If uncertainty in position and momentum of an electron are equal, then uncertainty in its velocity is

If uncertainty in position and momentum of an electron are equal, then uncertainty in its velocity is
  1. $\frac{1}{2 m} \sqrt{\frac{h}{\pi}}$
  2. $\frac{1}{m} \sqrt{\frac{h}{\pi}}$
  3. $\sqrt{\frac{h}{\pi}}$
  4. $m \sqrt{\frac{h}{\pi}}$

Solution

$\begin{aligned} & \Delta \mathrm{x} \times \Delta \mathrm{p}=\frac{\mathrm{h}}{4 \pi} \\ & \text { Given, } \Delta \mathrm{x} \times \Delta \mathrm{p} \\ & (\Delta \mathrm{p})^2=\frac{\mathrm{h}}{4 \pi} \Rightarrow \Delta \mathrm{p}=\sqrt{\frac{\mathrm{h}}{4 \pi}} \\ & \Rightarrow \mathrm{~m} \Delta \mathrm{v}=\frac{1}{2} \sqrt{\frac{\mathrm{~h}}{\pi}} \\ & \Rightarrow \Delta \mathrm{v}=\frac{1}{2 \mathrm{~m}} \sqrt{\frac{\mathrm{~m}}{\pi}}\end{aligned}$

Asked in: AP EAMCET 2024 (20 May Shift 2)

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