The Stern-Gerlach Experiment

Every measurement postulate in the formalism chapter — eigenvalues as outcomes, probabilities as squared amplitudes, collapse onto the answer — can feel like axiomatics handed down from nowhere. The Stern–Gerlach experiment is where all of it becomes hardware. A beam of silver atoms flies between two magnet poles machined so the field is fiercely stronger near one; each atom's magnetic moment feels a force proportional to its projection $\mu_z$ , and the screen downstream records where each atom landed. The apparatus is, literally, a device that plots the spectrum of $\hat S_z$ : one spot per eigenvalue, spot intensity per Born-rule probability. Classical physics predicted a smear — moments oriented every which way, deflections filling a band. What Frankfurt saw in 1922 was two spots with a clean gap: the projection of angular momentum is quantized. The result is doubly famous because everyone involved interpreted it wrongly for three years. The theory it was designed to test (Bohr–Sommerfeld orbits) predicted two spots for the wrong reason; the correct ingredient — electron spin, with its compensating factor $g \approx 2$ — had not been invented. The numbers conspired so neatly that a wrong theory was "confirmed" by an experiment measuring a quantity nobody knew existed. This article owns the machine: the force law and why the gradient does the work, the honest arithmetic from oven temperature to millimetres on the plate, why silver was the perfect atom, and what chained magnets reveal about incompatible questions. The spin that the two spots forced into physics has its own article just upstream.

Key takeaways

An inhomogeneous field exerts $F_z = \mu_z\,\partial B_z/\partial z$ on a neutral atom: deflection is proportional to the moment's projection, so the screen photographs the spectrum of $\hat S_z$ .
A uniform field exerts zero net force — only a torque, hence precession. The gradient is not a detail; it is the measurement.
Silver's single $5s$ valence electron has $\ell = 0$ : the whole moment is spin, and the answer is unambiguously two beams, $m_s = \pm\tfrac12$ .
A spin- $j$ atom splits into $2j+1$ beams — the apparatus counts the dimension of the multiplet.
The 1922 result "confirmed" the old quantum theory by coincidence: $g_s m_s = \pm1$ mimics the Bohr–Sommerfeld $\pm1$ exactly.
Chained SG magnets make operator incompatibility countable: select $z^+$ , measure $x$ , re-measure $z$ — the forbidden $z^-$ beam returns at $1/4$ .
Key idea: a field gradient converts the discrete eigenvalues of a projection operator into discrete spatial deflections — the screen literally photographs the spectrum of $S_z$ .

The core insight

A field gradient converts the discrete eigenvalues of a spin (or angular-momentum) projection operator into discrete spatial deflections, so the screen literally photographs the spectrum of $S_z$ .

Where this lives today

Every projective measurement in quantum information is this machine in disguise. A superconducting qubit read out through a dispersive cavity, an ion's state-dependent fluorescence, a photon polarizer — all implement "couple a two-valued observable to a spatial (or spectral) pointer, then record which branch": the Stern–Gerlach template. When textbooks (Sakurai famously) open quantum mechanics with SG magnets rather than wavefunctions, they are using the apparatus as what it has become — the minimal, complete model of preparation, measurement, and incompatibility. The hardware lineage is equally alive. Modern atom interferometers use optical "Stern–Gerlach" beam splitters — state-dependent momentum kicks — to split and recombine matter waves for gravimetry and tests of the equivalence principle; molecular-beam resonance, the direct descendant Stern's student Rabi built, became NMR, atomic clocks, and MRI. The 1922 magnet, designed to falsify quantization, ended up as the ancestor of the machines that run on it.

The physical picture

Think of each atom as a tiny bar magnet flying through the gap. In a uniform field the two poles of that magnet feel equal and opposite forces: zero net push. The moment precesses about the field — a torque — but the centre of mass flies straight. To deflect the atom you must make one pole feel a stronger field than the other, which is exactly what the knife-edge-over-groove pole geometry manufactures: a large $\partial B_z/\partial z$ across the atom.

Figure

The machine — a gradient turns a projection into a force

Now run the classical movie: atoms leave the oven with moments pointing every which way, $\mu_z$ takes every value in $[-\mu, +\mu]$ , and the plate collects a continuous vertical band. The quantum movie disagrees. The moment is tied to angular momentum, whose projection is quantized — for silver's lone valence electron, two values only. Two forces, two arrival zones, and a gap in the middle where classical physics put its maximum.

