Search

The Mathieu–Schrodinger equation, which describes the behavior of a quantum pendulum, depending on the value of the parameter l (pendulum filament length), can have the symmetry of the Klein’s four-group or its invariant subgroups. The paper shows that the mean values of z-components of the angular momentum of nondegenerate quantum states (the symmetry region of the four-group) tend to zero and their root mean square fluctuations are non-zero. Consequently, in this region of parameter values, the fluctuations overlap the mean values of the angular momentum and they become indistinguishable. Therefore, it can be argued that if, with an increase in the parameter, the system goes into a non-degenerate state, then after the inversion of the parameter change and the transition to the region of degenerate states, the initial states will not be restored. This behavior of the average values of angular momenta is caused by the combined actions of two factors: discontinuous change in the system at the points of change of its symmetry and the presence of quantum fluctuations in nondegenerate states.