Degeneracy (quantum mechanics)

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In quantum mechanics , one speaks of degeneracy when several, linearly independent eigenstates exist for the same measured value ( eigenvalue ) of an observable .

The degree of degeneracy or degeneracy factor n is the number of linearly independent eigenstates with the same eigenvalue. These span the n -dimensional subspace to the same eigenvalue. The eigenvalue is then called n times degenerate.

Two states that belong to the same degenerate eigenvalue of an observable can consequently not be distinguished from one another by measuring these observables. However, for every n -fold degenerate eigenvalue, a second observable can be found that is commensurate with the first and has exactly n eigenstates with n different eigenvalues in the subspace belonging to the degenerate eigenvalue .

In many cases, degeneracy is the result of a symmetry of the physical system. Thus leading rotational symmetry of the Hamiltonian about any axis to an energy degeneration. The degenerate states can usually be distinguished here by their various eigenvalues ​​to form an angular momentum component . Conversely, it always follows from the degeneracy of an eigenvalue of an observable that it is invariant under every unitary transformation of the associated eigenspace.

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 .

Degeneration to the first three energy eigenvalues ​​of the hydrogen atom

Principal quantum number ${\displaystyle n}$	Angular momentum QZ ${\displaystyle l=0\ldots n-1}$	Orbital	magnetic QZ ${\displaystyle m_{l}=-l\ldots 0\ldots +l}$	total degeneration: -fold ${\displaystyle \sum _{l=0}^{n-1}{(2l+1)}=n^{2}}$
1	0	s	0	1
2	0	s	0	4th
	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

• Exchange degeneracy