A particle rolls at a wall it hasn't the energy to climb. Classically it always bounces back. But its wavefunction doesn't stop at the wall — it decays through the barrier, and a sliver leaks out the far side. That sliver is a real probability of passing through: the particle tunnels. Raise the energy toward the barrier top and the leak grows; lower it, or widen the wall, and it collapses exponentially.
Drag Energy below the barrier top — classically T should be zero, yet the number is alive: the wave leaks through. Now widen L and watch T crash — tunnelling is exponentially fragile, the reason α-decay half-lives span 24 orders of magnitude. Push E above V₀ and ripples appear: at resonances (kL = nπ) the barrier turns perfectly transparent, T = 1. The self-test proves the live curve is the exact closed form, that R + T = 1 to machine precision, and that an independent transfer-matrix wave-solve agrees.
↖ next drift over · this T(E) made dynamical The Wave Packet → draws this same barrier as a number with no picture. There, a Gaussian packet is fired at the wall and a transmitted lobe peels off — and its ∫|ψ|² past the wall is checked to land inside the energy-spread band of this bench's exact T(⟨E⟩). The static curve, finally seen happening.