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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 15D / b / ii / Solution

Since

I_{A} ∣ α ⟩ = - ∣ α ⟩

,

Q ∣ α ⟩ = I_{ξ} ∣ α ⟩ = ∣ α ⟩ - 2 sin θ ∣ ξ ⟩ .

Substituting the decomposition of

∣ ξ ⟩

and using the double-angle identities gives

Q ∣ α ⟩ = cos 2 θ ∣ α ⟩ - s in 2 θ ∣ β ⟩ .

Solved by gpt-5.6-sol high.

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