Q50 / 170 · Manipal (MET) 2018 · Chemistry

If uncertainty in position and velocity are equal, then uncertainty in momentum will be

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

Solution

$\begin{aligned} \because x & =\Delta v \\ \Delta x \times \Delta p & \geq \frac{h}{4 \pi} \\ \Delta x \times m \cdot \Delta v & =\frac{h}{4 \pi} \\ (\Delta v)^2 & =\frac{h}{4 \pi m} \quad(\because \Delta x=\Delta v) \\ \Delta p & =m \cdot \Delta v \\ & =m \sqrt{\frac{h}{4 \pi m}}=\sqrt{\frac{m h}{4 \pi}} \\ \Delta p & =\frac{1}{2} \sqrt{\frac{m h}{\pi}}\end{aligned}$

Hence, the correct answer is option: 'A'.

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