Most of Schrdinger 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 Schrdinger 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 Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger 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 Schrdinger equations.Most of Schrdinger 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 Schrdinger 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 Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger 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 Schrdinger equations.

Chongqing Shifan Daxue xuebao. Ziran kexue banJan 1, 20080 citations

AN APPROACH TO SOLUTION OF THE SCHRÖDINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

Seung Jun Chang, Jae Gil Choi, Hyun Soo Chung

In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ${\diamond}$ -convolutions. Further, we give an approach to solution of the Schr $\ddot{o}$ dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential. The Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ${\diamond}$ -convolutions. Further, we give an approach to solution of the Schr $\ddot{o}$ dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential. The Schr $\ddot{o}$ dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

Journal of the Korean Mathematical SocietyMar 1, 201310 citations

In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the

{\diamond}

-convolutions. Further, we give an approach to solution of the Schr

\ddot{o}

dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr

\ddot{o}

dinger equations for harmonic oscillator and double-well potential. The Schr

\ddot{o}

dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.