Fine structure: relativistic, Darwin and spin-orbit

The Schrödinger Hamiltonian is the leading term of the Dirac Hamiltonian, and fine structure is the next one. Reduce the Dirac equation to two components — the Foldy–Wouthuysen transformation, or equivalently second-order elimination of the small spinor components — and to order (v/c)² exactly three operators appear beyond p²/2m + V(r). The kinetic one is arithmetic: √(p²c² + m²c⁴) − mc² = p²/2m − p⁴/8m³c² + …, so Hᵣₑₗ = −p⁴/8m³c². The Darwin term (ħ²/8m²c²)∇²V has no classical ancestor at all. The spin-orbit term ξ(r)L⋅S couples the two angular momenta the electron already carries. For a hydrogenic orbital ⟨v⟩/c ≈ Zα/n, so every one of them is down on H₀ by (Zα/n)²: at Z = 1 the gross structure is 13.6 eV, fine structure is a few hundred μeV, and that ratio of about 5 × 10⁻⁵ is your licence to treat all three at first order.

Hydrogen's n-manifold is 2n²-fold degenerate — n² orbital states times two spin states — so this is degenerate perturbation theory, and the rule is to work in a basis that already diagonalises the perturbation inside the manifold. Check the commutators. Hᵣₑₗ and H_D are built from p², r and V alone, so they commute with L², S², L_z and S_z, and are diagonal in either basis. L⋅S is the awkward one: [L⋅S, J²] = [L⋅S, J_z] = [L⋅S, L²] = 0, but [L⋅S, L_z] = iħ(L×S)_z ≠ 0, and similarly for S_z. So mₗ and mₛ are not good quantum numbers, while n, l, j and m_J are. Couple with Clebsch–Gordan coefficients — |j = l ± ½, m_J⟩ is a two-term combination of |mₗ = m_J − ½, ↑⟩ and |mₗ = m_J + ½, ↓⟩ — and L⋅S = (J² − L² − S²)/2 is diagonal by construction. States of different l never mix, because L² still commutes with the whole of H_fs; that is why l survives as a label even though mₗ does not.

The Darwin term is the one with no classical picture, so read it as a potential sampled over a smeared position: ⟨V(r + δr)⟩ ≈ V(r) + ⅙⟨δr²⟩∇²V, and matching the coefficient ħ²/8m²c² gives ⟨δr²⟩ = ¾(ħ/mc)², a smear of order the reduced Compton wavelength ħ/mc = 3.86 × 10⁻¹³ m. That is Zitterbewegung — interference between the positive- and negative-energy components the Dirac spinor mixes. For a Coulomb potential ∇²(1/r) = −4πδ³(r), so ∇²V = Ze²δ³(r)/ε₀ and ⟨H_D⟩ = (ħ²Ze²/8m²c²ε₀)|ψ(0)|². Only l = 0 states have ψ(0) ≠ 0, with |ψₙ₀₀(0)|² = Z³/πn³a₀³ — for hydrogen 1s that is 2.15 × 10³⁰ m⁻³, giving +725 μeV. Set against the relativistic term's −906 μeV it leaves −181 μeV, which is α²/4 of the 13.606 eV binding energy. Two large numbers cancelling down to a fifth of the larger is the whole character of fine structure.

Boost to the electron's rest frame, watch the proton orbit it, and the magnetic field that orbit produces has magnitude B = (1/emc²)(1/r)(dV/dr)|L|. Pair it with μ = −gₛ μ_B S/ħ and gₛ ≈ 2 and you get (1/m²c²)(1/r)(dV/dr) L⋅S — twice what experiment shows. The reason is that the electron's frame is not inertial. It accelerates, and the composition of two non-collinear boosts is a boost times a rotation, so the rest frame itself precesses at ω_T ≈ a × v/2c². Thomas precession contributes exactly half the naive interaction with the opposite sign, leaving ξ(r) = (1/2m²c²)(1/r)(dV/dr), which for a Coulomb potential is Ze²/8πε₀m²c²r³. Hydrogen 2p's resulting 45.28 μeV interval reads as an effective internal field of 45.28 μeV / 2μ_B = 0.39 T. One caution: ⟨1/r³⟩ = Z³/[a₀³n³l(l+½)(l+1)] diverges as l → 0 while ⟨L⋅S⟩ vanishes there, so l = 0 is not a limit of this formula — it is the Darwin term's territory.

Do hydrogen n = 2 in units of u = E₂²/2mc² = (3.4014 eV)²/1.0220 MeV = 11.32 μeV. For 2s₁/₂ (l = 0): relativistic −u[4⋅2/½ − 3] = −13u = −147.17 μeV, Darwin +4ν = +8u = +90.56 μeV, spin-orbit zero, total −5u = −56.60 μeV. For 2p₁/₂ (l = 1, j = ½): relativistic −u[8/1.5 − 3] = −(7/3)u = −26.41 μeV, Darwin zero, spin-orbit −(8/3)u = −30.19 μeV, total −5u = −56.60 μeV. The same answer, from a different pair of operators. For 2p₃/₂ the spin-orbit term flips sign to +(4/3)u, giving −u = −11.32 μeV. In general E_fs = (Eₙ²/2mc²)[3 − 4n/(j + ½)] — l has cancelled. The 2p₃/₂ − 2p₁/₂ interval is 4u = 45.28 μeV, or 10 949 MHz against a measured 10 969 MHz; the 0.2% is reduced mass, higher orders in α, and QED.

Three limits. First, the l-degeneracy is an artefact of stopping at α²: quantum electrodynamics adds self-energy and vacuum polarisation at order α(Zα)⁴mc² with a logarithm, raising 2s₁/₂ above 2p₁/₂ by 1057.8 MHz — 4.37 μeV, or 9.7% of the fine-structure interval, small but resolved with room to spare. Second, the nucleus here is a spinless point; hyperfine structure, smaller by roughly mₑ/mₚ, splits hydrogen 1s by 1420 MHz. Third, the expansion parameter is Zα, not α. The exact Dirac–Coulomb eigenvalue is Eₙⱼ = mc²[1 + (Zα)²/(n − j − ½ + √((j+½)² − (Zα)²))²]^(−½), and everything above is its first two terms. At Z = 1 the next term is a 10⁻⁹ correction; at Z = 82, Zα = 0.60 and first order already misses a fifth of the 1s relativistic shift — use the exact eigenvalue, or diagonalise the Dirac Hamiltonian on a radial grid once the nucleus has finite size.