Quantum Tunneling Time — the Hartman Effect

Time inside a quantum tunneling event is one of the strangest open questions in quantum mechanics: how long does a particle spend crossing a barrier it classically shouldn't be able to cross at all? This simulator solves the exact stationary-state Schrödinger equation for a rectangular potential barrier, computes the Büttiker–Landauer phase-delay time from the energy-derivative of the transmission phase, and plots the resulting probability density and phase in 3D. Widen the barrier at fixed sub-barrier energy and the phase time saturates instead of growing — the Hartman effect — so the naive effective speed L/τ climbs past the speed of light even though no signal actually beats it, a genuinely counter-intuitive corner of quantum time distinct from wave-packet tunneling animations.