The Analytic Solution To The Schr(?)dinger Equation And Dirac Equation For The Superposition Potential Of The Power Functions

Analytic solution and approximate solution for solving the stationary state Schr?ddinger equation are presented . And the radial Schr?dinger equation with the high order power and inverse power potential function is studied. According to quantum system wave function must meet the standard conditions of single value, continuous and limited, First obtained radial coordinates r→0and the r→∞analytical solution, Then use neighborhood near the singular point of the series and the method of asymptotic solution obtained by combining indicators s power function, and the coefficient of the constraint, an analytic solution and energy level of the superposition potential V(r)=α₁r⁸+α₂r³+α₃r²+β₃r^-1 +β₂r^-3+β₁r^-4 has been obtained and some conclusions are presented. and using MATLAB software distribution wave function graphics.Generally speaking, more than five kinds of potential under the conditions of superposition only approximate solution of the Schrodinger equation, and there is no analytical solution. But if the power coupling function of the close relationship between the cases, there may be analytical solution. This paper discussed the existence of the power potential function of the conditions and analytical solutions are obtained analytical solution and the corresponding energy level structure.By using the usual method of variable separation , bound states of Dirac equation with scalar and vector potentials are solved. Properties of the system relate to three quantum numbers ( nr , m, s )and parameters ( K , A,β,γ)of potential1 The normalized angle wave function expressed interms of the universal associated-Legendre polynomial and normalized radial wave function expressed in terms of the confluent hypergeometric function are presented.The exact energy spectrum equations are obtained.