 Solution

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

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

For

- j \leq m < j

, the matrix element of

[O, J_{+}]

between

∣ j, m + 1 ⟩

and

∣ j, m ⟩

gives

C_{+} (j, m) (⟨ j, m + 1∣ O ∣ j, m + 1 ⟩ - ⟨ j, m ∣ O ∣ j, m ⟩) = 0.

The ladder coefficient is nonzero, so adjacent diagonal entries agree. Induction makes the diagonal entry independent of

m

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