The Second-Order Approximation of Superpotentials Based on SUSYQM

This paper is based on the shape invariance of the solvable superpotentials and uses the series expansion method to study the approximate expansion forms of these superpotentials. Firstly, this paper examines the differential equations satisfied by the first-order approximations of the superpotentials. Through an example, namely Rosen–Morse (trigonometric) superpotentials, the specific forms of these first-order approximations are analyzed. Based on these simple first-order approximations, this paper then delves into the ground-state wave functions of the superpotential. Secondly, this paper derives the differential equations satisfied by the second-order approximations with the first-order approximations. Using the harmonic oscillator superpotentials as an example, similarly, non-unique forms for the second-order approximations are obtained. By selecting simpler forms for the first- and second-order approximations, the authors further investigate the ground-state wave functions of the superpotential with the second-order approximation. Thirdly, the authors discuss the Hamiltonians of the potential with the first- and second-order approximations, concluding that the additional term originates from the corrections to the superpotential. Finally, conclusions and prospects are provided.