Illustrated Physics
physics, drawn
Theme

Only some values fit

Continuous objects producing discrete answers. A string that will sound some frequencies and not others, three permitted exponents, a speed no composition can exceed — the first sightings of quantisation, in a subject that has not reached it yet.
Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths. Waves

When two waves meet, they simply add

Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.

 Harmonics on a fixed string. Standing-wave patterns on a string clamped at both ends, at n = 1, 2, 3, 4. Only whole numbers of half-wavelengths fit, which is why the allowed frequencies are discrete. Waves

Only some notes fit, and that is where discreteness comes from

A string clamped at both ends can vibrate at some frequencies and not others. A continuous object producing a whole-number list is the oldest quantisation in physics.

 The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all. Optics

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.

 Response against driving frequency. The steady-state amplitude of a driven oscillator against driving frequency, at three damping ratios. Lighter damping gives a taller and narrower peak, and the peak sits slightly below the natural frequency. Waves

The frequency that gets an answer, and the quarter cycle nobody mentions

Push an oscillator at its own frequency and the response grows enormously. The reason is not that the push is in step with the motion — at resonance it is a quarter cycle out, and that is precisely why it works.

 Single-slit diffraction at three slit widths. Intensity against angle behind a single slit, evaluated from the integral across the aperture, for slits two, six and twenty wavelengths wide. A wide slit throws a nearly sharp shadow; a narrow one spreads light through a wide angle. Optics

Where rays stop being enough, and a shadow acquires a bright centre

Light going through a narrow gap spreads. No amount of ray tracing predicts it, the size of the spreading is set by one ratio, and taking that ratio to zero is exactly what the ray model is.

 The same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law. Electromagnetism

The shape decides the falloff, and the force law never changes

A point charge gives an inverse square, a line gives an inverse, a plane gives a constant. All three come from the same law, and the exponent belongs to the geometry of the source rather than to the physics.

 Heating 1 kg of water from -20°C to 130°C. Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey. Thermodynamics

The heat that changes no temperature, and where it actually goes

A kettle reaches a hundred degrees in a minute and takes five more to boil dry. The heat going in during those five minutes changes nothing a thermometer can see, and it is most of the energy in the whole process.

 Composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back. Relativity

Speeds that refuse to add, and the quantity that does

Run at half the speed of light, throw something forward at half the speed of light, and the result is not the speed of light. It is four-fifths of it, and there is a variable in which the arithmetic is still simple addition.

 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. Quantum

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 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. Quantum

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. Quantum

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.

 Which wavelength wins. The growth rate of a disturbance on a liquid thread against kR, the circumference divided by the wavelength. Everything to the right of one decays; the maximum sits at kR = 0.697, which is a wavelength of 9.01 radii or 4.51 diameters. That number, and not a property of any particular liquid, is what sets the spacing of the drops a tap breaks into. Fluids

The thread that cannot stay a thread

A stream of water from a tap breaks into drops, and it does so at a spacing that is always about four and a half diameters. Nothing chooses that number — it is the wavelength that grows fastest out of a competition between all of them, and it can be computed before any water is poured.

 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. Quantum

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 staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant. Thermodynamics

Half a kT for every way of moving

A heat capacity ought to be a count. Every quadratic term in a system's energy carries half a kT of it, so warming a gas is a matter of enumerating the ways its molecules can move — and the fact that the count comes out wrong for hydrogen is how a thermometer measured Planck's constant.

 The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else. Mechanics

The two pendulums that will not stop swapping

Coupled oscillators joined by a weak spring appear to hand energy back and forth. Nothing is handed anywhere: the system has only a pair of motions that keep their shape, at √(k/m) and √((k+2kc)/m), and the apparent traffic is the beat between them — 51 swings from one handover to the next at a coupling of one part in fifty. Extend the same arithmetic to N masses and it produces a dispersion relation with a hard ceiling, near 7 THz in copper.

