Collective motion from various aspects

Three methods to describe collective motion, the Random Phase Approximation (RPA), Wigner Function Moments (WFM), and Green’s Function (GF) methods, are compared in detail and their physical content analyzed in the example of a simple model, a harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulae for eigenfrequencies and transition probabilities of all collective excitations of the model. The exact relation between the RPA and WFM variables and the respective dynamical equations is established. The transformation of the RPA spectrum into one of WFM is explained. The very close connection of the WFM method with the GF method is demonstrated. A differential equation describing the current lines of RPA modes is established and the current lines of the scissors mode are analyzed as a superposition of rotational and irrotational components. The orthogonality of the spurious state to all physical states is proved rigorously. The text was submitted by the author in English.