Illustrated Physics
physics, drawn
Field

Quantum

Where the continuous picture runs out, and what replaces it.
The blackbody spectrum, against what classical physics predicted. Spectral exitance against wavelength for a blackbody at 3000, 4000, 5000 kelvin, in kilowatts per square metre per nanometre. Each curve peaks at the wavelength Wien's displacement law gives — 966 nm at 3000 K, 724 nm at 4000 K, 580 nm at 5000 K — and falls to nothing at short wavelengths.

The curve that would not come down

Classical physics predicted that a warm object radiates infinite power at short wavelengths. Every step of the derivation was correct, the prediction was absurd, and closing the gap required assuming that energy comes in lumps.

 Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption.

Light arrives in lumps, and brightness only changes how many

Shine dim blue light on a metal and electrons come out. Shine intense red light and none do, however long the wait. The frequency decides whether anything happens; the intensity decides only how much.

 The wavelength shift against scattering angle. How much longer a scattered photon's wavelength is, against the angle it scattered through. The shift runs from nothing at 0 degrees to 4.853 picometres straight back, passing through the electron's Compton wavelength of 2.4263 picometres at 90 degrees. Nothing about the incident light or the target material appears anywhere on this axis.

A photon with a momentum, and a collision that proves it

X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.

 An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.

Everything has a wavelength, and almost nothing shows it

If light with a momentum can behave like a particle, a particle with a momentum can behave like a wave. The wavelength is Planck's constant over the momentum, which for anything larger than a molecule is a number too small to have consequences.

 The states a box allows, drawn on their energies. A particle confined between two walls one unit apart. States 1, 2, 3 are drawn, each riding on a line at its own energy — 1E₁, 4E₁, 9E₁ — because the energies go as n². Each wavefunction has n − 1 places where it crosses zero inside the box: 0 for n = 1, 1 for n = 2, 2 for n = 3. Nothing about the particle's mass or the depth of the well appears in the shapes; only the count of half-wavelengths that fit does.

The box that allows only some energies

Confine a wave between two walls and only the shapes that fit survive. That is a fact about strings, organ pipes and drumheads, and applying it to a matter wave produces quantisation with no new assumption at all.

 Hydrogen's emission lines, where they are actually seen. Every transition down to level 1, 2, 3 in hydrogen, drawn at the wavelength it emits, on a logarithmic axis in nanometres. The Lyman series begins at 121.5 nm and crowds toward its limit at 91.1 nm; The Balmer series begins at 656.1 nm and crowds toward its limit at 364.5 nm; The Paschen series begins at 1874.6 nm and crowds toward its limit at 820.1 nm. Only the Balmer series has lines in the visible band, which is why it was the one found first.

The spectrum is a subtraction, not a list of values

An atom emits a handful of sharp wavelengths and nothing in between. They are not the atom's energies — they are the differences between them, which is why the lines come in families that crowd onto a limit.

 Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found.

Where the electron probably is

The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

 A packet, and the wavenumbers it is made of. Above: a wave packet built by adding a continuum of plane waves centred on wavenumber 12 with a spread of 1.6. Below: the weight given to each wavenumber. The packet's width, measured as the standard deviation of its probability, is 0.442; the spread of wavenumbers is 1.131; their product is 0.500, which is a half and cannot be less. Narrowing one bracket widens the other by exactly as much. Nothing quantum has been used to draw either panel.

Sharpness has to be paid for

A wave with one exact wavelength has no beginning and no end. Making it short requires adding wavelengths, and the two widths trade against each other exactly — which is a fact about waves, with Planck's constant added only to convert the units.

 A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

The wall that is not quite a wall

A particle without enough energy to climb a barrier sometimes appears on the other side of it. The probability falls exponentially with the barrier's width, which is why the effect is invisible at ordinary scales and why it can be turned into a microscope.

 The interference pattern arriving one particle at a time. The same double slit — 100 micrometres apart, slits 40 micrometres wide, lit at 633 nm, screen 1 metre away — recorded after 20, 200, 1000 arrivals, with the intensity that governs them plotted underneath. Each arrival is a single dot in one place, drawn at a position sampled from that intensity. After 20 there is no pattern to see; after 1000 the fringes are unmistakable, with the dark ones exactly 6.33 millimetres apart — the wavelength times the screen distance over the separation. The bright ones are not evenly spaced, because the single-slit envelope pulls each maximum toward the centre; its first zero is at 15.8 millimetres and is set by the width of one slit alone.

One arrival at a time, and the pattern still appears

Send particles through a double slit slowly enough that only one is ever in the apparatus, and each arrives as a single dot in one place. Wait, and the dots assemble into fringes that no dot knew about.

 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.

 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.

 One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

What happens when the wells get close

Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window.

 Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.

A nucleus with no clock

A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.

 Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises.

Where the quantum picture hands back the old one

A confined particle's probability density oscillates violently at every quantum number, and never stops. What makes the classical answer come back is not that the oscillations die away — it is that nothing can resolve them.

 Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

 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.

 Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

 How much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

The pressure that is not a temperature

Copper's conduction electrons are at a temperature of eighty thousand kelvin, in a wire that is at room temperature. That is not a figure of speech, it is what the exclusion principle does to a mole of particles, and it explains the largest unexplained number in the theory of metals.

 Twenty-four decades of lifetime from a factor of two in energy. The half-lives of 7 alpha emitters against the reciprocal square root of the alpha's energy — the Geiger–Nuttall coordinates — with the measured values as points and a one-line tunnelling model as the open ones. The energies span a factor of 2.2 and the half-lives span 24 decades, which is what an exponent does. The model has no fitted parameter in it and reproduces every lifetime to within 0.5 decades — bad arithmetic by any ordinary standard, and a hundred-thousandth of the range it is predicting.

A wall that a factor of two makes impassable

Polonium-212 lives three tenths of a microsecond. Thorium-232 lives fourteen billion years. The alpha particles they emit differ in energy by a factor of two, and the lifetimes differ by twenty-four decades — because the quantity that decides is not the energy but an exponent built from it, and an exponent is where small differences go to become enormous.

 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.

 A quarter of the time, an object is found without being touched. The outcomes of sending one photon into a balanced interferometer, with and without an opaque object in one arm, computed from the same amplitudes. With the arm clear, every photon leaves by the bright port and the dark port receives nothing. With the object in place, 50 per cent of photons are absorbed by it, 25 per cent reach the bright port and say nothing, and 25 per cent reach the dark port — which is impossible unless something is in the arm, and which happens with the object still sitting there unabsorbed and unlit. The photon that produced that click did not go through the blocked arm, because a photon that goes through a blocked arm is absorbed. So an object has been located by light that never met it. The price is that 50 per cent of attempts destroy the thing being looked for: an efficiency of 33 per cent, which the Zeno version of the apparatus takes as close to one as one likes.

The measurement that never touched it

A balanced interferometer sends every photon to one output and none at all to the other. Put an object in one arm and the empty port starts clicking — and a click there is caused by a photon that cannot have gone near the object, because a photon that goes near it is absorbed. The object has been found by light that never met it.

 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.

 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.

 A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

 Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

 Where 4s goes below 3d. The energies of five orbitals of a nucleus of charge 19, solved in a screened Coulomb potential, against how far out the screening extends. Nothing about the ordering is assumed: each level is found by integrating the radial equation outward and bisecting on the energy until the solution has the number of nodes that state is supposed to have, and the same solver returns hydrogen's −1/2n² to 7.3e-12 hartree when the screening is switched off. With no screening the three n = 3 levels would lie on top of one another. With screening they separate, always in the same order — s lowest, then p, then d — because a low angular momentum has no centrifugal barrier keeping it out of the core, so it spends part of its time inside the other electrons where the nuclear charge is unscreened. And at a screening length of 0.41 bohr the 4s level crosses below 3d, which is the fourth row of the periodic table: potassium and calcium put their electrons in 4s before anything goes into 3d, so they are an alkali metal and an alkaline earth rather than the first two transition metals. The model is a caricature — one screening length for every electron, no self-consistency, and no exchange — and it gets the ordering right anyway, which is the argument that the ordering is about penetration and nothing subtler.

The order the shells fill

In hydrogen every state with the same principal number has the same energy, and 4s and 3d differ by nothing. In every other atom they do not, and 4s is below 3d — which is why potassium is an alkali metal rather than the first transition metal. The difference is a small piece of probability that an s orbital has inside the innermost shell and a d orbital does not.

 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.

 What is lost is exactly what is recorded. Fringe visibility against the distinguishability of the record left in the environment, for six couplings between the interferometer and a marker. The points lie on the quarter circle V² + D² = 1, computed here to a part in 10¹² — the visibility from the output probabilities with the marker traced out, and the distinguishability from the overlap of the marker's two states, with nothing shared between the two calculations. The relation is the quantitative form of complementarity, and it is stronger than the usual statement: interference is not lost because something was disturbed, and not lost only when a measurement is made. It is lost exactly to the extent that the environment could in principle say which way the particle went, whether or not anybody looks at the environment. That is why decoherence is a matter of correlation rather than of disturbance: the coherence has not been destroyed but relocated, into a correlation between the particle and something else.

Where the interference goes

A superposition does not stop being a superposition when something interacts with it. What happens is that the coherence moves — out of the system and into a correlation between the system and its surroundings — and the interference disappears from any measurement made on the system alone. For a dust grain in air the move takes 10⁻²⁸ seconds, which is why nothing large has ever been seen in two places.

