WKB approximation

The WKB approximation is the way to solve a wave-like differential equation when the medium changes slowly compared with the wavelength — the regime where a wave behaves almost like a ray of light bending gently through a varying material. It answers: how does a rapidly oscillating (or rapidly decaying) solution evolve when its local wavelength drifts from place to place? The answer is to track an accumulating phase plus a slowly adjusting amplitude.

Consider an equation like epsilon^2 y'' + Q(x) y = 0 with a small parameter epsilon (in quantum mechanics epsilon plays the role of Planck's constant). The WKB ansatz writes the solution as an exponential of a series in epsilon: y ~ exp((1/epsilon)(S_0(x) + epsilon S_1(x) + ...)). Substituting and matching powers of epsilon gives, at leading order, S_0' = plus or minus sqrt(-Q), an accumulated phase like the integral of sqrt(Q) (the eikonal), and at next order an amplitude factor that goes like Q^(-1/4). So where Q > 0 you get oscillation with a local wavenumber sqrt(Q) and amplitude tapering as Q^(-1/4); where Q < 0 you get exponential growth or decay. The approximation is excellent precisely when Q varies slowly on the scale of the local wavelength.

WKB is the bridge between wave mechanics and classical mechanics (the semiclassical limit), it gives the Bohr-Sommerfeld quantisation rule for energy levels, the exponential rate of quantum tunnelling through a barrier, and the propagation of waves in slowly varying media in optics, acoustics, and plasma physics. The honest caveat: the simple formulas blow up where Q(x) = 0 — the turning points where a classical particle would stop and reverse. There the amplitude prefactor Q^(-1/4) is singular, and you must patch the oscillatory and exponential regions together using a local Airy-function solution, the WKB connection formulae.