Quantum Tunneling Through a Barrier

Fire a quantum particle at a wall too tall to climb and it sometimes comes out the other side. Classically a particle with energy below the barrier height is simply turned back, every time. The wave is not: it does not stop dead at the wall, it bends into an exponential that decays across the barrier, and if the barrier is thin enough a sliver of amplitude survives to the far side and resumes travelling. That leaked fraction is the transmission probability $T = [1 + V_0^2\sinh^2(\kappa L)/(4E(V_0-E))]^{-1}$, which shrinks fast, exponentially, as the barrier thickens or rises, the reason tunneling is a delicate affair that nonetheless runs alpha decay, the scanning tunneling microscope, and fusion in the Sun. Above the barrier the story flips: the particle is not guaranteed through either, the step acts as a partial mirror, and at special energies where exactly a half-integer of wavelengths fits the barrier the reflections cancel and transmission is perfect. The top panel animates the real part of the wave flowing in, the standing-wave ripple of incident plus reflected on the left, the decay inside, the smaller transmitted wave leaving on the right, all inside the static probability envelope. The bottom panel tracks $T$ and $R$ against energy, the tunneling tail below $V_0$ and the resonance comb above, always summing to one. Slide the energy, height, and width and watch the leak respond.