The ground state: weak singularity and small component

Work in SI units, quoting energies in keV and lengths in fm; where four-vectors appear the metric is (+, −, −, −). The nucleus is a point charge of infinite mass, V = −Zαħc/r, and topic 7.4's spinor, G(r)/r times the spinor harmonic of κ above and iF(r)/r times that of −κ below, obeys ħc(G′ + κG/r) = (E + mc² − V)F and ħc(F′ − κF/r) = −(E − mc² − V)G. For n = 1, topic 7.6 leaves nᵣ = 0 and κ = −1, so the series has a single term: try G = A r^γ e^(−λr) and F = B r^γ e^(−λr). Collect the 1/r terms. The first equation gives (γ − 1)A = ZαB, the second (γ + 1)B = −ZαA. These are two values of B/A, (γ − 1)/(Zα) and −Zα/(γ + 1), and they agree only if γ² = 1 − (Zα)². Topic 7.5's exponent at the origin returns, now as the condition that one term solve both equations. Dimensions: γ and Zα are pure numbers, and ħc/r, mc² and V are energies, as each equation requires.

The terms without 1/r give −ħcλA = (E + mc²)B and −ħcλB = −(E − mc²)A. Their product gives (ħcλ)² = m²c⁴ − E², and eliminating B/A with the 1/r result gives E = γmc² and ħcλ = Zαmc², so λ = Z/a₀ with a₀ = ħ/(mcα) = 52 918 fm. The decay length is Bohr's; everything relativistic is in γ. For hydrogen the binding mc²(1 − γ) is 13.6059 eV against Bohr's 13.6057 eV, and γmc² = mc² − (Zα)²mc²/2 − (Zα)⁴mc²/8 − … reproduces Quantum Mechanics II's 1s fine-structure shift, −(Zα)⁴mc²/8 = −0.18 meV. Scaled by the reduced mass the Dirac value is 13.59847 eV, 34 μeV above the measured ionisation energy, 13.59843 eV; that gap is the quoted 1S Lamb shift, 8.17 GHz. At Z = 92 the binding is 132.28 keV, against 115.16 keV from Bohr and 128.13 keV through order (Zα)⁴.

Because F/G is one constant at every radius, G² + F² = G²[1 + (1 − γ)/(1 + γ)] = 2G²/(1 + γ), and normalisation needs a single integral, ∫₀^∞ r^(2γ) e^{(−2Zr/a₀)} dr = Γ(2γ + 1)/(2Z/a₀)^(2γ+1). Hence A² = (1 + γ)(2Z/a₀)^(2γ+1)/[2Γ(2γ + 1)], in m^(−2γ−1) so that G² dr is a probability. At γ = 1, Γ(3) = 2 and A = 2(Z/a₀)³⁄², Schrödinger's 1s, as it must be. The share in the lower component is a ratio of identical integrals, Pₛₘₐₗₗ = [(1 − γ)/(1 + γ)] ÷ [2/(1 + γ)] = (1 − γ)/2. Hydrogen: 1.33 × 10⁻⁵. Iron, Z = 26: 0.91%. Uranium: 12.9%. Unit 6's order-v/c estimate ⟨T⟩/2mc² = (Zα)²/4 agrees with hydrogen's value to a part in 75 000, but gives 0.113 for uranium, 13% short, because (1 − γ)/2 = (Zα)²/4 + (Zα)⁴/16 + … and the higher terms are no longer small.

Quantum Mechanics II found the virial theorem from ⟨[H, r⋅p]⟩ = 0 in a stationary state; repeat it for H = cα⋅p + βmc² + V. Since [pᵢ, rⱼ] = −iħδᵢⱼ, [cα⋅p, r⋅p] = −iħ cα⋅p, while [V, r⋅p] = iħ r⋅∇V and β commutes with r⋅p. For V ∝ 1/r, r⋅∇V = −V, so ⟨cα⋅p⟩ = −⟨V⟩ and E = mc²⟨β⟩ for every bound state of the point Coulomb field. In the Dirac representation β is +1 on the upper pair and −1 on the lower, so ⟨β⟩ = 1 − 2Pₛₘₐₗₗ. With E = γmc², Pₛₘₐₗₗ = (1 − γ)/2 follows without an integral, and the two routes check each other. The same wavefunction gives ⟨1/r⟩ = Z/(γa₀), so ⟨V⟩ = −(Zα)²mc²/γ: in uranium −310.8 keV against Schrödinger's −230.3 keV, the electron sitting 1/γ = 1.35 times deeper in the potential.

The upper radial function is g = G/r ∝ r^(γ−1), so for γ < 1 the Dirac 1s wavefunction diverges at the nucleus where Schrödinger's is finite. The density ρ = (G² + F²)/(4πr²) goes as r^(2γ−2), yet stays integrable because 4πr²ρ ∝ r^(2γ). Both theories share e^{(−2Zr/a₀)}, so their ratio is exact and simple: ρ_D/ρ_S = 2(2Zr/a₀)^(2γ−2)/Γ(2γ + 1), which at γ = 1 is 2/Γ(3) = 1 at every r, as it must be. On a log–log plot the Dirac density near the origin is a line of slope 2γ − 2: −5.3 × 10⁻⁵ for hydrogen, −0.518 for uranium. At the proton's charge radius, 0.841 fm, the hydrogen ratio is 1.0006, and doubling it would take r ≈ 10^(−5653) a₀. In uranium the ratio is 10.0 at the nuclear surface, 7.56 fm, and falls below 1 only beyond 1.13 a₀/Z ≈ 650 fm, so the extra density near the nucleus is drawn from farther out: ⟨r⟩ = (2γ + 1)a₀/(2Z) is 714 fm against Schrödinger's 863 fm.

Quantum Mechanics II estimated the nuclear-size shift of an s level from the Schrödinger contact density: for 1s, ΔE = (2/3)(Zα)⁴mc²(rᵣₘₛ/ƛ_C)², with ƛ_C = ħ/mc = 386.16 fm. For hydrogen, with rᵣₘₛ = 0.841 fm, that is 4.6 neV, or 1.1 MHz, and relativity alters it by the 6 × 10⁻⁴ just found, about 0.7 kHz: small, though large enough that precision hydrogen theory includes it. For uranium, rᵣₘₛ = 5.857 fm gives 15.9 eV, but the density that estimate assumes is about ten times too small across the nucleus, and the full Dirac calculation for a finite nucleus, quoted here, gives about 199 eV, 12.5 times more. That shift is part of the 1s Lamb shift of U⁹¹⁺ measured as 460 ± 5 eV (Gumberidze and co-workers, 2005); the rest is almost all radiative. Inside a real nucleus the potential is finite, g tends to a constant and the r^(2γ−2) rise stops. The point-nucleus spinor is a good description outside the nucleus and the wrong one inside it; topic 7.9 takes it from there.