Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 10 / a

For an eigenfunction in the operator domain, $H{\psi }_n=E_n{\psi }_n$. Substitution into the Rayleigh quotient givesThe cancellation only needs a finite nonzero norm; orthonormalization makes both denominator and norm equal to one.

Assume $H$ is self-adjoint on its specified operator domain, and the admissible variations preserve that domain and any boundary conditions, so no boundary terms spoil self-adjointness. Let $N=⟨\varphi ,\varphi ⟩>0$. The first variation of the Rayleigh quotient isAt $\varphi ={\psi }_n$, the residual vanishes, so $\boxed{\delta F[{\psi }_n]=0}$. More explicitly, for a normalized ${\psi }_n$ and $\varphi ={\psi }_n+\eta$,For a fixed admissible direction scaled by a small parameter, the change is second order. This is stationarity, and is not necessarily a minimum for an excited state.

Conversely, vanishing first variation for every admissible complex direction yields the stationary equationAt the original stationary point, this is $(H-F[{\psi }_n]){\psi }_n=0$. The printed equation with an arbitrary perturbed $\varphi$ in the final factor is not generally true. Taking $\varphi ={\psi }_n+ϵ{\psi }_m$ with $E_m\ne E_n$ gives $(H-E_n)\varphi =ϵ(E_m-E_n){\psi }_m\ne 0$, even though the first-order change in $F$ is zero. Thus the valid claim is the stationary equation at the extremal state, not an exact eigenvalue equation for every variation.

Assume a complete orthonormal eigenbasis and a self-adjoint operator bounded below, with the trial state in its operator domain or associated energy-form domain. The expansion and orthogonality giveConsequentlyIt is a weighted average of eigenvalues, all at least $E_0$. This is the Rayleigh-Ritz variational principle; equality holds precisely when the trial state lies in the ground-state subspace. A continuous part of the spectrum would require the corresponding spectral integral in place of part of the sum.