Stern-Gerlach Experiment

A force on a magnetic moment in a non-uniform field

A neutral atom with a magnetic moment µ moving through a uniform magnetic field experiences torque (which precesses the moment) but no net force — the field's gradient is zero, so the +pole and −pole of the dipole feel equal and opposite forces. Inside an inhomogeneous field, the gradient is nonzero, and the dipole sees a net force F = ∇(µ·B). If B points mostly along z and varies along z, the force is approximately:

F_z = µ_z · (∂B_z / ∂z)

Classical picture: an ensemble of atoms emerging from a hot oven has random orientations of µ, so µ_z takes a continuous range of values from −|µ| to +|µ|. The detector should show a single smeared streak whose ends correspond to fully aligned and fully anti-aligned atoms.

Quantum picture: µ_z = g_s µ_B m_s, where m_s takes only the discrete values −s, …, +s. For a spin-½ system, m_s ∈ {−½, +½}. There are exactly two possible deflections, separated cleanly. The detector should show two spots.

Stern and Gerlach saw two spots. The classical theory was wrong about the world.

The 1922 apparatus

The setup was clever and minimal. A heated tantalum oven held metallic silver near 1300 K. Silver vapor escaped through a small aperture, then through a series of slits to define a thin pencil-shaped beam (~1 mm wide, vacuum better than 10⁻⁵ mbar to keep the mean free path much longer than the apparatus). The beam passed between the two pole faces of a specially shaped electromagnet — one pole face concave with a sharp edge, the other convex — engineered to produce a strong gradient ∂B_z/∂z. The pole geometry shaped the field so |B| was both large (saturating the moment direction) and steeply varying along z within the beam's cross-section.

The deflection length was about 35 cm. Atoms hit a cold glass plate at the end. Because silver oxidizes only weakly, individual atoms left almost no visible trace. Stern and Gerlach reportedly developed the deposit by "fortuitous" exposure to cigar smoke from a colleague — the sulfur in the smoke turned the silver into Ag₂S, a dark stain. The dark stain showed two distinct lines. The story is half-apocryphal but the photographic plates from 1922 do show the doublet.

The order-of-magnitude numbers are worth knowing:

oven temperature              T ≈ 1300 K
silver thermal velocity       v ≈ √(3kT/m) ≈ 580 m/s
apparatus length              L ≈ 0.35 m
flight time                   t ≈ L/v ≈ 6 × 10⁻⁴ s
B field gradient              ∂B/∂z ≈ 10 T/m
silver magnetic moment        µ_z ≈ µ_B = 9.27 × 10⁻²⁴ J/T
force                         F = µ_z (∂B/∂z) ≈ 10⁻²² N
acceleration (m_Ag = 1.79 × 10⁻²⁵ kg)
                              a ≈ 5 × 10² m/s²
deflection                    Δz = ½ a t² ≈ 0.1 mm

So the two spots were separated by about a tenth of a millimeter on the detector — small but resolvable on a glass plate.

Why silver?

Silver was not a random choice. The ground-state electron configuration of Ag is [Kr] 4d¹⁰ 5s¹. The 4d shell is full, contributing zero net orbital and spin angular momentum. The 5s valence electron has ℓ = 0 (so no orbital magnetic moment) but s = ½. The atom's net angular momentum is just the spin of the single unpaired electron. Its magnetic moment is therefore one Bohr magneton, all of it spin.

This is the simplest possible single-spin system available in a thermal atomic beam. Hydrogen would have worked too in principle but is harder to handle (and its single-electron magnetic moment was poorly understood in 1922). Silver gave Stern and Gerlach a clean, well-collimated, easily-condensed atomic-magnetic-dipole probe.

In retrospect, the experiment is a strange historical accident: the 1922 prediction the team set out to test was quantization of orbital angular momentum (Bohr's ℓ = 1 case), expected to give three spots. Instead they saw two — exactly what spin-½ predicts. They had stumbled into the discovery of spin three years before the concept was named, and the data sat in plain sight as a quantitative match for the spin-½ Zeeman splitting that Uhlenbeck and Goudsmit would propose in 1925.

Sequential Stern-Gerlach: non-commuting measurements

The single-shot experiment is striking but the cascade version is foundational. Imagine three Stern-Gerlach magnets in series, each with adjustable orientation:

SG_z   →  block −z output, keep +z output
SG_x   →  measure along x
SG_z   →  measure along z again

The first SG_z purifies the beam: every atom emerging is in the eigenstate |+z⟩. Send it into SG_x. Quantum mechanics decomposes |+z⟩ = (1/√2)(|+x⟩ + |−x⟩), so SG_x splits the beam 50/50. Block the |−x⟩ output and keep the |+x⟩ output. Now send that beam into a second SG_z. Naively you might expect 100 % |+z⟩ — after all, you measured +z and never disturbed the atoms. Instead you get 50 % |+z⟩ and 50 % |−z⟩. The intermediate measurement of S_x has destroyed the definite value of S_z.

The reason is that S_x and S_z do not commute: [S_x, S_z] = iℏ S_y, so they share no common eigenstates. Measuring one collapses the wavefunction onto an eigenstate of that one, and the other becomes a 50/50 superposition. The cascade is the cleanest demonstration of non-commuting observables in pedagogy, and it is the spin-version of position–momentum complementarity.

In a real teaching experiment with cold atoms or NV-centre spins one can reproduce all three legs of the cascade and see the populations behave exactly as quantum mechanics predicts. No classical theory of magnetic moments — even one allowing only two discrete orientations — can reproduce the 50/50 splits at intermediate angles.

