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.

This article examines the resolution of a problem in thermodynamics and statistical physics, demonstrating the derivation of physical processes under the standard dimension analysis, and presenting results in a standard form. The paper provides a brief elucidation of the Π theorem and furnishes examples of dimensional analysis employing the Π theorem for physical systems. Given the significance of dimensional analysis in physics, it warrants ample emphasis in the instruction of foundational physics courses. Additionally, the appendix of the paper offers standard representations of formulas sourced from the sixth edition of “Thermodynamics and Statistical Physics” written by Wang Zhi-Cheng.

In 1926, E.Schrödinger found an analytical solution for the time evolution of the one-dimensional harmonic oscillator Schrödinger Equation. This solution is not in the form of a summation of energy eigenstate wave functions, which provides a new perspective on the time evolution in quantum mechanics. This paper mainly analyzes this analytical time evolution solution and presents two derivation methods. Then the paper further generalizes to obtain more non-series analytical solutions for time evolution oscillator, and finds the recursive relation to obtain these analytical solutions. Finally, the educational significance and applications of these analytical solutions are discussed.

The standing and tumbling of a nail in a magnetic field is a daily phenomena, We calculated potentials of the finite size nail in the fields of a infinite long wire, the parallel infinite long wires and the circular current loop, respectively. The stable points of the potential determine the stable configurations of the nail. Involved in the intrinsic magnetic moment of the nail, the standing point of nail will change from a meta-stable point to a unstable point while moving in the magnetic field, then tumbling down. Landau theory of phase transition can be used to analyze the stable configuration of the nail. The numerical results shows that the critical index are exactly consistent with the prediction of Landau theory. making the standing and tumbling of a nail to be a perfect model of Landau theory. It is very charming that the abstract concept in theoretical physics are demonstrated in a simple classical phenomena.