The analytic solution of the radial Schr?dinger equation for the superposed potential of high-order power and inverse-power potential functions

Abstract

When the Schrdinger equation involves high-order power and inverse power potential functions or the superposed potential function of high-order anharmonic oscillatory potentials, introduced by the presence of electric dipole moment potential, molecular crystal potential, or the polarized equivalent potential, the solution of the Schrdinger equation becomes very complicated. In this paper, with the help of a combination of series solutions and asymptotic solutions utilized near the singular points, a series analytic solution of the wave functions of stationary state for radial Schrdinger equation with potential function V(r)=a1r6+a2r2+a3r-4+a4r-6 and the corresponding energy level structure are obtained under the tightly-coupled condition of the interacting power potential functions. Meanwhile, the paper gives a proper discussion and some important conclusions are drawn.