I've been studying quantum mechanics and classical mechanics for a little while now, and I still don't feel as though I fully understand the motivation for some of our choices in Heisenberg mechanics. For example, it clearly isn't a coincidence that the classical observables (functions of coordinates and their conjugate momenta) and the quantum observables (Hermitian operators) seem to form analogous Lie algebras with the Poisson bracket and commutator respectively. But it isn't clear to me why this is true. Is there some deep meaning contained in this statement? Or is it more indicative of the fact that in constructing a quantum model of the universe we took substantial inspiration from our intuition and previous study of classical mechanics?
Along these same lines, what motivates the move from classical functions on phase space to Hermitian operators? I understand why operators corresponding to observables must be self-adjoint (the eigenvalues must be real), but I don't understand what motivates the move to operators in general. Why would we expect that operators on a Hilbert space would give physical predictions? Part of my confusion here may also come from the fact that it isn't entirely clear to me what exactly these operators do in all cases. For example, I get that $\langle \psi | \hat{x} | \psi \rangle$ corresponds to the expected position of a particle in state $|\psi\rangle$, but it's much less obvious what the $\hat{x}$ operator does to a state in general. In some cases (such as $J_\pm$ when considering angular momentum), it's clear what the operator does to a state (raises or lowers eigenstates of $J_z$), but in all these cases the operator is non-Hermitian. Perhaps the answer to this question is simply that the model gives accurate predictions and so we use it, but I'm wondering if there's a better way to think about these things.
