Dynamical Functional Particle Method applied to the Schrödinger Equation : Exact solutions of three-body exotic ions

Abstract: The aim of this master thesis has been to solve the fully correlated non-relativistic Schrödinger Equation for three-particle systems by using the Dynamical Functional Particle Method (DFPM) and control the solutions by comparing with the results from the literature. The three-particle systems that has been analyzed is helium, a muon-based ion and the negative ion of positronium. The Schrödinger equation for S-states has been transformed and derived, and the wavefunction has been substituted to a wavefunction that treated the Cusp conditions when two particles approach each other as boundary condition by using the knowledge that the wavefunction is limited everywhere. The discrete grids for the distances were transformed so that the outer boundary condition could be placed further from the origin and obtain better description of the wavefunction. By using the chain rule and quotient rule, the Schrödinger Equation was transformed again and finite differences resulted in a discrete dynamical system that was programmed in C-code and iteratively stepped to the solution of the systems energy. The continuum energy was determined by applying Richardson extrapolation with the values of the energy for different step sizes. The resulting energies were -2.903304 (-2.90330456) a.u. in ground state and -2.174934 (-2.17493019) a.u. in first excited state for helium, -97.57 (-97.5669834) a.u. in ground state for the muon-based ion and -0.258405 (-0.262) a.u. in ground state for the negative ion of positronium, where the values in the parentheses are values from the literature, and the values were consistent with the results from literature. But the values obtained here were in general more exact due to the non-approximated method. There were limited precision during the calculations and the condition number of the matrices with the step sizes was high for small step sizes and results in a uncertainty and not more than 5-6 significant figures can be used for the value of the energies. A better value had been obtained if for example Multi Precision had been used because it can handle a high condition number. The curse of dimensionality is hard to beat and methods for treating bigger systems with DFPM has to be developed in order to have a higher order of the differential equation with respect to time and to increase the maximum time step to make the calculations faster. Keywords: Dynamical Functional Particle Method, Schrödinger Equation, three-particle systems, helium, muon, positronium