 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/10/a/i/solution

H ψ_{n} = E_{n} ψ_{n}

. Substitution into the Rayleigh quotient gives

F [ψ_{n}] = \frac{E _{n} ⟨ ψ _{n} , ψ _{n} ⟩}{⟨ ψ _{n} , ψ _{n} ⟩} = E_{n} .

The cancellation only needs a finite nonzero norm; orthonormalization makes both denominator and norm equal to one.

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