Hyperfine structure of the hydrogen and evidence for dark matter in spiral galaxies

Dark matter has a crucial role in our understanding of modern cosmology, since it dominates the mass budget of the universe. The present work introduces all the basic tools needed to understand how the study of the hydrogen atom has been a turning point in astronomy in the 60s. Indeed, the advent of radio astronomy allowed to collect robust evidences of the existence of dark matter. In the first chapter, we summarize the standard results on the ideal hydrogen atom. The emphasis is particularly put on the symmetries of the system and how they can justify the high degeneracy of the energy levels. In the second chapter, we systematically introduce all the formalism necessary to take into account the fine structure terms. We present the Lorentz group, its algebra, its spinorial representation and the Dirac equation. Finally we derive the relativistic wave equation for the electron. Using the latter equation, in the third chapter, we first discuss the physical origin of the fine structure terms and why they partially break the degeneracy of the energy levels. Then we compute the energy correction through perturbation theory. Even if we do not take into account the second quantization, we qualitatively present the Lamb shift and in more detail the hyperfine structure. In particular, we show how the famous 21cm line emerges from an hyperfine transition. Finally in the fourth chapter, we review the principal dark matter evidences from observational astronomy. In particular, we mention Zwicky’s pioneering work and other more convincing evidences obtained studying the galactic rotational curves. Moreover, we will introduce the most significant dark matter characteristics that are fundamental in order to set up a proper experiment for its detection.