 A grating of 20 slits. Intensity against angle behind a grating of 20 slits spaced 4 wavelengths apart, from I = [sin(Nu)/(N sin u)]² with u = π d sinθ/λ. The principal maxima sit at sinθ = mλ/d — -14.48°, 0.00°, 14.48° for m = -1, 0, 1 — and those angles contain no N at all, so they are exactly where two slits put them. What N changes is the width: the central maximum measures 0.635 degrees between its half-maximum points, measured off the drawn curve, against 0.635 degrees from bisection on the pattern itself. That width goes as 1/N — N times it is 12.692°, 12.691°, 12.691° at N = 64, 256, 1024, and 12.705° here, which is more, because the 1/N law is asymptotic and few slits are the far end of it: two slits give a peak 13 per cent wider than the limit. So a grating's resolving power R = mN is bought with the number of lines illuminated and with nothing else. In first order this grating resolves λ/Δλ = 20; the sodium doublet needs 982. Away from the maxima the pattern stays below 4.8 per cent of one, because there it is bounded by 1/(N sin u)². Optics

What a thousand slits buy that two cannot

The bright directions behind a grating are fixed by its ruling pitch and the wavelength alone, and no count of lines appears in them. What the count changes is the width of each maximum, which falls as 1/N — so resolving power is mN, and 1,200 illuminated lines separate the sodium D lines with a dip of 53.4 per cent where 300 show one line and no dip at all.

 6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled. Waves

The drum that has no harmonics

A string's allowed frequencies are 1, 2, 3, 4 times its lowest, because counting half-wavelengths is arithmetic. Clamp a membrane round a circle and the same reasoning returns 1.000, 1.593, 2.136, 2.295 instead — zeros of Bessel functions, not integers. And those numbers belong to the shape of the boundary alone, which raises a question nobody could answer until 1992.

 Where modulating a system sets it going. The regions of the modulation plane in which an oscillator with a damping ratio of 0.02 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 8.0%, against 40.0% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything. Waves

The swing that is pumped, not pushed

Nobody pushes a swing they are sitting on. They stand up at the bottom and sit down at the ends, which changes the pendulum rather than forcing it — and does so twice per period. The equation that describes it has no forcing term at all, so standing still is always a solution, and what the pumping changes is whether standing still is stable.

 The surface a spin decides. The free surface of a liquid in a dish of radius 0.5 m turning at 10, 20, 40 revolutions a minute. In the rotating frame the surface is a level set of gz − ½ω²r², so it is a paraboloid exactly and not to some approximation, and nothing about the liquid appears in its shape: the same curve is got with mercury, water or oil. The rim stands 14.0 mm, 55.9 mm, 223.6 mm above the centre at those rates. A parabola z = r²/4f has focal length f, so these surfaces are mirrors of focal length g/2ω² — 4.47 m at 10 rpm, 1.12 m at 20 rpm, 0.28 m at 40 rpm. That is checked here on the drawn curves rather than quoted: a vertical ray reflected off the surface at a quarter, a half, three quarters and the whole of the radius crosses the axis at the same height to 0.00%, which is what a mirror with no spherical aberration means. Doubling the spin quarters the focal length, and there is no other adjustment: the dish can only ever look straight up. Fluids

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

 The entropy of mixing, and the entropy of not mixing. The entropy gained when two ideal gases at the same temperature and pressure are allowed to mix, per particle and in units of Boltzmann's constant, against the proportion of the mixture that is the first gas. The curve has no property of either gas in it — not their masses, not their sizes, not how strongly they interact, since ideal gases do not — and it is largest at 0.500, where it reaches ln 2 = 0.6931. Below it is the same quantity for two samples of the SAME gas, which is zero at every proportion: removing the partition between two halves of a box of nitrogen changes nothing that can be measured, and putting it back recovers the original state. The two results are correct and they do not join up. Make the two gases more and more alike — two isotopes, then two nuclear spin states, then nothing at all — and the upper curve does not descend to meet the lower one; it stays exactly where it is until the two species become identical, and then jumps. What the figure is really about is that the jump is in the counting and not in the gas. Thermodynamics

Mixing what is already mixed

Let two different gases into each other's halves of a box and the entropy rises by a fixed amount that contains nothing about either gas. Do it with the same gas on both sides and it rises by nothing. Make the gases more and more alike and the answer does not converge — it jumps.