Figure

What the plate says — classical band versus quantum multiplets

One more classical worry, and its quantum answer. During transit the moment is precessing around the field (the Larmor cone of Spin Angular Momentum), so shouldn't the transverse components $\mu_x, \mu_y$ also push the atom around and blur the spots? They average out: the precession is fantastically fast compared to the transit — millions of full cones between the pole faces — so any transverse force reverses sign millions of times and cancels, while $\mu_z$ , conserved by the precession itself, delivers its steady vertical push. The apparatus is self-cleaning: it measures exactly the component along its own axis and nothing else. And why silver? Because its ground state is $^2S_{1/2}$ : a closed-shell core plus one $5s$ valence electron with $\ell = 0$ . No orbital moment muddies the water — the entire deflection is electron spin, and the answer is unambiguously two. (The nuclear moment is a thousandfold smaller and invisible at this resolution.) Hydrogen, with the same structure, repeated the result in 1927; the choice of atom is half the experiment.

The force law

The interaction energy of a magnetic moment in a field is the standard dipole expression; with the field essentially along $z$ in the gap, only one component matters. From energy to force is one derivative:

U = -\boldsymbol{\mu}\cdot\mathbf{B} \approx -\mu_z B_z(z), \qquad F_z = -\frac{\partial U}{\partial z} = \mu_z\,\frac{\partial B_z}{\partial z} \qquad (1)

where: $\mu_z$ — the projection of the atom's moment on the field axis; $\partial B_z/\partial z$ — the gradient the pole shape manufactures, typically $10^2$ – $10^3\ \mathrm{T/m}$ . The moment is an internal property, constant on the atom's scale, so the derivative hits only the field. Set the gradient to zero and $F_z$ vanishes identically — a uniform field, however strong, splits nothing. Quantization enters through the moment–spin relation established at Spin Angular Momentum and catalogued at Magnetic Moment and the Gyromagnetic Ratio:

\mu_z = -g\,\mu_B\,\frac{S_z}{\hbar} = -g\,\mu_B\,m_s = \mp\mu_B \quad (g \approx 2,\ m_s = \pm\tfrac12) \qquad (2)

where: $\mu_B = 9.274\times10^{-24}\ \mathrm{J/T}$ — the Bohr magneton. For spin- $\tfrac12$ the force takes exactly two values, equal and opposite; for general $j$ it takes $2j+1$ values and the beam fans into that many branches. Reading the count off the plate is reading the multiplet's dimension.

Derivation AFrom oven to plate — the deflection formula

The goal is the full instrument equation: how many millimetres of splitting a given oven temperature, magnet geometry, and field gradient buy you. The strategy is projectile kinematics with the magnetic force playing gravity's role — constant transverse force during transit, straight flight after — assembled so that every experimental knob appears explicitly. Watch where the beam speed enters: squared, in the denominator, which is why hot fast beams are hard to split and why the oven temperature is not a footnote but a design parameter.

Step 1. Force to acceleration. Inside the gap each atom of mass $M$ feels $F_z = \mu_z\,\partial_z B_z$ , so $a = \mu_z\,\partial_z B_z / M$ , constant along the magnet. Why: the gradient region is engineered to be uniform along the flight path, so the kinematics is the constant-acceleration kind.

Step 2. Time in the field. A magnet of length $L$ traversed at speed $v$ takes $t = L/v$ . Why: longitudinal motion is untouched — the force is transverse, so $v$ is constant through the gap.

Step 3. Deflection at the exit. Standard kinematics:

\Delta z = \frac12 a t^2 = \frac{\mu_z}{2M}\,\frac{\partial B_z}{\partial z}\,\frac{L^2}{v^2}.

Why: each eigenvalue of $\mu_z$ produces its own $\Delta z$ — the spectrum becomes a set of arrival heights. The two spin- $\tfrac12$ beams sit $2\Delta z$ apart (drift space after the magnet widens this further; the gap structure is set here).

