I have the mathematical background to compute things in quantum mechanics. If you give me a standard QM problem to solve I could probably do it. However, I have an extreme lack of understanding of why the postulates of QM are set up the way they are - that the complete description of a particle is given by an element of a Hilbert space and that there are these operators that represent position, momentum, angular momentum, and so on that seem like they were pulled out of a hat. Now the basis for any physical theory is grounded in experiment, so I would like to know the observations that establish the way quantum mechanics is set up. Thought experiments with results would be fine, I don't need the actual data from real experiments. In other words, I would like to understand why the QM formalism is set up the way it is, from a mathematical modeling perspective.