Abstract: We will review properties of the linear Schrödinger equation, in particular stating the Strichartz estimates. I will introduce a nonlinearity of the form $|u|^{alpha-1}u$ into the problem. Formulating this new PDE as an integral equation using Duhamel's principle, we can prove existence of a solution by a contraction mapping argument. The hard part is showing the mapping we introduce is really a contraction mapping and Strichartz's inequalities provide the necessary technical glue. The result is local, with length of existence depending on the size of the initial data.