1. The problem statement, all variables and given/known data Suppose that the Hamiltonian is invariant under time reversal: [H,T] = 0. Show that, nevertheless, an eigenvalue of T is not a conserved quantity. 2. Relevant equations 3. The attempt at a solution Using Kramer's Theorem. Consider the energy eigenvalue equation, [tex]H|\Psi\rangle = E|\Psi\rangle[/tex] for a time-reversal-invariant Hamiltonian, TH = HT. Therefore [tex]HT|\Psi\rangle = TH|\Psi\rangle = ET|\Psi\rangle[/tex], so both [tex]|\Psi\rangle[/tex] and [tex]T|\Psi\rangle[/tex] are eigenvectors with energy eigenvalue E. This implies two possibilities. 1. [tex]|\Psi\rangle[/tex] and [tex]T|\Psi\rangle[/tex] are linearly dependent, and so describe the same state, or 2. They are linearly independent, and so describe two degenerate states. It can further be shown that case 1 is not possible in certain circumstances. How can I show that there is no conserved quantity?