When treating scattering theory in (non relativistic) quantum mechanics, one at first considers the simple setup in which a plane wave impinges on a scattering potential, and one considers a scattered wave of the form (assuming cylindrical symmetry)
$$ psi(vec x) = f(theta) frac{e^{ikr}}{r} $$
One can then perturbatively solve Schrödinger’s equation up to first order in $ V(vec x) $ in order to find $ f(theta) $.
This seems fine, but I am bothered by the interpretation of this physical situation. Textbooks often state that
- The plane wave is a "beam" of many particles. This seems wrong to me because plane waves are non-physical (one should consider packets); furthermore the hamiltonian is that of a single particle. I can’t see how a many-particle setup (which should, by the way, take into account particle exchange symmetry related issues) can be described by a non-physical state of a single particle.
- In order to compute the differential cross-section, one calculates the outward scattered current of probability in the $ hat r $ direction. This is interpreted as the "number of particles" going in that particular direction, per unit surface per unit time. Then, this current is divided by the incident current in order to normalize the result (to make independent of the "number of particles" per unit surface per unit time going towards the scattering potential). Again, I can’t see how the probability current should have the interpretation of number flux of particles, or in other terms, how simply changing the normalization of a (actually non normalizable) state could be interpreted as changing the overall number of particles, since this is not how multiple particles are normally taken into account in quantum mechanics.