Spin-orbit coupling with the Thomas factor built in

SI units throughout; where four-vectors appear the metric is (+, −, −, −). Topic 6.06 left one spin term at order (v/c)², −(qħ/(8m²c²)) σ⋅(E × π − π × E), quoted from there. Specialise in three steps. Static fields: ∇ × E = −∂B/∂t = 0, and since (π × E)ψ = −iħ(∇ × E)ψ − (E × π)ψ, the antisymmetric pair collapses to 2E × π. No magnetic field: π = p. Central potential energy V(r) = qΦ(r): qE = −∇V = −(1/r)(dV/dr) r. Then −(ħ/(4m²c²)) σ⋅(qE × p) = (ħ/(4m²c²))(1/r)(dV/dr) σ⋅(r × p) = (1/(2m²c²))(1/r)(dV/dr) S⋅L, derived, with S = ħσ/2 and L = r × p. (1/r)(dV/dr) depends on r alone and commutes with every component of L, so the operator is Hermitian. Dimensions: J m⁻² × J² s² ÷ (kg² m² s⁻²) = J. Limit: as c → ∞ it vanishes, as a relativistic correction must.

For hydrogen V = −e²/(4πε₀r), so (1/r)(dV/dr) = e²/(4πε₀r³) and H_SO = (e²/(8πε₀))(1/(m²c²r³)) S⋅L, derived. For energies take Quantum Mechanics II's expectation values, quoted: ⟨1/r³⟩ = 1/(a₀³n³l(l + ½)(l + 1)) and ⟨S⋅L⟩ = (ħ²/2)[j(j + 1) − l(l + 1) − ¾]. For 2p, ⟨1/r³⟩ = 1/(24a₀³), and ⟨S⋅L⟩ is +ħ²/2 at j = 3/2 and −ħ² at j = 1/2, so 2P₃/₂ rises by mc²α⁴/96, which is h × 3.65 GHz, and 2P₁/₂ falls by mc²α⁴/48, h × 7.30 GHz; weighted by 2j + 1 the shifts cancel, 4 × 3.65 = 2 × 7.30. The doublet mc²α⁴/32 is h × 10 949 MHz, or h × 10 943 MHz with the reduced mass (quoted), against the measured 10 969.0 MHz. The kinetic correction depends only on n and l, and the Darwin term vanishes for l = 1, so this interval is the spin-orbit term alone: a clean test of its coefficient.

Quantum Mechanics II took another road. In the electron's instantaneous rest frame the nucleus moves, and its field transforms into B′ = −v × E/c²; with v = p/m and qE = −(1/r)(dV/dr) r, the moment μ = g(q/2m)S has energy −μ⋅B′ = (g/(2m²c²))(1/r)(dV/dr) S⋅L, derived. At g = 2 that is exactly twice the Dirac operator, and QM-II halved it by quoting Thomas. The half is not cosmetic: unhalved, the operator predicts a 2P doublet of 21.9 GHz, double the measured 10.97 GHz. The Dirac reduction needed no halving because it never left the nucleus's frame; the 1/(2m²c²) came out of topic 6.06's generator and its double commutator. The rest-frame argument is not wrong, only incomplete: the rest frame of an accelerating electron is itself rotating relative to the lab, and the energy of that rotation must be added before the two roads can agree.

Topic 6.04 carried Pauli's term through the same reduction and found the spin-orbit coefficient multiplied by 1 + 2a, derived there: the term's odd electric half, crossed with cα⋅p, adds 2a times topic 6.06's spin term. Since g = 2(1 + a), 1 + 2a is g − 1, and that rewriting is the point. The rest-frame count above gives a magnetic energy g/(2m²c²) for every g, so subtracting it from the Dirac result leaves −1/(2m²c²), a term with no g in it. It cannot be magnetic, because it survives when the moment is switched off. It is kinematic, the precession of the accelerating rest frame, which topic 6.09 derives from non-commuting boosts. Neither piece was inserted: the reduction produced their sum, and the dependence on the moment sorts them. Hydrogen agrees: (g − 1) × 10 943.32 MHz = 1.002 319 × 10 943.32 MHz = 10 968.70 MHz, against the measured 10 969.0 MHz. Limits: at g = 1 magnetism and Thomas cancel and the coupling vanishes; at g = 0 a spinning body with no moment still precesses, with coefficient −1.

All of this assumed V = qΦ, the time component of a four-vector. A Lorentz-scalar potential W(r), written W so that S stays the spin, enters as a position-dependent mass: H = cα⋅p + β(mc² + W) + V. Eliminate the small component as in topic 6.01: χ = [2mc² + E′ − V + W]⁻¹ cσ⋅p φ, with E′ = E − mc². Expanded once, the spin-dependent part is −(1/(4m²c²)) σ⋅p U σ⋅p with U = W − V, and σ⋅p U σ⋅p = Up² − iħ∇U⋅p + ħσ⋅(∇U × p) gives H_SO = (1/(2m²c²))(1/r) d(V − W)/dr S⋅L, derived. The renormalisation topic 6.05 needed adds only spin-independent terms, so the coefficient stands. The bookkeeping agrees: a scalar field has no rest-frame magnetic field, but it accelerates the particle, so Thomas's −1 acts on V + W while g acts on V alone. Nuclear mean fields, a scalar near −400 MeV and a vector near +350 MeV, quoted, add in V − W, which is why nuclear p₃/₂ lies about 6 MeV below p₁/₂ in ¹⁶O while hydrogen's 2P₃/₂ lies above 2P₁/₂.

The operator is order (v/c)² relative to the Bohr energies: hydrogen's 2P doublet over its n = 2 binding is (mc²α⁴/32)/(mc²α²/8) = α²/4 = 1.3 × 10⁻⁵. The reduction kept that order and nothing else, so it holds while Zα is small. For hydrogen-like uranium Zα = 0.67, and the leading-order doublet Z⁴mc²α⁴/32 = 3.24 keV falls 29% short of the 4.57 keV that Unit 7's exact Dirac levels give, quoted. For s states L annihilates the wavefunction, so the operator gives zero even though ⟨1/r³⟩ diverges; the s-state energy the n- and j-only formula needs comes from the Darwin term, topic 6.08. And the anomaly is quoted, never derived here: Schwinger's α/(2π) = 0.001 161 and the measured 0.001 159 652 come from the quantised field, which this operator does not contain. At hydrogen's precision the 0.34 MHz left after (g − 1) × 10 943.32 MHz is the size of the terms this order drops, α² × 10 943 MHz = 0.58 MHz.