I have a pretty broad set of questions I'm hoping that someone can shed light on how/if they're connected.
In quantum mechanics, the momentum operator is $\hat{p} = -i \hbar \partial_x$. So, it seems that if we have a manifold description of we should be able to associate the momentum operator with the vector field $\partial_x$ that lies in the tangent bundle
Furthermore in Hamiltonian classical mechanics, dynamics occurs on a $2d+1$ dimensional phase space, $d$ for each position and momentum degree of freedom, and 1 for time. I've heard that Hamiltonian dynamics occurs on the tangent bundle of position space. This would add up with the observation that the momentum operator looks like $\partial_x$, and from the fact that tangent vectors intuitively represent velocities/directions. But, I always see that Hamiltonian dynamics is described by the symplectic form $dp \wedge dx$ on the $2d+1$ dimensional space without making reference to any tangent bundle.
So I guess my question would be is there any way to relate the quantum momentum operator, the tangent bundle of position space and the symplectic form from classical mechanics, or do these all stem from something more fundamental?