A secondary solution of the Schrödinger and Klein-Gordon equations in modeling atomic, molecular and electrodynamic systems

Abstract

We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schrödinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schrödinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.