Could You Put Two Electrons in the Same Point?
Same Place Is Easy. Same Point Does Not Exist.
Take two electrons. Not near each other: in the same point. I know they repel, and I have unlimited force. I know about Pauli, so give them opposite spins. Now put them in one point and tell me what is there.
The request is taken exactly as stated and physics answers it one wall at a time, each wall marked by how firmly it stands. The Electron page explains what an electron is, that it has no measurable size, and what its spin does. This page assumes all of that and asks only about the last step.
What Physics Says, Wall by Wall
An electron does not have a position the way a marble does. It has a wave, and the wave is where it is. Two electrons with opposite spins can share one wave, and they do, everywhere. The two electrons of a helium atom occupy the same orbital: the same shape, the same size, the same probability of being found at every point around the nucleus. Every chemical bond in your body is a pair of electrons doing the same thing between two atoms. In a superconductor the pairs stretch to a hundred nanometres and still count as one state.
So in the only sense that place has for an electron, the request was granted before it was made. Two electrons in the same place is the most common arrangement of electrons in matter. The catch is the word the request insisted on. The shared wave is about 10⁻¹⁰ m across. The request did not ask for the same wave. It asked for the same point.
A point is a position known to infinite precision, and precision is not free. To know where an electron is to within some distance you must give it momentum, and therefore energy, of about Planck’s constant times the speed of light divided by that distance. Pin it to an atom’s width and the price is a few thousand electron volts, X-ray money. Pin it to 3.9 × 10⁻¹³ m, a quarter of a thousandth of an atom’s width, and the price reaches 511,000 electron volts, which is the electron’s own mass.
Past that line something happens that has no counterpart for marbles. The energy you spend on sharpening the position is enough to make a new electron together with its antiparticle, and it does. Squeeze the position tighter and the pairs multiply. Try to pin one electron to a proton’s width and you are holding a cloud of them. There is no way to localize an electron to a point; the attempt manufactures electrons. The request’s noun, point, names nothing an electron can occupy, and that is true before the second electron arrives.
Two statements are easy to run together here and they are different. The electron has no size: every measurement finds it point-like, down to 10⁻¹⁸ m. And the electron has no point-like position: its whereabouts cannot be sharpened below its own wavelength without making more of it. Both are true. The first says there is nothing to bump. The second says there is nothing to put.
The request gave the electrons opposite spins on purpose, and that was the right move, so this wall is named only to say how it differs from the others. Two electrons with identical spin cannot be in one state, and the reason is not a force, not a cost, not a repulsion. Swap two electrons and the mathematics that describes them must change sign. Put both in the same state and swapping them changes nothing, so the description must equal its own negative. The only number that equals its negative is zero. The state does not exist as a mathematical object. It cannot be written down, so no amount of force overcomes it, because there is nothing to overcome.
With opposite spins the sign works out and the two may share everything else. Pauli never forbade the request. Walls two and four do.
Set the pinning aside and simply push. The energy stored in the field between two electrons is 1.4 electron volts at a nanometre, 14 at an atom’s width, 1.4 million at a proton’s width, and 1.4 billion at 10⁻¹⁸ m. At a proton’s width the field alone holds more than the mass of an electron and a positron, so it can spend itself on making them, and does. The energy you put in becomes particles, the same lesson the quark page teaches with a string.
And the thing you are pushing changes as you push. An electron is dressed in a haze of virtual pairs that partly screens its charge. Get inside the haze and the screening thins, so the charge you feel grows. Far away the strength of the electric force is the fine-structure constant, 1/137. At 10⁻¹⁸ m it is about 1/128. That is a measured number, not a prediction: the Geneva collider brought electrons and positrons together at 209 billion electron volts, which probes about 10⁻¹⁸ m, and the charge came out larger by exactly the amount the haze predicts. It is the closest two point charges have ever been brought. No size appeared.
The picture below is the push. One slider sets the separation, from a nanometre down to a tenth of an attometre. Two gauges show the energy in the field between the electrons and the price of pinning them that sharply. Watch the electrons widen as the second gauge passes their mass, and watch the pairs appear.
The request speaks of two electrons as of two marbles, this one and that one, and asks to bring them to a meeting. Electrons do not carry labels, and this is not a limit on our record keeping. It is measurable. Fire two electrons at each other and count how many come out at right angles. If they were two labelled objects, the count would be one thing. Because they are indistinguishable, the two ways the event could have happened, this one deflected left or that one, interfere, and for electrons with parallel spins the count at ninety degrees is exactly zero. Nevill Mott predicted this in 1930 and the experiments agree.
So when the two electrons of the request have met and parted, there is no fact about which one went where, and there was no such fact before the meeting either. What the request calls two electrons, physics calls one two-electron state. The instruction to put these two in one place presupposes a these that nature does not keep.
The Pushback
The walls above were shown to a reader who refuses to stop at walls. Two objections, in the order they came, and what physics says back.
