Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 10 / b / i / Solution

For a spherically symmetric function, the Laplacian in spherical coordinates reduces toMultiplying the Helmholtz equation by $r^2$ givesThe endpoint at $r=0$ is singular in Sturm-Liouville terminology; bounded regular radial fields select the physical solution and eliminate the $1/r$ branch. At $r=a$ the field has the imposed zero Dirichlet value. For regular solutions and admissible trial functions, the endpoint term $r^2\psi {\psi }^{\prime }$ vanishes at the origin and at the wall.