Nuclear Collective Motion from a Group Theoretical Standpoint

Abstract

6'are classified using the method of the induced representations and the imprimitivity theorem. The wave function and the wave equation are obtained by this method. We introduce a Hamiltonian and give a formula for calculating the energy spectrum that depends on. deformations and rotations. basis, has developed primarily from symmetry concepts since the very beginning of nuclear physics. Hence one strongly relies on group theory in building new models which describe the nuclear collective motion by either introducing groups or exploit­ ing techniques based on group theory which are not customary in nuclear physics. The present paper exploits a fundamental technique well-known to mathematical physicists but not often used in nuclear physics. This technique is that of induced representations and imprimitivity theorem, originated by Wignei for the Poincare group, and is very well adapted to the collective motion group CM(3), since both the same intrinsic structure (semi-direct product). The use of this technique allows us to obtain several results, the most important one being the classification of all unitary irreducible representations of CM(3). The Hilbert space that carries an irreducible representation of CM(3) is to be regarded as the space of collective nuclear wave functions. Such a Hilbert space, for each irreducible representation, is characterized by an equation which assumes the role of a wave equation. The wave functions are defined on the space of mass quadrupole tensors, which allows an intuitive physical interpretation of the model. Using physical intuition and the symmetry of the model we define a Hamiltonian which yields a concise and easily interpreted formula for the energy spectrum. The latter depends on both deforma­ tions and rotations and is a continuum spectrum. One could add a potential, depend­ ing also on the mass quadrupole coordinates, in order to have a discrete energy sp~ctrum. This possibility remains open, since we have not pursued it further here. The outline of the paper is the following. In § 2 we present a survey on the subject of nuclear collective motions aimed at singling out CM(3) as nuclear collective motion group. Section 3 is very technical and yields the classification of the unitary irreducible representations of CM(3). In § 4 we introduce a possible Hamiltonian for , our model and derive the energy spectrum.