While a plane-wave approximation in high-energy physics works well in amajority of practical cases, it becomes inapplicable for scattering of thevortex particles carrying orbital angular momentum, of Airy beams, of theso-called Schr\"odinger cat states, and their generalizations. Such quantumstates of photons, electrons and neutrons have been generated experimentally inrecent years, opening up new perspectives in quantum optics, electronmicroscopy, particle physics, and so forth. Here we discuss the non-plane-waveeffects in scattering brought about by the novel quantum numbers of these wavepackets. For the well-focused electrons of intermediate energies, alreadyavailable at electron microscopes, the corresponding contribution can surpassthat of the radiative corrections. Moreover, collisions of the cat-likesuperpositions of such focused beams with atoms allow one to probe effects ofthe quantum interference, which have never played any role in particlescattering.
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