超对称量子力学在求解薛定谔方程的精确解时是一种非常重要的方法.库仑势作用下的薛定谔问题的求解是量子力学中的经典问题.基于超对称量子力学研究库仑势约束下的定态薛定谔方程的势代数相关问题无疑具有重要意义.本论文研究库仑势约束下薛定谔方程的势代数相关问题.首先依据径向方程以及角向方程的超势得到了伴随势的形状不变性关系，求得本征值和零能基态波函数.然后依据势代数法，讨论了径向方程以及角向方程的势代数形式的形状不变关系.最后讨论了系统出现的超对称自发破缺问题，并通过调整参数的两步法解决了径向波函数的自发破缺的问题.

Supersymmetric quantum mechanics is a very important method in solving the exact solution of Schrödinger's equation. The solution of Schrödinger equation with the Coulomb potential is also a classic problem in quantum mechanics. Based on supersymmetric quantum mechanics, it is also undoubtedly of great significance to study the potential algebra related problems of the stationarySchrödinger equation with Coulomb potential. The potential algebraic correlation problems of the stationary Schrödinger equation with Coulomb potential are studied based on Supersymmetric Quantum Mechanics. First, the shape invariance of partner potential is deduced from the superpotential of radial equation and angular equation, and the eigenvalue and the zero energy ground state wave function of those equations are acquired. Then, the shape invariance is discussed with potential algebraic method. Finally, starting from the radial wave function is not normalizable, we discuss the supersymmetry spontaneous broken.