Kramers' degeneracy 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. The law is named for the physicist Hans Kramers.

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

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

then for every energy eigenstate ${\displaystyle |n\rangle }$, the time reversed state ${\displaystyle T|n\rangle }$ 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).

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