Four states, and one of them is odd

An electron’s angular momentum is not a rotation of anything, and what survives of the analogy is the algebra. This is what the algebra is for. Two of them combine, and the way they combine decides a surprising amount of chemistry.

Three and one #

Both up and both down are already states of definite total spin, with $S^2 = 2\hbar^2$ and therefore $s = 1$. One up and one down is not, and the reason is worth stating carefully: it does not specify which particle is up, and the total spin is a property of the pair rather than of a labelling.

The combinations that do have a definite total spin are the sum and the difference. Applying $S^2$ as a matrix in the product basis and reading the result off gives $2\hbar^2$ for the sum and exactly zero for the difference.

So the four states split three and one: a triplet with $s = 1$ and three orientations, and a singlet with $s = 0$ and one. That is not a decomposition anybody would guess from the counting, and it is the reason spin cannot be thought of as two independent labels.

The other column is the operator that swaps the two particles. All three triplet states come back unchanged; the singlet comes back with a minus sign. That sign is the whole of the difference between them, and everything below follows from it.

The consequence in space #

Electrons are fermions, so the total state — spatial and spin together — must change sign when the two are exchanged. That ties the two halves together: a symmetric spin state forces an antisymmetric spatial one, and vice versa.

So the triplet, whose spins are symmetric, has an antisymmetric spatial state — and an antisymmetric function of two coordinates vanishes when they are equal. The two electrons of a triplet are never in the same place, and the hole in the middle of the curve is called the Fermi hole.

Nothing in that calculation contains a force. The electrons do not repel, do not attract, and know about each other only through the symmetry of the state they share. The root-mean-square separation comes out ten per cent larger for the triplet than for the singlet, and that difference is present before any interaction has been mentioned.

That is the honest statement of what the exclusion principle does, and it is worth insisting on because the usual shorthand — “electrons of parallel spin repel” — attributes to a force something that is a property of the state space. The exclusion principle produces a pressure without a force for exactly the same reason.

The energy it costs, once a repulsion exists #

Switch the Coulomb repulsion on and the separation becomes an energy: two electrons that are further apart pay less.

The energy difference between the triplet and the singlet is twice the exchange integral,

$$K = \iint \phi_a^*(1)\phi_b^*(2)\frac{e^2}{r_{12}}\phi_b(1)\phi_a(2)\,d^3r_1 d^3r_2$$

which involves the product of the two orbitals at both points and therefore needs them to overlap. The computed logarithm of $K$ tracks the logarithm of the overlap with slope 2.00, which is the statement that exchange is a two-orbital effect while direct repulsion is not.

It is also positive at every separation, and it has to be: the Coulomb kernel is positive definite, so the integral of a squared quantity against it cannot be negative. That sign is Hund’s first rule. The triplet lies below the singlet, so a configuration with two electrons in different orbitals puts them in the state of highest total spin — which is why oxygen’s ground state is a triplet, why oxygen is paramagnetic, and why the molecule’s chemistry is as sluggish as it is.

For the 1s2s configuration of helium the splitting is 0.80 electron-volts, which is an energy nobody would guess was the consequence of a minus sign in front of one term of a wavefunction.

Why the sign has to be there at all #

It is worth asking where the antisymmetry requirement comes from, because everything above rests on it and it is usually presented as a rule to be memorised.

The weak version is a statement about indistinguishability. Two electrons are not merely similar; there is no fact about which is which, so swapping them must leave every probability unchanged — and that requires the state to come back multiplied by a phase whose square is one, hence $+1$ or $-1$. That much is elementary and it says nothing about which.

Which one it is, is the spin-statistics theorem, and it is not elementary. It says that particles of half-integer spin take the minus sign and particles of integer spin the plus, and proving it requires relativistic quantum field theory: the argument runs through the requirement that fields at spacelike separation commute, which is causality, and shows that the wrong choice makes either the energy unbounded below or causality fail.

