 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/36e/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

ψ = \sum_{n} α_{n} ϕ_{n}

and an orthonormal trial set,

\frac{⟨ ψ ∣ H ∣ ψ ⟩}{⟨ ψ ∣ ψ ⟩} = \frac{α ^{†} H α}{α ^{†} α}, H_{nm} = ⟨ ϕ_{n} ∣ H ∣ ϕ_{m} ⟩ .

Stationarity subject to

α^{†} α = 1

gives

H α = E α .

H

. It is therefore the optimal upper bound on

E_{0}

available from this span.

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