Wave-Particle Duality

Chem 3240 · Lecture 1.5

Davit Potoyan

The Big Idea

  • Particles and waves are not mutually exclusive

  • Every quantum object shows both behaviors

  • An electron has a wavelength; a photon has momentum

  • Which behavior dominates depends on the experimental conditions

Diffraction and Interference

  • Diffraction: waves spread around obstacles or through openings
  • Interference: combined waves add (in phase) or cancel (out of phase)
  • Double slit: two slits produce interference bands on the screen

Bragg’s Law: X-rays Scatter Off Crystals

  • X-rays scatter off lattice atoms
  • Path differences give constructive (left) or destructive (right) interference

\[\boxed{2d \sin\theta = n\lambda}\]

  • \(d\): plane spacing, \(\lambda\): wavelength, \(n\): diffraction order
  • Interference was the hallmark of wave behavior

Electrons Also Diffract

  • Davisson and Germer (1925): intensity peaks in scattered electron beams
  • Peaks fit Bragg’s law (until then only for X-rays)
  • Electrons behave as waves

Compton Scattering: Light Has Momentum

  • X-rays scatter off free electrons like billiard balls
  • Conservation of momentum gives longer outgoing wavelength
  • Makes sense only if the photon is a particle with momentum

The de Broglie Relation

  • Wave-like and particle-like traits are inversely proportional

\[\boxed{\lambda = \frac{h}{p}}\]

  • \(h\): Planck’s constant, \(p\): momentum, \(\lambda\): wavelength

  • Heavy/fast objects: tiny wavelength (particle-like)

  • Light/slow objects: large wavelength (wave-like)

  • With \(E = T + V\): \(\quad \lambda = \dfrac{h}{\sqrt{2m(E - V)}}\)

  • Wavelength changes as a particle moves through different potentials (key for bonding)

The Double-Slit Puzzle

  • Electrons build an interference pattern
  • It persists even with one electron at a time
  • Each electron interferes with itself

Which Slit?

  • Try to detect which slit the electron took
  • The interference pattern disappears
  • Measurement changes the outcome (resolved later via QM postulates)

Heisenberg’s Uncertainty Principle

  • Cannot know exact position and momentum at once
  • Narrow the slit (localize \(x\)): momentum spreads out
  • A direct consequence of wave-particle duality

\[\sigma_x \sigma_p \geq \hbar/2\]

Takeaway

Every quantum object is both wave and particle, with wavelength set by \(\lambda = h/p\), so position and momentum can never both be sharp: \(\sigma_x \sigma_p \geq \hbar/2\).