Dipole matrix elements and selection rules

The Bohr frequency condition ħω = Eᵢ − E_f says where a line would sit. Whether it appears at all is a separate question, answered by Fermi's golden rule: the rate carries |⟨f| H′ |i⟩|², so a vanishing matrix element means no line no matter how good the energy match. The coupling to a plane-wave field is (e/mₑ) A⋅p with A carrying e^(ik⋅r), and across an atom that exponential barely moves: for Lyman-α, k a₀ = 2π(0.0529 nm)/(121.6 nm) = 2.7×10⁻³. Set e^(ik⋅r) ≈ 1 and the interaction becomes −d⋅E with d = −e r. Name the operator and its home: r acts on the radial and angular factors of the state space H_radial ⊗ Hₛₚₕₑᵣₑ ⊗ Hₛₚᵢₙ, and is the identity on the spin factor. Everything that follows is a statement about that one operator.

Cartesian components x, y, z mix messily under rotation. Recombine them as r¹₀ = z and r¹_(±1) = ∓(x ± iy)/√2 and the three components transform among themselves exactly as the l = 1 spherical harmonics do — that is what rank one means. Two facts follow immediately. First, the label q on r¹_q is the angular-momentum projection the photon carries along the quantisation axis, so σ⁺ light drives q = +1 and light linearly polarised along z drives q = 0. Second, r is odd under the parity operator: Π r Π† = −r, whereas H_Coulomb commutes with Π, so every basis state |n l m⟩ is a parity eigenstate with eigenvalue (−1)^l, though the l-degeneracy lets a mixture such as 2s + 2p be an energy eigenstate with no definite parity at all. Rank and parity are independent pieces of information about the same operator, and each one cuts the spectrum differently.

For any rank-k tensor, ⟨n′l′m′| T^k_q |n l m⟩ = ⟨l m; k q | l′ m′⟩ ⟨n′l′‖T^k‖n l⟩ / √(2l′+1). The Clebsch-Gordan coefficient depends only on the angular labels and is fixed by rotational symmetry; the reduced element depends on n, n′, l, l′ and nothing else — no m, no m′, no q. Count what that saves for 2p → 1s: three initial substates × one final state × three components of r is nine integrals, and all nine are one number, 1.290 a₀, times coefficients. Three of them are non-zero, each with |⟨1 m; 1 −m|0 0⟩|² = 1/3, and summing |⟨l′m′|C⁽¹⁾_q|lm⟩|² over m, m′ and q, with C⁽¹⁾_q = r¹_q/r the angular part of r, recovers the standard angular line strength |⟨l′‖C⁽¹⁾‖l⟩|² = max(l, l′), here equal to 1.

The Clebsch-Gordan factor is non-zero only when |l − 1| ≤ l′ ≤ l + 1 and m′ = m + q, giving Δl = 0, ±1 and Δmₗ = 0, ±1. That is the whole content of "the photon carries one unit of angular momentum", and it does not by itself remove Δl = 0 — though it does kill l = l′ = 0 outright, since coupling 0 to 1 can only return 1. Parity removes the rest, and it acts on the other factor: inserting Π†Π around an odd operator gives ⟨n′l′‖r‖nl⟩ = (−1)^(l+l′+1) ⟨n′l′‖r‖nl⟩, so the reduced element vanishes unless l + l′ is odd. Combine the two and only Δl = ±1 survives. The cuts are genuinely different: 3p → 2p (l = 1 → 1) passes the triangle rule and dies on parity, while 4f → 2s (l = 3 → 0) passes parity and dies on the triangle rule. Spin needs no argument at all — r is the identity on the spin factor, so ⟨mₛ′|mₛ⟩ = δ and Δmₛ = 0. Once spin-orbit coupling ties l to s, or several electrons share the angular momentum, the same theorem is restated for J, L and S, with ΔJ = 0, ±1 and 0 ↛ 0, in Selection rules from the Wigner-Eckart theorem, in Multi-Electron Atoms.

Apply them to hydrogen and the Grotrian diagram draws itself. The Lyman series is np → 1s only: Lyman-α at 121.6 nm is 2p → 1s, and the 2s level, sitting at the same energy, has no electric-dipole line at all. Balmer-α at 656.3 nm is not one transition but three — 3s → 2p, 3p → 2s and 3d → 2p, every one of them Δl = ±1 — while 3s → 2s and 3d → 2s are dead. In the pure Coulomb Hamiltonian all three components coincide, because energy depends on n alone; fine structure splits them into two main clusters about 0.33 cm⁻¹ apart, which is what a Fabry-Pérot etalon resolves and a school grating does not. Note the limits of the rules: they say allowed or forbidden, never bright or faint. Intensity needs the radial integral, the ω³ factor and the population of the upper level. Matching measured lines to these rules, intensities and fine structure included, is the work of Atomic Spectroscopy.

In the Coulomb model the 2s level has only 1s beneath it (the Lamb shift puts 2p₁/₂ just 1.06 GHz lower, a gap far too small for ω³ to matter), and 2s → 1s is Δl = 0, so no electric-dipole channel exists. The next multipoles do little better. E2 is shut exactly, since a rank-two operator cannot connect J = ½ to J = ½. M1 is not shut by symmetry: spin makes both levels J = ½, the triangle (½, 1, ½) closes, and M1 is parity-even. But its non-relativistic operator −μ_B(L + 2S)/ħ never touches the radial factor, so it meets ∫R₁₀R₂₀r²dr = 0, and relativistic corrections leave only 2.5×10⁻⁶ s⁻¹. What actually empties the level is two-photon emission, at about 8.23 s⁻¹ — a lifetime of 0.12 s against 1.6 ns for 2p, a factor of 7.6×10⁷. A different escape route runs the 21 cm line: the 1s hyperfine transition F = 1 → 0 also has Δl = 0, but both levels share the 1s radial function, so the spin part of the M1 operator meets an overlap of one instead of zero, and M1 is genuinely open — at A = 2.88×10⁻¹⁵ s⁻¹, τ ≈ 11 Myr, visible only because a galaxy supplies enough atoms to make an absurdly slow rate into a bright line. And because the 2s rule rests on a symmetry, breaking the symmetry repeals it: a static field of a few volts per centimetre mixes the near-degenerate 2s and 2p states, the mixed state inherits 2p's dipole amplitude, and the metastability is quenched.