In this work, the fractional Schrödinger equation with a specific potential \(V(x)=0\) is analyzed using both Caputo and Caputo-Fabrizio fractional derivatives. The order of the derivative considered is \(0 < \alpha \le 1\) for the Caputo derivative and \(0 \le \alpha \le 1\) for the Caputo-Fabrizio derivative. To address dimensional consistency, the equation was non-dimensionalized, and solutions were obtained in terms of the Mittag-Leffler function. In both cases, the conventional solutions are recovered when \(\alpha =1\) , demonstrating consistency with the classical Schrödinger equation.