 Three sources, and the number that separates them. The chance of detecting a second photon a delay τ after a first, divided by the chance if the two were independent, for three kinds of light. At long delay every curve is one, which is what independence means. At zero delay they are 2, 1 and 0, and those three numbers are three different physical situations. Thermal light is bunched: its intensity fluctuates, and a photon is more likely to be found where the intensity happened to be high, so it arrives with company for as long as the fluctuation lasts — 4 nanoseconds here. A laser is flat, because a coherent state has no intensity fluctuation to correlate with. And a single emitter is antibunched: it gives zero, exactly, because after emitting it is in its ground state and cannot emit again until it has been re-excited, which takes 12 nanoseconds. The zero is the important one. Every classical field, of every possible intensity distribution, has g²(0) at least one — the inequality follows from the fact that the mean square of a real positive quantity is at least the square of its mean. A measurement below one is not merely evidence for photons; it is a result no wave theory can produce.

The experiment a wave cannot pass

The photoelectric effect is offered everywhere as the proof that light is quantised, and it is not one — a classical wave falling on quantised matter reproduces every feature of it. The measurement that no wave can pass is a different one: send single photons at a beam splitter and count how often both detectors fire.

 Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are.

The crossing that never happens

Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything.

 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.

 A current that falls by a decade for every ångström. Tunnelling current against the width of a vacuum gap, on a logarithmic scale, for three work functions covering the range of clean metal surfaces. The curves are straight because the transmission is an exponential in the gap, and their slopes are 0.833, 0.944, 1.044 decades per ångström — fitted to the drawn curves and agreeing with 2κ/ln 10 to a part in a million. At 4.5 electronvolts a change of ten picometres, a tenth of an atomic radius, changes the current by 24 per cent. That is the sensitivity a scanning tunnelling microscope lives on, and it is why the instrument measures height by holding the current fixed and recording what the piezo had to do: the current is far too steep a function of height to be read as one.

The last atom does all the seeing

A tunnelling rate falls by a factor of eight for every tenth of a nanometre of extra barrier, which is normally quoted as the reason nothing ever tunnels anywhere. Read the other way it is a microscope: the second-nearest atom of a blunt metal tip carries a two-hundredth of the current the nearest one does, so a tip nobody sharpened resolves a single atom, and the resolution comes from an exponential rather than from any piece of engineering.

 Three copiers, and what each of them costs. How well three copying machines reproduce a qubit, against the state's angle from the pole. The first is a linear machine built to copy the two pole states perfectly; linearity then fixes what it does everywhere else, and on an equal superposition it produces an entangled pair whose overlap with the two copies wanted is exactly 0.500. That is the no-cloning theorem written as a number rather than as an argument: no adjustment is available, because the machine's behaviour on superpositions was decided the moment its behaviour on the basis was. The second measures in a fixed basis and prepares two copies of what it found, which is perfect at the poles and averages 0.6667 over the sphere — two thirds, exactly. The third is the best machine there is, and it manages 0.8333 on every state alike: five sixths, and not one.

The state that cannot be copied

Every measurement in this collection disturbs what it measures, and the obvious way round that is to make a spare first. It cannot be done, and the reason is not a practical difficulty or a limit on how good an apparatus can be: a copier is a linear machine, so fixing what it does to two states fixes what it does to their superpositions, and what it then does is not a copy.

 An average that follows Newton's law exactly. The centre of a wavepacket in a harmonic well, and the classical orbit started from the same place, drawn on top of each other. They agree to 5.5e-6 over 2.2 periods, which is the split-step integrator's own error and not a physical gap. This is exact and it is exact for every state of a harmonic oscillator, however wide, however lumpy, however far from classical: the theorem needs ⟨−V′(x)⟩ = −V′(⟨x⟩), which for a linear restoring force is true term by term. It is worth being suspicious of how strong that looks. The harmonic oscillator is the one potential where the average force over a state and the force at the state's centre cannot differ, so it is the worst possible example from which to conclude that quantum averages follow classical paths.

The average that obeys Newton

Ehrenfest's theorem says the centre of a wavepacket moves according to the average force over the state. That is exact, it is often quoted as the reason classical mechanics survives, and the two statements are not the same — because the average of a force is the force at the average only when the force is linear, which is to say almost never.

 Two barriers, and the energies at which they stop being barriers. Transmission against energy on a logarithmic scale, for one barrier of width 0.9 and height 20, and for two of them separated by a well of width 2. Below the top of the barrier the single one transmits an exponentially small amount and does so smoothly. The pair does not: at particular energies the transmission climbs by orders of magnitude and reaches 1.000000 — one, to every digit the arithmetic has — even though each barrier on its own passes 3.9e-2 of what arrives. Multiplying the two single-barrier transmissions would give 1.5e-3, which is wrong by 6e+2: amplitudes are what compose, not probabilities, and the region between the barriers is a box whose quasi-bound levels are where the amplitudes reinforce. The resonances sit at 6.438, 13.915 and the levels of an infinite well of the same width are at 2.467, 9.870 — near, and not equal, because a well with leaky walls has states that are not quite bound.

Two walls that let more through than one

Put a second barrier behind the first and the transmission does not fall — at particular energies it rises to exactly one, through a pair of walls each of which stops all but a few per cent. Probabilities cannot do that. Amplitudes can, and the region between the barriers is a box whose levels say where.

 The bands bend by the amount the Fermi levels differed. A junction between 1e+17 cm⁻³ p-type and 1e+16 cm⁻³ n-type silicon at equilibrium, with the Fermi level flat by construction — that is what equilibrium means — and the two band edges carrying the whole of the 0.774 V drop. The bending happens over 331.8 nm, and it is not symmetric: the depletion reaches 301.6 nm into the lightly doped side and only 30.2 nm into the heavily doped one, because the same exposed charge is reached sooner where there is more of it. An electron in the n-side conduction band therefore faces an uphill barrier of 0.774 V to reach the p side, while an electron already on the p side rolls downhill without any barrier at all — which is the asymmetry the whole device is, and it is drawn here before any current has been mentioned.

One level, and the field that bends the bands

Two pieces of the same crystal doped differently have their Fermi levels at different heights. Joining them cannot leave both, because a difference in electrochemical potential is precisely what makes charge move — and everything a diode does is the accounting of what had to happen for that one difference to reach zero.

 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.

 Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all.

The phase a magnet leaves on a path it never touched

An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

 A bundle that swings and never spreads. A Gaussian of the ground state's own width, released at rest from x = 3 in a harmonic well and propagated on a grid by split-step Fourier, drawn at 0 of a period, 0.25 of a period, 0.5 of a period. The packet slides from side to side and its shape does not change: over 2.2 full periods the width moves by 6.6e-5 per cent, and its centre tracks x₀cos t to 8.6e-6. Every other initial width breathes. This one is the displaced ground state, and it is the closest a quantum state comes to being a classical oscillator — a definite thing at a definite place, moving on the classical trajectory, staying the size it was.

The state that swings like a pendulum

Most quantum states of an oscillator look nothing like a swinging weight. One family does: it follows the classical trajectory exactly, never spreads, and sits at the uncertainty minimum for ever — and it is the state a laser and a driven circuit actually produce.

 The delay that stops caring how thick the wall is. The Wigner phase time — how late the transmitted packet's peak arrives, compared with a free particle covering the same distance — for an electron under a 3 eV barrier, at 0.2 nm, 0.5 nm, 1 nm thick. 0.2 nm: 0.3728 fs at half the barrier height; 0.5 nm: 0.4372 fs at half the barrier height; 1 nm: 0.4388 fs at half the barrier height. The widths span a factor of 5.0 and the times span 1.18. A thicker barrier is exponentially harder to cross and the delay in crossing it barely moves — which is either a remarkable fact about tunnelling or a sign that the phase time is not a traversal time, and the rest of the essay is about which.

How long the crossing takes

Tunnelling has a probability, and asking how long it takes turns out to be a different kind of question. The delay a transmitted packet shows stops growing once the barrier is opaque, so a thicker wall is crossed in the same time — and several defensible clocks give several different answers.

 Two pairs, and only one of them may cheat. The CHSH value a party shares with a second, against the value the same party shares with a third. Every quantum state lies inside a quarter circle whose radius is Tsirelson's bound, 2.8284, because the sum of the two squared values cannot exceed eight. The classical limit is 2 on each axis, and the square that would hold both violations sticks out of the circle everywhere except at its corner: the best both can manage at once is exactly 1.999998, which is the classical value and no violation at all. So a party maximally entangled with one other is correlated with everybody else exactly as a classical object would be. Nothing about the measurement or the apparatus was assumed; this follows from the state alone.

What two have they cannot give a third

Entanglement will not be shared. A pair that violates a Bell inequality is correlated with everything else exactly as a classical object would be, and the trade is exact enough to be drawn: two CHSH values must fit inside a circle of radius 2√2.

 Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more.

Why an atom is the size it is

A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

 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.

 A line is what two bodies give. The electron energy spectrum of tritium, against the vertical line a two-body decay would produce. A nucleus emitting one particle has no choice about how to share the energy: momentum conservation fixes it, and every electron comes out at the same energy. The observed spectrum is a continuum running from zero to the full available energy, with a mean at 0.31 of the endpoint. The shape is a count of the ways the energy can be divided between an electron and something else, and the something else was proposed for no other reason than that the shape requires one.

The energy that did not all arrive

A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.

 Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore.

