Kramers theorem

In quantum mechanics, the Kramers degeneracy theorem states that for every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is at least one more eigenstate with the same energy. In other words, every energy level is at least doubly degenerate if it has half-integer spin.

In theoretical physics, the time reversal symmetry is the symmetry of physical laws under a time reversal transformation:

$T: t \mapsto -t.$

If the Hamiltonian operator commutes with the time-reversal operator, that is

$[H,T]=0,$

then for every energy eigenstate $|n\rangle$, the time reversed state $T|n\rangle$ is also an eigenstate with the same energy. Of course, this time reversed state might be identical to the original state, but that is not possible in a half-integer spin system since time reversal reverses all angular momenta, and reversing a half-integer spin cannot yield the same state (the magnetic quantum number is never zero).

For instance, the energy levels of a system with an odd total number of fermions (such as electrons, protons and neutrons) 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 the 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.

See also

• Degeneracy
• Hendrik Anthony Kramers
• T-symmetry