The authors provide a numerical method to approximate solutions of the time-dependent Schrödinger equation \(i\partial_t u-\Delta u=u^3+f\) in a domain \(\Omega\), \(u=0\) on \(\partial\Omega\). The numerical method uses the finite difference approximation combined with Witt basis elements. First, the authors construct a discrete fundamental solution for the linear time-dependent Schrödinger operator. Unlike the discrete Green function, the main advantage of authors’ approach is that the discrete fundamental solution has an explicit expression. Next, the convergence of the numerical scheme is derived. At the end of the paper, various numerical examples are presented that show the consistency and stability of the algorithm.