Degenerate energy level

Degenerate energy level

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, two or more different quantum states are said to be degenerate if they are all at the same energy level
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

 for the system having more than one linearly independent
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

 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.

In quantum theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 this usually pertains to electronic configurations and the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

's energy levels, where different possible occupation states for particles may be related by symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. For example, in the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

, for a fixed energy eigenvalue, there exist several states which have that energy, but differ in the eigenvalues of angular momentum , spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

 component and so on. The usage comes from the fact that degenerate eigenstates correspond to identical eigenvalues of the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

. Since eigenvalues correspond to roots of the characteristic equation, degeneracy
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....

here has the same meaning as the common mathematical usage of the word.

If the symmetry is broken by a perturbation
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

, caused, for example, by applying an external magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

 or electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

 field, then the energies of the states can be changed, causing energy level splitting
Energy level splitting
Energy level splitting occurs in physics when the degenerate energy levels of two or more states are split because of external fields or other effects...

.

In electromagnetics
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

, degeneracy refers to modes of propagation which exist at the same frequency and longitudinal propagation constant. As an example, for a rectangular waveguide
Waveguide (electromagnetism)
In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves...

, the TEmn mode is degenerate to the TMmn mode if m and n are the same for both of them.