WKBApproximation

WKB approximation, named after Wentzel, Kramers, and Brillouin, is a semiclassical method for solving linear differential equations with a small parameter, typically Planck’s constant ħ. In quantum mechanics it provides approximate solutions to the Schrödinger equation in the limit of large action compared with ħ, linking wave behavior to classical trajectories. The method is closely related to the Liouville–Green (eikonal) approximation and has applications in wave propagation and other wave systems.

In one dimension, for the time-independent Schrödinger equation - (ħ^2/2m) ψ''(x) + V(x) ψ(x) = E ψ(x), the WKB ansatz expresses the wavefunction as ψ(x) ≈ A(x) exp(i S(x)/ħ). Solving to leading order yields the eikonal equation (S')^2 = 2m(E - V). In classically allowed regions where E > V, the WKB form is ψ(x) ≈ C/√p(x) exp(± i ∫ p(x)/ħ dx), with p(x) = √(2m(E - V(x))). In classically forbidden regions where E < V, the solutions decay or grow exponentially: ψ(x) ≈ C/√|κ(x)| exp(−∫ |κ(x)|/ħ dx), with κ(x) = √(2m(V(x) - E)).

Turning points, where E ≈ V(x), require connection formulas to match solutions across regions. This leads to the Bohr–Sommerfeld quantization condition ∮ p dx = 2πħ(n + 1/2) for bound states in smooth potentials.

Limitations include breakdown near turning points and caustics, and the method assumes slowly varying potentials and, in multiple dimensions, careful treatment of Maslov indices. Extensions cover multi-dimensional WKB, JWKB, and semiclassical quantization methods. WKB remains a foundational tool for understanding quantum tunneling, molecular spectra, and semiclassical approximations in various physical contexts.