# Step 1: **Understanding the Uncertainty Principle** When the electron passes through the **13.7 μm diameter hole** in the barrier, its position in the **x-direction** is localized to about the size of the hole. This introduces uncertainty in its momentum in the x-direction (\(\Delta p_x\)) due to the **Heisenberg Uncertainty Principle**: \[ \Delta x \cdot \Delta p_x \gtrsim \frac{\hbar}{2} \] Where: - \(\Delta x\) = diameter of the hole = 13.7 μm = \(13.7 \times 10^{-6}\) m - \(\hbar\) = reduced Planck's constant = \(1.055 \times 10^{-34}\) J·s **Explanation:** - The electron's x-position is uncertain by about the hole's diameter (\(\Delta x\)), which means its x-momentum (\(\Delta p_x\)) becomes uncertain by at least \(\frac{\hbar}{2\Delta x}\). --- # Step 2: **Calculating the Range of \(v_x\)** The uncertainty in x-momentum translates to a range for \(v_x\): \[ \Delta p_x \gtrsim \frac{\hbar}{2 \Delta x} \] \[ \Delta p_x = m \Delta v_x \] So, \[ \Delta v_x \gtrsim \frac{\hbar}{2 m \Delta x} \] Where: - \(m\) = mass of electron = \(9.11 \times 10^{-31}\) kg Plug in the values: \[ \Delta v_x \gtrsim \frac{1.055 \times 10^{-34}}{2 \times 9.11 \times 10^{-31} \times 13.7 \times 10^{-6}} \] Calculate the denominator: \[ 2 \times 9.11 \times 10^{-31} \times 13.7 \times 10^{-6} = 2 \times 9.11 \times 13.7 \times 10^{-37} \approx 249.554 \times 10^{-37} = 2.49554 \times 10^{-35} \] Now, plug in the numbers: \[ \Delta v_x \gtrsim \frac{1.055 \times 10^{-34}}{2.49554 \times 10^{-35}} \approx 4.23 \] So, \[ \Delta v_x \gtrsim 4.23 \text{ m/s} \] This is the uncertainty, so the likely range for \(v_x\) is about: \[ -4.23 \text{ m/s} < v_x < +4.23 \text{ m/s} \] --- ## **Final Answer** **After passing through the hole, the electron's likely range for \(v_x\) is:** \[ \boxed{-4.23\ \text{m/s} < v_x < +4.23\ \text{m/s}} \] This matches one of your previously tried answers, but **be sure to use the correct significant figures and sign conventions as required by your course or system**. This is the correct range based on the uncertainty principle and the size of the hole.