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Communications in Theoretical Physics

A Method to Solve the Schrödinger Equation for Any Power Hypercentral Potentials

ALI AKBAR RAJABI

Published Jul 1, 2007• 2 citations
Publisher

Abstract

An exact closed form of solution to the hyperradial Schrödinger equation is constructed for any general case comprising any hypercentral power and inverse-power potential. The hypercentral potential depends only on the hyperradius, which itself is a function of Jacobi relative coordinates that are functions of particle positions ( r 1 , r 2 ,…, r N ). This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ = powerseries×exp(polynomial) = [ f ( x )exp ( g ( x ))] is potentially a solution of the Schrödinger equation, where the polynomial g( x ) is an ansatz depending on the interaction potential.

DOI
10.1088/0253-6102/48/1/031

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An exact closed form of solution to the hyperradial Schrödinger equation is constructed for any general case comprising any hypercentral power and inverse-power potential. The hypercentral potential depends only on the hyperradius, which itself is a function of Jacobi relative coordinates that are functions of particle positions ( r 1 , r 2 ,…, r N ). This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ = powerseries×exp(polynomial) = [ f ( x )exp ( g ( x ))] is potentially a solution of the Schrödinger equation, where the polynomial g( x ) is an ansatz depending on the interaction potential.
The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V (r) = �1r 8 + �2r 3 + �3r 2 + �3r 1 + �2r 3 + �1r 4 . Generally speaking, there is only an approximate solution, but not analytic solution for Schrodinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r ! 1 and r ! 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions ofthe stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schrodinger equation; and lastly, they discuss the solutions and make conclusions.
We find the form of the potential depending on the coordinates and the time such that a solution, SSS, of the Hamilton--Jacobi equation yields an exact solution, exp⁡(iS/ℏ)\exp ({\rm i} S/\hbar)exp(iS/ℏ), of the corresponding Schrödinger equation.
Finding an analytical solution to the Schrödinger equation with power function superposition potential is essential for the development of quantum theory. For example, the harmonic oscillator potential, Coulomb potential, and Klazer potential are all classed as power function superposition potentials. In this study, the general form of the power function superposition potential was used to decompose the second-order radial Schrödinger equation with this potential into the first-order Ricatti equation. Furthermore, two forms of the power function superposition potential are constructed with an exact analytical solution, and the exact bound-state energy level formula is obtained for these two potentials. Finally, the energy levels of some of the diatomic molecules were determined through calculation. And our results are actually consistent with those obtained by other methods.
The goal of the dissertation is to find new method of solving two-dimensional Schrödinger equation in such cases, when the separation of the variables is not applicable. The results are applied to the two-dimensional Schrödinger equation with the potentials of the form of the quartic polynomial, of the sextic polynomial and of the quartic Morse potential. For these cases, the analytical formulae for the ground state wave functions and the corresponding energies have been found. For the specific class of the potentials of the form of the quartic polynomial, analytical formula for one of the excited states and for the corresponding energy have been found.
The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.