The Stern–Gerlach experiment

A dipole in a magnetic field has energy U = −μ⋅B. In a uniform field that energy is the same everywhere, so there is no net force — only a torque, under which the moment precesses about the field at the Larmor frequency while the atom flies straight on. Force needs a gradient: F_z = μ_z(∂B_z/∂z). That is why Gerlach machined one pole to a knife edge facing a grooved pole — the geometry buys a gradient around 1.4 × 10³ T m⁻¹ across a beam a slit-width wide. The precession is also why only the z-component matters: μₓ and μ_y swing round of order a million times during the transit, average to zero, and deflect nothing. Put in μ_z = μ_B and the force is 1.3 × 10⁻²⁰ N — feeble, yet 7 × 10³ times the silver atom's weight, and applied to a neutral atom, so there is no Lorentz force qv × B to drown it. That last clause is why the experiment works with atoms and fails with free electrons.

What lands on the plate is a histogram of μ_z. Classically an oven randomises orientation, so μ_z = μ cos θ with cos θ uniformly distributed: every value between −μ and +μ is equally represented, and the prediction is one continuous band. Quantum mechanics instead allows only discrete projections, so a moment of quantum number j lands in 2j + 1 traces. Run the count for orbital angular momentum: l = 0 gives one undeflected trace, l = 1 gives three, l = 2 gives five — always an odd number, and always with a trace at the centre, because integer l includes m = 0. The plate showed two stripes and an empty centre. That single photograph rules out the classical band and every integer-l ladder at once: whatever splits the beam has 2j + 1 = 2, hence j = 1/2, and half-integers never occur for orbital motion.

Inside the magnet the atom is a projectile: constant transverse acceleration a = μ_zB′/m ≈ 7.3 × 10⁴ m s⁻² for μ_z = μ_B — about 7400g — held for a time t = L/v. A 550 m s⁻¹ atom spends 64 μs in a 3.5 cm magnet and exits z = ½at² ≈ 0.15 mm off axis, so the two spin states sit about 0.3 mm apart, the scale of the historic traces. A field-free drift D then multiplies the effect through the exit slope: z = (μ_zB′L/mv²)(L/2 + D), and 10 cm of drift stretches 0.3 mm to about 2 mm. Note what the formula punishes: deflection goes as 1/v², and the oven supplies a Maxwell–Boltzmann spread of speeds, so each trace smears into a streak — fast atoms crowd the centre, slow ones feather outward. The two-lobed 1922 pattern was resolvable because the collimating slit was narrower than the splitting, not because the atoms shared one speed.

Silver was luck as much as design, and the luck is in its electron count. Forty-six of the 47 electrons close into [Kr] 4d¹⁰, and a filled subshell has zero total orbital and zero total spin angular momentum. Everything is carried by the lone 5s electron, and s states have l = 0: the ground term is ²S₁/₂, so silver has no orbital moment to deflect at all. The nuclear moment is real but scales with the proton mass where μ_B has the electron mass, and silver's is about 10⁴ times smaller than μ_B — invisible at this resolution. The beam is effectively one free electron spin per atom, chaperoned by 108 u of neutral ballast that kills the Lorentz force. A two-way split of an l = 0 atom is maximally awkward for orbital explanations, which is why the decisive repeat was Phipps and Taylor's 1927 run with ground-state hydrogen: one electron, certainly 1s, certainly l = 0 — and still two traces.

Be strict about what the plate measures: μ_z, nothing else, and it came out close to ±μ_B. Since μ_z = −g m μ_B, a deflection fixes only the product g⋅m. Old quantum theory factorised it as g = 1, m = ±1 with m = 0 mysteriously absent — that is the space quantisation Stern and Gerlach announced in 1922, three years before spin was proposed. Modern mechanics factorises the same number as g ≈ 2, mₛ = ±1/2. The deflections cannot tell these apart; what breaks the tie is outside evidence. Spectroscopy gives silver an l = 0 ground state, so orbital m = ±1 is unavailable, and the anomalous Zeeman analysis — later Dirac's equation — supplies gₛ ≈ 2. Only with that imported g does ±μ_B become s = 1/2. The habit generalises: a measurement returns an eigenvalue of the operator it couples to, and every further conclusion is inference that must be paid for elsewhere.