Quantum Tunneling

Roll a ball at a hill taller than its energy and it always comes back. There is no version of that story where the ball turns up on the far side — it simply does not have what it takes to reach the top, and everything after that is downhill in the wrong direction.

Set the energy below the barrier height and watch. The dashed sky line sits below the top of the amber block, so by every classical account nothing should ever reach the right-hand side. Some of the wave appears there anyway. The particle was never a ball.

Inside the barrier the wave does not vanish. It stops oscillating and starts decaying — exponentially, at a rate set by how far the energy falls short of the top. That is the whole mechanism, and you can see it in the faint teal curve as it crosses the amber block.

If the wall is thin enough, there is still amplitude left when you reach the other side, and that leftover amplitude is a real probability of finding the particle there. Widen the barrier and watch transmission collapse exponentially — a modest change in width costs orders of magnitude. Compare the measured number with the exact plane-wave transmission as you go. The measurement sits somewhat higher (roughly 10% against 7% at the default settings) because a finite wave packet carries a spread of energies, and the components above the mean tunnel far more easily than the ones below it. Widen the packet σ — which narrows its energy spread — and the gap closes.

Tunnelling is not a curiosity. It is how a scanning tunnelling microscope images single atoms: hold a sharp tip a fraction of a nanometre above a surface and the current that leaks across the gap is so violently sensitive to distance that individual atoms stand out as hills.

It is also how electrons leak out of flash memory cells, which is why an SSD left unpowered for years eventually forgets, and how protons in the Sun’s core fuse at temperatures far too low to overcome their electrical repulsion. Without tunnelling, stars would not shine.