Quantization at different spins and field axes

System	Total spin s	Number of spots	m values	g-factor
Silver atom (5s¹)	½	2	±½	~2.0
Hydrogen ground state	½	2	±½	2.0023
Sodium 3s¹	½	2	±½	2.0
Helium-4 ground state (¹S₀)	0	1	0	—
Vanadium (³d⁴ ⁴s¹, ⁶S₅/₂)	5/2	6	±½, ±3/2, ±5/2	~2.0
Iron 3d⁶4s² (⁵D₄)	2	9 (with orbital)	−4..+4	~1.5 (Landé)
Phosphorus 3p³ (⁴S₃/₂)	3/2	4	±½, ±3/2	~2.0

The number of spots equals 2j + 1 where j is the total angular momentum quantum number (orbital plus spin coupled). For atoms with closed shells outside one valence electron, the count gives a direct readout of that electron's spin. Many beam-source experiments in the 1920s and 30s exploited this to map ground-state term symbols.

Where Stern-Gerlach physics shows up

• Every introductory quantum-mechanics course. The Stern-Gerlach setup is the canonical first example used to motivate spin, two-state systems, and Hilbert-space measurement axioms in textbooks from Sakurai to Griffiths to Townsend.
• Nuclear magnetic resonance (NMR) and MRI. NMR exploits the same underlying Zeeman splitting that Stern-Gerlach revealed. A typical 1.5 T MRI scanner lifts the proton spin-up/spin-down degeneracy by 64 MHz; resonant RF flips the populations and the precessing macroscopic moment is detected. Both experiments rely on µ·B coupling and on the discrete projection of µ_z.
• Atomic clocks and laser cooling. Hyperfine-state-selective Stern-Gerlach magnets (or their optical equivalents) are used to load only the desired clock state into atomic-fountain or optical-lattice clocks. NIST-F2 and similar instruments use exactly this kind of magnetic state preparation.
• Polarized neutron and proton sources. Hot atomic beams of hydrogen passed through SG-style magnets produce polarized hydrogen for nuclear-physics targets. The same idea underlies polarized ³He sources used at electron-scattering facilities like Jefferson Lab.
• Quantum information demonstrations. The first published "qubit measurement" pictures in textbooks are typically Stern-Gerlach cascades — the |+z⟩ vs |+x⟩ vs |+y⟩ projections form the surface of the Bloch sphere, and the SG cascade is the most direct visualization of a projective measurement on a single qubit.

Modern updates and re-runs

Stern-Gerlach has been re-performed in many forms. In the 1990s Friedrich and Herschbach reanalysed Stern's notebooks and concluded that the original 1922 deposit was a marginal pair of strands rather than two cleanly separated spots — but unambiguous nonetheless. In 2002 a group at Frankfurt repeated the experiment with modern detectors and confirmed the two-spot structure with millikelvin angular resolution.

Cold-atom versions in optical molasses can use weakly inhomogeneous fields to spatially sort spin populations of the same atomic species along different axes simultaneously, generating multi-arm interferometers. Single-NV centres in diamond can play the same role as the silver atom: a single spin-½ probe measured along arbitrary axes shows perfect SG cascade statistics on a single-quantum scale, with optical readout replacing the photographic plate.

What has not changed is the conceptual content. Two spots is two spots, and a 50/50 split after a 90° rotation of the analyzer is what spin-½ does. Every modification of the experiment recovers the same predictions of Pauli's spin-½ representation of SU(2).

Variants and extensions

• Sequential Stern-Gerlach cascades. Two or three magnets in series with rotated axes demonstrate the non-commutation of spin operators along different directions and the projection postulate of quantum measurement.
• Continuous-variable Stern-Gerlach. Replacing the discrete spin with a continuous quantum number (e.g. atomic position in an optical lattice) gives spatial superpositions whose Stern-Gerlach analogue is the field-gradient interferometer, used in atom-interferometry gravimetry.
• Stern-Gerlach with higher j. Atoms with larger j (e.g. chromium with J = 3) split into 2j + 1 = 7 sub-beams, an unmistakable visual map of the multiplet.
• Molecular Stern-Gerlach. Polar molecules in supersonic beams can be deflected by inhomogeneous electric fields (Stark effect) — same physics with E replacing B and dipole moment replacing magnetic moment. The basis for cold-molecule trapping and Stark decelerators.
• Single-particle Stern-Gerlach with nitrogen-vacancy centres. Modern quantum-information demonstrations use a single NV centre spin in diamond, optically read out, to reproduce the SG cascade on a one-quantum scale with arbitrary field-axis rotations.

Common pitfalls

• Treating SG output as classical "definite-direction" spins. The output beam is in an eigenstate of S_z (for an SG_z magnet), not a classical magnet pointing up. Asking "but which way is its S_x pointing?" has no answer — the question is meaningless until you measure S_x.
• Forgetting that uniform B fields do not deflect. A common student error is to assume the splitting comes from the field magnitude. It comes from the gradient. A homogeneous magnetic field (no matter how strong) produces only torque and Larmor precession, never net force.
• Reading 2j + 1 as 2s + 1. For an atom with both orbital and spin angular momentum the splitting is into 2j + 1 sub-beams along the total angular momentum, not 2s + 1. The silver case is special because ℓ = 0, so j = s and the two values coincide.
• Confusing spin measurement with spin precession. Stern-Gerlach is a position-correlated measurement of S_z, not a precession of the spin. Larmor precession happens inside the field but does not cause the splitting; the gradient force does.
• Assuming the atom "had a definite spin all along". Sequential Stern-Gerlach refutes this. After a SG_x measurement, the atoms emerging from the +x port have S_x = +ℏ/2 with certainty, but their S_z is uniformly random — they cannot have had a definite S_z value before the SG_x measurement.