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: \(\sigma_x \sigma_p \geq \hbar/2\).