Mathematical analysis of a Bohr atom model

Bohr proposed in 1913 a model for atoms and molecules by synthesizing Planck’s quantum hypothesis with classical mechanics. When the atom number $Z$ is small, his model provides good accuracy for the ground-state energy. When $Z$ is large, his model is not as accurate in comparison with the experimental data but still provides a good trend agreeing with the experimental values of the ground-state energy of atoms. The main objective of this paper is to provide a rigorous mathematical analysis for the Bohr atom model. We have established the following: (1) An existence proof of the global minimizer of the ground-state energy through scaling. (2) A careful study of the critical points of the energy function. Such critical points include both the stable steady-state electron configurations as well as unstable saddle-type configurations. (3) Coplanarity of certain electron configurations. Numerical examples and graphics are also illustrated.