Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 34D (Applications of Quantum Mechanics)

For a normalized trial function in the quadratic-form domain of the Schrödinger operator, integration by parts gives $⟨E⟩=⟨\psi ,H\psi ⟩$. Expand in the complete orthonormal energy eigenstates, $\psi ={\sum }_n{c}_n{\psi }_n$ with ${\sum }_n{c}_n^2=1$. ThenMinimizing over a trial family therefore gives the best upper bound within that family, the quantum variational principle.

For $V=|x|$, choose ${\psi }_{\alpha }=(\alpha /\pi {)}^{1/4}{e}^{-\alpha x^2/2}$ with $\alpha >0$. The Gaussian integrals giveThus $E(\alpha )=\alpha /2+(\pi \alpha {)}^{-1/2}$, whose derivative vanishes at $\alpha ={\pi }^{-1/3}$. It tends to infinity at both parameter endpoints and has positive second derivative, so this is its minimum. $\boxed{E_0\le \frac{3}{2}{\pi }^{-1/3}\simeq 1.0242}$ is the resulting estimate.

The potential is even; the unique nodeless ground state is even and the first excited state is odd, by the one-dimensional Sturm oscillation theorem. To estimate $E_1$, minimize the same Rayleigh quotient over odd trial functions, for example normalized $x{e}^{-\alpha x^2/2}$, or the compactly supported odd family $C_{a}x(1-|x|/a)$ for $|x|\le a$ and zero otherwise. Oddness gives exact orthogonality to the true ground state, not merely to an approximate trial ground state. The quantum variational principle restricted to the odd sector supplies an upper bound for $E_1$. No further integration is needed.