# Confusion regarding factor 2 in uncertainty principle [duplicate]

I have encountered a problem regarding Heisenberg's uncertainty principle. Somewhere I found the expression

$$\Delta (x) \Delta (p) \geq \hbar$$

Whereas in some other places,

$$\Delta (x) \Delta(p) \geq \frac{h}{4\pi}$$

Here the second expression yields different expression than the first one. Why are the two expressions of uncertainty principle different? Which one is the correct form?

• – Sanya Aug 1 '16 at 16:54
• This principle is almost never used for exact calculations but rather to estimate orders of magnitude so you can just as well write $\Delta x \Delta o \geq \pi^{0.3} \hbar$ and it will still be valid. – Blazej Aug 1 '16 at 22:07
• Possible duplicates: physics.stackexchange.com/q/69604/2451 , physics.stackexchange.com/q/103208/2451 and links therein. – Qmechanic Aug 2 '16 at 9:31

## 1 Answer

See Ballentine: Quantum mechanics p. 224 (chapter 8): the uncertainty principle applied to position and momentum yields, in 1 dimension: $$\Delta_X \Delta_P \geq \frac{\hbar}{2}=\frac{h}{4 \pi}$$ But the other version is often used when it's mainly about an estimation of magnitude and the factor of $0.5$ does not really matter.

Edit: On Wikipedia it says that the first unproofed heuristically motivated version of the principle given by Heisenberg himself was without the factor of $0.5$. Maybe that is a historical explanation.

• i wonder it matters for very precise calculation....i had encountered the one.... – Lamichhane88 Aug 1 '16 at 17:03
• @Lamichhane88 well, if you want to be precise - above formulation is what can be proofed mathematically; open any book on theoretical QM and search for uncertainty principle ... – Sanya Aug 1 '16 at 17:06
• i guess that helped... – Lamichhane88 Aug 8 '16 at 14:56