Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 33

For a self-adjoint Hamiltonian bounded below, every nonzero admissible trial wavefunction satisfies the quantum variational principleHere $E_0$ is the lowest exact energy, and the trial function must obey the physical boundary conditions and have finite relevant integrals. Equality holds for a wavefunction in the ground-state eigenspace. The quotient is unchanged by normalization or any nonzero overall scalar.

Expand the trial wavefunction in a complete orthonormal set of exact eigenfunctions, $\psi ={\sum }_n{c}_n{\varphi }_n$, with $\hat{H}{\varphi }_n=E_n{\varphi }_n$ and $E_n\ge E_0$. The coefficients may depend on any chosen variational parameters. Orthogonality removes the cross terms in both integrals, givingConsequentlyThis proves the quantum variational principle. The denominator is positive because the trial function is nonzero. Equality requires every occupied exact state to have $E_n=E_0$, allowing for ground-state degeneracy. With a continuous part of the spectrum, the corresponding nonnegative spectral integral gives the same conclusion.

Write $g=e^{-q^2/2}$ and $f=2q^2-a$. Product differentiation for the reduced quantum harmonic oscillator givesFor an eigenfunction, this must equal $E(2q^2-a)g$ at every $q$. Matching the $q^2$ coefficients gives $E=5/2$, and matching constants then gives $5a/2=2+a/2$. Hence $a=1$ and $E=5/2$. This is the even $n=2$ oscillator state. Its normalized wavefunction is ${\varphi }_2=(2q^2-1)e^{-q^2/2}/(\sqrt{2}{\pi }^{1/4})$.

Direct evaluation of the Rayleigh quotient exposes a problem with the requested minimization. Using the supplied Gaussian integrals,ThusThe eigenfunction value $a=1$ is a maximum, not a minimum. Indeed $\mathcal{E}(1)=5/2$, whereas $\mathcal{E}(0)=11/6$ is already lower. There is no finite minimizing $a$ in this family: its infimum is $1/2$, approached as $|a|\to \infty$, when the normalized trial wavefunction approaches the ground state. This is consistent with the quantum variational principle. The original printed instruction in this subpart is therefore inconsistent as an unconstrained minimization; a stationary Rayleigh quotient need not be a minimum.

The intended excited state can instead be obtained by imposing an excited-state variational constraint. Orthogonality to the ground state requireswhich fixes $a=1$. Every member of the family is even, so it is already orthogonal to the odd $n=1$ state. With the ground state excluded, the lowest allowed oscillator energy is therefore $E_2=5/2$. In the exact oscillator basis, $\psi$ is proportional to $\sqrt{2}{\varphi }_2+(1-a){\varphi }_0$, making both the maximum and the orthogonality correction transparent.

Exact solutions of the Schrödinger equation are unavailable for most interacting or realistic molecular systems. The quantum variational principle turns the search for a ground-state wavefunction into optimization of a computable energy quotient in a tractable trial family. The result is an upper bound: enlarging a nested trial space can only lower or preserve the optimized energy, providing a systematic convergence check. Trial functions can incorporate boundary conditions and known symmetry before calculation. Near an exact normalized ground state, a small orthogonal wavefunction error changes the energy only to second order, which explains why useful energies can result from imperfect trial functions. This does not ensure that every other observable is equally accurate, nor that a restricted trial family captures correlation adequately. Excited states require orthogonality or appropriate symmetry restrictions; unrestricted minimization seeks the ground state, as the preceding calculation demonstrates.