Illustrated Physics
physics, drawn
Concept

Degeneracy — where it appears

Two or more distinct states sharing the same energy or frequency. Degeneracies are where weak perturbations do their work, because a coupling that changes little elsewhere must split states that would otherwise be indistinguishable in energy.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

The same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

The orbit that does not come back to itself

A bounded orbit under any central force oscillates between a smallest and a largest radius for ever. That it should also return to the same point is a further demand, and only two force laws in existence meet it — the inverse square, and the linear spring.

astrophysics · Orbit stability 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.

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.

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

The line that is really two

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

quantum · Atomic spectra Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

thermodynamics · Third law 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.

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.

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

The questions that can be asked together

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

quantum · Uncertainty The arrow the orbit cannot turn. A Kepler orbit of eccentricity 0.6, integrated for two revolutions, with the Laplace–Runge– Lenz vector constructed from the position and velocity at five points along it. Every one of the five is the same arrow: its length varies by 3.7e-11 over the whole run and its direction by 2.1e-10 radians. It points at the perihelion and its length is 0.600000, which is the orbit's eccentricity measured independently from the closest and furthest radii as 0.600000. Energy and angular momentum fix the size and shape of an orbit and say nothing about which way it points; this vector is the missing statement, and only an inverse square has one.

The arrow that says which way the orbit points

Energy and angular momentum fix the size and shape of an orbit and say nothing about its orientation. The inverse-square force has a third conserved quantity that supplies it — a vector pointing at the perihelion whose length is the eccentricity — and no other force law does.

astrophysics · Orbit stability Where a small weight is felt, and where it is not. The fractional change in the frequency of harmonic 3 of a stretched string when a point mass of 0.06 of the string's own mass is placed at each position along it. The solid curve is the exact answer, found by solving the string's frequency equation for the loaded string at each position; the dashed curve is the first-order prediction, minus the mass fraction times the square of the mode shape. The two agree to 10.4 per cent at the antinode, where the shift is largest. Every node is a place where the exact shift is zero to better than a part in a thousand million, and that is not an approximation: a mass at a node is never moved by the mode, so it takes no part in the motion and cannot change its rate. Between the nodes the shift follows the square of the displacement, which is the square of the amplitude — the mass is felt in proportion to the kinetic energy the mode was already keeping there.

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

waves · Standing waves 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.

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.

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

The field an atom calls strong

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

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

A few cycles that are only mass and spin

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

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

Two states where the counting says three

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

quantum · Spin

Named alongside it

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

SpinAngular momentumEigenvalueEnergy levelsPerturbationSymmetryAtomic spectraBoundary conditionCentral forceEntropyEquilibriumFine structure constant

All concepts