The wall you cannot climb but can pass through
Roll a ball at a hill. If it does not have enough energy to reach the top, it rolls partway up, stops, and rolls back. Every time. Forever. This is one of the most ironclad rules of the everyday world: you cannot get to the other side of a barrier taller than your energy allows. A skateboarder who cannot reach the lip of a ramp never lands on the far side — not occasionally, not rarely, never.
Quantum particles break this rule. A particle fired at a barrier it does not have the energy to climb sometimes — not always, but a definite, calculable fraction of the time — simply appears on the far side, having crossed a region it could never have stood in. It did not go over the top and it did not smash through. It tunnelled. This is quantum tunnelling, and it is among the strangest things the quantum world genuinely does. The barrier in question is a potential barrier: a region of high potential energy, too tall for the particle to enter by classical rights.
Why it happens: the tail reaches the other side
Tunnelling is not a new rule bolted onto the particle. It is the leaky wall from the last guide, taken one step further. Recall that when a quantum wave meets a wall it cannot classically enter, it does not stop dead — it pokes in as a drooping evanescent tail that fades the deeper it goes. In a thick wall, that tail fades all the way to nothing long before reaching the far side, and the particle is, for all practical purposes, sealed out.
But make the wall thin enough, and the tail has not finished fading by the time it reaches the far face. A small but non-zero piece of the wave emerges on the other side — and once it emerges, it resumes being a normal travelling wave again. That surviving sliver is a real, ongoing wave, which means there is a real, non-zero chance of finding the whole particle over there. The particle never had enough energy to be inside the wall; it was never observed mid-wall; yet the wave's smooth insistence on not snapping to zero carries a fragment of it through.
What makes a wall easy or hard to tunnel
How often a particle gets through is the tunnelling probability, and it depends very sharply on the wall. The faster the evanescent tail dies inside the wall, and the further it has to reach, the less of it survives to the far side. Three knobs control it.
- Wall thickness. A thinner wall is dramatically easier to tunnel through — the probability falls off exponentially as the wall gets wider, so even a small extra thickness can shut tunnelling down almost completely.
- Wall height. The taller the barrier compared to the particle's energy, the faster the tail dies inside it, and the less gets through.
- Particle mass. Lighter particles tunnel far more readily than heavy ones. An electron slips through walls that a much heavier proton would treat as nearly solid.
That exponential sensitivity is the crucial fact. Tunnelling is not a sometimes-large, sometimes-small effect — it switches from "happens constantly" to "essentially never" over a tiny change in wall width. This razor edge is exactly why tunnelling can be both invisible in daily life and the engine behind real technologies. You will never tunnel through a real door: you are far too massive and the door is unimaginably thick on the atomic scale, so your tunnelling probability is a decimal point followed by an absurd number of zeros — a once-in-many-lifetimes-of-the-universe event. But shrink to an electron and a wall a few atoms thick, and tunnelling goes from impossible to routine. The remainder of the wave, the part that does not get through, simply bounces back — the reflection side of the same ledger.
What tunnelling is not
It is easy to over-romanticize this, so let us be honest about what does and does not happen. The particle does not borrow energy and pay it back; it does not get a magic speed boost; it does not exist "inside" the wall as a little ball squeezing through a gap. Throughout, the only thing that exists is the wavefunction, and the wavefunction simply has a small amplitude on the far side. When you finally look, you find the whole, ordinary particle either on the near side or the far side — never half-stuck in the wall. Tunnelling is a statement about probabilities, not about a particle squeezing through a tube.
And crucially, energy is never violated. The particle that emerges on the far side has exactly the energy it started with — tunnelling does not let it gain or lose any. What looked impossible was impossible only under the classical assumption that a particle is a hard little ball with a definite position. Once you accept that it is a wave whose tail refuses to snap to zero, nothing impossible has happened at all. With that honest picture in hand, the next guide leaves the toy walls behind and shows where tunnelling actually runs the universe.