That is a deeply unsatisfying situation pedagogically and an interesting one physically. A rule about two electrons in an atom turns out to be a consequence of relativity and causality, and there is no known route to it that stays inside non-relativistic quantum mechanics. Feynman said several times that he could not find an elementary explanation of it and that this meant it was not fully understood.

What can be said without the theorem is that the choice is consistent: given the minus sign for electrons, everything in this essay follows, and every prediction it makes is confirmed. The alternation in the next figure is one of them, and it is a direct measurement of the sign for protons.

The same arithmetic in a spectrometer #

The protons in a hydrogen molecule are spin-halves too, and they obey the identical algebra — with a consequence that is directly visible.

The molecule’s total state must change sign when the two protons are swapped. Rotating the molecule by half a turn is that swap, and it multiplies the rotational state by $(-1)^J$. So the nuclear triplet, of weight three, goes with odd rotational levels and the singlet with even.

The spectrum therefore counts the states, for anybody with a spectrometer: the ratio is exactly three, and the three is the number of states in a triplet. It is the most direct measurement of a spin multiplicity there is, and it was observed before anybody knew what it meant.

The same arithmetic runs a spectrometer, and it runs the most famous experiment in the subject. The singlet’s correlation between two analysers, plotted against the angle between them, is a cosine — and no list of instructions carried by the two particles in advance can reproduce it. Everything peculiar about that curve descends from the minus sign in the state identified at the top of this essay. The oddness of the fourth state is not a bookkeeping convention; it is the thing the experiment measures.

The gas that is two gases #

The same fact has a thermodynamic face which took fifteen years to sort out.

Hydrogen is two gases, and its heat capacity says so. The rotational contribution, computed for a sample in equilibrium between the two forms, would show a tall peak as the levels come into play. What is actually measured is flatter, and it corresponds to three parts ortho to one part para whatever the temperature — because the conversion between them requires flipping a nuclear spin, which almost nothing does. The three-to-one is the same counting as the three symmetric states against the one antisymmetric one, sitting in a bottle and refusing to equilibrate.

Ortho-hydrogen — the nuclear triplet — occupies odd rotational levels only; para-hydrogen, the singlet, even ones only. Prepare the gas at room temperature and the ratio is three to one. Cool it, and it stays three to one, because turning a nuclear spin over requires something to couple to it and an ordinary collision does not.

So the low-temperature heat capacity is that of a frozen mixture rather than of an equilibrium one, and the two curves are quite different. Measurements from 1912 disagreed with the equilibrium calculation and nobody could say why; Dennison identified the cause in 1927.

The practical residue is large. The ortho-to-para conversion releases 523 kilojoules per kilogram, which is more than hydrogen’s latent heat of vaporisation — so liquid hydrogen stored as the room-temperature mixture boils itself away as it slowly converts. Every hydrogen liquefier therefore runs the gas over a paramagnetic catalyst, which supplies the magnetic coupling the collisions do not, and delivers nearly pure para-hydrogen.

A tank of rocket fuel is arranged around the symmetry of a two-particle wavefunction.

The exchange energy is not an interaction #

A point of language causes more confusion here than anything else in the subject, and it is worth settling.

The quantity $K$ is routinely called the exchange interaction or the exchange force, and it appears in Hamiltonians as a term $-2J\,\mathbf{S}_1\cdot\mathbf{S}_2$ that looks exactly like a coupling between two spins. Whole models of magnetism are built on it.

But no term in the underlying Hamiltonian couples the spins at all. The Coulomb repulsion between two electrons does not mention spin; it is a function of positions only. What produces the energy difference is that the spin state, through the antisymmetry requirement, decides which spatial state is allowed, and the two allowed spatial states have different Coulomb energies.

So the spin-dependent term is an effective one: correct, useful, and derived rather than fundamental. It is written as a spin coupling because in a two-state situation the projection onto the triplet and singlet can be expressed with $\mathbf{S}_1\cdot\mathbf{S}_2$, and that form is convenient — not because anything is coupling to anything.

