In this paper, a finite difference-based numerical approach is developed for time-fractional Schrodinger equations with one or multidimensional space variables, with the use of fractional linear multistep method for time discretization and finite difference method for spatial discretization. The proposed method leads to achieve second order of accuracy for time variable. Stability and convergence theorems for the constructed difference scheme is achieved via z-transform method. Time-fractional Schrodinger equation is considered in abstract form to allow generalization of the theoretical results on problems which have distinct spatial operators with or without variable coefficients. Numerical results are presented on one and multidimensional experimental problems to verify the theoretical results.