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?


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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


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.

  • $\begingroup$ 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

Not the answer you're looking for? Browse other questions tagged or ask your own question.