Degenerate energy levels

From degeneration is called in quantum mechanics, when two or more states of a quantum system at the same energy exist.

The degree of degeneracy n is the number of linearly independent solutions to the same energy eigenvalue. N states have the same energy, it is called n -fold degeneracy. (Caution: the symbols n for the degree of degeneracy should not be confused with the symbols n is the principal quantum number! )

The degenerate states differ in the values ​​of other observables (eg the train or the total angular momentum, or spin ); one says that the states are degenerate in these observables.

A degeneration is generally a result of a symmetry of the physical system. Thus rotational symmetry leads to a degeneration in any component of the angular momentum of the same for a fixed amount, the translational symmetry in crystals leads to degenerate lattice vibrational states, so that one when considering the lattice vibrations may be limited to the first Brillouin zone (see phonon ).

Example: degeneracy in the hydrogen atom

In the non-relativistic description of the hydrogen atom, all states with the same principal quantum number are degenerate. This degeneracy is due to the symmetry of the Kepler problem.

The consideration of the electron spins ( the so-called fine structure ) lifts this degeneracy partially. Corrections due to the interaction with the core ( hyperfine structure) and on the basis of quantum electrodynamics ( Lamb shift ) further reduce the degeneration and the degeneration in the component of the total angular momentum, which is retained owing to the rotational symmetry.