Question

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

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

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 \(168km/h \times \frac{1m/s}{3.6km/h} = 46.67m/s\).

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 \(p = mv = 0.145kg \times 46.67m/s = 6.767kg.m/s\).

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: \( \lambda = 9.795 \times 10^-27 nm\).

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.

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and light on the atomic and subatomic scale. It introduces concepts that are quite different from classical physics. One such concept is wave-particle duality, highlighted by the de Broglie wavelength.

Gaps between the classical and quantum worlds are bridged by concepts like de Broglie's hypothesis. This hypothesis proposes that every particle has a wave associated with it, known as the de Broglie wavelength. However, the effects are significant only at the atomic level. Larger objects, like a baseball, have negligible wavelengths, thus indistinguishable in our macroscopic world.

Understanding quantum mechanics offers insight into why these wave-like properties are crucial for small particles, influencing behavior and interactions at the quantum level. It revolutionized science, explaining phenomena that classical physics could not and leading to advancements in technology like semiconductors and lasers.

Momentum

Momentum, in classical mechanics, is the quantity of motion of a moving body. It is calculated as the product of an object's mass and its velocity. Represented by the letter \(p\), the formula is expressed as \(p = mv\), where \(m\) is mass and \(v\) is velocity. This concept, while straightforward, gains depth in quantum mechanics.

In the exercise, we calculated the momentum of a baseball to determine its de Broglie wavelength. Though momentum itself is a classical concept, it intertwines beautifully with quantum principles. Momentum in quantum mechanics is not just a measure of motion but part of a duality; it is also linked to wave characteristics like wavelength.

This duality means that every particle with momentum has an associated wavelength. This relationship is a pillar of quantum mechanics, showing that understanding classical mechanics is fundamental before diving into the quantum realm.

Planck's Constant

Planck's constant \(h\), named after physicist Max Planck, is a fundamental constant in quantum mechanics. It is a tiny quantity, \(6.62607004 \times 10^{-34} \; \text{m}^2 \text{kg/s}\), reflecting the scale at which quantum effects become significant. It is central to the de Broglie wavelength calculation, where \( \lambda = \frac{h}{p} \), linking the quantum world to momentum.

Planck's constant represents the proportion of the energy of a photon to its frequency, establishing a fundamental scale in quantum mechanics. Its discovery marked the birth of quantum physics, allowing scientists to describe the behavior of electrons in atoms, among other phenomena.

In the context of the exercise, applying Planck's constant to a macroscopic object like a baseball illustrates how minuscule quantum wavelengths become when dealing with large masses. This helps explain why quantum effects are not observed in everyday objects, as their associated wavelengths are too tiny to detect.