  We develop a formalism for the scattering of a particle on the $q$-deformed
Euclidean space. We write down $q$-versions of the Lippmann-Schwinger equation.
Their iterative solutions for a weak scattering potential lead us to
$q$-versions of the Born series. With the expressions for the wave functions of
the scattered particle, we can write down S-matrix elements. We show that these
S-matrix elements satisfy unitarity conditions. Considerations about the
interaction picture for a quantum system in the $q$-deformed Euclidean space
and a discussion of a $q$-version of time-dependent perturbation theory
conclude our studies.
