This book presents a detailed analytical description of the derivation of the Lamb shift in the Hydrogen atom using quantum field theory. This shift in energy levels, relative to what is predicted in traditional particle physics theory, is due to the phenomena of vacuum fluctuations and vacuum polarization which are accounted for in the quantum field theory as applied in particle physics. The derivation reported is done using perturbation theory and extends up to 4th order. This derivation has been carried out to higher orders by scientists all over the world throughout several decades and agrees exceptionally well with experimental data demonstrating the physical reality of vacuum fluctuations. This book also includes a historical overview of the understanding of the atom and its interaction with electromagnetic radiation as it was in the first half of the 20th century, when the calculation of the Lamb shift was first attempted.