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Imagine a billiard table that's is covered we can't see what's happening under the cover.

Now imagine we throw in a ball whose throw in time, mass, size, position and velocity is unknown.

To measure it we shoot in balls of known mass, size and position and velocity.

And all the measurement we can do is listen to the sounds of balls hitting the walls. We can measure the exact position and time a ball hit the wall, and that's all.

Using these measurements only can we find out the throw in position and the momentum of the original unknown ball we threw in?

Note this setup is very much like when we try to measure something using a particle beam, all we can see is the hits on the detector, but not the particles themselves.

The central point of the question is whether something like the uncertainty principle can arise in a classical setup.

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  • $\begingroup$ You'll need to throw a good amount of test balls to make sure you hit the goal several times. But chaos theory says that, over time, you need increasingly good resolution of your measurements to keep the initial conditions known to a fixed resolution. If you add that your balls are not perfect and all collisions are not perfect, you'll have at some point a knowledge of the problem worse than the size of the table. Anyways eventually you always reach the quantum limit $\endgroup$
    – fffred
    Commented Mar 14, 2016 at 23:32
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    $\begingroup$ You've described a very complicated problem. Certainly there is information contained in the fact that after the original ball and the 1st ball after that collide that the first ball hits the wall at a certain place and time - but it is incomplete information. Throwing further balls in and recording wall impact and times provide further constraints. Feed all that data into a supercomputer and, after enough balls, my feeling is that there should be enough info in principle to fix the original ball's position and momentum fairly accurately assuming that everything is fully deterministic. $\endgroup$
    – user93237
    Commented Mar 14, 2016 at 23:32
  • $\begingroup$ @fffred "But chaos theory says that, over time, you need increasingly good resolution of your measurements to keep the initial conditions known to a fixed resolution." - That's a good point. Would help a lot if we could, for example, choose the "probing" balls to have infinitesimally small masses compared to the original ball so that the act of measuring the state of the original ball with each new ball collision would have an infinitesimal effect on the state of the original ball. $\endgroup$
    – user93237
    Commented Mar 14, 2016 at 23:38

2 Answers 2

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In this situation, in principle, yes we should be able to predict the motion of the billiard balls given enough information about the setup. The balls are classical, even though we can't look at them with our eyes, we can still listen to the sound of them hitting the sides of the table. Even though these sounds are the "measurements", the measurements don't change the state of the system, or collapse it into a particular state. Our uncertainty in this situation would only be a reflection of the amount of missing information we have about the system. Once the system was specified, it would remain so for all time.

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It is not clear what question you are asking. Are you asking:

A. According to classical mechanics, can we determine the exact position and momentum of the ball?

Or are you asking

B. Is the answer we get from classical mechanics perfectly accurate?

Re A, this seems like a hard question (per Samuel Weir's comment) though I wouldn't be shocked to learn that some clever insight makes it easy. Certainly it is not true that the uncertainty principle intervenes, because, according to classical mechanics, position does in fact commute with momentum.

Re B, the answer is: No, the answer we get from classical mechanics is not perfectly accurate. Nor, as far as I know, is the answer we get from classical mechanics to any other question regarding the real world. Nor, as far as I know, is the answer we get from any physical theory to this or any other question about the real world.

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