Chapter 1 - Connection Between Schrődinger and Hamilton–Jacobi Equations in the Case of Stationary Atomic and Molecular Systems

Abstract

Our approach is based on the equivalence between the Schrödinger and wave equations, which is valid for stationary atomic and molecular systems. We prove that the characteristic surface of the wave equation, which has the significance of a wave surface, and its normal curves are periodic solutions of the Hamilton-Jacobi equation, written for the same system. We prove that the motion of the wave surface is periodic, and that the normal curves are closed. The Bohr generalized relation is valid for these curves, resulting that de Broglie relations are valid for multidimensional systems. We show that the wave surfaces and their normals depend on the same constants of motion as those resulting from the Schrödinger equation. Consequently these geometric elements can be used as analytical tools for calculating the energetic values of the system and for determining its symmetry properties, as detailed in concrete examples in a second book.