Quantum Tunneling

Tunneling overview

Classically: particle with KE < V barrier reflects. Quantum: small but nonzero probability of passing through.

Wave function inside the barrier (where E < V) has the form Ψ ∝ exp(-κx), where κ = √(2m(V-E))/ℏ. This decays exponentially but doesn't vanish.

Tunneling probability

For rectangular barrier, height V, width d:

T ≈ exp(-2 · κ · d) = exp(-2d · √(2m(V−E)) / ℏ)

(Approximation; exact formula has small correction.) Key dependence:

• Higher barrier (V) → less tunneling.
• Wider barrier (d) → exponentially less tunneling.
• Heavier particle (m) → less tunneling.
• Particle energy closer to V → more tunneling.

Real-world tunneling

System	Tunneling effect
Stellar fusion	Protons fuse despite Coulomb barrier; rate ~ 10⁻²² per second per pair
Alpha decay	Alpha particle tunnels through nuclear barrier
Scanning tunneling microscope	Electrons tunnel from tip to surface; image atoms
Flash memory	Electrons tunnel into/out of floating gate; store bits
Tunnel diode	Negative differential resistance via tunneling
Josephson junction	Cooper pairs tunnel coherently in superconductor
Field emission	Electrons tunnel from cold cathodes
Enzyme catalysis	H tunneling in biological reactions

JavaScript — tunneling calculations

const h_bar = 1.055e-34;
const m_e = 9.11e-31;
const eV = 1.602e-19;

// Tunneling probability for rectangular barrier
function tunnelingProbability(mass, V_barrier_J, E_particle_J, d_meters) {
  const dE = V_barrier_J - E_particle_J;
  if (dE <= 0) return 1;  // Classically allowed
  const kappa = Math.sqrt(2 * mass * dE) / h_bar;
  return Math.exp(-2 * kappa * d_meters);
}

// Electron through 1 nm barrier of 1 eV
console.log(`e through 1nm × 1eV: ${tunnelingProbability(m_e, eV, 0, 1e-9).toExponential(2)}`);
// ~10⁻¹⁰ — tiny but nonzero

// Wider barrier (10 nm)
console.log(`e through 10nm × 1eV: ${tunnelingProbability(m_e, eV, 0, 10e-9).toExponential(2)}`);
// ~10⁻⁹⁹ — utterly tiny

// Higher barrier (4 eV)
console.log(`e through 1nm × 4eV: ${tunnelingProbability(m_e, 4*eV, 0, 1e-9).toExponential(2)}`);
// Much less than 1 eV barrier

// STM tip-surface gap exponential dependence
function stmCurrent(gap_nm, base_gap_nm = 0.5) {
  // Current ∝ exp(-2κ(gap - base_gap))
  // For typical work function 5 eV, κ ≈ 11.4 /nm
  const kappa_per_nm = 11.4;
  return Math.exp(-2 * kappa_per_nm * (gap_nm - base_gap_nm));
}

console.log(`Current at 0.5 nm: ${stmCurrent(0.5)}`);    // 1 (reference)
console.log(`Current at 0.6 nm: ${stmCurrent(0.6).toExponential(2)}`);  // ~10⁻¹ — 10× less
console.log(`Current at 1.0 nm: ${stmCurrent(1.0).toExponential(2)}`);  // ~10⁻⁵
// Tiny gap changes → huge current changes — exquisite atomic resolution

// Alpha decay (rough): U-238
function alphaDecayRate(Q_alpha_MeV, Z, mass_alpha = 6.64e-27) {
  // WKB approximation; very approximate
  // T = exp(-2π·η) where η is Sommerfeld parameter
  // Skip detailed calc; rough exponential dependence
  // Real Geiger-Nuttall law: ln(λ) ≈ a + b·Z/√Q
  return null;  // Use literature values
}

// Half-lives — sensitivity to Q (energy released)
const halflives = {
  'Polonium-212': '0.3 µs (Q = 8.95 MeV)',
  'Polonium-218': '3.1 min (Q = 6.11 MeV)',
  'Radium-226': '1600 years (Q = 4.87 MeV)',
  'Uranium-238': '4.5 Gyr (Q = 4.27 MeV)',
};
console.log(halflives);
// Note: 12% reduction in Q → 17 orders of magnitude change in half-life!
// Geiger-Nuttall law shows enormous sensitivity to barrier conditions

// Stellar fusion (rough): proton-proton chain
function ppFusionRate(T_K) {
  // Gamow factor: exp(-2π·Z₁Z₂·e²·m/(ℏ·v·... ))
  // Highly approximated; full formula has Gamow energy
  // Rate proportional to exp(-bT^(-1/3))
  return Math.exp(-25 * Math.pow(1.5e7 / T_K, 1/3));
}

console.log(`Solar core (1.5e7 K): rate ~ ${ppFusionRate(1.5e7).toExponential(2)} (relative)`);
console.log(`Cooler star (1e7 K): rate ~ ${ppFusionRate(1e7).toExponential(2)}`);
// Highly T-sensitive — that's why stars are stable thermometers

Where tunneling matters

• Stellar fusion. Stars don't have enough thermal energy to fuse without tunneling.
• Radioactive decay. Alpha decay rates set by tunneling probability.
• Microscopy. STM resolves individual atoms via tunneling current.
• Memory. Flash memory writes/erases by electron tunneling through oxide.
• Quantum computing. Some qubit designs (transmons) use Josephson junctions (tunneling-based).
• Biology. Enzyme catalysis sometimes uses proton/electron tunneling.
• Modern electronics. Tunneling becomes dominant leakage mechanism in nano-scale transistors (challenging for further miniaturization).

Common mistakes

• Thinking it's just probability without explanation. Tunneling has clear quantum origin: wave function extends through barriers.
• Linear vs exponential dependence. Tunneling is EXPONENTIAL in barrier width; small width changes → huge probability changes.
• Treating particles as classical. Macroscopic objects don't tunnel meaningfully (mass too large; tunneling probability tiny). Quantum tunneling matters at atomic scales.
• Forgetting energy conservation. Tunneling doesn't violate it — particle has same energy on both sides of barrier. Energy-time uncertainty allows brief borrowing.
• Confusing tunneling with photoelectric. Photoelectric — photon gives energy > barrier (work function). Tunneling — particle with E < barrier passes through anyway.
• Treating it as rare exception. Very common at nano scales — fundamental to chemistry, biology, electronics, astrophysics.