In classical electrodynamics, accelerated charges emit electromagnetic radiation. The associated energy loss of the particle can be modeled as a radiation reaction force, proportional to the change in acceleration. Including this force in the usual dynamical equation for a charged particle yields the Abraham-Lorentz-Dirac (ALD) equation. For point particles, solutions to the ALD equation are problematic. First, they require additional initial conditions beyond those needed in regular Newtonian dynamics, but more concerning are the presence of unphysical run-away solutions and acausal trajectories. These problems arise in the point-like limit of the theory, so it is possible that quantum mechanics is the more appropriate framework with which to address the problem. In order to explore the similarities and differences of the classical and quantum frameworks, we compute classical trajectories of charged particles in the presence of a Coulomb field, including radiation reaction. We compare the associated classical cross-section with the quantum field-theoretic cross-section for Coulombic scattering, including radiative corrections. With these calculations, we will be able to assess when the field theoretic formulation of single-photon emission can approximate the classical radiation reaction.