The Zeeman effect and Paschen-Back

Put the atom in a uniform B and choose the symmetric gauge A = ½ B × r. Minimal coupling p → p + eA turns p²/2m into p²/2m + (e/2m)(A⋅p + p⋅A) + e²A²/2m. The cross term collapses to (e/2m)B⋅L = (μ_B/ħ)L⋅B, the spin magnetic moment adds (μ_B/ħ)gₛ S⋅B with gₛ = 2.0023 imported from the Dirac equation and QED rather than derived here, and the last term is diamagnetic: H_dia = (e²B²/8mₑ)(x² + y²). Its size relative to the linear term is eB⟨ρ²⟩/4ħ, about 1.06 × 10⁻⁶ per tesla when ⟨ρ²⟩ ≈ a₀². That is why it goes — and the same estimate says when you may not drop it. Since ⟨ρ²⟩ grows as n⁴, the two terms are comparable near n = 30 at 1 T, and in a neutron-star field of 10⁸ T the diamagnetic term dominates outright and the whole level scheme has to be rebuilt.

Write L + gₛ S = J + (gₛ − 1)S, so H_Z = (μ_B B/ħ)[J_z + (gₛ − 1)S_z] for B along z. Now ask what would happen if gₛ were 1. The second term vanishes, H_Z is proportional to J_z alone, and it is already diagonal in |n l j mⱼ⟩ with shift μ_B B mⱼ for every level whatever its l, j or s — equal spacing everywhere, and, since Δmⱼ = 0, ±1, exactly three lines in any transition. That is the 'normal' Zeeman effect, and it is precisely what Lorentz's classical orbiting electron predicts. Every complication in this topic — the unequal g_J values, the six-component sodium D2 pattern, the bent intermediate-field curves, the Paschen-Back crossover — traces back to the single fact that gₛ ≈ 2 rather than 1. Singlet levels have S = 0, so (gₛ − 1)S_z has nothing to act on and they show the normal pattern; that is the only sense in which 'normal' is normal.

While μ_B B ≪ Δ_fs, treat H_Z as a perturbation on the fine-structure eigenstates and diagonalise it inside one j. J_z is already diagonal there. For S_z use the projection theorem — the Wigner-Eckart theorem specialised to a vector operator acting within a fixed-j subspace — which gives ⟨j mⱼ|S_z|j mⱼ⟩ = ⟨J⋅S⟩ ħ mⱼ / [ħ² j(j+1)]. With J⋅S = (J² + S² − L²)/2 = (ħ²/2)[j(j+1) + s(s+1) − l(l+1)], this returns the Landé factor g_J = 1 + [j(j+1) + s(s+1) − l(l+1)]/[2j(j+1)] and the shift g_J μ_B B mⱼ. So ²S₁/₂ gets g_J = 2, ²P₁/₂ gets 2/3, ²P₃/₂ gets 4/3, and any singlet gets 1. Note what the theorem is not: it is not the classical claim that S precesses so fast about J that only its parallel component survives. The vector model reproduces the same fraction, but the real statement is an exact one about matrix elements inside one irreducible subspace.

'Weak' means μ_B B small compared with the fine-structure interval, not compared with the transition energy. For sodium's 3p term Δ_fs = 17.196 cm⁻¹ = 2.132 meV, while μ_B B = 57.88 μeV per tesla, so the two are equal at 36.8 T: a 1 T lab magnet sits at a ratio of 0.027, comfortably weak. Hydrogen's 2p is a different story. Its interval is only 45.3 μeV (11.0 GHz), so the crossover falls at 0.783 T and a 2 T superconducting magnet has already left the Landé regime behind. The difference is spin-orbit strength, which scales roughly as Z_eff⁴/n³. Push down one more level of structure: hydrogen's 1s hyperfine interval is 5.87 μeV, so its Back-Goudsmit crossover sits near 0.05 T. That is why hyperfine Breit-Rabi diagrams bend at a few hundred gauss while sodium's fine-structure fan stays straight to several tesla.

With B along z, [J_z, H] = 0 at every field, so mⱼ stays a good quantum number throughout; [J², H] ≠ 0 once H_Z is present, so j does not. H therefore block-diagonalises by mⱼ, and for s = ½ each block is at most 2 × 2, spanning |l+½, mⱼ⟩ and |l−½, mⱼ⟩. Its diagonal entries are the spin-orbit energies Aħ²l/2 and −Aħ²(l+1)/2 plus the Landé shifts; its off-diagonal entry, for gₛ = 2, is ⟨l−½, mⱼ|H_Z|l+½, mⱼ⟩ = −μ_B B √[(l+½)² − mⱼ²]/(2l+1). Two consequences follow. First, the coupling leaves the trace untouched, so the two eigenvalues are pushed apart symmetrically — level repulsion, not a common drift. Second, the coupling vanishes at mⱼ = ±(l+½): the stretched states have no partner to mix with, so they are exact eigenstates and stay exactly linear in B at any field. For hydrogen 2p at 0.500 T the mⱼ = +½ eigenvalues are +37.60 and −23.75 μeV, while Landé predicts +34.39 and −20.55 — each off by 3.20 μeV, or 9.3%.

When μ_B B ≫ Δ_fs, swap the roles: H_Z becomes the zeroth-order term and spin-orbit the perturbation. L and S then precess about B independently, mₗ and mₛ are separately good, and E = μ_B B(mₗ + gₛ mₛ) + Aħ² mₗ mₛ, because ⟨L⋅S⟩ reduces to ⟨L_z S_z⟩ = ħ² mₗ mₛ once the ladder terms average away. Now run a dipole transition through it. The photon does not couple to spin, so Δmₛ = 0 sits on top of Δmₗ = 0, ±1, the spin term cancels between upper and lower levels, and the field-linear shift is just μ_B B Δmₗ — three lines at 0 and ±μ_B B. The Lorentz triplet comes back, which is what Paschen and Back saw and why the limit carries their names. What survives of the fine structure is the residual Aħ² mₗ mₛ, splitting each outer component into a close doublet of separation Aħ² = 2Δ_fs/(2l+1): the spin-orbit constant, read off at a field that has otherwise beaten the coupling flat.