In this thesis the aim is to use symplectic integrators to solve spatially discretized Schr?dinger equation. The whole process, in a few words, is as follows: In Chapter ?, two popular techniques, namely composition and splitting, have been reviewed owing to the fact that structure preserving methods of desired order can be constructed by means of them. Then, Hamiltonian systems and their symplectic flows are discussed. In Chapter ?, symplectic splitting methods are applied to one dimensional harmonic oscilator and stability of the resulting numerical flow (which is called stability matrix and possesses polynomial entries) is investigated in detail.
