Time-Reversal Symmetry of Quantum Systems with Quasi-energy Spectrum

Abstract

In this section, it will be shown how the wave function of a non-stationary state of a quantum system is transformed under the action of the time reversal operator $\mathbf {T}$. Despite the fact that in the case of non-stationary states the total energy of the system is uncertain, it does not mean violating the law of energy conservation, since the average energy is conserved. It is found that the wave functions $\Psi \left( \xi , t\right)$ and $\mathbf {T}\Psi \left( \xi , t\right)$, on average, belong to the same energy level and are linearly dependent, if $\mathbf {T}^{2}=\mathbf {1}$. However, if $\mathbf {T}^{\mathbf {2}}\mathbf {=}-\mathbf {1}$, then the wave functions $\Psi \left( \xi , t\right)$ and $\mathbf {T}\Psi \left( \xi , t\right)$ belonging in average to the same level, are orthogonal (quasi-degeneracy of energy levels due to time-reversal symmetry).