Heisenberg Uncertainty Principle

Statement

For any quantum state, the standard deviations (uncertainties) of position and momentum satisfy:

Δx · Δp ≥ ℏ / 2

where ℏ = h/(2π) ≈ 1.055 × 10⁻³⁴ J·s.

This is fundamental — not due to measurement disturbance, but a property of quantum states themselves.

Other uncertainty relations

Pair	Relation
Position - Momentum	Δx · Δp ≥ ℏ/2
Energy - Time	ΔE · Δt ≥ ℏ/2
Angular momentum - Angle	ΔL · Δθ ≥ ℏ/2
Number - Phase (photons)	ΔN · Δφ ≥ 1/2

General — for any non-commuting observables [A, B] ≠ 0, exact simultaneous knowledge impossible.

Numerical examples

System	Δx	Min Δp	Min Δv
Electron in atom (~1 Å)	10⁻¹⁰ m	5.3 × 10⁻²⁵ kg·m/s	~6 × 10⁵ m/s
Electron in nucleus (~10⁻¹⁵ m)	10⁻¹⁵ m	5.3 × 10⁻²⁰ kg·m/s	~6 × 10¹⁰ m/s (relativistic!)
Bullet (1 cm precision)	10⁻² m	5.3 × 10⁻³³ kg·m/s	~5.3 × 10⁻³¹ m/s (negligible)
Excited atom (10 ns lifetime)	—	—	ΔE ≥ 5.3 × 10⁻²⁶ J ≈ 3 × 10⁻⁷ eV (linewidth)

JavaScript — uncertainty calculations

const h_bar = 1.055e-34;

// Position-momentum: minimum Δp for given Δx
function minMomentumUncertainty(deltaX) {
  return h_bar / (2 * deltaX);
}

// Velocity uncertainty for particle of mass m
function minVelocityUncertainty(deltaX, mass) {
  return minMomentumUncertainty(deltaX) / mass;
}

// Electron localized to atom (~1 Å)
const m_e = 9.11e-31;
console.log(`Electron in atom: Δv ≥ ${(minVelocityUncertainty(1e-10, m_e)).toExponential(2)} m/s`);
// ~5.8e5 m/s — comparable to actual orbital velocity

// Electron in nucleus
console.log(`Electron in nucleus: Δv ≥ ${(minVelocityUncertainty(1e-15, m_e)).toExponential(2)} m/s`);
// ~5.8e10 m/s — much greater than c! Means electron CAN'T be confined to nucleus

// Energy-time uncertainty
function minEnergyUncertainty(deltaT) {
  return h_bar / (2 * deltaT);  // J
}

const eV = 1.602e-19;
function minEnergyEv(deltaT) { return minEnergyUncertainty(deltaT) / eV; }

// Excited atom with 1 ns lifetime
console.log(`1 ns excited state: ΔE ≥ ${minEnergyEv(1e-9).toExponential(2)} eV`);
// ~3.3e-7 eV — natural linewidth

// Hydrogen 2p → 1s lifetime ~ 1.6 ns; gives Lyman-α natural linewidth
console.log(`H Lyman-α linewidth: ${minEnergyEv(1.6e-9).toExponential(2)} eV`);

// Macroscopic check: 1 kg ball localized to 1 mm
const m_ball = 1;  // kg
console.log(`1 kg ball Δv: ${(minVelocityUncertainty(1e-3, m_ball)).toExponential(2)} m/s`);
// 5.3e-32 m/s — utterly negligible

// Compton wavelength: when uncertainty starts requiring relativity
function comptonWavelength(mass) {
  return h_bar / (mass * 3e8);
}

console.log(`Electron Compton: ${(comptonWavelength(m_e) * 1e12).toFixed(2)} pm`);
// ~ 0.4 pm — at this scale, relativistic effects become important

// Confinement energy: minimum KE for particle in box
function confinementEnergy(L, mass) {
  // From uncertainty: p ~ ℏ/L, KE ~ p²/(2m)
  const p = h_bar / L;
  return p * p / (2 * mass);
}

// Electron in 1 nm box
console.log(`Electron in 1 nm box: KE ~ ${(confinementEnergy(1e-9, m_e) / eV).toFixed(2)} eV`);

// Atomic radius from uncertainty + Coulomb
function bohrRadiusEstimate() {
  // Confinement energy = Coulomb at atom radius
  // ℏ²/(2m·a²) = ke²/a → a = ℏ²/(m·k·e²) ≈ 5.3 × 10⁻¹¹ m
  return h_bar * h_bar / (m_e * 8.99e9 * 1.602e-19 * 1.602e-19);
}

console.log(`Bohr radius from uncertainty: ${(bohrRadiusEstimate() * 1e10).toFixed(2)} Å`);

Where uncertainty matters

• Atomic structure. Why electrons don't collapse to nucleus; sets atom size.
• Quantum tunneling. Energy-time uncertainty enables tunneling through barriers.
• Particle physics. Virtual particles (very short Δt → large ΔE allows them); particle widths.
• Atomic clocks. Linewidth limits precision; cool atoms to reduce velocity uncertainty.
• Heisenberg microscope thought experiment. Pedagogical aid for understanding measurement disturbance.
• Quantum cryptography. Eavesdropping disturbs states; can't be done without detection (BB84 protocol).
• Cosmology. Vacuum fluctuations from energy-time uncertainty; quantum origin of structure in Big Bang.

Common mistakes

• Treating it as measurement limitation. NO — it's intrinsic. Quantum states can't have arbitrarily precise x AND p simultaneously.
• Using ℏ instead of ℏ/2. Strict bound is ℏ/2 (Heisenberg's original was ~ℏ; Robertson tightened to ℏ/2).
• Confusing Δx with measurement error. Δx is standard deviation of position distribution — even if measured perfectly, the state has spread.
• Applying to macroscopic. ℏ so small that uncertainty is negligible at human scales.
• Misusing energy-time form. ΔE·Δt ≥ ℏ/2 has subtle interpretation — t isn't an observable but a "duration" parameter. Different from x-p which both have operators.
• Treating it as "nature is fuzzy." Quantum systems have well-defined states (with their own uncertainty). It's not vagueness, it's a precise mathematical structure.