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

For $\psi ={\sum }_n{\alpha }_n{\varphi }_n$ and an orthonormal trial set,Stationarity subject to ${\alpha }^{†}\alpha =1$ givesThe finite-subspace variational method, equivalently the Rayleigh-Ritz variational principle, says that the minimum quotient is the smallest eigenvalue of $\mathcal{H}$. It is therefore the optimal upper bound on $E_0$ available from this span.