Question

A particle of mass one microgram is confined to move along one direction (x-axis) within a region $1$ mm in extension. What is the uncertainty in its velocity?

• A. $3.313\times {10}^{-20}cm$ per sec
• B. $5.012\times {10}^{-20}cm$ per sec
• C. $8.325\times {10}^{-20}cm$ per sec
• D. $5.27\times {10}^{-21}cm$ per sec

Answer 1

Answer: (D)

$5.27\times {10}^{-21}cm$ per sec

$m=1{\mu }_g={10}^{-6}g={10}^{-9}kg.$

$\Delta x=1mm={10}^{-3}m$

uncertainity principle is $\left(\Delta x\right)\left(\Delta p\right)\ge \frac{h}{4\Pi }$

$\left(\Delta x\right)\left(m\right)\left(\Delta V\right)\ge \frac{h}{4\Pi }$

$\Rightarrow \Delta V=\frac{h}{\left(4\Pi \right)\left(m\right)\left(\Delta x\right)}$

$=\frac{6.628\times {10}^{-34}}{\left(4\Pi \right)\left({10}^{-9}\right)\left({10}^{-3}\right)}$

$=5.27\times {10}^{-23}m$

$=5.27\times {10}^{-21}cm$

uncertainity in velocity is $5.27\times {10}^{-21}$per sec.