Quasar absorption line studies have matured into a modern science that has contributed to the development of our contemporary cosmological paradigm, ranging from the Big Bang, across Cosmic Noon, to the Present Epoch. Researchers focus on key ions, transitions, and absorption lines because they are most common in the Universe. Each of these lines has a unique cosmic visibility in that there is a strong relationship between the observed wavelength of a redshifted line, the cosmic era in which it originated, and the type of astrophysical environment it probes. In this chapter, we outline the main eras of the evolution of the Universe, describe the phases and ionization conditions of the gas in the Universe, and show the connection between ions/transitions and the cosmic era and gas phases they probe.

Chapter 2 explains how in constructing quantum mechanics, old ideas had to be discarded and well-established principles had to be modified or even abandoned. First it was necessary to be convinced of the physical reality of the atomic structure of matter; then to show that Newtonian mechanics is not directly applicable to the study of the atom; later to show that Maxwell’s electrodynamics alone does not describe all the elementary processes of interaction between an atom and other quantum particles. The chapter takes us from Planck’s energy quantum to Einstein’s quantization as a universal phenomenon, from Bohr’s quantum atom model, and the early quantization rules, to de Broglie’s waves associated with corpuscles in motion. It ends with a corollary on the electrodynamic nature of quantum mechanics.

Hydrogen is the most abundant element in the Universe and neutral hydrogen, HI, is present in virtually all astrophysical structures ranging from the filamentary cosmic web to the inner regions of galaxies to the intracluster medium. The absorption transition from ground state to the lowest excitation state in neutral hydrogen gives rise to the countless optically thin Ly α forest lines and, in the highest column density structures, the damped Ly α absorption lines (DLAs). In optically thick structures, radiative ionization creates sharp “breaks” in quasar spectra called Lyman-limit systems (LLSs). HI correlates with the overdensity of the astrophysical environment, but this relationship evolves with redshift. HI also traces the mass density of neutral gas and the ionization history of the Universe. In this chapter, we describe the cosmic evolution of Ly α absorbers as recorded in quasar spectra from the Epoch of Reionization to the present epoch. At the highest redshifts, the transition from a dense Ly α forest to Ly α spikes to the famous Gunn-Peterson trough is described.

In this chapter, we begin by writing out the full reaction rate matrix accounting for the radiative and collisional processes presented in Chapter 34. The radiation field is assumed to originate externally and is thus not in equilibrium with the gas. We then derive the closed-form equilibrium solution for a pure hydrogen gas. Important to achieving equilibrium are the photoionization and recombination timescales. The industry standard ionization code is Cloudy; we describe how one uses this code to create model clouds. Important concepts such as the ionization parameter, cloud ionization structure, and shelf shielding of ionizing photons are discussed in detail. The building of grids of models is explained and example grids showing predictions of ionic column densities and ionization corrections are presented for commonly observed ions. Non-equilibrium collisional ionization models are described, and grids are presented. Sensitivities of the models to variations in the ionizing spectrum are explored. Finally, homology relationships useful for scaling cloud models to infer cloud densities, sizes, masses, and cloud stability are derived.

Chapter 17 begins by approaching the general problem of a quantum system subject to a time-dependent perturbation. We then apply perturbation theory to the interaction of atomic matter with the radiation field and derive the most important radiative corrections. Atomic transitions – both induced and spontaneous – are discussed in detail, with some important applications. The formal introduction of field quantization allows us to make contact with quantum electrodynamics and quantum optics and discuss the Jaynes–Cummings model. Stochastic electrodynamics, in which the matter–field interaction is not a perturbation but is present ab initio, is used to explain atomic stability and its breakdown leading to radiative transitions. In the same spirit, the quantization of the field is physically justified by deriving the conventionally postulated field operators and their basic commutators. The chapter ends with a brief introduction to second quantization, an elegant and powerful formalism for describing and analyzing quantum many-body systems.

Chapter 7 is devoted to the free particle and the derivation of the time-dependent Schrödinger equation. The free particle might be thought to be the simplest quantum problem of all, but a detailed treatment of it, including both Born and Dirac normalization, serves to highlight some surprising features. Powerful tools are introduced that are of wider application, notably the Green function method and the Feynman propagator. This material paves the way for the study of the time-dependent Schrödinger equation, the introduction of the current density of particles and the distinction between pure states and mixtures; the equation is applied specifically to the study of coherent states. The chapter ends with an introduction to the propagator in the general case.

In this chapter, we apply the formalism of hydrogenic and multi-electron atoms and build the periodic table of ground-state elements. Examination of the table shows that all elements in a given column share the same Russell-Saunders state symbol; they have identical orbital and total angular momentum states and valence electron multiplicities. These columns are formally grouped, and we show how each group shares the same spectral characteristics (the transition energies differ, but the relationships between transitions are identical from one element to another in a group). We then introduce the idea of iso-electronic sequences, which neatly explain the many lithium-like and sodium-like ions (CIV, NV, OVI, NeVIII, MgII, etc.) that have hydrogenic-like spectral series, including zero-volt resonant fine-structure doublets. We then provide accurate tables of ionization potentials and describe the physical reasons for the ion-to-ion trends in these potentials. We conclude the chapter with a complete suite of Grotrian diagrams (visual representations of the energy states and allowed electron transitions) for ions commonly studied using quasar absorption lines.

Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.

In the 1950s, Lyman Spitzer predicted that a hot gaseous medium surrounded the Milky Way in a halo/corona and that this gas should be detectable in strong absorption from highly ionized oxygen and nitrogen. It was confirmed in the 1970s using the Copernicus satellite. In the early 1990s, the first hydrodynamic cosmological simulations predicted that a warm-hot intergalactic medium (WHIM) was pervasive and extended out to the mildly overdense regions in the Universe. At low redshifts, the WHIM was predicted to harbor most of the baryons in the Universe. This was a bold prediction in which five-, six-, and seven-times oxygen (OVI, OVII, and OVIII) was predicted to trace this gas in absorption. The latter two require the X-ray spectroscopy, which has its challenges. The WHIM is also believed to be the source of the so-called broad Ly α absorbers (BLAs) in the Ly α forest and can be probed using fast radio bursts. In this chapter, we describe the discovery and confirmation of the WHIM and its characteristic properties. This includes a review of cooling flows, astrophysical plasmas, shocks, and interfaces.