The energy ladder of hydrogen
At its defaults it draws the energy ladder of hydrogen. The first 6 energy levels of hydrogen, drawn to scale in eV, at -13.61, -3.40, -1.51, -0.85, -0.54, -0.38. The levels crowd toward zero rather than spreading out, so the levels have a top and an atom has an ionisation energy. The arrow marks a transition: 3 to 2 releases 1.890 eV, a photon at 656.1 nm.
energy-levels is one function in lib/figures/atomic.js —
what happens once one is bound by the other. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
The first 6 energy levels of hydrogen, drawn to scale in eV, at -13.61, -3.40, -1.51, -0.85, -0.54, -0.38. The levels crowd toward zero rather than spreading out, so the levels have a top and an atom has an ionisation energy. The arrow marks a transition: 3 to 2 releases 1.890 eV, a photon at 656.1 nm.
The energy ladder of an oscillator
The options are the ones Half a kT for every way of moving passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The first 6 energy levels of an oscillator, drawn to scale in ħω, at 0.5, 1.5, 2.5, 3.5, 4.5, 5.5. The levels are evenly spaced, which is why an oscillator absorbs one frequency and not a series. The arrow marks a transition: 3 to 2 releases 1.000 ħω.
The energy ladder of a box
The options are the ones Half a kT for every way of moving passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The first 5 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 3 to 2 releases 5.000 E₁.
The energy ladder of a box
The options are the ones No two in the same state, and why matter has volume passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
The energy ladder of a box
The options are the ones No two in the same state, and why matter has volume passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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: 3 to 2 releases 5.000 E₁.
The energy ladder of a box
The options are the ones No two in the same state, and why matter has volume passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The first 8 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 3 to 2 releases 5.000 E₁.
What checks it
physicscheck asserts something about energy-levels that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
QuantumNo 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.
QuantumThe atom the size of a bacterium
Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.
QuantumThe atom with no nucleus
An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.
QuantumThe 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.
QuantumThe 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.
QuantumThe 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.
QuantumThe electrons that pay in fixed amounts
Fire electrons through mercury vapour and raise the voltage that drives them, and the current they carry should simply rise. In 1914 James Franck and Gustav Hertz found it rising, then collapsing, then rising again, over and over, every 4.9 volts. Below that energy an electron bounces off a mercury atom and keeps almost everything; above it, it can hand the atom exactly 4.9 electronvolts and no other amount. The atom will accept energy only in the size of the gap between its levels — shown here by collisions rather than by light, and misread by its discoverers as something else entirely.
QuantumThe 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.
QuantumThe ion that exists because its electrons keep apart
Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.
QuantumThe 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.
QuantumThe line that takes eleven million years
A hydrogen atom in its ground state can flip its electron's spin relative to the proton's and emit a photon of 21 centimetres. Left alone, one atom takes eleven million years to do it. The Galaxy is nonetheless bright at that wavelength, because there is so much hydrogen, and because the flip is so slow that the line's brightness counts atoms without caring how hot they are.
QuantumThe lines a warm vapour hides
Shine a tunable laser through a cell of rubidium vapour and the absorption is one smooth hump half a gigahertz wide, though the atoms' own lines are eighty times narrower and three of them lie inside it. The atoms are moving, each sees the light Doppler-shifted by its own speed, and their lines are smeared together. Send a second, stronger beam back the other way and narrow dips appear in the hump at exactly the frequencies of the hidden lines — because only atoms standing still can be talked to by both beams at once. The vapour stays at room temperature and the motion is not removed; it is selected against.
QuantumThe metal that is yellow because it is heavy
Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.
QuantumThe 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.
QuantumThe nucleus that would make positrons from nothing
Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.
QuantumThe 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.
QuantumThe 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.
QuantumThe X-ray line that counts the protons
Strike a metal with fast electrons and it gives off X-rays at a few sharp energies characteristic of the metal. In 1913 Henry Moseley measured the strongest of them for element after element and found that the square root of its energy rose in equal steps, one step per element — as if something inside each atom were being counted. It was the charge on the nucleus. The periodic table had been ordered by weight; Moseley's line ordered it by protons, fixed the elements it was missing, and showed that an electron deep inside a heavy atom behaves like the electron in hydrogen, with the charge turned up.
QuantumWhere 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.
QuantumWhere 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.
QuantumWhy 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.