We know that an electron behaves like a point charge, and the probability density of its position is given by the Born rule. Now suppose that we shoot an electron toward an atom, so the electron gets scattered due to the Coulomb forces the nucleus and the electrons of the atom push on it.
How is the direction of this coulomb force determined? If the electrons are acting as point charges, and not smeared charges inside the orbitals, the coulomb force vector on the shot electron should have a random but certain direction changing by time. Is this true?
Is this how the scattering of the electrons work? How is the probability density for the deflection angle of the shot electron calculated?