Born approximation for scattering of wave packets on atoms. I. Theoretical background for scattering of a wave packet on a potential field

Laser photons carrying non-zero orbital angular momentum are known and exploited during the last twenty years. Recently it has been demonstrated experimentally that such (twisted) electrons can be produced and even focused to a subnanometer scale. Thus, twisted electrons emerge as a new tool in atomic physics. The state of a twisted electron can be considered as a specific wave packet of plane waves. In the present paper-I we consider elastic scattering of the wave packets of fast non-relativistic particles on a potential field. We obtain simple and convenient formulae for a number of events in such a scattering. The equations derived represent, in fact, generalization of the well-known Born approximation for the case when finite sizes and inhomogeneity of the initial packet should be taken into account. To illustrate the obtained results, we consider two simple models corresponding to scattering of a Gaussian wave packet on the Gaussian potential and on the hydrogen atom. The scattering of twisted electrons on atoms will be considered in the next paper-II.