Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 33B / b / i

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- Solution i

Solution

The unperturbed ground state is

∣0, 0 ⟩

ℏ ω

. Since

H_{int} ∣0, 0 ⟩ = \frac{λ ℏ}{2 ω} ∣1, 1 ⟩,

the first-order energy correction vanishes and first-order nondegenerate perturbation theory gives

∣Ω ⟩ = ∣0, 0 ⟩ - \frac{λ}{4 ω ^{2}} ∣1, 1 ⟩ + O (λ^{2}), E_{Ω} = ℏ ω + O (λ^{2}) .

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