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International Journal of Modern Physics B

EXACT NUMERICAL SOLUTION OF SCHRÖDINGER'S EQUATION FOR A PARTICLE IN AN INTERACTION POTENTIAL OF GENERAL SHAPE

M. Rieth, Wolfram Schommers, Sotirios Baskoutas

Published Oct 30, 2002• 45 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979202014802

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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.
A new 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 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 have been discussed critically.
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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 paper I have drawn out the steps to be followed in order to derive the exact Schr\"odinger wave function for a particle in a general one-dimensional time-dependent linear potential. To this end I have used the so-called Lewis and Riesenfeld invariant method, which is based on finding an exact quantum-mechanical invariant in whose eigenstates the exact quantum states are found. In particular, I have obtained the wave functions of a particle in the linear potential well, driven by a monochromatic electric field.
An important technique for teaching introductory wave mechanics is provided by the numerical calculation of stationary state solutions of the Schrödinger equation. Realistic problems in wave mechanics that do not require sophisticated computer techniques are very valuable for the instruction of undergraduates. In the central-field model for atoms, the calculation of electron states reduces to the solution of a one-dimensional differential equation whose numerical solutions are no more difficult to obtain than those for a finite square well. A method used by our students is discussed in this paper for the calculation of the electron wavefunctions in a central-field potential. By working this problem, students obtain meaningful results that give them insight into the theory of atomic structure. Results of their experience are included, and possibilities of extending this example are discussed.
The time-dependent and time-independent Schrödinger equations have been numerically solved for several quantum mechanical problems with known analytical solutions using the basis-spline collocation algorithm. This algorithm has been demonstrated to be efficient and versatile. The results from these calculations illustrate the necessary numerical considerations required for solving both time-independent and time-dependent Schrödinger equations and form a basis that could be used for solving these and similar problems.