The eigenvalue problem for second-order ordinary differential equation (SOODE) in a finite interval with the boundary conditions of the first, second and third kind is formulated. A computational scheme of the finite element method (FEM) is presented that allows the solution of the eigenvalue problem for a SOODE with the known potential function using the programs ODPEVP and KANTBP 4M that implement FEM in the Fortran and Maple, respectively. Numerical analysis of the solution using the KANTBP 4M program is performed for the SOODE exactly solvable eigenvalue problem. The discrete energy eigenvalues and eigenfunctions are analyzed for vibrational-rotational states of the diatomic beryllium molecule solving the eigenvalue problem for the SOODE numerically with the table-valued potential function approximated by interpolation Lagrange and Hermite polynomials and its asymptotic expansion for large values of the independent variable specified as Fortran function. The efficacy of the programs is demonstrated by the calculations of twelve eigenenergies of vibrational bound states with the required accuracy, in comparison with those known from literature, and the vibrational-rotational spectrum of the diatomic beryllium molecule.
The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1D atom with short-range interaction potential modelled by a δ-function is considered. The model is related to the up-to-date laser techniques providing the duration of pulses as short as a few attoseconds and the intensities higher than 10<sup>14</sup> W/cm<sup>2</sup>.
We present new calculation schemes using high-order finite element method implemented on unstructured grids with triangle elements for solving boundary-value problems that describe axially symmetric quantum dots. The efficiency of the algorithms and software is demonstrated by benchmark calculations of the energy spectrum, the envelope eigenfunctions of electron, hole and exciton states, and the direct interband light absorption in conical and spheroidal impenetrable quantum dots.
Three-body model comprising a diatomic homonuclear molecule and an atom, the solutions of which are necessary for modelling interactions of three-body systems with laser radiation and spectroscopy, is formulated in the collinear configuration of the adiabatic representation. The mapping of the relevant 2D boundary-value problems (BVPs) in the Jacobi coordinates and in polar (hyperspherical) coordinates is reduced to a 1D BVP for a system of coupled second-order ordinary differential equations (ODEs) by means of the Kantorovich expansion in basis functions of one of the two independent variables, depending on the other independent variable parametrically. The efficiency of the proposed approach and software is demonstrated by benchmark calculations of the discrete spectrum of Be<sub>3</sub> trimer in the collinear configuration.
Proc. SPIE. 9917, Saratov Fall Meeting 2015: Third International Symposium on Optics and Biophotonics and Seventh Finnish-Russian Photonics and Laser Symposium (PALS)
KEYWORDS: Modeling, Radon, Chemical species, Matrices, Electrons, Ions, 3D modeling, Numerical modeling, Helium, Light-matter interactions
3D boundary-value problem (BVP) that arises in modelling the interactions of two-electron quantum systems (atoms or ions) with laser radiation, is formulated in the internal coordinate frame with the electrons placed in the focal points of the spheroidal system of coordinates with mapping on a hyper-sphere.<sup>1, 2</sup> The wave function is sought for in the form of a decomposition in the basis of surface functions of the Coulomb two-center problem having the purely discrete spectrum and parametrically depending on the hyper-sphere hyper-radius. Surface functions are sought for in the form of an expansion in the basis of Legendre polynomials with unknown components calculated as solutions of the reduced problem. Efficiency of the approach in comparison with that of the conventional hyper-spherical one<sup>3, 4</sup> is demonstrated by the example of helium atom.
Proc. SPIE. 9448, Saratov Fall Meeting 2014: Optical Technologies in Biophysics and Medicine XVI; Laser Physics and Photonics XVI; and Computational Biophysics
Basing on Dirac equation for interacting massless fermions, we propose a nonlinear model that describes a possible mechanism of ferromagnetism in graphene structures, resulting from electron-electron interaction and spontaneous breaking of spin symmetry of valence electrons. Qualitative predictions of the model are important for practical applications in spintronics. Localized kink-antikink patterns of valence electron spin density on the graphene surface are calculated, their interaction is described, and, finally, the formation of their quasi-bound metastable states (breathers) is investigated. The spectrum of breathers is calculated in both the analytical and the numerical form. Once created, the inverted population of the appropriate states may be used to generate quantum coherent nonlinear spin waves that can find practical applications in nanoelectronics and spintronics. Optical excitation and possible lasing transitions between the breather discrete states are discussed.
