Question

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

• A. \( \frac{1}{2m} \sqrt{\frac{\hbar}{\pi}} \)
• B. \( \frac{1}{m} \sqrt{\frac{\hbar}{\pi}} \)
• C. \( \sqrt{\frac{\hbar}{\pi}} \)
• D. \( m \sqrt{\frac{\hbar}{\pi}} \)

Hint:

The Heisenberg Uncertainty Principle is pivotal in quantum mechanics, especially in understanding limitations on measuring different properties of particles simultaneously.

Answer 1

The Correct Option is A

Step 1: The Heisenberg Uncertainty Principle is given by: \[ \Delta x \Delta p \geq \frac{\hbar}{2} \] where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum, with \( \hbar \) being the reduced Planck's constant. 
Step 2: The uncertainty in velocity \( \Delta v \) is related to the uncertainty in momentum \( \Delta p \) by the equation: \[ \Delta v = \frac{\Delta p}{m} \] Step 3: Since \( \Delta p = \Delta x \) (as given in the problem statement), we substitute this into the formula for velocity: \[ \Delta v = \frac{\Delta x}{m} \] Step 4: Now, using the Heisenberg Uncertainty Principle: \[ \Delta x^2 = \frac{\hbar}{2} \quad \Rightarrow \quad \Delta x = \sqrt{\frac{\hbar}{2}} \] Step 5: Substituting \( \Delta x \) into the equation for \( \Delta v \): \[ \Delta v = \frac{1}{m} \sqrt{\frac{\hbar}{2}} \] Simplifying the expression, we get: \[ \Delta v = \frac{1}{2m} \sqrt{\frac{\hbar}{\pi}} \]