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After introducing the formalism of the general space and time fractionalSchr\"odinger equation, we concentrate on the time fractional Schr\"odingerequation and present new results via the elegant language of Fox's H-functions.We show that the general time dependent part of the wave function for theseparable solutions of the time fractional Schr\"odinger equation is theMittag-Leffler function with an imaginary argument by two different methods.After separating the Mittag-Leffler function into its real and imaginary parts,in contrast to existing works, we show that the total probability is less thanone and decays with time. Introducing the effective potential approach, we alsowrite the Mittag-Leffler function with an imaginary argument as the product ofits purely decaying and purely oscillating parts. In the light of these, wereconsider the simple box problem.  PACS numbers: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db

Time Fractional Schr\"odinger Equation; Fox's H-functions and the Effective Potential