Abstarct: In recent years, the idea of fractional derivative in quantum mechanics can be considered as one of the most creative ideas in physics. Given this idea, the Schrodinger equation can be rewritten in a new form. Such a formulation of quantum mechanics has significant applications in mathematics, physics, and other fields of science. Mr. Hermann rewrote the collective behavior patterns of the nuclei and could describe the complex behavior of the nuclei by using this formulation. Also he was able to justify different nuclear states by changing the order of the derivative. There are several definitions for the fractional derivatives, that each of them expresses it in a different way, including the definitions of Riemann-Liouvile and Caputo, which can be described as an exact definition of fractional derivative. Recently, researchers found a model from the fractional derivative model that presents a new definition with an adaptive form and leads to interesting results in this field. In the thesis, we want to introduce the fractional derivative and as well as to describe the relativistic and non-relativistic systems by using the fractional derivatives in the presence of specific potentials.