From Fermi's Golden Rule to the Decay and Scattering of Fundamental Particle

This paper commences with the Fermi's Golden Rule in quantum mechanics, employing concepts corresponding to quantum principles to elucidate fundamental notions in field theory such as particle decay widths and scattering cross-sections. It further establishes the intrinsic relationships among these concepts. By analyzing the transition rate under time-dependent perturbation theory, the physical meaning of the transition matrix element and density of states in decay and scattering processes is elucidated. Furthermore, this paper also shows the relativistically covariant formulas for decay width and scattering cross-section by using phase-space integral formulations. A simplified model, which describes interactions among three scalar particles, is constructed for a demonstration of particle decay and scattering in one-dimensional space. This approach intuitively reveals the kinematic and dynamic characteristics of particle scattering. It facilitates a natural transition for learner from quantum mechanics to the framework of quantum field theory, offering instructive insights for understanding collider physics.