The angular momentum that is not a rotation

The word is a disaster and it is two hundred years too late to change it. An electron has an intrinsic angular momentum of $\hbar/2$ and an intrinsic magnetic moment to go with it, and both were named by analogy with a spinning charged ball because that is the only object in classical physics that has both. The analogy fails, and it fails by a margin large enough that following it through is the fastest route to understanding what is actually being described.

There is no radius at which the picture survives. Making the electron bigger to slow the equator down runs into the fact that scattering experiments see no structure at all down to $10^{-18}$ metres, and making it smaller makes the arithmetic worse.

It is worth being clear about what that calculation does and does not settle. It does not prove that the electron is a point; nothing does, and the bound is experimental. What it proves is that if the angular momentum were the angular momentum of rotating matter, the electron would have to be enormous by nuclear standards — hundreds of times the size of an atomic nucleus, and a substantial fraction of an atom — and it demonstrably is not. The failure is not a matter of the analogy being imprecise. It is off by five orders of magnitude in a quantity that has a hard ceiling.

What a classical angular momentum requires #

The contrast is worth setting out, because the classical quantity is not merely a different value of the same thing.

What a classical angular momentum requires is three ingredients, and every one of them is a property of an extended object: a mass, the square of how far that mass sits from an axis, and a rate of turning. Take any of the three to zero and the angular momentum goes with it. An electron has no measured size at all — the experimental bound is small enough that a classical sphere with its angular momentum would have to turn its surface far faster than light — so the classical recipe does not merely give the wrong answer here. It has no ingredients to work with.

A classical angular momentum needs mass distributed at a distance from an axis and a rate of turning. It can take any value, it points along an axis that can be any direction, and measuring its component along a direction returns a number that varies continuously with angle.

An electron’s spin has none of those properties. Its magnitude is fixed at $\hbar/2$ and cannot be changed by any means whatever. Its component along any axis returns one of two values and never anything between them. And there is no rate and no radius in it.

What does survive the analogy #

Something does survive, and it is the reason the name is not merely wrong. A magnetic moment goes with the angular momentum, in a fixed ratio, exactly as it does for a rotating charge — and a moment in a field feels a torque and no net force, which is what the deflecting apparatus below has to be built around.

What does survive the analogy is the magnetic moment. A current loop behaves at a distance exactly like a small bar magnet, with a moment that is a current times an area; a classical spinning charge has a moment proportional to its angular momentum with a ratio $q/2m$; and an electron has one too — with the ratio twice as large. That factor of two had no explanation for two decades and came out of Dirac’s equation without being put in, which is the strongest evidence that spin is a relativistic property rather than a rotation.

And the algebra survives entirely. Spin obeys the same commutation relations as orbital angular momentum, adds to it to give a total, and generates rotations in exactly the way angular momentum does in classical mechanics. That is not a small residue: in modern physics, angular momentum is defined as whatever generates rotations, and spin does, so spin is angular momentum by the only definition that survives. The same reversal has already been made once in this collection for the quantity that survives a change of shape — the conserved thing is identified by what it does rather than by what it is made of.

What has been given up is the picture, not the quantity. The quantity is conserved, it adds, it torques in a field, and it is carried away by emitted photons in the amounts a conservation law requires.

The measurement that has two answers #

Send atoms through a field gradient and they are deflected by an amount proportional to the component of their moment along the gradient.

That is the apparatus in which an answer that was not there before is manufactured, and two things in the figure are the whole of the subject. The output is two beams and not a smear, which is quantisation. And the third analyser produces both of its outputs, which means the middle one destroyed the information the first one established — a filter that removes something cannot put it back.

The half-angle #

The relationship between the preparation axis and the measurement axis carries a factor that no arrow has.

The half-angle is the single most consequential feature of a spin-½. It says that two states are orthogonal — meaning that one never gives the other’s answer — when they point in opposite directions rather than at right angles. A vertical arrow and a horizontal arrow have zero projection on one another; a vertical spin measured horizontally gives up and down with equal probability, which is as far from a definite answer as it is possible to be.

And the classical answer is not lost: it is the average. That is the pattern that recurs everywhere in this collection — the old description survives as an average or a limit of the new one — and it is why nobody noticed for so long that the projection of a compass needle and the reading of a spin analyser are different kinds of quantity.

The turn that has to be made twice #

The half-angle has a consequence that sounds like bookkeeping and is not.

An overall sign on a state is unobservable, so the minus sign after one turn cannot be detected by any measurement on the spin alone. It can be detected by comparison. Split a beam, rotate the spin in one arm only, recombine, and the two arms interfere with a relative phase that has period $4\pi$.

The experiment was done with neutrons in 1975, with a silicon crystal interferometer and a magnetic field in one arm, and the period is 720 degrees. That is a fact about the world rather than about notation, and it is the sharpest available demonstration that a spin-½ is not a little arrow: an arrow returns to itself after one turn, necessarily, because it is a direction in space.

