Introduction: When Eigenvalues Repeat
Degeneracy occurs when multiple linearly independent states share the same eigenvalue. It's usually a sign of symmetry—the more symmetric the system, the more degeneracy.
Definition
An eigenvalue is \(g\)-fold degenerate if there are \(g\) linearly independent eigenvectors with that eigenvalue. The degenerate eigenvectors span a degenerate subspace.
Sources of Degeneracy
- Symmetry: Rotational symmetry → angular momentum degeneracy
- Hidden symmetry: Hydrogen's \(n^2\) degeneracy (beyond rotational)
- Accidental: Rare, no underlying symmetry reason
Worked Examples
Example 1: 2D Harmonic Oscillator
With \(E_{n_x,n_y} = \hbar\omega(n_x + n_y + 1)\), the first excited level has \(n_x + n_y = 1\):
States: \(|1,0\rangle\) and \(|0,1\rangle\) — 2-fold degenerate
Example 2: Hydrogen Atom
Energy depends only on \(n\): \(E_n = -13.6/n^2\) eV
For each \(n\), there are \(n^2\) states (including spin: \(2n^2\))
\(n = 2\): degeneracy = 4 (states \(2s, 2p_x, 2p_y, 2p_z\))
Example 3: Lifting Degeneracy
Adding a perturbation that breaks symmetry "splits" degenerate levels. A magnetic field splits hydrogen's \(m\)-degeneracy (Zeeman effect).
The Quantum Connection
Degeneracy is physically important: within a degenerate subspace, any linear combination is also an eigenstate. Perturbations select specific linear combinations, and atomic spectra show split lines when degeneracy is lifted. Understanding degeneracy is essential for spectroscopy and quantum chemistry.