Bound states & barriers

quantum tunneling

Quantum tunnelling is the passage of a particle through a barrier that, by the rules of classical physics, it does not have enough energy to cross. Imagine a ball rolling toward a hill too high to climb; classically it always rolls back. Quantum mechanically, there is a real chance the ball appears on the other side without ever going over the top — it tunnels straight through.

The explanation lies in the wave nature of matter. A particle is described by a wavefunction, a spread-out probability amplitude, not a hard pellet at a single point. When this wave meets a barrier it does not abruptly halt; it decays exponentially inside the barrier rather than instantly vanishing. If the barrier is thin enough, the wave is still nonzero at the far edge, which means a nonzero probability of detecting the particle beyond. Nothing is borrowed and energy stays conserved; the particle simply has a chance of being found across.

Tunnelling is not an exotic edge case but a workhorse of nature and technology. It is how the Sun shines, since protons fuse only by tunnelling through their mutual repulsion; how some radioactive nuclei decay; and how enzymes shuttle electrons in living cells. Engineered versions power scanning tunnelling microscopes, flash memory, and tunnel diodes. The effect is real, routine, and exploited daily.

transmission probability T ∝ e^(−2κL) (E below barrier height V₀)

Even with too little energy, a particle has an exponentially small but real chance of passing through.

Tunnelling is sometimes described as 'borrowing energy' via the uncertainty principle, but that picture is misleading. Energy is conserved; the effect comes from the wavefunction's exponential decay, and the particle emerges with the same energy it entered.

Also called
tunnellingquantum tunnelling隧道效应