 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. Quantum

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.

 The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves. Optics

The crystal that answers twice

Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

 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. Quantum

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.

 The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw. Waves

The pipe that will not carry a low note

A wave squeezed sideways acquires a lowest frequency. Below it nothing travels — the field is there, it is large, and it goes nowhere. Above it the guide is dispersive whether or not anything in it is, and the pattern inside runs faster than light while the signal does not.

 The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is. Waves

The gap a repeat opens

Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

 A lens with two flat faces. Rays entering a rod whose refractive index falls parabolically from the axis outward, parallel to the axis and at -2.4, -1.2, 0, 1.2, 2.4 mm from it. The ray equation in such a medium is the harmonic oscillator's, so each path is a cosine of the same period whatever height it started at — which is exactly the condition for a focus, and the reason all of them cross the axis together at 12 mm. Nothing is curved anywhere: the faces are flat and the bending is done by the inside of the glass. Cut the rod at a quarter of the period and it images; cut it at half and it relays the beam parallel again, inverted. The period is a property of the profile alone, which is why a rod like this is specified by a length rather than by a curvature. Waves

The channel with no walls

A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well.

 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. Quantum

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.

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

The magnetism classical physics forbids

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

 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. Quantum

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. Quantum

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.

 Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at. Fluids

The whirlpool that comes in one size

Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself.

 Nine planes behind a grating, with no lens anywhere. The intensity across two periods of a 20 µm grating, at nine planes between it and the Talbot distance z_T = 2d²/λ = 1.26 millimetres, illuminated at 633 nm. The top profile is the grating itself and the bottom is the plane at z_T, and they agree to 2.8e-14: free space has reproduced the object with no imaging element of any kind. The middle profile, at half the Talbot distance, is the object shifted sideways by half a period, to 1.1e-14. At a quarter of the way the grating's own period has vanished entirely — its amplitude there is 1.3e-16 — because the odd orders have all turned by the same right angle and the even ones have not. With this grating open for half of each period there are no even orders either, so the plane is uniform: 9.6e-3 at twice the frequency as well, and a screen there shows no grating at all. None of this is interference between two beams; it is the whole spectrum of the object arriving with the phases exp(−iπλzm²/d²), which are all multiples of 2π when z is z_T. Optics

The grating that photographs itself

Put a grating in a beam of light and hold a screen behind it. At one particular distance the screen shows the grating again — sharp, at full contrast, right way up, with no lens anywhere in the apparatus. Half way there it shows the grating shifted sideways by half a period. A quarter of the way there, a half-open grating shows nothing at all.

 Entropy that is still there at absolute zero, counted and measured. Four substances whose entropy does not go to zero when they are cooled as far as anybody can cool them, with the entropy counted from the arrangements they froze into beside the entropy measured by integrating their heat capacities. Ice is the famous one: every oxygen has four hydrogen bonds and the rule is that two hydrogens sit near it and two far, which leaves six of the sixteen placements legal, and Pauling's count of the whole crystal collapses to R ln(3/2) = 3.371 J per mole per kelvin against a measured 3.41 — an agreement to one per cent from an argument on one line. Carbon monoxide and nitrous oxide are the easy cases, molecules that can lie either way round in the lattice and have too little to gain by choosing. The worst of the four is off by 25 per cent, which is the honest state of this subject: the counts are crude, they ignore the correlations between neighbouring choices, and they still land within sight of a calorimeter. What the figure is really about is that the third law has an escape clause and the escape clause is measurable. A perfect crystal has one arrangement and zero entropy; a crystal that ran out of time while it had many has the logarithm of however many it stopped at, permanently. Thermodynamics

The entropy that is still there at zero

The third law says a perfect crystal has no entropy at absolute zero. Ice has 3.41 joules per kelvin per mole left over, and the number can be recovered from one line of counting — two hydrogens near each oxygen and two far, six legal arrangements out of sixteen, R ln(3/2). The law has an escape clause and the escape clause is measurable.

