What exactly are the axioms of non-relativistic QM of one spin-0 particle? The mathematical model we're working with is the Hilbert space [itex]L^2(\mathbb R^3)[/itex] (at least in one formulation of the theory). But then what? Do we postulate that observables are represented by self-adjoint operators? Do we say that a measurement of an operator [itex]A[/itex] on a system prepared in state [itex]|\psi\rangle[/itex] yields result an and leaves the system in the eigenstate [itex]|n\rangle[/itex] with probability [itex]|\langle n|\psi\rangle|^2[/itex]? Then how do we handle e.g. the position and momentum operators, which don't have eigenvectors? Can the problem of unbounded operators be solved without the concept of a "rigged Hilbert space"? Is it easy to solve when we do use a rigged Hilbert space? What is a rigged Hilbert space anyway? I think I brought this up a few years ago, but apparently I wasn't able to understand it even after discussing it. I think I will this time, because of what I've learned since then. Don't hold back on technical details. I want a complete answer, or the pieces that will help me figure it out for myself.