The distinction matters in practice for two reasons. It explains the size: exchange energies are electrostatic in scale, tenths of an electron-volt, while true magnetic dipole interactions between two electron moments a few ångströms apart are around $10^{-4}$ electron-volts. Nothing magnetic could hold iron’s spins aligned against room temperature, and the electrostatic mechanism can. And it explains the sign changes: whether $J$ comes out positive or negative depends on the orbital geometry, so the same “interaction” gives ferromagnets and antiferromagnets with no change of principle.

An effective interaction that arises from a constraint rather than from a term in the Hamiltonian is a recurring shape, and recognising one is usually the difference between predicting its size and merely fitting it.

Where the same three-and-one appears #

In the helium spectrum, as two apparently separate elements. The singlet and triplet systems of helium have very few transitions between them, so nineteenth-century spectroscopists catalogued them as two substances and called them parahelium and orthohelium. They are one element in two spin systems, and the near-absence of transitions between them is a selection rule about the spin.

In ferromagnetism. The exchange integral between neighbouring atoms in iron is positive and large, so parallel spins are favoured — and the alignment is a consequence of electrostatics acting through the symmetry, not of the magnetic interaction between the moments, which is a thousand times too small to hold at room temperature. Ferromagnetism is a phenomenon classical physics forbids for precisely this reason.

And in entanglement. The singlet is the standard entangled state: measuring one spin along any axis fixes the other along the same axis, with a correlation no shared instruction list can reproduce. The minus sign that makes it a singlet is what makes the correlation the same on every axis.

The same three-and-one turns up wherever spins are sorted rather than counted. Send a beam through an analyser and then through a second one, and what emerges is a set of counts rather than an algebra — but the counts are the algebra. Two such analysers, one on each half of a prepared pair, are what report the singlet’s perfect anticorrelation: whatever direction the first is set to, the second finds the opposite, and it finds it every time.

The temperature that separates the two pictures #

One number decides whether any of this is visible, and it is worth extracting.

Nuclear-spin statistics show themselves only where the rotational levels are resolved — that is, where the spacing $\theta_{\text{rot}} = \hbar^2/2Ik$ is comparable with the temperature. For hydrogen that is 85.4 kelvin, which is why hydrogen’s rotational heat capacity has structure at accessible temperatures and why the ortho–para distinction is a laboratory fact.

For deuterium the moment of inertia is twice as large, so the rotational temperature is 43 kelvin — and the nuclei are spin-one bosons, so the symmetry requirement reverses and the ratio is 2:1 the other way. For nitrogen it is 2.9 kelvin, and for iodine 0.05, so their rotational levels are classical at any ordinary temperature and the alternation is unobservable in a heat capacity while still being present in the spectrum.

The general shape of the rule is that a quantum statistical effect becomes thermodynamically visible when the level spacing it concerns reaches $kT$, which is the same criterion that freezes out a degree of freedom. Hydrogen is the one molecule light enough for that criterion to be met somewhere convenient, which is why every discussion of this subject is a discussion about hydrogen.

The three to one that limits a display #

The triplet lies below the singlet and there are three of it. Both halves of that sentence have consequences outside atomic physics, and the second one currently sets a ceiling on a manufacturing technology.

Excite a molecule with light and the electron that moves keeps its spin, so the excited state is a singlet and can drop back to the singlet ground state freely: that is fluorescence, and it is over in nanoseconds. Some molecules instead cross over into the triplet, and the triplet cannot drop back, because the ground state is a singlet and the transition would have to flip a spin. It is spin-forbidden, so it happens slowly — milliseconds to minutes rather than nanoseconds — and that is phosphorescence, which is why a glow-in-the-dark toy glows after the light is off.

“Forbidden” here means suppressed rather than impossible. Spin–orbit coupling mixes a little singlet character into the triplet, and its strength climbs steeply with atomic number, so a phosphor is made to work by putting a heavy atom into it. A material’s afterglow is tuned by choosing an element for its nucleus’s charge.

