Question

A German physicist gave a principle about the uncertainities in simultaneous measurement of position and momentum of small particles. According to the physicist:
It is impossible to measure simultaneously the position and momentum of small particle with absolute accuracy or certainty. If an attempt is made to measure any of these two quantities with higher accuracy, the other becomes less accurate. The product of the uncertainty in position $\left(\Delta x\right)$ and uncertainty in position $\left(\Delta p\right)$ is always constant and is eual to or greater than $\frac{h}{4\pi }$, where $h$ is Plank's constant i.e. $\left(\Delta x\right)\left(\Delta p\right)\ge \frac{h}{4\pi }$.

Uncetainty in postion is twice the uncertainty in momentum, uncertainty in velocity is:

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

Answer 1

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

According to the question,
$\Delta x=2\Delta p$
$\Delta x.\Delta p=\frac{h}{4\pi }\Rightarrow 2(\Delta p{)}^2=\frac{h}{4\pi }$
$\Rightarrow \Delta p=\frac{1}{2\sqrt{2}}\sqrt{\frac{h}{\pi }}$
$\Rightarrow \Delta v=\frac{1}{2\sqrt{2}m}\sqrt{\frac{h}{\pi }}$