 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1/a1/a/ii/1/solution

Let

Q ψ_{n} = q_{n} ψ_{n}

with

ψ_{n} \neq = 0

. The Hermitian operator identity and conjugate linearity of the first slot of the inner product imply

q_{n} ⟨ ψ_{n}, ψ_{n} ⟩ = ⟨ ψ_{n}, Q ψ_{n} ⟩ = ⟨ Q ψ_{n}, ψ_{n} ⟩ = q_{n}^{*} ⟨ ψ_{n}, ψ_{n} ⟩ .

The squared norm is positive, so

q_{n} = q_{n}^{*}

: every such eigenvalue is real.

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