Aimed at applications to the photonics of composite two-electron quantum systems like a helium atom in hyper spherical coordinates, the boundary value problem (BVP) for a system of coupled self-adjoined 3D elliptic partial differential equations of the Schrodinger type with homogeneous third-type boundary conditions is formulated in coupled-channel adiabatic approach. The Kantorovich reduction of the problem to BVPs for ordinary second-order differential equations (ODEs) with respect to functions of a single hyper-radial variable is implemented by expanding the solution over a set of surface (angular) functions that depend on the hyper-radial variable as a parameter. Benchmark calculations are presented by the example of the ground and first excited states of a Helium atom. The convergence of the results with respect to the number of the surface functions and their components is studied. The comparison with the known results is presented.
The model for quantum tunneling of a diatomic homonuclear molecule is formulated as a 2D boundary-value problem (2D BVP) for the Schrodinger equation with homogeneous boundary conditions of the third type. The molecule is considered as a pair of identical particles coupled via the effective potential. For short-range barrier potentials the Galerkin reduction to BVP for a set of closed-channel second-order ordinary differential equations (ODEs) is obtained by expanding the solution in a basis of transverse variable functions. Benchmark calculations of quantum tunneling through Gaussian barriers are presented for a pair of identical nuclei coupled by Morse potential. The results are compared with the direct numerical solution of the original2D BVP obtained using the Numerov scheme. The effect of quantum transparency, i.e., the resonance behavior of the transmission coefficient versus the energy of the molecule, is shown to be a manifestation of the barrier metastable states, embedded in the continuum below the dissociation threshold, as well as quantum diffusion. The possibility of controlling the dynamics of atom-ion collisions by laser pulses is analyzed using a lD BVP two-center model with Poschl-Teller potentials.
In the effective mass approximation for electron (hole) states of a spheroidal quantum dot with and without
external fields the perturbation theory schemes are constructed in the framework of the Kantorovich and adiabatic
methods. The eigenvalues and eigenfunctions of the problem, obtained in both analytical and numerical forms,
are applied to the analysis of the absorption coefficient for an ensemble of spheroidal quantum dots with random
dimensions of the minor semiaxis and with parabolic or non-parabolic dispersion laws for holes and electrons,
affected by a homogeneous electric field, i.e., the quantum-confined Stark effect.
To enhance the laser-stimulated recombination of antihydrogen from cold antiproton-positron plasma in a trap
we propose to use a new resonance mechanism involving the quasi-stationary states of the positron that arise
from the joint action of the Coulomb field of the antiproton and the strong magnetic field of the trap. The
recombination rate is expressed via the cross-section of laser ionization of the atom that has strongly non-monotonic
frequency dependence due to the presence of quasi-stationary states merged into the continuum
background. The estimates using previously calculated ionization cross-section show the possibility to enhance
the laser-stimulated recombination by means of the optimal laser frequency choice.
A new efficient method of calculating the photoionization of a hydrogen atom in a strong magnetic field is
developed basing on the Kantorovich approach to the parametric boundary problems in spherical coordinates.
The progress as compared with our previous paper [SPIE Proc. <b>6165</b>, p. 66−82, (2006)] consists in computation
of the wave functions of continuous spectrum, including the quasi-stationary states imbedded in the continuum.
The photoionization cross sections for the ground and excited states are in good agreement with the calculations by other authors.
A new effective method of calculating the wave functions of discrete and continuous spectra of a hydrogen atom in
a strong magnetic field is developed based on the Kantorovich approach to the parametric eigenvalue problems in
spherical coordinates. The two-dimensional spectral problem for the Schrodinger equation with fixed magnetic
quantum number and parity is reduced to a spectral parametric problem for a one-dimensional equation for
the angular variable and a finite set of ordinary second-order differential equations for the radial variable. A
canonical transformation is applied to approximate the finite set of radial equations by means of a new radial
equation describing an open channel. The rate of convergence is examined numerically and illustrated with a set
of typical examples. The results are in good agreement with calculations by other authors.
Proc. SPIE. 6165, Saratov Fall Meeting 2005: Laser Physics and Photonics, Spectroscopy and Molecular Modeling VI
KEYWORDS: Oscillators, Chemical species, Matrices, Hydrogen, Magnetism, Numerical analysis, Finite element methods, Erbium, Chemical elements, Information operations
We present theoretical calculations for the evolution of Zeeman states in a train of short electric half-cycle pulses
(kicks). For the numerical solution of the corresponding time-dependent Schrodinger equation (TDSE) the high accuracy
splitting scheme based on the unitary approximations of the evolution operator is developed. The
finite element method is used for determining the spatial form of the solution. The efficiency and stability of the
developed computational method is shown for 1D models in the cases of second-, forth-, and sixth-order accuracy
with respect to the time step. Numerical calculations for the kicked hydrogen atom in the presence of magnetic
field are performed using the scheme of the sixth-order accuracy with respect to a time step and both Galerkin
and Kantorovich reductions of the problem with respect to the angular variables. For a particular choice of the
electric- and magnetic-field parameters and the initial Zeeman state the corresponding results exhibit a two-state
resonance picture.
