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Chapter 28: Problem 1

What is the de Broglie wavelength of a basketball of mass \(0.50 \mathrm{kg}\) when it is moving at \(10 \mathrm{m} / \mathrm{s} ?\) Why don't we see diffraction effects when a basketball passes through the circular aperture of the hoop?

Short Answer

Expert verified
Answer: The diffraction effects become noticeable when the size of the aperture (i.e., the hoop) is comparable to the de Broglie wavelength of the particle. In this case, the de Broglie wavelength of the basketball is \(1.33 \times 10^{-34}\,\mathrm{m}\), while the diameter of a standard basketball hoop is \(0.46\,\mathrm{m}\). The wavelength of the basketball is many orders of magnitude smaller than the diameter of the hoop, which means the diffraction effects are negligible and not noticeable when the basketball passes through the hoop.

Step by step solution

01

Understanding de Broglie's wavelength formula

The de Broglie wavelength is determined by the following formula: $$ \lambda = \frac{h}{mv} $$ where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\)), \(m\) is the mass of the particle, and \(v\) is the velocity of the particle.
02

Calculating the wavelength of the basketball

We are given the mass \(m = 0.50\,\mathrm{kg}\) and the velocity \(v = 10\,\mathrm{m/s}\) of the basketball. We now use the de Broglie's wavelength formula to calculate the wavelength : $$ \lambda = \frac{h}{mv} $$ $$ \lambda = \frac{6.626 \times 10^{-34}\, \mathrm{J} \cdot \mathrm{s}}{(0.50\, \mathrm{kg})(10\, \mathrm{m/s})} $$ $$ \lambda \approx 1.33 \times 10^{-34}\, \mathrm{m} $$
03

Understanding why diffraction effects aren't seen with the basketball

The diffraction effects become noticeable when the size of the aperture (i.e., the hoop) is comparable to the wavelength of the particle. The diameter of a standard basketball hoop is \(0.46\,\mathrm{m}\), while the de Broglie wavelength of the basketball is \(1.33 \times 10^{-34}\,\mathrm{m}\). We can see that the wavelength is many orders of magnitude smaller than the diameter of the hoop. Consequently, the diffraction effects are negligible and not noticeable when the basketball passes through the hoop.

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