• WKB (Wentzel–Kramers–Brillouin): approximate solutions of the time-independent Schrödinger equation; used for bound-state energies and tunnelling rates.

• Validity: \(V(x)\) varies slowly on the scale of the local wavelength.

  • Classically allowed (\(E>V\)): \(\lambda=\dfrac{2\pi\hbar}{p}\), \(p(x)=\sqrt{2m(E-V)}\); require \(\Big|\dfrac{d\lambda}{dx}\Big|\ll1\).

  • Forbidden (\(E<V\)): decay length \(\dfrac{\hbar}{\gamma}\), \(\gamma=\sqrt{2m(V-E)}\); require \(\Big|\dfrac{d}{dx}\dfrac{\hbar}{\gamma}\Big|\ll1\).

WKB wave function in the allowed region

• \(\psi''=-\dfrac{p^2}{\hbar^2}\psi\); try \(\psi=A(x)e^{i\phi(x)}\) and separate real and imaginary parts: