  Quantum Rutherford scattering and scattering of classical waves off
Coulomb-like potentials have similar formal structures and can be studied using
the same mathematical techniques. In both contexts, the long-range nature of
the interaction leads to a divergent total cross-section, which has been
interpreted and regularized in various ways in the past literature. We review
in detail the origin of this divergence, in both real and multipole spaces, and
show that it arises from incorrectly using approximations out of their domain
of validity. We also stress that although classical and quantum Rutherford
scattering share the same formalism, the natures of the associated physical
observables differ. We comment on the role of interference: while interference
can be safely neglected in a quantum context (due to the fact that the
observable quantity is a flux, and the incoming flux is collimated), in a
classical context one expects to see a superposition of transmitted and
scattered waves in a broad region downstream of the target, hence a
cross-section is not connected to any physically observable quantity.
