In this work, we solve time, space and time-space fractional Schrödinger equations based on the non-singular Caputo–Fabrizio derivative definition for 1D infinite-potential well problem. To achieve this, we first work out the fractional differential equations defined in terms of Caputo–Fabrizio derivative. Then, the eigenvalues and the eigenfunctions of the three kinds of fractional Schrödinger equations are deduced. In contrast to Laskin’s results which are based on Riesz derivative, both the obtained wave number and wave function are different from the standard ones. Moreover, the number of solutions is finite and dependent on the space derivative order. When the fractional orders of derivatives become integer numbers (one for time derivative or/and two for space), our findings collapse to the standard results.