vectors which are eigenvectors of the Hamiltonian of the linear harmonic oscillator
It's also possible to represent these basis vectors themselves in a certain basis. When they are represented in the basis consisting of state vectors having a precisely defined position, the components of the above becoming respectively Dirac delta functions, complex exponentials and Hermite polynomials.
So the key (and new) idea here for me is that state vectors are abstract objects living in the Hilbert space and everything from quantum mechanics I've learned before (configuration and momentum space wave functions in particular) are specific representations of these state vectors.
The part that confuses me, is the fact that my lecture notes keep talking about the Hilbert space of 'square integrable wave functions'. But that means we are talking about a Hilbert space of components of a state vector instead of a Hilbert space of state vectors?
If there is anybody who read this far and can tell me if my understanding of Hilbert spaces which I described is correct and how a 'Hilbert space of square integrable wave functions' fits into it all, I would be very grateful.
Best Answer
This is a good question, and the answer is rather subtle, and I think a physicist and a mathematician would answer it differently.
Mathematically, a Hilbert space is just any complete inner product space (where the word "complete" takes a little bit of work to define rigorously, so I won't bother).
But when a physicist talks about "the Hilbert space of a quantum system", they mean a unique space of abstract ket vectors $\{|\psi\rangle\}$, with no preferred basis. Exactly as you say, you can choose a basis (e.g. the position basis) which uniquely maps every abstract state vector $|\psi\rangle$ to a function $\psi(x)$, colloquially called "the wave function". (Well, the mapping actually isn't unique, but that's a minor subtlety that's irrelevant to your main question.)
The confusing part is that this set of functions $\{\psi(x)\}$ also forms a Hilbert space, in the mathematical sense. (Mumble mumble mumble.) This mathematical Hilbert space is isomorphic to the "physics Hilbert space" $\{|\psi\rangle\}$, but is conceptually distinct. Indeed, there are an infinite number of different mathematical "functional representation" Hilbert spaces - one for each choice of basis - that are each isomorphic to the unique "physics Hilbert space", which is not a space of functions.
When physicists talk about "the Hilbert space of square-integrable wave functions", they mean the Hilbert space of abstract state vectors whose corresponding position-basis wave functions are square integrable. That is:
$$\mathcal{H} = \left \{ |\psi\rangle\ \middle|\ \int dx\ |\langle x | \psi \rangle|^2 < \infty \right \}.$$
This definition may seem to single out the position basis as special, but actually it doesn't: by Plancherel's theorem, you get the exact same Hilbert space if you consider the square-integrable momentum wave functions instead.
So while "the Hilbert space of square integrable wave functions" is a mathematical Hilbert space, you are correct that technically it is not the "physics Hilbert space" of quantum mechanics, as physicists usually conceive of it.
I think that in mathematical physics, in order to make things rigorous it's most convenient to consider functional Hilbert spaces instead of abstract ones. So mathematical physicists consider the position-basis functional Hilbert space as the fundamental object, and define everything else in terms of that. But that's not how most physicists do it.