Fractional Dynamical Behavior in Quantum Brownian Motion

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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 Schrodinger equation. Particularly, the square of mean displacement which is ensemble-averaged over our configuration is found to be proportional approximately to tδ in the long time limit, where δ = 0.96 ± 0.02. The power-law behavior with scaling exponents ε = 0.98 ± 0.02 and θ = 0.51 ± 0.01 is estimated for < p(t) >² and < f(t) >², and the result presented is compared with other numerical calculations.