Radial Schrodinger Equation Research Articles

Family of Twelve Steps Exponential Fitting Symmetric Multistep Methods for the Numerical Solution of the Schrödinger Equation

In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrodinger equation. This application shows the efficiency of the new family of methods.

A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrodinger equation

An exponentially-fitted Runge-Kutta method for the numerical integration of the radial Schrodinger equation is developed. Theoretical and numerical results obtained for the well known Woods-Saxon potential show the efficiency of the new method.

Semiclassical approximation and 1/n expansion in quantum-mechanical problems

The semiclassical approximation and the technique of 1/n expansion are used to calculate the eigenenergies and the wave functions for the radial Schrodinger equation. It is shown that the expressions that are asymptotically exact in the limit n=nr+l+1 → ∞ and which describe the above eigenenergies and the asymptotic coefficients at the origin and at infinity ensure a satisfactory precision even for states characterized by modest values of the quantum numbers nr and l, including the ground state.

Theoretical study of the A30+ ← X10+ and B31 ← X10+ transitions in the Cd-rare gas van der Waals molecules

Excitation spectra arising from A 3 0 - ← X 1 0 + and B 3 1 ← X 1 0 + electronic transitions in the Cd-rare gas (RG) van der Waals molecules are calculated using newly obtained theoretical potential curves for these species. In the molecular structure calculations, Cd 20+ and RG 8+ cores are simulated by energy-consistent pseudopotentials which also account for scalar-relativistic effects and spin-orbit (SO) interaction within the valence shell. Potential energies in the AS coupling scheme have been obtained by means of ab initio complete-active-space multiconfiguration self consistent-field (CASSCF)/complete-active-space multireference second-order perturbation theory (CASPT2) calculations with a total 28 correlated electrons, while the SO matrix has been computed in a reduced Cl space restricted to the CASSCF level. The final Ω potential curves are obtained by diagonalization of the modified SO matrix (its diagonal elements before diagonalization substituted for the corresponding CASPT2 eigen-energies). The spectroscopic parameters for the ground and several excited states of the Cd-RG complexes deduced from the calculated potential curves are in quite reasonable agreement with available experimental data. In addition, the radial Schrodinger equation for nuclear motion was solved numerically with the calculated potentials to evaluate the corresponding vibrational levels and radial wavefunctions. The latter have been used in the calculation of the appropriate Franck-Condon factors to yield information on relative intensities of the vibrational bands of the Cd-RG complexes. The theoretical vibrational progressions are discussed in the context of experimental spectra.

Radial Schrödinger equation: The spectral problem

Using the integral transformation method involving the investigation of the Laplace transforms of wave functions, we find the discrete spectra of the radial Schrodinger equation with a confining power-growth potential and with the generalized nuclear Coulomb attracting potential. The problem is reduced to solving a system of linear algebraic equations approximately. We give the results of calculating the discrete spectra of the S-states for the Schrodinger equation with a linearly growing confining potential and the nuclear Yukawa potential.

A Theoretical Study of theA 2Σ+–X 2Π System of the SiP Molecule

The A 2Σ+ and X 2Π electronic states of the SiP species have been investigated theoretically at a very high level of correlation treatment (CASSCF/MRSDCI). Very accurate potential energy curves are presented for both states, as well as the associated spectroscopic constants as derived from the vib-rotational energy levels determined by means of the numerical solution of the radial Schrodinger equation. Electronic transition moment function, oscillator strengths, Einstein coefficients for spontaneous emission, and Franck-Condon factors for the A 2Σ+-X 2Π system have been calculated. Dipole moment functions and radiative lifetimes for both states have also been determined. Spin-orbit coupling constants are also reported. The radiative lifetimes for the A 2Σ+ state, taking into account the spin-orbit diagonal correction to the X 2Π state, decrease from a value of 138 ms at v' = 0 to 0.48 ms at v' = 8, and, for the X 2Π state, from 2.32 s at v'' = 1 to 0.59 s at v'' = 5. Vibrational and rotational transitions are expected to be relatively strong.

Quantum theory of the relativistic oscillator from the HOOP (higher-order one-particle) viewpoint

Expanding on earlier work, it is shown how to rewrite the non-relativistic problem of two particles coupled by a spring as one higher-order one-particle (HOOP) problem, which then can be elevated to be Lorentz covariant. The higher-order Lagrangian is then developed into an Ostrogradsky Hamiltonian format, from which canonical commutation rules are invoked and a relativistic wave equation produced. Within the centre-of-momentum frame of reference, the latter is reduced to a radial Schrodinger equation that is solved numerically, with a display of radial eigenfunctions that are compared with non-relativistic ones. The details of this example illustrate conclusively how the HOOP approach to direct-interaction many-particle relativistic dynamics works.

Variationally optimized numerical orbitals for molecular calculations

The problem of variational optimization of the atomic orbitals used in molecular calculations is investigated. It is shown that the variational principle leads to an equation similar to the radial Schrodinger equation but containing an inhomogeneous term. As an example, the equations are solved for the minimum basis set orbitals for the methane molecule. The results show a substantial improvement over those of a previous calculation optimizing in a minimum basis of Slater orbitals.

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrodinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrodinger equation and of coupled differential equations arising from the Schrodinger equation indicate that this new approach is more efficient than other well known methods.

Parametric potential determination by the canonical function method

The canonical function method (CFM) is a powerful means for solving the Radial Schrodinger Equation (RSE). The mathematical difficulty of the RSE lies in the fact it is a singular boundary value problem. The CFM turns it into a regular initial value problem and allows the full determination of the spectrum of the Schrodinger operator without calculating the eigenfunctions.Following the parametrisation suggested by Klapisch and Green-Sellin-Zachor we develop a CFM to optimise the potential parameters in order to reproduce the experimental Quantum Defect results for various Rydberg series of He, Ne and Ar as evaluated from Moore's data.