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

 The half-life that depends on the electrons. Half-lives of 3 nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring where fully stripped ions can be kept circulating for months. The changes are not corrections. ¹⁸⁷Re goes from 41.6 billion years to 32.9 — a factor of 1.26 billion — because a beta decay whose available energy is only 2.5 keV cannot put an electron into the continuum but can put one into an empty K orbital, and stripping the atom opens that channel. ¹⁶³Dy is stable as an atom and decays in 47 days as a bare nucleus, for the same reason. The rate is a property of the nucleus and of what surrounds it, not of the nucleus alone.

The half-life that chemistry can change

A nucleus has no clock, the usual account goes, and nothing outside it touches the rate. That is very nearly true and it is not exactly true. Electron capture takes an electron from the nucleus's own position, so its rate is proportional to how many electrons are there — which is chemistry, worth a per cent. And a nucleus stripped of every electron can gain a decay channel it did not have: rhenium-187 goes from forty-two billion years to thirty-three.

 A swing that dies away and comes back. The envelope of the mean position of a coherent state with 9 quanta on average, in an oscillator whose levels carry a small quadratic term, Eₙ = n + n²/240, over one revival time of 240 oscillator periods. The swing collapses within about 9.1 periods, when the packet has spread round its orbit, and it stays at zero for most of the run. It returns whole at half the revival time, on the opposite side, and whole again at the full revival time; the envelope is summed from the state's energy components and checked against α·exp(−2n̄ sin²χt) to a part in a hundred million. The dashed curve is a classical ensemble started from the same distribution, each member orbiting at the frequency its own energy gives: it collapses in the same way and never returns, its swing at the half and full revival times 0.1% and 0.1% of the start.

The return a classical cloud never makes

Put a swinging quantum packet in a well whose frequency depends a little on the amplitude and it spreads round its orbit until its swing has vanished. That part is not quantum at all: a cloud of classical oscillators does exactly the same. What no classical cloud can do is come back — and the quantum packet reassembles whole, on schedule, splitting into copies on the way, because its energies are discrete.

 A quasi-probability that goes negative. The Wigner function of the oscillator's state at a quarter of its revival time, with 4 quanta on average: two copies of the packet, at x = ±2.83, in equal superposition. It is computed from the wavefunction by the Wigner transform on a lattice, and it integrates to one. The two copies are the two positive blobs. Between them lies a pattern of stripes with no classical counterpart, running from 0.289 down to −0.289, where the shaded warm regions and their outlines mark negative values. A negative probability is not a probability, so this distribution cannot describe a cloud of classical particles. The stripes are also taller than the blobs, so the interference carries more structure than the copies themselves: the highest stripe reaches 0.289 and the centre of a blob 0.159.

The probability that goes below zero

Classical mechanics describes an uncertain state as a cloud of points in the plane of position and momentum. Quantum mechanics has an exact counterpart, the Wigner function, whose shadows are the true position and momentum distributions — and which goes negative. It goes negative for a single photon, for every superposition of two packets, and for every pure state that is not a Gaussian. Where it is negative no classical cloud can imitate the state, and losing energy to the surroundings erases the negative regions first.

 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.

 The floor a state's survival cannot go below. The probability that a quantum state is still found in its initial state, against time measured as its energy spread times time over ħ, for four states with the same spread. The shaded region under cos²(ΔE t/ħ) is forbidden by Mandelstam and Tamm's theorem, and every curve — each a sum of phases over the state's energies — stays out of it. Two equally weighted levels run along its edge and reach an orthogonal state at exactly π/2, the fastest any state with this spread can. The same two levels driven off resonance, with the same spread, never get further than a survival of 0.500. Three equally spaced levels become orthogonal only at 1.7101, 1.0887 times the limit, and a coherent state never does, bottoming out at 0.0183. A spread of energy is permission to change, not an obligation.

The fastest a state can stop being itself

Time has no operator, so the energy–time relation cannot be the commutator inequality it resembles. What stands in its place is sharper: a state whose energy is spread by ΔE cannot become a different, orthogonal state in less than πħ/2ΔE, and cannot do it faster than its mean energy above the ground state allows either. Two equally weighted levels reach both limits exactly. Nothing else does.

 Four outcomes, one of which happened. The correlation between the two outer particles' measurements, against the angle between their analysers, for each of the four results the middle measurement can give. Every one of the four leaves the outer pair maximally entangled — each curve reaches one and minus one — so every outcome is as good as any other, and the outer parties have a perfect Bell pair whichever it was. What differs is which correlation they have, and the four are shifted and reflected versions of each other. The flat line is their average, which is zero at every angle, checked here at seven hundred and twenty angles to twelve figures. That vanishing is the whole reason the operation cannot be used to send anything: until the middle party's two classical bits arrive by an ordinary channel, the outer parties' data is indistinguishable from noise, and no correlation appears at all. The entanglement is created instantly and is useless until a message travelling no faster than light says which of the four it is.

A link between two that never met

Take two entangled pairs sharing no particle, measure the two inner particles jointly, and the two outer ones — which have never interacted, never been in the same place, and have no history in common — are entangled. Nothing travelled between them. What has to travel is two classical bits saying which of four results occurred, and until those arrive the outer parties see nothing at all.

 How much a block knows about the rest. The entanglement between a block of a one-dimensional chain and everything outside it, against how long the block is, for three states of the same number of particles. The straight line is a randomly chosen state, whose entanglement is the block's length times the logarithm of two — a volume law, and what almost every state in Hilbert space does. The flat curve is the ground state of a chain with a gap: it saturates, varying by less than a twentieth of a per cent from a block of eight to one of forty, because the boundary of a one-dimensional block is two points however long the block is. The middle curve is the ground state of a gapless chain, which grows as the logarithm of the size with a coefficient measured here as 0.333 against the third that conformal field theory gives. Both ground states are enormously less entangled than a random state, and that is not a detail about chains: it is why a ground state can be written down at all.

The corner of Hilbert space that is ever visited

Monogamy between three parties says how much of a correlation a pair may hold. Read across a boundary in a many-body system it says something much stronger: the entanglement between a region and the rest scales with the boundary rather than the volume, for the ground state of anything with local interactions. That is why such a state can be written down at all — and why almost every state in Hilbert space is one that nothing ever prepares.

 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.

 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.

 Three identical photons in a three-way splitter: some outcomes are forbidden. One photon enters each input of a symmetric three-way splitter, which sends each photon to each output with equal probability. Across: the ten ways three photons can leave, written as how many leave by each output. Bars, for each outcome: photons that can be told apart, identical photons, and identical fermions. Distinguishable photons spread over all ten, as independent coins would. Identical photons never produce 210, 201, 120, 102, 021, 012 — the 6 outcomes whose output labels do not add to a multiple of three — and pile into the rest: 300 with 0.222, 111 with 0.333, 030 with 0.222, 003 with 0.222. Identical fermions leave one per output every time (probability 1.000). Each probability is a sum of amplitudes over the distinct ways to reach the outcome, and each set sums to one.

The outcomes identical photons refuse

Two identical photons meeting at a beam splitter always leave together, and that one fact carries three more. No classical light can empty the coincidence dip more than halfway, so the depth is a test of what light is; the depth measures how identical two photons are, however they differ; and with three photons in a three-way splitter whole classes of outcome become impossible — the first case of a sum over paths that no known algorithm can evaluate quickly as the photons multiply.

 Four states of light, each with the same area of noise. The field of a single mode of light drawn as a point in a plane whose two axes are its two quadratures — the parts of the wave in step with a reference and a quarter-cycle out of step. The distance from the centre is the amplitude and the angle is the phase. Each state is a cloud of 400 sampled measurements with its two-standard-deviation outline, in units where the vacuum's noise is one in every direction. The vacuum is a round cloud at the centre. Steady laser light is the same round cloud moved away from the centre. Light squeezed by 4 dB is an ellipse of the same area: narrower than the vacuum by a factor of 0.63 in one direction and wider by 1.58 in the other. Pointed along the direction from the centre it is quiet in amplitude; pointed across it, quiet in phase. The sampled spreads were checked against each state's widths.

The noise pushed below the floor

A perfectly steady laser beam still flickers, by an amount set by the vacuum itself, and for most of the twentieth century that flicker was treated as the floor of any optical measurement. It is a floor only for one shape of noise. Light can be made quieter than the vacuum in one property by being made louder in another — and the price, the fragility and the use of that trade are all visible in how the noise is shaped, which is why the world's gravitational-wave detectors now run on it.

 Two paths through a neutron interferometer at two heights. A neutron interferometer cut from one silicon crystal, tilted by 30 degrees about its incoming beam so that one path runs higher than the other. Left: the two paths, split at the first slab, turned at the second and recombined at the third, enclosing 10.1 square centimetres; the heights are drawn exaggerated. Taken as a rectangle of the same area, the upper path runs 15.8 mm higher for 3.2 cm. There a neutron of wavelength 1.445 Å, moving at 2738 m/s, is slower by 56.5 micrometres per second, so its wavelength is longer by 20.6 parts per thousand million. Over 3.2 cm that accumulates 28.7 radians less phase than the lower leg — 4.6 whole fringes — computed by integrating the local wavenumber along both legs and checked against 2πm²gλA sin α / h².

The fall that leaves the mass in the phase

Every body falls the same way whatever its mass, and a neutron is no exception. But a neutron is also a wave, and the phase that wave accumulates while falling depends on the mass — as its square, at a fixed wavelength. Tilt a neutron interferometer so that one path runs a centimetre higher than the other and the neutrons swing between its two detectors, which in 1975 was the first measurement in which gravity and quantum mechanics both had to be right at once.

