新环状势的狄拉克与薛定谔方程束缚态解

Bound States Solution of Dirac Equation and Schrdinger Equation for a New Ring Potential

本文提出了一种新的环状非球谐振子势V(r,θ)=K/2r2+A/r2+β/(r2sin2θ)+(γcos2θ)/(r2sin2θ)。在标量势与矢量势相等的条件下,给出了Dirac方程和薛定谔方程的束缚态波函数解u(β'r)=1/Γ(L+3/2)((2β'.Γ(n+L+3/2))/nr)～(1/2)·(β'r)L+1·e(-(β'r)2)2·F(-nr,L+3/2(β'r)2)。通过分离变量法得到相应的角向波函数方程和径向波函数方程,得出用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数(d2F(ρ))/dp2+(2s+1/2-ρ).dF(ρ)/dρ-(E/2-s-1/4)F(ρ)=0,进而由径向波函数满足的束缚态边界条件获得精确的束缚态能谱方程En=1/2+2n+2s=2n+L+3/2。

Under strong coupling conditions,the non-relativistic motion of particles and spin zero or half-integer the nature of relativistic particles in the system with the Schrdinger equation and Dirac equation can respectively describe.A new ring-shaped harmonic oscillator potential is proposed in this paper.The exact bound solution of Dirac equation and Schrdinger equation for the above potential is obtained under the condition of equal scalar potentials and vector potentials.They can be separated into an corresponding to the wave function equations of angular and radial wave function equation with the method of separation variables.The normalized angle wave function and radial wave function were expressed respectively in terms of the generalized associated-Legendre function and the confluent hyper-geometric function are presented,that the normalized radial wave function,and then by the radial bound state wave function are satisfied with the boundary conditions,obtains the exact bound state energy spectrum equations.The exact energy spectrum equations are obtained and meanwhile,proper discussion and some important conclusions are presented.