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

WriteTaking the stated double-commutator identity between the two eigenstates gives, if the matrix element is nonzero,For nonnegative integers $\ell ,{\ell }^{\prime }$, this equation has precisely the possibilitiesIndeed, ${\ell }^{\prime }=\ell$ forces $\ell =0$, while for unequal values the difference
$({\ell }^{\prime }-\ell )({\ell }^{\prime }+\ell +1)$ satisfies the equation only when its first factor has magnitude one.

Under parity, $z\mapsto -z$, while a hydrogen state of orbital angular momentum $\ell$ has parity $(-1{)}^{\ell }$. Therefore a nonzero matrix element requiresso ${\ell }^{\prime }$ and $\ell$ must have opposite parity. This excludes every case ${\ell }^{\prime }=\ell$, including ${\ell }^{\prime }=\ell =0$. Combining the results gives