Illustrated Physics
physics, drawn
Concept

Pauli exclusion — where it appears

The requirement that a state of identical fermions change sign when two labels are swapped, which forbids two of them from occupying the same state. Its consequences are not small: it gives matter its volume, fixes the order the shells fill, and holds up a white dwarf against gravity with no temperature at all.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

The angular momentum that is not a rotation

An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed.

quantum · Spin The hole that holds exactly one particle. How likely a second particle is to be found a distance away from a first, relative to a gas with no correlation at all, for three cases that differ in nothing but the symmetry of the state under swapping the two labels. There is no interaction anywhere in this calculation: no Coulomb term, no potential, no force. Distinguishable particles give a flat line, which is what no interaction ought to give. Identical fermions dig a hole that reaches exactly zero at zero separation and fills back in over about a wavelength. Identical bosons do the opposite and pile up to twice the density. Integrating the fermion hole gives 0.9992 particles missing from around each one — exactly one, and the sum rule holds at any density, because raising the density narrows the hole in exact proportion. That is what makes the effect worth a name of its own. It is often called an exchange force and it is not a force: nothing carries momentum between the particles, and no term in the energy is proportional to a distance. It is a statement about which states exist. What follows from it is most of chemistry — the reason two atoms with filled shells repel, the reason a metal's electrons cost so much less Coulomb energy than a random arrangement would, and the reason matter takes up room.

The force with no force in it

Two identical fermions keep apart and two identical bosons crowd together, and neither is being pushed. The Hamiltonian contains no interaction at all: what produces the hole and the pile is which many-particle states exist, and the hole it digs around each electron holds exactly one particle at any density whatever.

quantum · Exclusion The interaction that orders a magnet is not the magnetic one. For six ordered magnets, two temperatures on a logarithmic scale: the energy of the magnetic interaction between two neighbouring moments, expressed as a temperature, and the temperature at which the material actually orders. iron orders at 1043 K against a dipolar scale of 0.201 K, a factor of 5182; cobalt orders at 1394 K against a dipolar scale of 0.117 K, a factor of 11961; nickel orders at 627 K against a dipolar scale of 0.015 K, a factor of 42312; gadolinium orders at 293 K against a dipolar scale of 0.790 K, a factor of 371; europium oxide orders at 69 K against a dipolar scale of 0.615 K, a factor of 112; lithium holmium fluoride orders at 1.53 K against a dipolar scale of 1.230 K, a factor of 1.24. The five ferromagnets order between a hundred and forty thousand times above the only interaction their moments have with each other, so whatever aligns them is not magnetism. The sixth is the control: lithium holmium fluoride is a magnet whose ordering really is dipolar, and its two temperatures agree.

What holds a magnet together is not magnetism

Two neighbouring moments in iron interact magnetically with an energy worth a fifth of a kelvin, and iron keeps its order to 1,043 kelvin. Whatever aligns them is five thousand times stronger than the only force they exert on one another — and it is electrostatic, with the exclusion principle deciding which of two spatial arrangements two electrons may use.

electromagnetism · Magnetisation

Named alongside it

The objects these essays reach for when they reach for this one.

Exchange interactionFerromagnetismAngular momentumAntisymmetryBosonCovalent bondCurie temperatureDipole dipole interactionExchange holeFermionHubbard modelIdentical particles

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