pionic atom localized states under the relativistic effect and memory model

The presented work is innovative in proposing a novel equation for the velocity of constituent particles in relativistic localized states, derived from a unique synthesis of quantum field theory, memory models, and harmonic-oscillator dynamics. this approach offers a new way to model the internal dynamics of hadronic systems, with potential applications in nuclear physics and exotic atoms. its primary strength lies in integrating disparate theoretical frameworks to describe complex quantum phenomena effectively. we introduce a significant physical equation for calculating the velocity of constituent particles in a localized state, derived from quantum field theory, the relativistic behavior of interactions, and a memory model for moving mass. this equation is obtained from the ratio of the relativistic mass to the rest mass of the constituent particles, together with boltzmann's constitutive equations for a quantum harmonic-oscillator system that incorporates the kelvin–voigt memory model and the features of quantum field theory characteristic of the strong interaction within a hadronic localized state. initially, we determine the relativistic masses of the constituent particles in pionic-atom localized states (bound states) by solving a modified schrödinger equation in a symplectic space constructed from two intertwined subspaces generated by the creation and annihilation operators. we then employ the kelvin-voigt memory model equation to determine the velocities of the particles in the relativistic regime within the framework of the strong interaction among hadrons. the theoretical predictions for the pionic atom are in good agreement with current experimental data.