 The cross a shaken cylinder leaves in a stratified fluid. The beams radiated by a small body oscillating in a fluid of buoyancy frequency 1.053e-2 per second — a period of 9.9 minutes — at 0.3, 0.6, 0.9 times that frequency. The disturbance does not spread in circles. It leaves along four rays, and the angle of those rays to the horizontal is fixed entirely by the ratio of the driving frequency to the buoyancy frequency: 17.5° at 0.3N, 36.9° at 0.6N, 64.2° at 0.9N. Nothing about the size of the body, the amplitude of the shaking or the wavelength enters. Drive it faster and the beams stand up; drive it slower and they lie down; drive it above N and there are no beams at all, because the dispersion relation ω = N cos φ has no solution. The short arrows across each beam are the wavevector, which is perpendicular to the beam — the dot products drawn here are 6e-17 — so the crests travel sideways across the ray while the energy travels along it, and a fluid doing this looks, in a photograph, as though its waves are moving at right angles to where they are going. Fluids

The wave that picks an angle

Shake a rod slowly in a tank of salty water layered by density and the disturbance leaves along four straight beams, at an angle fixed entirely by how fast the shaking is. Change the wavelength and the angle does not move. Change the frequency and it does. Above the buoyancy frequency there is no wave at all.

 Cold matter, and the mass above which nothing holds it up. The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it. Astrophysics

The mass no cold matter can hold up

A white dwarf gets smaller as it gets heavier, which no ordinary object does. Follow that curve upward and the radius reaches zero at 1.46 solar masses — because once the electrons are relativistic the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball does not depend on its radius at all.

 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. Quantum

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.

 How many modes a square has below a given wavenumber. The number of vibration modes of a square with wavenumber below k, counted exactly — every eigenvalue of this region is a closed form, so the staircase is the true count and not an estimate. There are 265 of them below k = 60. The smooth curves are what Weyl's law predicts. The upper one is the leading term alone, the area times k² over 4π, and it is too high by 21.5 modes at the right-hand edge; the lower one subtracts the perimeter term, the perimeter times k over 4π, and is out by 2.4. The content of the law is that the count depends on the region through its area and its perimeter and — to this order — through nothing else at all: not through its shape, not through where its corners are, not through whether it is convex. The staircase's steps are the individual modes, and they cluster where two different pairs of indices give the same wavenumber. That the count is smooth in the large while being a staircase in the small is what makes a mode count usable in thermodynamics, where it appears as a density of states and never as a list. Waves

How many ways there are to vibrate

A drum has infinitely many modes, and below any given frequency it has a finite number of them. That number turns out to depend on the drum's area and the length of its rim and — to the accuracy anybody uses — on nothing else about its shape. Almost every result in thermal physics that involves waves is an application of that count.

 Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains. Fluids

The heap that becomes a solid

Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else.

 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. Quantum

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. Quantum

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 temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative. Thermodynamics

Hotter than any temperature there is

A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

 How far into the ground a surface wave goes. The horizontal and vertical displacements of a Rayleigh wave against depth, in wavelengths, at Poisson's ratio 0.25, each divided by the largest displacement anywhere. Both fall off exponentially, and at one wavelength down the vertical component is 19.3 per cent of its surface value — so the wave is confined to a skin about a wavelength thick, and the thickness is set by the wavelength rather than by anything about the material. The horizontal component changes sign at 0.193 wavelengths, which is not a node of a standing wave but a reversal: above that depth the ground moves one way round its elliptical orbit and below it the other. A long-period wave therefore samples deep rock and a short-period one samples only the surface, which is what makes a seismogram's dispersion a measurement of the structure underneath. Waves

The wave a surface is enough to hold

A pipe guides with walls and a fibre guides with a slower core. A solid needs neither: one free surface binds a wave that is not a bulk wave bouncing but a separate solution, travelling slower than any wave in the material, dying away exponentially into it, and carrying its energy round a circle instead of over a sphere — which is why it is the part of an earthquake that knocks buildings down.

 Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn. Fluids

The angles a film has no choice about

A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

 Five rays, and the one that decides which side everything falls on. Light traced past a non-rotating mass at five impact parameters, by integrating the exact null geodesic rather than the weak-field formula. The shaded disc is the horizon and the dashed circle is the photon sphere at three masses. Rays passing closer than 5.196 masses are captured — 2 of the five here — and rays passing further escape, however far they are bent on the way. The two nearest the critical value behave in the most striking way: one wraps 1.06 times round before leaving and the other wraps round before falling in, and between them lies a ray that would circle for ever on the unstable orbit. Far away the same integration reproduces the familiar weak-field deflection: at sixty masses it gives 0.07021 radians against 4M/b = 0.06667, so the strong field and the weak one are one calculation. Astrophysics

The circle light cannot leave

A black hole has three radii and they are all sometimes called its size. The horizon is where nothing can come back from; the photon sphere, half again as far out, is where light can circle and cannot keep circling; and the shadow a distant observer sees is larger than either, because the ray that just escapes was bent on its way out.

 Rings whose radii go as the square root of their number. A 20-zone plate for 550 nanometres at 200 millimetres, drawn to scale, beside its zone radii against zone number. The outermost is 1.483 millimetres and the innermost 0.332, and the curve is a square root because the radii come from making each zone's extra path exactly half a wavelength longer than the last. The consequence worth noticing is on the drawing rather than in the formula: every ring has the same area, to 0.003 per cent, so each contributes about equally to what arrives on the axis and the rings get thinner outwards to keep it so. Alternate rings are opaque. Half the light is thrown away and the axis gets brighter, because what is thrown away is the half that would have arrived out of phase with the rest. Optics

The lens that is a set of rings

A lens focuses by delaying the light at its centre until every path takes the same time. A zone plate does the opposite: it changes nothing about the light that gets through, and paints out the light that would have arrived out of step. Half the aperture is thrown away and the axis gets brighter, which sounds like a contradiction and is the whole idea.

 The size of the quantum says what is carrying the current. What the flux quantum would be for each candidate carrier charge, in units of 10⁻¹⁵ webers, against the measured value drawn as a line. A single electron would give 4.1357, a pair 2.0678, a triple 1.3786. The measurement is 2.0678, which picks the pair and excludes the others by a factor of two — not by a few per cent, so no question of experimental accuracy arises. The whole of the argument is that the condensate's wavefunction must come back to itself round the ring, which makes the enclosed flux a multiple of h over the carrier's charge; measuring the multiple therefore measures the charge, without any charge ever being measured. That is how the pairing was established in 1961, four years after it was proposed and by an experiment that looks nothing like a measurement of a charge. Electromagnetism

The two in the flux quantum

A superconducting ring cannot hold whatever flux is applied to it. It holds a whole number of quanta and drives a current to make up the difference, and the size of that quantum is Planck's constant divided by twice the electron's charge. The factor of two was measured in 1961, four years after somebody predicted that the carriers are pairs — by an experiment in which no charge is measured at all.

 Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up. Thermodynamics

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

 Four modes of a string that is heavier in one place. The first four modes of a string whose mass per unit length rises to 5 times the light end's over a smooth bump centred 62 per cent of the way along, drawn beneath them. Nothing is symmetric any more: the shapes bunch up over the heavy region, where the local wavelength is shorter, and the amplitudes there are smaller. The frequencies are 1.70, 4.01, 6.10, 8.12, which are in the ratios 1.000, 2.361, 3.590, 4.779 rather than 1, 2, 3, 4 — this string has no harmonics and would sound like a bell rather than a violin. What has not changed is the one thing an ordering needs: the interior zeros, marked, run 0, 1, 2, 3 exactly as they do on a uniform string. That is Sturm's theorem, and the nodes here are counted on the computed shapes rather than assumed. Waves

The count that cannot be cheated

A string with a lump in it has no harmonics, no symmetry and no obvious order to its modes. It has one thing left: the nth mode crosses the axis exactly n−1 times, whatever the string is made of. Ordering by frequency and ordering by node count turn out to be the same operation, and in two dimensions the equality quietly becomes an inequality.

