Introduction to SO(4) group with applications to Physics: Galilean transformation and hydrogen atom

This work aims to present an introduction to the group $SO(4)$ and two applications in physics: one in Classical Mechanics and other in Quantum Mechanics. The generators of the $SO(4)$ group will be determined, as well as their Lie algebra. The application in Classical Mechanics will be in obtaining the homogeneous Galilei transformation and in Quantum Mechanics will be in obtaining the energy spectrum of the hydrogen atom, in the non-relativistic regime. The quantum version of the Laplace-Runge-Lenz vector will be fundamental for the construction of Bohr’s formula for the energy levels of the respective atom.

Keywords:
SO(4) group; Galilean transformation; Laplace-Runge-Lenz vector; hydrogen atom