Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 33A / a / Solution

In position representation,which differentiates only with respect to $x$ and $y$. Multiplication by $z$ therefore commutes with it:Taking the matrix element of this commutator between angular-momentum eigenstates givesThus the matrix element can be nonzero only whenThis is the magnetic part of the electric-dipole selection rules for hydrogen.