Question

An electron is confined to a box of length \( L \). If the length of the box changes to \( 2L \), how would the uncertainty of momentum of the electron change?

• A. It will be twice
• B. It will be half
• C. It will remain the same
• D. It will be one-fourth

Hint:

Increasing the confinement region reduces the momentum uncertainty due to the Heisenberg principle.

Answer 1

The Correct Option is B

From Heisenberg's uncertainty principle:
\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \] For a particle in a box, the uncertainty in position is approximately the box length:
\[ \Delta x \approx L \] If the box length doubles (\( 2L \)), the uncertainty in position increases:
\[ \Delta p \propto \frac{1}{\Delta x} = \frac{1}{L} \] Since \( L \) becomes \( 2L \), the uncertainty in momentum reduces by half:
\[ \Delta p' = \frac{1}{2} \Delta p \]