# Degeneracy

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## Degeneracy (quantum mechanics)

A term referring to the fact that two or more stationary states of the same quantum-mechanical system may have the same energy even though their wave functions are not the same. In this case the common energy level of the stationary states is degenerate. The statistical weight of the level is proportional to the order of degeneracy, that is, to the number of states with the same energy; this number is predicted from Schrödinger's equation. In quantum mechanics and in other branches of mathematical physics, the term degeneracy is employed also to characterize the eigenvalues of operators other than the energy operator. See Eigenvalue (quantum mechanics)

## Degeneracy

(in quantum mechanics), a phenomenon that consists in the fact that some value ƒ that describes a physical system (an atom or molecule) has an identical value for various states of the system. The number of such states to which the same value of f corresponds is called the degeneracy ratio of the given quantity.

The instance of degeneracy most frequently dealt with is degeneracy of the energy levels of a system when the system has a certain energy value but may nonetheless be in several different states. For example, the degeneracy ratio for the energy of a free particle is infinite: the energy of the particle is defined only by the numerical value of its momentum; the momentum may have any direction (that is, it may be chosen by an infinite number of methods). In the example given above, the link between degeneracy and the physical symmetry of the system is clearly apparent—here this symmetry is the equality of all directions in space.

Upon the motion of a particle in an external field, degeneracy is essentially linked with the structure of this field—with the symmetry properties of the field. If the field is spherically symmetrical—that is, if the equality of directions is preserved in it—then the directions of the orbital moment of momentum, the magnetic moment, and the spin of the particle (for example, the electron in an atom) may not affect the energy value (the atom). Consequently, degeneracy of energy also exists here. However, if such a system is placed in a magnetic field H, the direction of the magnetic moment μ begins to exert an influence on the energy value; the energy levels of the various states (with various directions μ), which formerly coincided, are now different. As a result of the interaction of the magnetic moment of the particle with this field, the particle receives the additional energy μHH, whose value depends on the mutual orientation of the magnetic moment and the field (μH is a projection of μ on the direction of the field H, which in quantum mechanics may take on only a discrete number of values). The “breakup” of the energy levels—that is, the relaxation of degeneracy, which may be complete or partial (when the degeneracy ratio only decreases)—depends on concrete conditions. The breakup of the levels of atoms, molecules, or crystals in a magnetic field is called the Zeeman effect. The breakup of levels may also occur under an external electric field (the Stark effect).

Thus, the relaxation of degeneracy is brought about by the “turning on” of the proper interactions. Since the presence of degeneracy attests to the existence in the system of certain symmetries, the relaxation of degeneracy takes place when the physical conditions in which the system is located change in such a way that the order of these symmetries decreases. In the example mentioned above, the system at first had spherical symmetry (there were no separate directions in it), but the inclusion of an external constant magnetic field singled out a direction—the direction of the field. The symmetry of the system decreased and became axial—that is, it became a symmetry relative to an axis directed along the field.

If the inclusion of interactions leads to a decrease in symmetry and the relaxation of degeneracy, the reverse is also true. In case of the “turning off” of the interactions, an increase in the symmetry of the system and the appearance of degeneracy will occur. This is important for the classification of elementary particles. For example, if the electromagnetic (and weak) interactions are disregarded (“turned off”), the properties of the neutron and proton will be identical, and they may be regarded as two different (charged—that is, distinguished only by electrical charge) states of the same particle—the nucleón. Consequently, the state of the nucleón in this case has twofold degeneracy.

V. I. GRIGOR’EV and V. D. KUKIN

## degeneracy

[di′jen·ə·rə·sē]
(mathematics)
The condition in which two characteristic functions of an operator have the same characteristic value.
(physics)
The condition in which two or more modes of a vibrating system have the same frequency; a special case of the mathematics definition.
(quantum mechanics)
The condition in which two or more stationary states of the same system have the same energy even though their wave functions are not the same; a special case of the mathematics definition.
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