Open AccessThe analytic solution of the radial Schr?dinger equation for the superposed potentials
Author(s) -
HU Xian-quan,
Lipeng Cui,
Guang Luo,
Yu Ma
Publication year - 2009
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.199
H-Index - 47
ISSN - 1000-3290
DOI - 10.7498/aps.58.2168
Subject(s) - power series , physics , series (stratigraphy) , schrödinger equation , gravitational singularity , mathematical analysis , bounded function , wave function , analytic function , convergent series , function (biology) , integral equation , mathematics , quantum mechanics , paleontology , evolutionary biology , biology
The method of solving the radial Schrdinger equation is studied by under the tight coupling condition of several positivepower and inversepower potential functions. The precise analytic solutions and the conditions that determine the existence of analytic solution are searched when the potential of the radial Schrdinger equation is Vr=α1r8α2r3+α3r2+β3r-1β2r-3β1r-4. According to the single valued, bounded and continuous stipulations of wave function in a quantum system, firstly, the asymptotic solution is solved when the radial coordinate r→∞ and r→0; secondly, the asymptotic solutions are combined with the series solutions in the neighborhood of irregular singularities; and then the power series coefficients are compared. A series of analytic solutions of the stationary state wave function and the corresponding energy level structure are deduced by tight coupling between the coefficients of potential functions for the radial Schrdinger equation. And the solutions are discussed and the conclusions are made.