We present an algorithm for solving the time-dependent Schrödinger equation that is based on the finite-difference expression of the kinetic energy operator. Students who have some knowledge of linear algebra can understand the theory used to derive the algorithm. This is because the finite-difference kinetic energy matrix and the Hückel matrix for linear conjugated hydrocarbons have similar forms. We offer 3 modules that can be used to solve 1-dimensional time-dependent Schrödinger equations, in real time. Module 1 contains two worksheets designed to show quantum dynamics in bound potentials. Module 2 contains 2 worksheets designed to illustrate tunneling. Module 3 contains 2 worksheets set up to illustrate the effect of time-dependent external fields. Further, modules 1 and 3 also include the Verlet algorithm so students can compare the quantum dynamical evolution of the systems to their classical counterpart in real time.