Degenerate energy levels

This article is about different quantum states having the same energy. For other uses, see Degeneracy.

"Quantum degeneracy" redirects here. It sometimes refers to a degenerate matter.

In quantum mechanics, a branch of physics, two or more different states of a system are said to be degenerate if they are all at the same energy level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same eigenvalue. Conversely, an energy level is said to be degenerate if it contains two or more different states. The number of different states at a particular energy level is called the level's degeneracy, and this phenomenon is generally known as a quantum degeneracy.

From the perspective of quantum statistical mechanics, several degenerate states at the same level are all equally probable of being filled.

Mathematics

The term comes from the fact that, for a point spectrum Hamiltonian H, degenerate eigenstates correspond to identical eigenvalues. Since eigenvalues correspond to roots of the characteristic equation, degeneracy here has the same meaning as the common mathematical usage of the word.

The eigenvalue λ is called nondegenerate (or simple) when its corresponding eigenvector is unique to within a constant factor, or, the same, the corresponding eigenspace is one-dimensional.

Indeed, the eigenspace { ψ : H | ψ ⟩ = λ | ψ ⟩ }  (in bra-ket notation) is not necessarily one-dimensional. If there exist at least two linearly independent ket-vectors in it, then this eigenvalue is said to be degenerate. Its degree of degeneracy is then the dimension of the eigenspace, which is the same as the number of distinct (linearly independent) quantum states associated with it.

Examples

In atomic physics, electron's energy levels are often degenerate, where different possible occupation states for particles may be related by symmetry. For example, in the hydrogen atom, for a fixed energy eigenvalue, there exist several states which have that energy, but differ in the eigenvalues of angular momentum L², spin component S_z and so on. The eigenvalue of an operator which distinguishes between degenerate states is called a quantum number.

Perturbation

If the symmetry is broken by a perturbation, caused, for example, by applying an external magnetic or electric field, then the energies of the states can be changed, causing energy level splitting.

See also

• Density of states

Further reading

• Quantum Mechanics (Volume 1), Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe