Illustrated Physics
physics, drawn
Concept

Spin — where it appears

An intrinsic angular momentum a particle carries without rotating, quantised, and the property the exclusion principle divides matter by. It cannot be a rotating ball: reproducing an electron's angular momentum in a sphere of its measured size would need a surface moving faster than light.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

The answer that was not there before

Send a beam through an analyser and it splits in two. Send one half through a second analyser turned sideways, then through a third pointing the way the first did, and the property the first analyser removed has come back.

quantum · Measurement The energy ladder of a box. The first 6 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0, 36.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 4 to 3 releases 7.000 E₁.

No two in the same state, and why matter has volume

Nothing in the energy levels of an atom says how many electrons may occupy each one. The answer is one per state, it is not derived from any force, and it is the reason a table holds a cup up.

quantum · Exclusion The correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

The correlation no instructions can produce

A pair of gloves in two boxes agrees perfectly and needs no physics, because the answers were settled at packing. What no packing can imitate is the shape that appears as the two analysers are turned relative to each other, and the shape is a number — 2.828 where every list of pre-agreed answers is stuck at 2.

quantum · Entanglement The rotation two boosts leave behind. The angle through which a frame's axes are turned after two boosts of equal size, against the angle between the two boosts, for 4 speeds. Two boosts in the same direction compose to a boost and nothing else, which is the zero at the left; two in different directions do not. What is left over is a rotation, and it is not small at large speeds: at β = 0.3 it peaks at 2.7° when the boosts are 91° apart, at β = 0.6 it peaks at 12.8° when the boosts are 96° apart, at β = 0.85 it peaks at 36.1° when the boosts are 108° apart, at β = 0.95 it peaks at 63.2° when the boosts are 122° apart. Each curve here is computed by multiplying the two boost matrices and pulling the rotation out of the product, not by evaluating a formula; the closed form for perpendicular boosts is used to check the extraction and appears nowhere in the drawing. The consequence is that the Lorentz boosts do not form a group by themselves — compose two and you leave the set — and that an object carried round a closed path in velocity space comes back turned.

The turn that two pushes leave behind

Two boosts in different directions do not compose to a boost. The product carries a rotation, so a frame carried once round a closed path comes back turned — and the size of that turn was the factor of two standing between the calculated and the measured splitting of a spectral line.

relativity · Velocity addition 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 field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

electromagnetism · Magnetisation Four predictions no list of answers can keep. The four measurements a three-particle GHZ state predicts with certainty, and what the best possible list of pre-agreed answers does with them. Each row is a choice of which quantity to measure on each of the three particles; the quantum value is not an average but a certainty, so a single run of any row has a determined outcome. Every list of answers assigns a value to X and to Y at each particle, which is sixty-four lists in all, and the enumeration finds that 32 of them get three of the four right and 32 get one. None gets four, and none can: multiplying the four left-hand sides together gives every X and every Y twice, so the product is +1 for any list whatever, while the product of the four quantum values is −1. The column on the right is one such list, which agrees with the first three rows and is then forced into the opposite of the fourth. That is the whole argument, and it needs no inequality, no average and no repetition: measure the first three rows on three copies, and the fourth is predicted with certainty and comes out the other way.

The disagreement that one run settles

Bell's argument is a statistical one — a correlation of 2.828 where a pre-agreed list of answers is stuck at 2, dug out of hundreds of coincidences. Add a third particle and the argument stops being about how often. Three measurements predict a fourth with certainty, every list of answers that gets the three right gets the fourth exactly backwards, and one run of the experiment is enough.

quantum · Entanglement One level, split by the speed of the electron in it. Hydrogen's n = 2 level on the left at the scale of the whole spectrum, and the same level on the right at a scale ten thousand times finer. The gross structure puts it 3.4014 electronvolts below the ionisation limit; the relativistic and spin–orbit corrections then split it into a lower pair, j = ½, and an upper, j = 3/2, separated by 45.3 microelectronvolts. That gap is 1.331e-5 of the level's own binding energy, which is exactly α²/4 — so the size of the splitting is a measurement of how fast the electron is going, α being its speed in units of c on the innermost orbit. The same physics in sodium is larger by four orders of magnitude: its two D lines at 588.995 and 589.5924 nanometres differ by 2.133 millielectronvolts, which is 1.01e-3 of the transition energy, because the outer electron of sodium penetrates to where the nuclear charge is far from screened and the correction goes as the fourth power of the charge it sees.

