Quantum Tunnelling: The WKB Approximation and the Gamow Factor

The exact rectangular barrier

For a particle of energy E < V₀ incident on a rectangular barrier of height V₀ and width L, matching the wavefunction and its derivative at both boundaries gives a closed-form transmission coefficient. Three regimes emerge depending on how large κL is:

κ = √(2m(V₀−E)) / ħ
T = [1 + V₀²·sinh²(κL) / (4E(V₀−E))]⁻¹
κL ≫ 1 (thick/tall barrier):  T ≈ 16·E(V₀−E)/V₀² · e^(−2κL)

The thick-barrier row is the key result: T falls off exponentially with κL — a factor of e in T per extra unit of width — which is why tunnelling is negligible for macroscopic objects but dominant for electrons and light nuclei.

WKB: generalising to any barrier shape

Real barriers — the Coulomb tail around a nucleus, the vacuum gap under an STM tip — aren't rectangular. The Wentzel–Kramers–Brillouin (WKB) approximation treats V(x) as locally rectangular at each point, defining a position-dependent decay constant κ(x), then integrates across the full classically forbidden region between turning points a and b (where E = V(a) = V(b)):

κ(x) = √(2m(V(x)−E)) / ħ
T ≈ exp[ −2∫ₐᵇ κ(x) dx ]     ← the Gamow tunnelling integral

George Gamow applied exactly this formula in 1928 to explain alpha decay: an alpha particle trapped by the strong nuclear force still faces a long-range Coulomb barrier on the way out. The Gamow integral through that barrier reproduces the Geiger–Nuttall law spanning over twenty orders of magnitude in half-life — for a 5 MeV alpha, T can be as small as 10⁻³⁸, yet the particle "attempts" the barrier ~10²¹ times per second, so decay still happens on human timescales for many isotopes.

Where WKB breaks down

WKB assumes the local de Broglie wavelength varies slowly. It fails right at the turning points, where κ(x) → 0 and the naive WKB wavefunction diverges — fixed with Airy-function connection formulas — and for very thin or low barriers (κL ≲ 1), where the exact solution should be used instead. Despite these caveats, WKB is accurate to a few percent for essentially every barrier relevant to nuclear and semiconductor physics: alpha decay, STM tunnelling currents ∝ e^(−2κd), Fowler-Nordheim tunnelling in flash memory, and negative-differential-resistance tunnel diodes all rest on this single integral.

Frequently asked questions

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