Question

Uncertainity in position is twice the uncertainity in momentum uncertainity in velocity is :

• A. $\frac{h}{\pi }$
• B. $\frac{1}{2m}\sqrt{\frac{h}{2\pi }}$
• C. $\frac{1}{2\sqrt{2m}}\sqrt{h}$
• D. $\frac{h}{4\pi }$

Answer 1

The correct option is B $\frac{1}{2m}\sqrt{\frac{h}{2\pi }}$

From Heisenberg’s uncertainty principle,
$\Delta x.\Delta p\ge \frac{h}{4\pi }$
(or) $\Delta x.m.\left(\Delta v\right)\ge \frac{h}{4\pi }$
where h = Planck’s constant
Δx = uncertainty in position
Δv = uncertainty in velocity
Δp = uncertainty in momentum
m = mass of a particle

Given: $\Delta x=2\Delta p$
Putting the values, we get
$2\Delta p.\left(\Delta p\right)\ge \frac{h}{4\pi }=2(\Delta p{)}^2$
We know that, $\Delta p=m\Delta v$
$\Delta v=\frac{1}{2m}\sqrt{\frac{h}{2\pi }}$