The line that is really two

Sodium's yellow line is two lines six-tenths of a nanometre apart, and the gap is not a property of the light. It is a splitting of the atom's own level, caused by the electron's magnetic moment sitting in the field it sees because it is moving — and its size, relative to the level it splits, is exactly α²/4.

quantum · Atomic spectra 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 Which pairs have anything between them. The commutator of every pair of 4 observables of a spin-½, multiplied out in 2×2 complex arithmetic and shown by the size of AB − BA. The diagonal is exactly zero: an observable commutes with itself, which is why measuring the same thing twice gives the same answer. S², the total angular momentum, is a multiple of the identity here and so commutes with everything — its row and its column are zero, and a spin can have a definite total angular momentum and a definite component at the same time. The largest entry is 0.7071 in units of ħ². Sz: 0.000 with Sz, 0.707 with Sx, 0.707 with Sy, 0.000 with S²; Sx: 0.707 with Sz, 0.000 with Sx, 0.707 with Sy, 0.000 with S²; Sy: 0.707 with Sz, 0.707 with Sx, 0.000 with Sy, 0.000 with S²; S²: 0.000 with Sz, 0.000 with Sx, 0.000 with Sy, 0.000 with S². A zero cell is a promise that the two quantities can be sharp together; a non-zero one is an obstruction whose size sets how badly they cannot.

The questions that can be asked together

Two quantities can have definite values at once exactly when their operators commute. That is a piece of arithmetic about matrices, and everything the uncertainty principle forbids follows from it — including the fact that most of the time it forbids nothing at all.

quantum · Uncertainty The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting.

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

relativity · Relativistic dynamics What comes back after one turn, and what needs two. A spin-½ pointing along z and rotated about the x axis through 720°, with the rotation integrated step by step rather than evaluated from a formula. The direction of the spin — the quantity a Stern–Gerlach magnet, a compass or any other instrument reports — is back where it started after 360°, exactly as the orientation of any other object would be. The state is not: its overlap with the state it began in has reached −1 there, and returns to +1 only after 720°. At 360° the overlap is -1.000 and ⟨σz⟩ is 1.000; At 720° the overlap is 1.000 and ⟨σz⟩ is 1.000. Both curves come off one integration of dψ/dθ = −(i/2)σx ψ whose norm is checked before anything is drawn, so the factor of two between their rates is a property of the propagation rather than of two separate formulae that were chosen to differ.

The turn that has to be made twice

Turn a spin-½ through a full circle and it does not come back. The direction it points in does, and every measurement on it does, but the state itself has changed sign — and a second full turn is needed before anything is where it started. The sign is invisible on one spin and measurable the moment a superposition has one branch turned and the other not.

quantum · Spin The quantity that went down, and the one that went up. The books for GW150914: two black holes of 36 and 29 solar masses merging into one of 62, with a final spin of 0.67. 3.0 solar masses left as gravitational waves, so the mass fell by 4.6 per cent. The total horizon area rose, from 107417 to 168330 in units of the Sun's gravitational radius squared — an increase of 57 per cent. The two progenitors are taken as non-spinning, which is the assumption that makes the test hardest to pass: a spinning hole of the same mass has a smaller horizon, so any spin they actually had would only widen the gap. Mass is the quantity that behaves the way energy usually does and it is not the one with a direction. Area is, and it is the reason the area has been read as an entropy ever since.

