Orbital moments and the normal Zeeman effect

Let an electron of charge −e circle at speed v on radius r. It passes any point once per period T = 2πr/v, so the loop carries current I = −e/T = −ev/(2πr), and a loop's moment is current times area: μ = IA = −(ev/2πr)(πr²) = −evr/2. Multiply and divide by the mass and the orbit's geometry disappears: μ = −(e/2mₑ)(mₑvr) = −(e/2mₑ)L. Radius and speed have cancelled; nothing survives but the charge-to-mass ratio, which is why the same relation holds for any orbit once L is read as the angular momentum. The minus sign is physical — a negative charge circulating one way is a current circulating the other — so μ points opposite L. And because the result says one operator is a constant multiple of another, it passes into quantum mechanics unchanged: no correction factor appears, gₗ = 1 exactly, and every eigenstate of L_z is automatically an eigenstate of μ_z.

Project onto the field direction: μ_z = −(e/2mₑ)L_z = −mₗ μ_B, one Bohr magneton per unit of mₗ. A moment in a field carries energy U = −μ⋅B, so with B along z each sublevel shifts by ΔE = mₗ μ_B B: the 2l+1 values of mₗ, degenerate in zero field, spread into a ladder of rungs μ_B B apart. The degeneracy was never an accident — with no field there is no preferred axis, and states differing only in orientation must share an energy. The field supplies the axis, and the symmetry argument dies with it. Scale matters: at 1.0 T the spacing is 5.79 × 10⁻⁵ eV, about 450 times smaller than room-temperature kT = 0.0259 eV, so the rungs stay essentially equally populated, and about 3 × 10⁻⁵ of a 2 eV optical transition, so the splitting is a fine perturbation on the line, not a rearrangement of the atom.

Take a singlet d → p line. The upper level fans into five rungs, the lower into three, and the dipole rules Δl = ±1, Δmₗ = 0, ±1 allow nine distinct transitions. The photon energy is hν = hν₀ + (mₗ′ − mₗ)μ_B B, because both ladders — whatever their n and l — share the identical spacing μ_B B. So the frequency depends on the difference Δmₗ alone, and nine transitions deliver exactly three frequencies: ν₀, and ν₀ ± μ_B B/h. Nothing here is approximate; it is a strict coincidence enforced by gₗ = 1 being the same number in every orbital level. That is also its fragility. Give the two levels different g factors — say 1.2 and 1.0 — and hν picks up g′m′ − gm, which takes up to nine distinct values: the triplet fans apart. The number of Zeeman lines counts the distinct values of the energy shift, never the sublevels and never the arrows between them.

The three lines are distinguishable by more than frequency. A Δmₗ = 0 transition leaves the z angular momentum unchanged: its charge oscillation runs along B, and an oscillating dipole radiates strongest broadside and not at all along its own axis. So the π line is linearly polarised parallel to B, brightest viewed transverse to the field, and absent when you sight along it. A Δmₗ = ±1 transition removes ±ℏ of z angular momentum, carried off by charge circulating in the plane perpendicular to B: viewed along the field these σ lines are circularly polarised with opposite senses; viewed transverse they appear linearly polarised perpendicular to B. Geometry is therefore a prediction: transverse observation shows three lines, longitudinal shows only the two σ components. Both checks pass, and the sense of the circular polarisation seen along B told Zeeman the radiating charge is negative — a sign determined before the electron was isolated.

Put numbers on cadmium's 643.85 nm singlet line at 1.0 T. The spacing is Δν = μ_B B/h = 14.0 GHz against an optical frequency ν = c/λ = 4.66 × 10¹⁴ Hz — a shift of three parts in 10⁵. In wavelength, Δλ = λ²Δν/c ≈ 19 pm, demanding resolving power λ/Δλ ≈ 3 × 10⁴: a large grating or a Fabry–Pérot étalon, and hopeless for a prism spectroscope. Run the logic backwards and the experiment measures a fundamental constant: the observed Δν per tesla is μ_B/h, and since μ_B = eℏ/2mₑ, the charge-to-mass ratio follows as e/mₑ = 4π(μ_B/h) = 4π × 1.40 × 10¹⁰ ≈ 1.76 × 10¹¹ C kg⁻¹. That is how a wavelength shift of a few parts per hundred thousand delivered e/mₑ in 1896, a year ahead of Thomson's cathode-ray deflections.

Every step above assumed the moment is purely orbital, g = 1. Spin breaks that: the intrinsic moment carries gₛ ≈ 2.002, so any term with S ≠ 0 has a Landé factor g_J depending on l, s and j, and the upper and lower ladders of a transition no longer share a spacing. The three-line coincidence collapses and the pattern fans out — sodium's D₁ line shows four components, D₂ six. Only singlet terms, where paired spins cancel the spin moment — bright lines of cadmium, zinc, mercury — show the clean triplet. The name is a historical accident: Lorentz's classical theory predicted exactly this triplet with exactly the μ_B B/h spacing, so it was 'normal' and everything else 'anomalous'. The joke is on the labels. Hydrogen, the simplest atom, is anomalous, because one electron cannot pair its spin; and the anomalous majority is the pattern classical physics can never produce — the one whose explanation had to wait for spin.