Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 34A / d

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- Solution d

Solution

The quantum harmonic oscillator Hamiltonian is

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

-eigenstate, with eigenvalue

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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