The area that is not allowed to shrink

Two black holes merge and the result weighs less than the sum, because three solar masses left as gravitational waves. The horizon area went up by more than half. Mass is the quantity that behaves like energy and it is not the one with a direction; area is, and the theorem saying so has been tested against a real merger.

astrophysics · Horizons The rectangle a spin tilts. The laboratory energy of the pion from a tau into a pion and a neutrino moving at 0.8 of the speed of light, for samples of 60,000 decays whose parents have αP = +1, 0, −1 along their line of flight. Every sample fills the same interval, 313 to 2667 MeV: the edges are set by the masses and the speed, and they do not move. Inside that interval an unpolarised sample is flat and a polarised one is tilted, its height at each energy 1 + αP times the cosine of the rest-frame angle that energy corresponds to. Reading the slope back from each sample gives +0.993, −0.010, −1.006, each with a standard error near 0.006. A parent spinning along its flight throws the pion forwards in its own frame, and the boost turns forwards into more energetic, so the tilt of an energy spectrum measures a polarisation without any angle being measured at all.

The slope a spin leaves in a spectrum

An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.

relativity · Relativistic dynamics Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

relativity · Relativistic dynamics The levels that refuse to cross. The six energies of sodium 3p against an applied field from zero to 4 crossover fields, 147 T, from diagonalising spin–orbit coupling and the field together. At zero field there are two levels, 515.5 GHz apart. The two states with the largest |mⱼ| are straight lines at every field, because nothing else shares their mⱼ. The other four bend. The two states with mⱼ = −½ approach within 486.0 GHz near 12.3 T and then separate again — √2 ζ exactly, since the coupling between them sets how close they may come — while the mⱼ = −3/2 state falls straight through the lower mⱼ = +½ state at 16.6 T as if it were not there. States that the field cannot mix cross freely; states it can mix exchange character instead, and that exchange is the passage from one pattern to the other.

The field an atom calls strong

A magnetic field splits sodium's two yellow lines into ten, at spacings set by Landé's factors — until the field grows past the one the electron already feels from its own motion, when the ten reorganise into the three a theory without spin predicts. Nothing about the magnet decides which pattern appears. The atom does: the same 45 tesla is a strong field for hydrogen, a middling one for sodium and a weak one for caesium.

quantum · Atomic spectra A few cycles, and everything about them is two numbers. The strain radiated by a remnant of 62 solar masses spinning at 0.68 of its maximum, as it settles down, with the decaying envelope of its fundamental mode drawn over it. The fundamental rings at 274 hertz and decays in 3.7 milliseconds, which is 1.0 cycles — this is not a bell and it does not sustain. Every frequency and every decay time in the sum is fixed by the mass and the spin alone; nothing about what made the remnant survives into them. What does depend on the collision is how loudly each mode is excited, and the relative amplitudes here are the rough values a merger of two comparable masses produces rather than a prediction.

A few cycles that are only mass and spin

After the orbit is gone there is one object left, distorted, and it settles down by radiating at frequencies that belong to it rather than to the collision. For a black hole those frequencies are fixed by the mass and the spin and by nothing else — so the first mode measured is a measurement and every mode after it is a test, and the test is that four curves in one plane pass through one point.

astrophysics · Gravitational waves The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all.

Two states where the counting says three

A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

quantum · Spin No beam is narrow enough and wide enough at once. Three lengths at the far end of a Stern-Gerlach magnet 10 centimetres long with a gradient of 1000 tesla a metre, against the width of the beam entering it, for an electron at 100 electronvolts. The spin splitting is a horizontal line at 2.9e-6 metres: it does not depend on the beam's width. The Lorentz blurring rises in proportion to the width, because the field a particle sees depends on where in the beam it is, and the divergence-free condition ties a gradient in one component to a gradient in another. It overtakes the splitting at 1.0e-9 metres. Narrower than that and diffraction has already spread the beam by 3.9e-3 metres, which is larger still. There is no width at which the splitting is the largest of the three, and the magnet's length and gradient cancel out of the comparison entirely — so no magnet helps.

The experiment that defines spin and cannot be done on it

A Stern–Gerlach magnet separates magnetic moments and is how spin was discovered. It cannot be made to work on a free electron, and the obstruction is not the apparatus: the field gradient that splits the beam also deflects the charge by an amount that varies across it, and the ratio of the splitting to that blurring comes out as the de Broglie wavelength over the beam width — with the magnet's length and gradient cancelling exactly.

quantum · Spin

Named alongside it

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

MeasurementAngular momentumSuperpositionDegeneracySymmetryInterferenceMagnetic momentAtomic spectraEnergy levelsThe Lorentz transformationPolarisationQuantisation

All concepts