  The dynamical behavior for a quantum Brownian particle is investigated under
a random potential of the fractional iterative map on a one-dimensional
lattice. For our case, the quantum expectation values can be obtained
numerically from the wave function of the fractional Schr$\ddot{o}$dinger
equation. Particularly, the square of mean displacement which is
ensemble-averaged over our configuration is found to be proportional
approximately to $t^{\delta}$ in the long time limit, where $\delta$ $=$ $0.96
\pm 0.02$. The power-law behavior with scaling exponents $\epsilon$ $=$ $0.98
\pm 0.02$ and $\theta$ $=$ $ 0.51 \pm 0.01$ is estimated for $ \bar {{< p(t)
>}^2}$ and $ \bar {{< f(t) >}^2}$, and the result presented is compared with
other numerical calculations.
