Connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the conservative bound systems

Abstract

It is well known that the Schrodinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrodinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.