It has long irked me that quantum mechanics appears to be so bound up in mysticism, if that isn't too strong a word.
It shouldn't be like that: it should be possible to follow the reasoning all the way back to some intuitively obvious axioms - "intuitively obvious" , at least if you invest enough in understanding the mathematics.
The way I was introduced to quantum mechanics years ago, it went something like: "First, you calculate the Hamiltonian, then mumble mumble and then you have the Hamiltonian Operator". I have never come across a good, intuitive explanation for why we should expect the apparently magical process of translating the classical Hamiltonian to an operator on (a dense subset of) a Hilbert space, $L^2(R^d)$, to work.
Where did the idea come from, that the physical state of a system should be described by a complex-valued "wave-function" and the observables are differential operators that act on wave-functions - and what was the intuitive reasoning behind? What is the mathematical justification - other than "it seems to work"?
The maths don't scare me - I have some understanding of manifolds, bundles, Lie algebras and -groups, I understand that wave-functions are sections of the base manifold into the complex line bundle and that you can, locally, consider them simply to be complex valued functions.