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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 36E / b / Solution

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 .

The finite-subspace variational method, equivalently the Rayleigh-Ritz variational principle, says that the minimum quotient is the smallest eigenvalue of

H

. It is therefore the optimal upper bound on

E_{0}

available from this span.

Solved by gpt-5.6-sol high.

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