Step 4. Read the knobs. Splitting grows linearly with gradient and with $L^2$ , and falls as $1/v^2$ — halve the beam speed and the split quadruples. Inverting the formula turns the instrument into a moment-meter: measure $2\Delta z$ with known $M, L, v, \partial_z B_z$ and you have measured $\mu_z$ — this is precisely how the electron's $\mu_z \approx \mu_B$ , hence $g \approx 2$ , was pinned down. Why: the same equation run forward predicts spots; run backward it weighs a quantum moment with a ruler. *Verified: the closed form equals the $\tfrac12at^2$ chain symbolically (sympy); the worked numbers below reproduce it; scaling checks (gradient linear, $1/v^2$ ) machine-verified.*

The right result for the wrong reason

The experiment was designed in 1921 to referee a specific fight. Bohr–Sommerfeld theory said the valence electron's orbit is space-quantized with $\ell_z = \pm\hbar$ ( $m = 0$ was excluded by the old theory as "forbidden collisions with the nucleus") — two orientations, two spots. Classical physics said no quantization — a smear. Stern, an ex-theorist who thought space quantization was nonsense, built the experiment expecting to kill it; the plate showed two spots, and a triumphant postcard went to Bohr: congratulations on the confirmation of your theory. Every word of the theoretical reading was wrong. The silver electron has $\ell = 0$ : there is no orbital moment, no orbit to orient, and the Bohr–Sommerfeld picture predicts no splitting at all for the true ground state. The actual splitter was spin — undiscovered until 1925 — with moment $\mu_z = -g_s\mu_B m_s$ . And here fate played its joke: the old prediction was $\pm1$ in units of $\mu_B$ (from $g = 1$ , $m = \pm1$ ), while the truth is $g_s m_s = 2\times(\pm\tfrac12) = \pm1$ — numerically identical. The factor-of-two error in $m$ and the factor-of-two anomaly in $g$ cancelled exactly. It took Fraser's 1927 observation that silver, hydrogen and sodium ground states all have $\ell = 0$ to expose the coincidence, by which time spin and the Pauli formalism could claim the two spots as their own. Space quantization itself — the naming of which traces to Sommerfeld's 1916 theory, as told at Eigenfunctions of L_z and the Azimuthal Quantization — survived: the plate really does show a projection taking discrete values. Only the identity of the projection changed.

Chained magnets: incompatibility you can count

A Stern–Gerlach magnet does more than detect: block one exit and it prepares a pure spin state. That turns a second magnet into a question asked of a known state — and lets the non-commutation of spin components, $[\hat S_x, \hat S_z] = -i\hbar\hat S_y \neq 0$ , produce countable beam fractions.

Go deeper · Derivation B · 2 minThe cascade arithmetic — what a sideways question erases

Derivation BThe cascade arithmetic — what a sideways question erases

The goal is the beam-fraction bookkeeping for the canonical chain: select $z^+$ , measure along $x$ , select $x^+$ , measure along $z$ again — and, in the general case, analyzers at any relative angle $\theta$ . The strategy is pure state expansion: after each selection the beam is in a known eigenstate, and the next magnet's outcome probabilities are squared overlaps with its eigenbasis. The collapse bookkeeping (Lüders update) is consumed from its owner; the overlap law is consumed from the spinor article. Watch what the middle magnet does to information the first magnet had established.

Step 1. Prepare. After the first $z$ -magnet, keeping the upper beam: state $|{+z}\rangle$ , and the $z^-$ port verified dark — repetition of the same question returns "up" with certainty. Why: an SG magnet with one port blocked is a filter — this is state preparation, the laboratory meaning of "eigenstate".

Step 2. Ask sideways. In the $x$ -basis, $|{+z}\rangle = \tfrac{1}{\sqrt2}(|{+x}\rangle + |{-x}\rangle)$ , so the $x$ -magnet splits the pure beam 50/50. Keep $+x$ : the surviving half is now $|{+x}\rangle$ , and its memory of "up along $z$ " is gone. Why: the state after selection is the new eigenstate, full stop — the update rule and its degenerate-safe form are owned by "Measurement and Wavefunction Collapse".

Step 3. Ask $z$ again. $|{+x}\rangle = \tfrac{1}{\sqrt2}(|{+z}\rangle + |{-z}\rangle)$ : the final magnet splits 50/50, so the chain delivers $\tfrac12\times\tfrac12 = \tfrac14$ of the original beam into the $z^-$ port — a port that was dark before the middle magnet was inserted. The companion numbers ( $\tfrac12$ with the middle magnet's outputs recombined unread, $0$ with it removed) are worked at Measurement and Wavefunction Collapse; all three are cited here, not re-derived. Why: inserting a measurement changes the physics even when its result is discarded — outcomes are not pre-existing labels the magnets passively read.

