Solutions of time- dependent Schrödinger equations for various systems

The aim of this study is to introduce approximate and exact solution methods of the Schrödinger wave equation for several explicitly time explicitly dependent Hamiltonians. There is a general theory for the solutions of the Schrödinger wave equation for systems that has time independent Hamiltonians. Thanks to this theory, solutions of the Schrödinger equation can be found for all such systems, that is the state function can be determined for all the times for the systems whose state function is known for an instant. However, there is no general theory for the solutions of the Schrödinger equation for the explicitly time dependent Hamiltonians. While an exact solution can be obtained for some Hamiltonian classes, approximate solutions have been developed for several other Hamiltonians. In this thesis, first of all, approximate solution methods are introduced and then several exact solution methods will be considered: The exact solutions are obtained by using the integral transform methods for short-ranged time dependent Hamiltonians, Galilei coordinate transformation for systems moving with a constant velocity and the invariant matrix method for spin (1/2) systems evolving under time dependent magnetic fields.