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?


1 Answer 1


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.... $\endgroup$ Aug 1, 2016 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, 2016 at 17:06
  • $\begingroup$ i guess that helped... $\endgroup$ Aug 8, 2016 at 14:56

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