The "throwing a stone" problem is one of the most elementary classical mechanics problem a student encounters when starts learning physics.
Given an initial position $\mathbf{x}_0$ and an initial velocity $\mathbf{v}_0$, find the trajectory of an object, given a force field $\mathbf{F}(\mathbf{x},\mathbf{v},t)$. For example, $\mathbf{F}=\mathbf{g}=(0,0,-g)$, assuming a homogenous newtonian gravitational field.
I wish to understand the classical limit of QM, so I am curious as to how one may solve such a problem in QM. I am specifically interested in the case when the "stone" is considered of course, point-like, but as a macroscoping object, whose mass is close to that of an actual stone.
Basically, my line of thought is, even if such a problem is difficult to solve directly, since, based on our current understanding of the world, it is fully quantum-theoretic in nature, classical problems in theory, should be soluble in quantum mechanical ways.
Of course, I am aware of Schrodinger's and Heisenberg's equations, I am more interested in specific solutions, since the main thing that's unclear to me here is the initial-value problem.
The classical equation can be solved uniquely given an initial position and an initial velocity.
However, if we switch to QM, this method breaks down. For a stone to be thrown, from point $\mathbf{x}_0$, with velocity $\mathbf{v}_0$, one would have to assume that an inital state $|\psi_0\rangle$ for the stone is given, which is simultaneously a position and momentum eigenstate: $$ \hat{x}_i|\psi_0\rangle=x_i|\psi_0\rangle \\ \hat{p}_i|\psi_0\rangle=p_i|\psi_0\rangle, $$ but this is, of course, impossible.
I guess the classical limit arises from the fact that $\hbar$ is very small compared to any action/angular momentum dimension quantity that appears in this problem, and as such, the $\hat{x}$ and $\hat{p}$ operators "nearly commute", hence there are states which are "nearly position eigenstates" and "nearly momentum eigenstates" simulataneously.
I, however, have absolutely no idea how to formulate these "nearlies" mathematically.
Question: How to treat mathematically the quantum mechanical problem of throwing a macroscopic stone? My main motivation is to show through a specific example that the behaviour quantum mechanics ascribes to a macroscopic object is effectively the same as classical behaviour, however I have no idea how to treat the initial value problem. How can we determine the initial state of the stone?
Note: I am aware that the classical-quantum mechanical correspondance can be shown the most easy way using path integrals, but I have always gotten the impression that it was a quite general procedure, not related to any specific problem. I am not interested in answers in that direction. I am only interested in finding a suitable initial state for this stone.