 The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

 Helium's excitations, and the line that decides how fast it can flow. The energy of an elementary excitation in superfluid helium-4 against its momentum, as a smooth fit through published neutron-scattering values: a phonon branch rising at the speed of sound, 238 m/s, a maximum near 13.9 K, and the roton minimum of 8.62 K at 1.92 Å⁻¹. A straight line from the origin has a slope that is a velocity. The steepest line that just touches the spectrum is Landau's critical velocity, 57 m/s, and it touches at 1.98 Å⁻¹, just past the roton minimum. A body moving slower than that cannot make any excitation at all and so cannot lose energy to the liquid; the dashed line is the speed of sound, far above it.

The speed below which nothing can be made

An object moving through an ordinary liquid always feels drag, because it can always pass some of its energy to the liquid. In a superfluid there may be nothing it can pass it to. Landau's argument turns the question into geometry: an excitation of momentum p and energy ε can be made only by a body moving faster than ε/p, so a liquid flows without friction below the smallest value of that ratio. For helium the smallest value is not the speed of sound but 57 m/s, set by the dip in its spectrum called the roton — and an ideal gas, with no dip and no sound, has no such speed at all.

 Why a flip of the spin takes eleven million years. Spontaneous emission rates against frequency on logarithmic axes. The two lines are the rates an electric dipole of e a₀ and a magnetic dipole of one Bohr magneton would give, both rising as the cube of the frequency; they differ by α²/4, 1.3 × 10⁻⁵, the square of the ratio of the two moments. Measured lines sit near them: Lyman α at 6.3 × 10⁸ s⁻¹, Hα and sodium's D line near 10⁸, and hydrogen's 21 cm line at 2.87 × 10⁻¹⁵ s⁻¹ — a lifetime of 11.0 million years. Its slowness is two factors multiplied: a magnetic moment rather than an electric one, and a frequency nearly a million times lower, cubed.

The line that takes eleven million years

A hydrogen atom in its ground state can flip its electron's spin relative to the proton's and emit a photon of 21 centimetres. Left alone, one atom takes eleven million years to do it. The Galaxy is nonetheless bright at that wavelength, because there is so much hydrogen, and because the flip is so slow that the line's brightness counts atoms without caring how hot they are.

 An atom made of powers of n. Five properties of a hydrogen atom in a level of principal quantum number n, each divided by its value in the ground state, against n from 1 to 300 on logarithmic axes. The radius of a circular orbit rises as n², to 0.53 µm at n = 100; the orbital period as n³, to 1.5 × 10⁻¹⁰ s; the radiative lifetime of the circular state, from the Larmor loss on its orbit, as n⁵ — slope 5.000 — to 0.9 s. The binding energy falls as n⁻², to 1.36 meV, and the electric field that pulls the electron away falls as n⁻⁴, to 3.2 V/cm. Across n = 1 to 300 the lifetime changes by 2 × 10¹² and the ionising field by 8 × 10⁹.

The atom the size of a bacterium

Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

 Each unit of angular momentum costs a factor of tens of thousands. Weisskopf's estimates of the half-life of a gamma-ray transition in a nucleus of mass number 100, against the photon's energy, for electric multipoles of order 1 to 5 — the photon carrying away 1 to 5 units of angular momentum. At 1 MeV the estimates are 10⁻¹⁵ s for E1, 10⁻¹¹ s for E2, 2.0 µs for E3, 292.5 ms for E4, 17.3 h for E5. Each extra unit multiplies the half-life at 1 MeV by between 60,000 and 210,000: about a thousand of that is (λ/2πR)², the square of the photon's reduced wavelength over the nucleus's radius, which grows as the energy falls, and the rest is a numerical factor that grows with the order. A transition that must carry away five units at 100 keV has an estimated half-life longer than the age of the universe; the same energy carried off with one unit takes less than a nanosecond.

The spin a photon has to carry away

An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.

 A temperature the circuit refuses to go below. The escape temperature — the temperature that would give the measured escape rate if escape were purely thermal — against the actual temperature of the junction, for a 10 µA, 5 pF junction at biases of 0.985 and 0.99 of its critical current. Hot, the two are equal: the phase is kicked over the barrier by thermal noise. Cooled below the crossover, 39 mK at 0.985 and 36 mK at 0.99, the escape temperature stops falling and levels off at 42 mK and 44 mK: the rate no longer depends on the temperature at all. This levelling-off, measured by Michel Devoret, John Martinis and John Clarke in 1985 with the junction's electrical environment carefully characterised, was the evidence that a macroscopic electrical variable tunnels.

The circuit that goes through a wall

Tunnelling is usually told about electrons and alpha particles — single, small, quantum things passing through barriers they cannot climb. In 1985 it was seen in an electrical circuit: the phase difference across a Josephson junction, a coordinate shared by all the electrons in two pieces of superconducting metal, trapped in a well and escaping from it through the barrier rather than over it. The evidence was a temperature below which the circuit's escape rate stopped caring how cold it was. The same circuit showed discrete energy levels, and its descendants are the qubits of superconducting quantum computers.

 The triangle of collisions that survive. Every collision of a particle ε above a filled Fermi sea with a partner from the sea, at zero temperature: the partner's energy across, one product's energy up, both from the Fermi level, the other product's fixed by energy conservation. Without exclusion every point in the plotted square would be an allowed collision. With it, the partner must come from below the Fermi level, and both products must land above it: the partner no deeper than ε below, the first product above zero, the second product above zero too. What survives is the shaded triangle, whose area is half of ε squared, and the collision rate is proportional to it. Halve the particle's excess energy and the triangle shrinks to a quarter; at the Fermi level itself it vanishes, and a particle there cannot scatter at all.

The collisions the exclusion principle forbids

A copper wire holds one free electron per atom, packed a quarter of a nanometre apart and repelling one another with the full Coulomb force. It seems impossible that an electron could travel any distance through that crowd without colliding. It travels micrometres — thousands of atoms — before another electron deflects it. The collisions are not weak. They are forbidden: to collide, both electrons need empty states to move into, and in a sea filled to the brim almost none exist. The same rule makes cold helium-3 thicken as the square of how cold it gets.

 Each wavelength sees from a different height. The weighting functions of 7 infrared channels in a carbon-dioxide band — how much each height contributes to the radiance a satellite measures in that channel — for absorption strengths chosen so that the peaks sit at 3, 8, 14, 20, 27, 35, 43 km. Each is scaled to its own maximum. A channel's peak is where the carbon dioxide above that height amounts to one optical depth: deeper, its emission is absorbed on the way out; higher, there is too little gas to emit. Each function is about 17 km wide at half its peak, the same for every channel, because the absorber's density falls with one scale height, 7 km. The dotted curve is the temperature profile of the standard atmosphere, on its own scale from 180 to 300 K, whose cold tropopause and warm stratosphere the channels between them straddle.

The spectrum that is a thermometer at every height

A satellite looking down at the Earth in the infrared sees, at each wavelength, the glow of whatever layer of air that wavelength escapes from. In the wings of carbon dioxide's great absorption band the air is transparent and the satellite sees the ground; towards the band's centre the gas grows opaque and each wavelength escapes from higher up. A single spectrum is therefore a stack of thermometers at different heights, and weather forecasts are built on reading it. What cannot be read from it is anything much thinner than the ten to twenty kilometres of air each of those thermometers averages over.

 A barrier the metal was supposed to choose. The barrier height at a metal contact on n-type silicon against the metal's work function, from Bardeen's model with interface-state densities of none, 10¹² per cm² per eV, 10¹³ per cm² per eV, 10¹⁴ per cm² per eV, a 0.5 nm gap and a neutrality level a third of the gap above the valence band. With no interface states the barrier is Schottky and Mott's φₘ − χ, rising one for one with the work function: from aluminium to platinum, 0.23 to 1.60 eV. With 10¹² per cm² per eV the slope is 0.92 and the same metals give 0.27 to 1.53 eV; with 10¹³ per cm² per eV the slope is 0.53 and the same metals give 0.48 to 1.20 eV; with 10¹⁴ per cm² per eV the slope is 0.10 and the same metals give 0.71 to 0.84 eV. Dense enough interface states pin every barrier near 0.76 eV, and the metal hardly matters.

The barrier the metal cannot choose

Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.

 A sharp angle costs a spread of turning. Left: the probability of finding a rotor at each angle, for three states. Right: the probability of each value of its angular momentum, in units of ħ, for the same states. A state bunched to a circular spread of 0.025 has an angular-momentum spread of 2.21ħ; a state bunched to a circular spread of 0.302 has an angular-momentum spread of 0.59ħ; a state with no preferred angle has an angular-momentum spread of 0.00ħ. The angular momentum can only take whole-number values, so its probabilities are bars; the angle is continuous. The state with no preferred angle has exactly one value of angular momentum, and its angle is spread evenly round the circle — where the naive rule would say that its angle's spread times ħ/2 should be infinite, it is π/√3.

The angle the uncertainty principle cannot be written for

Position and momentum trade exactly: squeeze one and the other spreads, with a product never below ħ/2. Angle and angular momentum look like the same pair and are not. A rotor in a state of definite angular momentum has zero spread in angular momentum and a finite spread in angle, π/√3, and their product is zero. Nothing is wrong with the rotor. The relation was written with a commutator that does not hold, because an angle lives on a circle and a line has no ends to wrap round. The relation that does hold measures the angle's spread by how far its average position on the circle falls short of the rim.

