 WKB approximation

The WKB approximation separates rapid phase variation from a slowly varying amplitude in a linear ordinary differential equation. Foron an interval with smooth $q>0$, its leading oscillatory form isSubstitution of $y=A{e}^{iS/\epsilon }$ gives $S^{\prime 2}=q$ at leading order and $2S^{\prime }{A}^{\prime }+S^{\prime \prime }A=0$ at the next order, hence $A\propto q^{-1/4}$. This is the mathematical origin of the flux-normalized WKB amplitude.

Where $q<0$, the leading branches instead grow and decay exponentially, with $|q{|}^{-1/4}$ amplitude and exponents $\pm {\epsilon }^{-1}\int \sqrt{|q|}dx$. A wave turning point at $q=0$ makes this formula singular and requires an Airy turning-point connection formula for a simple zero. Slow variation and an error estimate are essential; the displayed expression is not an exact solution for arbitrary $q$. The real exponential version has a rigorous error-control theorem.^[ref]

The WKB approximation, short for the Wentzel-Kramers-Brillouin approximation, is a mathematical technique used primarily in quantum mechanics to find approximate solutions to the Schrödinger equation in the semiclassical limit, where quantum effects can be approximated by classical trajectories. The WKB method arises when studying quantum systems with a potential that varies slowly compared to the wavelength of the particle.