Quantum tunneling

Classically, a ball rolled at a hill it hasn't the energy to climb rolls back. Quantum mechanically, the particle's wavefunction does not stop dead at the barrier face. Inside the barrier it stops oscillating and instead decays exponentially — but exponential decay never quite reaches zero, so if the barrier is thin enough, a small but nonzero amplitude emerges on the far side. Square it and you have a real probability of finding the particle through.

The popular explanation, that the particle "borrows" energy to get over the top, is worth discarding. Nothing is borrowed and the particle never has more energy than it started with; it does not go over the barrier at all, and it is never found part-way inside it. What crosses is the wavefunction, and the energy on the far side is exactly the energy on the near side.

The exponential is the practically important part. Probability falls off steeply with both the width of the barrier and the square root of its height, so tunnelling is overwhelming at atomic scales and utterly negligible at everyday ones — which is why you cannot walk through a wall but an alpha particle escapes a nucleus. That same steepness is what makes the scanning tunnelling microscope work: a current that changes by orders of magnitude over a fraction of an atomic diameter turns tunnelling into a height sensor sharp enough to image single atoms. Tunnelling also runs the Sun, letting protons fuse at temperatures far below what classical repulsion would demand.