Wave-Particle Duality

Two faces of one thing

Behavior	Wave-like	Particle-like
Light	Diffraction, interference	Photoelectric, Compton
Electrons	Diffraction (Davisson-Germer)	Cathode rays, individual clicks
Atoms	Atomic interferometry	Mass spectrometry
Molecules	Buckyball interference	Chemistry, mass measurement

de Broglie wavelength

λ = h / p = h / (m·v)

Every particle has an associated wavelength. h = 6.626 × 10⁻³⁴ J·s (very small).

Object	Mass (kg)	Velocity (m/s)	Wavelength
Electron in atom	9.11e-31	~2.2e6	~3.3 Å (X-ray scale)
Electron in TEM	9.11e-31	~10⁸	~7 pm
Hydrogen atom (room T)	1.67e-27	~2e3	~2 Å
Buckyball C60 (slow)	1.2e-24	~100	~5 pm
Pollen grain	10⁻¹³	~10⁻³	~10⁻¹⁸ m (impossibly small)
Baseball	0.145	30	10⁻³⁴ m
Person walking	70	1	~10⁻³⁵ m

JavaScript — duality calculations

const h = 6.626e-34;
const c = 3e8;

// de Broglie wavelength
function deBroglie(mass, velocity) {
  return h / (mass * velocity);
}

// Electron at 10⁶ m/s
const m_e = 9.11e-31;
console.log(`Electron λ: ${(deBroglie(m_e, 1e6) * 1e10).toFixed(2)} Å`);  // ~7.3

// Photon: λ = h·c/E (massless, p = E/c)
function photonWavelength(energy_J) {
  return h * c / energy_J;
}

// 1 eV photon
const eV = 1.602e-19;
console.log(`1 eV photon: ${(photonWavelength(eV) * 1e9).toFixed(0)} nm`);  // 1240 nm (IR)

// Particle wavelength from kinetic energy
function deBroglieFromKE(mass, KE) {
  // p = √(2mKE)
  const p = Math.sqrt(2 * mass * KE);
  return h / p;
}

// 100 eV electron
console.log(`100 eV electron: ${(deBroglieFromKE(m_e, 100 * eV) * 1e9).toFixed(2)} nm`);

// Resolution limit comparison: electron microscope vs optical
function microscopeResolution(wavelength) {
  // Diffraction limit: ~ λ/2 for high-NA imaging
  return wavelength / 2;
}

console.log(`Visible light (550 nm): ${microscopeResolution(550e-9) * 1e9} nm (200 nm)`);
console.log(`100 keV electron:`,
  microscopeResolution(deBroglieFromKE(m_e, 100e3 * eV)) * 1e12, 'pm');
// Massive improvement — 1.85 pm vs 200 nm

// Determine if quantum effects matter (de Broglie ~ size of system)
function quantumMatter(particle_lambda, system_size) {
  return particle_lambda / system_size;  // if > 0.01 or so, quantum effects matter
}

// Electron in atom
console.log(quantumMatter(3.3e-10, 1e-10));  // ~3 — definitely quantum

// Electron in 1m wire
console.log(quantumMatter(3.3e-10, 1));  // ~3.3e-10 — classical

// Bullet vs system size
console.log(quantumMatter(deBroglie(0.005, 800), 0.01));  // ~10⁻³⁵ — utterly classical

Where wave-particle duality matters

• Quantum mechanics fundamentals. Underlies Schrödinger's equation, all quantum predictions.
• Electron microscopy. Electron wavelength much smaller than visible light → atomic-scale imaging.
• X-ray crystallography. X-rays have atomic-scale λ → diffract off crystals → reveal atomic structure.
• Quantum interference experiments. Test foundations of quantum mechanics; double-slit with molecules etc.
• Atomic clocks. Use atomic interferometry; precision time measurements.
• Quantum computing. Qubits exploit superposition (related to wave nature) and discrete states (particle).
• Particle physics. Detector design relies on particle aspect; beam guidance on wave aspect.

Common mistakes

• Treating wave and particle as simultaneous. They're complementary aspects revealed by different experiments. Bohr's complementarity principle — observing one excludes the other.
• Asking "is it really a wave or particle?" Neither. It's a quantum object. Both wave and particle are imperfect classical analogies.
• Forgetting particle aspect for visible objects. Macroscopic objects always look like particles (λ utterly small). Wave nature only matters at quantum scales.
• Misusing classical wave properties. Quantum waves are probability amplitudes (complex). Classical waves are physical (real-valued). Don't apply directly.
• Treating measurement as passive. Measurement disturbs quantum state. "Trying to see which slit" eliminates interference. Observer effect is real.
• Confusing duality with super-heroic physics. Photons aren't both ON and OFF; particles aren't both alive and dead. Duality is a precise mathematical property, not magic.