Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 10 / ii / Solution

Here $p=w=1$ and $q=x^2$ on the whole real line. The quantum harmonic oscillator energy form is $Q[\psi ]={\int }_{\mathbb{R}}(|{\psi }^{\prime }{|}^2+x^2|\psi {|}^2)dx$. Admissible trials have a square-integrable weak derivative and finite potential energy; they need not have a classical second derivative everywhere.