What is the de Broglie wavelength (in \(\mathrm{nm}\) ) associated with a 2.5 -g Ping-Pong ball traveling 35 mph?

Understanding unit conversion is essential for solving physics problems, especially in quantum mechanics where the scales of measurement differ greatly from everyday experiences. To start, we need to comprehend that different physical quantities, like mass and speed, are often measured in various units, and converting them to a standard unit of measurement is crucial for accurate calculations.

When dealing with mass, the standard unit in the International System of Units (SI) is the kilogram (kg), but in everyday scenarios, we might encounter grams (g). To convert grams to kilograms, we multiply by 0.001, since there are 1000 grams in a kilogram. Similarly, speed is typically measured in meters per second (m/s) in scientific contexts, though it may be quoted in miles per hour (mph) in daily life. To convert mph to m/s, we multiply by the conversion factor 0.44704.

In the given exercise, we converted a Ping-Pong ball's mass from grams to kilograms and its speed from mph to m/s before applying the de Broglie wavelength formula. Accurate unit conversion is crucial to ensure that the computed wavelengths are correct and in the expected order of magnitude.

Quantum mechanics is a fundamental branch of physics that provides the theoretical framework for understanding the behavior of particles at the atomic and subatomic levels. It operates under principles that are strikingly different from classical physics, describing phenomena that classical physics can't, such as the wave-particle duality of matter.

This duality is where the concept of de Broglie wavelength arises. Proposed by Louis de Broglie, it states that particles also have wave-like properties, and the wavelength of this 'matter wave' is inversely proportional to the particle's momentum. The smaller and faster a particle is, the shorter its de Broglie wavelength will be. Although in everyday objects like a Ping-Pong ball, its de Broglie wavelength is so small due to its large mass and relatively slow speed that it has no observable wave-like behavior. However, understanding these concepts in quantum mechanics is vital for interpreting the behavior of electrons, photons, and other subatomic particles that exhibit both wave-like and particle-like properties depending on the experiment.

The wavelength formula presented in the context of quantum mechanics is associated with the de Broglie hypothesis, which connects wave-like properties to particles of matter. The de Broglie wavelength of any particle is given by the formula \(\lambda = \frac{h}{mv}\), where \(\lambda\) represents the wavelength, \(h\) is the Planck constant (a value of roughly \(6.626 × 10^{-34} J•s\)), \(m\) stands for mass, and \(v\) is the velocity of the particle.

Understanding the de Broglie wavelength formula and how to apply it is crucial in quantum mechanics. It enables the calculation of the wavelength of any moving particle, providing insights into the wave-like behavior of matter. By plugging in the converted mass and velocity of the Ping-Pong ball into the formula, we get the de Broglie wavelength in SI units of meters, which can then be converted to nanometers or other desired units. Remember that this calculation is primarily applicable to microscopic particles, where their wave properties are significant enough to influence their physical behavior.