 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1/a1/a/ii/2/solution

For two eigenfunctions, apply the same Hermitian operator identity across the two states:

q_{n} ⟨ ψ_{m}, ψ_{n} ⟩ = ⟨ ψ_{m}, Q ψ_{n} ⟩ = ⟨ Q ψ_{m}, ψ_{n} ⟩ = q_{m} ⟨ ψ_{m}, ψ_{n} ⟩ .

Hence

(q_{n} - q_{m}) ⟨ ψ_{m}, ψ_{n} ⟩ = 0

. Distinct eigenvalues therefore imply orthogonal eigenfunctions. Within a degenerate eigenspace, orthogonality is a possible basis choice, rather than a consequence for every pair of vectors.

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