Dipole selection rules and hydrogen spectra

Light is a travelling wave, so the perturbation the atom feels is E₀ ε cos(k⋅r − ωt), and Fermi's golden rule makes the transition rate proportional to |⟨f| e^(ik⋅r) ε⋅p |i⟩|². The exponential is the awkward factor, and one number disposes of it. For Lyman-α the wavelength is 121.6 nm while the electron cloud is about a₀ = 0.0529 nm across, so the phase changes by only k a₀ = ω a₀/c = 2.7 × 10⁻³ from one side of the atom to the other. More generally, a transition of order one Rydberg gives k a₀ ≈ α/2 = 3.6 × 10⁻³. Set e^(ik⋅r) = 1 and the field is uniform over the whole atom; what survives, after the standard commutator step that converts ε⋅p into r, is the electric-dipole operator d = −e r. Every rule in this topic is a statement about that single vector. The price is stated in the same breath: the next term in the expansion, i k⋅r, is smaller by 10⁻³, and the rates it produces are smaller by (k a₀)² ≈ 10⁻⁵. Nothing is being forbidden here — it is being postponed.

Under the parity operation r → −r a hydrogen eigenfunction picks up a sign fixed entirely by l: P ψₙₗₘ = (−1)^l ψₙₗₘ, because the radial factor is untouched and Yₗₘ(π − θ, φ + π) = (−1)^l Yₗₘ(θ, φ). The dipole operator is itself odd, since r → −r by construction. So the integrand of ⟨f|r|i⟩ carries the overall sign (−1)^(l_f) × (−1) × (−1)^(lᵢ), and the integral of an odd function over all space is zero. The matrix element can survive only if l_f + lᵢ is odd, that is, only if Δl is odd. That one line — Laporte's rule — kills every Δl = 0 transition at once: s → s, p → p, d → d, and with them 2s → 1s. It has done half the work of the topic without evaluating a single integral, and it costs nothing beyond noticing that a central potential is invariant under inversion, so parity is a good quantum number in the first place.

Parity says Δl is odd; it does not say Δl = ±1. For that, write the components of r in the same language as the wavefunctions: z = r √(4π/3) Y₁₀, and x ± iy = ∓ r √(8π/3) Y_(1±1). Each component is r times a Y₁q with q = 0 or ±1, so the angular part of ⟨f|r|i⟩ is ∫ Y*_{(lf mf)} Y₁q Y_{(lᵢ mᵢ)} dΩ — an integral of three spherical harmonics. It vanishes unless l_f, lᵢ and 1 satisfy the triangle inequality |lᵢ − 1| ≤ l_f ≤ lᵢ + 1, and unless m_f = mᵢ + q. The triangle rule permits l_f = lᵢ − 1, lᵢ or lᵢ + 1; parity has already removed the middle case. What is left is Δl = ±1 and Δm = 0, ±1, exactly. Read physically, the photon is a spin-one object of odd parity, and the atom must hand over precisely the one unit of angular momentum it carries away. The value of q is the polarisation — q = 0 is light linearly polarised along z, q = ±1 is circular — which is why the components of a Zeeman-split line are polarised differently.

The radial factor ∫₀^∞ R_{(nf lf)}(r) r R_{(nᵢ lᵢ)}(r) r² dr is where every quantum number except l has been sent, and it enforces nothing. It is tempting to hope orthogonality will kill some of them, but radial functions are orthogonal only within the same l, and Δl = ±1 guarantees the two belong to different l — different radial equations, no orthogonality to invoke. The integral is therefore generically non-zero for any Δn, including Δn = 0: in hydrogen 3p → 3s radiates nothing only because the two are degenerate, while in sodium, where screening lifts that degeneracy, the very same Δn = 0, Δl = −1 transition is the yellow D lines at 589 nm. What the radial integral does control is strength. It shrinks as the two radial extents diverge, going as nᵢ^(−3/2) once nᵢ is large, and A goes as ω³ times its square. Across the Lyman series the rates run 6.26 × 10⁸, 1.67 × 10⁸ and 6.82 × 10⁷ s⁻¹ for 2p, 3p and 4p → 1s — a ratio of 1 : 0.27 : 0.11. Weak and forbidden are different words, and only one of them is a selection rule.

Point the rules at hydrogen and the observed series fall out. The Lyman series is np → 1s and nothing else, because 1s has l = 0 and only l = 1 is one step away; a 3d atom therefore cannot reach the ground state by one dipole photon and must cascade 3d → 2p → 1s. The Balmer series is richer, because n = 2 offers both parities: ns → 2p, nd → 2p and np → 2s are all allowed. So Balmer-α, the red line at 656.3 nm, is not one transition but a blend of 3s → 2p, 3p → 2s and 3d → 2p, degenerate in the pure Coulomb solution and separated only at the 0.01 nm level by fine structure. The three are far from equal in rate — 6.47 × 10⁷ s⁻¹ for 3d → 2p, 2.24 × 10⁷ s⁻¹ for 3p → 2s and 6.31 × 10⁶ s⁻¹ for 3s → 2p — so how much each contributes to the line depends on those rates and on how the n = 3 sublevels are populated. Now switch on a magnetic field and the Δm rule becomes visible. Ignore spin for a moment: the emitted photon energy shifts by (mᵢ − m_f) μ_B B, and Δm is restricted to 0 and ±1, so however many sublevels the two levels carry, the line splits into exactly three — the normal Zeeman pattern. Balmer-α's fifteen separate dipole channels produce only three frequencies.

The 2s state is the rules' own test case. It cannot reach 1s by an electric-dipole photon, since both states have l = 0. Electric quadrupole is closed too: that operator goes as Y₂q, and its angular integral between two spherically symmetric s states vanishes. Magnetic dipole is nearly closed, because the operator L + 2S does not touch the radial functions and ⟨1s|2s⟩ = 0; only a relativistic correction cracks it open, at about 10⁻⁶ s⁻¹. What remains is two photons emitted at once, sharing the 10.2 eV in any proportion, at a rate of 8.23 s⁻¹ — a lifetime of 0.122 s against 1.60 ns for 2p, a factor of 7.6 × 10⁷. That factor is what a selection rule is worth. It is also observable twice over: 2s hydrogen in a nebula lives long enough to radiate a smooth two-photon continuum instead of a line, and a stray electric field that mixes a little 2p into 2s quenches the state at once.