Up to a <global phase>, $x(a-x)$ is real, positive throughout the interior, has no interior <nodes of a wavefunction>, and is symmetric about $x=a/2$. It is smooth inside the box and has one central maximum. These are the shape and <reflection symmetry> properties of the lowest <particle in a box> <eigenfunction>, $\sin(\pi x/a)$; they make it a plausible <variational method> trial without making it exact.

Solution

Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A2 / c / ii / Solution

 Solution (source code)