  We investigate how the normal energy transport is realized in one-dimensional
quantum systems using a quantum spin system. The direct investigation of local
energy distribution under thermal gradient is made using the quantum master
equation, and the mixing properties and the convergence of the Green-Kubo
formula are investigated when the number of spin increases. We find that the
autocorrelation function in the Green-Kubo formula decays as $\sim t^{-1.5}$ to
a finite value which vanishes rapidly with the increase of the system size. As
a result, the Green-Kubo formula converges to a finite value in the
thermodynamic limit. These facts strongly support the realization of Fourier
heat law in a quantum system.
