Mean-field approaches for the characterization of many-body systems: study of electron scattering and nuclear decay processes in different environments

In this thesis, two theoretical-computational methods for the study and analysis of many-body systems are presented. The protagonist of the first part of this writing will be the Dirac-Hartree-Fock (DHF) method, a relativistic quantum method that defines a mean-field (MF) approximation of the many-particle Dirac equation. This approach, implemented using Hermite Gaussian functions (HGFs) basis sets and radial mesh, will constitute the pivotal element of the first and second chapters. In the first chapter, in particular, after having introduced the scattering theory and Gaussian functions, we will discuss the results obtained for electron elastic scattering in water. In the second chapter, we will treat nuclear beta decays and describe the approach, based on the use of radial basis sets and Fermi-Dirac (FD) statistics, for the modeling and analysis of decay rates under terrestrial and stellar conditions. Set aside the DHF method and its applications, in the third chapter, we will introduce the second and last approach, a non-relativistic variational method through which have been calculated the bound and scattering states, the energies, the photodetachment cross sections, the decay rates and the scattering phases of ions, atoms and molecules, such as the positronium negative ion (Ps− ) and hydrogen anion (H− ). The model adopts optimized multi-dimensional Gaussian basis functions (OMGBFs). An advantage of utilizing Gaussian basis functions consists in deriving analytically the one- and two-body integrals essential for the calculation of matrix elements. The ultimate purpose of this thesis will be to show the efficiency and versatility of the two methods.