Applications of a property of the Schrödinger equation to the modeling of conservative discrete systems. III (Q2718289)

We have demonstrated in previous papers [J. Phys. Soc. Japan 67, No. 8, 2645--2652 (1998; Zbl 0974.35102), 68, No. 3, 763--770 (1999; Zbl 0992.81018)] the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He\(_2\), Be\(_2\) and B\(_2\) molecules.