 A pointer that ends up where no eigenvalue could put it. The distribution of a measuring pointer's position, in units of its own initial spread, after it has been coupled to a spin whose two values would move it by ±g. Strongly coupled, g = 3, the pointer splits into two separate spots at ±3: an ordinary measurement. Weakly coupled, g = 0.1, the two possible shifts overlap almost entirely and the pointer's centre moves by only 0.020, the average of the spin times g. Keeping only the 0.99 per cent of runs in which the spin is afterwards found pointing along x, the kept pointer is centred at 0.802 — 8.0 times g, far outside the range ±g that either value of the spin could produce, and approaching the weak value 10, which it would reach exactly for a still weaker coupling.

The value a gentle measurement returns

Couple a pointer to a spin so weakly that the pointer's shift is far smaller than its own blur, and a single run learns almost nothing about the spin. Now keep only the runs in which the spin is later found in a chosen, nearly opposite state. The kept pointers move as though the spin's value were ten, or a hundred, though it can only be plus or minus one. That weak value was predicted in 1988 and has been measured many times. It is not a hidden value of the spin, and it does not break the uncertainty principle: it is paid for exactly in discarded runs. What it does give is a way to watch a quantum system without disturbing it.

 What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything.

The ion that exists because its electrons keep apart

Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.

 Three identical quarks in one state, and the label that rescues them. The Δ⁺⁺ particle, mass 1,232 MeV and charge +2, drawn as the quark model describes it: three up quarks, all in the lowest orbital state, all with their spins pointing the same way, giving spin 3/2. Left: with nothing else to tell them apart, the state is unchanged by swapping any two quarks — symmetric in where they are, in their spins and in their flavour — which the exclusion principle forbids for identical fermions. Right: give each quark a label that takes three values, and put the three in the one combination of the three values that changes sign under every swap. The whole state is then antisymmetric, as it must be, and the particle is allowed. The label is called colour, and three is the smallest number of values that can do this for three quarks.

The label three identical quarks forced

In 1964 the quark model had one embarrassment it could not explain away. A particle called the Δ⁺⁺, well measured and entirely ordinary, had to be three identical up quarks in the same state with their spins pointing the same way — which the exclusion principle forbids outright. The way out was to give quarks a hidden property that takes three values, so that the three could be different after all. Nobody could see the property, and for years it looked like a bookkeeping trick. Then three independent measurements counted it, and every one came out at three. The exclusion principle, which keeps electrons in shells, had predicted the colour of the strong force.

 Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.

The grating made of light

A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

 The heat two surfaces exchange when they almost touch. The radiative heat-transfer coefficient between two parallel half-spaces at 300 K, in W/m²K on a logarithmic axis, against the width of the vacuum gap between them, also logarithmic, from fluctuational electrodynamics: silicon carbide (solid) and gold (dashed), with two perfect black bodies, 4σT³ = 6.12 W/m²K, for comparison (dotted). Far apart, silicon carbide exchanges 3.3 W/m²K — less than black — and gold, a good mirror, only 0.024. Closer than the thermal wavelength, about 17 µm, waves that cannot leave either surface begin to tunnel across the gap. At 100 nm silicon carbide exchanges 137 W/m²K, at 10 nm 9338, at 1 nm 9.3·10⁵ — rising as the inverse square of the gap, 1.5·10⁵ times the black-body value at 1 nm. Gold exceeds black only below a few hundred nanometres and then grows much more slowly: 1284 at 10 nm, 2561 at 1 nm.

The heat that crosses a gap too narrow for light

Stefan and Boltzmann's law is usually read as a ceiling: no surface can radiate more heat than a perfect black body at the same temperature, 6.1 watts per square metre for every degree of difference at room temperature. It is a ceiling only for surfaces that are far apart. Bring two of them closer than the wavelength of their thermal glow and they begin to exchange waves that never leave either surface — waves that cling to it and die away within a wavelength — and those waves tunnel across the gap. Two plates of silicon carbide ten nanometres apart exchange fifteen hundred times what two black bodies can, almost all of it in a single colour, and the colour is the one in which silicon carbide is too shiny to glow at all.

 Which atoms in a Fermi sea are allowed to take a photon's kick. A slice through the occupied momentum states of a gas of identical fermions at zero temperature — a filled disc whose radius is the Fermi momentum — with each state marked by whether it can absorb a recoil of 0.8 of the Fermi momentum to the right, the kick a scattered photon gives. A state can scatter only if its kicked momentum lands outside the sea, in an empty state. The kicked sea is the second circle; the atoms in the overlap of the two (grey) land on occupied states and cannot scatter, and only those in the crescent (blue) can. In this slice 50 per cent of the states are allowed; over the whole sphere the allowed share is S(q) = 3q/4 − q³/16 = 0.568. A kick larger than the sea's diameter, twice the Fermi momentum, is never blocked, which is why the effect needs a sea whose Fermi momentum is comparable with the photon's.

The light a full sea will not scatter

An atom that scatters a photon recoils, and the recoil has to put it into a new state of motion. In a cold, dense gas of identical fermions most of the states it could recoil into are already taken, and the exclusion principle forbids the scattering outright. Nothing about the atom has changed — not its electrons, not its transition, not its cross-section — yet the gas becomes dimmer than the same number of atoms scattered one at a time. The effect was predicted in the 1990s and seen in 2021, and it turns transparency into a measurement of how full a quantum gas is.

 The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm.

The bounce that comes in fixed heights

A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

 Position written as frequency. Four tubes of water side by side in a 1.5 T magnet, their protons all precessing at 63.87 MHz, until a gradient of 10 mT/m is switched on along the row. Then the field, and so the precession frequency, rises steadily across the row, by 425.8 Hz for every millimetre. The signal the receiving coil picks up is a sum of waves at different frequencies, and its spectrum is a picture: the amount of signal at each frequency is the number of protons at the corresponding position. The axis below is position and the axis above it frequency, and they are the same axis. Nothing has been focused. The radio wave at 64 MHz has a wavelength of 4.7 m in air, and the tubes are centimetres apart.

The image made of frequencies

A hydrogen nucleus in a magnetic field precesses at a frequency proportional to the field — 63.87 megahertz in a 1.5 tesla scanner. Make the field rise steadily across the body and every position precesses at its own frequency, so the radio signal the body gives off is a chord whose spectrum is a map of where the protons are. A magnetic resonance image is never focused. It is computed, as a Fourier transform of a signal recorded one number at a time, and its resolution, a millimetre, is set by how strong the gradient is and how long the signal is listened to — not by the wavelength of the radio waves, which is several metres.

 The oldest age a trace of modern carbon allows. The age a radiocarbon measurement would report against the sample's true age, when a fraction of the carbon in it is modern contamination — from groundwater, roots, handling or the chemistry of preparation — of 5, 1 and 0.1 per cent; dashed, no contamination. Young samples are barely affected. Old ones are ruled by the contamination, because their own carbon-14 has dwindled to less than it: with 1 per cent, a sample of any age beyond about 60,000 years reads 38.1 thousand years, and 5 per cent caps it at 24.8 thousand. Even 0.1 per cent, a microgram of modern carbon in a milligram sample, caps it at 57 thousand. The limit of radiocarbon dating, usually quoted as about fifty thousand years, is a limit on cleanliness, not on the half-life.

The clock read by counting atoms

Radiocarbon dating reads a clock that ticks slowly: a gram of living carbon holds sixty thousand million carbon-14 atoms, and only thirteen and a half of them decay each minute. Counting those decays to date a milligram of bone would take years. Counting the atoms themselves, one by one in an accelerator, takes an hour, because in any measurement shorter than the clock's mean life almost every atom is still there, undecayed and countable. And the method's famous limit of about fifty thousand years turns out not to be set by the half-life at all. It is set by how much modern carbon a sample has picked up, because a trace of contamination outweighs whatever carbon-14 a very old sample has left.

 The hyperfine structure, read off a room-temperature cell. The difference between the probe's absorption with the pump off and on, which removes the Doppler hump and leaves only the Doppler-free features, against detuning. The three transitions, labelled by the excited level F′, sit at −423.6, −266.65 and 0 MHz, and the three crossovers midway between them at −345.1, −211.8, −133.3 MHz. In this model, which empties the ground level only by exciting it, the crossover between F′ = 2 and 3 is the second strongest feature and the weak transitions to F′ = 1 and 2 are small. In real rubidium spectra the crossovers are often the strongest of all, because excited atoms can fall to the other ground level and be lost to the transition altogether — optical pumping, which the model leaves out and which deepens every hole the pump burns. The positions do not depend on any of that: the separations between the features are the hyperfine splittings of rubidium's excited state, read to a fraction of a megahertz in a glass cell on a bench.

The lines a warm vapour hides

Shine a tunable laser through a cell of rubidium vapour and the absorption is one smooth hump half a gigahertz wide, though the atoms' own lines are eighty times narrower and three of them lie inside it. The atoms are moving, each sees the light Doppler-shifted by its own speed, and their lines are smeared together. Send a second, stronger beam back the other way and narrow dips appear in the hump at exactly the frequencies of the hidden lines — because only atoms standing still can be talked to by both beams at once. The vapour stays at room temperature and the motion is not removed; it is selected against.

