WKB Approximation — Semiclassical Bound States
Quantization condition ∮p·dq = (n+½)h and tunneling through barriers
WKB approximation (Wentzel-Kramers-Brillouin 1926): ψ(x) ≈ A/√|p(x)| · exp(±i∫p·dx/ℏ) where p(x)=√(2m(E−V(x))).
Bohr-Sommerfeld quantization: ∮ p·dq = (n+½)h — area enclosed by classical orbit in phase space is quantized in units of h (with Maslov correction ½).
Valid when: |dλ/dx| ≪ 1, i.e., de Broglie wavelength changes slowly. Breaks down at turning points (Airy function connections).
Tunneling: In classically forbidden regions V>E: ψ ~ exp(−∫|p|dx/ℏ). Transmission T ~ exp(−2∫√(2m(V−E))dx/ℏ).
Accuracy: Improves as ℏ→0. Harmonic oscillator energy levels exact to all orders by WKB due to special structure.