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Analysis And Simulation Of Schrődinger Eigenfunctions Of Several Potentials With Numerov Method

FendaRizky Pratama

Published Aug 22, 2016• 0 citations
OpenAlex

Abstract

The eigenfunctions as the solutions of Schrodinger equation for particle in the box, simple harmonic and perturbed oscillators, as well as in the case of Coloumb, linear, Morse, Yukawa, and modified Woods-Saxon potentials are simulated with fourth-order Numerov algorithm. The simulation techniques for initialization, which is aimed to acquire qualitative behaviour of eigenfunctions are derived by solving Schrodinger equation in particular region directly or by WKB approximation and Bohr-Sommerfeld quatization rule. The simulated eigenfunctions possess good accuracy, with maximum relative error in the order 10─3 compared to the analytical eigenfunctions or are in the good agreements with the analytical eigenfunctions calculated in the previous works. This work also provides some illustrations of how approximation methods are able to improve the accuracy in computational simulations.

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The eigenfunctions as the solutions of Schrodinger equation for particle in the box, simple harmonic and perturbed oscillators, as well as in the case of Coloumb, linear, Morse, Yukawa, and modified Woods-Saxon potentials are simulated with fourth-order Numerov algorithm. The simulation techniques for initialization, which is aimed to acquire qualitative behaviour of eigenfunctions are derived by solving Schrodinger equation in particular region directly or by WKB approximation and Bohr-Sommerfeld quatization rule. The simulated eigenfunctions possess good accuracy, with maximum relative error in the order 10─3 compared to the analytical eigenfunctions or are in the good agreements with the analytical eigenfunctions calculated in the previous works. This work also provides some illustrations of how approximation methods are able to improve the accuracy in computational simulations.
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.
In this study, we present analytical solutions of the Schrödinger equation with the Multiparameter potential containing the different types of physical potential via the asymptotic iteration method (AIM) by applying a Pekeris-type approximation to the centrifugal potential. For any n and l (states) quantum numbers, we get the bound state energy eigenvalues numerically and the corresponding eigenfunctions.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.
In this work, aiming to solve numerically the Schrödinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schrödinger equation with potentials of the form V (x) + αδp(x), where δp(x) is a pseudo-delta function, a very high and thin barrier.The numerical results show good agreement with analytical results found in the literature.Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta barrier in the center.