Coalescence and inflection properties of Coulomb wavefunctions

We discuss some pedagogically useful properties of the energy eigenfunctions for the Coulomb potential, in particular the coalescence property when the particle is close to the origin, and the inflection property at the nodal points. With the choice of an appropriate form for the wavefunctions, these properties allow us to obtain the energies and the eigenfunctions of the free hydrogen atom, and develop accurate wavefunctions for the confined hydrogen atom in some domains of confinement. We also consider an extension of these ideas to the two-dimensional analogue of the - 1/r potential, and to other potentials