Arezu - - Department of Physics and Engineering Sciences, Buein Zahra Technical University, Qazvin, Iran
- - - Department of Physics, University of Agriculture and Environmental Sciences Umuagwo, Imo State, Nigeria

This paper studies the important concepts in the fundamental optimization of mass and thermal properties of strong interactions based on quantum approximate operators. It explores and analytically calculates the radial part of the Schrödinger equation at finite temperature using the two intertwined spaces based on the Wick ordering method in the Bernoulli Potential. We provide analytical expressions for the ground state energy eigenvalues to define the zeroth approximation. Bernoulli potential refers to some of the potential types that are returned to the Bernoulli series. The method presented in this study allows us to rewrite some of the exponential potentials simply to even power levels in the Bernoulli series. Given that the Hulthen potential is a modified form of main hadronic interaction potentials, it is used to calculate the bound state mass and properties of hadronic atoms such as π-atoms, κ-atoms or various hadronic structures with positive/negative and heavy/light quark bound states such as 〖D_s〗^+,〖B_c〗^+,〖〖〖B_ 〗^+,D^+ ,D〗_ 〗^(*+) at finite temperatures. The main goal of this research is to the combined quantum operators and two intertwined spaces within the Bernoulli expansion to determine the best approximation of upsilon meson mass and thermal properties.

Bound state, Relativistic correction, Bernoulli Potential, Strong interaction