Question

A free particle of mass $m$ confined within the walls of a one-dimensional box of length $a$ (with the potential outside the box being infinity), is in the eigenstate having quantum number $n = 4$The magnitude of uncertainty in the measurement of momentum of the particle is $Y \times \frac{h}{a}$The value of $Y$ is _ _ _. (answer in integer)

Hint:

For particle in a box, $\Delta p = \frac{nh}{2a}$ and $\langle p \rangle = 0$Always use variance definition for uncertainty

Answer 1

Correct Answer: 2

Step 1: Momentum in a box.
For a stationary state the average momentum is zero, so the uncertainty equals the root of the mean square momentum.

Step 2: Use the allowed magnitude.
\[ \Delta p = \frac{nh}{2a} \]

Step 3: Put in $n = 4$.
\[ \Delta p = \frac{4h}{2a} = \frac{2h}{a} \Rightarrow Y = 2 \]

Step 4: Answer.
\[ \boxed{2} \]