Transmission: —
The WKB (Wentzel-Kramers-Brillouin) approximation gives the tunneling probability through a classically forbidden region. Where E < V(x), the wave function decays as ψ ~ exp(−∫κ dx) where κ(x) = √(2m(V−E))/ℏ. The transmission coefficient T ≈ exp(−2∫κ dx) — a single integral over the barrier. For a rectangular barrier: T ≈ exp(−2κa) with κ = √(2m(V₀−E))/ℏ. The oscillating wave outside the barrier is shown in blue (incident + reflected region) and yellow (transmitted region), with the exponential decay (red) inside the barrier.