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The Heisenberg uncertainty principle states that

$$\Delta x\Delta y\geq\hbar/2$$

Since the magnitude of $\hbar$ is $10^{-34}$ we could measure both $x$ and $p$ with an uncertainty magnitude of $10^{-17}$, but we could also measure $x$ down to an uncertainty magnitude of $10^{-25}$, causing $\Delta p$ to be bounded by $10^{-9}$. For all I know, we could also have $\Delta x$ about $10^{-100}$, at least in principle, thus causing $\Delta p$ to be no better than $10^{66}$, that is, the values the momentum can take are wildly spread out and nearly-completely uninformative.

Now the question: what is the smallest $\Delta x$ (or $\Delta p$) ever achieved in experiments? Is there any theoretical lower bound, except for $0$?

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There are theoretical limits, they happen in regimes where classical (as in non-relativistic) quantum mechanics, respective special relativistic quantum mechanics break down.

The first is Zitterbewegung, an effect of relativistic quantum mechanics, which predicts that the position of a particle jitters with a frequency of $\frac{2mc^2}h$ with an amplitude of the order of the Compton wavelength of the particle.

When the momentum uncertainty becomes to big there will be quantum gravitational effects (semi-classically the particle will have so much momentum uncertainty that it can reach energies where it forms a black hole). Since we do not know the theory of quantum gravitation, we do, however not know exactly what happens if we approach this limit. Intuitively, this limit is reached at an uncertainty of the order of the Planck length.

On the momentum side, the problem is, that the measurement apparatus must grow with the precision of the measurement, here we again run in a limit with classical quantum mechanics (where no limit is set on the propagation of effects) and in a relativistic setting we will get problems with synchronizing the measurement apparatus as it grows (I know of no detailed analysis of this, hints are welcome). The latest time we run into trouble is when our measurement apparatus reaches the size of the observable universe.

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  • $\begingroup$ So let's focus on position and consider an electron. Since the Compton wavelength is given by $\frac{h}{mc}$, $m$ is about $10^{-31}$, and $c$ is about $10^8$, the jitter amplitude should be about $10^{-11}$. Assuming we are in the electron system, we should localize the electron in a spherical spatial region with a radius of $10^{-11}$, and we can't do any better because the electron is jittering inside there. Did I understand well? If yes, why can't we localize the electron during the jittering just like we do inside the orbital? $\endgroup$ – Vanni Rovera May 7 '18 at 19:50
  • $\begingroup$ Yes, if you have a very fast sensor, then you can resolve the position at a short moment of time (that is within the zitterbewegung), but then you get the problem of a high energy uncertainty. The qunatum-gravitation limit, however, is real, and we do not know the solution (quantum loop gravitation and string theories do make some predictions, but we don't know which theory is physical). $\endgroup$ – Sebastian Riese May 7 '18 at 20:06
  • $\begingroup$ Ok, you provided me very interesting theoretical insights, thank you very much. And what about actual measurements? Beside the fact that we never reached a $\Delta x$ so little for quantum gravitational effects to come into play, what is the smallest $\Delta x$ we have reached? Any clue about? $\endgroup$ – Vanni Rovera May 7 '18 at 20:43
  • $\begingroup$ Well, if you do not specifically look for small objects, then we get the best position measurements for macroscopic objects (which of course still follow quantum mechanics) since $p = mv$ and so for large masses we can measure more precisely without transferring large impulses. The best position measurement with the highes precision for macroscopic objects is probably the interferometric measurement of the distance of the test masses in the Advanced LIGO detectors (which is on the order of $10^{-18}\,m$ according to Wikipedia, but is not an absolute position measurement). $\endgroup$ – Sebastian Riese May 7 '18 at 21:23
  • $\begingroup$ A side note: We of course run into perhaps fundamental engineering problems long before we get even close to the theoretical limits (since the energy transfer needed to make the measurement increases dramatically). $\endgroup$ – Sebastian Riese May 7 '18 at 21:25

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