This question might sound very silly, so I'm sorry if that's the case. I'll try my best to make my point clear here. Before explaining, just to make clear, I'm not confused because of the Math involved. I've started now to study Functional Analysis, but I have a reasonable background in Math. What I'm confused is what is the overall idea of Quantum Mechanics.
Before I've started the course of Physics I've always heard people saying that Quantum Mechanics was all about describing microscopic phenomena (electrons, atoms and so on) so that we are able to understand the structure of matter.
Since I've started the Physics course some years ago I took some introductory courses in modern Physics and Quantum Mechanics. In those courses the main thing that was stressed were two points:
The need for Quantum Mechanics, i.e. the situations on which Classical Mechanics failed to describe phenomena and predict things, was seem mostly on experiments studying the structure of matter. In other words, the need for Quantum Mechanics was only seem when dealing with microscopic phenomena.
The basic idea upon which Quantum Mechanics is based is the wave-particle duality. So it is seem that particles on these microscopic phenomena behave as waves. Those matter waves have a direct interpretation in terms of probability amplitudes.
In other words, those introductory courses led me to think that Quantum Mechanics is all about dealing with matter waves governed by Schrödinger's equation in order to study microscopic phenomena.
On the other hand, this semester I'm taking a more serious course on Quantum Mechanics. One of the main things that has been stressed up to now is the sharp distiction between wavefunctions and kets and also between the space of functions and the state space.
I've already asked about wavefunctions and kets here and about the space of functions and state space here. I think I got the overall idea: a ket is a state vector. That is, it is one object that encodes all the available information of a system. The main point is that the ket is not a wavefunction, although it can be related to one. In other words, we have abstracted the idea of state contained inside the wavefunction picture.
Although this is quite nice, I see now a gap between that old picture I had about Quantum Mechanics and this new one. I thought Quantum Mechanics was all about wave-particle duality and dealing with matter waves. But now, we are simply talking about abstract states of a system.
More than that, I can't see anymore the connection with the microscopic phenomena. In truth, this language of "abstract states of a system" could, IMHO, be used in Classical Mechanics as well. In other words, things seem so general, that I'm not yet being able to connect with what already learned before. In truth, if someone asked "what Quantum Mechanics is all about?" today I would be unsure of what to say.
Considering all this, my question (which I believe to be quite silly) is: what Quantum Mechanics is all about? How to bridge the gap between the abstract language of state vectors living in Hilbert Spaces and the more "intuitive" picture of wave-particle duality and microscopic phenomena?