Rayleigh-Ritz Variational Method Research Articles

Quantum and classical dynamics of Langmuir wave packets

The quantum Zakharov system in three spatial dimensions and an associated Lagrangian description, as well as its basic conservation laws, are derived. In the adiabatic and semiclassical cases, the quantum Zakharov system reduces to a quantum modified vector nonlinear Schrödinger (NLS) equation for the envelope electric field. The Lagrangian structure for the resulting vector NLS equation is used to investigate the time dependence of the Gaussian-shaped localized solutions, via the Rayleigh-Ritz variational method. The formal classical limit is considered in detail. The quantum corrections are shown to prevent the collapse of localized Langmuir envelope fields, in both two and three spatial dimensions. Moreover, the quantum terms can produce an oscillatory behavior of the width of the approximate Gaussian solutions. The variational method is shown to preserve the essential conservation laws of the quantum modified vector NLS equation. The possibility of laboratory tests in the next generation intense laser-solid plasma compression experiment is discussed.

Inner-shell excited quartet states in Li-like O5+ and Ne7+ ions

The Rayleigh-Ritz variational method and a configuration-interaction wave function are used to obtain the energies, fine structures and hyperfine structures of high-lying core excited quartet states \(1s2lnl^\prime\,\!^{4}{\rm P}^{0}(n)~(n=1\!-\!5)\) and \(1s2pnp\,\!^{4}{\rm P}(m)~(m=1\!-\!5)\) in O5+ and Ne7+ ions, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The wavelengths, oscillator strengths, and lifetimes are also explored. The results compared with other theoretical and experimental data in the literature.

RESONANT COMPTON SCATTERING IN NONTHERMAL ASTROPHYSICAL PLASMAS

The nonthermal effect on the resonant Compton scattering process is investigated in Lorentzian astrophysical plasmas. The screened atomic wave functions and energies for the hydrogenic ion in Lorentzian plasmas are obtained by the Rayleigh–Ritz variational method. The resonance Compton scattering cross section in Lorentzian plasmas is also obtained as a function of the spectral index and plasma parameters by the photon-perturbation Hamiltonian in the dipole representation. It is shown that the nonthermal character of the Lorentzian plasma enhances the resonant Compton scattering cross section. In addition, the nonthermal effect on the cross section is found to be more significant in plasmas with higher densities.

Nonthermal and screening effects on photoionizations in Lorentzian plasmas

The nonthermal and plasma screening effects on the x-ray photoionization process are investigated in astrophysical Lorentzian plasmas. The screened atomic wave function and energy eigenvalue of the target ion in Lorentzian plasmas are obtained by the Rayleigh–Ritz variational method. The x-ray retardation and screened Coulomb corrections are considered to obtain a photoionization cross section as a function of the spectral index and plasma parameters. It is shown that the nonthermal character of the Lorentzian plasma suppresses the photoionization cross section. Hence, the photoionization cross sections in nonthermal plasmas are found to be always smaller than those in thermal plasmas. It is also shown that the plasma screening effect is more significant for small spectral indices. In addition, the nonthermal effect on the photoionization cross section is found to be decreased with an increase in the Debye length.

Rayleigh–Ritz variation method and connected‐moments polynomial approach

AbstractWe show that the connected‐moments polynomial approach proposed recently is equivalent to the well known Rayleigh–Ritz variation method in the Krylov space. We compare the latter with one of the original connected‐moments methods by means of a numerical test on an anharmonic oscillator. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Energy and relativistic correction of the 1sns configuration in helium

A Mathematica program is developed to calculate the energy and relativistic correction of helium-like atoms in 1sns configuration. A new set of trial functions for 1sns configuration in helium has been suggested. By virtue of the Rayleigh-Ritz variational method, the non-relativistic energies of 1sns configuration in helium (n=2—5) have been calculated, the relativistic corrections, which include mass correction, one- and two-body Darwin correction and spin-spin contact interaction, have been further calculated with the theory of perturbation. The calculated results are in good agreement with experimental data.

Small amplitude, transverse vibrations of circular plates with an eccentric rectangular perforation elastically restrained against rotation and translation on both edges

The Rayleigh–Ritz variational method is applied to the determination of the first four frequency coefficients for small amplitude, transverse vibrations of circular plates with an eccentric, rectangular perforation that is elastically restrained against rotation and translation on both edges. Coordinate functions are used which identically satisfy the boundary conditions at the outer circular edge, while the restraining boundary conditions at the inner edge of the cutout are dealt with directly through the energetic terms in the functional expressions. The procedure seems to show very good numerical stability and convergence properties. As an added bonus, the method allows for increased flexibility in dealing with boundary conditions at the edge of the cutout.

