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.