Quantum Physics & Wave-Particle Duality Notes

Wave Properties and Electromagnetism

• Wave Attributes:

  • Wavelength ($\lambda$): Distance per complete wave cycle.

  • Period ($T$): Time required for one complete cycle.

  • Frequency ($f$): Number of cycles per unit time ($f = \frac{1}{T}$).

  • Velocity ($v$): Speed of wave propagation ($v = \lambda f$); determined solely by the properties of the medium.

  • Amplitude ($A$): Maximum displacement from equilibrium; energy ($E$) and intensity ($I$) scale proportionally to amplitude squared ($E, I \propto A^2$).

• Wave Classifications:

  • Transverse waves: Oscillation is perpendicular to the direction of wave travel.

  • Longitudinal waves: Oscillation is parallel to the direction of wave travel.

• Unification of Electromagnetism and Light:

  • 1670: Olaus Roemer measures the speed of light using observations of Jupiter's moons, obtaining $\approx 2.3 \times 10^8\,\text{m/s}$.

  • 1820: Hans Christian Oersted discovers that electric currents in a wire produce magnetic fields.

  • 1831: Michael Faraday demonstrates electromagnetic induction, showing that a changing magnetic field induces an electric voltage.

  • 1873: James Clerk Maxwell publishes equations of electromagnetism, establishing that light is an electromagnetic wave.

Wave Phenomena: Diffraction and Interference

• Diffraction:

  • Spreading and bending of waves as they pass around obstacles or through narrow openings.

  • Pronounced diffraction occurs when the opening size is smaller than or comparable to the wavelength ($\lambda$).

  • The Poisson spot (observed by Fresnel and François Arago after Siméon Poisson's mathematical derivation) is a bright spot formed by constructive interference at the precise center of a circular object's shadow.

• Interference Principles:

  • Constructive interference: Wave crests align, resulting in added amplitude when path length difference $\varepsilon = \lambda, 2\lambda, 3\lambda, \dots$

  • Destructive interference: Wave crest aligns with a trough, canceling out amplitude when path length difference $\varepsilon = \frac{\lambda}{2}, \frac{3\lambda}{2}, \frac{5\lambda}{2}, \dots$

  • Applications: Active noise cancellation uses out-of-phase sound generation; anti-reflective coatings utilize destructive cancellation with thickness $d = \frac{\lambda}{2}$.

Particle Nature of Light and the Photoelectric Effect

• Photoelectric Effect:

  • Emission of electrons from a metal surface exposed to light.

  • Kinetic energy of ejected electrons increases linearly with light frequency ($f$), following $E = h f$.

  • No electrons are emitted below a specific threshold frequency, regardless of light intensity.

  • Experimental determination by Millikan (1916) yields Planck's constant $h = 4.1 \times 10^{-15}\,\text{eV}\cdot\text{s}$.

Wave-Particle Duality of Matter

• de Broglie Relation:

  • Matter exhibits wavelike behavior with wavelength $\lambda = \frac{h}{p}$, where momentum $p = m v$.

  • Photons possess zero rest mass, move at speed $c \approx 3.0 \times 10^8\,\text{m/s}$, and carry momentum $p = \frac{h}{\lambda}$ and energy $E = h f$.

• Experimental Evidence for Matter Waves:

  • 1897: J. J. Thomson identifies cathode rays as negatively charged particles (electrons).

  • 1912: Lawrence Bragg and William Bragg demonstrate X-ray diffraction off crystal lattices.

  • 1920–1927: Clinton Davisson, Charles Kunsman, and Lester Germer observe electron diffraction off an annealed, crystallized nickel target.

  • 1927: Davisson and Germer, along with George Thomson and Andrew Reid, publish decisive proof of electron diffraction (Davisson and G. P. Thomson shared the 1937 Nobel Prize).

  • 1961: Claus Jönsson performs double-slit interference using electrons.

  • 1988: Anton Zeilinger demonstrates double-slit interference with slow-moving neutrons.

  • 1991–2002: Double-slit interference confirmed for helium/sodium atoms (1991) and $C_{70}$ fullerene molecules (2002).

• Single-Particle Interference:

  • Even when microscopic entities (photons, electrons, neutrons) pass through double slits one particle at a time, an interference pattern accumulates over time, proving that individual quantum entities possess intrinsic wave properties.