Physics · Module 23
The last module used the electron's wave nature to build the atom. This one turns the same wave picture on the electron's position itself — why it can never be pinned down exactly, why that is what keeps matter from collapsing, and why a wave can leak through a wall it has no business crossing.
Here is a piece of physics you have probably heard misdescribed. The usual version — "measuring a particle disturbs it, so you cannot know both its position and its momentum" — makes it sound like a limitation of our instruments. It is not. It is a property of waves, and you met it in Module 12 without knowing.
A beam of electrons passes through a single narrow slit and lands on a screen. You now make the slit narrower, so that you know more precisely where each electron was as it went through.
What happens to the patch where the electrons land?
It gets wider. This is single-slit diffraction, Module 12, and you can do it tonight with a torch and two fingers: the narrower the gap, the more the light fans out. Module 12 even gave you the formula — the first dark fringe sits at θ, the angle off straight ahead, where sin θ = λ/a, so a smaller slit width a means a bigger angle.
Now read that as a statement about the electron rather than about the beam.
The product sits at a few times ħ and will not go below it, however you set the controls. It is a few times ħ rather than exactly ħ/2 because "the width of the central patch" is a rough way of defining a spread; do it properly with standard deviations and you land on the textbook inequality. The physics is that the product has a floor.
Now a question that ought to have bothered you back in Module 13, when we said the electron is attracted to the nucleus by a force that grows as it gets closer, as 1/r².
Why does the electron not simply fall in? Classically it should. It is attracted, there is nothing in the way, and worse — an accelerating charge radiates (Module 18), so an orbiting electron should be pouring out energy and spiralling inward. Do the calculation with classical physics and a hydrogen atom collapses in about 10−11 seconds. Matter should not exist for the length of a blink.
Uncertainty is the answer, and it is a properly beautiful argument.
So the size of an atom is a truce. And since atoms are what everything is made of, this is the reason a table is solid, the reason you have a volume, and the reason the floor holds you up. Matter takes up space because you cannot squeeze a wave into a point without paying an energy bill that rises faster than anything can afford.
Push hard enough and you can win the argument — but you need a star to do it. In a white dwarf, gravity has compressed all the electrons into such a small volume that squeezing further would force each one onto a higher energy rung — Pauli's rule (Module 22) bars two electrons from the same state, so compressing the whole star costs enormous confinement energy. That rising cost is what holds the star up. Push past even that, with enough mass, and the electrons are crushed into the protons and you get a neutron star, held up by exactly the same effect acting on neutrons. Beyond that, nothing holds, and you get a black hole. The stability of matter and the fate of a dying star are the same calculation.
Roll a ball at a hill. If it has less energy than the height of the hill, it rolls partway up, stops, and comes back. Always. This is Module 5's energy ledger, and it does not negotiate.
Now send an electron wave at a barrier of potential energy it does not have enough energy to cross.
Before you answer: recall what waves do at boundaries. In Modules 7, 10 and 12, wherever a wave met an edge, part of it passed through and part came back. The electron here is a wave.
An electron with 3 eV of energy arrives at a barrier 5 eV high and half a nanometre thick.
What happens?
Some of them get through, and they arrive on the far side with exactly the 3 eV they started with. No energy is borrowed and none is repaid; the ones that arrive have the same energy as the ones that bounce.
The reason is what a wave does at a barrier. Outside, the electron has kinetic energy to spare and its wave oscillates — crest, trough, crest — as every wave in this course has. Inside the barrier, every joule of energy is held by the barrier and there is none left to oscillate with, so the wave decays instead: each step further in, the amplitude shrinks by the same fraction. That is an exponential. If the barrier is thick, the exponential has died away by the far side and nothing emerges. If the barrier is thin, the exponential has not collapsed completely, and a small but real wave continues on the other side. That surviving wave, squared, is the probability of finding the electron there.
Nothing was ever "inside" the barrier in the sense of a ball halfway up a hill. There is a wave in there, but the electron is only ever detected on one side or the other, at full energy. Asking where it was during the crossing is Module 21's forbidden question in a new costume.
