ADDITIONAL METHODS FOR TEACHING THE QUANTUM THEORY OF THE HYDROGEN ATOM - Avdonin, Kostiantyn (Physical and Mathematical Education, 2025)
This paper proposes the possibility of applying integral equations corresponding to the Schrödinger equation to find the radial component of the wave function of a quantum system of two particles and the permissible values of the principal quantum number of the system. The relevance of the work is due to the need to supplement the methodology for presenting such an important key problem of quantum mechanics as the quantum theory of the hydrogen atom, which consists of expanding the set of methods for finding solutions to the Schrödinger equation for the hydrogen atom and their systematization, which will contribute to a better understanding of this topic. The proposed sequence of presenting the material takes into account the latest trends in the development of methods for finding solutions to the stationary Schrödinger equation in analytical form, demonstrating the possibility of finding the energy spectrum of a quantum system of particles using integral equations.Formulation of the problem. The quantum theory of the hydrogen atom in physics textbooks and manuals is most often taught based on the single-particle Schrödinger equation for the electron, switching to a reference frame associated with the atomic nucleus, neglecting the motion of the nucleus, or part of the material is taught when considering the motion of the electron in a central force field. To generalize the teaching methodology, it is better to immediately consider a quantum system of two particles: an electron and a hydrogen atom nucleus.Materials and methods. The main methods for solving the problem are: means of finding solutions to ordinary, linear, and partial differential equations, the theory of multidimensional integral equations, and the theory of Green's functions.Results. It was found that the radial components of the wave functions of the hydrogen atom, which determine the energy spectrum, can be obtained without transitioning to a spherical system, which simplifies the presentation of the topic material. A generalization of the sequence of presentation of the material on the quantum theory of the hydrogen atom is proposed.Conclusions. The proposed generalization of the method for finding solutions to the Schrödinger equation for the hydrogen atom allows for a more complete presentation of the material on the topic "Quantum Mechanics", increases the degree of provability of the presentation of the material, and facilitates its understanding.
Keywords: hydrogen atom; integral equation; quantum mechanics; methodology