Perturbation theory without wave functions for the Zeeman effect in hydrogen

We apply perturbation theory to the hydrogen atom in a magnetic field by means of a recurrence relation for properly chosen moments of the wave function. The method is suitable for both numerical and symbolic computation and allows the treatment of classes of states with common symmetry properties. We derive analytic expressions in terms of the principal quantum number for the perturbation corrections to the energy of the unperturbed states with quantum numbers n-1=m=l, n-2=m=l-1, n-2=m=l, n-3=m=l-1, and (n-3=m=l,n-3=m=l-2) some of which have not been reported before. Disagreements with some previous results are pointed out.