Time-Dependent 3D Oscillator with Coulomb Interaction: An Alternative Approach for Analyzing Quark–Antiquark Systems

Abstract In this work, the dynamics of quark–antiquark pair systems is investigated by modeling them as general time-dependent 3D oscillators perturbed by a Coulomb potential. Solving this model enables the prediction of key mesonic properties such as the probability density, energy spectra, and quadrature uncertainties, offering theoretical insights into the confinement of quarks via gluon-mediated strong interactions. To tackle the mathematical difficulty raised by the time dependence of parameters in the system, special mathematical techniques, such as the invariant operator method, unitary transformation method, and the Nikiforov–Uvarov functional analysis are used. The wave functions of the system, derived using these mathematical techniques, are expressed analytically in terms of the Gauss hypergeometric function whose mathematical properties are well characterized. Our results provide the quantum-mechanical framework of quark–antiquark systems that are essential for exploring the nonperturbative aspects of quantum chromodynamics. In addition, the underlying mathematical structure may serve as a foundation for addressing broader challenges in particle physics, including the origin of mass and its connection to the Higgs mechanism.