Quantum Tunneling — WKB Transmission

Quantum Barrier

Particle energy E: 0.40 (V₀ units) Barrier height V₀: 1.0 Barrier width d: 1.0 (λ_dB units) Mass m/mₑ: 1.0 Barrier shape:

WKB T—

Exact T (square)—

Decay κ·d—

λ_dB—

WKB tunneling: T ≈ exp(−2∫√(2m(V(x)−E)/ℏ²)dx) for E < V. Exact square barrier: T = [1 + (κ²+k²)²sinh²(κd)/(4k²κ²)]⁻¹. Resonances at integer half-wavelengths in well. Gamow factor (α-decay), Fowler-Nordheim (field emission), STM operation all use WKB tunneling.