The multistate Rayleigh–Ritz variational method with the principle of minimal sensitivity for anharmonic oscillators

A theoretical scheme to employ the principle of minimal sensitivity for choosing the optimal values of nonlinear parameters is proposed for the multistate Rayleigh–Ritz variational method. Anharmonic oscillators are particularly considered in this paper. Applications of the present scheme to the one-dimensional Morse and two double-well potentials indicate that it provides much more accurate and faster convergent approximations to the exact energy eigenvalues than several schemes existing in the literatures.

Bound States and Resonance States of H^- and He Embedded in Debye Plasma Environments

The doubly excited S-, P- resonance states of H i and the doubly-excited S-, P-, D-resonance states of He embedded in Debye plasma environments have been investigated for by calculating the density of resonance states using stabilization method. The ground states of H i and the bound excited states of He atom in plasmas for various Debye lengths have also been estimated in the framework of Rayleigh-Ritz variational method and by using a quasi-random process. Highly correlated wave functions have been used to represent the correlation efiects between the charged particles. Some of our calculated results on the bound states and resonance states are highlighted in this paper and the detail results will be presented at the conference. DOI: 10.2529/PIERS060907063228

Bound D‐states of helium atom under Debye screening

AbstractWe have investigated the 1snd1,3D (3 ≤ n ≤ 7) state energies of helium atom embedded in weakly coupled plasma environments using the Rayleigh–Ritz variational method. The effect of the plasma environment is taken care of using a Debye screening model. A correlated wave function involving exponential expansion has been used to represent correlation between the charge particles. The bound 1snd1,3D (3 ≤ n ≤ 7) state energies of helium for various Debye lengths along with the excitation energies of few singlet and triplet states are reported. Our results are useful references to atomic physics, plasma physics, and astrophysics research communities. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

The [formula omitted] rearrangement

In this paper, we present a summary of our work in progress on calculating cross sections for the He + H ¯ → He p ¯ + e + rearrangement process in He H ¯ scattering. This has involved a study of the system He p ¯ within the Born–Oppenheimer (BO) approximation using the Rayleigh–Ritz variational method. This work has been reported in [A.C. Todd, E.A.G. Armour, J. Phys. B 38 (2005) 3367] and is summarised here. Similar calculations are in progress for the He + H ¯ entrance channel. We intend to use the entrance channel and rearrangement channel wave functions to obtain the cross sections for the rearrangement using the distorted wave Born approximation T-matrix method described elsewhere in these proceedings [E.A.G. Armour, S. Jonsell, Y. Liu, A.C. Todd, these Proceedings, doi:10.1016/j.nimb.2006.01.049].

Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field

Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigen- states of a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external mag- netic field is discussed. Being smooth at the QD boundaries and of finite depth and range, this potential can only confine a finite number of excess electrons thus forming a realistic model of a QD with smooth inter- face between the QD and its embedding environment. It is argued that the method of intermediate problems, which provides convergent improvable lower bound estimates for eigenvalues of linear half-bound Hermitian operators in Hilbert space, can be fused with the classical Rayleigh-Ritz variational method and stochastic variational method thus resulting in an efficient tool for an alytical and numerical studies of the energy spec- trum and eigenstates of the Gaussian quantum dots, confining small-to-medium number of excess electrons, with controllable or prescribed precision. Various theoretical and experimental aspects of the physics of nano-sized and low-dimensional systems have been under steady uninterrupted development for decades, but only in the last ten- fifteen years of research activities in the field, considerable extra momentum has been gained which can be partly ascribed to the progress in nanofabrication of these systems, partly to the development of new experimental techniques but, most of all, to a general continuous trend and desire to diminish the size of components of integrated electronic circuits down to the so-called mesoscopic scale. Among low-dimensional systems of various kinds, quantum dots are potentially fit for many practical applications and are currently thought to be promising building blocks for novel electronic, spintronic and optoelectronic devices. Reliable estimation of the energy spectrum and eigenstates of a quantum dot is a typical purpose of almost every theoretical study because their properties crucially stipulate the relevant physical characteristics of the quantum dot standing alone as a part of electric circuits or interacting with the environment through its various interfaces. As a matter of fact, nearly all the mathematical methods, developed within the domain of quan- tum mechanics so far, have already been employed in the theory of quantum dots in various specific applications though on a varying scale. The most frequently applicable methods of approximate calculation of eigenvalues and eigenstates of realistic physical models of low-dimensional quantum systems, quantum dots among them, are various numerical methods which permit either direct calculations of the magnitudes in question without proper error estimates or, at best, provide the calculations with nonincreasing or even convergent upper bounds for the eigenvalues. The widely applicable Rayleigh-Ritz method does it but, again, without error estimates. To control the error of the approximations provided by the upper bounds for some quantity it would be enough to derive the corresponding lower bounds which are highly desirable to be

Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge

Rayleigh–Ritz variational method is applied to the determination of the first four frequency coefficients for the title problem by making use of coordinate functions which identically satisfy the boundary conditions at the outer edge. Good stability and convergence properties are found. The mathematical model seems to be realistic and accurate, within the realm of the classical theory of vibrating plates.

