In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LOG) finite element method for solving the time-fractional coupled Schrodinger system. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Through analysis we show that our scheme is unconditionally stable, and the L-2 error estimate has the convergence rate O(h(k+1) + (Delta t)(2) + (Delta t)(alpha/2) h(k+1/2)) for the linear case. Extensive numerical results are provided to demonstrate the efficiency and accuracy of the scheme. (C) 2012 Elsevier Ltd. All rights reserved.