WKB vs Exact Transmission — Rectangular Barrier

Exact quantum-mechanical transmission (RED) versus WKB approximation (SLATE dashed) for a rectangular potential barrier. Below the barrier: both shown. Above: resonances where T→1 (only exact shown).

log₁₀(T) vs Energy E (eV)

V₀ (eV) 1.00 eV

L (nm) 1.00 nm

T_exact (both tunneling and above-barrier)

T_WKB = exp(−2κL) (tunneling only, E < V₀)

T_exact / T_WKB ratio (tunneling regime only)

T_exact / T_WKB

Ratio = 1 reference (WKB exact match)

E probe = 0.50 eV

At Probe Energy E

E (probe) 0.50 eV

E / V₀ 0.500

T_exact —

T_WKB —

T_exact / T_WKB —

κ or k₂ (barrier) —

κL (or k₂L) —

Resonance Energies (E > V₀)

T→1 when k₂L = nπ, i.e.:
E_n = V₀ + n²π²ħ²/(2mL²)

Formulas

ħ²/2m = 0.03813 eV·nm²
k₁ = √(E/0.03813) nm⁻¹
Tunneling (E < V₀):
κ = √((V₀−E)/0.03813)
T_exact = 1/(1 + γ²sinh²(κL))
γ = (k₁²+κ²)/(2k₁κ)
T_WKB = e^(−2κL)
Above barrier (E > V₀):
k₂ = √((E−V₀)/0.03813)
T_exact = 1/(1 + γ²sin²(k₂L))
γ = (k₁²−k₂²)/(2k₁k₂)

Physics Notes

WKB accuracy: WKB works best when κL ≫ 1 (deep tunneling). As E→V₀ from below, κ→0, κL→0, and WKB underestimates T by a factor of ~4 for a symmetric barrier.

Resonances: Above the barrier, T oscillates between 0 and 1. WKB cannot predict these — it gives T≈1 for all E>V₀ (no reflection from a classically allowed region).

Energy conservation: T + R = 1 at all energies (checked numerically).