The collection

Every essay — page 3

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Quantum

Where the continuous picture runs out, and what replaces it.

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

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

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

5 figures · part 2 on Photon $An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.$

Everything has a wavelength, and almost nothing shows it

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

4 figures · part 1 on Matter waves

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.

4 figures · part 2 on Standing waves

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

The spectrum is a subtraction, not a list of values

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

4 figures · part 1 on Atomic spectra

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

Where the electron probably is

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

4 figures · part 1 on Atomic structure

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.

4 figures · part 1 on Uncertainty

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

The wall that is not quite a wall

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

4 figures · part 1 on Tunnelling

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

One arrival at a time, and the pattern still appears

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

4 figures · part 2 on Matter waves

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

The answer that was not there before

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

5 figures · part 1 on Measurement

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.

4 figures · part 1 on Exclusion

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

What happens when the wells get close

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

4 figures · part 1 on Bands $Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.$

A nucleus with no clock

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

4 figures · part 1 on Decay

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.

4 figures · part 1 on Correspondence

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

The pressure that only knows depth

A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely.

5 figures · part 1 on Hydrostatics

Force multiplied, distance paid. Two pistons on one body of fluid, of areas in the ratio 16 to 1. A force of 200 N on the small one holds 3.20 kN on the large one, and pushing the small piston 16 cm raises the large one by 10.0 mm. The two products are the same number: nothing is gained except the shape of the bargain.

Force multiplied, and nothing gained

A push on a small piston becomes a much larger push on a large one, in the ratio of their areas, with no machinery in between except the liquid. What the liquid will not do is give anything away — the distances shrink by the same factor the forces grow by, and the product is untouched.

4 figures · part 2 on Hydrostatics

Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

The weight of the water that is not there

A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.

4 figures · part 1 on Buoyancy

A heeled hull, and the couple it makes. A rectangular hull of beam 3 m heeled 18°, with the waterline solved so that it displaces the same volume it did upright. The centre of buoyancy has moved 0.244 m to the low side, and weight and buoyancy now act along two lines 0.059 m apart — a couple that turns the hull back upright.

Why a ship comes back upright

Whether a floating body rights itself or rolls over is not decided by its weight, its density or how deep it sits. It is decided by the shape of the slice the water cuts through it, and the number that settles it can be worked out before the vessel is built.

4 figures · part 2 on Buoyancy

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

The skin that is not a skin

A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.

4 figures · part 1 on Surface tension

The small bubble empties into the large one. Two soap bubbles of radius 4 mm and 12 mm joined by an open tube. The excess pressure inside each is 4γ/R — 72.8 Pa and 24.3 Pa — so the 4 mm bubble is at the higher pressure and blows itself into the other. The smaller a bubble gets the harder it pushes, so the process runs away rather than settling: there is no equilibrium anywhere except both bubbles equal.

The small bubble blows up the big one

Connect two soap bubbles of different size and the small one empties into the large one. Everybody expects the opposite, and the reason it happens is one equation with a radius in the denominator — which also means the process runs away rather than settling.

5 figures · part 2 on Surface tension

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.

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.

4 figures · part 3 on Surface tension

Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

How high water will climb

Water rises up a narrow tube against gravity, and the narrower the tube the higher it goes. The height is set by a curved surface pulling on a circumference while gravity pulls on an area, and the two scale differently — which is the whole of it.

4 figures · part 1 on Capillarity

The straight line between two plates. A layer of water 10 mm deep with its top plate drawn along at 1 m/s. In the steady state the velocity is a straight line from zero at the fixed surface to the plate's speed at the moving one, and the stress needed to keep it going is μ times that slope: 0.100 Pa. The fluid at each wall is at rest with respect to it, which is an experimental fact rather than a consequence of anything above.

Momentum going sideways

Viscosity is usually described as friction between layers of fluid, which gets the effect right and the mechanism wrong. What is actually happening is that momentum is being conducted across the flow — by the same equation, with the same solutions, as a drop of ink spreading.

4 figures · part 1 on Viscosity

The fourth power in a pipe

Halve a pipe's radius and the flow through it falls to a sixteenth. The exponent is four rather than two, because narrowing a pipe both removes cross-section and slows what is left — and one law with that exponent in it governs a blood vessel, a hypodermic needle and a water main.

4 figures · part 2 on Viscosity

Four answers to one push. Shear stress against shear rate for four fluids. The straight line through the origin is the Newtonian definition and is the only one of the four for which the word viscosity names a number. The Bingham fluid does not move at all until the stress passes 0.4, which is why toothpaste holds a shape on a brush and why wet concrete can be stood in a heap.

The fluid that answers back

For water, stress is proportional to how fast it is sheared, and the constant of proportionality is its viscosity. For paint, blood, ketchup and cornflour in water, it is not — and once the proportionality goes, so does the idea that viscosity is a number a substance has.

4 figures · part 1 on Rheology

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