Fractional Dynamical Behavior in Quantum Brownian Motion

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.