Q. The uncertainties in the velocities of two particles $A$ and $B$ are 0.05 and 0.02 $ms^{- 1}$ respecively. The mass of $B$ is five times to that of mass $A$. What is the ratio of uncertainties $\left( \frac{\mathrm{\Delta}x_{A}}{\mathrm{\Delta}x_{B}} \right)$ in their positions?
A) 2
B) 0.25
C) 4
D) 1
Explanation
(a)
According to Heisenberg
$\mathrm{\Delta}x \times m \times \mathrm{\Delta}v = \frac{h}{4\pi}$
where, $\mathrm{\Delta}x$=uncertainty in position.
$m$=mass of particle
$\mathrm{\Delta}v$=uncertainty in velocity.
According to question
$\mathrm{\Delta}x_{A} \times m \times 0.05 = \frac{h}{4\pi}\ \ \ \ \ \ \ \ldots(i)$
$\mathrm{\Delta}x_{B} \times 5m \times 0.02 = \frac{h}{4\pi}\ \ \ \ \ldots(ii)$
Eq. (i) divided by Eq. (ii), then
$\frac{\mathrm{\Delta}x_{A} \times m \times 0.05}{\mathrm{\Delta}x_{B} \times 5m \times 0.02} = 1$
or $\frac{\mathrm{\Delta}x_{A}}{\mathrm{\Delta}x_{B}} = 2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.