The S-matrix encodes all scattering information: S = [[r, t'], [t, r']], where t is the transmission amplitude and r the reflection amplitude. Unitarity (|t|² + |r|² = 1) follows from probability conservation. For a rectangular barrier of height V₀, quantum tunneling occurs even when E < V₀ — the wave function decays exponentially inside but has nonzero amplitude on the far side. At E = V₀ + (nπ/d)² (resonance energies), the transmission reaches 1.0 exactly (Ramsauer-Townsend effect). The S-matrix in quantum field theory generalizes this to encode all particle interaction amplitudes.