Heisenberg's uncertainty principle states that we cannot know the position and momentum of subatomic particles simultaneously...but what exactly is the boundary of size of such a particle? Does such a boundary even exist or is it simply defined as all particles in the standard model?
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asked Jul 9, 2019 at 18:03
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3$\begingroup$ HUP is still valid for billiard balls and planets, just not as useful. $\endgroup$– safesphereCommented Jul 9, 2019 at 18:19
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$\begingroup$ What do you mean ?We can know both their momenta and positions simultaneously.. $\endgroup$– Schwarz KugelblitzCommented Jul 9, 2019 at 18:20
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1$\begingroup$ No, you cannot, but it's irrelevant for macroscopic objects, because your margin of error is much worse. $\endgroup$– safesphereCommented Jul 9, 2019 at 18:30
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2 Answers
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You can use uncertainy principle for everything, what determines the error of measurement though is the $mass$ of an object not its size. In the classical limit (a very rough estimation indeed) we can write HUP like this: $$\Delta x \Delta p\geq \hbar/2 \rightarrow \Delta x \Delta v\geq \hbar/(2m) $$ As the mass increases, the right side of inequality decreases. In the case of macroscopic objects it's safe to assume that it will become zero (i mean just look at $\hbar$ scale) so according to HUP you can measure velocity and poisition of an object simultaneously, without a noticeable error. In the case of microscopic object though, the right side will become big enough to make us believe that if we measure position or velocity, we will "mess with" the other greatly. In other words, momentum of macroscopic objects is big enough (because of mass) that we don't care for errors in scale of $\hbar$.
Do note that this was not a technical answer, but it should be good enough for laymans in my opinion. The truth is you should solve Schrödinger equation for macroscopic objects and find true values of $\Delta x$ and $\Delta p$. You will see that both of them will be tiny (most of the times at least) due to the mass, or other classical limits.
Also you might find this helpful: https://en.wikipedia.org/wiki/Correspondence_principle
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There is no limit. It valid for all objects irrespective of their sizes and shapes.
For objects of practical size this uncertainty is irrelevant as measurement error is much greater than the uncertainty. For example consider a ball of mass 1kg moving at 1 m/s. Using the uncertainty principle we get uncertainty in position to be of the order of $10^{-36}m$. This is such a small quantity that the error in measurement will be order of magnitudes greater and therefore the uncertainty principle is irrelevant.
Uncertainty principle is valid for all objects but only relevant for objects with very small momentum.
