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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?

Qmechanic
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Lamichhane88
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1 Answers1

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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.

Sanya
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