gen-define-quantum-tunneling-and-give-one-short-worked-example-5151

Define quantum tunneling and give one short worked example.

# Quantum Tunneling **Definition:** Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential energy barrier that it classically could not surmount because it has insufficient energy. This occurs because the particle's wavefunction doesn't drop to zero within the barrier, giving a non-zero probability of finding the particle on the other side. ## Worked Example: Electron Tunneling Through a Barrier **Problem:** An electron with energy E = 5 eV encounters a rectangular potential barrier of height V₀ = 10 eV and width a = 0.5 nm. What is the approximate transmission probability? **Solution:** For a rectangular barrier where E < V₀, the transmission coefficient is approximately: $$T \approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa a}$$ where $\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}$ **Calculate κ:** $$\kappa = \frac{\sqrt{2(9.11 \times 10^{-31}\text{ kg})(5 \text{ eV} \times 1.6 \times 10^{-19}\text{ J/eV})}}{1.055 \times 10^{-34}\text{ J·s}}$$ $$\kappa = 1.14 \times 10^{10} \text{ m}^{-1}$$ **Calculate transmission:** $$2\kappa a = 2(1.14 \times 10^{10})(0.5 \times 10^{-9}) = 11.4$$ $$T \approx 16\left(\frac{5}{10}\right)\left(\frac{5}{10}\right)e^{-11.4} = 4(1.1 \times 10^{-5}) \approx \boxed{4.4 \times 10^{-5}}$$ The electron has about a 0.0044% chance of tunneling through.