Electron transmission through potential barrier

A particle with E < V₀ can still pass through a barrier of width a (quantum tunnelling). For a wide barrier (κa ≳ 1), the transmission coefficient is

T ≈ 16 (E/V₀)(1 − E/V₀) e^{−2κa}, where κ = √(2m(V₀ − E))/ħ.

Given: E = 3 eV, V₀ = 4 eV, a = 20 Å = 2×10⁻⁹ m, m = 9.109×10⁻³¹ kg, ħ = 1.055×10⁻³⁴ J·s.

V₀ − E = 1 eV = 1.602×10⁻¹⁹ J
κ  = √(2 × 9.109×10⁻³¹ × 1.602×10⁻¹⁹) / 1.055×10⁻³⁴
   = √(2.919×10⁻⁴⁹) / 1.055×10⁻³⁴
   = 5.403×10⁻²⁵ / 1.055×10⁻³⁴
   = 5.12×10⁹ m⁻¹

2κa = 2 × 5.12×10⁹ × 2×10⁻⁹ = 20.48
e^(−20.48) = 1.27×10⁻⁹

Prefactor: 16 × (3/4) × (1/4) = 3

T = 3 × 1.27×10⁻⁹ = 3.8×10⁻⁹

Percentage transmission = 3.8×10⁻⁹ × 100 ≈ 3.8×10⁻⁷ %.

If a textbook uses the simpler form T ≈ e^(−2κa), the result is 1.27×10⁻⁹, that is, about 1.3×10⁻⁷ %.

Answer: T ≈ 3.8×10⁻⁹, i.e. about 3.8×10⁻⁷ % of the electrons are transmitted.