Ground-State Energy
Ask why an electron doesn’t fall into the nucleus and you will usually be told that it sits on the lowest rung of a ladder, and there is no rung below. That is true. It is also a restatement of the question: it says the falling stops without saying what stops it.
Something has to make the bottom rung the bottom. It turns out to be a race between two costs, and the winner is decided by an exponent – which is why matter has a size, why you can lean on a table, and why the periodic table runs out of room long before infinity.
The catastrophe that never happened
By 1911 the atom looked like a tiny solar system: a heavy nucleus with electrons going round it. The picture died almost immediately, and it died of something very simple. Anything electrically charged that changes direction throws off light, and throwing off light costs energy. An electron going round a nucleus is changing direction constantly. It should radiate, spiral inward, and be swallowed in about a hundredth of a billionth of a second.
Not “eventually”. Not “in the early universe”. Every atom, everywhere, before you could finish reading this sentence. Instead the hydrogen atom has been the same size for thirteen billion years.
The usual rescue is that the electron lives on a ladder of allowed rungs and simply cannot go below the bottom one. That is correct, and it is worth noticing how little it explains. Why is there a bottom rung? What would go wrong if the ladder just kept going down?
Confinement sends a bill
Here is the fact everything else rests on: a wave squeezed into a smaller space has to wiggle harder.
You already know this from a guitar. Press a string against a fret and you shorten it; the wave that fits into the shorter length has to bend more sharply, and the note goes up. Pressing further up the neck raises the pitch further. Nothing about the string changed except how much room its wave was given.
An electron is a wave too, and for a wave, wiggling is energy. So confining an electron is never free. Push it into half the space and you do not pay twice as much – you pay four times as much, because the bill goes as the square of how tightly you squeeze. Hold on to that “square”. It is the whole argument.
A race between two exponents
Now put the two costs side by side.
Moving the electron closer to the nucleus earns energy, because opposite charges attract and falling toward an attraction always pays. Halve the distance and you collect twice as much. That is the prize.
But halving the distance also halves the room the wave has, and we just established that the bill for that goes as the square: halve the distance and you owe four times as much.
So the two sides of the ledger grow at different rates, and that difference is the entire answer. Far out, the prize is bigger than the bill and the electron keeps falling inward. Close in, the bill overtakes the prize and falling further would cost more than it earns. Between them is one distance where the two exactly balance, and that is where the electron stops.
Nothing is holding it up. It stops because going lower is a bad deal.
Drag the slider below to change how fast the prize grows and watch the floor form, flatten, and then disappear. When the prize starts growing faster than the bill, there is no balance point at all, and the electron falls forever.
Hydrogen, in two lines
Put real numbers into that balance for a real electron and a real proton and the crossover lands half an ångström out – about five billionths of a centimetre. The energy there is 13.6 electron-volts below freedom, which is the amount of energy it takes to tear hydrogen apart.
Both numbers were measured long before anyone could explain them, and both come out of the balance without solving anything at all. No orbits, no ladder, no equation harder than finding where a curve turns.
That distance is why matter takes up room. Atoms are not solid, and they are mostly empty, but they cannot be compressed past the point where the bill overtakes the prize. When you lean on a table you are leaning on that refusal.
Where the floor runs out
Every argument has an edge, and this one has a spectacular edge.
The whole race assumed the electron is slow. That is a very good assumption in hydrogen, where the innermost electron travels at less than a percent of light speed. It gets worse as nuclei get heavier: a bigger positive charge pulls the innermost electron closer and whips it faster. In lead it is moving at about 60% of the speed of light. In the heaviest elements ever made, around 85%.
Near light speed the confinement bill changes shape. It stops growing as the square of the squeeze and starts growing in step with the prize – the two sides of the ledger begin rising together instead of one outrunning the other. The race becomes a tie, and a tie is decided by which side is bigger rather than which grows faster.
Work out where the prize finally wins outright and you get a nuclear charge of about 137. Beyond it, for a nucleus small enough, the innermost electron has no lowest state to sit in. The floor that has held matter up for the whole of this page simply runs out.
Real nuclei are not that highly charged, and real nuclei are not points, which pushes the true limit out to around 173. But the edge is there, and finding it is what hands this story over to relativity.