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The most common form of Heisenberg's uncertainty principle I've seen online is $$ \Delta x \Delta p ~\geq~ \dfrac{\hbar}{2}.$$

However, I also regularly see $$\Delta x \Delta p ~\geq~ \hbar. $$

Sadly, I used the latter one in a project recently and I'm afraid it's incorrect. Obviously the upper one one is true if the latter one is, but why are these two versions used instead of only one of them?

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Heisenberg initially published an approximation: $\Delta x \Delta p \gtrsim h$ and only later was it properly proven to be $\Delta x \Delta p \geq \frac{\hbar}{2}$.

Note that they only differ by $4 \pi$.

There are some details with statistics that get glossed over. In the proof it isn't really the $\Delta$ (because what does that really mean without any sort of confidence interval anyways?) but rather the standard deviation $\sigma$.

See the History section of Wikipedia on the principle for more details.

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    $\begingroup$ The last point is actually something that has always bothered me tremendously. I'm sometimes not sure whether the 'popular' usage is even correct, when it should be the standard deviation that appears in the equation. $\endgroup$
    – Danu
    Commented Mar 12, 2014 at 22:32
  • $\begingroup$ Well, in these formulae, $\Delta x$ means the standard deviation $\sigma_x$ or whatever is your preferred other notation. $\endgroup$
    – Luboš Motl
    Commented Mar 13, 2014 at 6:08

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