Quantum Tunneling: the WKB Approximation

Exact solution: the rectangular barrier

Consider a particle of energy E < V₀ incident on a rectangular barrier of height V₀ and width L. Inside the barrier the TISE −ℏ²/(2m)ψ'' + V₀ψ = Eψ has exponential, not oscillatory, solutions, because E − V₀ < 0:

Matching ψ and ψ' at both boundaries x = 0 and x = L (continuity is required because the potential is finite) gives a unique solution with a non-zero amplitude on the far side — the particle has a non-zero probability of appearing beyond the barrier, something strictly forbidden classically.

The transmission coefficient

The transmission coefficient T = |ψ_transmitted|² / |ψ_incident|² for the rectangular barrier has an exact closed form:

The last row is the key result: for a wide or high barrier, T falls off exponentially with κL. This exponential sensitivity — a factor of e in T per (ℏ/√(2m(V₀−E))) of extra width — is why tunneling is negligible for everyday macroscopic objects but dominant for electrons and light nuclei.

The WKB idea: a slowly-varying barrier

Real barriers (a nucleus's Coulomb + strong-force potential, the gap between an STM tip and a surface) are not rectangular. The WKB (Wentzel–Kramers–Brillouin) approximation handles arbitrary V(x) by treating it as locally rectangular: at each x define a local decay constant κ(x) and assume the wave function still looks locally exponential, provided V(x) varies slowly on the scale of the local wavelength:

The 1/√κ(x) prefactor is required by probability-current conservation as the local "wavelength" changes; it is the same trick used in the WKB treatment of slowly-varying oscillatory potentials for bound states (Bohr-Sommerfeld quantisation).

The Gamow tunneling integral

Applying WKB across the full classically forbidden region — from the inner turning point a (where E = V(a)) to the outer turning point b (where E = V(b)) — gives the Gamow factor:

George Gamow used exactly this formula in 1928 to explain alpha decay: the alpha particle inside a nucleus is trapped by a strong nuclear well but faces a long-range Coulomb barrier on the outside. The tunneling integral through the Coulomb tail predicts the Geiger–Nuttall law — an exponential relationship between decay half-life and alpha energy that spans over twenty orders of magnitude in half-life, matched almost perfectly by the Gamow factor.

Validity: when does WKB break down?

WKB assumes the local de Broglie wavelength changes slowly, formally |dλ/dx| ≪ 1. It fails in two specific places:

• Near the turning points a and b, where E = V(x) and κ(x) → 0 — the WKB wave function itself diverges there. The fix is the Airy-function connection formulas, which patch the oscillatory and exponential WKB solutions smoothly across each turning point.
• For very thin or low barriers (κL ≲ 1), where the exponential approximation loses accuracy and the exact solution (or a numerical one) should be used instead.

Despite these limitations, WKB is accurate to a few percent for essentially all barriers relevant to nuclear and semiconductor physics, because those barriers are, in the relevant sense, "wide" compared to the particle's wavelength.

Applications: decay, STM, tunnel diodes

Computing T numerically

For an arbitrary barrier V(x), the Gamow integral is evaluated by numerical quadrature over the classically forbidden region:

// Gamow / WKB transmission for an arbitrary barrier V(x)
function wkbTransmission(V, E, a, b, hbar, m, steps = 2000) {
  let integral = 0;
  const dx = (b - a) / steps;
  for (let i = 0; i < steps; i++) {
    const x = a + (i + 0.5) * dx;
    const Vx = V(x);
    if (Vx > E) {
      // classically forbidden: kappa(x) is real
      const kappa = Math.sqrt(2 * m * (Vx - E)) / hbar;
      integral += kappa * dx;
    }
  }
  return Math.exp(-2 * integral);   // T ≈ e^{-2∫κ dx}
}

// Example: parabolic barrier V(x) = V0 * (1 - (x/L)^2)
const T = wkbTransmission(
  x => V0 * (1 - (x / L) ** 2),
  E, -L, L, HBAR, MASS
);