Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 10

Take real admissible functions with finite $F,G$, positive $G$, and a self-adjoint domain for the Sturm-Liouville operator. Compactly supported variations $\eta$ avoid any boundary contribution at infinity. Formal self-adjointness givesFor $\Lambda =F/G$, the quotient rule therefore givesVanishing for arbitrary variations requiresConversely an eigenfunction in the self-adjoint domain makes this variation zero. Multiplying its equation by $y$ and integrating also shows directly that its stationary quotient is $\boxed{\Lambda [y_n]={\lambda }_n}$.

Thus the stationary values are the admissible eigenvalues. Under the usual positive-weight, semibounded hypotheses the lowest is the global minimum, while higher eigenfunctions generally give saddle points. The assumption $y\to 0$ alone is not a complete domain specification for arbitrary $\rho ,\sigma$: the required integrals and boundary form must exist. The rapidly decaying oscillator trials in part (b) meet these conditions.

Normalize the Gaussian trial first. The Gaussian integral givesIts second and fourth moments are $⟨x^2⟩=1/(2\alpha )$ and $⟨x^4⟩=3/(4{\alpha }^2)$. Since ${\psi }^{\prime }=-\alpha x\psi$, the kinetic term is $\int ({\psi }^{\prime }{)}^2={\alpha }^2⟨x^2⟩=\alpha /2$. At the specified coupling, the energy is thereforeThe stationary condition isThe quadratic factor is positive for real $\alpha$, so the unique positive stationary width is $\alpha =2$. Also $E^{\prime \prime }={\alpha }^{-3}+9{\alpha }^{-4}>0$ and $E$ diverges at both ends of $\alpha >0$, proving a global minimum within this trial family. Its value isThe associated Sturm-Liouville problem isHere $\rho =1$, $\sigma =x^2+2x^4$ and $w=1$. The Rayleigh-Ritz variational principle makes the exact lowest eigenvalue the minimum over all normalized admissible states. Restricting to Gaussians can only raise this minimum, hence $\boxed{E_0\le 13/8}$. The inequality is strict because $H{\psi }_{\ast }/{\psi }_{\ast }=2-3x^2+2x^4$ is not constant: the trial is not an exact eigenfunction. This is the Gaussian trial for a quadratic-plus-quartic oscillator.

Enlarge the trial space while retaining the optimized Gaussian. For example, usenormalize it and minimize the Rayleigh quotient over its coefficients, optionally optimizing $\alpha$ too. This even trial space respects the even ground-state parity of the confining symmetric potential and adds independent shape freedom beyond a Gaussian width.

The original Gaussian remains included by setting the later coefficients to zero, so the optimized upper bound cannot increase. Finite-basis optimization is the Rayleigh-Ritz method, with matrix entries given by the Hamiltonian and overlap integrals. Increasing the basis size systematically improves the available approximation.

For any real coupling, the same normalized Gaussian family has energyIf $ϵ<0$, then as $\alpha \downarrow 0$ the negative ${\alpha }^{-2}$ term dominates the positive ${\alpha }^{-1}$ and $\alpha$ terms. Each trial still has finite energy, is normalized and decays at infinity, but its energy tends to $-\infty$. ThereforeThis variational sequence makes the instability of the negative quartic potential explicit, without mistaking any local well near the origin for a global ground-state bound.