 Three curves that can only meet at a point. Water's phase diagram within a kelvin of its triple point, with each boundary drawn at the slope Clausius and Clapeyron give it from measured latent heats and densities: 44.4 Pa/K for boiling, 50.3 for sublimation, and -135 bar per kelvin for melting — negative, and so steep that it is drawn vertical here, because ice is the less dense phase and pressure therefore melts it. Two things follow that no amount of measurement could adjust. The three slopes are not independent: the sublimation and vaporisation curves differ in slope by 5.92 Pa/K, which is exactly what the fusion latent heat predicts, because going solid → gas directly and solid → liquid → gas must cost the same enthalpy. And the sublimation curve is the steeper of the two, so the two cross rather than touch — which is why the triple point is a crossing and not a tangency, and why ice sublimes below it and melts above it. Thermodynamics

Why the triple point is a point

Three phases of one substance coexist at one temperature and one pressure and nowhere else, and the reason is arithmetic rather than chemistry: count the numbers that describe the state, count the conditions equilibrium imposes, and subtract. The same subtraction says four phases of one substance are impossible, and it says so without knowing what the substance is.

 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. Quantum

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. Quantum

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.

 The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all. Waves

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

 The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream. Fluids

The wave that is required to stand still

A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

 The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers. Thermodynamics

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

 An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium. Waves

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

 A current with no voltage, and then a voltage that is a frequency. Current against mean voltage for a Josephson junction shunted by a resistance, in units of its critical current and of the critical current times the resistance. Up to the critical current the junction carries a supercurrent at exactly zero voltage, set by the difference of the two superconductors' phases. Above it a mean voltage appears, growing as R√(I² − Ic²) and approaching the ohmic line at large current; the curve is integrated from the junction equation and checked against that result at three currents. The voltage is not steady. It is a train of pulses, each the phase slipping by one turn, at a frequency of 483.6 GHz per millivolt: for a junction with Ic = 1 mA and R = 1 Ω, one unit of the voltage axis is 1 mV and 484 GHz. Electromagnetism

The voltage that is a frequency

Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.

 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. Quantum

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.

 Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends. Waves

The end that knows how the middle was cut

A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

 A gap that every angle from air falls into. The stop bands of a stack of quarter-wave layers of index 4.6 and 1.6 — tellurium and polystyrene, the pair of the first such mirror — against frequency, in units of the design frequency, and the parallel index β = n₀ sin θ₀ the light brings along the layers; TE polarisation on the right, TM on the left. Shaded regions are gaps. Light from air can only have β between −1 and 1, the vertical lines; within those lines the gaps for both polarisations overlap between f = 0.848 and 1.321, the band marked across the figure, so every angle of incidence and both polarisations are reflected: a relative width of 43.6%. The TM gap narrows as β grows and closes at the internal Brewster index, 1.51, which light from air cannot reach. Waves

The mirror that works from every direction

A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.

 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. Quantum

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 étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for. Optics

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

 The condition three modes never quite satisfy. By how much three modes of a thirty-two mass chain fail to be in resonance — the sum of two mode frequencies minus the frequency of their sum, on a logarithmic scale, against the second of the two modes for four choices of the first. The leading nonlinear term couples modes in threes and the exchange accumulates only where this quantity is zero. It never is: the chain's dispersion is a sine, a sine is concave, and the sum of two of its values always exceeds the value at their sum. The smallest mismatch anywhere on the chain is at the two lowest modes and equals 2.156e-4 — which the scan finds and which is the cube of pi over four times the cube of one more than the mode count, checked here on chains from eight masses to two hundred and fifty-six. That closed form is the whole of why this is a finite-chain problem: the mismatch falls as the cube of the length, so a long enough chain is arbitrarily close to resonant and the continuum limit is exactly resonant, which is where the solitary waves come from. Thermodynamics

The condition three modes never meet

Whether two modes of a chain can hand energy to a third is arithmetic on the dispersion relation, and for a chain of masses the answer is never: a sine is concave, so the sum of two frequencies always exceeds the frequency of their sum. The smallest shortfall anywhere on a chain of N masses is π³/4(N+1)³ — never zero, and never far from it — and a shortfall turns a transfer into a beat.

 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. Quantum

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. Quantum

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.

 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. Quantum

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.

 A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵. Waves

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

 A pendulum with unequal steps. The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone. Electromagnetism

The circuit that forgets its charge

A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.

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