What two-valuedness forces #

Once a particle has a two-valued internal quantity, several other things follow, and they are the reason spin matters far beyond the measurement of magnetic moments.

Two-valuedness forces the fine structure, and that was measured long before it was understood. Nearly every line in an atomic spectrum is doubled at high resolution, by an interaction between the electron’s spin and the magnetic field it experiences in its own orbital motion. The splitting is a fraction of a per cent, it was catalogued for decades under a name that described it rather than explained it, and it is one of the earliest pieces of evidence that the electron carries an angular momentum with two settings.

It forces the shape of matter, too. Two electrons per state rather than one is what makes a periodic table with the row lengths it has, and it is what makes a metal’s electrons fill a sea up to a definite ceiling rather than crowding into the lowest level. Both consequences follow from a count of two, and the count is two because a spin-half has two settings and no more.

The connection between the half-integer value and the rule that no two may share a state is not obvious and it is not a coincidence: the spin–statistics theorem says that half-integer-spin particles must have antisymmetric joint states and integer-spin particles symmetric ones. The 720-degree periodicity above and the antisymmetry are the same fact viewed twice — exchanging two identical particles is, topologically, half a rotation.

And it forces the correlations that make Bell’s theorem possible. A pair of spin-half particles in a singlet has a correlation between distant measurements steeper than any list of shared instructions can produce, prepared in advance can reproduce — and the minus sign that does the work in that state is available only because each particle has exactly two settings. A three-valued object would not produce it.

Two more places the two values decide something large #

A magnet. The magnetism of iron is almost entirely spin rather than orbital motion, and the reason iron is magnetic and copper is not has nothing to do with the strength of any magnetic force. It is the exchange interaction: because the joint state of two electrons must be antisymmetric, electrons with parallel spins are kept apart in space, which lowers their electrostatic repulsion. Aligning spins is therefore electrostatically favourable, and the energy involved is an electronvolt rather than the microelectronvolt a magnetic dipole interaction would supply. A magnet is held together by the exclusion rule and Coulomb’s law, and the magnetic field is the effect rather than the cause.

The two values decide something in two more places worth naming. A magnetic resonance experiment drives transitions between exactly those two settings, and the frequency it needs is what makes an imaging machine possible. And a ferromagnet’s moment is those same two settings aligned in bulk — a macroscopic quantity built from a two-way choice, which is why a magnet has a saturation and cannot be made stronger by trying harder.

A measurement of time. Every atomic clock in the world counts a transition between two states that differ only in how a nuclear spin is oriented relative to an electron spin. The caesium standard’s 9,192,631,770 hertz is a hyperfine splitting, an energy difference of forty microelectronvolts produced by two magnetic moments noticing each other, and the second is defined by it. A quantity with no classical existence sets the unit of time.

Where the value comes from #

Spin was not derived; it was measured, then postulated, and only then derived. The order is worth recording, because it is the usual order.

Stern and Gerlach fired silver atoms through a gradient in 1922 and got two beams. A gradient rather than a uniform field, because a uniform field gives a torque and no force at all and would have turned the moments without moving the atoms. They were testing space quantisation of orbital angular momentum and their result was, for that purpose, wrong — silver’s ground state has no orbital angular momentum, so there should have been one beam. The two beams were an unexplained fact for three years.

Uhlenbeck and Goudsmit proposed the intrinsic angular momentum in 1925 and were told by Lorentz that the surface of the electron would have to move faster than light — the calculation drawn at the top of this essay. They published anyway.

And Dirac’s relativistic equation for the electron, in 1928, produced the spin, its value, and the factor of two in its magnetic moment without any of the three having been put in. That is the strongest kind of confirmation available: a property that had to be added by hand to a theory falls out of the next theory as a consequence of requiring it to be consistent with relativity.

What a spin does in a field, which is precess #

The essay has been about measurement, and has said nothing about what a spin does when it is simply left in a magnetic field and not interrogated. The answer is the one place where the discarded picture is exactly right, which makes it worth stating carefully.

A magnetic moment in a field feels a torque equal to the moment crossed into the field. For an ordinary object that torque would tip the moment toward alignment. For a moment tied to an angular momentum it cannot, because a torque changes angular momentum in the direction the torque points — which is perpendicular to both the moment and the field — so the moment sweeps round the field direction instead of falling toward it. That is precession, and it is the same reason a leaning gyroscope circles rather than topples.

Working out the rate gives a result with a pleasant absence in it. Because the moment and the angular momentum are proportional, the equation reduces to the angular momentum rotating about the field at a rate

$$\omega_L = \gamma B,$$

where $\gamma$ is that ratio. The angle between the spin and the field does not appear, so every orientation precesses at the same rate, and the magnitude of the spin does not appear either.

The remarkable part is that this is exactly true quantum mechanically. The expectation value of a spin in a uniform field obeys precisely the classical precession equation, with no corrections and no limits taken — the one place in this essay where the little-arrow picture is not an approximation but the answer. The component along the field is constant, which is why the two outcomes of a measurement along that axis are stable in time; the components across it rotate, which is why a measurement along a transverse axis gives an answer that oscillates.

