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## Main Question or Discussion Point

The cross section for scattering by a Coulomb potential 1/r is the same for both classical and quantum mechanics, and the total cross section is infinite. I understand this classically as saying that no matter how large an impact parameter an incoming particle has, it will still be deflected at least a little bit by the potential, so the cross sectional area in which incoming particles are scattered at some angle > 0 is infinite.

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?

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?