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Suppose we have two small masses both with the same velocity. Their mass is $m$ and velocity is $v$ and $-v$ respectively. So they collide at some point and the collision is a perfect elastic one. We mark this point as the origin.

Now we can know the velocity of any particle which will be $v$. So its momentum will be $mv$. Now after $t$ seconds its displacement from the origin will be $vt$.

Now we know the momentum and position of the small mass at the same time. Isn't it the violation of the uncertainty principle? Where am I wrong?

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  • $\begingroup$ As of right now youve stated a classical mechanics problem, is this system supposed to be quantum mechanical? If so you should indicate that as in classical mechanics there is no violation here since the uncertainty principle has to do with quantum interactions. $\endgroup$ – Triatticus Jul 17 '18 at 12:13
  • $\begingroup$ @Triatticus Excuse me but why isn't my problem quantum mechanical instead of classical mechanical? $\endgroup$ – Theoretical Jul 17 '18 at 12:37
  • $\begingroup$ @AsifIqubal The phrase "small masses both with the same velocity" doesn't suggest a system in which quantum effects are noticeable. $\endgroup$ – user191954 Jul 17 '18 at 12:53
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    $\begingroup$ Whay is a collision? Having both particles at the same place? That would mean that their position is a Dirac delta function, and so their momenta are completely undefined. $\endgroup$ – FGSUZ Jul 17 '18 at 12:59
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    $\begingroup$ Also, if you say that their initial momentum is $mv$, their position is uncertain. You can think that it is "everywhere" until you measure it; the probability density is uniform. Hence you don't know if it's colliding... $\endgroup$ – FGSUZ Jul 17 '18 at 13:00
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In quantum mechanics, a particle is not located at a single place. Rather, the particle has a state that gives the probability of finding it in any given region. Likewise the particle doesn't have a single momentum. Rather, the state gives probabilities for finding the particle has momentum in any given range of momenta.

The uncertainty principle constrains the probability distribution of the momentum of the particle given the distribution for its position and vice versa. Your thought experiment makes assumptions that are false in quantum theory and also in reality. Those assumptions happen to be a good approximation for the sort of thing you see in everyday life, but they have been broken in many experiments that test quantum theory.

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  • $\begingroup$ Is Newton's momentum conservation law valid for quantum interaction? $\endgroup$ – Theoretical Jul 17 '18 at 13:14
  • $\begingroup$ If two particles interact you will not find a state in which their total momentum after the interaction is different from the momentum before the interaction. Quantum mechanics contradicts Newton's laws and so the specific laws he wrote down don't hold in quantum mechanics except as an approximation in some circumstances. For example, in Newtonian mechanics there is only version of each particle and that is not true in quantum mechanics. $\endgroup$ – alanf Jul 17 '18 at 14:40

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