 A beaker that empties at a steady rate. The height of liquid helium inside a beaker 1 cm across, held 1 cm above the level of the bath it stands in, against time, as the film carries liquid up the inner wall, over the rim and down into the bath. The film carries a fixed volume per centimetre of rim — 7.5·10⁻⁵ cm³/s at 1.4 K — whatever the difference of levels, so the level falls in a straight line, 180 μm a minute, and the beaker is level with the bath in 56 minutes. At 1.9 K, with less superfluid, it takes 95 minutes; at 2.1 K, 279. Dashed, a flow driven by the difference of levels, like a siphon, starting at the same speed: it slows exponentially as the difference shrinks and never quite finishes. The film's straight line is the signature of a flow set by a critical speed, not by the push.

The film whose thinnest point sets the flow

Leave a beaker of superfluid helium standing in a bath and it empties itself, the liquid creeping up the inside wall as a film eighty atoms thick, over the rim and down the outside. The film itself is nothing special: ordinary liquid helium coats the wall just as thickly. What is special is that it flows without friction, and so its flow is set not by how hard it is pushed but by how fast it can go. The level falls in a straight line rather than slowing as it nears the bath's, and the whole flow is decided at one place, the rim, where the film is thinnest and has to run fastest.

 A current that grows thirty decades with the field. The current density of electrons tunnelling out of a cold metal against the field at its surface, on a logarithmic scale, from the Fowler–Nordheim law, for work functions of 2.7 eV (lanthanum hexaboride), 4.5 eV (tungsten) and 5.6 eV (a surface barely willing to let electrons go). For tungsten the current is 0.009 A/m² at 2 GV/m, 5·10⁵ at 4 GV/m and 6·10⁹ at 8 GV/m — rising by more than twenty orders of magnitude between 1 and 4 GV/m. The work function sits in the exponent as a three-halves power, so lowering it from 4.5 to 2.7 eV raises the current at 3 GV/m by a factor of 2·10⁵. A current density of a million amperes per square metre, an ordinary working value for a field emitter, needs about 4.2 GV/m on tungsten.

The electrons a field pulls from cold metal

A metal holds its electrons behind a step a few electronvolts high, and to get them out the usual way is to heat the metal white-hot until some climb over it. Put a strong enough field outside instead and the step tilts into a triangle a couple of nanometres wide, and electrons leave a cold metal by tunnelling through it. The current grows thirty decades as the field rises tenfold, a sharp needle at a thousand volts reaches the field that a flat plate would need a million volts for, and the electrons that come out all have nearly the same energy — which is why the brightest, sharpest electron microscopes are fed from a single etched tungsten point.

 Four operations on one half, four states of the pair. The sender holds one half of an entangled pair prepared in the state |Φ+⟩ = (|00⟩ + |11⟩)/√2, the receiver the other. To send two bits she applies one of four operations to her half alone — do nothing, flip it (X), shift its phase (Z), or both — and sends it. The table gives, computed from the state vectors, the probability that the receiver's joint measurement of both halves in the Bell basis returns each of the four Bell states: one on the diagonal, zero everywhere else. Four perfectly distinguishable outcomes from one transmitted qubit is two bits. Sent without the pair, a qubit can carry at most one.

The qubit that carries two bits

A single quantum two-state system, sent from one person to another, can carry at most one bit of message — a theorem, not an engineering limit. If the two already share half each of an entangled pair, the sender can do one of four things to her half alone, send just that half, and the receiver can tell which of the four she did. Two bits arrive on one qubit. Nothing is free: the shared pair had to be delivered, and counted honestly the rate is still a bit per qubit. What entanglement buys is the freedom to send half the message before the message exists, and it also buys privacy, since the qubit in transit carries nothing at all on its own.

 Ramsey fringes: two short pulses and a long wait. The fraction of atoms found in the upper state against the detuning of the drive from the atoms' frequency, in cycles per pulse length, for two π/2 pulses of length τ separated by a free time T = 10τ (solid), computed from the exact two-level evolution. Fringes of period 1/T ride under an envelope set by the short pulses (dotted, the response to one pulse of length 2τ). The central fringe is 0.0444 cycles per τ wide at half height, close to 1/2T. A single long pulse covering the same total time, 12τ (dashed), gives a peak 0.0666 wide: 1.50 times wider, and with no way of telling a large detuning from a small one by its side fringes alone.

The wait that sets the precision

To measure an atom's frequency, the obvious method is to drive it and look for the response, and the sharpness of the answer is set by how long the drive lasts. Norman Ramsey found in 1949 that most of the drive is wasted. Two short pulses with a long wait between them do better than one long pulse, because during the wait the atom keeps its own time undisturbed, and any difference from the drive's time piles up as an angle the second pulse can read. Every primary clock in the world runs on that idea, and its precision is set by two numbers: how long the atoms can be left alone, and how many of them there are.

 Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

The hydrogen atom inside a crystal

Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

 Two lifetimes a thousand times apart. The fraction of positronium atoms surviving in the ground state against time since formation, on a logarithmic time axis, for the spin singlet, para-positronium (red), and the spin triplet, ortho-positronium (blue), in vacuum. Para lives 125.1 ps and ortho 142.1 ns, a ratio of 1135. The lowest-order formulas give 124.5 ps, from α⁵mc²/2ħ, and 138.7 ns, from 2(π² − 9)/9π · α⁶mc²/ħ; the ratio is almost entirely one extra power of the fine-structure constant, the price of a third photon.

The atom with no nucleus

An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.

 The square root of an X-ray's energy, one step per element. The square root of the energy of each element's Kα X-ray line, in units of the Rydberg energy, against its atomic number, for 24 elements from aluminium to lead (dots), with Moseley's law, √(3/4)·(Z − 1), drawn as a line. The points march up in equal steps, one per element, because each step adds one proton to the nucleus and one unit to the charge the falling electron feels. When Moseley first did this, in 1913–14, the line had gaps where no known element fell: the numbers marked, 43, 61, 72 and 75, were found later as technetium, promethium, hafnium and rhenium. At high atomic number the points rise above the line, by 6 per cent for lead, because the innermost electrons there move fast enough for relativity to deepen their binding.

The X-ray line that counts the protons

Strike a metal with fast electrons and it gives off X-rays at a few sharp energies characteristic of the metal. In 1913 Henry Moseley measured the strongest of them for element after element and found that the square root of its energy rose in equal steps, one step per element — as if something inside each atom were being counted. It was the charge on the nucleus. The periodic table had been ordered by weight; Moseley's line ordered it by protons, fixed the elements it was missing, and showed that an electron deep inside a heavy atom behaves like the electron in hydrogen, with the charge turned up.

 A current that falls every 4.9 volts. The share of electrons reaching the anode of a mercury-vapour tube against the accelerating voltage, from a simulation of 3000 electrons at each voltage: the electrons must arrive at the grid with more than 1.5 eV to climb the last small retarding voltage, and lose 4.886 eV — the energy of mercury's first excited state — whenever they hit an atom with at least that much, after a random further distance averaging 0.03 of the gap. The current rises with voltage, then collapses each time the electrons have gained just enough to excite an atom and too little left to reach the anode: minima at 6.3, 11.2, 16.1, 21.0, 25.8 V, about 4.9 V apart. Energy was being handed to the atoms in lumps of 4.886 eV and in no other amount — the first direct evidence, in 1914, that an atom's internal energy is quantised.

The electrons that pay in fixed amounts

Fire electrons through mercury vapour and raise the voltage that drives them, and the current they carry should simply rise. In 1914 James Franck and Gustav Hertz found it rising, then collapsing, then rising again, over and over, every 4.9 volts. Below that energy an electron bounces off a mercury atom and keeps almost everything; above it, it can hand the atom exactly 4.9 electronvolts and no other amount. The atom will accept energy only in the size of the gap between its levels — shown here by collisions rather than by light, and misread by its discoverers as something else entirely.

 What an amplifier does to a signal's uncertainty. A signal drawn in the plane of its two quadratures — the in-phase and out-of-phase parts of a wave — as a disc one standard deviation across: a coherent input (black), with the vacuum's own uncertainty; and three outputs at a gain of 4: what a noiseless amplifier would give, the input disc scaled up by √G (dashed), which quantum mechanics forbids; what the least noisy phase-insensitive amplifier gives (blue), the same centre with a disc 1.323 times wider again; and what a phase-sensitive amplifier gives (red), the in-phase quadrature stretched by √G and the other squeezed by 1/√G, with no noise added. Referred back to the input, the phase-insensitive amplifier has added 0.375 of a quantum of noise; the signal's distance from the origin, measured in its own spread, has fallen by 0.756.

The noise every amplifier must add

An amplifier that magnifies a signal without caring about its phase must also add noise, and not because of hot transistors or bad design. However perfect it is, it adds at least half a quantum of noise referred to its input — exactly as much as the vacuum fluctuations the signal already carried — so a faint coherent signal always leaves with half its signal-to-noise ratio. The reason is the same rule that makes position and momentum uncertain, applied to the two halves of a wave, and an amplifier that broke it would be a machine for copying quantum states.

 A resistance counted in whole numbers. The Hall resistance of a gallium-arsenide layer with 3 × 10¹⁵ electrons per square metre, against the perpendicular field (solid), beside the classical B/ne (dotted); and the resistance along the current, on an arbitrary scale (red). Classically the Hall resistance rises in proportion to the field. In the layer it climbs in steps and stays on flat plateaus at h/νe² — 25813, 12906, 8604, 6453 ohms — each centred where the classical line crosses it, at integer filling. On each plateau the resistance along the current falls to zero: the current flows with no dissipation at all. Between plateaus both change. The model drawn here gives each Landau level a width 0.18 of the level spacing and lets only the middle 0.16 of it carry current across the sample.

