Experiment: Stern–Gerlach

Few experiments in the history of physics have delivered so unambiguous a verdict with so modest an apparatus. In the winter of 1921–1922, in a cramped laboratory of the Institute for Theoretical Physics in Frankfurt am Main, Otto Stern and Walther Gerlach sent a beam of neutral silver atoms through a strongly inhomogeneous magnetic field and found that the beam split cleanly into two — not into the continuous fan that every classical intuition demanded. The experiment made space quantization visible on a glass plate: the projection of an atomic magnetic moment onto a field axis takes only discrete values. That the discreteness was at first attributed to the wrong angular momentum — orbital rather than spin — takes nothing away from the observation itself, which stands to this day as the prototype of a quantum measurement. This chapter presents the experiment in full phenomenological detail; the quantum-mechanical derivations it calls for are developed in Angular Momentum and Spin and are marked pending here.

Historical context, 1921–1922

By 1921 the old quantum theory was an uneasy patchwork of classical mechanics and quantization rules. Bohr's planetary model of 1913 had been generalized by Sommerfeld, who quantized not only the size and shape of electron orbits but also their orientation: in the presence of a magnetic field defining an axis, the component of the orbital angular momentum along that axis was restricted to a discrete set of values, a doctrine Sommerfeld called Richtungsquantelung — direction quantization, or space quantization [Sommerfeld:1916]. The rule accounted economically for the normal Zeeman splitting of spectral lines, but it carried an outrageous corollary: an atom entering a magnetic field must instantaneously find its angular momentum pointing only along the quantized directions, however the atom was oriented beforehand. To most physicists trained in the classical tradition this seemed a bookkeeping fiction, useful for spectroscopy but surely not a statement about the actual disposition of atoms in space.

Otto Stern, then at Frankfurt, was among the sceptics — and he possessed the one instrument capable of settling the matter. In the molecular-beam method he had been developing since 1919, atoms fly through high vacuum in nearly collision-free straight lines, so that a force acting on a single atom is recorded directly as a deflection of the beam. In 1921 Stern published the proposal in full: pass a beam of neutral atoms through an inhomogeneous magnetic field; if the atomic magnetic moments are oriented at random, as classical mechanics holds, the beam broadens into a continuous smear, whereas if space quantization is real the beam must split into discrete components [Stern:1921]. The experiment was conceived as a decisive test between two sharply distinct predictions — and, by Stern's own account, he expected and hoped to refute the quantum prediction. The documented recollections collected by Friedrich and Herschbach preserve the flavour of the wager: Stern is reported to have vowed to quit physics if Bohr's “nonsense” of spatial quantization turned out to be correct, and even sympathetic colleagues regarded the quantized orientations as a computational timetable rather than physical reality [Friedrich:2003]. It should be remembered that both sides of the wager were arguing within the old quantum theory: the modern resolution — electron spin — was not on anyone's table.

The circumstances were as unfavourable as the idea was sharp. Postwar Germany was sliding into inflation, and the Frankfurt institute had almost no money for equipment. The experiment survived on improvised patronage: Max Born, Stern's chief, raised funds by charging admission to popular lectures on relativity; private donations arrived from abroad; and a grant arranged through the committee of Einstein's Kaiser Wilhelm Institute for Physics — which Einstein then directed, largely as a fund-dispensing body — helped pay for the electromagnet [Friedrich:2003]. Gerlach, an accomplished experimentalist who had joined Frankfurt in 1920, carried the brunt of the laboratory work through repeated failures of vacuum, alignment, and detection, with Stern contributing the crucial design insight that the field gradient, not the field strength, produces the deflecting force.

