Quantum Tunneling — WKB Approximation

WKB Tunneling: When E < V₀, the wavefunction decays as ψ ~ e^{−κx} inside the barrier, where κ = √(2m(V₀−E))/ℏ. The WKB transmission coefficient is T ≈ exp(−2∫κdx) = exp(−2κa) for a rectangular barrier. This exponential sensitivity makes tunneling profound: nuclear fusion in stars, scanning tunneling microscopes, alpha decay (Gamow 1928), and tunnel diodes all rely on it. The packet is a Gaussian wavepacket |ψ⟩ evolved using split-operator FFT method.