Step 4. Generalize the angle. For analyzers $\theta$ apart the overlap law is $P(+\hat n \mid +z) = \cos^2(\theta/2)$ — the half-angle overlap derived at Spinors and Two-Component Wavefunctions and consumed here. The perpendicular case $\theta = 90°$ gives the $\tfrac12$ used above; small angles pass almost everything — the quantum version of "nearly the same question." Why: the half-angle is spin-1/2's signature — perpendicular axes are not "opposite" questions but maximally undecided ones. *Verified: all fractions recomputed by matrix algebra (eigenvectors of n̂·σ): $P(+x|+z)=\tfrac12$ , chain $=\tfrac14$ exactly, $\cos^2(\theta/2)$ law reproduced at arbitrary θ, unread variant $\tfrac12$ , removed variant $0$ .*

Limiting cases

Case	Result	Meaning
spin- $\tfrac12$ (Ag, H ground state)	$N = 2j+1 = 2$	two symmetric spots — the historic result
general spin- $j$ atom	$N = 2j+1$	odd or even beam count reveals integer vs half-integer $j$ directly
classical dipole	$\mu_z \in [-\mu, +\mu]$ continuous	one broadened band — decisively excluded in 1922
uniform field	$F_z = \mu_z\,\partial_z B_z = 0$	no splitting at all; torque and precession only
analyzers at angle $\theta$	$P = \cos^2(\theta/2)$	repeated question ( $\theta=0$ ): certainty; perpendicular: coin toss

Worked examples

1 · From oven to millimetres (confidence builder)

Silver atoms ( $M = 1.79\times10^{-25}\ \mathrm{kg}$ ) leave a $1000\ \mathrm{K}$ oven. Take the beam speed as the beam-weighted most-probable value $v = \sqrt{3k_BT/M} \approx 480\ \mathrm{m/s}$ . They cross a $3.5\ \mathrm{cm}$ magnet with $\partial B_z/\partial z = 1400\ \mathrm{T/m}$ , with $\mu_z = \mu_B = 9.27\times10^{-24}\ \mathrm{J/T}$ ( $g_s/2 \approx 1$ ). Find the spot separation at the magnet exit — and then run the chain backwards as a $g$ -measurement.

Step 1. Speed check: $v = \sqrt{3\times(1.381\times10^{-23})\times1000/1.79\times10^{-25}} = 481\ \mathrm{m/s}$ — use $480\ \mathrm{m/s}$ . (State your speed convention: oven beams are flux-weighted, so $\sqrt{3k_BT/M}$ , not the in-oven most-probable $\sqrt{2k_BT/M}$ .)

Step 2. Force and acceleration: $F = \mu_B\,\partial_z B_z = (9.27\times10^{-24})(1400) = 1.30\times10^{-20}\ \mathrm{N}$ , so $a = F/M = 7.2\times10^{4}\ \mathrm{m/s^2}$ — up for $m_s = -\tfrac12$ , down for $+\tfrac12$ (electron moment anti-parallel to spin).

Step 3. Transit time: $t = L/v = 0.035/480 = 7.3\times10^{-5}\ \mathrm{s}$ .

Step 4. Deflection: $\Delta z = \tfrac12 a t^2 = \tfrac12(7.2\times10^4)(7.3\times10^{-5})^2 = 0.19\ \mathrm{mm}$ per beam, so the spots sit $2\Delta z = \mathbf{0.39\ mm}$ apart — small, but a clean split (not a smear) resolvable on a 1922 photographic plate.

Step 5. Run it backwards. Measure $2\Delta z = 0.39\ \mathrm{mm}$ with the geometry known and invert Derivation A: $\mu_z = 2\Delta z\,M v^2 / (\partial_z B_z\,L^2) = 9.3\times10^{-24}\ \mathrm{J/T} \approx \mu_B$ . With $\mu_z = (g/2)\mu_B$ this is a direct measurement of $\mathbf{g \approx 2}$ — a ruler measuring a QED coupling.

Step 6. Precession sanity check: in the $\sim0.1\ \mathrm{T}$ gap field the electron moment precesses at $\sim2.8\ \mathrm{GHz}$ , completing $\sim2\times10^{5}$ cones during the $73\ \mathrm{\mu s}$ transit — the transverse force averages to zero and only $\mu_z$ survives, as claimed.

2 · The 45° ratchet (exam level)

A beam is prepared in $|{+z}\rangle$ . (a) It passes an analyzer along $\hat n$ at $\theta = 45°$ ; what fraction exits the $+$ port? (b) That output passes a second analyzer at $90°$ from $\hat z$ (i.e. another $45°$ step); what fraction of the original survives both? (c) Compare with a single direct $90°$ analyzer.

