degeneracy

Degeneracy is what we call it when two or more genuinely different states happen to share the same eigenvalue. Most often the word refers to energy: several distinct quantum states that all sit at exactly the same energy level are said to be degenerate, and the number of such states is the degree of degeneracy. They are different states yet indistinguishable by that one observable alone.

Degeneracy is usually a fingerprint of symmetry. In the hydrogen atom, for example, states with the same principal quantum number but different shapes have equal energy because of the special form of the Coulomb force; rotational symmetry likewise makes the differently oriented states of a given angular momentum share an energy. When the symmetry is broken — by an electric or magnetic field, say — the degenerate level splits into separate, slightly different levels.

Degeneracy matters because it changes the bookkeeping of states without changing which energies are allowed. It complicates perturbation theory, which must be handled with special care when levels coincide, and it governs how many electrons can occupy a shell. The lifting of degeneracy by a field is exactly what produces the split spectral lines seen in effects like the Zeeman effect.