One-dimensional Time-independent Schrodinger Equation Research Articles

A new implicit high-order six-step singularly P-stable method for the numerical solution of Schrödinger equation

In this paper, we present a new implicit six-step singularly P-stable method with vanished phase-lag and its derivatives up to fifth order for the numerical integration of the one-dimensional radial time independent Schrodinger equation. The periodicity region of the method is plotted and the numerical stability and phase properties of the new methods are analyzed. The advantage of the new method in comparison with similar methods—in terms of efficiency, accuracy and stability—have been shown by implementing them in the radial time-independent Schrodinger equation during the resonance problems with the use of the Woods–Saxon potential.

Computation of energy eigenvalues of the anharmonic Coulombic potential with irregular singularities

The present contribution concerns the computation of energy eigenvalues of a perturbed anharmonic Coulombic potential with irregular singularities using a combination of the Sinc collocation method and the double exponential transformation. This method provides a highly efficient and accurate algorithm to compute the energy eigenvalues of one-dimensional time-independent Schrodinger equation. The numerical results obtained clearly illustrate the efficiency and accuracy of the proposed method. The codes are written in Julia and are available on github at https://github.com/pjgaudre/DESincEig.jl .

Reanalysis of an open problem associated with the fractional Schrödinger equation

It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrodinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.

A family of explicit linear six-step methods with vanished phase-lag and its first derivative

A family of explicit linear sixth algebraic order six-step methods with vanished phase-lag and its first derivative is obtained in this paper. The investigation of the above family of methods contains: The theoretical study of the above mentioned family of methods contains: The application of the new family of linear six-step sixth algebraic order methods to the resonance problem of the one-dimensional time independent Schrodinger equation is used for the computational study of the new family of methods. The result of the above mentioned theoretical and computational investigation is that the new proposed family of linear explicit schemes are computationally and theoretically more effective than other well known methods for the approximate solution of the radial Schrodinger equation and related initial or boundary value problems with periodic and/or oscillating solutions.

A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution

We develop a family of six methods for the numerical integration of the Schrodinger equation and related initial value problems with oscillating solution. Three of the methods are constructed so that they are P-stable, using the methodology of Wang (Comp Phys Comm 171(3):162–174, 2005). Also two of these three methods are trigonometrically fitted with trigonometric orders one and two. The other three methods are constructed so that they are trigonometrically fitted with orders one, two and three. We show that there is an equivalence between the three pairs of methods, as if the property of P-stability can be substituted by an extra trigonometric order, that is the P-stable method is equivalent to the method with trigonometric order one, the P-stable method with trigonometric order one is equivalent to the method with order two, and the P-stable method with order two is equivalent to the method with order three. There is a condition that we choose the same frequency for the P-stability test problem y′′ = −θ 2 y and the functions that the method has to integrate exactly, in order to be trigonometrically fitted: {cos(ω x), sin(ω x), x cos(ω x), x sin(ω x), x 2 cos(ω x), x 2 sin(ω x)}. A stability analysis and a local truncation error analysis are performed on the methods and also the v–s diagrams are produced, where v = ω h and s = θ h. Finally the methods are applied to IVPs with oscillating solutions, such as the one-dimensional time independent Schrodinger equation and the nonlinear problem.

Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

The solution of the one-dimensional time-independent Schrodinger equation is considered by exponentially fitted symplectic integrators. The Schrodinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the doubly anharmonic oscillator.

New Second-order Exponentially and Trigonometrically Fitted Symplectic Integrators for the Numerical Solution of the Time-independent Schrödinger Equation

The solution of the one-dimensional time-independent Schrodinger equation is considered by trigonometrically and exponentially fitted symplectic integrators. The Schrodinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator, doubly anharmonic oscillator and the exponential potential.

A Family of Exponentially-fitted Runge–Kutta Methods with Exponential Order Up to Three for the Numerical Solution of the Schrödinger Equation

We have constructed three Runge–Kutta methods based on a classical method of Fehlberg with eight stages and sixth algebraic order. These methods have exponential order one, two and three. We show through the error analysis of the methods that by increasing the exponential order, the maximum power of the energy in the error expression decreases. So the higher the exponential order the smaller the local truncation error of the method compared to the corresponding classical method. The difference is higher for higher values of energy. The results confirm this, when integrating the resonance problem of the one-dimensional time-independent Schrodinger equation.

A variable-step Numerov method for the numerical solution of the Schr�dinger equation

Numerov’s method is one of the most widely used algorithms for solving second-order ordinary differential equations of the form y’’ = f(x,y). The one-dimensional time-independent Schrodinger equation is a particular example of this type of equation. In this article we present a variable-step Numerov method for the numerical solution of the Schrodinger equation.

Exponentially fitted symplectic methods for the numerical integration of the Schr�dinger equation

The solution of the one-dimensional time-independent Schrodinger equation is considered by exponential-fitting symplectic integrators. The Schrodinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the hydrogen atom.

Numerical solution of one-dimensional time-independent Schr�dinger equation by using symplectic schemes

The symplectic schemes are extended to the solution of one-dimensional time-independent Schrodinger equation. The Schrodinger equation is first transformed into a Hamiltonian canonical equation by means of the Legendre transformation, and then two methods are developed to solve the numerical solution of the one-dimensional time-independent Schrodinger equation: the symplectic scheme-matrix eigenvalue method and the symplectic scheme-shooting method. Both methods are applied to the calculations of a one-dimensional harmonic oscillator, the hydrogen atom, and a double-well anharmonic oscillator. It is shown that the numerical results of the two methods are nearly the same and are in good agreement with the exact ones when the step length is taken to be properly small. The computation with the symplectic scheme-shooting method spends less computer time than that with the symplectic scheme-matrix eigenvalue method. And thus the symplectic scheme-shooting method is a better numerical method for the calculation of the eigenvalue problem. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 343–349, 2000

Application of the exponential propagation matrix method to the numerical solution of the Schrödinger equation

A method is proposed for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent Schrodinger equation. The eigenvalue problem is transformed into a system of two first-order ordinary linear differential equations. The propagation matrix for the solution is written as the exponential of a traceless matrix which is then approximated by the Magnus expansion. The bound-state eigenvalues and eigenfunctions are obtained by requiring that the latter satisfy the appropriate boundary conditions. The Morse potential is considered as an illustrative example.

Solving the Schrodinger equation in arbitrary quantum-well potential profiles using the transfer matrix method

A simple, accurate, and fast algorithm for solving the one-dimensional time-independent Schrodinger equation is presented. The algorithm is based on the transfer matrix method. This makes it possible to calculate all bound and quasi-bound energy levels and the corresponding wave functions for an arbitrarily shaped potential profile. The results of calculations are compared with those obtained by other authors for various types of problems. A central part of this study deals with solving the Schrodinger equation in quantum-well structures. The results show that the transfer matrix method is as accurate as other methods, but it is easier to implement and, hence, is superior for calculations on small computer, such as a PC. >