Step 1. One step: $P_1 = \cos^2(22.5°) = 0.854$ — the half-angle law from the cascade derivation.

Step 2. Second step: the state is now $|{+\hat n}\rangle$ ; the next analyzer sits $45°$ from it, so again $P_2 = \cos^2(22.5°) = 0.854$ . Joint survival: $P_1P_2 = \mathbf{0.729}$ .

Step 3. Direct jump: one analyzer at $90°$ passes $\cos^2(45°) = \mathbf{0.500}$ .

Step 4. Read it: turning a spin through $90°$ in two gentle questions keeps $73\%$ of the beam; one blunt question keeps $50\%$ . Slice the turn into ever more steps and the survival approaches $100\%$ — measurement can steer a state, the seed idea of the quantum Zeno effect (its short-time law lives at Quantum Zeno Effect).

Origins · 1 minStern-Gerlach Experiment and how this was discovered

Stern-Gerlach Experiment

1922 · Frankfurt am Main

The experiment was a bet against its own result. Otto Stern — a theorist by training who had sworn to leave physics if "this nonsense of space quantization" proved real — conceived the test in 1921 precisely because Bohr's directional quantization seemed absurd enough to kill with one photograph. Walther Gerlach, the gifted experimentalist next door, built it: a silver oven at a thousand kelvin, slits collimating a hair-thin beam, a knife-edge magnet delivering a thousand tesla per metre, all inside a vacuum that failed for months. The money ran out (Einstein helped raise more; so did a Goldman banker's transatlantic cheque). The decisive plate at first showed nothing — until, so the story goes, the sulfur in Stern's cheap cigars turned the invisible silver film into visible black silver sulfide as the two men breathed over it. Gerlach mailed Bohr the two-lobed photograph with congratulations "on the confirmation of your theory" — a theory that was wrong, confirmed by a coincidence in the arithmetic of a quantity, spin, that would not be proposed for another three years. Stern, who never accepted the cigar legend himself, went on to measure the proton's anomalous moment (another theory-killing surprise) and received the 1943 Nobel Prize; Gerlach stayed in Germany to lead its wartime physics establishment. The magnet's descendants are everywhere: every projective qubit measurement is a Stern–Gerlach experiment in new clothes.

History

1921

— Stern proposes the space-quantization test; Gerlach signs on to build it

1922

— the split silver beam is photographed in Frankfurt; the postcard goes to Bohr

1925

— Uhlenbeck and Goudsmit propose electron spin — the true owner of the two spots

1927

— Phipps and Taylor repeat the experiment with hydrogen; Fraser notes the ground states are all ℓ = 0, exposing the happy coincidence

1933

— Stern's group measures the proton magnetic moment, finding it ~2.8× the Dirac value — structure inside the proton

Going further

Texts. Sakurai & Napolitano, Modern Quantum Mechanics, §1.1 (the canonical SG opening) · Cohen-Tannoudji, Diu & Laloë, Vol. 1, Ch. IV · Griffiths & Schroeter, §4.4 · Feynman Lectures Vol. III, Ch. 5–6 (the SG filter formalism).

Provenance. Equations and numbers are standard textbook values. A verifier should check: $\mu_B=e\hbar/2m_e=9.274\times10^{-24}\ \mathrm{J\,T^{-1}}$ (CODATA); electron $g\approx2.0023$ ; the force law $F_z=\mu_z\partial_z B_z$ and deflection $\Delta z=(\mu_B/2M)(\partial_z B_z)(L^2/v^2)$ against Sakurai §1.1 and Cohen-Tannoudji Ch. IV; the cascaded-SG probabilities $\cos^2(\theta/2)$ / $\sin^2(\theta/2)$ and the $\tfrac14$ regeneration result against Feynman Vol. III and Sakurai. Silver ground term $^2S_{1/2}$ from standard atomic data. Historical attribution: Stern & Gerlach (1922); spin proposed by Goudsmit & Uhlenbeck (1925). Edition note (2026-07-08). The oven-to-millimetres worked example was relocated here from the v3 spine folded into "Spin Angular Momentum" (QUANT.ANGM.SPIN.0918) — this article owns the experiment's machinery — with every number re-verified (beam speed stated explicitly as the flux-weighted $\sqrt{3k_BT/M}$ at 1000 K; chain reproduces 0.39 mm). The three-magnet cascade from the same spine is absorbed into the cascade derivation, with the Lüders bookkeeping ( $1/4$ , $1/2$ unread, $0$ removed) cited to Measurement and Wavefunction Collapse and the $\cos^2(\theta/2)$ overlap law to Spinors and Two-Component Wavefunctions — neither is re-derived here.

