The discovery of quantum theory at the start of the 20th century revealed an en-tirely new and unsuspected level of phenomena that govern the behavior of physical systems at the atomic and molecular level. Building on the discoveries of Planck, Einstein and Bohr, Erwin Schrödinger developed a wave equation for matter, and demonstrated that extremely light and small particles such as electrons and protons obey this wave equation rather than Newtonian mechanics. Schrödinger's wave eqeuation and Dirac's relativistic generalization of it, have been extrememly successful in explaining the properties of matter. The invention of transistors (for example) flowed from the insights obtained from the Schrödinger equation. In principle, all of the properties of matter, including color, reactivity, tensile strength, electrical properties and so on, could be calculated from first principles by solving the wave equation, except that its solution for many-particle systems presents computational problems that are formidable even for the most powerful modern computers.The Generalized Sturmian Method, explored in my thesis, is a promising new approach to the solution of many-particle Schrödinger equations.