Wigner Function Moments versus RPA in a simple model

Two complementary methods to describe the collective motion, RPA and Wigner Function Moments (WFM) method, are compared using an example of a simple model—harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulas for eigenfrequencies and transition probabilities of all collective excitations of the model, including the scissors mode, which is a subject of our special attention. The exact relation between the variables of the two methods and the respective dynamical equations is established. The transformation of the RPA spectrum into the WFM one is explained.