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I have seen the Born approximation of the quantum mechanical scattering cross section for a Yukawa potential e^(-mr)/r. Here the total cross section is finite. In the classical case, though, I feel like the same argument as was used for the Coulomb potential applies: although the potential falls off quickly, there is always some force at arbitrarily large distances from the origin, so all incoming particles should be deflected at least a little bit, no matter how large their impact parameters. So I expect that in classical mechanics, the total cross section for the Yukawa potential is infinite.

I'm somewhat uncomfortable with this cross section being infinite in classical mechanics, but finite in quantum mechanics. Is there a conflict here?