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I was originally surprised to see that,

$$\Delta x \cdot \Delta p \gt {{\hbar} \over 2}$$

But, then I realized that $\hbar/2=5.27 \cdot 10^{-35}$. According to this other question, the smallest length ever measured was on the order of $10^{-18}$. Of course at that point, I bring the Planck length into consideration. It's order of magnitude is $10^{-35}$. I was quite shocked to see that the uncertainty is so small compared to this unit and our practical probing unit.

My question is this. How relevant is the Heisenberg Uncertainty Principle in the lab? Does it really limit what can be probed at a practical level, or is it a theoretical limit still? In addition, if the Planck scale is shown to be the shortest meaningful length, is having a limit on uncertainty only 5 times larger than that fundamental length really that inconvenient?

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    $\begingroup$ Your numbers lack units. Plug in units and typical length and momentum scales and you see it's quite relevant at e.g. the scales we have for atoms. $\endgroup$
    – ACuriousMind ♦
    Commented Aug 16, 2015 at 0:40
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    $\begingroup$ In a hydrogen atom the kinetic energy is on the order of 5 eV, from $ T = p^2/2m$ we get that the typical momentum is about 7 keV/c. On the other hand $\hbar/2a_0 \approx 12$ keV/c where $a_0$ is the Bohr radius. Since these quantities are of similar magnitude the Heisenberg principle is certainly very relevant. $\endgroup$
    – Robin Ekman
    Commented Aug 16, 2015 at 1:31
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    $\begingroup$ It is also important to note that the Heisenberg principle is neither an axiom of quantum mechanics nor a statement about measurement uncertainty. It's a consequence of the founding principles of quantum mechanics so that you can't have the equations of QM without it. It is important precisely when quantum effects are. It is not always straightforward to determine the latter though, you need to make estimates like the above, which are not always obvious. $\endgroup$
    – Robin Ekman
    Commented Aug 16, 2015 at 1:39
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    $\begingroup$ Anna and Robin, I am pretty sure that the founders of QM would disagree with your claim that the uncertainty principle is not an axiom, or that it is not a postulate of quantum mechanics. It is, and it is arguably the most important new one. Sources on the Internet agree, too, for example, en.wikipedia.org/wiki/Copenhagen_interpretation says that HUP is a detailed formulation of "another axiom" of QM. And various books sensibly says that Heisenberg "postulated" HUP, see abarim-publications.com/HeisenbergUncertaintyPrinciple.html Why do you find "axiom" or "postulate" bad? $\endgroup$
    – Luboš Motl
    Commented Aug 16, 2015 at 6:39
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    $\begingroup$ @LubošMotl, the Wikipedia page says that the "incompatibility [of conjugate variables] is expressed quantitatively by Heisenberg's uncertainty principle", but I think the proper statement here is that it is the commutation relations that express incompatibility and are postulated or axiomatic. By finding a representation of the CRs you can do all of quantum mechanics, e.g. calculate the hydrogen spectrum, but just the HUP is not enough for that. $\endgroup$
    – Robin Ekman
    Commented Aug 16, 2015 at 15:56

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In a hydrogen atom the kinetic energy is on the order of $8$ eV. From $T = \frac{p^2}{2m}$ we get that the typical momentum is about $3$ keV/c ($m = 511$ keV/$c^2$). On the other hand $\hbar/(2a_0)\approx 1.2$ keV/c where $a_0$ is the Bohr radius, which is about the size of a hydrogen atom. Since these quantities are of similar magnitude the Heisenberg principle is certainly very relevant.

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