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first I'd like to state the postulates as I'm familiar of them for the sake of clearance

1. all the information about a system is represented by ## |\psi\rangle## a member of Hilbert space.

2. the possible measurement results of the system are the eigenvalues of hermitian operators in the Hilbert space, after measurement the state of the system is ##|\phi\rangle## which is the eigenvector corresponding to the measurement result.

3. the probability of measuring a result ##\lambda## is ##\langle\phi|\psi\rangle^2## where ##\langle\phi|## is the complex conjugate of the eigenvector corresponding to the eigenvalue ##\lambda## (this is assuming normalization, if not the vectors should be normalized)

4. the time evolution of the system is governed by the Schroedinger equation.

after we got that out of the way here are my questions:

1. in hindsight, representing the information of a system by some general mathematical object instead of forcing it to be the path of a particle (as is the case in classical mechanics) seems like the more logical route, what I don't get is why should the group of these objects form a Hilbert space, what physical intuition is there to support this notion?

2. why the requirement for the results to be eigenvalues of operators? more importantly what do the operators represent, is it the actual act of measurement?

3. again what is the intuition behind the measurement being probabilistic, I mean how from the interference of electrons do you come to the conclusion that the measurement should be a probabilistic result?

lastly, this is not a question but more of my point of view on QM and I would be delighted if someone could give me some direction to whether this point of view is sensible or whether is contradicts some aspect of QM, so from my point of view QM seems to be a theory about what information is available to us more so than being a description of the physical system under consideration, that is it seems to state ( in postulates 1+2) that there is the info I have on the system, I measure it and then the info I have of the system is the measurement result, as such for the purposes of further calculations/predictions the system is now in the state representing the information I've obtained, this does not imply anything about what happens "under the hood" or what physically changed in the system if anything, it's just the information I have is updated.

I know this is a long post with some questions that might seem stupid to some of the people with more familiarity with QM than I, so I'd like thank anyone who bothered to read this far and would be truly grateful to anyone who takes the time to reply.