Algebraic approach to matrix elements: Recurrence relations and closed formulas for hydrogenlike wave functions.

In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = rk with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.

Asymptotic behaviours and general recurrence relations

Recursiveness in matrix rational interpolation problems

CHAPTER 4 - Recurrence Relations and Algebraic Equations

Several scattering problems arising at diffraction of plane wave on simple shape bodies are often solved by expansion of the solution in series of eigenfunction of one-dimensional Sturm-Liouville problem. In order to evaluate the coefficients of these expansions numerical values of matrix elements are needed. In the simplest cases these are multipole matrix elements of the form V/sup (k)/ /sub nm/=/spl int//sup 1//sub -1/x/sup k/y/sub n/(x)y/sub m/(x)dx. The general approach to the problem was proposed by Yu. Slavyanov (see Theor. and Phys., no.120, p.473, 1999). We give an example including numerical computations.

We present an algorithm for DFT calculations employing Gaussian basis sets for the wave function and a Fourier basis for the potential representation. In particular, a numerically very efficient calculation of the local potential matrix elements and the charge density is described. Special emphasis is placed on the consequences of periodicity and explicit $\mathbit{k}$-vector dependence. The algorithm is tested by comparison with more straightforward ones for the case of adsorption of ethylene on the silicon-rich $\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ surface clearly revealing its substantial advantages. A complete self-consistency cycle is speeded up by roughly one order of magnitude since the calculation of matrix elements and of the charge density are accelerated by factors of 10 and 80, respectively, as compared to their straightforward calculation. Our results for ${\mathrm{C}}_{2}{\mathrm{H}}_{4}:\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ show that ethylene molecules preferentially adsorb in on-top positions above Si dimers on the substrate surface saturating both dimer dangling bonds per unit cell. In addition, a twist of the molecules around a surface-perpendicular axis is slightly favored energetically similar to the case of a complete monolayer of ethylene adsorbed on the $\mathrm{Si}(001)\text{\ensuremath{-}}(2\ifmmode\times\else\texttimes\fi{}1)$ surface.

Some properties of generalized secant integrals: extended definition and recurrence relations

We discuss how colour flows can be used to simplify the computation of matrix elements, and in the context of parton shower Monte Carlos with accuracy beyond leading-colour. We show that, by systematically employing them, the results for tree-level matrix elements and their soft limits can be given in a closed form that does not require any colour algebra. The colour flows that we define are a natural generalization of those exploited by existing Monte Carlos; we construct their representations in terms of different but conceptually equivalent quantities, namely colour loops and dipole graphs, and examine how these objects may help to extend the accuracy of Monte Carlos through the inclusion of subleading-colour effects. We show how the results that we obtain can be used, with trivial modifications, in the context of QCD+QED simulations, since we are able to put the gluon and photon soft-radiation patterns on the same footing. We also comment on some peculiar properties of gluon-only colour flows, and their relationships with established results in the mathematics of permutations.

The eigenspace of the total angular and spin momentum operator Ĵ2 in the j — j coupling and of the isospin operator T2 is the most appropriate for calculation of the nuclear properties. Due to the nature of nuclear forces solutions of nuclear equations are sought in full many-particle spaces built from primitive functions, called in context of nuclear shell-model calulations’orbits’, that are almost always taken as harmonic oscillator functions (cf Wong 1981; Brussaard and Glaudemans 1977). The full spaces have very high dimensions and therefore shell-model calculations in nuclear physics are concentrated mainly in the sd shell, with only a modest studies of other shells (McGrory and Wildenthal 1980). A program determining the dimensionalities of such model spaces has recently been published (Draayer and Valdes 1985). Calculation of matrix elements in (Ĵ2, T2)-adapted space is not simple, therefore computer programs performing calculations of nuclear structure work frequently in the ‘M-scheme’ or determinantal spaces where calculation of matrix elements is simpler but the dimension of the space is much bigger (Duch 1986b).

Group theoretical techniques similar to those employed for transition-metal complexes are applied to diatomic molecules. Expressions are given for reduced matrix elements between configurations with one, two and three electrons outside closed shells. The results agree with previous calculations but predict deviations from simple applications of the hypothesis of pure precession.

The unknown ladder operators for the Wei Hua potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The annihilation and creation operators have been obtained with the use of the factorization method. The coherent states for the Wei Hua anharmonic oscillator, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed in the framework of the supersymmetric quantum mechanics. The constructed ladder operators can be applied to compute the matrix elements in theoretical spectroscopy due to the fact that this potential is widely used to a proper description the large-amplitude vibrations of diatomic molecules.

Physically measurable quantities in quantum mechanics are associated with expectation values or matrix elements of the corresponding operators. It is therefore of fundamental importance to elaborate efficient methods for the calculation of such matrix elements. One of the most attractive features of the second quantized approach is its simplicity in evaluating matrix elements. To appreciate this, one must get some practice in the formalism.

A recurrence relation is presented which connects the general (n' not=n) hydrogenic radial matrix element of rbeta , where beta is a complex number to matrix elements involving exponents beta +2, beta -1, and beta -2 but the same sets of quantum numbers. This identity is found solely from the boundary conditions on the wavefunctions and the radial differential equation and should be of use in simplifying rk forms given elsewhere. For n'=n, the recurrence relation reveals a proportionality between the - beta -2, beta -1 matrix elements that includes the Pasternack-Sternheimer identity as a special case. Further results may be obtained by beta -differentiation, and new identities for matrix elements of rk log (r) are given. The n'=n matrix elements of rk are studied for k>or=-1; closed-product expressions are found for k-values near the sum and difference of the angular momenta, so that simple prescriptions can be given for the iterative calculation of n-dependent explicit forms. It is suggested that complex-variable theory be applied to a study of these results.

Nuclear matrix elements of neutral currents in muonic atom decays

The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.