Based on the missed exact solution special case of the newly discovered spherical harmonic partial differential equation, the analytical expression of the spherical harmonic function is regressed, and the wave function of the zero magnetic quantum number that overturns the traditional understanding of the Schrödinger equation theory for hydrogen atoms is briefly introduced. At the same time, the answer to the question of the existence and uniqueness of the probability density function definition based on the statistical interpretation of Born in quantum mechanics is clarified. Whether it is the inherent causal relationship hidden in the basic definition or the completely accurate solution of the wave equation, new conclusions clearly indicate that scientific theories described using partial differential equations, especially quantum mechanics, are facing significant changes, and all related theories will inevitably be rewritten after improving the theory of partial differential equations.