Angular momentum, spin addition and selection rules

A rotation by φ about n̂ acts on states as Û = exp(−iφ n̂⋅Ĵ/ħ); the three Hermitian generators Ĵₓ, Ĵ_y, Ĵ_z are defined by that role, and their algebra [Ĵᵢ, Ĵⱼ] = iħ εᵢⱼₖ Ĵₖ is the statement that rotations about different axes do not commute. If the physics is the same in every orientation, Û†ĤÛ = Ĥ for every φ, which for infinitesimal φ is [Ĥ, Ĵᵢ] = 0. The Heisenberg equation d⟨Ĵᵢ⟩/dt = (i/ħ)⟨[Ĥ, Ĵᵢ]⟩ then vanishes, and so does the one for Ĵ² = Ĵ⋅Ĵ, which commutes with every component. Ĥ, Ĵ² and Ĵ_z share an eigenbasis, and a state that starts as |J M⟩ stays inside that (J, M) sector under any interaction contained in Ĥ — strong, electromagnetic or weak. Two things must be said about which Ĵ. It is the total for the whole system, Ĵ = Ĵ₁ ⊗ 1 + 1 ⊗ Ĵ₂ + L̂ for two bodies, and only the total commutes with an interaction that couples the parts. And because Ĵₓ, Ĵ_y and Ĵ_z do not commute among themselves, only j and one m label a state, which is why a ledger balances J and M and never a vector.

The states |j₁ m₁⟩|j₂ m₂⟩ span (2j₁+1)(2j₂+1) dimensions, and Ĵ² for the total is not diagonal in them. Build the coupled basis directly. The top state |j₁ j₁⟩|j₂ j₂⟩ has M = j₁ + j₂ and is the only state with that M, so it must be |J = j₁+j₂, M = j₁+j₂⟩. Apply Ĵ₋ = Ĵ₁₋ + Ĵ₂₋ to walk down its ladder; at each M the state orthogonal to the ladder starts the next multiplet, J = j₁ + j₂ − 1, and the process stops when the dimensions are used up, at J = |j₁ − j₂|. For 1 ⊗ ½ that is 6 = 4 + 2: J = 3/2 and J = 1/2, nothing else. For ½ ⊗ ½ it is 4 = 3 + 1, a triplet and a singlet. Two facts ride on this construction. J changes in integer steps, so whether the total is integer or half-integer is fixed by how many half-integer j's are coupled: an odd number gives a half-integer total, an even number an integer one, whatever orbital motion adds. And the maximum J = j₁ + j₂ is not the classical parallel sum: √(J(J+1)) is always less than √(j₁(j₁+1)) + √(j₂(j₂+1)), so even in the stretched state the two spins are not aligned — for ½ ⊗ ½ the triplet has ⟨Ŝ₁⋅Ŝ₂⟩ = +¼ħ², an angle of 70.5°, not 0°.

The coefficient ⟨j₁ m₁ j₂ m₂|J M⟩ is the overlap between a product state and a coupled one, so its square is the probability that a pair prepared with definite m₁ and m₂ is found with total J. Work 1 ⊗ ½ explicitly. Lowering |3/2, 3/2⟩ = |1,1⟩|½,½⟩ with Ĵ₋ gives √3 |3/2, ½⟩ = √2 |1,0⟩|½,½⟩ + |1,1⟩|½,−½⟩, so |3/2, ½⟩ = √(2/3)|1,0⟩|½,½⟩ + √(1/3)|1,1⟩|½,−½⟩, and the orthogonal combination is |½, ½⟩ = √(2/3)|1,1⟩|½,−½⟩ − √(1/3)|1,0⟩|½,½⟩ with the Condon–Shortley sign. Invert: a spin-1 particle in m₁ = +1 meeting a spin-½ in m₂ = −½ is in J = 3/2 with probability 1/3 and J = 1/2 with probability 2/3. That number sets a relative rate. If a reaction proceeds only through a J = 3/2 resonance, this spin combination feeds it at one third the strength of m₁ = +1, m₂ = +½, which is pure J = 3/2. The same coefficients run in every direction: a decaying J = 3/2, M = ½ state emits its two products with m₁ = 0, m₂ = +½ two thirds of the time and m₁ = 1, m₂ = −½ one third of the time, which is how a polarisation measurement reads a parent's spin.

