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 another eigenstate with the same energy related by time-reversal. In other words, the degeneracy of every energy level is an even number if it has half-integer spin. The theorem is named after Dutch physicist H. A. Kramers.

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

${\displaystyle T:t\mapsto -t.}$

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

${\displaystyle [H,T]=0,}$

then, for every energy eigenstate ${\displaystyle |n\rangle }$, the time reversed state ${\displaystyle T|n\rangle }$ is also an eigenstate with the same energy. These two states are sometimes called a Kramers pair.^[1] In general, this time-reversed state may be identical to the original one, but that is not possible in a half-integer spin system: since time reversal reverses all angular momenta, reversing a half-integer spin cannot yield the same state (the magnetic quantum number is never zero).