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当微优按Hermite多项式Hk的收敛级数作如下展开时,V(X)=b2X2+Σk CkHk(b1/2X), 则可将其微扰梯度算子方法应用于微扰谐振子波动方程的求解中.发现若将Hermite多项式基与二项式系数函数依量子数一起使用,则可大大简化微扰梯度与因子分解函数.因此,在不增加其复杂性的情况下,便可求得任意级微扰的本征值与本征函数的分析表示式.通过计算,本文给出了X的偶性微扰势函数V(X),为了说明如何应用改进后的微扰梯度算子方法,本文重新研究了其势函数为V(x)=x2+λX2/(1+gX2),且g>0时的Schr?dinger方程的求解过程.The interaction potential with the form as V(x)= x2+λx2/(1+gx2) where g >0, appears in several areas of laser theory, quantum field theory, atom and nuclear physics. One could consider that the solution of the eigenequation either by the classical Rayleigh-Schr?dinger perturbation scheme or by the perturbed ladder operators scheme. Nevertheless, the perturbation series does not converge for any values of λ and g. In the present paper, it is shown that this difficulty can be overcome as long as the potential function can be expanded in a convergent series on the basis ofthe Hermite polynomials. Therefore, the eigenequation ((d2)/(dx2)-V(x)+ξ)φ(x)=0,∞2)/(dX2)-b2X2-Σkc2kH2k(b1/2X)+ξ)φ(X)=0.
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