Question

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

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

Hint:

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Heisenberg's uncertainty principle: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$.
Step 2: Detailed Explanation:
$\Delta x = \Delta v$. $\Delta p = m \Delta v = m \Delta x$. So $(\Delta x)^2 \geq \frac{h}{4\pi m}$, $\Delta x = \sqrt{\frac{h}{4\pi m}}$. Then $\Delta p = m \Delta x = m \sqrt{\frac{h}{4\pi m}} = \sqrt{\frac{mh}{4\pi}} = \frac{1}{2}\sqrt{\frac{mh}{\pi}}$.
Step 3: Final Answer:
$\Delta p = \frac{1}{2}\sqrt{\frac{mh}{\pi}}$.