the WKB approximation
/ double-yoo-kay-BEE /
When a potential varies slowly compared with the particle's wavelength, quantum mechanics is nearly classical, and the WKB approximation (after Wentzel, Kramers, and Brillouin) is the systematic way to exploit that. The picture: the wavefunction is a wave whose local wavelength adjusts to the local momentum, oscillating fast and short where the particle moves fast, stretching out where it moves slowly. It is the bridge between the wave description and the smooth classical trajectory.
In a classically allowed region where E > V(x), define the local momentum p(x) = sqrt(2 m (E - V(x))). WKB writes psi(x) approximately proportional to (1/sqrt(p)) exp(±(i/hbar) integral p(x) dx) -- an oscillation whose phase accumulates the classical action and whose amplitude 1/sqrt(p) keeps the probability current constant (the particle lingers where it is slow). In a classically forbidden region where E < V, the momentum becomes imaginary and the same formula turns into a decaying or growing exponential exp(±(1/hbar) integral |p| dx). Two payoffs follow. First, Bohr-Sommerfeld quantization: integral p dx over a full classical orbit equals (n + 1/2) 2 pi hbar, which reproduces bound-state energies without solving the differential equation. Second, the tunnelling factor: transmission through a barrier is approximately exp(-(2/hbar) integral sqrt(2 m (V - E)) dx) across the forbidden region, generalizing the rectangular-barrier result to any smooth barrier.
The method's power is that it turns quantum problems into integrals along classical paths, and it is exact in the limit hbar -> 0. But it is an approximation with a sharp failure mode: it breaks down exactly at the classical turning points, where E = V(x), because there p(x) -> 0 and the 1/sqrt(p) amplitude blows up. Stitching the oscillating and exponential solutions across a turning point requires the 'connection formulas' (obtained from a linear approximation and Airy functions), and it is this careful patching that delivers the crucial +1/2 in the quantization condition.
Applied to alpha decay, WKB integrates sqrt(2 m (V - E)) across the Coulomb barrier surrounding the nucleus. Because that integral grows steeply as the alpha energy drops, the predicted half-life swings over roughly 20 orders of magnitude for only a factor-of-two change in alpha energy -- the Geiger-Nuttall law, which WKB explains quantitatively.
The WKB barrier integral turns a modest change in decay energy into a huge change in half-life.
WKB is a semiclassical expansion in hbar, valid when the wavelength varies little over its own length; it fails at turning points, where the naive formula diverges and connection formulas are mandatory. The famous (n + 1/2) quantization is exact for the harmonic oscillator by luck, not in general -- for other potentials it is only leading order.