 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/34a/d/solution

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

H = ℏ ω (N + \frac{1}{2})

. Therefore

e^{- i H t /ℏ} ∣ z ⟩ = e^{- iω t /2} ∣ z e^{- iω t} ⟩,

so the evolved state remains an

A

z e^{- iω t}

The coherent-state overlap gives

⟨ z ∣ z e^{- iω t} ⟩^{2} = exp [- 2∣ z ∣^{2} (1 - cos ω t)] = exp [- 4∣ z ∣^{2} sin^{2} \frac{ω t}{2}] .

For the classical period

T = 2 π / ω

, this probability equals one whenever

t = k T

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