Using an ansatz for the eigenfunction,we obtain the analytic solution to the schrdinger equation for the superposition potential V(r)=B_6r6+B_5r5+B_4r4+B_3r3+B_2r~2+B_1r.
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 V(r)=α1r8+α2r3+α3r2+β3r-1+β2r-3+β1r-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.
Most of Schrdinger equations in quantum mechanics haven't analytic solutions except the problems of well with infinite depth,one-dimensional linear harmonic oscillator and Coulomb's potential as well as three-dimensional isotropic harmonic oscillator.These Schrdinger equations include the various equations in the non-spherical ring shaped oscillator potential that haven't analytic solutions. We can only solve these problems approximately in our research activities, but the solving process of these problems is often exceptional complex.The limitation handicaps us to lucubrate the idiographic problems.This paper adopts NU Method(A.F.Nikiforov-V.B.Uvarov Method) to solve the angular equation of the Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger equation in the non-spherical ring shaped oscillator potential step by step until we obtains the analytic solutions of the equation,thus we obtain the analytic solution of the angular eigenfunction,and simplify the process of solutions.Moreover,this paper adopts special functions in the radial equation at the same time to broaden our study in solving the Schrdinger equations.
The solution of the Schrodinger equation with the physical potential is important part in quantum mechanics. The anharmonic potential is one of the physical potential that is interesting to be explored. Many methods have been developed to solve the Schrodinger equation. The Ansatz wave function method is one of the many methods that used to solve the Schrodinger equation. The Ansatz wave function method was used to solve the radial part of Schrodinger Equation. In this study we present the solution of the radial part Schrodinger equation in D-dimensional system with potential V(r)=ar2+br+cr-1+dr-2+er-3+fr-4. The important part in Ansatz wave function method is to determine the wave function Ansatz corresponding to the shape of the anharmonic potential that used. The Ansatz wave function determined by using supersymmetric approximation. By substituting the wave function Ansatz into Schrodinger equation and using algebraic we obtain the radial eigenfunction and the energy, and then the eigenfunction visualized.