Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 17B / d / Solution

Apply the adjoint operator identity of part (c) with $\chi =\hat{A}\varphi$ and use part (a):For a nonzero eigenfunction, positivity of the norm in particular forces $E\ge \hslash \omega /2$. Hence the requested norm isIt vanishes exactly at the ground-state energy.