The resistance counted in whole numbers

Pass a current along a thin layer of electrons in a strong magnetic field and measure the voltage across it. Classically the ratio rises steadily with the field. In fact it climbs in steps, and on each step it sits at the fundamental constant h/e² divided by a whole number — 25,812.807 ohms, then half that, then a third — the same to better than a part in a thousand million in every sample, every material and every laboratory, however dirty the layer and however irregular its shape. The steps exist because of the dirt, and they are exact because of a count that disorder cannot change.

 The peak that rises out of a cloud. A slice through the shadow a released cloud of a million atoms casts after it has expanded, the measurement by which condensates are seen, at 1.1, 0.8 and 0.5 times the transition temperature: the thermal atoms, whose spread of speeds sets the cloud's width, give a broad rounded profile; the condensed atoms, all in one state with only the small spread of momenta their trap allowed, give a narrow peak at the centre, drawn here with the shape repulsion between the atoms gives it. Above Tc there is only the broad cloud. At 0.8 Tc a sharp peak stands up from its middle, holding 47 per cent of the atoms; at 0.5 Tc the peak holds 87 per cent and the thermal wings have shrunk. The two-component shape — a narrow peak on a broad base — was the signature that announced the first gaseous condensates in 1995.

The condensate a trap makes

A gas of atoms held in a magnetic trap and cooled to a few hundred billionths of a degree does something no gas in a box does at any temperature it can reach: a large share of its atoms falls into the trap's single lowest state, and a sharp peak rises out of the cloud. The reason is counting. The trap's excited states can hold only so many atoms at a given temperature, and the number grows as the cube of the temperature — so below one temperature the atoms run out of room, and every atom that does not fit has only one place to go.

 Fringes that identical nuclei add. The cross-section for two identical spinless nuclei repelling each other, divided by the classical answer — the Rutherford counts for the nucleus hit and the nucleus thrown, added — against the scattering angle in the centre-of-mass frame, for Sommerfeld parameters η = 1, 4 and 16 (η is half the closest approach in units of the wavelength over 2π). Because the detector cannot tell which nucleus it caught, the two amplitudes add before squaring, and the cross term oscillates in angle: at 90° the two paths are mirror images and the rate is exactly twice the classical one at every energy. Between 30° and 90° the curve completes about 2 oscillations for η = 4 and 7 for η = 16 — their number grows in proportion to η, which is to say in proportion to 1/ħ.

The scattering quantum mechanics returns unchanged

Rutherford worked out how alpha particles bounce off a nucleus by treating them as tiny planets on hyperbolic orbits, two years before Bohr's atom and fifteen before quantum mechanics. He had no right to be exactly correct, and he was: the full quantum calculation returns his formula without a single correction, at every energy. Coulomb's force is the one force for which that happens. Make the two nuclei identical, though, and quantum mechanics adds fringes no orbit can produce — doubling the rate at right angles for one kind of particle and halving it for the other.

 Two kinds of flicker in one heat bath. The variance of the number of photons in one mode of black-body radiation, against hν/kT, both on logarithmic axes, split into its two parts: n̄, the variance a shower of independent particles would have, and n̄², the variance of the intensity of randomly phased waves; their sum is what thermodynamics demands of Planck's law. Where the photons are rare, at hν much more than kT, the particle term wins; where they are crowded, the wave term wins; they are equal at hν/kT = ln 2 = 0.693, where each mode holds one photon on average. At hν = 5kT the wave term is 0.68 per cent of the particle term; at hν = kT/10 the particle term is 10.5 per cent of the wave term.

The flicker that is half wave and half particle

In 1909, sixteen years before quantum mechanics, Einstein asked how much the energy in a small corner of a hot oven should fluctuate, and answered with nothing but thermodynamics and Planck's law. The answer has two terms. One is exactly what a spray of independent particles would give; the other is exactly what randomly overlapping waves would give. Neither theory of light could produce the sum, and the sum is what the oven does. The two terms are equal where each mode of the radiation holds one photon on average — a dividing line that runs through every hot thing, from the microwave sky to the Sun.

 The innermost level dives towards the sea. The energy of an atom's innermost (1s) electron, in units of the electron's rest energy mc², against the nuclear charge Z: dashed, Dirac's formula for a point nucleus, √(1 − (Zα)²); solid, the same equation solved for a nucleus of realistic size. Above +mc² lie the free electrons; below −mc², the filled sea of negative-energy states that Dirac's theory requires. For a point nucleus the level stops at Z = 137, where the formula fails. A real nucleus keeps it: it crosses zero near Z = 148 and reaches −mc² at Z ≈ 170, binding the electron by twice its own rest energy. Beyond that the level has nowhere left to go but into the sea.

The nucleus that would make positrons from nothing

Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.

 A hydrogen atom tilted by a laser field. The potential energy of hydrogen's electron along the direction of a laser's electric field at the field's peak, in atomic units (one unit of energy is 27.2 eV, one of length 0.053 nm), for peak intensities of 3 × 10¹³ and 10¹⁴ W/cm² and at the barrier-suppression intensity, 1.37 × 10¹⁴; the dashed line is the bound level, 13.6 eV below the ionisation limit. The field tips the Coulomb well into a barrier on one side. At 3 × 10¹³ W/cm² the barrier is 15 atomic units thick at the level — 0.8 nm — at 10¹⁴, 6; at the suppression intensity its top has fallen to the level and the electron can leave without tunnelling at all. The barrier is real only for the fraction of the laser's period that the field stays near its peak.

The atom a light wave opens like a wall

Red light cannot free an electron from hydrogen: each photon carries 1.55 electronvolts and the electron is bound by 13.6. Make the light intense enough, though, and the atom ionises anyway — by absorbing nine photons at once, or, at higher intensity, by having its potential tipped over by the light's electric field until the electron tunnels out through the side. Which of the two happens is decided by a single ratio, found by Leonid Keldysh in 1964: how long the electron needs to get through the barrier, against how long the field holds still. The tunnelling side of that ratio is where attosecond physics begins.

 A signal that dies, and comes back when the spins are turned over. The size of the signal from an ensemble of proton spins whose precession rates are spread by 20 Hz (root mean square) about the mean, with a true relaxation time T₂ of 80 ms, against time after they are tipped. The free signal falls to 1/e in about 10 ms and to 0.03 by 20 ms: not because any spin has relaxed but because they have fanned out. A π pulse at 30 ms (marked) turns every spin over, and at 60 ms the signal returns to 0.472 — exactly e^(−2τ/T₂), the dashed envelope, since every static difference in rate has cancelled. Without the pulse the signal at 60 ms is below one part in a million.

The echo that gives back everything but the wandering

Tip a sample of protons in a slightly uneven magnetic field and their signal dies within milliseconds. None of them has relaxed: they have fanned out, each precessing at the rate its own patch of field sets. Turn every spin over halfway through and the fan closes again, and the signal comes back as an echo, smaller only by the decay that was real. What the echo cannot return is a phase gained while a spin moved, and that one loss, which grows as the cube of the waiting time, is how a scanner finds a stroke within minutes.

 A round uncertainty sheared into a thin one. The uncertainty in the direction of 100 atoms' combined spin, seen along the spin, drawn out to one standard deviation in each direction. Left: the atoms each in an independent superposition — a disk of variance N/4 = 25 in every transverse direction, so a rotation shows only once it moves the spin by √N/2 = 5.0 units. Right: after one-axis twisting, which turns the spin about the vertical axis at a rate proportional to its height, so that the upper half of the disk is pushed one way and the lower half the other (arrows). The disk becomes an ellipse with variances 1.22 and 535, its long axis tilted 12.2° from the horizontal. Its narrow direction, turned to lie along the rotation to be measured, is 12.0 dB quieter than the disk, allowing for the 11.8% the twisting costs in the length of the spin.

The atoms that measure better together

A clock read with a hundred independent atoms locates its phase no better than the scatter of a hundred coin tosses allows, because each atom answers the final question by itself. Let the atoms interact for a moment before they are read and that scatter can be shaped: the round uncertainty of their combined spin is sheared into a thin ellipse, and a phase error invisible against the coin-toss noise becomes four standard deviations wide. The gain grows with the number of atoms, and it is real only if the detector can count atoms nearly one by one.

 Four regimes along one line of mixed states. The weight p of the singlet in Werner's mixture of a singlet and complete noise, from 0 (pure noise) to 1 (a perfect singlet), divided by what is known about it. Below p = 1/3 the state is separable: it can be made by handing out unentangled pairs. Above 1/3 it is entangled, by the partial-transpose test. Yet up to at least p = 1/2 — Werner's explicit model of 1989 — and, by a later argument through Grothendieck's constant, up to about 0.66, a local hidden-variable model reproduces the outcomes of every pair of projective measurements on it, so no Bell experiment of that kind can reveal the entanglement. Above 1/√2 ≈ 0.707 the CHSH inequality is violated. Between about 0.66 and 0.71 (and, by more recent searches, between about 0.68 and 0.70) it is not known whether any Bell inequality is broken.

