Calculation of ground- and excited-state energies of confined helium atom

We calculate the energies of ground and three low lying excited states of confined helium atom centered in an impenetrable spherical box. We perform the calculation by employing variational method with two-parameter variational forms for the correlated two-particle wave function. With just two variational parameters we get quite accurate results for both ground and excited state energies.Comment: 13 pages, No figur

Improved strategies for variational calculations for helium

The aim of this work is to apply trial functions constructed from Hylleraas functions with three independent sets of nonlinear scale factors to variational calculations for helium and helium-like ions. The ground state and low-lying Rydberg energy levels of these ions have been calculated to several orders of magnitude greater accuracy than previous work in this area while using an equal, or in most cases, a reduced number of basis functions. Each of the three sectors of the basis set is found to describe a different scale of coordinate space corresponding to the asymptotic, intermediate, and close-ranged distances between particles. The incorporation of the third, close-ranged sector, allows the basis set to better model complex correlation effects between the nucleus and the two electrons in the atomic three-body problem. Optimization of the basis set parameters is achieved through standard variational techniques and the validity of the wave functions near the electron-nucleus and electron-electron coalescence points is tested using the Kato cusp conditions. The tripled basis set is also applied to the 1/ Z perturbation expansion as a case study. A multiple-precision package, MPFUN90 written by David H. Bailey, was used to alleviate numerical instabilities which arose for certain states.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2004 .N57. Source: Masters Abstracts International, Volume: 43-01, page: 0225. Adviser: G. W. F. Drake. Thesis (M.Sc.)--University of Windsor (Canada), 2004

Theoretical studies of positronium formation in positron collisions with lithium and hydrogen atoms

The Kohn variational method has been used to study elastic scattering and positronium (Ps) formation in positron collisions with atomic hydrogen and lithium, in the energy region where only these two channels are open. In common with other alkali metals, lithium is interesting in that its valence electron is sufficiently weakly bound that positronium formation is exothermic, and hence an open channel for incident positrons of zero energy. For such a process, Wigner's threshold theory predicts an s-wave cross section which has an inverse dependence on the wavenumber, k, of the projectile as k→0. Using a model potential and very elaborate trial functions, a detailed investigation of s- and p-wave positron-lithium scattering has been made in the energy range 0-1.84eV, and preliminary results have also been obtained for d-wave scattering. The s-wave Ps formation cross section, as calculated variationally, appears to be in accordance with the Wigner theory, although this partial wave contributes negligibly to the Ps channel across most of the energy range considered. The p and d partial waves make a much more substantial contribution to the rearrangement process. New cross sections for positron-hydrogen scattering have been calculated for the energy region close to the positronium formation threshold, and results have been compared with the predictions of R-matrix threshold theory

Theoretical studies of positronium formation in low energy positron-helium collisions

Theoretical investigations of the collisions of low energy positrons with helium atoms have been made in the energy region where only the elastic scattering and the positronium formation channels are open. We have used the two channel Kohn variational method and new accurate numerical procedures have been developed to perform the many six dimensional integrations. The effect of the inexactness of the target wavefunction on the final results has been investigated and consequently we have used very elaborate helium wavefunctions in our calculations. The s-wave positronium formation cross section has been calculated using trial functions containing 502 short-range terms and is found to be very small. The s-wave elastic scattering cross section is found to be the main contributor to the total elastic cross section and a detailed investigation of the behaviour of the cross section at the positronium formation threshold reveals a 'rounded step' feature which is predicted by Wigner's threshold theory. The positronium formation cross sections for p- and d-wave scattering have been calculated, and we find the d-wave component to be dominant for energies greater than 1eV above threshold. The total positronium formation cross section is evaluated using the s-, p- and d-wave results of this work and the first Born approximation for the higher partial waves. A difference is found between theory and experiment which is attributed to the uncertainty in the absolute values of the experimental data and the convergence of the theoretical results. The p- and d-wave elastic scattering cross sections have been calculated and are found to be ≈30% of the total elastic scattering cross section for energies above the positronium formation threshold. We have also investigated the annihilition of positrons with the electrons in the helium atom at energies corresponding to room temperature. The angular correlation function has been calculated and is found to agree very well with the latest experimental measurements

Natural orbitals and their occupation numbers in a non-interacting two-anyon system in the magnetic gauge

We investigate the properties of natural orbitals and their occupation numbers of the ground state of two non-interacting anyons characterised by the fractional exchange parameter $\alpha$ and confined in a harmonic trap. We work in the boson magnetic gauge where the anyons are modelled as composite bosons with magnetic flux quanta attached to their positions. We derive an asymptotic form of the weakly occupied natural orbitals, and show that their corresponding (ordered descendingly) occupation numbers decay according to the power law $n^{-(4+2\alpha)}$, where $n$ is the index of the natural orbital. We find remarkable numerical agreement of the theory with the natural orbitals and their occupation numbers computed from the spectral decomposition of the system's wavefunction. We explain that the same results apply to the fermion magnetic gauge.Comment: 10 pages, 3 figure

Stability of Few-Charge Systems in Quantum Mechanics

We consider non-relativistic systems in quantum mechanics interacting through the Coulomb potential, and discuss the existence of bound states which are stable against spontaneous dissociation into smaller atoms or ions. We review the studies that have been made of specific mass configurations and also the properties of the domain of stability in the space of masses or inverse masses. These rigorous results are supplemented by numerical investigations using accurate variational methods. A section is devoted to systems of three arbitrary charges and another to molecules in a world with two space-dimensions.Comment: 101 pages, review articl

Recent Developments in the Dynamic Stability of Elastic Structures

Dynamic instability in the mechanics of elastic structures is a fascinating topic, with many issues still unsettled. Accordingly, there is a wealth of literature examining the problems from different perspectives (analytical, numerical, experimental etc.), and coverings a wide variety of topics (bifurcations, chaos, strange attractors, imperfection sensitivity, tailor-ability, parametric resonance, conservative or non-conservative systems, linear or nonlinear systems, fluid-solid interaction, follower forces etc.). This paper provides a survey of selected topics of current research interest. It aims to collate the key recent developments and international trends, as well as describe any possible future challenges. A paradigmatic example of Ziegler's paradox on the destabilizing effect of small damping is also included

A fast iterative algorithm for near-diagonal eigenvalue problems

We introduce a novel iterative eigenvalue algorithm for near-diagonal matrices termed iterative perturbative theory (IPT). Built upon a "perturbative" partitioning of the matrix into diagonal and off-diagonal parts, IPT computes one or all eigenpairs with a complexity per iteration of one matrix-vector or one matrix-matrix multiplication respectively. Thanks to the high parallelism of these basic linear algebra operations, we obtain excellent performance on multi-core processors and GPUs, with large speed-ups over standard methods (up to $\sim50$x with respect to LAPACK and ARPACK). For matrices which are not close to being diagonal but have well-separated eigenvalues, IPT can be be used to refine low-precision eigenpairs obtained by other methods. We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices. In a real-world application from quantum chemistry, we find that IPT performs similarly to the Davidson algorithm.Comment: Based on arXiv:2002.1287

Time integration and steady-state continuation for 2d lubrication equations

Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyse steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.Comment: 33 pages, 16 figure