Question 40.5: Wavelengths for Microscopic and Macroscopic Objects (A) Calc......
Wavelengths for Microscopic and Macroscopic Objects
(A) Calculate the de Broglie wavelength for an electron \left(m_{e}=9.11 \times 10^{-31} \mathrm{~kg}\right) moving at 1.00 \times 10^{7} \mathrm{~m} / \mathrm{s}.
(B) A rock of mass 50 \mathrm{~g} is thrown with a speed of 40 \mathrm{~m} / \mathrm{s}. What is its de Broglie wavelength?
(A) Conceptualize Imagine the electron moving through space. From a classical viewpoint, it is a particle under constant velocity. From the quantum viewpoint, the electron has a wavelength associated with it.
Categorize We evaluate the result using an equation developed in this section, so we categorize this example as a substitution problem.
Evaluate the wavelength using Equation 40.15:
\lambda=\frac{h}{m_{e} u}=\frac{6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}}{\left(9.11 \times 10^{-31} \mathrm{~kg}\right)\left(1.00 \times 10^{7} \mathrm{~m} / \mathrm{s}\right)}=7.27 \times 10^{-11} \mathrm{~m}
The wave nature of this electron could be detected by diffraction techniques such as those in the Davisson-Germer experiment.
(B) Evaluate the de Broglie wavelength using Equation 40.15:
\lambda=\frac{h}{m u}=\frac{6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}}{\left(50 \times 10^{-3} \mathrm{~kg}\right)(40 \mathrm{~m} / \mathrm{s})}=3.3 \times 10^{-34} \mathrm{~m}
This wavelength is much smaller than any aperture through which the rock could possibly pass. Hence, we could not observe diffraction effects, and as a result, the wave properties of large-scale objects cannot be observed.