  We study the linear fractional Schr\"odinger equation on a Hilbert space,
with a fractional time derivative of order $0<\alpha<1,$ and a self-adjoint
generator $A.$ Using the spectral theorem we prove existence and uniqueness of
strong solutions, and we show that the solutions are governed by an operator
solution family $\{U_{\alpha}(t)\}_{t\geq 0}$. Moreover, we prove that the
solution family $U_{\alpha}(t)$ converges strongly to the family of unitary
operators $e^{-itA},$ as $\alpha$ approaches to $1$.
