numerical integration

matrix elements

fractional equation

hydrogen atom

harmonic oscillator

Computer Physics Communications

The  Fortran  subroutine package   radial  for the numerical solution of the Schrodinger and Dirac wave equations of electrons in central potentials    V   (  r  )     is described. The considered potentials are such that the function    V   (  r  )   ≡  r  V   (  r  )     is finite for all    r    and tends to constant values when    r  →  0    and    r  →  ∞   . This includes finite-range potentials as well as combinations of Coulomb and finite-range potentials. The function    V   (  r  )      used in the numerical calculation is the natural cubic spline that interpolates a table of values provided by the user. The radial wave equations are solved by using piecewise exact  power series expansions   of the radial functions, which are summed up to the prescribed accuracy so that  truncation errors  can be completely avoided. The    radial    subroutines compute radial wave functions,  eigenvalues   for bound states and phase shifts for free states. Specific subroutines are also provided for computing the radial functions and phase shifts for free states of complex  optical potentials   having a finite-range absorptive imaginary part. The solution subroutines are accompanied by example main programs, as well as with specific programs that perform calculations relevant in atomic, nuclear, and radiation physics (the self-consistent solution of the Dirac–Hartee–Fock–Slater equations for  neutral atoms  and     positive ions  , and the calculation of cross sections for elastic scattering of high-energy electrons and  positrons   by atoms and of  nucleons  by nuclei). The distribution package includes a detailed manual with a description of the basic physics and the mathematical formulas implemented in the subroutines.       Program summary  Program Title:  radial   Program Files doi:  http://dx.doi.org/10.17632/h6nbws5h6s.1   Licensing provisions: CC by NC 3.0  Programming language: Fortran 90/95  Supplementary material: Manual in pdf format. Includes a description of the basic physics, and the complete set of formul...

radial: A Fortran subroutine package for the solution of the radial Schrödinger and Dirac wave equations