 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/35a/b/solution

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

Write

c = cos (θ /2)

and

s = sin (θ /2)

. Then

∣ ↑_{θ} ↓_{θ} ⟩ - ∣ ↓_{θ} ↑_{θ} ⟩ = (c^{2} + s^{2}) (∣ ↑↓ ⟩ - ∣ ↓↑ ⟩) .

Thus

∣ ψ ⟩

is unchanged. It is the spin-one-half singlet state, so joint rotations leave it invariant. Equivalently, direct use of the Pauli matrices shows

(S_{i}^{(1)} + S_{i}^{(2)}) ∣ ψ ⟩ = 0 (i = x, y, z) .

It is therefore an eigenstate of every Cartesian component of the combined spin, with eigenvalue zero.

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