This paper discusses the numerical solution of the 1-dimensional Schrödinger equation using method of lines approach (MOL) where the spatial dimension is discretize using some finite difference approximation leaving the time dimension to be the only independent variable in the resulting system of initial value problems. The effect of changing the discretization size on the stability of the solution procedure with respect to the absolute stability property of the numerical method used has been studied. In the study the use of Simpson’s rule in MATLAB has also been incorporated. The result indicates that there is some relationship between the discretization size and the convergence property. Further work will look into the modeling this equation to the application in Physics.