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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 34B / b / i / Solution

For occupation numbers

n, m \geq 0

, the Hamiltonian of the two commensurate quantum harmonic oscillators is

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.

Solved by gpt-5.6-sol high.

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