Two products flying apart have a relative wavefunction whose angular part is a spherical harmonic Y_L^m, with integer L and parity (−1)^L. The ledger for a decay A → B + C therefore couples three things: s_B ⊗ s_C gives the possible channel spins S, then S ⊗ L must contain J_A. For a strong or electromagnetic decay parity is conserved too, P_A = P_B P_C (−1)^L, which usually pins L to one value. ρ⁰ (Jᴾ = 1⁻) → π⁺π⁻: S = 0, so L = 1, and (−1)(−1)(−1)¹ = −1 matches. Δ⁺⁺ (3/2⁺) → pπ⁺: S = ½, and 3/2 ∈ ½ ⊗ L needs L = 1 or 2, while parity (+1)(−1)(−1)^L = +1 needs L odd, so the decay is a pure p-wave. η (0⁻) → π⁺π⁻ would need S = 0, L = 0 and parity +1 against the η's −1: forbidden, and it is not seen, the η going to 3π and γγ instead. Identical bosons add an exchange rule: π⁰π⁰ must be symmetric, so L is even, and ρ⁰ → π⁰π⁰ is forbidden at L = 1 even though the ρ⁰ → π⁺π⁻ ledger balances. Weak decays drop the parity line but keep every angular-momentum line.

Orbital angular momentum is integer, so it can never turn a half-integer total into an integer one. Whether each side of a reaction carries a half-integer J is therefore set by its count of half-integer spins alone — odd gives half-integer, even gives integer — and rotational invariance demands the two sides agree. This is the argument that made the neutrino necessary. A neutron (½) decaying to a proton (½) and an electron (½) would take a half-integer J to an integer one, since ½ ⊗ ½ ⊗ L is integer for every L: the reaction is closed by angular momentum, not only by the continuous energy spectrum. Add a third spin-½ product and ½ ⊗ ½ ⊗ ½ ⊗ L reaches ½ and 3/2 already at L = 0, and the ledger closes: n → p + e⁻ + ν̄ₑ. The same count runs through π⁻ → μ⁻ ν̄_μ (0 fermions to 2), K⁺ → π⁺ π⁰ (0 to 0) and Λ → p π⁻ (1 to 1), where the Λ's spin ½ reaches ½ ⊗ 0 ⊗ L with L = 0 or 1, and the weak interaction, ignoring parity, uses both — the interference of those two waves is how the Λ came to reveal parity violation.

An electric-dipole photon carries one unit of angular momentum, so the transition operator is a rank-1 tensor and the Wigner–Eckart theorem writes ⟨J′ M′|T̂¹_q|J M⟩ = ⟨J M 1 q|J′ M′⟩ × ⟨J′‖T̂¹‖J⟩. The Clebsch–Gordan factor vanishes unless J′ ∈ J ⊗ 1, so ΔJ = 0, ±1 with J = 0 → J′ = 0 excluded because 0 ⊗ 1 contains no 0, and M′ = M + q with q = 0, ±1. The dipole operator also has odd parity, so parity must flip: Δℓ = ±1 for a single electron. Everything about the rate that depends on orientation sits in the coefficient; everything about the atom sits in the reduced matrix element, one number per pair of levels. Hydrogen 2p → 1s satisfies every line and lives 1.6 ns; 2s → 1s fails Δℓ = ±1 and the parity flip, and the 2s state survives 0.12 s until two-photon emission, a second-order process, takes it. That factor of 7.5 × 10⁷ in lifetime is what the dipole selection rule looks like in the laboratory: not an absolute prohibition, only the loss of the leading term.