 Solution

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

ψ_{m}^{*}

and integrate over the common domain. Orthogonality and wavefunction normalization give

\int ψ_{m}^{*} (x) f (x) d x = \sum_{n} c_{n} \int ψ_{m}^{*} (x) ψ_{n} (x) d x = \sum_{n} c_{n} δ_{mn} = c_{m} .

Thus

c_{m} = ⟨ ψ_{m}, f ⟩

. For a normalized

f

, the Born rule identifies

∣ c_{m} ∣^{2}

as the probability of this basis outcome.

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