Rayleigh-Ritz variational principle

The extremal Rayleigh quotients of a self-adjoint operator are the extremal spectral values. In particular, a Hamiltonian with a discrete spectrum satisfies $R[\psi ]\ge E_0$, with equality precisely on its ground-state eigenspace. More generally, minimizing over states orthogonal to the first $j$ eigenspaces gives an upper bound on $E_j$.