One further story belongs here, explicitly flagged as anecdote. Stern recounted, decades later, that when he and Gerlach first examined the detector plate they saw nothing — until, as they peered at it, the silver trace slowly blackened before their eyes. Stern's explanation was that the sulfurous smoke of their cheap cigars (professorial salaries being too thin for good ones) had converted the invisibly thin silver deposit into visible black silver sulfide, in the manner of a photographic developer. The tale is charming and was told by Stern himself, but its literal role in the discovery is uncertain: the plates were in any case developed chemically as a matter of routine, and Friedrich and Herschbach, who traced the story and even re-enacted the chemistry, present it as a delightful embellishment rather than an established causal link in the chain of discovery [Friedrich:2003]. We repeat it in that spirit.

Stern–Gerlach (1921–1922)

The experiment ran in Frankfurt from 1921 to the decisive plates of February 1922. Its logic is worth stating before its hardware: a neutral atom carries no net charge, so the Lorentz force cannot deflect it; but if the atom carries a magnetic moment \(\vect{\mu}\), an inhomogeneous field exerts a net translational force on it. The deflection of a beam of neutral atoms therefore reads off the distribution of magnetic-moment components along the field gradient — a direct image, atom by atom, of the quantity that space quantization declares to be discrete.

Apparatus

An electrically heated oven evaporates metallic silver at a temperature of about \(1300\,\mathrm{K}\); atoms effuse from a small aperture into high vacuum, which must be maintained continuously over runs lasting several hours (the working pressures were of the order of \(10^{-3}\,\mathrm{Pa}\); a single cracked gasket could destroy a night's run). Two collimating slits, a few hundredths of a millimetre wide, select from the effusing vapour a thin ribbon-shaped beam. The beam then passes along the edge of a deflecting electromagnet whose pole pieces have the now-famous asymmetric profile — one pole machined to a sharp knife edge, the facing pole carrying a channel-shaped groove — so that the field between them, of magnitude on the scale of \(B \approx 0.1\,\mathrm{T}\) near the beam, varies strongly across the gap, with a vertical gradient of order

sustained over a beam path through the field of about \(3.5\,\mathrm{cm}\). Detection is cumulative: the silver atoms condense on a cold glass plate placed beyond the magnet, and the invisibly thin deposit built up over hours is developed photographically to render the trace visible [Gerlach:1922a] [Gerlach:1922b].

Procedure

The oven temperature fixes the velocity distribution of the beam: silver atoms at \(1300\,\mathrm{K}\) leave the source with a broad thermal spread of speeds whose beam-averaged value is of order \(660\,\mathrm{m}/\mathrm{s}\) — a kinetic-theory estimate, its precise value depending on how the effusive flux weights the Maxwell distribution. With the magnet off, a run records the undeflected beam profile, a single narrow line reproducing the collimator geometry. With the magnet energized, the atoms traverse the gradient region and each acquires a transverse deflection determined by the component \(\mu_z\) of its magnetic moment along the gradient. The classical expectation is unequivocal: magnetic moments emerging from a hot oven point in random directions, and in the field they precess about the field axis at the Larmor frequency, which preserves \(\mu_z\); the ensemble therefore presents a continuum of \(\mu_z\) values from \(-\abs{\vect{\mu}}\) to \(+\abs{\vect{\mu}}\), and the recorded trace should be a single trace, broadened into a smear that is densest at the centre and fades toward maximal deflection. Any discrete multiplicity of traces is irreconcilable with a classical moment distribution.

Observations and data

The developed plates of February 1922 showed the beam split into two discrete traces, with no detectable intensity at the undeflected centre where the classical smear would peak. The pattern has the shape of a pair of lips: at the centre of the beam ribbon, where the gradient is strongest near the knife edge, the two traces reach their maximal separation of about \(0.2\,\mathrm{mm}\), and they close toward the ends of the ribbon where the field is more nearly uniform [Gerlach:1922a]. On 8 February 1922 Gerlach mailed a postcard to Niels Bohr carrying a photograph of the split trace and congratulations on the confirmation of his theory — an image that has become one of the icons of quantum physics [Friedrich:2003].

