Illustrated Physics
physics, drawn
Concept

Singlet — where it appears

The state of two spin-half particles whose total spin is zero, antisymmetric under exchanging them and carrying no direction at all. Its antisymmetry forces the spatial part of the state to be symmetric, which costs electrostatic energy — and taken far apart it is the pair that violates a Bell inequality.

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

Four product states, and the four combinations that have a total spin. The four ways two spin-halves can be arranged, on the left, and the four combinations of them that are eigenstates of the total spin, on the right. Two of the products — both up and both down — are already eigenstates. The other two are not: one spin up and the other down does not specify a total spin, because it does not say which spin is which, and the states that do are the sum and the difference. The eigenvalues printed beside them are computed by applying S² as a matrix in the product basis and reading the result off, then solving s(s + 1) for s: |↑↑⟩ gives 2ħ², so s = 1; (|↑↓⟩ + |↓↑⟩)/√2 gives 2ħ², so s = 1; |↓↓⟩ gives 2ħ², so s = 1; (|↑↓⟩ − |↓↑⟩)/√2 gives 0ħ², so s = 0. The column on the far right is the eigenvalue of the operator that swaps the two particles: the three states with s = 1 come back unchanged and the one with s = 0 comes back with a minus sign. That sign is the whole of the difference. It is why the three are called a triplet and the one a singlet, why they behave differently in a magnetic field, and — through the requirement that the total state of two electrons be antisymmetric — why the two families occupy space differently before any force between them has been mentioned.

Four states, and one of them is odd

Two spin-halves make four states, and they split three and one rather than into four of a kind. Three come back unchanged when the two particles are swapped and one comes back with a minus sign — and that single sign decides how far apart two electrons sit before any force between them has been mentioned, and why hydrogen gas is two gases that do not interconvert.

quantum · Spin 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 interactionCurie temperatureDipole dipole interactionFermi holeFerromagnetismHubbard modelHunds rulesMagnetisationMean-field theoryOrtho paraPauli exclusionSpin

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