Question

Calculate the de Broglie wavelength for each of the following. a. an electron with a velocity \(10 . \%\) of the speed of light b. a tennis ball \((55 \mathrm{~g})\) served at \(35 \mathrm{~m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})\)

Short answer

The de Broglie wavelength for an electron traveling at 10% of the speed of light is approximately \(2.43 \times 10^{-12}\) m, and the de Broglie wavelength for a tennis ball served at 35 m/s is approximately \(3.43 \times 10^{-34}\) m.

Step-by-step solution

Part a: Electron at 10% speed of light

Given: - velocity of electron, \(v = 0.10c = 0.10 \times 3 \times 10^8 \, \text{m}/\text{s}\) (since 10% of the speed of light) - mass of electron, \(m_e = 9.11 \times 10^{-31} \, \text{kg}\) - Planck's constant, \(h = 6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\) Calculating the wavelength using the de Broglie's formula: \(λ = \frac{h}{mv}\) Plug in the values: \(λ = \frac{6.63 \times 10^{-34} \, \text{J} \cdot \text{s}}{(9.11 \times 10^{-31} \text{kg})(0.10 \times 3 \times 10^8 \, \text{m}/\text{s})}\) Calculate: \(λ = 2.43 \times 10^{-12} \, \text{m}\) So, the de Broglie wavelength for an electron with a velocity of 10% of the speed of light is approximately \(2.43 \times 10^{-12}\) m.

Part b: Tennis ball served at 35 m/s

Given: - mass of tennis ball, \(m_t = 55 \, \text{g} = 0.055 \, \text{kg}\) - velocity of tennis ball, \(v = 35 \, \text{m}/\text{s}\) - Planck's constant, \(h = 6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\) Calculating the wavelength using the de Broglie's formula: \(λ = \frac{h}{mv}\) Plug in the values: \(λ = \frac{6.63 \times 10^{-34} \, \text{J} \cdot \text{s}}{(0.055 \, \text{kg})(35\, \text{m}/\text{s})}\) Calculate: \(λ = 3.43 \times 10^{-34} \, \text{m}\) So, the de Broglie wavelength for a tennis ball served at 35 m/s is approximately \(3.43 \times 10^{-34}\) m.

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that explores the behavior of the microscopic world. It is especially focused on particles like electrons, photons, and other subatomic structures. At the heart of quantum mechanics lie some core principles that can seem strange compared to classical physics. One of these is that particles exhibit both wave and particle characteristics simultaneously. This is known as wave-particle duality, a fundamental concept that is explored further using de Broglie's wavelength.

Quantum mechanics revolutionized our understanding by proposing that energy levels within an atom are quantized. This idea can be understood through experiments showing that an electron can behave like a wave. Therefore, their properties are described using probabilities, rather than definitive paths. Quantum mechanics relies heavily on mathematical formulations to predict the behavior of these tiny particles, which are somewhat counterintuitive to our everyday experiences.

Overall, quantum mechanics provides a new lens through which we study the universe, breaking away from classical explanations and venturing into probabilities and uncertainties of the atomic and subatomic world.

Electron Velocity

The velocity of an electron is a critical factor when calculating its de Broglie wavelength. As tiny particles, the behavior and speed of electrons can drastically impact their interaction with other particles. In the context of the problem here, the velocity of the electron is set as 10% of the speed of light.

Electrons, because they are incredibly light, achieve high velocities even with small amounts of energy. The speed of light is approximately \(3 \times 10^8 \text{ m/s}\), making 10% of that about \(3 \times 10^7 \text{ m/s}\).

Understanding an electron's velocity helps physicists determine not only its wavelength but also its energy and the way it interacts with magnetic and electric fields. High velocity electrons are significant in areas like electron microscopy and particle accelerators, where they are used to probe and analyze materials or subatomic particles in more detail.

Planck's Constant

Planck's constant is a key quantity in quantum mechanics that appears in various formulas, including the calculation of the de Broglie wavelength. It is denoted by \(h\) and has a value of \(6.63 \times 10^{-34} \text{ J} \cdot \text{s}\).

Named after Max Planck, this constant serves as a bridge between the macroscopic and quantum worlds. It is integral in defining the smallest action or quantum of action that can be observed—a fundamental limit in the laws of physics. Planck's constant is used in multiple equations, including the formula for energy of a photon \(E = hu\), where \(u\) is the frequency of the photon.

The role of Planck's constant in calculating wavelengths via the de Broglie equation highlights its importance in expressing the wave nature of particles. It's this constant that helps translate the effects that aren't observable in everyday life into concrete measurements and predictions at the atomic and subatomic level.

Wave-Particle Duality

Wave-particle duality is the concept in quantum mechanics where particles like electrons exhibit both wave and particle properties. This is a key idea that was highlighted by Louis de Broglie, leading to the concept of the de Broglie wavelength. His theory suggests that every particle has an associated wavelength, determined by its momentum.

For instance, in this exercise, calculating the de Broglie wavelength of an electron (a classically considered particle) at a significant fraction of the speed of light shows its wave-like behaviors. Wave-particle duality is at the heart of many experimental observations, such as electron diffraction, where electrons create interference patterns typical of waves.

This duality isn't just theoretical—it has direct applications. Technologies like electron microscopes exploit wave-particle duality to visualize objects at incredibly small scales. Thus, understanding how and when an electron or any particle behaves like a wave or a particle can significantly impact our manipulation of materials and technology at the nano-level.