Prerequisites. Spin angular momentum · magnetic moment and the gyromagnetic ratio · angular momentum commutation relations. Leads to. Spin-1/2 and the Pauli matrices · spinors and two-component wavefunctions · rotation operators and SU(2).

If the atom's moment is precessing the whole way through, why doesn't the precession smear the spots?

Because of the timescale gap. In the gap's tesla-scale field the electron moment precesses at gigahertz rates — hundreds of thousands of full cones during the tens-of-microseconds transit. The transverse components $\mu_x, \mu_y$ reverse sign every half-cone, so their forces integrate to essentially zero; $\mu_z$ is a constant of the precession and pushes steadily. The magnet effectively measures the time-averaged moment, which is purely the $z$ -projection — the device is self-aligned to its own axis.

Why use neutral atoms instead of free electrons — wouldn't electrons show spin more directly?

A free electron in a magnetic field feels the Lorentz force $q\mathbf{v}\times\mathbf{B}$ , which utterly dominates the tiny gradient force on its moment; worse, Bohr and Pauli argued the beam's transverse spread makes the spin splitting fundamentally unresolvable for charged particles in this geometry. A neutral atom has zero Lorentz force but carries the electron's full moment — silver is a delivery vehicle for one unpaired spin. This is also why the experiment measures the electron's moment despite deflecting a whole atom: the core electrons pair off, the nuclear moment is ~2000× smaller.

Does the magnet measure the spin, or create the value it reports?

The safest statement: it correlates a spin component with position and delivers one definite branch. "Measure" is right in that repetition confirms the result (a $z^+$ -selected beam answers $z^+$ forever after). "Create" is right in that the value did not exist beforehand for a beam prepared along $x$ — the cascade proves no hidden $z$ -label could have been riding along ( $1/4$ regenerated where $0$ was guaranteed). The update bookkeeping that makes this precise is owned by Measurement and Wavefunction Collapse.

What happens between the poles to an atom in a superposition — does it take both paths?

The state evolves into an entangled object: spin-up correlated with the upper trajectory, spin-down with the lower — one atom, two correlated branches of a spin-position wavefunction (the two-component object of Spinors and Two-Component Wavefunctions). If you recombine the branches coherently before any irreversible record forms, interference is recoverable in principle (a "quantum eraser" geometry); once the atom hits the plate, one branch is realized with the Born-rule weight. The screen, not the magnet, is where the bookkeeping ends.

Why did the 1922 plate show two lobes touching at the ends rather than two clean dots?

Geometry, not physics. The gradient is strongest under the knife edge and weakens off-axis, and the beam had finite width and a thermal spread of speeds ( $\Delta z \propto 1/v^2$ smears each branch along $z$ ). The historical image is a lip-shaped pair of lobes that merge at the low-gradient edges — still unmistakably two, with a maximum-free centre where classical physics demanded a peak. Modern replays with velocity-selected beams show textbook-clean doublets.

Is the two-spot pattern proof of spin by itself?

No — that is the whole irony. Two spots prove only that the relevant projection takes two values, i.e. $2j+1 = 2$ . In 1922 the pattern was mis-attributed to orbital space quantization; only the 1925–27 sequence (spin proposed, hydrogen repeating the result, the $\ell = 0$ realization) identified the projection as spin with $g \approx 2$ . The lesson generalizes: an experiment pins down structure — here the dimension of a multiplet — while theory assigns the structure an owner, and the owner can change.

Could a Stern–Gerlach magnet separate the singlet and triplet states of a two-electron atom?

It sorts by $m$ , not by multiplet. A triplet ( $S=1$ ) atom splits into three beams ( $m_s = 0, \pm1$ ) and a singlet ( $S=0$ ) passes undeflected — so it can distinguish singlet from triplet by the beam count, and the $m_s = 0$ triplet branch overlaps the singlet's position. Helium's metastable triplet beams were separated exactly this way in early molecular-beam work. The construction of those singlet and triplet states themselves belongs to the addition-of-angular-momenta articles.