 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/34b/b/i/solution

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

For occupation numbers

n, m \geq 0

H_{0} = N_{A} + 2 N_{B} + \frac{3}{2}, E_{n, m} = n + 2 m + \frac{3}{2} .

The five lowest distinct levels and their product-state bases are

E \frac{3}{2} \frac{5}{2} \frac{7}{2} \frac{9}{2} \frac{11}{2} states ∣ n, m ⟩ ∣0, 0 ⟩ ∣1, 0 ⟩ ∣2, 0 ⟩, ∣0, 1 ⟩ ∣3, 0 ⟩, ∣1, 1 ⟩ ∣4, 0 ⟩, ∣2, 1 ⟩, ∣0, 2 ⟩ degeneracy 11223

The repeated levels exhibit quantum degeneracy.

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