# Heisenberg's uncertainty principle - Planck's (reduced) constant divided by $2$ or not? [duplicate]

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?

## marked as duplicate by Brandon Enright, Qmechanic♦Mar 13 '14 at 0:58

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 $\sigma$.
• Well, in these formulae, $\Delta x$ means the standard deviation $\sigma_x$ or whatever is your preferred other notation. – Luboš Motl Mar 13 '14 at 6:08