We present and analyze a numerical method to solve the time-dependent linear
Pauli equation in three space-dimensions. The Pauli equation is a
"semi-relativistic" generalization of the Schr\"odinger equation for 2-spinors
which accounts both for magnetic fields and for spin, the latter missing in
predeeding work on the linear magnetic Schr\"odinger equation. We use a four
operator splitting in time, prove stability and convergence of the method and
derive error estimates as well as meshing strategies for the case of given
time-independent electromagnetic potentials (= "linear" case), thus providing a
generalization of previous results for the magnetic Schr\"odinger equation.
Some proof of concept examples of numerical simulations are presented.