![]() To avoid the difficult calculation of phase shifts from the numerical solution of the radial wave equation, we adopt a simplified strategy that combines the (first) Born approximation, for both the scattering amplitude and the phase shifts, and the Wentzel–Kramers–Brillouin (WKB) approximation for the phase shifts. Accurate DCSs for potential scattering can be computed by using the partial-wave expansion method, at the expense of considerable numerical work. We consider the elementary quantum formulation based on the relativistic Schrödinger (or Klein–Gordon) wave equation obtained from the correspondence principle. This scheme qualifies as semi-relativistic, because it accounts for relativistic kinematics in a rigorous way, but disregards the differences between the interactions observed from the laboratory and the center-of-mass frames. Solution method: A relativistic extension of the classical trajectory method is formulated on the assumption that the interaction in the center-of-mass frame is central, which is a fundamental requirement of the adopted calculation schemes the DCS in the laboratory frame is then obtained from the relativistic (Lorentz) transform of the DCS calculated in the center-of-mass frame. A Java graphical user interface allows running the program and visualizing the results interactively. Calculation results are written in a number of output files with formats suited for visualization with a plotting program. The user is allowed to select the atomic number of the target atom, the potential model, the kind of projectile and its kinetic energy. The program eccpa provides DCSs computed with four different approaches: the classical trajectory method, the Born approximation, the partial-wave expansion method with approximate phase shifts, and the eikonal approximation. Calculations are performed within the static-field approximation with screened Coulomb potentials expressed as a sum of Yukawa terms with their parameters fitted to approximate the atomic electrostatic potentials resulting from the Thomas–Fermi model and from self-consistent Dirac–Hartree–Fock–Slater calculations. Nature of problem: The program computes differential cross sections (DCSs) for elastic collisions of charged particles (electrons, positrons, muons, antimuons, protons, antiprotons, and alphas) with neutral atoms. ![]() The program eccpa is useful for assessing the validity and the relative accuracy of the various approximations, and as a pedagogical tool. Various approximate solution methods are described and applied to a generic potential represented as a sum of Yukawa terms, which allows a good part of the calculations to be performed analytically. Collisions of spin 1/2 projectiles are also described by solving the Dirac wave equation. The wave equation for the relative motion, as obtained from the correspondence principle, is formally identical to the non-relativistic Schrödinger equation with the reduced mass and the effective potential, and it reduces to the familiar Klein-Gordon equation when the mass of the target atom is much larger than that of the projectile. The equation of the relative motion in the center-of-mass frame is shown to have the same form as in the non-relativistic theory, with a relativistic reduced mass and an effective potential. Although this assumption neglects the effect of relativity on the interaction, it allows using strict relativistic kinematics. ![]() To allow the use of fast and robust calculation methods, the interaction is assumed to be the same in the center-of-mass frame and in the laboratory frame. ![]() The collisions are described within the framework of the static-field approximation, with the interaction between the projectile and the target atom represented by the Coulomb potential of the atomic nucleus screened by the atomic electrons. Instead, we can do this much more generally.The Fortran program eccpa calculates differential and integrated cross sections for elastic collisions of charged particles with atoms by using the classical-trajectory method and several quantum methods and approximations. Like Photon suggested in the comments, do not plug in the position basis form of $H$.
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