关键词: 分离变量, 哈密顿-雅可比方程, 定态薛定谔方程, 对称性与守恒量

Abstract: This paper systematically analyzes the physical significance of the particular solutions constructed via the “direct product” and “direct sum” forms of separation of variables in solving partial differential equations. Taking the central force field problem as an example, it compares and contrasts the approaches used in the Hamilton Jacobi equation (HJE) and the stationary Schrdinger equation. The study shows that the “direct product” form leads to a complete set of eigenfunctions, allowing the general solution to be expanded as a linear superposition of these eigenfunctions, while the “direct sum” form yields Hamiltons characteristic function, which is related to the eigenfunctions via exponential mapping. Examples demonstrate that the constants introduced through separation of variables correspond to conserved quantities of the system, which originate from symmetries. This work establishes a unified framework of “symmetry conserved quantity-separability of equations”, clarifies the physical essence and applicable scope of the two methods, and provides a clear theoretical basis for both teaching and research.

Key words: separation of variables, Hamilton Jacobi equation, stationary Schrdinger equation, symmetry and conserved quantities