Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A1 / b / iii / Solution

Separation of variables makes the angular factor a spherical harmonic. Applying the Quantum Hamiltonian givesConsequently $R{Y}_{lm}$ is an energy eigenfunction provided the radial factor satisfiesThis is the radial Schrödinger equation for a central potential, here with the attractive Coulomb potential. An arbitrary $R$ does not give an eigenfunction; regularity, decay and wavefunction normalization select admissible bound quantum states.