alpha decay tunneling
Alpha decay is a kind of radioactivity in which a heavy nucleus expels a clump of two protons and two neutrons — an alpha particle, the nucleus of helium. The puzzle that baffled early physicists was that the alpha particle escapes even though it does not have nearly enough energy to climb over the powerful wall of repulsion surrounding the nucleus. By the rules of classical physics, it should stay trapped forever.
In 1928 George Gamow, and independently Ronald Gurney and Edward Condon, solved the riddle with quantum tunnelling. The alpha particle, treated as a wave rattling around inside the nucleus, does not have to surmount the barrier; it has a small probability of tunnelling straight through it on each attempt. With the particle striking the wall billions upon billions of times a second, even a minuscule per-attempt chance eventually adds up to escape.
This was one of the first triumphant applications of quantum mechanics to the nucleus, and it explains a striking fact: alpha-emitting isotopes have half-lives ranging from microseconds to billions of years. Because the tunnelling probability depends so steeply on the barrier, a tiny change in the alpha particle's energy translates into an enormous change in how long the nucleus survives. Gamow's theory tied that vast spread of lifetimes to a single, elegant quantum mechanism.
Because tunnelling depends so steeply on energy, tiny energy shifts swing half-lives across many orders of magnitude.
Tunnelling does not let the alpha particle 'cheat' energy conservation. It emerges with exactly the energy released by the decay; the barrier governs how likely escape is per attempt, not how much energy the particle ends up with.