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: If the Hamiltonian operator commutes with the time-reversal operator, that is then, for every energy eigenstate , the time reversed state is also an eigenstate with the same energy. These two states are sometimes called a Kramers pair. 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). Mathematical statement and proof In quantum mechanics, the time reversal operation is represented by an antiunitary operator acting on a Hilbert space . If it happens that , then we have the following simple theorem: Theorem. If is an antiunitary operator acting on a Hilbert space satisfying and a vector in , then is orthogonal to . Proof. By the definition of an antiunitary operator, , where and are vectors in . Replacing and and using that , we get which implies that . Consequently, if a Hamiltonian is time-reversal symmetric, i.e. it commutes with , then all its energy eigenspaces have even degeneracy, since applying to an arbitrary energy eigenstate gives another energy eigenstate that is orthogonal to the first one. The orthogonality property is crucial, as it means that the two eigenstates and represent different physical states. If, on the contrary, they were the same physical state, then for an angle , which would imply To complete Kramers degeneracy theorem, we just need to prove that the time-reversal operator acting on a half-odd-integer spin Hilbert space satisfies . This follows from the fact that the spin operator represents a type of angular momentum, and, as such, should reverse direction under : Concretely, an operator that has this property is usually written as where is the spin operator in the direction and is the complex conjugation map in the spin basis. Since has real matrix components in