Wave-Particle Duality

Chem 3240 · Lecture 1.4

Davit Potoyan

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 or destructive interference

2d \sin\theta = n\lambda

  • d: plane spacing, n: diffraction order

Electrons Also Diffract

  • Davisson and Germer (1927): 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

p_{photon} = \frac{h}{\lambda}

  • Wavelength shift: \Delta\lambda = \dfrac{h}{m_e c}(1-\cos\theta)

The Pattern So Far

  • X-rays (waves) scatter like billiard balls: light has momentum
  • Electrons (particles) show Bragg peaks: matter has a wavelength
  • Particles and waves are not mutually exclusive; every quantum object shows both
  • Which behavior dominates depends on the experimental conditions

The de Broglie Relation

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

\lambda = \frac{h}{p}

  • Heavy/fast objects: tiny wavelength (particle-like); light/slow objects: large wavelength (wave-like)
  • With E = T + V: \lambda = \dfrac{h}{\sqrt{2m(E - V)}}, so the wavelength changes with the potential

Waves Have to Fit

  • Wrap the wave around a loop: only a whole number of wavelengths survives
  • 2\pi r = n\lambda = nh/p \;\Rightarrow\; mvr = n\hbar: quantization, from a wave

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.