Behaviour at infinity and at the origin: γ = √(κ² − (Zα)²)

Units are SI, with ħc = 197.327 MeV fm and α = e²/(4πε₀ħc) = 1/137.036; the metric signature is (+, −, −, −), and it enters here only through E² − p²c² = m²c⁴. E includes the rest energy, and κ follows topic 7.02: −1 for s₁/₂, +1 for p₁/₂, −2 for p₃/₂. Topic 7.04 derived, for any central V, dG/dr = −κG/r + (E + mc² − V)F/(ħc) and dF/dr = κF/r − (E − mc² − V)G/(ħc), normalised by ∫(G² + F²) dr = 1. Here V = −Zαħc/r, a point nucleus. Quantum Mechanics II solved its radial equation by finding the behaviour as r → ∞ and r → 0, factoring both off, and terminating a series in between; topic 7.06 does the last step. What changes is the technique at each end. The equations are first order and coupled, so a trial form turns them into a 2 × 2 linear system for the two amplitudes, and the exponent comes from requiring a non-zero solution: a vanishing determinant.

As r → ∞, κ/r and V/(ħc) fall away, leaving dG/dr = (E + mc²)F/(ħc) and dF/dr = (mc² − E)G/(ħc). Try G = a e^(−λr) and F = b e^(−λr): −ħcλ a = (E + mc²)b and −ħcλ b = (mc² − E)a, and a non-zero (a, b) needs (ħcλ)² = (mc² + E)(mc² − E) = m²c⁴ − E², derived. A normalisable solution needs real, positive λ, so a bound state has |E| below mc²; outside that window λ is imaginary and the solutions oscillate, the positive- and negative-energy continua. Dimensions: ħcλ is an energy. Limit: with E = mc² − B, (ħcλ)² = B(2mc² − B) → 2mc²B as B/mc² → 0, so λ → √(2mB)/ħ, QM-II's value; for hydrogen's 13.6 eV that is 1/a₀ = 1.89 × 10¹⁰ m⁻¹, and the relativistic factor √(1 − B/2mc²) differs from 1 by 6.7 × 10⁻⁶. The first equation fixes the ratio: F/G = b/a → −ħcλ/(E + mc²) = −√((mc² − E)/(mc² + E)). The 1/r terms dress e^(−λr) with a power of r, which topic 7.06's series carries.

As r → 0, Zαħc/r outgrows E ± mc² while κ/r grows at the same rate, so both 1/r terms stay: dG/dr = (−κG + ZαF)/r and dF/dr = (κF − ZαG)/r. Try G = a r^s and F = b r^s. Then (s + κ)a − Zα b = 0 and Zα a + (s − κ)b = 0, and a non-zero (a, b) requires (s + κ)(s − κ) + (Zα)² = 0: s² = κ² − (Zα)², so s = ±γ with γ = √(κ² − (Zα)²), derived. Dimensions: every term is a pure number. Limit Zα → 0: γ → |κ|, the smaller orbital label plus one, l + 1 for negative κ, where the upper component leads, and l′ + 1 for positive κ, where the lower one does; so the dominant component starts as QM-II's r^(l+1), with its own orbital label. What is new is that γ is not an integer once Zα ≠ 0: for |κ| = 1 it is 1 − 2.66 × 10⁻⁵ in hydrogen and 0.7411 in uranium, and the two roots ±γ differ by 2γ rather than by the integer 2l + 1.

The eigenvector of the same determinant fixes the ratio at the origin: for the r^γ root, F/G → (γ + κ)/(Zα), derived. For s₁/₂, κ = −1, this is −(1 − γ)/(Zα) ≈ −Zα/2: −0.0036 in hydrogen and −0.386 in uranium, so the upper component leads, as non-relativistic intuition expects. For p₁/₂, κ = +1, it is (1 + γ)/(Zα): 274 in hydrogen and 2.59 in uranium. The small component of a p₁/₂ state is the large one at the nucleus. Topic 7.04's estimate F ≈ (ħ/2mc)(dG/dr + κG/r), with G ∝ r², puts the crossover at r ≈ 1.5ħ/mc ≈ 580 fm in hydrogen. F carries l′ = 0, and with it a density |ψ|² ∝ r^(2γ−2) that does not vanish at the origin, which is why p₁/₂ levels, unlike p₃/₂, feel the nuclear size and parity-violating contact interactions. For |κ| = 2, γ is close to 2 and |ψ|² vanishes as r^(2γ−2) ≈ r², as a p₃/₂ density should; a d₃/₂ density does the same, through its l′ = 1 lower component, not as QM-II's r⁴.

Near r = 0 a solution can mix r^γ and r^(−γ), and one must go. Since G² + F² ∝ r^(2s), the norm ∫ r^(2s) dr converges at 0 only if 2s exceeds −1. The r^γ root passes at every Zα below |κ|, where γ is real; r^(−γ) fails, and is excluded, only while γ ≥ ½. For |κ| ≥ 2 that always holds, but for |κ| = 1 it needs Zα ≤ √3/2, that is Z ≤ 118.7, just beyond oganesson, Z = 118, the heaviest named element. From Z = 119 to 137 both roots are square-integrable. The condition that still decides is a finite mean potential energy: ⟨V⟩ = −Zαħc ∫ (G² + F²) r⁻¹ dr has integrand r^(2γ−1) for the r^γ root, finite for any positive γ, and r^(−2γ−1) for r^(−γ), infinite for every γ, derived. QM-II met the same situation in the s wave, where ψ ∝ 1/r is square-integrable and was excluded because ∇²(1/r) = −4πδ³(r). Operator theory, quoted, agrees: up to Zα = √3/2 the Hamiltonian has a single self-adjoint form, and beyond it finite ⟨V⟩ picks the distinguished one, as topic 7.09's finite nucleus confirms.

The near-origin power is not hidden: an observable that weights small r exposes it. The magnetic hyperfine energy of an s₁/₂ electron involves ∫ G F r⁻² dr, quoted, and with the exponents above its integrand near the origin goes as r^(2γ−2): it converges only for γ above ½, the same line at which r^(−γ) turns normalisable, derived. For hydrogen-like bismuth, ²⁰⁹Bi⁸²⁺, Zα = 0.6057 and γ = 0.7957. The point-nucleus Dirac 1s spinor multiplies the non-relativistic Fermi value, 2.748 eV, by 1/(γ(2γ − 1)) = 2.125, quoted, giving 5.839 eV; the measured splitting is 5.084 eV (Klaft and co-workers, 1994), 1.85 times the Fermi value. Relativity accounts for the bulk of that enhancement. The point theory overshoots by 15%, mostly because r^(γ−1) is steepest inside the nucleus, where the real charge and magnetisation are spread over about 5.5 fm, the finite-nucleus physics that topic 7.09 introduces; radiative corrections make up a small remainder. The far end is checked the same way: the measured 1s binding of U⁹¹⁺ fixes λ, as the second worked example shows.