P-stability, Trigonometric-fitting and the numerical solution of the radial Schrödinger equation

In this paper we will study the importance of the properties of P-stability and Trigonometric-fitting for the numerical integration of the one-dimensional Schrödinger equation. This will be done via the error analysis and the application of the studied methods to the numerical solution of the radial Schrödinger equation.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrödinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrödinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrödinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.     

Accurate basis set by the CIP method for the solutions of the Schrödinger equation

In this paper, we propose a basis set approach by the Constrained Interpolation Profile (CIP) method for the calculation of bound and continuum wave functions of the Schrödinger equation. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution by the constraints from the spatial derivative of the original equation. Thus the solution even on the subgrid scale becomes consistent with the master equation. By increasing the order of the polynomial, this solution quickly converges. The method is tested on the one-dimensional Schrödinger equation and is proven to give solutions a few orders of magnitude higher in accuracy than conventional methods for the lower-lying eigenstates. The method is straightforwardly applicable to various types of partial differential equations.     

Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.     

A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients

We propose a compact split-step finite difference method to solve the nonlinear Schrödinger equations with constant and variable coefficients. This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and Gross-Pitaevskii equation.     

The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge-Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed.     

Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r²+λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.     

Application of coordinate transformation and finite differences method in numerical modeling of quantum dash band structure

In this paper we propose an efficient and simple method for the band structure calculation of semiconductor quantum dashes. The method combines a coordinate transformation (mapping) based on an analytical function and the finite differences method (FDM) for solving the single-band Schrödinger equation. We explore suitable coordinate transformations and propose those, which might simultaneously provide a satisfactory fit of the quantum dash heterointerface and creation of an appropriate computational domain which encloses the quantum dash structure. After mapping of the quantum dash and the rest of computational domain, the Schrödinger equation is solved by the FDM in the mapped space. For the proposed coordinate transformations, we investigate and analyze applicability, robustness and convergence of the method by varying the FDM grid density and size of the computational domain. We find that the method provides sufficient accuracy, stability and flexibility with respect to the size and shape of the quantum dash and above all, extreme simplicity, which is promising and essential for an extension of the method to the multiband Schrödinger equation case.     

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.References: [1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission

We present a grid-based procedure to solve the eigenvalue problem for the two-dimensional Schrödinger equation in cylindrical coordinates. The Hamiltonian is discretized by using adapted finite difference approximations of the derivatives and this leads to an algebraic eigenvalue problem with a large (sparse) matrix, which is solved by the method of Arnoldi. By this procedure the single particle eigenstates of nuclear systems with arbitrary deformations can be obtained. As an application we have considered the emission of scission neutrons from fissioning nuclei.     

Application of the Feynman-Kac path integral method in finding excited states of quantum systems

Group theory considerations and properties of a continuous path are used to define a failure tree procedure for finding eigenvalues of the Schrödinger equation using stochastic methods. The procedure is used to calculate the lowest excited state eigenvalues of eigenfunctions possessing anti-symmetric nodal regions in configuration space using the Feynman-Kac path integral method. Within this method the solution of the imaginary time Schrödinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic atom that transform as Γ² and Γ⁴ irreducible representations. Numerical results are compared with exact analytical results.     

Highly precise solutions of the one-dimensional Schrödinger equation with an arbitrary potential

This work concerns the obtaining of a highly accurate solution of the one-dimensional Schrödinger equation with a practically arbitrary potential. The approach is based on the power series method and it is implemented in the ultra high precision mode. It is shown that such an approach yields not only highly precise values of the energies but also accurate wave functions.     

Parallel Split-Step Fourier Methods for the Coupled Nonlinear Schrödinger Type Equations

The nonlinear Schrödinger type equations are of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation.In this paper we introduce parallel split-step Fourier methods for the numerical simulations of the coupled nonlinear Schrödinger equation that describes the propagation of two orthogonally polarized pulses in a monomode birefringent fibers. These methods are implemented on the Origin 2000 multiprocessor computer. Our numerical experiments have shown that these methods give accurate results and considerable speedup.     

A method for solving the molecular Schrödinger equation in Cartesian coordinates via angular momentum projection operators

A method for solving the Schrödinger equation of N-atom molecules in 3N−3 Cartesian coordinates usually defined by Jacobi vectors is presented. The separation and conservation of the total angular momentum are obtained not by transforming the Hamiltonian in internal curvilinear coordinates but instead, by keeping the Cartesian formulation of the Hamiltonian operator and projecting the initial wavefunction onto the proper irreducible representation angular momentum subspace. The increased number of degrees of freedom from 3N−6 to 3N−3, compared to previous methods for solving the Schrödinger equation, is compensated by the simplicity of the kinetic energy operator and its finite difference representations which result in sparse Hamiltonian matrices. A parallel code in Fortran 95 has been developed and tested for model potentials of harmonic oscillators. Moreover, we compare data obtained for the three-dimensional hydrogen molecule and the six-dimensional water molecule with results from the literature. The availability of large clusters of computers with hundreds of CPUs and GBytes of memory, as well as the rapid development of distributed (Grid) computing, make the proposed method, which is unequivocally highly demanding in memory and computer time, attractive for studying Quantum Molecular Dynamics.     

Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method

The coupled nonlinear Schrödinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we derive a finite element scheme to solve this equation, we test this method for stability and accuracy, many numerical tests have been conducted. The scheme is quite accurate and describe the interaction picture clearly.     

On Chaos in Nonlinear Schrödinger Equation for Spin 1/2

Using Melnikov's method we find Smale horseshoes in orbit structure of a simple, externally driven nonlinear Schrödinger equation.     

Program for quantum wave-packet dynamics with time-dependent potentials

We present a program to simulate the dynamics of a wave packet interacting with a time-dependent potential. The time-dependent Schrödinger equation is solved on a one-, two-, or three-dimensional spatial grid using the split operator method. The program can be compiled for execution either on a single processor or on a distributed-memory parallel computer.