Selection rules from Wigner–Eckart

UPV-11.09 built the factorisation for hydrogen, and it carries over in one line: ⟨α′J′M′|T¹_q|αJM⟩ = (−1)^{(J′−M′)} ( J′ 1 J ; −M′ q M ) ⟨α′J′‖T¹‖αJ⟩. What changes in a many-electron atom is which angular momentum the theorem may be applied to. The atomic dipole d = −e Σᵢ rᵢ is a sum of one-electron vectors, and a sum of rank-one tensors is still rank one, so the theorem holds for the atom as a whole. But the electron–electron repulsion couples each ℓᵢ to the others, and spin–orbit couples L to S, so the only angular momentum the electrons conserve exactly is their total J. The exact rules are therefore the rules on J and M; rules on L, on S or on a single electron's ℓ hold only as far as those labels do. The saving is as large as in hydrogen: for sodium's D2 line, 4 upper M values × 2 lower M values × 3 values of q make 24 amplitudes, every one a 3j symbol times the single reduced element ⟨²S₁/₂‖d‖²P₃/₂⟩, of order e a₀. In a many-electron atom that one number also absorbs everything a central-field picture leaves out — configuration mixing, core polarisation — without touching a single ratio between the 24.

A 3j symbol vanishes unless two conditions hold. Its bottom row must sum to zero, giving m′ = m + q; since q takes only the values −1, 0 and +1 for a rank-one tensor, that is ΔM = 0, ±1, with q fixed by the light itself — q = 0 is π, linearly polarised along the quantisation axis, and q = ±1 is σ±, circular about it. Its top row must close a triangle with sides j, 1 and j′, which reads |j − j′| ≤ 1 ≤ j + j′, and both halves earn their keep. The left half is the familiar ΔJ = 0, ±1; the right half is what forbids J = 0 → J′ = 0, because no triangle has sides 0, 1 and 0. Notice what is not forbidden: ΔJ = 0 is perfectly allowed away from 0 → 0. One further exact zero hides inside ΔJ = 0, though — ( j 1 j ; 0 0 0 ) = 0 — so M = 0 → M′ = 0 never appears on a J → J line.

Rotational symmetry is blind to reflection, so no 3j symbol can see parity. UPV-13.04 derives the gate from P†rP = −r, and its result is all this lesson needs: a dipole element vanishes unless the two states have opposite parity. In a many-electron atom parity belongs to the configuration, π = (−1)^{(ℓ₁ + ℓ₂ + … + ℓ_N)}, so every term of one configuration shares it and no E1 line ever joins two of them — the ³P, ¹D and ¹S terms of a p² configuration can radiate among themselves only through M1 or E2. For one jumping electron the gate sharpens to Δℓ = ±1. The two gates are genuinely independent, and each catches what the other lets through. Sodium's 3s ²S₁/₂ → 4s ²S₁/₂ passes the triangle test (1/2, 1, 1/2 is a legal triangle) and passes ΔM, yet has no E1 amplitude, because s → s does not change parity. Mercury's 6s6p ³P₀ → 6s² ¹S₀ passes parity, odd to even, and in a spinless isotope has none because 0, 1, 0 is not a triangle.

The dipole operator acts on spatial coordinates only: it is d ⊗ 1 in the spin factor. In an LS-coupled basis |L S J M⟩ the spin overlap factors straight out as δ_(S′S), giving ΔS = 0, and applying Wigner–Eckart inside the orbital space gives ΔL = 0, ±1 with L = 0 ↛ L′ = 0. Both rest on LS coupling being a good description, which in turn needs the spin–orbit interaction Σᵢ ξ(rᵢ) ℓᵢ⋅sᵢ to stay small next to the residual electrostatic splitting between terms. That sum commutes with J but not with S², because it couples each electron's spin separately: inside one term it acts as ζ L⋅S and only splits the J levels, but between terms it connects S to S ± 1 at fixed J and parity. ξ grows roughly as Z⁴, so in a heavy atom the labels leak: a nominal ³P₁ carries a small ¹P₁ admixture and borrows strength through it. Mercury, Z = 80, is the standard case — 6s6p ³P₁ → 6s² ¹S₀ at 253.7 nm has A ≈ 8 × 10⁶ s⁻¹, against A ≈ 8 × 10⁸ s⁻¹ for the fully allowed ¹P₁ line at 184.9 nm. Suppressed a hundredfold, not absent. Put the same pair of levels in helium, Z = 2: the 1s2p ³P₁ → 1s² ¹S₀ branch runs at A ≈ 2 × 10² s⁻¹ against A ≈ 1.8 × 10⁹ s⁻¹ for the allowed 1s2p ¹P₁ → 1s² ¹S₀ at 58.4 nm, a suppression of 10⁷ rather than 10². Take that comparison from ³P₁ and not from helium's famous 7900 s metastable 2 ³S₁, tempting though it is: 1s2s → 1s² is s → s, so parity blocks E1 there whatever the spin character of the state, and the lifetime measures both gates at once and neither one on its own.

Specialise to k = 1 with j′ = j: a vector operator acting inside one multiplet. Both V and J are rank one on the same space, so their reduced elements differ only by a number, and ⟨jm′|V|jm⟩ = (⟨J⋅V⟩/[ħ²j(j+1)]) ⟨jm′|J|jm⟩. Apply that to μ = −μ_B(L + 2S)/ħ and the number that falls out is the Landé factor g_J = 1 + [J(J+1) + S(S+1) − L(L+1)]/[2J(J+1)]. For ²P₃/₂, 1 + (3.75 + 0.75 − 2)/7.50 = 4/3; for ²P₁/₂, 1 + (0.75 + 0.75 − 2)/1.50 = 2/3; for ²S₁/₂, 2. A weak-field component then shifts by (gᵤ Mᵤ − gₗ Mₗ) μ_B B, so D2 opens into six lines at ±1/3, ±1 and ±5/3 in units of μ_B B, while D1 opens into four at ±2/3 and ±4/3. At 0.50 T one unit is μ_B B/h = 6.998 GHz, so D2 spans 23.3 GHz — safely inside the weak-field regime, since sodium's 3p fine-structure interval is 515 GHz. How many lines there are is the ΔM rule; where they sit is the projection theorem.