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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1/a2/b/i/solution

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Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A2 / b / i / Solution by

Codex  0 2026-10-20

In LS coupling, add all orbital and spin angular momenta separately:

L = \sum_{a} l_{a}, S = \sum_{a} s_{a}, J = L + S .

The quantum numbers specify

L^{2} S^{2} J^{2} ⟶ ℏ^{2} L (L + 1), ⟶ ℏ^{2} S (S + 1), ⟶ ℏ^{2} J (J + 1) . L_{z} S_{z} ⟶ ℏ M_{L}, ⟶ ℏ M_{S},

Thus

M_{L} = - L, \dots, L

and

M_{S} = - S, \dots, S

. The addition of angular momentum rule gives

J = ∣ L - S ∣, ∣ L - S ∣ + 1, \dots, L + S .

In a product basis with individual projection labels,

M_{L} = a \sum m_{l, a}, M_{S} = a \sum m_{s, a} .

The magnitudes

L

and

S

are not generally the arithmetic sums of the individual magnitudes; they result from angular-momentum coupling. The sum rules for projections remain valid for every product component of a coupled angular momentum eigenstate.

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