Quantum Mechanics is considered as one of the most weird branches in physics with the most, although accurate, counter intuitive predictions. The problem lies not on the theory but on what it is trying to explain: the microscopic world. Moreover, due to the complicated form of the Schrödinger equation, there are only few examples where an analytical solution to the problem exists. Therefore, the motivation of this paper is to explore the effects of Quantum Mechanics solving the Schrödinger equation numerically for different potentials. Firstly, the algorithms and the numerical methods used to calculate the eigenvalues and the eigenstates are explained. Subsequently, on the results section, the numerical system is stressed to find out its precision and the method is used for two real models: the ammonia molecule and the hydrogen atom. Finally, an applet that uses the numerical system of this paper is discussed. 2017/2018