Symmetry of the quadratic Zeeman effect for hydrogen

The $O(4)$ symmetry of the hydrogen-atom degeneracy is exploited to obtain an exact separation of the quadratic Zeeman interaction on the Fock hypersphere in momentum space, and hence a complete classification of levels in the weak-field limit. The separation constant includes a double-minimum potential, and levels below the top of the barrier have an approximate O(2)×O(2) symmetry and vibrational structure. Levels above the top of the barrier have an approximate $O(3)$ symmetry, and a structure similar to that of an oblate symmetric top. The crossover between the two types of levels becomes sharper at high n, and is related to two different types of collective coupling of angular momentum and the Runge-Lenz vector.