How is the degeneracy calculated in quantum physics

Degeneracy (quantum mechanics)

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Of degeneration In quantum mechanics, one speaks when two or more states of a quantum mechanical system exist with the same energy.

The Degree of degeneration or Degeneracy factor$ \ textstyle n $ (often also referred to as $ \ textstyle g $) is the number of linearly independent solutions to the same energy eigenvalue. If $ \ textstyle n $ states have the same energy, one speaks of $ \ textstyle n $ -fold degeneracy. (Caution: the same symbol $ \ textstyle n $ is also used for the main quantum number!)

Analogously, one says for states that have the same eigenvalue for a certain observable (e.g. the orbital or total angular momentum or the spin): the states are in this observable degenerate. Accordingly, the energetically degenerate states are degenerate with respect to the observable "energy".

The states that have degenerated with respect to an observable can always be distinguished from another suitable observable by their eigenvalues.

In many cases, degeneracy is the result of a symmetry in the physical system. Rotational symmetry around any axis leads to a degeneracy of energy with regard to every component of the angular momentum with a fixed amount of angular momentum.

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 can be traced back to the symmetry of the Kepler problem.

Principal quantum number
$ n $
Angular momentum QZ
$ l = 0 \ ldots n-1 $
Orbital magnetic QZ
$ m_l = -l \ ldots 0 \ ldots + l $
total degeneration:
$ \ sum_ {l = 0} ^ {n-1} {(2l + 1)} = n ^ 2 $ times
1 0 s 0 1
2 0 s 0 4
1 p −1, 0, +1
3 0 s 0 9
1 p −1, 0, +1
2 d −2, −1, 0, +1, +2

Taking into account the electron spin (the so-called fine structure) partially eliminates this degeneracy. Corrections due to the interaction with the nucleus (hyperfine structure) and due to quantum electrodynamics (Lambshift) further reduce the degeneracy, except for the degeneracy in the components of the total angular momentum, which is retained due to the rotational symmetry.

See also