The Kramers degeneracy theorem states that the energy levels of systems with an odd total number of electrons, protons, and neutrons (i.e. the total number of fermions) remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). It was first discovered in 1930 by H. A. Kramers as a consequence of Breit equation.
As shown by Eugene Wigner in 1932, it is a consequence of the time reversal invariance of electric fields, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of fermions. The theorem is valid for any configuration of static or time-varying electric fields.
For example: the hydrogen (H) atom contains one proton and one electron, so that the Kramers theorem does not apply. The lowest (hyperfine) energy level of H is nondegenerate. The deuterium (D) isotope on the other hand contains an extra neutron, so that the total number of fermions is three, and the theorem does apply. The ground state of D contains two hyperfine components, which are twofold and fourfold degenerate.
Famous quotes containing the word theorem:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)