Matrix mechanics applications to vibrational and rotational collective motion in nuclei

The matrix-mechanics approach to quantum theory is applied to study three problems of interest in nuclear collective motion. The first phenomenon consists of the anharmonic vibrations in two simplified solvable nuclear models, possessing the symmetries R(5) and R(5) x R(5). The matrix elements and energies of the one- and two-phonon states are calculated using matrix mechanics, and compared with the results of an exact diagonalization. The comparison is fairly good. The next application is to the study of the asymmetric rigid rotor at a high angular momentum I. It is found that matrix elements can be expanded in inverse powers of I; the orders of magnitude of all the matrix elements can be ascertained. Explicit calculations are carried out up to third order. The Generalized Hartree-Fock theory is then used to derive from first principles the conditions under which a many-body system of nucleons behaves like an asymmetric rotor at large angular momentum. A microscopic theory of the triaxial nucleus is thereby obtained, and expressions for the three moments of inertia are derived in terms of single-particle quantities. Self-consistent cranking (SCC) model in leading order, and first-order corrections to it, are derived. To illustrate the application of the Generalized Hartree-Fock theory, and the development of the approximation scheme in the large-angular momentum limit, the caseof a two-dimensional system of nucleons rotating in a plane is considered