The entangled pair a list of answers can imitate

Entanglement and the violation of a Bell inequality are usually spoken of as the same thing, and they are not. Mix a perfect singlet with random noise and the pair stays entangled until the noise is two-thirds of the mixture. It stops violating the CHSH inequality much sooner, when the noise passes three-tenths. In between lies a family of genuinely entangled states whose every measurement can be reproduced by particles carrying instructions fixed in advance — Reinhard Werner wrote the instructions down in 1989. The entanglement in those states is real and can be spent, on teleportation for one thing. It simply cannot be seen by asking two distant analysers what they find.

 The same fringes from cooler and hotter molecules. The rate at which molecules arrive across a few periods of the interference pattern behind the last grating, relative to its mean, for molecules at 1000, 2500, 3000 K, each with the visibility the emission model gives: 46 per cent, 14 per cent, 4 per cent. The cooler molecules draw fringes nearly as strong as cold ones; at 3000 K the pattern is a faint ripple on a flat background. The average number arriving is the same in every case: the heated molecules are not lost, and none of them has been deflected out of the beam. They arrive in the same places on average and no longer interfere.

The molecule that interferes until it glows

A molecule of seventy carbon atoms, sent through a matter-wave interferometer, travels both arms at once and lands in fringes, as an electron or a neutron does. Heat it, and the fringes fade. Nothing has touched it from outside: a hot molecule glows, and each photon it gives off is a messenger that may carry away the information of which arm it was in. Whether it does depends on one comparison — the photon's wavelength against the separation of the arms. Long photons leave the fringes alone; short ones erase them. The loss of interference that is usually described as the onset of the classical world is here a measurement made by the molecule's own light, and its threshold can be computed.

 Reversing the magnetism in step with the swing. The rod hung from a fibre as a torsion pendulum with a period of 10 s and a quality factor of 200, its magnetisation reversed 40 times, once every half swing, in step with its motion, then left alone; the angle it turns through, integrated, in degrees. Each reversal adds the same small kick of angular momentum and the kicks add up, as pushes on a swing do: the swing grows to 15.46° if the magnetism is spin, 30.92° if it were orbital currents. A single reversal would turn the rod by about 0.45° — measurable in principle, buried in practice by the much larger twist that the coil's own field puts on any slight misalignment, which is why Einstein and de Haas drove the rod at its resonance and why their first result, g′ = 1, came out wrong.

The rod that turns when its magnetism flips

Hang an iron rod on a thin fibre inside a coil, reverse the current, and the rod turns a little. Nothing has pushed it sideways: reversing the magnetisation reversed the angular momentum the electrons inside it carry, and since nothing outside can twist the rod about its axis, the iron lattice has to take up the difference. Einstein and de Haas did it in 1915 to prove that magnetism is electric charge going round. They got the right effect and the wrong number, and the right number — twice what they expected — was evidence, years before anyone proposed it, that the electron's magnetism comes from something that is not a rotation at all.

 Two measures of how uncertain a split state is. For the two-peaked state, against half the separation of the peaks in units of their width: Heisenberg's product σₓσₚ divided by its minimum ħ/2 (solid), and the entropic sum Hₓ + Hₚ minus its minimum ln(πeħ) (dashed), both computed from the state's distributions. As the peaks separate the product grows without limit, 8.1 times the minimum at the right of the figure, because the variance counts the empty space between the peaks as uncertainty. The entropic excess levels off at 2 ln 2 − 1 = 0.386 (0.386 at the right): the position's entropy gains ln 2, one bit, for not knowing which peak, and the momentum's gives back 1 − ln 2 because its fringes pack it into narrow bands. The entropic relation says what is true of the state: it is a fraction of a coin toss away from a minimum-uncertainty packet, however far apart the two places are.

The sharpness a variance cannot see

Heisenberg's inequality is a statement about standard deviations, and a standard deviation is a poor measure of how sharply something is known. A particle that is certainly in one of two narrow places a metre apart has a position spread of half a metre, and satisfies the inequality with room to spare while being sharply localised. Measure uncertainty by information instead — by the Shannon entropy of each distribution — and the relation tightens: the sum of the two entropies has a floor that the split state cannot get under either, and the floor survives in places where the variance form says nothing at all.

 A chaotic rotor heats without limit, and its quantum version stops. The mean kinetic energy of a rotor kicked once a period by a potential of strength K = 5, against the number of kicks, starting at rest. Classically (an ensemble of 4,000 rotors with random angles) the motion is chaotic and the momentum random-walks, so the energy grows in a straight line, 6.3 per kick, near the K²/4 = 6.25 a random walk of kicks K sin θ gives. Quantum mechanically, with an effective Planck constant ƛ = 1 and 0.5, the energy follows the classical line for a while and then stops, fluctuating about 58 and 123: the smaller ƛ lets it follow the classical growth for longer and climb further before it stops. Nothing in the quantum rotor's equations damps it; the growth stops by interference.

The heating a quantum rotor refuses

Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.

 The cloud that stops shrinking. The rms size of a cloud of 10⁵ ideal atoms in a harmonic trap, in units of the Fermi radius, against temperature in units of the Fermi temperature, for fermions, for bosons and for a classical gas. The classical cloud shrinks as the square root of half the temperature ratio, all the way to nothing. The bosons follow it down to their condensation temperature, 0.52 of the Fermi temperature, and then collapse into the trap's ground state, whose rms size is 0.054 Fermi radii. The fermions part from the classical line near the Fermi temperature and level off at one over the square root of eight, 0.354 of the Fermi radius: at a quarter of the Fermi temperature they are 1.17 times the classical size, and cooling further barely changes them. Mixed together in one trap, as lithium-7 and lithium-6 were in 2001, the two clouds start the same size and end 6.5 times apart in width for ideal atoms of these numbers — further still in volume. The fermions are held up by nothing but the rule that no two of them may share a state.

The cloud that stops shrinking

Two kinds of lithium atom, cooled together in one magnetic trap to a few hundred billionths of a degree, start as clouds of the same size and end very differently. The lithium-7 atoms, which are bosons, shrink with the temperature and then collapse into a condensate. The lithium-6 atoms, which are fermions, stop shrinking at about a quarter of their Fermi temperature and refuse to get any smaller, however much colder they become. Nothing pushes them apart but the rule that no two identical fermions may share a state. The photographs taken in 2001 were the first time Fermi pressure — the pressure that holds up white dwarfs — was seen as a picture.

 The push that makes an electron swing. An electron in a band of width 4J, pushed by a constant force F = 0.25 J per lattice spacing, over two Bloch periods h/Fa. Above: its crystal momentum, which grows steadily as Newton's law says it should, and is folded back each time it reaches the edge of the Brillouin zone, ±π/a. Below: its position, in lattice spacings, which does not grow steadily at all. The electron's velocity is the slope of the band, and the slope reverses at the zone edge, so it moves forward, slows, stops 16 lattice spacings along — 4J/F — and comes back, oscillating instead of accelerating. A free particle with the band-bottom effective mass, pushed by the same force, is dashed: the two agree only for the first fraction of a period. The electron is not stopped by anything; it is turned round by the lattice, which takes the momentum the force gives it.

The push that makes an electron swing

Push a free electron with a steady force and it accelerates for as long as the force lasts. Push an electron in a crystal with the same force and, according to the band theory every transistor depends on, it should do something else entirely: move forward, slow, stop, come back and do it again, oscillating on the spot at a frequency set by the force and the lattice spacing. Felix Bloch's oscillation was predicted in 1928 and not seen for sixty-four years, because in a real metal the electron is knocked off course ten thousand times before it completes one swing. It was found in crystals built with a thirty-times-larger spacing, and in lattices made of light where nothing scatters at all.

 A lifetime that oscillates with a distance. The rate at which an excited atom emits a photon, relative to its rate in empty space, against its distance from a perfect mirror in wavelengths, for its dipole pointing across the mirror and along it. On the mirror itself the rate doubles for the first and vanishes for the second. As the distance grows both swing about 1 with a period of half a wavelength and a shrinking size: at half a wavelength the along-mirror rate is 0.96, and at a wavelength 0.99. Nothing in the atom has changed; its excited electron and its transition are the same in every position. An emission rate is a property of an atom and of the room it is in.

The lifetime that depends on where the mirror is

A half-life is treated as the most private fact about an atom or a nucleus, and for a nucleus that is nearly true. An excited atom beside a mirror decays faster or slower depending on how many half-wavelengths away the mirror stands, and between two mirrors closer than half a wavelength it does not emit at all. The atom has not changed. A decay is a transition to a state with a photon in it, and the room decides whether that state exists.

 The spectrum two rates would make. Balance absorption against spontaneous emission alone, with level populations in the Boltzmann ratio, and the radiation comes out as x³ times e to the minus x, Wien's law; add stimulated emission, and the same balance gives x³ over e to the x minus one, which is Planck's. The two agree on the right, where a photon is much more energetic than the thermal energy and stimulated emission is negligible, and disagree on the left, where only Planck's approaches the classical count of waves. At x = 1 Planck's is 1.58 times Wien's. Measurements around 1900 at long wavelengths had already found that Wien's law failed, so experiment required the third rate seventeen years before anyone saw it in a laser or a maser.

The third process equilibrium demands

In 1916, with general relativity just finished, Einstein asked how atoms and light could reach equilibrium at all. Absorption and emission, balanced against each other, gave the wrong spectrum. A third process was needed, in which light makes an excited atom emit a copy of itself. Nobody had seen it, and it took forty years to be put to use. It also fixes a ratio that goes as the cube of the frequency, which is why X-ray lasers are so hard.

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