The coupled radial equations for G(r) and F(r)

SI units throughout, with E the total energy including mc²; where four-vectors appear the metric is (+, −, −, −), though nothing here needs one. This unit's earlier topics supplied the labels, quoted here: in a central V(r), J², J_z, Dirac parity and K = β(Σ⋅L + ħ) commute with H, and K's eigenvalue −κħ fixes j = |κ| − ½ and the orbital l of the upper pair. So the upper two components are a radial function times the spinor harmonic χ_κ^m. The lower two share j and m but, because Dirac parity is β times spatial inversion, have the opposite orbital parity, so they carry χ_(−κ)^m with l′ = 2j − l. Write ψ = ((G/r)χ_κ^m, i(F/r)χ_(−κ)^m)ᵀ. The 1/r does the job of QM-II's u = rR, and the i is a choice that will make both equations real. The harmonics are orthonormal on the sphere, so ∫ψ†ψ d³r = ∫₀^∞ (G² + F²) dr: G and F carry m^(−1/2), and one integral normalises the pair. Four complex functions of three coordinates have become two real functions of one.

The only derivative in H = cα⋅p + βmc² + V sits in cσ⋅p, which mixes radius and angle. Insert (σ⋅r̂)² = 1 on the left and use (σ⋅a)(σ⋅b) = a⋅b + iσ⋅(a × b), valid for operators if their order is kept: σ⋅p = (σ⋅r̂)[r̂⋅p + iσ⋅(r̂ × p)]. With r̂ standing to the left of p, r̂⋅p = −iħ∂/∂r and r̂ × p = L/r, so σ⋅p = (σ⋅r̂)(−iħ∂/∂r + iσ⋅L/r), derived. The angular part is already diagonal: since (σ⋅L + ħ)χ_κ^m = −κħχ_κ^m, σ⋅L is −(1 + κ)ħ on χ_κ^m and (κ − 1)ħ on χ_(−κ)^m. And σ⋅r̂, which commutes with J but is odd under inversion, swaps the two harmonics, σ⋅r̂ χ_κ^m = −χ_(−κ)^m in the previous topic's phase convention. On (f/r)χ_κ^m the bracket gives −iħ[f′/r − f/r² + (1 + κ)f/r²] = −(iħ/r)(f′ + κf/r), and σ⋅r̂ then supplies −χ_(−κ)^m. In one line: σ⋅p[(f/r)χ_(±κ)^m] = (iħ/r)(f′ ± κf/r)χ_(∓κ)^m.

Call the upper pair φ = (G/r)χ_κ^m and the lower pair χ = i(F/r)χ_(−κ)^m. The lower row of Hψ = Eψ is cσ⋅pφ = (E + mc² − V)χ: its left side is (iħc/r)(G′ + κG/r)χ_(−κ)^m and its right side i(E + mc² − V)(F/r)χ_(−κ)^m, so the i, the 1/r and the harmonic cancel and dG/dr = −(κ/r)G + (E + mc² − V)F/ħc. The upper row, cσ⋅pχ = (E − mc² − V)φ, gives i × (iħc/r)(F′ − κF/r) = −(ħc/r)(F′ − κF/r) on χ_κ^m, so dF/dr = (κ/r)F − (E − mc² − V)G/ħc. Both are derived, for any central V, with nothing dropped. Dimensions: κ/r and an energy over ħc are both m⁻¹. Every coefficient is real, so G and F can be taken real; the radial current cψ†(α⋅r̂)ψ ∝ iGF(|χ_(−κ)^m|² − |χ_κ^m|²) then vanishes, since σ⋅r̂ is unitary and the two angular densities are equal. The system's matrix is traceless, so G₁F₂ − F₁G₂ is constant for two solutions at one E: two solutions regular at r = 0 are proportional, one per κ and E, as in QM-II.

With V = 0 the pair decouples exactly. Differentiate the first equation, substitute the second, and replace (E + mc²)F/ħc by G′ + κG/r: the κG′/r terms cancel and G″ + [k² − κ(κ + 1)/r²]G = 0, with k² = (E² − m²c⁴)/(ħc)². That is QM-II's free radial equation for u = rR, and κ(κ + 1) is l(l + 1) for both signs of κ. The same steps give F″ + [k² − κ(κ − 1)/r²]F = 0, and κ(κ − 1) = l′(l′ + 1): F carries the other orbital. The regular solutions are G = kr jₗ(kr) and F = B kr j_(l′)(kr), and the first equation fixes B = sign(κ) cp/(E + mc²), derived. As p → 0, B → p/2mc, Unit 6's v/2c; as p → ∞, B → 1, and the small component of an ultrarelativistic electron is as large as the other. At a kinetic energy T = 1.00 MeV, B = 0.703 and F holds T/2E = 33% of the probability. Since E² = p²c² + m²c⁴ is exact, this case tests every sign in the derivation with no approximation to hide behind.

Solve the first equation for F, still exactly: F = ħc(G′ + κG/r)/(2mc² + ε − V), with ε = E − mc². For a slow electron ε and V are electron-volts against 2mc² = 1.022 MeV, so F ≈ (ħ/2mc)(G′ + κG/r): F has no freedom of its own. Put this into the second equation. The κG′/r terms cancel as in the free case, and multiplying by −ħc leaves −(ħ²/2m)G″ + [V + ħ²κ(κ + 1)/2mr²]G = εG, derived. That is QM-II's radial equation for u = rR, and κ(κ + 1) is l(l + 1) whether κ = −(l + 1) or κ = l, so G is QM-II's uₙₗ for either j. For hydrogen it returns ε = −13.6 eV/n², whose 2p → 1s line at 10.204 eV matches the measured Lyman-α, 10.199 eV at 121.567 nm, to 5 parts in 10⁴, the reduced-mass correction. Integrating F² by parts with the same cancellation gives ∫F² dr = ⟨T⟩/2mc² for every κ: 3.33 × 10⁻⁶ for any n = 2 level of hydrogen, Unit 6's result recovered from the radial equations.

κ(κ + 1) = 2 for both κ = −2 and κ = +1, so at leading order 2P_(3/2) and 2P_(1/2) share a G and an energy. Hydrogen's measured splitting is 10.969 GHz, or 4.536 × 10⁻⁵ eV, which is 1.33 × 10⁻⁵ of the 3.40 eV binding, α²/4: the relative size, (v/c)², of the (ε − V)/2mc² dropped from the denominator. Keep that term to first order and the radial equation gains two things: a factor 1 − (ε − V)/2mc² on its kinetic and centrifugal terms, and a new term −(ħ²V′/4m²c²)(G′ + κG/r). The new term's piece −κħ²V′G/4m²c²r is the only one that tells the two signs of κ apart, and writing −κ = 1 + σ⋅L/ħ shows it is Unit 6's spin–orbit coupling, Thomas factor included, now emerging from the radial equations; the Hard example evaluates it. The expansion itself fails where |V| is not small beside 2mc²: for Coulomb, inside r = Zαħ/2mc = Z × 1.41 fm, which is 1.41 fm in hydrogen but 130 fm in uranium, far outside its nucleus. There F is no longer slaved to G, and the exact equations must be solved as they stand, the next topic's work at the origin and at infinity.