In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in quantum mechanics 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.
In quantum theory this usually pertains to electronic configurations and the electron's energy levels, 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, spin component and so on. The usage comes from the fact that degenerate eigenstates correspond to identical eigenvalues of the Hamiltonian. Since eigenvalues correspond to roots of the characteristic equation, degeneracy here has the same meaning as the common mathematical usage of the word.
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.
In electromagnetics, degeneracy refers to modes of propagation which exist at the same frequency and longitudinal propagation constant. As an example, for a rectangular waveguide, the TEmn mode is degenerate to the TMmn mode if m and n are the same for both of them.
Famous quotes containing the words degenerate, energy and/or levels:
“When I think of some of the Persians, the Hindus, the Arabs I knew, when I think of the character they revealed, their grace, their tenderness, their intelligence, their holiness, I spit on the white conquerors of the world, the degenerate British, the pigheaded Germans, the smug self-satisfied French.”
—Henry Miller (1891–1980)
“A government deriving its energy from the will of the society, and operating, by the reason of its measures, on the understanding and interest of the society ... is the government for which philosophy has been searching and humanity been fighting from the most remote ages ... which it is the glory of America to have invented, and her unrivalled happiness to possess.”
—James Madison (1751–1836)
“The word which gives the key to the national vice is waste. And people who are wasteful are not wise, neither can they remain young and vigorous. In order to transmute energy to higher and more subtle levels one must first conserve it.”
—Henry Miller (1891–1980)