Question

After you tell your \(60.0-\mathrm{kg}\) roommate about de Broglie's hypothesis that particles of momentum \(p\) have wave characteristics with wavelength \(\lambda=h / p\), he starts thinking of his fate as a wave and asks you if he could be diffracted when passing through the \(90.0-\mathrm{cm}\) -wide doorway of your dorm room. a) What is the maximum speed at which your roommate could pass through the doorway and be significantly diffracted? b) If it takes one step to pass through the doorway, how long should it take your roommate to make that step (assuming that the length of his step is \(0.75 \mathrm{~m}\) ) for him to be diffracted? c) What is the answer to your roommate's question? (Hint: Assume that significant diffraction occurs when the width of the diffraction aperture is less than 10.0 times the wavelength of the wave being diffracted.)

Short answer

Based on the given information, the maximum speed at which your roommate could pass through the doorway and be significantly diffracted is approximately \(1.22 \times 10^{-35}\, \mathrm{m/s}\). To achieve this, it would take your roommate approximately \(6.15 \times 10^{35}\, \mathrm{s}\) to make a single step. This time scale is extremely long and not practically possible. Therefore, the answer to your roommate's question of whether or not they could be significantly diffracted when passing through the doorway is "No".

Step-by-step solution

Identify known and unknown variables

Given: Mass of the roommate (m) = \(60.0\,\mathrm{kg}\) Doorway width (D) = \(90.0\,\mathrm{cm}\) = \(0.9\,\mathrm{m}\) Planck's constant (h) = \(6.626 \times 10^{-34}\, \mathrm{Js}\) We need to find the maximum speed (v) at which your roommate could pass through the doorway and be significantly diffracted.

Determine significant diffraction condition

According to the hint, significant diffraction occurs when the width of the diffraction aperture is less than 10.0 times the wavelength of the wave being diffracted. So, \(D < 10\lambda\)

Apply de Broglie's wavelength formula

The de Broglie wavelength formula is \(\lambda = \dfrac{h}{p}\). From the momentum formula \(p = mv\), the formula becomes: \(\lambda = \dfrac{h}{mv}\) Using \(D < 10\lambda\), the formula becomes \(D < 10\dfrac{h}{mv}\).

Solve for maximum speed

We can solve for v: \(v > \dfrac{10h}{mD}\) Plug in the values: \(v > \dfrac{10(6.626 \times 10^{-34}\, \mathrm{Js})}{(60.0\, \mathrm{kg})(0.9\, \mathrm{m})}\) \(v > 1.22 \times 10^{-35}\, \mathrm{m/s}\) The maximum speed at which your roommate could pass through the doorway and be significantly diffracted is approximately \(1.22 \times 10^{-35}\, \mathrm{m/s}\). #b) Calculate the time to make a step for diffraction#

Determine the time it takes to make a step for diffraction

We are given the length of your roommate's step, which is \(0.75\,\mathrm{m}\), and we need to find the time it would take to make that step for diffraction to occur. From the maximum speed found in part a: \(t = \dfrac{0.75\,\mathrm{m}}{1.22 \times 10^{-35}\, \mathrm{m/s}}\) \(t \approx 6.15 \times 10^{35}\, \mathrm{s}\) So it would take your roommate approximately \(6.15 \times 10^{35}\, \mathrm{s}\) to make that step for him to be diffracted. #c) Answer to roommate's question#

Consider the time scale

The time found in part b is an extremely long period, and it's not practically possible for your roommate to take that long to make a step.

Conclusion

It is very unlikely that your roommate could be significantly diffracted when passing through the doorway. So, the answer to your roommate's question is "No".