The deflection is governed by the force on a magnetic dipole in an inhomogeneous field, which for the geometry of the experiment reduces to

Taking Equation (80.2) together with the measured geometry, gradient, and thermal velocity, the observed splitting yields the magnitude of the silver atom's magnetic moment. As an order-of-magnitude check: a moment of one Bohr magneton in the gradient Equation (80.1) feels a force \(F_z \approx 9\times 10^{-21}\,\mathrm{N}\), which acting on a silver atom (mass about \(1.8\times 10^{-25}\,\mathrm{kg}\)) moving at \(660\,\mathrm{m}/\mathrm{s}\) over \(3.5\,\mathrm{cm}\) produces a deflection of order a tenth of a millimetre — consistent with the observed trace separation. Gerlach and Stern's quantitative analysis, published later in 1922, concluded that the moment equals one Bohr magneton within their experimental uncertainty of about ten percent [Gerlach:1922b]; in modern units,

by the current CODATA adjustment [Tiesinga:2021].

Interpretation

The immediate conclusion was announced in the title of the discovery paper: “experimental proof of direction quantization in the magnetic field” [Gerlach:1922a]. The projection of the atomic magnetic moment onto the field axis is not continuously distributed; it takes exactly two values, \(\mu_z = \pm\mu_{\mathrm{B}}\), for silver. Space quantization is a fact of nature, and Stern lost his wager.

Yet the 1922 reading of that fact was wrong in nearly every detail. Within the Bohr–Sommerfeld theory the splitting was ascribed to an orbital angular momentum of one unit, whose projection was held (on Bohr's argument) to take the values \(\pm\hbar\) with the value zero forbidden — hence two traces. The modern account is entirely different: the ground state of silver is \(^2S_{1/2}\); its single valence electron has orbital angular momentum \(\ell = 0\) and hence no orbital magnetic moment at all. What the plates recorded is the intrinsic spin of that electron, \(s = 1/2\), whose projection \(S_z = \pm\hbar/2\) couples to the field through a gyromagnetic factor \(g_s \approx 2\), so that \(\mu_z = \mp g_s \mu_{\mathrm{B}}/2 \approx \mp\mu_{\mathrm{B}}\). The numerical coincidence \(g_s s \approx 1\) made the spin-\(1/2\) doublet mimic exactly the (doubly erroneous) old-quantum-theory prediction, and so the experiment was celebrated for years as confirming a model that could not have produced its result. Spin was proposed only in 1925 and the Stern–Gerlach pattern correctly reinterpreted in its terms by 1927 [Friedrich:2003] [Sakurai:2017]. The observation outlived its first explanation — a pattern this treatise will meet again.

Primary references

[Stern:1921], the 1921 proposal; [Gerlach:1922a] [Gerlach:1922b], the observation.

The phenomenon: space quantization

Stripped of its 1922 vocabulary, the experimental fact is the following.

Two features of Phenomenon 80.1 deserve emphasis, because they carry the whole weight of the quantum theory that Angular Momentum and Spin builds. First, the discreteness is two-fold. Any orbital angular momentum \(\ell\) yields an odd number \(2\ell + 1\) of projections; an even multiplicity cannot arise from orbital motion at all and forces a half-integer angular momentum, \(s = 1/2\), of no classical provenance. Second, the discreteness holds for any choice of axis. The magnet's orientation is at the experimenter's disposal, yet along whatever direction one measures, the answer is \(\pm\hbar/2\) — never a continuum, never a projection factor \(\cos\theta\). No classical vector quantity can behave this way, and the resolution — that the measured component is an operator with a two-point spectrum, not a pre-existing number — is the substance of the angular-momentum algebra developed in Angular Momentum and Spin and of the measurement postulates of The Postulates of Quantum Mechanics.

