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

For any normalized trial state $|\psi ⟩$, the spectral decomposition gives the variational method boundChoose a family $|\psi (\alpha )⟩$, form its Rayleigh quotient, and minimize over the variational parameters. Every resulting value is an upper bound, and the minimum is the best bound within that family.