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2: Wave-Particle Duality

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    In classical mechanics, waves and particles are two completely distinct types of physical entity. Waves are continuous and spatially extended, whereas particles are discrete and have little or no spatial extent. However, in quantum mechanics, waves sometimes act as particles, and particles sometimes act as waves—this strange behavior is known as wave-particle duality. In this chapter, we shall examine how wave-particle duality shapes the general features of quantum mechanics.

    • 2.1: Wavefunctions
      This page discusses waves as periodic disturbances characterized by a wavefunction in one dimension, illustrating variations in properties like pressure and electric fields. It outlines key characteristics including wavelength, period, amplitude, wavenumber, angular frequency, and phase angle. Waves oscillate between extremities and propagate directionally, with their velocity determined by the relationship \(v=\omega/k\).
    • 2.2: Plane-Waves
      This page explains the characteristics of one-dimensional and three-dimensional plane waves, described by specific wavefunctions. The one-dimensional plane wave is represented as \(\psi(x,t)=A\,\cos(k\,x-\omega\,t+\varphi)\), with maxima along parallel planes moving in the positive \(x\)-direction. The three-dimensional wave is described by \(\psi({\bf r},t)=A\,\cos({\bf k}\cdot{\bf r}-\omega\,t+\varphi)\), where the wavevector \({\bf k}\) indicates direction and wavenumber.
    • 2.3: Representation of Waves via Complex Functions
      This page introduces complex numbers, composed of a real part and an imaginary part, represented as \(z = x + {\rm i}\,y\). It explains their visualization in an infinite plane and discusses modulus and argument. Euler's theorem links complex numbers with wavefunctions, facilitating the representation of real functions. The page concludes that this complex representation simplifies the combination of amplitude and phase, leading to a compact notation for wavefunctions across different dimensions.
    • 2.4: Classical Light-Waves
      This page explains the properties of a monochromatic, linearly-polarized plane light wave in a vacuum, detailing the electric field represented by a complex wavefunction. It outlines the dispersion relation connecting frequency and wavelength, emphasizing a constant phase-velocity \(c\). Additionally, it covers the wave's energy density and indicates that momentum density is proportional to energy density divided by the speed of light.
    • 2.5: Photoelectric Effect
      This page discusses the photoelectric effect, discovered by Heinrich Hertz in 1887, where light of a certain frequency causes electron emission from a metal surface. Key points include the requirement of a threshold frequency, proportionality between light intensity and photoelectron current, and the independence of electron energy from intensity.
    • 2.6: Quantum Theory of Light
      This page explains Einstein's quantum theory, which characterizes monochromatic light as a stream of massless particles called photons, with energy formulated as \(E = \hbar\,\omega\). It details that these photons move at light speed \(c\), aligning with special relativity's stipulation for massless particles. Additionally, it addresses the energy-momentum relationship for these particles, defining momentum as \(p = \frac{E}{c}\) and relating it to the wave number with \(p = \hbar\,k\).
    • 2.7: Classical Interferences of Light Waves
      This page explains the classical interference of light waves using the double-slit experiment. It details how monochromatic light creates an interference pattern due to path and phase differences from two slits. The intensity of the pattern is derived from the superposition of wavefunctions, resulting in alternating light and dark bands, spaced according to the formula \(\Delta y = \frac{D\,\lambda}{d}\).
    • 2.8: Quantum Interference of Light
      This page explores the quantum-mechanical interpretation of double-slit light interference, illustrating that photons exhibit both particle and wave characteristics. Even with single photons, an interference pattern forms, signaling a probability distribution tied to the wavefunction. However, measuring a photon's path eliminates the interference, highlighting the probabilistic nature of photons and the fundamental wave-particle duality that defies deterministic predictions.
    • 2.9: Particles
      This page examines the behavior of classical and quantum particles, particularly focusing on non-relativistic particles like electrons. It describes their motion under no external forces, connecting concepts of velocity, energy, and momentum. The introduction of wave-particle duality illustrates the wave-like properties of massive particles, demonstrated through de Broglie wavelengths and interference patterns.
    • 2.10: Wave-Packets
      This page explains the wavefunction of a massive particle, expressing it as a plane wave for equal probability across the x-axis. Localizing the particle uses superposition leading to a Gaussian distribution, emphasizing higher probability near a specific position. The relationship between position and momentum spaces is illustrated through the Fourier transform, showing Gaussian distributions in both.
    • 2.11: Evolution of Wave-Packets
      This page explores the time evolution of a particle's wavefunction through its Fourier transform, illustrating how it evolves and spreads over time, governed by dispersion relations. It introduces group velocity and contrasts light and particle wave-packet characteristics, noting that light packets maintain shape due to linear dispersion, while particle packets disperse, highlighting Heisenberg's uncertainty principle.
    • 2.12: Heisenberg’s Uncertainty Principle
    • 2.13: Schrodinger's Equation and Wavefunction Collapse
      This page discusses the wavefunction of a free particle and introduces Schrödinger's equation, which governs its time evolution. It explains the wavefunction's integral representation and its dynamics under varying potential energy. The text also addresses wavefunction collapse during measurement, outlining the difference between continuous evolution via Schrödinger's equation and the abrupt changes during measurement, emphasizing fundamental principles in quantum mechanics.
    • 2.14: Exercises
      This page covers multiple aspects of quantum mechanics, including photon emission from a He-Ne laser, hydrogen ionization energy, and photoelectron energy calculations that help establish Planck's constant and aluminum's work function. It further explores de Broglie wavelengths, electron diffraction, waveguide characteristics, and the uncertainty principle related to electron confinement, along with Schrödinger equation solutions for specific wavefunctions.

    This page titled 2: Wave-Particle Duality is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.