An explicit algorithm for the time-stepping solution of the Schrödinger equation is described, which is second-order accurate in time. It is a staggered-time algorithm, in which the real and imaginary parts of the wave function are defined at alternate times. The method combines the speed and simplicity of explicit methods with the accuracy and stability of second-order implicit methods. Because of this simplicity and speed, the algorithm is well suited for pedagogical applications on personal computers, as well as for computation-intensive research applications.