de Broglie Wavelength

de Broglie wavelength

λ = h / p

For non-relativistic particle of mass m at velocity v: p = mv, so:

λ = h / (m · v)

For particle with kinetic energy KE: p = √(2m·KE), so:

λ = h / √(2m·KE)

For photon: massless, p = E/c, so λ = hc/E.

Wavelengths of various particles

Particle	m, v	λ
Electron at 100 eV	9.11e-31 kg, ~6e6 m/s	0.12 nm
Electron in TEM (100 kV)	~10⁸ m/s (relativistic)	~4 pm
Thermal electron (300 K)	~10⁵ m/s	~7 nm
Hydrogen atom (300 K)	1.67e-27, ~2700 m/s	~1 Å
Buckyball C60 (slow)	1.2e-24, ~200 m/s	~3 pm
Bullet (10 g at 800 m/s)	0.01 kg	~10⁻³⁴ m
Tennis ball (60 g at 50 m/s)	0.06 kg	~10⁻³⁴ m
Person walking	70 kg, 1 m/s	~10⁻³⁵ m

Heavy or fast objects → tiny λ → utterly negligible quantum effects.

JavaScript — de Broglie calculations

const h = 6.626e-34;
const m_e = 9.11e-31;
const m_n = 1.675e-27;  // neutron
const m_p = 1.673e-27;  // proton
const eV = 1.602e-19;

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

function deBroglieFromKE(mass, KE_J) {
  return h / Math.sqrt(2 * mass * KE_J);
}

// Various electrons
console.log(`Thermal electron: ${(deBroglie(m_e, 1e5) * 1e9).toFixed(2)} nm`);
console.log(`100 eV electron: ${(deBroglieFromKE(m_e, 100*eV) * 1e9).toFixed(2)} nm`);
console.log(`10 keV electron: ${(deBroglieFromKE(m_e, 10000*eV) * 1e9).toFixed(3)} nm`);

// For relativistic electron, need full formula. Approximate non-relativistic up to ~50 keV.
function deBroglieRelativistic(KE_J, rest_mass = m_e) {
  // E_total = KE + m·c² ; p² = (E_total/c)² - (m·c)²
  const c_speed = 3e8;
  const E_total = KE_J + rest_mass * c_speed * c_speed;
  const p = Math.sqrt(E_total*E_total / (c_speed*c_speed) - rest_mass*rest_mass*c_speed*c_speed);
  return h / p;
}

// 100 keV electron (relativistic correction matters)
console.log(`100 keV electron (relativistic): ${(deBroglieRelativistic(100000*eV) * 1e12).toFixed(2)} pm`);

// Neutron at 0.025 eV (thermal)
console.log(`Thermal neutron: ${(deBroglieFromKE(m_n, 0.025*eV) * 1e10).toFixed(2)} Å`);
// ~1.8 Å — perfect for crystal diffraction

// Macroscopic object — to confirm utterly small
console.log(`Baseball at 30 m/s: ${deBroglie(0.145, 30).toExponential(2)} m`);  // 10⁻³⁴

// Microscope resolution comparison
function diffractionLimit(wavelength) {
  return wavelength / 2;  // very rough
}

console.log(`Visible light (550 nm) limit: ${(diffractionLimit(550e-9) * 1e9).toFixed(0)} nm`);
console.log(`100 kV electron limit: ${(diffractionLimit(deBroglieRelativistic(100000*eV)) * 1e12).toFixed(2)} pm`);

// What v does an electron need to have λ = 1 nm?
function velocityForWavelength(mass, wavelength) {
  return h / (mass * wavelength);
}

console.log(`Electron for 1 nm λ: ${velocityForWavelength(m_e, 1e-9).toExponential(2)} m/s`);
// ~7e5 m/s — about 0.2% c, non-relativistic

Where de Broglie wavelength matters

• Quantum mechanics foundation. Schrödinger's equation describes evolution of matter waves.
• Electron microscopy. Resolves atoms by exploiting electrons' tiny λ.
• Neutron diffraction. Crystal structure, magnetic materials, polymers.
• Atomic interferometry. Measure gravity, rotation, fundamental constants with cold atoms.
• X-ray crystallography. X-rays have de Broglie wavelengths ~atomic spacing.
• Quantum dots. Confinement at de Broglie scale gives discrete energy levels.
• Cold atom experiments. Ultracold atoms have macroscopically large λ — Bose-Einstein condensates show wave behavior.

Common mistakes

• Using non-relativistic formula at high speeds. p = m·v fails when v approaches c. Use p = γm·v or full relativistic formulas.
• Confusing momentum and velocity. p ≠ m·v in relativity. λ = h/p, not h/(mv) at high speeds.
• Thinking wave is "physical." de Broglie wave is a probability amplitude, not a physical oscillation. |Ψ|² gives where you'd find the particle.
• Treating macroscopic objects as quantum. Their λ is so small (10⁻³⁵ m or less) that quantum effects are invisible. Only quantum at atomic scales.
• Using just thermal velocity. For room-T thermal motion, KE = (3/2)kT. λ depends on mass (heavy molecules → smaller λ at same T).
• Forgetting massless photons. For light, λ = h/p where p = E/c. Photon de Broglie reduces to standard λ = c/f relation.