quasidegenerate

Quasidegenerate, often written quasi-degenerate, refers to a situation in which two or more eigenvalues of a system’s Hamiltonian are very close in value but not exactly equal. The concept is common in quantum mechanics, chemistry, and related fields, where energy levels, electronic configurations, or vibrational modes nearly coincide. It contrasts with exact degeneracy, where energies are identical due to symmetry or other constraints.

In perturbation theory, quasidegeneracy signals the breakdown of nondegenerate approaches because small perturbations can cause substantial mixing between nearly equal states. Practically, one treats a near-degenerate subspace more exactly by diagonalizing the perturbation within that subspace or by diagonalizing the full Hamiltonian restricted to the nearly degenerate states. This approach is the essence of degenerate perturbation theory or quasi-degenerate perturbation theory, and is often implemented via techniques such as Löwdin partitioning to produce an effective Hamiltonian for the near-degenerate manifold.

Applications are widespread. In chemistry, quasidegenerate electronic states occur when frontier orbitals are close in energy, necessitating configuration interaction among near-degenerate configurations. In spectroscopy, closely spaced lines reflect quasidegenerate states. In solid-state physics, bands with nearly identical energies can interact, affecting effective masses and transport properties. In nuclear and particle physics, mixing of near-resonant states is described using quasi-degenerate formalisms.

Numerical aspects include challenges in solving near-degenerate eigenproblems, where standard methods may be ill-conditioned. Careful treatment includes subspace-rotation techniques, higher-precision arithmetic, or introducing small symmetry-breaking perturbations to resolve levels and studying their behavior as the perturbation tends to zero.