Abstract: 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. 

Key words:  shape invariance, supersymmetry spontaneous broken, potential algebra, Coulomb potential