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This question already has an answer here:

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?

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marked as duplicate by ACuriousMind, knzhou, Gert, Qmechanic Aug 2 '16 at 9:33

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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.

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  • $\begingroup$ i wonder it matters for very precise calculation....i had encountered the one.... $\endgroup$ – Lamichhane88 Aug 1 '16 at 17:03
  • $\begingroup$ @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 ... $\endgroup$ – Sanya Aug 1 '16 at 17:06
  • $\begingroup$ i guess that helped... $\endgroup$ – Lamichhane88 Aug 8 '16 at 14:56

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