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Physical and Chemical Aspects of the Study of Clusters Nanostructures and Nanomaterials

THE COMPUTER – ASSISTED SOLUTION OF THE ONE – DIMENSIONAL SCHRÖDINGER EQUATION TAKING INTO ACCOUNT TUNNELING EFFECTS

Irina N. Belyaeva, Igor Kіrichenko, Олег Дмитрович Пташний, Наталья Николаевна Чеканова, Н.А. Чеканов, Тетяна Ярхо

Published Dec 16, 2019• 0 citations
Publisher

Abstract

The general scheme of the method of integration of the second order ordinary differential equations with regular singular points is outlined. The algorithm of finding a general solution of the second order differential equation with regular singular points is given. The Schr dinger equation for anharmonic oscillator with the potential of the fourth, sixth and eighth degrees of non-linearity is considered, the values of its energy spectrum are found.

DOI
10.26456/pcascnn/2019.11.283

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Modern Physics Letters A•Dec 20, 1990•148 citations
The general scheme of the method of integration of the second order ordinary differential equations with regular singular points is outlined. The algorithm of finding a general solution of the second order differential equation with regular singular points is given. The Schr dinger equation for anharmonic oscillator with the potential of the fourth, sixth and eighth degrees of non-linearity is considered, the values of its energy spectrum are found.
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.
A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.