Chapter 28, Problem 59GP

To determine

The speed and time taken while moving and also, the noticed diffraction effects

Answer to Problem 59GP

The speed is $\text{1}\text{.8}\text{×}{\text{10}}^{\text{-35}}\text{m/s}$ .

The diffraction effects are noticed.

The time is ${\text{10}}^{\text{34}}\text{s}$ .

Explanation of Solution

Given info: Mass, $\text{m = 75}\text{.0}\text{kg}$

Wavelength is $\text{0}\text{.50}\text{m}$

Formula used:

The de Broglie wavelength is calculated by $\text{λ=}\frac{\text{h}}{\text{mv}}$

  $\text{λ}$ is wavelength

  $h$ is Planck’s constant, $6.63\times {10}^{-34}J\cdot s$

  $m$ is mass

  $v$ is speed

Calculation:

The de Broglie wavelength is determined by $\text{λ=}\frac{\text{h}}{\text{mv}}\mathrm{...}\left(1\right)$

Substitute the value in equation $\left(1\right)$

  $\text{0}\text{.50}\text{m}\text{=}\frac{\text{6}\text{.63}\times {\text{10}}^{\text{-34}}\text{J}\cdot \text{s}}{\left(\text{75}\text{.0}\text{kg}\right)\text{v}}$

  $\text{v}\text{=}\text{1}\text{.8}\text{×}{\text{10}}^{\text{-35}}\text{m/s}$

Since $\text{λ}$ is comparable to the width of a typical doorway. So, yes, diffraction effects will be noticed.

However, letting that walking through the doorway requires travel through a distance of $\text{0}\text{.2 m}$ , the time $\text{Δt}$ needed will be calculated by the expression $\text{v}\text{Δt}\text{=}\text{d}\mathrm{...}\left(\text{2}\right)$

Substitute the value in equation $\left(\text{2}\right)$

  $\left(\text{1}{\text{.8×10}}^{\text{-35}}\text{m/s}\right)\text{Δt}\text{=}\text{0}\text{.2}\text{m}$

  $\text{Δt}{\text{=10}}^{\text{34}}\text{s}$

Conclusion:

All the required values are calculated.