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Chapter 8: Problem 41

Calculate the de Broglie wavelength, in nanometers, associated with a \(145 \mathrm{g}\) baseball traveling at a speed of \(168 \mathrm{km} / \mathrm{h} .\) How does this wavelength compare with typical nuclear or atomic dimensions?

Short Answer

Expert verified
The de Broglie wavelength associated with the baseball is approximately 9.795 x 10^-27 nm. This wavelength is extremely small and far less than typical atomic or nuclear dimensions, indicating that wave-like behaviour is not observable at the macroscopic level. It underlines the essence of the wave-particle duality concept in quantum mechanics that the wave nature of a particle becomes more noticeable as the scale approaches the quantum realm.

Step by step solution

01

Convert mass and velocity to appropriate units

The mass of the baseball is given in grams and needs to be converted to kilograms by using the conversion factor \(1kg = 1000g\). Therefore, the mass in kg is \(145g \times \frac{1kg}{1000g} = 0.145kg\). The speed of the baseball is given in km/h, and needs to be converted to m/s by using the conversion factor \(1m/s = 3.6km/h\). Therefore, the speed in m/s is \(168km/h \times \frac{1m/s}{3.6km/h} = 46.67m/s\).
02

Calculate the momentum of the particle

The momentum of a particle is defined as the product of its mass and velocity. Therefore, the momentum of the baseball is given by \(p = mv = 0.145kg \times 46.67m/s = 6.767kg.m/s\).
03

Calculate the de Broglie wavelength

The de Broglie wavelength is given by \(\lambda = \frac{h}{p}\), where \(h\) is the Planck's constant. Substituting the given Planck's constant (6.62607004 × 10^-34 m^2 kg / s) and the calculated momentum yields the de Broglie wavelength as: \( \lambda = \frac{6.62607004 \times 10^-34 m^2.\(kg\) / \(s)}{6.767kg.m/s} = 9.795 \times 10^-36 m\). To convert this into nanometers, multiply by \(10^9 nm/m\) to get: \( \lambda = 9.795 \times 10^-27 nm\).
04

Compare with typical nuclear or atomic dimensions

The obtained wavelength is extremely small, even compared to the size of an atom, which is typically on the scale of 0.1 nm. Its size is even small when compared to a nucleus, which has a size on the order of \(10^{-15}\) m (or 0.001 nm). Therefore, although all objects exhibit wave-like behavior, the effect is not observable at the macroscopic level because the wavelength is very small.

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