Time reversal symmetry

Summary

Time reversal symmetry is the invariance of the laws of motion under reversing the direction of time. In quantum mechanics it is implemented by an antiunitary operator $\Theta$, which reverses momenta and spins but keeps positions. A system is time-reversal symmetric when $\Theta$ commutes with its Hamiltonian. Then the time reverse of every solution of the Schrödinger equation is again a solution. For particles of half-integer spin $\Theta^2 = -1$, and this forces the degeneracy of energy levels stated by Kramers' theorem.