  This paper is about the fractional Schr\"odinger equation expressed in terms
of the Caputo time-fractional and quantum Riesz-Feller space fractional
derivatives for particle moving in a potential field. The cases of free
particle (zero potential) and a linear potential are considered. For free
particle, the solution is obtained in terms of the Fox $H$-function. For the
case of a linear potential, the separation of variables method allows the
fractional Schr\"odinger equation to be split into space fractional and time
fractional ones. By using the Fourier and Mellin transforms for the space
equation and the contour integrals technique for the time equation, the
solutions are obtained also in terms of the Fox $H$-function. Moreover, some
recent results related to the time fractional equation have been revised and
reconsidered. The results obtained in this paper contain as particular cases
already known results for the fractional Schr\"odinger equation in terms of the
quantum Riesz space-fractional derivative and standard Laplace operator.
