A Quantum Leaks Through the Wall
In the classical world a ball without enough energy cannot top a hill; it just bounces back. Yet in the quantum world the wavefunction does not stop inside the wall, it merely decays gently as exp(-κx). So a small amplitude leaks out the far side, and the chance the particle gets through, T, is not zero. In the first figure we watch the oscillation on the left fade inside the wall and then revive, smaller, on the right.
When the incoming wave on the left reaches the wall, it does not stop like a classical particle. Inside the wall the oscillation dies and the amplitude shrinks smoothly as exp(-κx). Then on the far side it revives as a small oscillation. Drag the slider to widen the wall and the transmitted amplitude on the right grows visibly smaller. The wavefunction ψ stays continuous and nonzero straight through the wall.
Here T is the transmission probability, a dimensionless number between 0 and 1, an entirely different quantity from torque T or period T. Raise the particle energy E with the slider toward the wall height V and T rises toward 1. Roughly T ≈ exp(-2κL), so a taller or thicker wall makes T plunge. Here κ = √(2m(V-E))/ℏ is the decay rate that sets how fast the wave fades inside the wall.
Setting the two worlds side by side makes the difference clear. A classical particle with energy below the wall has T = 0, so it always bounces straight back. A quantum particle in the very same situation has T > 0, so part of it gets through the wall. Raise the wall height with the slider and the quantum fraction shrinks but never reaches zero. The classically forbidden region is one the quantum particle enters and crosses.
Look closely at what happens inside the wall. The amplitude is a smooth tail falling as exp(-κx). Raise κ with the slider, that is, make the wall higher, and the tail nosedives far more steeply. The steeper the tail, the smaller the amplitude left at the wall's far edge, and so the less that leaks out. A thicker or higher wall means a faster decay and far less transmission.
This leakage hides throughout nature and technology. Use the segments to pick three scenes. In alpha decay an alpha particle inside the nucleus tunnels out through the nuclear barrier, and that transmission chance sets the half-life. One kind of electron microscope images atoms from the tunneling current that leaks across the gap between a tip and the surface. Flash memory stores information by tunneling electrons through a thin insulating barrier to trap charge.