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
Turning points, where E ≈ V(x), require connection formulas to match solutions across regions. This leads to
Limitations include breakdown near turning points and caustics, and the method assumes slowly varying potentials and,