Degeneracies

Degeneracy is the property of a system in which two or more distinct states share the same value of a particular observable, most often energy or an eigenvalue of a mathematical operator. In mathematics and physics, degeneracy refers to the dimension of the eigenspace associated with a given eigenvalue: many independent eigenvectors share the same eigenvalue.

In linear algebra, an eigenvalue can be repeated; its algebraic multiplicity is the number of times it appears as a root of the characteristic polynomial, and its geometric multiplicity is the dimension of the corresponding eigenspace. A full set of independent eigenvectors exists when the geometric multiplicity equals the algebraic multiplicity. If not, the operator is defective and the eigenvalue is highly degenerate.

In quantum mechanics, degeneracy often arises from symmetry. A Hamiltonian invariant under a symmetry group can have energy levels that are degenerate because different states transform into each other under the group. Degeneracy can be broken or lifted by perturbations that reduce symmetry, such as external magnetic or electric fields (Zeeman or Stark effects) or distortions in a crystal lattice.

Common examples include the hydrogen atom, where energy levels depend only on the principal quantum number n, yielding a 2n^2 degeneracy with respect to angular momentum quantum numbers. In chemistry, p orbitals are degenerate in isolated atoms, and crystal fields or molecular environments can remove such degeneracy.

Degeneracy is distinguished as symmetry-related or accidental, the latter arising at special parameter values without an underlying symmetry requirement. Understanding degeneracies aids in predicting spectra, selection rules, and the influence of perturbations.