One of the advantages of a quantum mechanical description of a bound states system like atoms, exotic atoms, or hadronic systems is determination of the mass and the energy spectrum in the ground and excited states as in excited states. Study of the energy spectrum in Coulomb potential of atomic systems, in relativistic conditions due to requirements of using higher grades of relativistic corrections have attracted physics theoreticians. The ability to create mono-electron ions of heavy, semi-heavy, strange atoms and/or hadronic atoms in laboratory conditions has boomed the need of more precise and meticulous corrections. In this article, we begin our discussion of the heavy ions with one-electron interactions in the spherical-symmetric potential (i.e. Coulomb), using quantum field theory, and quantum mechanic’s technics. These technics are the great triumphs of our technically description. Therefore, too many different models and methods explained the energy spectrum of the heavy ions. It has been determined, in the numerous energy intervals based on potential models and given corrections in the relativistic limit. This document contains a theoretical definition on the heavy ions with one-electron characteristics that describes the nature of interactions this limit. Relativistic nature of interactions directly effect on the mass of particles and it is a good reason to study how relativistic mass could effect on the energy spectrum for the n-body particles system that bounding to each other by electrostatic potential. Thus, the mass and energy spectrum of a coupled state are analytically determined. The mechanism of production of a constituent mass of particles forming a coupled state for the heavy ions with one-electron was described.

Loop Function, Green’s Function, Functional Integral, Hamiltonian, Constituent Mass