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Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

1971; Elsevier BV; Volume: 2; Issue: 6 Linguagem: Inglês

10.1016/0010-4655(71)90028-2

1879-2944

D. L. Moores,

Advanced Frequency and Time Standards

Today, the ‘hydrogen atom model’ is known to play its role not only in teaching the basic elements of quantum mechanics but also for building up effective theories in atomic and molecular physics, quantum optics, plasma physics, or even in the design of semiconductor devices. Therefore, the analytical as well as numerical solutions of the hydrogen-like ions are frequently required both, for analyzing experimental data and for carrying out quite advanced theoretical studies. In order to support a fast and consistent access to these (Coulomb-field) solutions, here we present the Dirac program which has been developed originally for studying the properties and dynamical behavior of the (hydrogen-like) ions. In the present version, a set of Maple procedures is provided for the Coulomb wave and Green's functions by applying the (wave) equations from both, the nonrelativistic and relativistic theory. Apart from the interactive access to these functions, moreover, a number of radial integrals are also implemented in the Dirac program which may help the user to construct transition amplitudes and cross sections as they occur frequently in the theory of ion–atom and ion–photon collisions.Title of program: DiracCatalogue number: ADUQProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUQProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: NoneComputer for which the program is designed and has been tested: All computers with a license of the computer algebra package Maple [1]Program language used: Maple 8 and 9No. of lines in distributed program, including test data, etc.:2186No. of bytes in distributed program, including test data, etc.: 162 591Distribution format: tar gzip fileCPC Program Library subprograms required: NoneNature of the physical problem: Analytical solutions of the hydrogen atom are widely used in very different fields of physics [2,3]. Despite of the rather simple structure of the hydrogen-like ions, however, the underlying ‘mathematics’ is not always that easy to deal with. Apart from the well-known level structure of these ions as obtained from either the Schrödinger or Dirac equation, namely, a great deal of other properties are often needed. These properties are related to the interaction of bound electron(s) with external particles and fields and, hence, require to evaluate transition amplitudes, including wavefunctions and (transition) operators of quite different complexity. Although various special functions, such as the Laguerre polynomials, spherical harmonics, Whittaker functions, or the hypergeometric functions of various kinds can be used in most cases in order to express these amplitudes in a concise form, their derivation is time consuming and prone for making errors. In addition to their complexity, moreover, there exist a large number of mathematical relations among these functions which are difficult to remember in detail and which have often hampered quantitative studies in the past.Method of solution: A set of Maple procedures is developed which provides both the nonrelativistic and relativistic (analytical) solutions of the ‘hydrogen atom model’ and which facilitates the symbolic evaluation of various transition amplitudes.Restrictions onto the complexity of the problem: Over the past decades, a large number of representations have been worked out for the hydrogenic wave and Green's functions, using different variables and coordinates [2]. From these, the position–space representation in spherical coordinates is certainly of most practical interest and has been used as the basis of the present implementation. No attempt has been made by us so far to provide the wave and Green's functions also in momentum space, for which the relativistic momentum functions would have to be constructed numerically. Although the Dirac program supports both symbolic and numerical computations, the latter one are based on Maple's standard software floating-point algorithms and on the (attempted) precision as defined by the global Digits variable. Although the default number, Digits = 10, appears sufficient for many computations, it often leads to a rather dramatic loss in the accuracy of the relativistic wave functions and integrals, mainly owing to Maple's imprecise internal evaluation of the corresponding special functions. Therefore, in order to avoid such computational difficulties, the Digits variable is set to 20 whenever the Dirac program is (re-)loaded.Unusual features of the program: The Dirac program has been designed for interactive work which, apart from the standard solutions and integrals of the hydrogen atom, also support the use of (approximate) semirelativistic wave functions for both, the bound- and continuum-states of the electron. To provide a fast and accurate access to a number of radial integrals which arise frequently in applications, the analytical expressions for these integrals have been implemented for the one-particle operators rk, e−σr, dm/drm, jL(kr) as well as for the (so-called) two-particle Slater integrals which are needed to describe the Coulomb repulsion among the electrons. Further procedures of the Dirac program concern, for instance, the conversion of the physical results between different unit systems or for different sets of quantum numbers. A brief description of all procedures as available in the present version of the Dirac program is given in the user manual Dirac-commands.pdf which is distributed together with the code.Typical running time: Although the program replies promptly on most requests, the running time also depends on the particular task.References:[1] Maple is a registered trademark of Waterloo Maple Inc.[2] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, Berlin, 1957.[3] J. Eichler and W. Meyerhof, Relativistic Atomic Collisions, Academic Press, New York, 1995.