Adding angular momenta and Clebsch-Gordan coefficients

Four commuting operators label the product states: Ĵ₁², Ĵ₂², Ĵ₁z, Ĵ₂z give the uncoupled basis |j₁ m₁⟩|j₂ m₂⟩, and for fixed j₁ and j₂ there are (2j₁+1)(2j₂+1) of them. The total Ĵ = Ĵ₁ + Ĵ₂ obeys the standard angular-momentum commutators, because the two act on different degrees of freedom and every component of Ĵ₁ commutes with every component of Ĵ₂, so its own labels j and m are also available — but Ĵ² = Ĵ₁² + Ĵ₂² + 2Ĵ₁⋅Ĵ₂, and the cross term contains Ĵ₁₊Ĵ₂₋ + Ĵ₁₋Ĵ₂₊, which shuffles m₁ against m₂. So Ĵ² commutes with Ĵ₁² and Ĵ₂² but not with Ĵ₁z or Ĵ₂z separately, and you must choose a set: (Ĵ₁², Ĵ₂², Ĵ₁z, Ĵ₂z) or (Ĵ₁², Ĵ₂², Ĵ², Ĵz). Which one is good is decided by the Hamiltonian. An internal coupling proportional to Ĵ₁⋅Ĵ₂ — spin-orbit, spin-spin, hyperfine — is diagonal in the coupled basis; a strong external field that grips each moment separately prefers the uncoupled one. That competition is exactly the weak-field versus Paschen-Back divide of the next topic.

The largest possible m is j₁ + j₂, and only one product reaches it: m₁ = j₁ with m₂ = j₂. One state means one multiplet passes through it, so jₘₐₓ = j₁ + j₂ and j never exceeds the sum. Step down to m = j₁ + j₂ − 1 and two products qualify; the jₘₐₓ ladder claims one combination, and the single leftover must start a new multiplet with j = j₁ + j₂ − 1. Each unit step inward adds one more product and opens one more ladder, until the smaller momentum runs out of projections at m = |j₁ − j₂|; below that the count is flat and nothing new opens. Hence j runs from |j₁ − j₂| to j₁ + j₂ in integer steps — 2 min(j₁, j₂) + 1 multiplets, never a half step between neighbours even when the j values are half-integers. The audit confirms it: for j₁ = 2, j₂ = 3/2 the totals are j = 7/2, 5/2, 3/2, 1/2 with 8 + 6 + 4 + 2 = 20 states, exactly the 5 × 4 products you started with. The name is the classical limit showing through: two classical vectors of lengths J₁ and J₂ add to any magnitude between |J₁ − J₂| and J₁ + J₂, and for large j₁ and j₂ the allowed totals, spaced by ħ, fill that same range almost continuously.

Ĵz = Ĵ₁z + Ĵ₂z is an operator identity, so m = m₁ + m₂ holds for every state with no exception — any Clebsch-Gordan coefficient violating it vanishes, which is why the tables are organised by total m. Nothing similar holds for j, because the magnitudes are not arrows of length jħ. Two electron spins each have |S| = (√3/2)ħ ≈ 0.87ħ; laid parallel as arrows they would reach 1.73ħ, yet the allowed totals are √2 ħ ≈ 1.41ħ for s = 1 and exactly 0 for s = 0 — neither is the arrow sum. The honest statement is weaker and stranger: a product state of definite m₁ and m₂ generally has no definite j at all. ↑↓ has m = 0 but is an equal superposition of s = 1 and s = 0. The only products that are simultaneously coupled states are the two stretched ones at m = ±(j₁ + j₂), the ends of the m range, where a single product has nothing to mix with.

Every Clebsch-Gordan table is built with one tool. Start at the unique top: for two spin-½ particles, |1, 1⟩ = ↑↑, forced because no other product has m = 1. Apply Ĵ₋ = Ŝ₁₋ + Ŝ₂₋. The left side gives ħ√(1⋅2 − 1⋅0) = √2 ħ times |1, 0⟩; on the right each one-particle lowering carries the factor ħ√(¾ + ¼) = ħ, producing ħ(↓↑ + ↑↓). Dividing through, |1, 0⟩ = (↑↓ + ↓↑)/√2 — both coefficients 1/√2, and no integral was ever computed. The m = 0 subspace is two-dimensional, so one combination is left over; orthogonality forces |0, 0⟩ = (↑↓ − ↓↑)/√2, with the overall sign fixed by the Condon–Shortley convention that the coefficient of the highest m₁ is positive. One more lowering gives |1, −1⟩ = ↓↓, completing the 4 × 4 unitary matrix. Larger tables need nothing new: lower until each ladder ends, orthogonalise whenever a new multiplet opens, and the phase convention keeps every entry real.

An electron in an orbital of quantum number l couples its orbit to its spin, giving j = l + ½ or l − ½ — two fine-structure levels — with j = ½ alone when l = 0, which is why s terms show no doublet. The general coefficients are worth knowing: |l+½, m⟩ = √((l+m+½)/(2l+1)) |mₗ = m−½⟩↑ + √((l−m+½)/(2l+1)) |mₗ = m+½⟩↓, and the orthogonal combination builds j = l−½. For a p electron, l = 1: |3/2, ½⟩ = √(2/3)|mₗ = 0⟩↑ + √(1/3)|mₗ = 1⟩↓. Read the structure off the square roots: at the stretched rungs m = ±(l+½) one of them vanishes and the state is a single product, while mid-ladder the blend is strongest. The dimension audit holds as always: 2(2l+1) = (2l+2) + 2l, so for l = 1 the six products regroup into four j = 3/2 states and two j = 1/2 states.

Each squared coefficient is a measurement probability. In the triplet |1, 0⟩ the amplitude on ↑↓ is 1/√2, so measuring S₁z returns +ħ/2 exactly half the time: coupling to definite total spin has made the individual projections maximally uncertain. That is the cost. The payoff is that in the coupled basis Ĵ₁⋅Ĵ₂ = (Ĵ² − Ĵ₁² − Ĵ₂²)/2 collapses to a number. For l ⊗ ½ it equals +lħ²/2 when j = l+½ and −(l+1)ħ²/2 when j = l−½; a p electron gives +ħ²/2 and −ħ², so a spin-orbit Hamiltonian ξ L̂⋅Ŝ splits the level into a doublet of separation ξ(2l+1)ħ²/2 — for l = 1, precisely 3ξħ²/2. No wavefunction integral was needed: the angular content of fine structure is pure coupling algebra, and the radial physics enters only through ξ. That is the hand-off to the next topic, where term symbols, the interval rule, and the Landé g factor all consume j, l and s through exactly this machinery.