Degeneracy

Degeneracy is a property of a system or model in which two or more distinct states share the same value of a given quantity, such as energy, probability, or a measurement. Degeneracy often arises from symmetry, redundancy, or constraints that reduce the effective variety of states. In physics and chemistry, degeneracy refers to the number of independent states that have the same energy or other property. The symbol g is commonly used to denote degeneracy. In quantum mechanics, energy level degeneracy means several eigenstates correspond to the same eigenvalue; for example, the hydrogen atom has energy levels dependent on the principal quantum number only, yielding orbital degeneracy, and electron spin adds a twofold degeneracy. Degenerate orbitals are orbitals with equal energy in an atom or molecule. Degeneracy pressure is a physical consequence of the Pauli exclusion principle, supporting dense astrophysical objects.

In mathematics, a degenerate case or degenerate object is one that lies on the boundary between standard types. A degenerate matrix has zero determinant and is not invertible; it has reduced rank. Degenerate conics include a point, a line, or a pair of lines rather than a proper ellipse, hyperbola, or parabola. In probability, a degenerate distribution concentrates all probability mass at a single point, giving zero variance.

Degeneracy is also relevant in numerical computation and statistics, where it can indicate linear dependence among variables or a failure of assumptions, often requiring reformulation or regularization.