A family of P-stable exponentially‐fitted methods for the numerical solution of the Schrödinger equation

A family of P‐stable exponentially‐fitted methods for the numerical solution of the Schrodinger equation is developed in this paper. An application to the resonance problem of the radial Schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially‐fitted methods of the same kind.

Accurate solutions of coupled radial Schrödinger equations

A constructive numerical-analytical method of solving coupled Schrodinger equations is presented when a Hamiltonian is a quadratic form of the momentum and contains a matrix potential energy term, which is, in particular, a superposition of Coulomb and polynomial potentials. A technique for solving coupled radial Schrodinger equations is developed. The method is based on the matching of exact solutions, constructed as algebraic combinations of power series, power functions, and a logarithmic function in the neighbourhood of regular singularity r = 0, and of the asymptotic expansions of solutions in the neighbourhood of irregular singularity r = . This method of matching allows us to calculate accurately eigenvalues with corresponding wavefunctions of a discrete spectrum, in(out)-solutions and an S-matrix for a given value of energy from a continuous spectrum, and resonance states. Wavefunctions derived are expressed in analytical form. The method is applied to solving the Schrodinger equation in the case of a matrix Hamiltonian with Coulomb potential.

Embedded eighth order methods for the numerical solution of the Schrödinger equation

A new method for the approximate numerical integration of the radial Schrodinger equation is developed in this paper. Phase-lag and stability analysis of the new method is included. The new method is called the embedded method because of a simple natural error control mechanism. Numerical results obtained for the phase-shift problem of the radial Schrodinger equation show the validity of the developed theory.

New variable-step algorithms for computing eigenvalues of the one-dimensional Schr�dinger equation

Two variable-step algorithms have been developed for computing eigenvalues of the radial Schrodinger equation. The eigenvalues are computed directly as roots of a function known in transmission line theory as the impedance. The novel numerical algorithms are based also on the piecewise perturbation analysis. The first new variable-step method is based on two methods, one with zeroth order solution and the other with first order perturbative corrections. The second variable-step method called “block method” is based on three methods, one with zeroth order solution, another with first order perturbative corrections and another with second order perturbative corrections. The eigenvalue problems are very common in many engineering problems contexts involving vibrations, elasticity and other oscillating systems.

Reply to the comment by Stahlhofen on “Factorization of the radial Schrödinger equation and four kinds of raising and lowering operators of hydrogen atoms and isotropic harmonic oscillators”

Abstract The equivalence of the raising and lowering operators derived from the factorization of the radial Schrodinger equation and the conserved quantities responsible for the closeness of classical orbits for hydrogen atoms and isotropic harmonic oscillators is physically connected with the dynamical symmetry, in reply to the comment by Stahlhofen [Phys. Lett. A 241 (1998) 298].

Realizations of U q ( su (1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrodinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(xi − xj).

New approach to the correlation spectrum near intermittency: A quantum mechanical analogy

The correlation spectrum of fully developed one-dimensional mappings is studied near and at a weakly intermittent situation. Using a suitable infinitematrix representation, the eigenvalue equation of the Frobenius-Perron operator is approximately reduced to the radial Schrodinger equation of the hydrogen atom. Corrections are calculated by quantum mechanical perturbation theory. Analytical expressions for the spectral properties and correlation functions are derived and checked numerically. Compared to our previous work, the accuracy of the present results is significantly higher owing to the controlled and systematic approximation scheme

Four kinds of raising and lowering operators ofn-dimensional hydrogen atom and isotropic harmonic oscillator

The factorization of the radial Schrodinger equation ofn-dimensional (n ≥ 2) hydrogen atoms and isotropic harmonic oscillators was investigated and four kinds of raising and lowering operators were derived. The relation betweenn-dimensional (n ≥ 2) and one-dimensional hydrogen atoms and harmonic oscillators was discussed.

High resolution Fourier transform infrared emission spectra of lithium iodide

The high resolution infrared emission spectrum of lithium iodide has been recorded with a Fourier transform spectrometer. A total of 1024 transmissions with Δ v = 1 for 7 LiI (from v = 1 → 0 to v = 8 → 7), 387 transitions with Δ v = 2(v = 2 → 0 to v = 6 → 4) and 504 transitions with Δ v = 1 for 6 LiI (v = 1 → 0 to v = 5 → 4) have been assigned. The infrared data have been used to determine improved mass-dependent Dunham Y lm coefficients for the ground X1 Σ+ electronic state. The mass-independent Dunham U lm coefficients were obtained along with Born-Oppenheimer breakdown constants. Finally, all of the data were fitted to the eigenvalues of the radial Schrodinger equation containing a modified Morse potential function to determine the parameters of an effective ground state potential.)

Calculation of the electronic structure of crystals using a pseudoorbital basis

We present a method for calculating the electronic structure of crystals in the context of a density functional theory using a pseudoorbital basis. The basis functions are obtained from a solution of the radial Schrodinger equation using pseudopotentials which retain their normalization. We propose an analytical form for the representation of a numerical solution. As an example, we calculate the band structure, total energy, and lattice constant for crystalline silicon. It is shown that in comparison with the traditional approach, using a plane wave basis, our method makes it possible to achieve equivalent precision in the results with a significant reduction in the size of the basis function set, which therefore shows promise for the consideration of compounds with more complex structure.