Does the Feynman diagram plane wave assumption produce incorrect results?

Let us assume that two electrons pass a nucleus. What is the probability that one of the electrons gains momentum p in the experiment?

We shoot the electrons from opposite directions toward the nucleus. We model the electron fluxes as plane waves with a definite momentum q or -q.

    q  e-  ------>      ● Z+      <-----  e-  -q

  

Let us draw some Feynman diagrams.

 -q  e- -----------------------

  q  e- -----------------------

                    |  p  virtual photon

      Z+ -----------------------

Above we have the basic case.

    q e- -------------------------

                       | p

   -q e- -------------------------

                   | p

     Z+ ------------------------

Above is another case.

The end result is the same in both diagrams: the nucleus and one of the electrons exchange momentum p. We measure the momenta of both electrons after the experiment.

Feynman diagrams calculate probability amplitudes in the momentum space. The amplitude is a function of the momenta of the output particles.

Feynman rules say that we must add the probability amplitudes (which are complex numbers) of the diagrams to get an approximation of the scattering probability. Typically, the probability amplitudes have a different phase. There is some destructive interference.

Adding probability amplitudes is correct if the output is a product of plane waves.

Note that the nucleus is an almost classical object. After the experiment, we can measure its path. Let us assume that the nucleus had very small momentum - it essentially stayed static in the experiment.

Electrons which gained the momentum p directly from the nucleus came from a relatively small spatial volume. They do not form a plane wave.

Electrons which gained the momentum via the other electron come from a much larger spatial volume.

Let us next do a thought experiment.

Let us assume that that we send two arbitrary waves which have a 180 degree phase shift. The wave A comes from a small spatial volume v. A weaker wave B comes from a large spatial volume V.

Why would there be a lot of destructive interference? There is not. We cannot destroy the energy that we used to create the wave A by putting energy to the wave B.

Feynman rules assume that the output of a collision is a product of plane waves. But often it is not. If the electron receives the momentum p in the first diagram, then it flew close to the nucleus. We gained knowledge about the relative position of the nucleus and the electron. Then we cannot model the output with plane waves.

We could calculate the experiment result using the Schrödinger equation because no new particles are created in it. Who thinks that a very crude plane wave model can replace the Schrödinger equation calculation? That would be a miracle.

How to fix Feynman rules? Obviously we have to be very careful if we try to add the probability amplitudes of two diagrams. If the output cannot be approximated with plane waves, then we cannot calculate any interference of the diagrams.

What about vacuum polarization? There Feynman did add the interference of two diagrams, the simplest one and the one-loop diagram. Is that an error?