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We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L 2 -norm with order O (τ + h k + 1) . Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L 2 -norm with order O (τ + h k + 1 + H k + 2) . The numerical experiment is provided to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]