Spin-orbit coupling and the Landé g factor

In the lab frame the electron orbits a stationary nucleus. Transform to the electron's instantaneous rest frame and the charged nucleus circulates instead — a current loop threading the electron's position with a magnetic field Bᵢₙₜ ≈ −(v × E)/c², built from the same Coulomb field E that binds the atom and pointing along the orbital angular momentum L. The electron's spin moment μₛ = −gₛ μ_B S/ħ then has an orientation energy −μₛ⋅Bᵢₙₜ, and once the Thomas half of the next section is included, gₛ ≈ 2 makes reversing the spin in that field cost about μ_B Bᵢₙₜ — so a measured splitting reads off the field as Bᵢₙₜ ≈ ΔE/μ_B. Its size is not small: hydrogen's 2p splitting of 4.5 × 10⁻⁵ eV gives close to 0.8 T, a solid laboratory electromagnet's worth, and sodium's 3p doublet of 2.1 × 10⁻³ eV gives about 36 T, beyond all but the strongest magnets on Earth — supplied free by the atom's own motion. Since Bᵢₙₜ points along L, the orientation energy is proportional to L⋅S: the atom couples its own spin to its own orbit, with no external apparatus anywhere in the story.

The rest-frame derivation hides an error: the electron's frame is not inertial. It accelerates continuously toward the nucleus, and in special relativity the composition of two non-parallel boosts is not a boost but a boost plus a rotation. The electron's frame therefore rotates — Thomas precession — at a rate exactly half the naive spin-precession rate and with the opposite sense, and the net effect is to halve the coupling: H_SO = (1/(2mₑ²c²))(1/r)(dV/dr) L⋅S, the ½ being Thomas's. History pivoted on this factor. In 1926 the doublet intervals came out twice too large from the naive calculation; gₛ ≈ 2 and Thomas's ½ conspire so the observed splitting looks as if the spin g were 1, and the spin hypothesis of Uhlenbeck and Goudsmit was rescued within months. The Dirac equation later delivers the ½ automatically; at this level you insert it by hand and know why it is there.

H_SO does not commute with L_z or S_z separately — the coupling makes L and S precess about their sum — so mₗ and mₛ stop being good quantum numbers. It does commute with J² = (L+S)², L², S² and J_z, so the coupled basis |l, s; j, mⱼ⟩ diagonalises it. Square J = L + S to get L⋅S = ½(J² − L² − S²), and every matrix element becomes arithmetic: ⟨L⋅S⟩ = (ħ²/2)[j(j+1) − l(l+1) − s(s+1)]. For a p electron (l = 1, s = ½): j = 3/2 gives +ħ²/2 and j = 1/2 gives −ħ². Two checks come free. Degeneracy-weighted, 4 × (+½) + 2 × (−1) = 0: the coupling splits the level about an unmoved centre of gravity, so an average over the multiplet feels nothing. And the gap between adjacent j levels is proportional to the larger j — the Landé interval rule, your first diagnostic for whether a measured multiplet really is spin-orbit split.

With H_SO switched on, the labels that still commute with the Hamiltonian for one electron are n, l, s, j and mⱼ, and spectroscopy compresses the level part into the term symbol n²ˢ⁺¹L_J: the superscript is the spin multiplicity 2s + 1, the letter S, P, D, F codes l = 0, 1, 2, 3, and the subscript is j. Sodium's 3p level becomes 3²P₃/₂ and 3²P₁/₂, both decaying to 3²S₁/₂, and every fine-structure diagram you will meet is drawn in this notation. Two readings of it trip students. The superscript counts spin orientations, not levels: ²S₁/₂ is a doublet term with a single level, because with l = 0 there is only j = ½ and no orbit for the spin to couple to. And the subscript is the only label that knows about the coupling — mₗ and mₛ are no longer good, so a term symbol never carries them. How far apart the levels sit is a separate, radial question: in a hydrogenic atom the doublet gap scales as Z⁴/n³, and the larger effective charge a penetrating valence electron feels is why sodium's 3p gap of 2.1 × 10⁻³ eV dwarfs hydrogen's 2p gap of 4.5 × 10⁻⁵ eV. UPIV-11.07 derives the gap along with the relativistic and Darwin terms that complete hydrogen's fine structure.

A level's magnetic moment is μ = −μ_B(L + 2S)/ħ — spin counts double because gₛ ≈ 2 — and that 2 means μ is not antiparallel to J = L + S. In a weak field, L and S precess rapidly about J at the fine-structure frequency while J precesses slowly about B. The component of μ perpendicular to J therefore averages to zero, and only the projection along J couples to the field: ⟨μ⟩ = −g_J μ_B J/ħ with g_J = ⟨(L + 2S)⋅J⟩/⟨J²⟩. Write L + 2S = J + S, so g_J = 1 + ⟨S⋅J⟩/[j(j+1)ħ²], and square L = J − S to get S⋅J = ½(J² + S² − L²). The result is g_J = 1 + [j(j+1) + s(s+1) − l(l+1)]/[2j(j+1)], and each level moves by g_J μ_B B mⱼ. Test it on limits you already trust: l = 0 leaves pure spin and g_J = 2; s = 0 leaves pure orbit and g_J = 1. For one electron it collapses to g_J = 1 ± 1/(2l + 1) for j = l ± ½, tending to the orbital value as l grows. For sodium: g(²S₁/₂) = 2, g(²P₁/₂) = 2/3, g(²P₃/₂) = 4/3 — three levels, three different magnetic prices per unit mⱼ. That is why alkali Zeeman patterns are anomalous: a normal three-line pattern needs upper and lower levels to shift at the same rate, which s = 0 guarantees by forcing g_J = 1 everywhere; the line patterns themselves are counted in UPIV-11.07.

Switch on a field B along z and H gains (μ_B B/ħ)(L_z + 2S_z), a term with its own preferences. H_SO keeps J² and J_z but not L_z or S_z, as the third section showed; the Zeeman term commutes with L_z and S_z but not with J², because the 2 on S_z makes it J_z + S_z rather than a multiple of J_z. Apart from L² and S², only J_z commutes with both, so mⱼ = mₗ + mₛ is the one label that stays exact at every field. Whether j or the pair mₗ, mₛ is approximately good is settled by the ratio μ_B B/ΔE_FS. Well below 1, j survives and the projection argument holds; well above 1, the field decouples L from S, mₗ and mₛ become good again, and each state shifts by (mₗ + 2mₛ)μ_B B — the Paschen-Back regime. The handover sits where the applied field matches the internal one from the first section, B ≈ ΔE_FS/μ_B: about 0.8 T for hydrogen 2p and 36 T for sodium 3p, so the same magnet is strong for one atom and weak for another. Between the limits only mⱼ labels anything; the Hard example diagonalises one mⱼ block exactly and finds g_J in its low-field slopes.