Question

The wave nature of matter was first proposed by Louis de Broglie, who suggested that the wavelength \((\lambda)\) of a particle was related to its mass \((m)\) and its velocity \((\nu)\) by the equation: \(\lambda=h / m \nu\), where \(h\) is Planck's constant \(\left(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)\). Calculate the de Broglie wavelength of: (a) a \(0.0459 \mathrm{~kg}\) golf ball traveling at \(95 \mathrm{~m} / \mathrm{s}\); (b) an electron traveling at \(3.88 \times 10^{6} \mathrm{~m} / \mathrm{s}\). Can you explain why the wave nature of matter is significant for the electron but not for the golf ball? (Hint: Express mass in kilograms.)

Short answer

The de Broglie wavelength for the golf ball is extremely small and not detectable, whereas the wavelength for the electron is significant enough to be observed. Wave nature of matter is important for particles like electrons due to their tiny mass leading to observable wavelengths; however, it's negligible for macroscopic objects like golf balls with large mass.

Step-by-step solution

Write Down Known Values for Golf Ball

Firstly, write down the known values for the golf ball: the mass (m) is 0.0459 kg, the velocity (v) is 95 m/s, and Planck's constant (h) is given as \(6.626 \times 10^{-34} J\cdot s\).

Calculate de Broglie Wavelength for Golf Ball

Next, use the de Broglie equation to calculate the wavelength for the golf ball: \(\lambda = \frac{h}{mu}\). Plugging in the known values, \(\lambda = \frac{6.626 \times 10^{-34}}{0.0459 \cdot 95}\), and solve for \(\lambda\).

Write Down Known Values for Electron

Now, write down the known values for the electron. Consider the mass (m) of the electron is approximately \(9.109 \times 10^{-31}\) kg, which is a standard constant, and the velocity (v) given is \(3.88 \times 10^{6} \) m/s.

Calculate de Broglie Wavelength for Electron

Use the same de Broglie equation to calculate the wavelength for the electron: \(\lambda = \frac{h}{mu}\). Plugging in the known values for the electron, \(\lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \cdot 3.88 \times 10^{6}}\), and then solve for \(\lambda\).

Explanation of Wave Nature Significance

Explain why wave nature is significant for the electron but not for the golf ball. The wavelength calculated for the electron is within the range of observable phenomena, while the wavelength for the golf ball is minuscule and therefore, its wave nature cannot be detected.

Key concepts

Wave-Particle Duality

The principle of wave-particle duality is a cornerstone of quantum mechanics, proposing that all particles exhibit both wave and particle characteristics. According to this concept, particles such as electrons and photons can exhibit behavior typical of waves under certain conditions. This duality was first brought to light by Einstein's explanation of the photoelectric effect and was further developed by Louis de Broglie.

De Broglie hypothesized that if light, known to have a wave nature, can display particle properties (photons), then particles could also exhibit wave properties. His theory led to the development of the de Broglie wavelength equation, which links a particle's mass and velocity to a corresponding wavelength. The ability to calculate the wavelength of particles like electrons can be curiosity-provoking and may challenge one's understanding of the micro-world. Exploration of this duality enables us to comprehend the behavior of quantum objects and opens a world where classic intuition does not always apply.

Quantum Mechanics

Quantum mechanics, a fundamental theory in physics, provides an intricate description of the behavior of particles at the subatomic scale. Unlike classical mechanics, which operates on the assumption that processes are deterministic and predictable, quantum mechanics introduces a probabilistic and uncertain view of the microcosmos. This shift from determinism to uncertainty was revolutionary.

The rules of quantum mechanics help explain the inherently wavy behavior of particles as they traverse space. This field tackles scenarios where objects can be in a superposition of states, a term engendered by the peculiar nature of particles behaving as both waves and particles. Quantum mechanics requires us to consider not just the position or speed of an object, but also its wave-like interactions with other quantum objects. This branch of physics has significant applications, from quantum computing to fundamental particle interactions, and it continues to be a rich area for research and discovery.

Planck's Constant

Planck's constant is a physical constant that plays a vital role in quantum mechanics, represented by the symbol 'h'. Its value is approximately \(6.626 \times 10^{-34} J \times s\text{,}\) which was first determined by Max Planck. This constant is central to the quantum description of nature and appears in many fundamental equations, including those describing the energy levels of atoms and the emission of radiation from black bodies.

Planck's constant essentially sets the scale for the quantization in the quantum realm and ties into the wave-particle duality by defining the relationship between the frequency of a photon (or any particle) and its energy. It also appears in the de Broglie wavelength equation, enabling the calculation of the wave nature of matter. This constant is a key to unlocking the mysterious and counterintuitive behaviors at the quantum level and is fundamental to understanding the behaviors of elementary particles and their interactions.

Particle Physics

Particle physics is a compelling field that studies the smallest known building blocks of the universe and their fundamental interactions. This branch of physics delves into the high-energy world of subatomic particles such as quarks, electrons, and neutrinos. Advanced particle colliders, like the Large Hadron Collider (LHC), are utilized to dissect the components and forces at play within an atom.

One key aspect of particle physics is understanding how the wave-particle duality principle applies to different particles. The field scrutinizes how particles like protons, electrons, and photons behave under various circumstances and form the backbone of matter and the universe. Utilizing theories from quantum mechanics, researchers aim to explain the inherently probabilistic nature of particles, while Planck's constant helps in defining interactions at the quantum level. Particle physics continually seeks to expand our understanding of the universe by exploring how these foundational elements interact and bind to form the complex structures visible in the cosmos today.