Do that last instruction, because the sensitivity is the whole story. Halving the barrier thickness does not double the transmission; it multiplies it by something like thirty, because the wave inside is dying exponentially. A change in thickness of one atom's width changes the current tenfold.
That extreme sensitivity is not a nuisance. It is an instrument.
The scanning tunnelling microscope. Bring a very sharp metal tip within a nanometre of a surface, put a small voltage across the gap, and electrons tunnel across the vacuum. Because the current depends exponentially on the gap, moving the tip by the width of a single atom changes the current tenfold. Scan the tip across and keep the current constant, and you trace out a map of individual atoms. Most of the pictures you have seen of individual atoms on a surface — including the famous ones where atoms have been pushed into position to spell words — were made this way.
Alpha decay. An alpha particle inside a heavy nucleus is trapped behind a barrier of electrical repulsion far higher than its energy. Classically it can never escape and uranium would be stable for ever. It tunnels — and because the probability is exponentially sensitive, small differences in nuclear size produce half-lives ranging from microseconds to billions of years. Module 24 picks this up.
The Sun. Two protons in the Sun's core repel each other electrically and, at the Sun's temperature, do not have nearly enough energy to get close enough to fuse. Classically the Sun should not shine. It shines because the protons tunnel through their mutual repulsion. Module 24 again.
The memory in your phone. Flash storage works by pushing electrons through a thin insulating layer onto an isolated island of silicon, where they sit and represent a bit. They get there by tunnelling, and they stay because the barrier is just thick enough that tunnelling back out takes years. Every photograph you have ever saved is sitting behind a barrier it should not have been able to cross.
One more consequence, and it is the one you can hold in your hand.
An atom sitting in an excited state will, sooner or later, drop down and emit a photon. That is spontaneous emission, and it is what a neon sign does: each atom decides for itself when to fall, so the photons come out at random times, in random directions, with random phases. The light is incoherent — a crowd all talking at once.
In 1917 Einstein, working out how light and matter must exchange energy to stay in thermal equilibrium, found that spontaneous emission alone did not balance the books. There had to be a third process nobody had noticed.
What comes out is unlike any other light. All the photons are copies of each other, so the beam has a single sharp wavelength, stays in step over long distances — that is coherence: a fixed, lasting phase relationship between the light arriving by one path and the light arriving by another. It is what interference needs, and it is the reason two ordinary lamps give no fringes: each atom in a filament emits an independent burst lasting about a hundred-millionth of a second, so the pattern re-forms millions of times a second and averages to a blur. The beam also spreads out only as much as diffraction forces it to. A laser aimed at the Moon — and this has been done, bounced off mirrors the Apollo missions left on the surface — cut so that they send light straight back the way it came, whatever angle it arrives at — arrives as a spot a few kilometres across after a quarter of a million miles. An ordinary torch would have spread across the sky.
Take a cheap laser pointer and shine it on a rough white wall from a few metres away. Look closely at the spot: it is not smooth. It is covered with a fine, restless granular sparkle that seems to move when you move your head. That is speckle, and it is pure interference — light scattered from thousands of tiny irregularities on the wall, arriving at your eye with different path lengths and adding up as Module 12 says. You only see it with laser light, because only laser light is coherent enough for the phases to still be related after that journey. Shine a torch at the same wall and the spot is perfectly smooth.
Then hold the pointer against a fine comb, or send it through two pinholes made with a needle in foil, and put Module 12's fringes on the wall — the same experiment Young did in 1801, now with a light source that Young could not have imagined and that exists only because of everything in this module.
| Idea | What it says |
|---|---|
| Uncertainty | ΔxΔp ≥ ħ/2 — a property of waves, derivable from Module 12's single slit, not a statement about clumsy instruments. |
| Why matter is solid | Confinement energy rises as 1/r² and beats the electrical gain of 1/r. The truce fixes the size of the atom at 0.053 nm — and the size of everything made of atoms. |
| Tunnelling and lasers | A wave leaks through a barrier it cannot climb, exponentially sensitively (the STM, alpha decay, the Sun, flash memory). And stimulated emission copies photons, which — with a population inversion and two mirrors — is a laser. |