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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.

Bence Racskó
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  • see my answer to a similar question here https://physics.stackexchange.com/q/330144/ – anna v May 01 '17 at 06:56
  • @annav For the purposes of this question, I am ignoring the fact that a stone is a composite. You can take the stone to be an electron with $m=1 kg$ or whatever and no EM interactions present. I meant "macroscopic" only in the sense that all possible $xp$ products are far greater than $\hbar$. – Bence Racskó May 01 '17 at 07:04
  • You then do not have a quantum mechanical problem. What is the potential that will define the quantum mechanical wavefunction? the gravitational is very very weak . what is throwing the stone? – anna v May 01 '17 at 07:24
  • @annav Is it not possible to encode the initial conditions fully in the state vector at $t=0$? – Bence Racskó May 01 '17 at 07:37
  • throwing a stone involves the classical gravitational potential, which controls the trajectory. A quantum mechanical trajectory is a probability distribution from a wavefunction solution in some potential problem. For example an electron scattering off an electron . then the probability can be calculated given the electromagnetic potential between them. http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/imgpar/feynm2.gif there are initial momenta and directions but also a potential – anna v May 01 '17 at 07:40
  • @annav So you are essentially saying that this problem is only well-posed if there is a potential that gives an initial nudge to the stone? I totally get that usual QM problems are not posed like mine is, but nontheless, situations involving propagating stones exist and since they exist (and are nongravitational in nature), they should be describable by QM.

    Imagine that someone comes up to you spouting quantum mysticism bs about QM making it possible for you to exist in two places at one and such, but this person is mathematically adequate enough to understand basic calculus and linear...

     – Bence Racskó May 01 '17 at 07:47
  • ... algebra. I would like to be able to make a comparision between a very simple classical mechanics problem (throwing a stone) and the corresponding quantum mechanical problem in a way that it is manifest that QM in these scales will simply reduce to classical mechanics. (For the record this is not a situation that has happened to me, but I defo would like to be able to make this argument if ever needed, that's my motivation for this question.) – Bence Racskó May 01 '17 at 07:49
  • it is the density matrix formalism that describes how the classical emerges from the quantum mechanical as I explain in the answer I linked. If there is no potential there is no interaction. a stone with a given momentum would travel to infinity if there were no potential between it and the earth – anna v May 01 '17 at 07:55

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You consider your initial state to be a wavepacket with some uncertainty in both, momentum and position, and as you mentioned, because of the size of the mass, you can make both uncertainties pretty small from a macroscopic point of view. For a 1kg mass you can have uncertainties of $10^{-17}m$ and $10^{-17}m/s$ For position and speed.

The solution can be found here

  • I believe the OP was asking, first, for an explicit example of such a wave packet. (And then, perhaps, for an explicit solution to the Schrodinger equation for that wave packet subjected to a potential due to gravity.) So this doesn't seem to be anything like an answer. – WillO May 01 '17 at 13:26
  • @WillO Yes, that's right, the explicit example of such a wave packet is what I was mainly interested is. A solution would also be nice, especially if I can show that the expectation value follows classical trajectories (but I think that can be shown generally, too) and that the square deviation is very very small, emphasis is on the latter.

    In the process of checking Willy's link, tho.

     – Bence Racskó May 01 '17 at 16:44