WKB Tunneling Through Shaped Barriers

The WKB approximation gives the semiclassical tunneling transmission T ≈ exp(−2∫|p(x)|dx/ℏ) where p(x) = √(2m(V(x)−E)) in the classically forbidden region. Different barrier shapes give dramatically different transmission coefficients, and a double barrier creates resonant tunneling peaks.

Barrier Shape

Type: Barrier height V₀ (units of E₀): 2.0 Barrier width a (units): 1.0 Particle energy E: 1.0

T = —

WKB integral: γ = ∫√(2m(V−E))/ℏ dx over classically forbidden region. T ≈ e^(−2γ). For double barrier, resonant tunneling occurs when quasi-bound state energy matches E.