Fall to the centre: Zα above l + ½

Units are SI with ħ and c kept, the metric diag(1, −1, −1, −1) of topic 3.01, and V = −Zαħc/r. Topic 3.06 derived the radial equation u″ + [(E² − m²c⁴)/(ħc)² + 2EZα/(ħc r) − (l(l + 1) − (Zα)²)/r²]u = 0, quoted here. As r → 0 the bracketed terms grow as r⁰, r⁻¹ and r⁻², so only the last can balance u″. Put u = r^s: s(s − 1) = l(l + 1) − (Zα)², so s = ½ ± √((l + ½)² − (Zα)²), derived; these are l′ + 1 and −l′ in topic 3.06's notation. The (Zα)² comes from V² in (E − V)², an attraction with the centrifugal term's shape that Schrödinger's equation never had. Checks: s is a pure number, and Zα → 0 returns Schrödinger's l + 1 and −l. For hydrogen's 1s, s₊ = 1 − 5.3 × 10⁻⁵, so the radial function u/r grows as r^{(−5.3 × 10⁻⁵)}: the exact Klein-Gordon ground state diverges at the origin, yet the factor r^{(−5.3 × 10⁻⁵)} is only 0.07% larger at r = 10⁻⁶ a₀ than at a₀. Harmless here, it is a warning of what large Zα will do.

Near r = 0, ∫|u|² dr converges only if 2 Re s + 1 > 0, that is Re s > −½. For l ≥ 1 the smaller root sits near −l and fails: at Z = 82, l = 1 gives s₋ = −0.875, and it passes only once (Zα)² exceeds 5/4, at Z ≥ 154. For s states below Zα = ½ both roots lie between 0 and 1: at Z = 30, s₊ = 0.950 and s₋ = 0.050, and u → 0 for each. The Klein-Gordon density carries an extra factor (E − V)/mc² ∝ 1/r near the nucleus, and even that integral converges, because 2s₋ > 0. Schrödinger's other argument fails too: there a constant u meant a radial function ∝ 1/r, whose Laplacian is a δ function, but r^(s₋) with s₋ ≠ 0 produces none. So the point-Coulomb equation leaves a one-parameter family of s waves, and keeping r^(s₊) is a boundary condition imposed from outside, stated rather than forced. The next section supplies the physics behind it.

Inside a nucleus of radius R the potential is finite, so the solution there is regular and reaches R with a logarithmic derivative λ = R u′/u. Once Zαħc/R dwarfs E and mc² the interior has no length but R, so λ is a pure number independent of R; for a charged shell, V is constant inside, the interior solution is sin(Zαr/R) and λ = Zα cot(Zα). Outside, u = a r^(s₊) + b r^(s₋), and matching at R gives b/a = R^{(s₊ − s₋)}(s₊ − λ)/(λ − s₋), derived. Since s₊ − s₋ = 2√(¼ − (Zα)²) is positive, b/a → 0 as R → 0: the point-nucleus limit is r^(s₊), and that is the reason for the choice. It converges slowly where the power is small. At Z = 10 it is 0.989, close to Schrödinger's 1; at Z = 68 it is 0.123, so a nucleus a million times smaller still leaves b at 0.18 of its value. Close to Zα = ½ the wavefunction remembers the nucleus.

When (Zα)² exceeds (l + ½)² the roots are s = ½ ± iσ with σ = √((Zα)² − (l + ½)²). Since r^(iσ) = cos(σ ln r) + i sin(σ ln r), the real solutions are u = A√r cos(σ ln(r/r₁) + δ), derived; the length r₁ is there because a logarithm needs a pure number, and changing it only shifts δ. Nodes sit where the cosine vanishes, at radii in the fixed ratio e^(π/σ), so infinitely many lie between any radius and the origin. Both conjugate solutions give |u|² ∝ r, equally normalisable, and the near-origin equation holds no length that could fix δ. With the measured α the thresholds are Z_c = 68.52 for s states, 205.55 for p and 342.59 for d, so only s states of real nuclei are in reach. At Z = 92, σ = 0.448: one node per factor 1110 in r, 0.33 per decade, so a point nucleus would add about one node between 0.3 fm and 300 fm and hide the rest at radii nothing can resolve.

Near the origin the equation is u″ + [(Zα)² − l(l + 1)]u/r² = 0, Schrödinger's zero-energy equation in an attractive −γħ²/(2mr²) with γ = (Zα)² − l(l + 1). That potential has no length of its own: if u(r) is a bound state of a pure r⁻² well at energy E, then u(ηr) is one at η²E for every η > 0, so one bound state implies levels without a floor. Below γ = ¼ it binds nothing; above it the zero-energy solution has infinitely many nodes, and by the oscillation theorem each marks a deeper state. This is Landau's fall to the centre, derived. Classically the particle falls once the attraction beats L²/2mr²; quantum mechanics adds ¼ to the margin, which is why the threshold is (l + ½)², not l(l + 1). The energy formula records it: E₁ₛ = mc²√(½ + √(¼ − (Zα)²)), simplified here from topic 3.07, reaches mc²/√2 at Zα = ½ and is complex beyond. A complex E makes e^(−iEt/ħ) grow or decay, so it labels no stationary state.

Give the nucleus a radius and the matching fixes δ: at fixed r the phase is σ ln(r/R) − φ, with tan φ = (λ − ½)/σ. The levels are then real but depend on R, and the point limit does not exist: each time R shrinks by e^(π/σ) another node squeezes out of the nucleus. For real nuclei, R ≈ 1.2A¹⁄³ fm from electron scattering (7.1 fm for lead, quoted), and the window between R and the edge of the near-origin region, about ħ/(2mcZα), spans only a few e-folds: too few for one node, so the levels are finite, and they shift with R. The spinless pion is the test. In pionic lead, Z = 82 and Zα = 0.598 is above ½, so the point formula is complex, yet a 1s level bound by about 7 MeV has been measured, quoted, its energy set by the nuclear size and the pion's strong interaction. The same log-periodic tower is the Efimov effect of three-body physics: ratio 22.7 predicted, about 21 measured in caesium, quoted.