Sequential Stern–Gerlach experiments

The full quantum-mechanical lesson of the apparatus emerges only when several Stern–Gerlach magnets are chained in series, with beam blocks used to select one output of each stage as the input of the next. In this form the experiment is best known as the opening thought experiment of modern quantum-mechanics pedagogy [Sakurai:2017] [CohenTannoudji:1977] — but it is a gedanken experiment made real: each stage is precisely the apparatus of Section 80.2, and versions of the cascade have been realized with atomic and neutron beams. We write \(\mathrm{SG}\hat{z}\) for a Stern–Gerlach magnet whose gradient axis is \(\hat{z}\), and label its two output beams by the measured spin component \(S_z = \pm\hbar/2\).

Three arrangements suffice to dismantle classical expectations. In the first, an \(\mathrm{SG}\hat{z}\) stage selects the \(S_z = +\hbar/2\) beam and feeds it to a second \(\mathrm{SG}\hat{z}\) stage: only the \(+\hbar/2\) output fires. Measurement is repeatable; the first stage has prepared a state that the second confirms with certainty. In the second arrangement the selected \(S_z = +\hbar/2\) beam enters an \(\mathrm{SG}\hat{x}\) stage: both outputs fire, with equal intensities. A definite value of \(S_z\) evidently does not carry with it any definite value of \(S_x\). The third arrangement is the decisive one: the \(S_x = +\hbar/2\) output of the second stage is passed to a final \(\mathrm{SG}\hat{z}\) magnet — and both \(S_z\) outputs fire, again with equal intensities. The beam entering the \(\mathrm{SG}\hat{x}\) stage contained, by construction, no \(S_z = -\hbar/2\) component whatsoever; the intervening measurement of \(S_x\) has destroyed the previously established value of \(S_z\). Table 80.1 summarizes the three cases.

In the language of the postulates of The Postulates of Quantum Mechanics, each magnet performs a projective measurement of one spin component; the transmitted beam is the corresponding eigenstate; and the probabilities in Table 80.1 are Born-rule overlaps between eigenstates of \(S_z\) and \(S_x\). The impossibility of jointly sharp values of \(S_z\) and \(S_x\) is the physical face of their non-commutativity,

so that no filtering protocol, however elaborate, can produce a beam that passes both an \(\mathrm{SG}\hat{z}\) and an \(\mathrm{SG}\hat{x}\) test with certainty [Sakurai:2017] [CohenTannoudji:1977].

Legacy

The Stern–Gerlach experiment founded a lineage of beam techniques whose descendants populate the rest of this treatise. Stern's Hamburg laboratory turned the molecular-beam method into precision physics: diffraction of helium and hydrogen beams from crystal surfaces confirmed the de Broglie wavelength for whole atoms, and beam deflection measurements yielded the proton's magnetic moment — anomalously large, a first hint of nucleon substructure. Rabi's molecular-beam magnetic-resonance method, built directly on Stern–Gerlach state selection, led through nuclear magnetic resonance to the atomic clock and the maser; a Stern–Gerlach magnet sorting state populations sat at the heart of the first ammonia maser. The historical thread from the Frankfurt magnet to these developments is traced in detail by Friedrich and Herschbach [Friedrich:2003].

The experiment has also been repeated and refined in modern settings — longitudinal and matter-wave-interferometric variants with neutrons and cold atoms among them — always with the same two-valued verdict; and its conceptual role has, if anything, grown. A Stern–Gerlach magnet acting on a spin-\(1/2\) atom is the prototype of a qubit measurement: a two-level quantum system, an apparatus that entangles the qubit with a macroscopically distinct pointer variable (here, the atom's transverse position), and a readout that yields one bit with Born-rule statistics. Every projective measurement in today's quantum technologies is, conceptually, a Stern–Gerlach experiment in different hardware, and the sequential cascades of Section 80.4 prefigure the state-preparation-and-measurement protocols of Entanglement and Bell Tests. It is fitting that the sharpest early test of quantum discreteness — undertaken by a sceptic hoping to refute it — should have become the standard mental model for the theory's most characteristic act, the measurement [Friedrich:2003].