Discrete algorithms for symbolic computation of topological phases and observables in optical interferometric systems are presented and illustrated using a set of test models. The calculation of the parameters of a birefringent plate that can be measured by means of Mach-Zehnder interferometer is implemented in terms of Maple and Mathematica. Near-field test models of the systems, that possess both geometrical and dynamical phases in the far-field region, are constructed beyond the the bounds of the ray approximation. These models imply a set of discrete sources with variable parameters and make use of the appropriate set of separable potentials.
Proc. SPIE. 5476, Saratov Fall Meeting 2003: Laser Physics and Photonics, Spectroscopy, and Molecular Modeling IV
KEYWORDS: Oscillators, Chemical species, Matrices, Hydrogen, Laser development, Magnetism, Numerical analysis, Finite element methods, Chemical elements, Algorithm development
High accuracy splitting algorithms based on the unitary approximations of the evolution operator for the time-dependent Schrodinger equation (TDSE) with a train of laser pulses are developed. The efficiency of the algorithms is shown using typical examples of a hydrogen atom affected by a train of laser pulses in the dipole approximation and an additional constant magnetic field. The stabilization effects are discussed.
A special class of separable potentials in the momentum space is proposed which is equivalent to 3D δ-functions in the configuration space. These potentials allow generation of a large number of exact solutions of time-dependent and time-independent Helmgoltz (Schroedinger) equations. It is shown numerically that in the 1D case δ-functions effectively approach the continuous potentials via a convergent algorithm.
Proc. SPIE. 5067, Saratov Fall Meeting 2002: Laser Physics and Photonics, Spectroscopy, and Molecular Modeling III; Coherent Optics of Ordered and Random Media III
KEYWORDS: Chemical species, Physics, Numerical analysis, Wave propagation, Ionization, Finite element methods, Mathematics, Molecular spectroscopy, Chemical elements, Pulsed laser operation
Stable adaptive methods for solving the time-dependent Schrodinger equation (TDSE)) are considered in the framework of conventional finite-element representation of smooth solutions over coordinate spaces of a projective type with long derivatives. Generalization of Cranck-Nicholson scheme of forth order in time step is implemented. Projective “hidden variable” representation of strongly oscillating solutions is realized to extract explicitly the strongly variable gauge phase factor and to evaluate only the “pilot solution” which is reduced to a smooth envelope of the solution under consideration. Such an approach corresponds to the known transformation from Euler space variables to Lagrangian ones and the inducing characteristic representation of self-similar solutions widely used in the flow propagation problems. We study both smooth and strongly oscillating solutions of TDSE describing conventional atomic models in the laser pulse field. It is shown that for short-range potentials the “pilot solution” can be naturally interpreted as the spectrum of the outgoing wave. The examples considered show the efficiency and stability of the elaborated methods.
Proc. SPIE. 5067, Saratov Fall Meeting 2002: Laser Physics and Photonics, Spectroscopy, and Molecular Modeling III; Coherent Optics of Ordered and Random Media III
KEYWORDS: Femtosecond phenomena, Chemical species, Electrons, Hydrogen, Laser applications, Ionization, Laser stabilization, Electromagnetism, Pulsed laser operation, Systems modeling
The soluble model of interaction of a finite series of zero - duration pulses with an atom is considered. The model is based on the nowadays laser techniques providing duration of pulses of a few femtoseconds and even less, and intensities higher than 10<sup>14</sup>-10<sup>20</sup> Wt/cm<sup>2</sup>.
A three-body scattering problem is formulated in an adiabatic representation as a multi-channel spectral problem for a system of one-dimensional integral equations using the Schwinger variational functional. The stable Newtonian iteration schemes for calculating the eigenfunctions and eigenvalues which are phase shift and energy for continuous and discrete spectrum, correspondingly, are elaborated. Convergence and efficiency of the proposed schemes are demonstrated using the exact solvable model of three identical particles (bosons) on a line with pair attractive (delta) -potentials.
Semiclassical model of double ionization of Helium atom by fast electron based on the classical trajectories for the ejected electrons is presented. The quantities needed to fit known experimental data are calculated. Typical examples of trajectories of the ejected electrons are presented also.