A certain pitcher's fastballs have been clocked at about \(100 \mathrm{mph}\). (a) Calculate the wavelength of a \(0.141-\mathrm{kg}\) baseball (in \(\mathrm{nm})\) at this speed. (b) What is the wavelength of a hydrogen atom at the same speed? ( 1 mile \(=1609 \mathrm{~m}\) )

The de Broglie equation is a fundamental concept in quantum mechanics that relates the wave-like properties of particles to their momentum. It is given by the formula \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength associated with a particle, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity. The equation proposes that every object has a wave associated with it, and the wavelength of this wave decreases with an increase in the object's momentum. This relationship is quite significant in the quantum realm for small particles like electrons and protons, but the wave characteristics of larger objects, such as baseballs, are imperceptible due to their relatively large mass compared to subatomic particles.

In the given exercise, the de Broglie equation allows us to calculate the theoretical wavelength of a baseball and a hydrogen atom when they are propelled at a certain speed. The application of the equation to everyday objects like baseballs illustrates the pervasiveness of quantum phenomena; however, the resulting wavelengths are extremely small, reinforcing the idea that quantum effects are negligible at human scales.

Planck's constant is one of the most crucial elements of quantum mechanics, represented by the symbol \( h \) and having a value of approximately \( 6.63 \times 10^{-34} \text{Js} \) (joule seconds). This constant is named after Max Planck who posited that energy is quantized, thus establishing the foundation of quantum theory. In the world of small particles, like atoms and photons, Planck's constant helps describe the granularity of energy exchange and the behavior of particles in terms of wave-particle duality.

This infinitesimally small number has vast implications: it sets the scale at which quantum effects become significant and serves as a fundamental boundary between classical and quantum physics. It is used in the de Broglie equation to calculate the wavelength of any particle, emphasizing the wave nature of matter. The tiny magnitude of Planck's constant is a reminder of why quantum effects are not observed in everyday objects and why a baseball's wavelength is imperceptibly small.

In physics problems, it's often necessary to convert between units of velocity to ensure that equations are dimensionally consistent. In our exercise, velocity is originally given in miles per hour (mph), which is a unit of speed commonly used in the United States for transportation. However, the de Broglie equation and most scientific formulas require the velocity to be in meters per second (m/s).

To convert from mph to m/s, we multiply the speed value by the conversion factor \( \frac{1609 \text{m}}{1 \text{mile}} \), which accounts for the number of meters in a mile, and then divide by the number of seconds in an hour, 3600. This gives us the velocity in the metric system, which can then be inserted into the de Broglie equation and other scientific formulas. This step is vital for ensuring accurate calculations, as mismatched units can lead to incorrect or meaningless results.

After using the de Broglie equation to calculate the wavelength of a particle, it is often necessary to convert the result into more convenient units. While the calculation may yield wavelengths in meters (\( m \)), scientists and engineers frequently use nanometers (\( nm \)) to measure such small distances. One meter is equivalent to \( 10^9 \) nanometers.

To convert from meters to nanometers, you simply multiply the result by \( 10^9 \). This scale adjustment makes the numbers more manageable and easier to compare with familiar objects in the quantum realm, such as atoms and subatomic particles, which are also measured on the nanometer scale. Understanding unit conversion is not only essential for accuracy but also helps in visualizing and contextualizing the scale of the phenomena being studied.