Date of Award

Spring 1-1-2017

Document Type


Degree Name

Master of Science (MS)


Applied Mathematics

First Advisor

Andreas Becker

Second Advisor

Agnieszka Jaron-Becker

Third Advisor

Bengt Fornberg


Many contemporary problems in theoretical atomic, molecular, and optical physics involve the solution of the time-dependent Schrödinger equation describing the dynamics of a many-body system interacting with external time-dependent fields. In order to perform meaningful simulations and analysis of the time-dependent results, it is important to accurately take into account the energy eigenstates of the atomic or molecular system on a spatial grid, within the constraints provided by the time-dependent simulation. To this end, efficient numerical methods of obtaining solutions of ground and excited bound states using the time independent Schrödinger equation are needed. In this thesis we analyze various algorithms based on iterative methods, including the popular imaginary time propagation method, and compare the numerical performance to that of an implicitly restarted Arnoldi algorithm. The analysis includes a comparison of different order finite difference schemes and applications of the methods to the hydrogen atom and several single-active-electron potentials. Our results reveal a superior efficiency of the Arnoldi method within the limits given by the time-dependent simulations with respect to computation time, accuracy, as well as the number of resolved eigenstates.