Question

In the text (Section 11.6 ) it was mentioned that current theories of atomic structure suggest that all matter and all energy demonstrate both particle- like and wave-like properties under the appropriate conditions, although the wave-like nature of matter becomes apparent only in very small and very fast- moving particles. The relationship between wavelength \((\lambda)\) observed for a particle and the mass and velocity of that particle is called the de Broglie relationship. It is $$ \lambda=h / m v $$ in which \(h\) is Planck's constant \(\left(6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right), * m\) represents the mass of the particle in kilograms, and \(v\) represents the velocity of the particle in meters per second. Calculate the "de Broglie wavelength" for each of the following, and use your numerical answers to explain why macroscopic (large) objects are not ordinarily discussed in terms of their "wave-like" properties. a. an electron moving at 0.90 times the speed of light b. a \(150-\mathrm{g}\) ball moving at a speed of \(10 . \mathrm{m} / \mathrm{s}\) c. a 75 -kg person walking at a speed of \(2.0 \mathrm{~km} / \mathrm{h}\)

Short answer

The de Broglie wavelengths of the various objects are: \(2.43 \times 10^{-12}\,\text{m}\) for the electron, \(4.42 \times 10^{-34}\,\text{m}\) for the 150 g ball, and \(1.98 \times 10^{-37}\,\text{m}\) for the 75 kg person. Since the wavelengths for the macroscopic objects (ball and person) are much smaller than typical light wavelengths, their wave-like properties are not noticeable in everyday life, making it unnecessary to discuss macroscopic objects in terms of wave-like properties. On the contrary, the electron has a larger wavelength relative to its size, which results in observable wave-like behavior.

Step-by-step solution

(Step 1: Identify the given information)

For each object, we are provided with its mass and velocity, which will be used to calculate the de Broglie wavelength using the formula \(\lambda = h/(mv)\). a. Electron: mass = \(9.11 \times 10^{-31}\,\text{kg}\) (mass of an electron), velocity = \(0.9c\), where \(c\) is the speed of light (\(3.0 \times 10^8\,\text{m/s}\)). b. 150 g ball: mass = \(0.150\,\text{kg}\) (converted from grams to kilograms), velocity = \(10\,\text{m/s}\). c. 75 kg person: mass = \(75\,\text{kg}\), velocity = \(2.0\,\text{km/h}\) = \((2.0 \times 1000)/3600\,\text{m/s}\) (converted from km/h to m/s).

(Step 2: Calculate the de Broglie wavelength for each object)

We will now use the identified values and the de Broglie relationship to find the wavelength for each object. a. Electron: \(\lambda = \frac{h}{mv}\) \(\lambda = \frac{6.63 \times 10^{-34}\,\text{Js}}{(9.11 \times 10^{-31}\,\text{kg})(0.9 \times 3.0 \times 10^8\,\text{m/s})}\) \(\lambda = 2.43 \times 10^{-12}\,\text{m}\) b. 150 g ball: \(\lambda = \frac{h}{mv}\) \(\lambda = \frac{6.63 \times 10^{-34}\,\text{Js}}{(0.150\,\text{kg})(10\,\text{m/s})}\) \(\lambda = 4.42 \times 10^{-34}\,\text{m}\) c. 75 kg person: \(\lambda = \frac{h}{mv}\) \(\lambda = \frac{6.63 \times 10^{-34}\,\text{Js}}{(75\,\text{kg})((2.0 \times 1000)/3600\,\text{m/s})}\) \(\lambda = 1.98 \times 10^{-37}\,\text{m}\)

(Step 3: Explain why macroscopic objects are not discussed in terms of wave-like properties)

Based on the calculations, it can be seen that the de Broglie wavelengths of the 150 g ball and the 75 kg person are incredibly small (on the order of \(10^{-34}\,\text{m}\) and \(10^{-37}\,\text{m}\), respectively). These wavelengths are much smaller than the typical wavelengths of light, which are in the range of nanometers (\(10^{-9}\,\text{m}\)) to micrometers (\(10^{-6}\,\text{m}\)). Due to this, the wave-like properties of macroscopic objects are not noticeable in everyday life. On the other hand, the electron's wavelength is in the picometer range (\(10^{-12}\,\text{m}\)) which is much larger compared to its size and is within the realm of wave-like behavior. As a result, only microscopic particles such as electrons are commonly described in terms of their wave-like properties.

Key concepts

Wave-Particle Duality

The concept of wave-particle duality lies at the heart of quantum mechanics and is epitomized in the de Broglie hypothesis, which suggests that every particle exhibits both wave-like and particle-like properties. Louis de Broglie proposed that not only does light have wave and particle characteristics—as demonstrated by phenomena like diffraction (wave) and photoelectric effect (particle)—but this duality also applies to material particles, such as electrons.

When it comes to observing wave-like behavior in particles, this is where scale matters. For very small entities, like electrons, the wavelength is significant enough to be observable under appropriate conditions. Using de Broglie's equation \(\lambda = \frac{h}{mv}\), we can compute the wavelength (\(\lambda\)) associated with a particle of mass (\(m\)) moving with velocity (\(v\)), where \(h\) represents Planck's constant. This calculation reveals that as objects become larger, their associated wavelengths become so minute that wave-like properties are no longer perceptible, a concept crucial to understanding why everyday items don't display quantum behaviors observable in the microscopic world.

Planck's Constant

Planck's constant, symbolized by \(h\), is a fundamental quantity in quantum mechanics. Its value is approximately \(6.63 \times 10^{-34} \text{ J} \cdot \text{s}\) and it represents the proportionality factor between the energy and frequency of a photon. Planck's constant is a bridge between the microscopic quantum world and the macroscopic classical physics, setting the scale at which quantum effects become significant.

When applied in the de Broglie relationship \(\lambda=\frac{h}{mv}\), Planck's constant allows us to determine the de Broglie wavelength of particles, thereby illustrating the inherently quantum nature of matter. Its very small magnitude emphasizes why quantum effects are not experienced in our day-to-day life with larger, slower-moving macroscopic objects, whose de Broglie wavelengths are infinitesimally small.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It challenges classical mechanics with its introduction of probabilistic outcomes, quantization, and concepts like superposition and entanglement.

In the context of the de Broglie wavelength, quantum mechanics dictates that all particles have wave-like properties. This is crucial in understanding phenomena such as electron diffraction and the formation of atomic orbitals, which cannot be explained by classical physics. Quantum mechanics reshapes our view of matter and energy, highlighting that at the quantum level, particles like electrons behave in ways that defy our macroscopic expectations, such as existing in multiple states simultaneously or being affected by observation.