A semiclassical (WKB) asymptotic method for linear differential equations with a small parameter (often ħ or 1/k) that seeks solutions in the form ψ(x)=A(x) e^{i S(x)/ħ} (or exponential analogues) so that at leading order S satisfies an eikonal (Hamilton–Jacobi) equation and A satisfies a transport equation; applicable when the coefficients vary slowly relative to the local wavelength.

Principle

Balance the highest-derivative term with the rapidly varying phase to obtain, at leading order, a local dispersion (eikonal) relation; amplitude follows from a transport equation ensuring conservation of flux. Validity requires slowly varying coefficients compared with the local oscillation scale and absence of nearby turning points.

Demonstration

Illustrative scenario: For the one-dimensional Schrödinger equation −(ħ^2/2m)ψ'' + V(x)ψ = Eψ with V(x) varying slowly, the WKB leading-order gives ψ(x) ≈ C [2m(E−V(x))]^{-1/4} exp(± i ∫^x √{2m(E−V(x'))}/ħ dx'), with exponential decay in classically forbidden regions obtained by analytic continuation of the phase integral.

Misapplication

Applying WKB across a turning point (where E−V(x)=0) without using connection formulas: the error is ignoring that the local wavelength diverges and the WKB ordering fails there, so naive matching gives incorrect amplitudes and phases.

Consequence

When applicable, WKB yields accurate local approximations, semiclassical quantization rules, and tunneling estimates derived from action integrals; misapplication near turning points or rapid variations gives qualitatively wrong transmission/reflection and incorrect quantization.

Reversal

Breaks down near turning points, caustics, or regions where the potential varies on the scale of the wavelength; use uniform approximations (Airy-function matching), connection formulae, or exact/numerical solutions in such regions. Multidimensional extensions require additional geometric considerations (Maslov indices, Lagrangian manifolds).

Boundary

Applies to linear second-order (and some higher-order) wave equations with a clear small parameter and smoothly varying coefficients; excludes strongly disordered media, rapid spatial variations on the wavelength scale, and inherently nonlinear wave equations.

Semantic Tension

Provides a direct link between classical action and wave behavior (semiclassical insight) but competes with exact or numerical methods where global accuracy or behavior near singularities matters; tension between locality of WKB and the need for global matching.

Synthesis

WKB translates local wave propagation into classical action language: phase is governed by the eikonal (classical) action and amplitude by transport, giving a practical semiclassical bridge between differential equations and particle-like trajectories while requiring careful treatment of singular regions.