Question

If uncertainty in position and momentum of a particle are numerically equal, then the uncertainty in the speed of the particle should be:

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

Answer 1

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

We know from Heisenberg's uncertainty principle:
$\Delta x.\Delta p\ge \frac{h}{4\pi }...\left(i\right)$
where $\Delta x:\text{uncertainty in position}\Delta p:\text{uncertainty in momentum}h:\text{Planck's contant}$

Since $\Delta p=\Delta \left(m\text{v}\right)$
$\Delta p=m\Delta \text{v}\because m\text{is constant}$
Using equation (i), we get
$(\Delta p{)}^2\ge \frac{h}{4\pi }\because \Delta x=\Delta p$

$\Rightarrow \Delta p\ge \sqrt{\frac{h}{4\pi }}$

$\Rightarrow \Delta \text{v}\ge \frac{1}{m}\sqrt{\frac{h}{4\pi }}=\frac{1}{2m}\sqrt{\frac{h}{\pi }}$