The numbers are large and are the basis of a technology. An electron precesses at 28 gigahertz per tesla, a proton at 42.58 megahertz per tesla, and those two frequencies are what electron and nuclear magnetic resonance are named after. Driving the system with an oscillating field at exactly that frequency tips the spin away from the field axis by an angle proportional to how long the drive is applied — which is how a spin is set to an arbitrary state rather than merely measured, and is the whole of quantum control for a two-level system.

Reading a spin in bulk #

A single spin is very hard to detect and an enormous number of them is easy, and how that works involves an arithmetic that looks at first as though it should not.

Put water in a field of 1.5 tesla at room temperature. The proton spins have two states split by the Larmor energy, and the population ratio follows the Boltzmann factor: the splitting is $h$ times 64 megahertz, which is $4.2\times10^{-26}$ joules, against a $kT$ of $4.1\times10^{-21}$. The exponent is a hundred-thousandth. So the excess of spins in the lower state over the upper one is about five parts in a million, and the sample is essentially unpolarised.

Five parts per million of nothing would be nothing. Five parts per million of Avogadro’s number is not: a cubic millimetre of water holds around $10^{20}$ protons, so the net excess is some $10^{14}$ spins, all precessing at the same frequency and — after a drive pulse has tipped them together — all in phase. A coherent, rotating magnetisation of $10^{14}$ moments induces a perfectly measurable voltage in a nearby coil.

That is magnetic resonance, and the reason it images anatomy is that the precession frequency is proportional to the field. Superimpose a gradient on the uniform field and every position precesses at a slightly different frequency, so the frequencies present in the received signal are a map of where the protons are. The picture is a Fourier transform of a signal whose frequency axis has been made into a spatial axis by a magnet.

Two things about that are worth carrying beyond the application. The first is that the feeble polarisation is not a defect to be engineered away but the ordinary condition of any thermal spin system — the reason higher-field magnets give better images is that the exponent, and therefore the excess, is proportional to the field. The second is what the measurement is actually of: not a spin state, but the coherence between two of them, which is the transverse component that has no classical counterpart in a two-valued quantity and behaves exactly like a classical rotating vector anyway.

Where the model stops #

The factor of two is not exactly two. Quantum electrodynamics gives 2.002319304…, and that anomaly is measured and computed to twelve significant figures in agreement. It is the most precisely tested prediction in physics, and it comes from the electron’s interaction with the fields it itself produces — which means the electron is not the bare object the elementary treatment describes.

Nothing here explains why the value is a half. It is a half because the electron transforms as a spin-½ representation of the rotation group, which is a statement about which mathematical object it is rather than an explanation. Why nature uses that representation for matter and the integer ones for forces is the spin–statistics theorem’s business, and the theorem needs relativity and quantum field theory to prove.

Orbital angular momentum is quantised too, and differently. It comes in whole units, its states are ordinary functions of position, and the electron’s distribution in an atom is what those functions describe. Spin has no such function, no position dependence and no whole-number values, which is why it is called intrinsic rather than orbital.

A neutron and a proton have anomalous moments too, and much larger ones — 2.79 and −1.91 nuclear magnetons against the 1 a point particle would give. That is a signature of internal structure, and it was one of the first pieces of evidence that they had some.

And the classical limit is not a large spin. A spin of 1,000ħ is still measured in discrete steps and still has a half-angle; what makes something behave classically is not a large quantum number alone but decoherence, which is a different subject.

What the pictures cannot show #

Every drawing of a spin as an arrow — including every drawing in every textbook, including the ones this essay’s figures imply — is the picture the essay says is wrong. There is no faithful alternative: a spin-½ state is a point on a sphere whose antipodes are orthogonal rather than opposite, and no drawing of a three-dimensional object has that property. The arrow is used because there is nothing better, and every use of it carries the half-angle as an unwritten footnote.

Nor can the interference figure show what is being interfered. The two arms of the neutron interferometer are separated by centimetres, the neutron goes through both, and the recombined intensity is a statement about a single particle whose two histories differ by a rotation. A drawing of two beams implies two neutrons, and there is one.

Where this ladder goes next #

What has been established is that a quantity can be angular momentum in every operational sense — conserved, additive, generating rotations, torquing in a field — without being the rotation of anything. The name is a historical accident and the object is not a small version of the classical one; it is a different kind of thing that reduces to the classical one only on average.

The habit worth carrying away is the arithmetic test that opened the essay. When a quantum quantity is explained by analogy with a classical mechanism, put numbers in the mechanism. A spinning ball, a planetary orbit, a vibrating string: each is a real analogy with real content, and each has a regime in which it produces an absurdity. Finding the absurdity is more informative than accepting the analogy, because the size of the failure says how much of the classical picture can be kept.

What is left on this ladder is what happens when spins are put together: how two of them add to a triplet and a singlet, why the singlet is the state that carries no direction at all, and how a magnetic material is built out of the exchange interaction that antisymmetry alone produces.