Exact Klein-Gordon Coulomb levels and pionic atoms

Units are SI with ħ and c kept, and the metric is diag(1, −1, −1, −1) as fixed in topic 3.01, though only the radial equation is needed now. Topic 3.06 brought the bound problem to u″ + [−κ² + 2ZαE/(ħc r) − l′(l′ + 1)/r²]u = 0, with κ = √(m²c⁴ − E²)/(ħc). Quantum Mechanics II's equation was the same with 2Zαmc/(ħr) in the middle, so with ρ = κr its analysis applies line for line: ρ₀ = 2ZαE/(ħcκ), u = ρ^(l′+1) e^(−ρ) v(ρ), and the recursion for v ends, as it must for a normalisable u, only if ρ₀ = 2(nᵣ + l′ + 1) with nᵣ = 0, 1, 2, … Integer l was never used, only the choice of the ρ^(l′+1) root at the origin, which topic 3.08 examines. So ZαE = N√(m²c⁴ − E²), where N = nᵣ + l′ + 1 = n − (l + ½) + √((l + ½)² − (Zα)²) and n = nᵣ + l + 1. The right side is positive, so E must be too, and squaring gives E²[N² + (Zα)²] = N²m²c⁴, that is E = mc²[1 + (Zα)²/N²]^(−1/2). Derived.

Dimensions: Zα and N are pure numbers, so E is mc² times a number. Limit: as Zα → 0, N → n and E → mc² − mc²(Zα)²/(2n²), the Bohr level plus the rest energy, with its l-degeneracy intact. What is new is that N depends on l. The (Zα)²/r² piece of the squared potential, an extra attraction that topic 3.06 folded into l′, matters most where the centrifugal barrier is weakest: for hydrogen's 1s, l′ = −½ + √(¼ − α²) = −5.3 × 10⁻⁵, while for 2p it falls only 1.8 × 10⁻⁵ below 1. Low-l states are drawn further in and bind more, and the accidental l-degeneracy of the Coulomb problem is gone. Now set Dirac's formula, quoted in Quantum Mechanics II and derived in Unit 7, beside it: its N is n − (j + ½) + √((j + ½)² − (Zα)²). The two are identical except that the spinless particle carries l + ½, a half-integer, where the electron carries j + ½, an integer. Sommerfeld's 1916 formula also had the integer, quoted, which is why it fitted the fine structure the spinless formula misses.

Write δ = l + ½ − √((l + ½)² − (Zα)²) ≈ (Zα)²/(2l + 1), so N = n − δ and 1/N² ≈ 1/n² + 2(Zα)²/((2l + 1)n³). Expanding [1 + y]^(−1/2) = 1 − y/2 + 3y²/8 with y = (Zα)²/N² gives E ≈ mc² − mc²(Zα)²/(2n²) − (mc²(Zα)⁴/(2n⁴))[n/(l + ½) − ¾], derived. Quantum Mechanics II evaluated ⟨−p⁴/(8m³c²)⟩ in the hydrogen state nl by writing p² = 2m(Eₙ − V), and found −(Eₙ²/(2mc²))[4n/(l + ½) − 3]; with Eₙ = −mc²(Zα)²/(2n²) that is the same expression, state by state. Topic 3.03 showed that the stationary equation shifts levels by exactly this operator at first order, and the exact solution now confirms that nothing else appears: no contact term and no l⋅s, because a scalar wave function has no spin to couple. The cost sits in the coefficient, n/(l + ½) where hydrogen's fine structure has n/(j + ½). For hydrogen's 1s the spinless term is −(1.25/2)mc²α⁴ = −906 μeV; Dirac's, with 1 − ¾ in the bracket, is −181 μeV.

Set the two n = 2 spectra side by side in units of mc²α⁴ = 1.449 meV, which is h × 350.4 GHz. Spinless: 2s at −13/128 (35.6 GHz below Bohr) and 2p at −7/384 (6.39 GHz below), so 2s lies 1/12 of the unit, 120.8 μeV or 29.2 GHz, below 2p. Dirac: 2s₁/₂ and 2p₁/₂ together at −5/128 (13.7 GHz below), 2p₃/₂ at −1/128 (2.74 GHz below). The gaps are Quantum Mechanics II's quoted terms, computed: 2s₁/₂ − 2s = +1/16, the Darwin term mc²α⁴/(2n³); 2p₃/₂ − 2p = +1/96 and 2p₁/₂ − 2p = −1/48, the spin-orbit term with its Thomas factor. Weighted 4 : 2 the spin-orbit shifts cancel, so the spinless 2p is the centroid of hydrogen's doublet and only the s level is wrong on average. Measurement decides, quoted: 2p₃/₂ − 2p₁/₂ = 10.97 GHz, and 2s₁/₂ sits 1.06 GHz above 2p₁/₂. Schrödinger met this clash with Sommerfeld's fine structure in the winter of 1925 to 1926, shelved the relativistic equation and published the non-relativistic one, quoted.

A negative pion has spin 0, quoted, so the Klein-Gordon equation is its equation, with the reduced mass μ = m_πM/(m_π + M) in place of m. Its orbits are small: in pionic hydrogen μc² = 121.5 MeV and the Bohr radius ħ/(μcα) is 223 fm, inside the electron's reduced Compton wavelength of 386 fm. There the pion sees the proton's charge less screened by virtual electron-positron pairs, and vacuum polarisation, which the course quotes rather than derives, adds 3.24 eV to the 1s binding against 0.22 eV from relativity. With both included, circular orbits, l = n − 1, which keep clear of the nucleus, give X-ray lines such as pionic nitrogen's 5g → 4f near 4.055 keV, measured with crystal spectrometers to a few ppm; lines like it fix m_πc² = 139.570 39(18) MeV, quoted. The lowest levels do reach the nucleus, and whatever the electromagnetic calculation leaves unexplained measures the strong interaction: pionic hydrogen's 1s is 7.086 eV more bound than electromagnetism predicts and 0.856 eV wide, quoted. Topic 3.09 explains why these energies survive although the one-particle reading does not.