Relativistic energies of the singly excited and doubly excited Rydberg series in helium

The Rayleigh-Ritz variational method is used with a multiconfiguration-interaction function to study the energies of singly excited and doubly excited Rydberg series in helium atom, including the mass polarization and relativistic corrections. The oscillator strengths, the transition rates, and the wavelengths are also calculated. The results are compared with other theoretical and experimental data in the literatures.

Energy, fine structure, and transition rate of the doubly excited 3Pe and 3De states for helium

Energies and fine-structure corrections for the doubly excited 3Pe and 3De states of the helium atom are calculated using the Rayleigh–Ritz variational method and the saddle-point variational method with a multiconfiguration-interaction function. The relativistic and mass polarization corrections are included. The oscillator strengths, transition rates, and wavelengths are also calculated. The doubly excited states are grouped into Rydberg series labeled by the quantum numbers K, T, and A to display the systematic regularity along the series. The results are compared with the theoretical and experimental data in the literature. PACS Nos.: 31.15.Pf, 31.25.Jf, 32.70.Cs

Fine structure and hyperfine structure of some excited states of helium

The fine structure and the hyperfine structure for some singly-excited and doubly-excited states of helium atom are calculated using Rayleigh-Ritz variational method with multi-configuration-interaction wave functions. The calculated results of the fine structure for the Rydberg series are in good agreement with other theoretical and experimental data. The hyperfine parameters and the hyperfine coupling constants of 3He are also obtained for this system.

ENERGY, OSCILLATOR STRENGTH AND HYPERFINE STRUCTURE OF THE LOW-LYING EXCITED STATES FOR Be-LIKE SYSTEM

The Rayleigh–Ritz variational method is carried out with a multiconfiguration-interaction wave function and restricted variational method to obtain the relativistic energies of the 1s22s2p 1Po , 1s22s2p 3Po , and 1s22p23P states for the beryllium-like isoelectronic sequence (Z=4–10), including the mass polarization and relativistic corrections. The oscillator strengths and transition rates are also calculated. The results are compared with other theoretical and experimental data in the literature. The hyperfine structure of the low-lying excited states for this system is also explored.

Elastic oscillations of cylindrical fuses

A fast current pulse causes a material to heat and undergo rapid expansion. We calculate the response of a cylindrically shaped material to a pulse on the microsecond time scale. The first step is to obtain the breathing modes of elastic oscillation of the cylinder. These modes are calculated using a Rayleigh–Ritz variational method introduced by Demarest for cubes. The boundary conditions are derived, which give the amplitude of each elastic mode in response to the sudden heating. The results are illustrated by calculations on a station arrester made of a ZnO ceramic.

Theoretical modeling of dark current and photo-response for quantum well and quantum dot infrared detectors

Dark currents (due to direct tunneling process) and photo-response for both quantum well (QW) and quantum dot (QD) systems are calculated. A stabilization method (SM) is used to study the dynamic behavior and photo-response of an electron in an isolated QW, coupled QW or QD (with conical shape) under uniform electric field. A stabilization graph is obtained by plotting the eigenvalues of a QW or QD embedded in a confining box made of barrier material as functions of the size of the box. The eigenvalues of the system are calculated within the effective mass approximation via the Rayleigh–Ritz variational method. Density of states (DOSs) and time-dependent wave functions associated with the quasi-bound state are constructed via the SM. It is shown that the DOSs have a Lorentzian profile and the width of the Lorentzian profile gives the tunneling rate. The photo-response for QWs and conical QDs are studied and implications for device applications are discussed.

RELATIVISTIC ENERGIES AND FINE STRUCTURES OF THE EXCITED STATES FOR BERYLLIUM-LIKE BⅡ

The Rayleigh-Ritz variational method is used with a multiconfiguration-interaction function and restricted variation method to obtain the energies of 1s22s2 1S, 1s22s2p1P0, 1s22s2p3P0 and 1s22p2 3P in beryllium-like boron, including the mass polarization and relativistic corrections. Fine structure, oscillator strengths, transition rates and wavelengths are also calculated. The results are compared with other theoretical and experiment data in the literature.