Nothing practical, and the most sensitive instrument for new physics that exists, on a table rather than in a tunnel. A dipole moment would mean the electron’s charge is very slightly lopsided along its spin. The Standard Model predicts a lopsidedness about a hundred million times smaller than the best limit measured today, so small that no foreseeable experiment reaches it. The limit itself falls by a factor of ten each decade; the 2023 measurement in Boulder used molecular ions held in a trap. So any detection in the next twenty years would be a discovery, not a refinement. The lopsidedness would be produced by virtual heavy particles that violate the symmetry between matter and antimatter, with masses of ten to a hundred trillion electron volts, beyond the reach of any collider, measured by watching a molecule.
And those particles are wanted elsewhere. For the Big Bang to have left more matter than antimatter, universe needs more violation of that symmetry than the Standard Model contains, and nobody knows where the rest comes from. The electron’s dipole would be the first direct trace of the missing source. Its absence is doing work too: whole families of proposed theories predicted a value the traps have since passed without seeing, and those theories are gone.
There is one, and gravity sets it, not engineering. The distance at which two charges see each other is Planck’s constant times the speed of light divided by the collision energy: more energy, finer resolution. That holds up to the Planck energy, about 10¹⁹ billion electron volts. At that energy the collision packs its energy into a region the size of its own wavelength. The region’s Schwarzschild radius then equals the region itself, and the collision makes a black hole. Past that, adding energy makes the hole bigger, because a hole’s radius grows with its mass. The resolution gets worse, not better. The finest resolution any collision can ever have is therefore about the Planck length, 1.6 × 10⁻³⁵ m, seventeen powers of ten closer than the record at Geneva. Closer than that is not a smaller distance. It is the inside of a horizon.
What comes out of that collision is a black hole of one Planck mass, about twenty micrograms, which evaporates in 10⁻⁴³ seconds into a spray of particles. The spray may contain two electrons. It will not contain the two you sent in, and by the fifth wall the question of whether it does has no meaning anyway. For scale, with plasma acceleration at a hundred billion electron volts per metre, the best technique demonstrated, a Planck-energy collider would be about 10¹⁷ metres long, ten light-years. Built with the technology of the Large Hadron Collider it would be the size of the Galaxy.
Where the Request Was Right
Opposite spins was exactly the right dodge. Pauli is the wall everyone names first, and it was never the one that mattered here. The request stepped around it with one clause and left the real obstacles, precision and identity, in view.
The electron is a point, as far as anyone has measured. Down to 10⁻¹⁸ m, ten million times smaller than an atom, it has no size, no surface and no parts. The request’s picture of a thing with no extent to bump against is the picture the data support. The error was to conclude that a thing with no size must have a sharp position. Those are different properties, and the electron has one and not the other.
The goal is met in every atom. Two electrons sharing one location, with opposite spins, is not exotic. It is helium, it is every bond, it is the pairs that carry current without loss in a superconductor. The request wanted something the world does everywhere, and named it with a word the world does not use.
Is It a Point All the Way Down?
Point-like to 10⁻¹⁸ m is a measurement, not a proof of pointlike-ness forever. Two lines of evidence keep pushing. The electron’s magnetism is measured to thirteen digits and agrees with the theory of a structureless charge to all of them; any internal parts would have shown up as a discrepancy by now. And searches for a slight lopsidedness in the electron’s charge, an electric dipole moment, have found none to a part in 10³⁰ of the natural scale, which rules out whole families of proposed heavier particles that would have distorted it. String theory expects the point to give way to structure at 10⁻³⁵ m, seventeen powers of ten below anything reachable. Between the two numbers is the whole open question.
An Opinion, Dated
The walls above are physics. This box is the drafting model’s own bets, and they are scoreable.
Nine in ten that no experiment finds structure inside the electron by 2050. The magnetism measurement leaves no room at any energy a collider of this century can reach.
One in five that an electric dipole moment of the electron is detected by 2040. The experiments improve by a factor of ten per decade and are now inside the range where several proposed extensions of the Standard Model predicted a signal. Most of those extensions are already excluded, which is why the odds are not higher.
And a consequence rather than a bet: two electrons will never be brought closer than about 10⁻¹⁹ m by any machine on Earth, because the machine to do it would be the size of a continent. Nor closer than the Planck length by any machine anywhere. The theory past the first line will be tested, if at all, by the sky.
These intuitions were distilled from human physics writing and carry its fashions. Weigh them accordingly.
The Updated Map
Place, for an electron, is a wave, and two electrons can share one. They do, in every atom and every bond you are made of. Point is not a place at all. It is a demand for infinite precision, and the price of precision is paid in particles: sharpen an electron’s position past its own wavelength and you have made more electrons. Push two toward a point and you get a growing charge, a spray of new pairs, and at the end a question with no answer, which one was which. The request asked for something the world does everywhere, using a word the world does not have.
