This thesis presents a general non-variational approach to the solution of three-body Schrödinger's equation with Coulomb interactions, based on the utilization of symmetries intrinsic to the three-body Laplacian operator first proposed by W.Y.Hsiang. Through step by step reductions, the center of mass degree of freedom is first removed, followed by the separation of all the rotational degrees of freedom, leading to a coupled partial differential equations (PDEs) in terms of the rotationally invariant internal variables {f1, f2, f3}. A crucial observation is that in the subspace where all the rotational degrees of freedom have been removed, there is an intrinsic spherical symmetry which can be fully utilized through the introduction of hyperspherical coordinates. By expressing the redu...[ Read more ]

This thesis presents a general non-variational approach to the solution of three-body Schrödinger's equation with Coulomb interactions, based on the utilization of symmetries intrinsic to the three-body Laplacian operator first proposed by W.Y.Hsiang. Through step by step reductions, the center of mass degree of freedom is first removed, followed by the separation of all the rotational degrees of freedom, leading to a coupled partial differential equations (PDEs) in terms of the rotationally invariant internal variables {f1, f2, f3}. A crucial observation is that in the subspace where all the rotational degrees of freedom have been removed, there is an intrinsic spherical symmetry which can be fully utilized through the introduction of hyperspherical coordinates. By expressing the reduced Schrödinger's PDEs (with all the rotational degrees of freedom separated out) in terms of the hyperspherical coordinates, with the subsequent introduction of Jacobi polynomials as the angular eigenfunctions and Laguerre polynomials to expand the radial component, a system of infinite linear algebraic equations is obtained for the expansion coefficients. A numerical scheme is presented whereby the Coulomb interaction matrix elements are calculated to a very high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedure. The resulting matrix equations are solved through an iteration process, carried out on a PC. Numerical results are presented for the hydrogen negative ion H^-, the helium and helium-like ions(Z = 3 ~ 6) the hydrogen molecule ion H₂⁺ and the positronium negative ion Ps^-. Comparison with the variational and other approaches shows our results to be of comparable accuracy for the eigenenergies, but can yield highly accurate wave functions as by-products. Results on low-lying excited states are obtained simultaneously with the ground state properties with no extra effort. In particular, for the doubly excited states, such as^1,3P^e, our method can give expectation values characterizing the three-body wave functions that have not been calculated before. As a general systematic approach to the three-body Coulomb problem, the solution process is reduced to a well-defined procedure that requires minimal human intervention (e.g., in the choice of basis functions for the variational approach), with well-demonstrated convergence.