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 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 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 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 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 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.
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.