838 research outputs found

    Recurrence relation for relativistic atomic matrix elements

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    Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired on the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non relativistic quantum mechanics. We obtain first the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use such relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.Comment: 10 pages, no figure

    Minimizing efforts in validating crowd answers

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    In recent years, crowdsourcing has become essential in a wide range of Web applications. One of the biggest challenges of crowdsourcing is the quality of crowd answers as workers have wide-ranging levels of expertise and the worker community may contain faulty workers. Although various techniques for quality control have been proposed, a post-processing phase in which crowd answers are validated is still required. Validation is typically conducted by experts, whose availability is limited and who incur high costs. Therefore, we develop a probabilistic model that helps to identify the most beneficial validation questions in terms of both, improvement of result correctness and detection of faulty workers. Our approach allows us to guide the experts work by collecting input on the most problematic cases, thereby achieving a set of high quality answers even if the expert does not validate the complete answer set. Our comprehensive evaluation using both real-world and synthetic datasets demonstrates that our techniques save up to 50% of expert efforts compared to baseline methods when striving for perfect result correctness. In absolute terms, for most cases, we achieve close to perfect correctness after expert input has been sought for only 20% of the questions

    Relativistically extended Blanchard recurrence relation for hydrogenic matrix elements

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    General recurrence relations for arbitrary non-diagonal, radial hydrogenic matrix elements are derived in Dirac relativistic quantum mechanics. Our approach is based on a generalization of the second hypervirial method previously employed in the non-relativistic Schr\"odinger case. A relativistic version of the Pasternack-Sternheimer relation is thence obtained in the diagonal (i.e. total angular momentum and parity the same) case, from such relation an expression for the relativistic virial theorem is deduced. To contribute to the utility of the relations, explicit expressions for the radial matrix elements of functions of the form rλr^\lambda and ÎČrλ\beta r^\lambda ---where ÎČ\beta is a Dirac matrix--- are presented.Comment: 21 pages, to be published in J. Phys. B: At. Mol. Opt. Phys. in Apri

    Relativistic Kramers-Pasternack Recurrence Relations

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    Recently we have evaluated the matrix elements ,where where O ={1,\beta, i\mathbf{\alpha n}\beta} arethestandardDiracmatrixoperatorsandtheangularbracketsdenotethequantum−mechanicalaveragefortherelativisticCoulombproblem,intermsofgeneralizedhypergeometricfunctions are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions _{3}F_{2}(1) $ for all suitable powers and established two sets of Pasternack-type matrix identities for these integrals. The corresponding Kramers--Pasternack three-term vector recurrence relations are derived here.Comment: 12 pages, no figures Will appear as it is in Journal of Physics B: Atomic, Molecular and Optical Physics, Special Issue on Hight Presicion Atomic Physic
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