Kramers degeneracy

States that all energy levels for a spin-1/2 particle come in degenerate pairs, since both \(\ket{E_n}\) and \(\Theta\ket{E_n}\) have the same eigenvalue.

Note that this does not mean that \(\Theta\ket{E_n}=\alpha\ket{E_n}\), since \(\Theta = -i\sigma_yK\), then \(Theta^2=-1\), and \(\Theta^2\ket{E_n}=|\alpha|^2\ket{E_n}\Rightarrow |\alpha|^2=-1\), which is not possible (i.e. \(\ket{E_n}\) and \(\ket{E_n'}\) are not the same state).