Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 36E (Applications of Quantum Mechanics)

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.

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.

The kinetic matrix is diagonal:For the linear potential,andThus, in units ${\hslash }^2/(m{a}^2)$,The lower eigenvalue gives the two-mode variational bound for a linearly tilted square well