Illustrated Physics
physics, drawn
Concept

Zero-point energy — where it appears

The energy a confined system retains in its lowest state, unavoidable because the longest wave that fits between its walls still carries a momentum. It cannot be removed by cooling, it grows as the confinement tightens, and it is what keeps helium liquid at absolute zero.

Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.

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

The box that allows only some energies

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

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

Sharpness has to be paid for

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

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

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

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

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

Where the quantum picture hands back the old one

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

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

The liquid that will not slow down

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

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

The motion that cannot be stopped

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

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

The pressure that is not a temperature

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

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

The state that swings like a pendulum

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

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

Why an atom is the size it is

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

quantum · Atomic structure The one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

The estimate that misses by a hundred and twenty

Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

astrophysics · Planck scale The area each quantum state is allowed. Phase-space orbits of the four lowest quantum levels, computed as eigenvalues of the discretised Schrödinger equation, for a harmonic well (left) and a box with hard walls (right), in units ħ = m = 1. Each orbit encloses an area that is a whole number of Planck constants plus a fixed offset: for the harmonic well 0.500, 1.500, 2.500, 3.500 h — n + ½ — and for the box 1.000, 2.000, 3.000, 4.000 h — n + 1. The harmonic orbit turns round smoothly at both ends and loses a quarter of a cycle at each; the box orbit is reflected sharply by each wall and loses half a cycle there.

The quarter cycle a turning point costs

The old quantum rule said an orbit's action is a whole number of Planck constants. It gets the harmonic oscillator wrong by exactly half a unit, the box by exactly one, and a neutron bouncing on a mirror by three quarters. Each is the same correction: a wave loses a quarter of a cycle every time it turns round smoothly, and half a cycle every time it hits a wall.

mechanics · Least action How much slower a warm atom ticks. The fraction by which a free atom's clock runs slow on average because of its thermal motion, (3/2)kT/mc², against temperature on logarithmic axes, for five atoms used in clocks or tests of relativity: hydrogen 4.1 × 10⁻¹¹ at 300 K, aluminium ion 1.5 × 10⁻¹² at 300 K, strontium 4.8 × 10⁻¹³ at 300 K, caesium 3.1 × 10⁻¹³ at 300 K, mercury ion 2.1 × 10⁻¹³ at 300 K. Every line falls in proportion to the temperature and to one over the mass. A clock aiming at one part in 10¹⁸ must hold strontium below 0.63 mK and hydrogen below 7.3 µK — or know their temperature well enough to correct for it. Pound and Rebka's gravitational shift across 22.5 m, 2.46 × 10⁻¹⁵, is marked for scale.

The clock that runs slow because it is warm

Every atom in a gas is moving, and every moving clock runs slow. The first-order Doppler shifts of a hot gas cancel on average; the time dilation does not, because it has only one sign. A hydrogen maser at 40 °C runs slow by four parts in a hundred billion from this alone, a nucleus in a crystal runs slow even at absolute zero, and the first test of gravitational time dilation nearly drowned in a one-kelvin temperature difference.

relativity · Time dilation The area under the spacings is the bond. The gap between each vibrational level and the next, against level number, for a Morse well and a Lennard-Jones well with the same depth and the same curvature at the bottom, Λ = 100, in units of the well depth. The sum of the gaps from the lowest level up is the energy needed to break the bond from there. For the Morse well the gaps fall in a straight line to zero, and a straight line through the first 5 of them, extended to zero, encloses 0.9473 against the true 0.9473. For the Lennard-Jones well the gaps start almost the same and fall slightly faster at first — which is what the straight line sees — then trail off in a long tail of small gaps near the top; the same extrapolation from the first 5 encloses 0.7988 against the true 0.9474 — 15.7 per cent short, because the straight line cannot see the crowded levels its data never reached.

The spectrum that weighs a bond

A vibrational spectrum measured near the bottom of a molecule's well gives its curvature, and the curvature says nothing about how deep the well is. Measured further up, it says everything. The levels of a real bond crowd together towards the top, the gap between neighbours falls towards zero, and the sum of all the gaps is the energy it takes to break the bond. A straight line drawn through the gaps that have been measured, and extended to zero, was for decades how bond energies were found — and for a bond with the tail of two neutral atoms it comes out a sixth short.

mechanics · Harmonic approximation A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is.

The molecule that lives outside its own well

The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

mechanics · Harmonic approximation London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm.

The attraction that weakens when light is too slow

Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

electromagnetism · Retardation The Casimir pressure between warm plates, from vacuum to heat. The attractive pressure between two parallel, perfectly reflecting plates at 300 K against their separation, on logarithmic axes, from the Lifshitz sum over Matsubara frequencies. Dashed: the zero-temperature law π²ħc/240a⁴. Dotted: the classical law ζ(3)kT/4πa³, which contains no Planck constant. At 1 μm the full pressure is 0.0013 Pa, within 0.2 per cent of the vacuum value; at 10 μm it is 3.96·10⁻⁷ Pa, 3.05 times the vacuum value and within 0.002 per cent of the classical law. The two laws cross at 3.28 μm; the thermal length ħc/kT is 7.63 μm. The second solid curve removes one term — the static magnetic fluctuation, which a metal of finite conductivity does not reflect — and sits 15 per cent lower at 1 μm and a factor of two lower far out. The plates are otherwise ideal mirrors: a model, chosen to isolate that one term.

The force that forgets Planck's constant

Two mirrors in a vacuum attract, with a pressure that contains Planck's constant and nothing else about the world but the speed of light. Warm the mirrors and pull them apart, and past a few micrometres at room temperature the Planck constant drops out of the answer. The force is still there and is now larger than the vacuum alone would give, but it has become classical: a count of thermal fluctuations of the static field, with kT of energy each. Half of that classical force comes from one fluctuation — a static magnetic field — and whether a real metal mirror reflects it is a question that has divided the people who measure the force for twenty years.

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

The bounce that comes in fixed heights

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

quantum · Matter waves The two lines a vibration adds to scattered light. The light scattered by silicon's optical vibration at 520 cm⁻¹, against its shift from the 532 nm laser line in cm⁻¹ (negative, lost to the crystal; positive, gained from it), on a logarithmic intensity axis, at 100 K, 300 K and 600 K; the strong unshifted line and its wings are cut out at the centre. The Stokes line, where the light gives up a quantum of vibration, barely changes: 1.00, 1.09 and 1.40 in relative strength. The anti-Stokes line, where it takes one, needs a vibration already excited and grows from 7·10⁻⁴ of the Stokes line at 100 K to 0.103 at 300 K and 0.36 at 600 K. The light comes back bluer only as often as the crystal is warm enough to pay for it.

The light that comes back bluer when it is warm

Shine a green laser on a crystal and a tiny fraction of the light comes back with a different colour: a little redder, having given a quantum of energy to one of the crystal's vibrations, or a little bluer, having taken one. Losing energy is always possible. Gaining it needs a vibration already excited, so the bluer light is fainter, and fainter still the colder the crystal. The ratio of the two is a Boltzmann factor, fixed by quantum statistics and nothing else. It is a thermometer that needs no contact and no calibration — and in a long glass fibre, it is thousands of thermometers at once.

optics · Scattering

Named alongside it

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

QuantisationUncertainty principleMatter waveAtomic structureBinding energyBound stateClassical limitCorrespondence principleDegeneracy pressureEquipartitionFluctuationsHarmonic approximation

All concepts