Quantum Mechanical Tunnelling

Roll a ball at a hill it hasn’t the energy to climb and you already know the ending: it slows, stalls short of the crest, rolls back. Everyday physics keeps strict accounts – speed buys height, and when the speed is spent the climb is over. The far side of the hill may as well not exist.

An electron heading for an energy wall taller than anything it carries should be the same dull story: a bounce, every single time. It isn’t. Now and then – about once in three hundred tries, for the wall this page plays with – the electron is simply found on the far side.

Quantum mechanics replaces the little ball with a wave – a spread-out ripple of maybe. And a wave meeting a wall it cannot climb doesn’t just stop. Inside the wall it can no longer ripple along – rippling is what having energy to spend looks like – so it does the one thing left to it: it fades, ferociously fast, losing a large slice of itself for every fraction of a nanometre it pushes deeper.

If the wall went on forever, that fade would be the end of the story. But real walls end – and whatever whisper of wave is still alive at the far face simply picks up and travels on. Quantum mechanics’ central rule does the rest: where there is wave, there is chance. The surviving whisper is the chance of finding the electron beyond the wall. Nothing drilled through. Nothing was borrowed. A wave just refused to be exactly zero.

Below is the whole event as a film. A packet of electron-wave sails in from the left, piles up against the wall and shudders – those stripes are the incoming wave criss-crossing its own reflection – while on the far side a whisper slips out and glides away. None of it is drawn by hand: a computer is solving the quantum equation of motion live, and the picture is simply what the solution looks like.

The whisper is tiny – about three escapes for every thousand attempts – so the far side is shown through a magnifying glass, with the zoom printed on screen. Squeeze the wall thinner with the slider and the whisper grows until no magnifier is needed. And mind the clock: the entire drama lasts about twenty-five femtoseconds – millionths of a billionth of a second.

How much gets through? The wave loses a fixed fraction for every step deeper into the wall, and compounding like that is merciless. Doubling the wall’s thickness doesn’t halve the odds – it multiplies the shortfall by itself. The electron that slips through this page’s wall once in three hundred tries would make it about once in four hundred thousand if the wall were merely twice as thick.

A rule that harsh sounds like it should make tunnelling a footnote to physics. Instead it is the making of some of the sharpest instruments ever built – because anything that sensitive to distance is, turned around, a superb ruler.

The scanning tunnelling microscope. Hover a metal needle a hair’s breadth above a surface – close enough for electron-waves to leak across the gap – and a tiny current flows with no contact at all. Because the leak is so sensitive to distance, a bump one atom tall swings the current enormously. Sweep the needle across the surface, keep the current steady, and the needle’s path traces the landscape atom by atom. The first true portraits of individual atoms, in 1981, were drawn exactly this way.

Radioactive decay. Deep inside heavy nuclei, clusters of particles sit trapped behind a barrier they could never classically escape. They leak out by tunnelling – and because the odds ride that merciless compounding, nearly identical nuclei live wildly different lives: one lasts microseconds, another outlives planets. George Gamow read it this way in 1928 – quantum mechanics’ first victory inside the nucleus.

The Sun. Two protons at the centre of the Sun carry nowhere near the energy needed to push through their electrical repulsion and touch – the wall is hundreds of times taller than their punch. They fuse anyway, by tunnelling through the last stretch. Without that, the Sun would simply go out. And closer to home: every photo on your phone was saved by electrons tunnelling through a whisper-thin layer of insulation onto a memory chip. The forbidden crossing, doing errands.

The equations say yes; the numbers say never. You are an astronomical crowd of particles, and every one of them would have to slip through the wall at the same instant. The combined odds are so small that walking into walls for the age of the universe would change nothing measurable. The same harshness that lets a microscope feel single atoms is what keeps you dependably in the room.

To watch the collision again with your hands on the controls – raising the wall, widening it, even lifting the electron clean over the top – head to chapter 4’s live simulation, which runs the same solver with every knob exposed.