 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/33b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The coupled basis consists of the triplet

∣1, 1 ⟩ = ∣ + + ⟩, ∣1, 0 ⟩ = \frac{∣ +- ⟩ + ∣ -+ ⟩}{2}, ∣1, - 1 ⟩ = ∣ - - ⟩,

and the singlet

∣0, 0 ⟩ = \frac{∣ +- ⟩ - ∣ -+ ⟩}{2} .

1

1/ 2

, or

- 1/ 2

as displayed, and

I = \sum_{m = - 1}^{1} ∣1, m ⟩ ⟨ 1, m ∣ + ∣0, 0 ⟩ ⟨ 0, 0∣.

Because each irreducible total-spin representation occurs once, an operator commuting with total angular momentum is

O = α P_{j = 1} + β P_{j = 0} .

In the product basis, the nonzero matrix elements are

⟨ + + ∣ O ∣ + + ⟩ = ⟨ - - ∣ O ∣ - - ⟩ = α,

⟨ + - ∣ O ∣ + - ⟩ = ⟨ - + ∣ O ∣ - + ⟩ = \frac{α + β}{2},

and

⟨ + - ∣ O ∣ - + ⟩ = ⟨ - + ∣ O ∣ + - ⟩ = \frac{α - β}{2} .

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