Degenerate energy levels

Degenerate energy levels are energy eigenvalues shared by two or more distinct quantum states. The states are distinct but cannot be distinguished by measuring energy alone.

Hamiltonian (quantum mechanics): The operator representing a quantum system’s total energy and governing its time evolution. Its eigenvalues are energy levels, and repeated eigenvalues signal degeneracy.
Eigenvalue: A scalar associated with a linear operator and one of its eigenvectors. An energy level is an eigenvalue of the Hamiltonian.
Hydrogen atom: The one-electron atom consisting of a proton and a bound electron. In the nonrelativistic Coulomb model, many states with different angular momentum share an energy.
Accidental degeneracy: Degeneracy not required by an evident symmetry of the system’s Hamiltonian. It contrasts with degeneracy directly enforced by a manifest symmetry.
Energy eigenstate: A quantum state with a definite value of the energy observable. A degenerate energy level contains multiple independent energy eigenstates.
Eigenvector: A nonzero vector that a linear operator maps to a scalar multiple of itself. Quantum energy eigenstates are eigenvectors of the Hamiltonian.
Quantum harmonic oscillator: A quantum system with a potential energy proportional to the square of displacement from equilibrium. Higher-dimensional versions have multiple states at the same total excitation energy.
Symmetry-protected degeneracy: Degeneracy enforced by a symmetry that prevents states from splitting in energy. It identifies the symmetry-enforced subset of degenerate levels.
Symmetry in quantum mechanics: An invariance of a quantum system under transformations represented by operators on its states. Symmetries can force different states to have identical energies.
Quantum state: A mathematical description of the possible outcomes of measurements on a quantum system. Distinct quantum states can share the same energy.

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