Question

If \( \Delta x \) is the uncertainty in position and \( \Delta v \) is the uncertainty in velocity of a particle and both are equal, then the correct expression for uncertainty in momentum for the same particle is:

• A. \( \frac{1}{4} \sqrt{\frac{mh}{\pi}} \)
• B. \( \frac{1}{3} \sqrt{\frac{mh}{2\pi}} \)
• C. \( \frac{1}{2} \sqrt{\frac{mh}{\pi}} \)
• D. \( \frac{1}{2} \sqrt{\frac{h}{m\pi}} \)

Hint:

Use \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \) and \( \Delta p = m \Delta v \) to derive momentum uncertainty.

Answer 1

The Correct Option is C

From Heisenberg uncertainty principle: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi}, \quad \text{and } \Delta p = m \Delta v \] Given \( \Delta x = \Delta v \), so: \[ \Delta x \cdot m \Delta x = m \Delta x^2 = \Delta p \cdot \Delta x \geq \frac{h}{4\pi} \Rightarrow \Delta p = m \Delta x = \sqrt{m \cdot \frac{h}{4\pi \Delta x}} = \frac{1}{2} \sqrt{\frac{mh}{\pi}} \]