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 approximately \(8.18 \times 10^{-35}\) meters, and for the electron it is approximately \(1.76 \times 10^{-10}\) meters. The wave nature is significant for the electron because its wavelength is comparable to atomic dimensions, unlike the golf ball's.

Step-by-step solution

Understanding de Broglie Wavelength

De Broglie proposed that every particle has a wave-like nature, and its wavelength can be calculated using the equation \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant, \(m\) is the particle's mass, and \(v\) is its velocity.

Calculate the Wavelength for the Golf Ball

Using the mass \(m = 0.0459 \text{ kg}\) and the velocity \(v = 95 \text{ m/s}\), we substitute the values into the de Broglie equation to find the wavelength of the golf ball.

Calculate the Wavelength for the Electron

Using the mass of an electron \(m = 9.109 \times 10^{-31} \text{ kg}\) and the given velocity \(v = 3.88 \times 10^{6} \text{ m/s}\), substitute these into the de Broglie equation to calculate the electron's wavelength.

Comparison and Explanation of Significance

Compare the calculated wavelengths of the golf ball and electron. The electron's wavelength is significant and measurable due to its small mass and high velocity. In contrast, the golf ball's wavelength is too small to be measured or have observable wave-like effects, thus showing why wave nature is significant in microscopic particles like electrons but not in macroscopic objects like golf balls.

Key concepts

Wave-Particle Duality

The concept of wave-particle duality is at the very heart of quantum mechanics, altering our understanding of fundamental physics. This revolutionary principle unveils that every particle exhibits both wave-like and particle-like properties. But why does this matter? Well, it challenges the classical view that particles and waves are distinct entities. An electron, for example, can interfere with itself when passing through a double-slit, creating an interference pattern characteristic of waves. However, when measuring, it appears as a discrete particle.

Physicist Louis de Broglie's hypothesis, which paved the way for wave-particle duality, states that all matter has a wavelength given by \(\lambda = \frac{h}{mv}\), translating the physical attributes of mass and velocity into wave phenomena. As a student delves into quantum mechanics, recognizing this duality becomes essential in understanding behavior at the quantum level, such as in the famous double-slit experiment. This experiment illustrates how particles like electrons show wave properties when not observed, and particle properties when measured.

Planck's Constant

Every student of physics encounters a fundamental constant: Planck's constant, symbolized by \(h\). With the numerical value of \(6.626 \times 10^{-34} \text{ J} \cdot \text{s}\), it's a cornerstone of quantum mechanics, setting the scale of quantum action. Think of it as a quantum of action; it represents the quantization of physical properties and sets the minimum value for any physical action in the universe.

Planck's constant is not just a numerical figure; it's instrumental in understanding quantum mechanics. It appears prominently in de Broglie's wavelength formula \(\lambda = \frac{h}{mv}\), the energy of photons \(E = hf\), where \(f\) is the frequency, and in the Heisenberg Uncertainty Principle \(\Delta x\Delta p \ge \frac{h}{4\pi}\). Recognizing the pivotal role of Planck's constant helps students to appreciate the discrete, or 'quantized', nature of energy levels in atoms and the phenomena of photoelectric effect, where light can eject electrons from a metal.

Quantum Mechanics

Quantum mechanics is a framework for the fundamental laws that govern the behavior and interaction of the smallest particles in the universe, such as electrons and photons. It's a domain where the intuitive perceptions from our macroscopic world do not always apply, often confounding our classical expectations. Phenomena at this microscopic level are quantized, meaning certain properties, like energy or angular momentum, can only take on specific, discrete values.

In quantum mechanics, we delve into the probabilities of finding a particle in a particular state rather than deterministically predicting its position and velocity. The famous Schrödinger's Cat thought experiment and the Heisenberg Uncertainty Principle articulate the intrinsic probabilistic nature of quantum phenomena. Students face the challenge of embracing the idea that an electron can exist in a superposition of states or possess wave-particle duality, transforming our understanding of the physical reality at the most fundamental level.