  In recent decades a lot of research has been done on the numerical solution
of the time-dependent Schr\"odinger equation. On the one hand, some of the
proposed numerical methods do not need any kind of matrix inversion, but source
terms cannot be easily implemented into this schemes; on the other, some
methods involving matrix inversion can implement source terms in a natural way,
but are not easy to implement into some computational software programs widely
used by non-experts in programming (e.g. Mathematica). We present a simple
method to solve the time-dependent Schr\"odinger equation by using a standard
Crank-Nicholson method together with a Cayley's form for the finite-difference
representation of evolution operator. Here, such standard numerical scheme has
been simplified by inverting analytically the matrix of the evolution operator
in position representation. The analytical inversion of the N x N matrix let us
easily and fully implement the numerical method, with or without source terms,
into Mathematica or even into any numerical computing language or computational
software used for scientific computing.