Now excite the molecule electrically instead, by bringing an electron and a hole together in a light- emitting diode. They arrive with uncorrelated spins, so the state they form is a singlet one time in four and a triplet three times in four — the counting of the first figure of this essay, in a device. A diode whose emitter can only fluoresce therefore throws away three quarters of everything injected into it, and that twenty-five per cent was the hard ceiling on organic displays for a decade.

The way past it was to make the triplets emit: complexes built round iridium or platinum, where the heavy atom’s spin–orbit coupling is strong enough to turn a millisecond phosphorescence into a microsecond one, which lifts the ceiling to a hundred per cent. Every phosphorescent pixel in a modern display is a piece of engineering against the ratio three to one.

The singlet that is a chemical species, and the one that is a signal #

Oxygen’s ground state is a triplet, which the essay used to explain the molecule’s sluggishness. The corollary is that oxygen has a singlet state lying about one electron-volt above it, and because the transition back down is spin-forbidden the state is remarkably long-lived — microseconds in water, and the better part of an hour in dilute gas.

That makes singlet oxygen a distinct chemical substance rather than a fleeting excitation, and a violently reactive one, since the spin obstruction that makes ordinary oxygen tolerable to be surrounded by has been removed. It is manufactured deliberately by shining light on a dye in the presence of oxygen, and used that way in photodynamic therapy, where the dye is delivered to a tumour and the light is aimed at it.

The nuclear singlet has a use of the same kind. Para-hydrogen has no net nuclear spin and therefore no nuclear magnetic resonance signal of its own — and if its two protons are added to a molecule in a single step, so that they end up inequivalent, the singlet order they were carrying reappears as a resonance signal enhanced by four orders of magnitude over the thermal one. The trick works because the ordinary NMR signal is pitifully weak, being the small excess of one spin orientation over the other in a thermal population, while para-hydrogen arrives with its spin order already perfect.

So the same 3:1 mixture that a liquefier spends energy destroying is, on a chemist’s bench, the starting material: the singlet fraction is separated, kept for weeks in a cold cylinder because nothing converts it, and used to light up a spectrum.

What the picture cannot show #

The pair-correlation calculation uses one pair of fixed orbitals. A real two-electron system has orbitals that adjust to each other, and the adjustment is comparable in size with the effect being drawn. What the figure establishes is that the symmetry alone produces a separation; how large it is in an actual atom requires a self-consistent calculation.

The exchange integral is computed in one dimension with a softened kernel. The scaling and the sign are what the figure is for, and the absolute values are in arbitrary units for that reason. The 0.80 electron-volts quoted for helium is a measured number, not this calculation’s.

The alternating intensities assume the rotational levels are those of a rigid rotor. Real hydrogen stretches as it spins, so the higher levels are a little closer together than the model puts them, and the envelope shifts. The alternation itself is exact and is a counting statement rather than a dynamical one.

And nothing here treats more than two particles. Combining three spin-halves gives a quartet and two doublets, and the pattern for many particles is a substantial piece of group theory. The two-particle case is unusually clean, and generalising from it by analogy is how most people get the three-particle case wrong.

The ladder from here #

Later rungs on this anchor: the addition of angular momenta in general, with the Clebsch–Gordan coefficients and the reason a quartet appears at three particles; Hund’s three rules and the ordering of a configuration’s terms; the exchange interaction as the origin of magnetic order, and the sign changes that give antiferromagnets; the spin-statistics theorem, which is where the connection between half-integer spin and antisymmetry actually comes from and which needs relativistic field theory to state; and the singlet as a resource, in entanglement and in quantum information.

The neighbouring ladders are the algebra of one spin, which is what this essay combines two of, and the exclusion principle, whose spatial content the Fermi hole makes visible. The correlation no instruction list produces is what the singlet becomes when the two halves are taken far apart.