The standard impact-parameter treatment for heavy-particle collisions is derived starting from a full quantal description and employing a linearised kinetic-energy operator associated with the relative motion of the atomic nuclei. The authors distinguish between direct and exchange collisions with the definition of appropriate interaction pictures. The differential scattering amplitude (and hence the differential cross section) for the various reaction channels is unambiguously obtained. By derivation, momentum-translation and spatial-translation operators are found.