In scattering cross sections we deal with $d\sigma/d\Omega$, incident area per scattered solid angle. When a particle scatters into a small finite $\Delta\Omega$, the incident particle was in a small finite area $\Delta\sigma$. However, in QM the incident state is a plane wave / asymptotic momentum eigenstate, so it's totally delocalized in position space. Isn't the probability for the incident particle to be found in the area $\Delta\sigma$ therefore zero (a small area out of an infinite plane)? If we integrate $d\Omega$ we'd find the total probability to be zero, which is absurd. Where did this reasoning go wrong?
It seems to me it would make more sense to define $dP/d\Omega$ instead of $d\sigma/d\Omega$. In a scattering there is some probability the final momentum angle is in some $d\Omega$. This would then integrate to 1. But obviously this is not done and the cross-sectional area $\sigma$ is somehow necessary.