Symmetry group of point transformations for the time-dependent Schrödinger equation: Harmonic interactions among nucleons.

Abstract

We have used Lie's method of extended group to obtain explicit forms of the generators and the structure of the maximal symmetry group of point transformations of the time-dependent Schr\"odinger equation for motions of nucleons interacting with two-body harmonic potential. The generators of the symmetry group correspond to different states of motion of the system. The maximal symmetry group is found to be a semidirect product of an infinite parameter Abelian invariant subgroup and a proper subgroup. For Z protons and N neutrons, this proper subgroup is a Lie group with 1/2[9Z(Z-1)+9N(N-1)+40] generators. Different nuclear modes of excitations have been assigned to the different generators. In particular the giant resonance mode and other collective modes of motion are shown to be consequences of the symmetry of the system.