A beam through a chain of analysers

At its defaults it draws 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.

stern-gerlach 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.

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 four ways two spin-halves can be arranged, on the left, and the four combinations of them that are eigenstates of the total spin, on the right. Two of the products — both up and both down — are already eigenstates. The other two are not: one spin up and the other down does not specify a total spin, because it does not say which spin is which, and the states that do are the sum and the difference. The eigenvalues printed beside them are computed by applying S² as a matrix in the product basis and reading the result off, then solving s(s + 1) for s: |↑↑⟩ gives 2ħ², so s = 1; (|↑↓⟩ + |↓↑⟩)/√2 gives 2ħ², so s = 1; |↓↓⟩ gives 2ħ², so s = 1; (|↑↓⟩ − |↓↑⟩)/√2 gives 0ħ², so s = 0. The column on the far right is the eigenvalue of the operator that swaps the two particles: the three states with s = 1 come back unchanged and the one with s = 0 comes back with a minus sign. That sign is the whole of the difference. It is why the three are called a triplet and the one a singlet, why they behave differently in a magnetic field, and — through the requirement that the *total* state of two electrons be antisymmetric — why the two families occupy space differently before any force between them has been mentioned.

The probability that two electrons in the same pair of orbitals are found a given distance apart, for the two ways their spatial state can be put together. There is no interaction in this calculation at all: the two electrons do not repel, do not attract and do not know about each other except through the symmetry of the state they share. The symmetric combination — which pairs with the antisymmetric spin state, the singlet — is largest at zero separation. The antisymmetric one, which pairs with the triplet, is exactly zero there, because swapping the two coordinates must change the sign of the wavefunction and a function equal to minus itself is nothing. The root mean square separation is 2.333 for the singlet and 2.512 for the triplet, in units of the orbital width. That gap is the origin of every exchange effect there is: it is not a force, it produces no term in any Hamiltonian, and it changes the energy the moment a repulsion is switched on — because two electrons that are further apart pay less for repelling each other. The hole in the middle of the triplet curve has a name, the Fermi hole, and it is the reason electrons of parallel spin behave as though they avoided one another.

The direct Coulomb integral and the exchange integral for two electrons in two overlapping orbitals, evaluated on a 220-point grid at 8 separations, on a logarithmic vertical axis. The direct integral is the ordinary repulsion between two charge clouds and falls slowly, because two clouds a long way apart still repel. The exchange integral involves the product of the two orbitals at both points, so it needs them to overlap, and it collapses as they separate — its logarithm tracks the logarithm of the overlap with a slope of 2.00, which is the statement that exchange is a two-orbital effect and direct repulsion is not. It is positive at every separation, and it has to be: the Coulomb kernel is positive definite, so the integral of a squared quantity against it cannot be negative. That sign is Hund's first rule. The triplet, whose electrons are already further apart before any repulsion is considered, lies below the singlet by twice this number — which for the 1s2s configuration of helium is 0.80 electron-volts, an energy nobody would guess was a consequence of a minus sign in front of one term of a wavefunction.

The relative strength of hydrogen's rotational lines at 300 K, computed as the nuclear-spin weight times the rotational degeneracy times the Boltzmann factor, with the rotational temperature 85.4 K. The envelope is the ordinary one — degeneracy climbing, Boltzmann factor falling, a maximum in between — but the lines do not sit on it. They alternate, odd J strong and even J weak, in a ratio that comes out exactly three when the degeneracy and the Boltzmann factor are divided back out. The three is the number of states in a triplet. The two protons are spin-halves and combine exactly as the two electrons of the first figure do; the total molecular state must change sign when they are swapped; rotating the molecule by half a turn is that swap, and it multiplies the rotational state by (−1)^J. So odd J must go with the antisymmetric-under-nothing nuclear triplet and even J with the singlet, and the spectrum counts the states for anybody with a spectrometer. It is the most direct measurement of a spin multiplicity there is, and it was made before anyone knew what it meant.

The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

physicscheck asserts something about stern-gerlach 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.