In this chapter, selected observational programs of merging galaxies, groups, and clusters are presented, and their reported results summarized. For each, neutral, low-, intermediate-, and high-ionization gas is examined separately. Various findings appear to indicate that comparing absorption to the “nearest galaxy,” the “most massive galaxy,” or the “central galaxy,” can strongly influence the inferred conclusions from the studies. The results that appear to agree between the various studies, show that compared to the CGM of member galaxies, the metal-line selected IGrM gas appears to be both relatively optically thin and kinematically quiescent in both its low- (MgII) and high-ionization (OVI) phases. Clusters, on the other hand, surprisingly appear to have neutral gas deep into their cores and, within the virial radius of the cluster, the sizes of the CGM of the individual member galaxies appears to be diminished compared to galaxies residing outside the virial radius. At the time of this writing, the study of the CGM in the IGrM and ICM environment is a developing area of study.

Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.

In this chapter, the taxonomy of the emission spectra of starbursts, active galactic nuclei (AGN), and quasars are compared. These spectra are discussed in terms of their emission line diagnostics as measured on Baldwin-Phillips-Terlevich (BPT) diagrams. The non-unique typing of AGN/quasars as Markarian galaxies, LINERs, Seyfert galaxies, radio galaxies, blazars, BL Lac objects, and flat-radio spectrum quasars is explained. The taxonomic subclassification of Seyfert galaxies and quasars based on the relative strengths of permitted broad lines and forbidden narrow lines are discussed. The quasar main sequence, which is based on the kinematics of the H β emission line and the luminosity ratio of the FeII/H β emission lines, is introduced. Insights into the nature of AGN/quasars can be gleaned from the fact that their luminosities and spectral energy distributions can be highly variable on timescales of hours to decades. Broad absorption lines (BALs) and narrow absorption lines (NALs) arise in strong outflows. The BALs may provide clues about viewing angles, leading to radio-quiet and radio-loud unified models of AGN and quasars.

Absorption line studies have shown that the circumgalactic medium (CGM) is an extended complex multiphase gas reservoir of galaxies. It is a kinematically diverse region that interfaces the baryon cycle activity within galaxies to the intergalactic environment in which the galaxies are embedded. In this chapter, selected observational programs and their reported results are presented. The focus is on empirical bivariate relations, such as absorption strength and covering fractions, versus impact parameter, stellar mass, star formation rate, etc. The CGM is presented as viewed through several commonly targeted ions, in particular HI, MgII, CIV, OVI, and NeVIII. Though this allows the various ionization stages of CGM gas to be examined in isolation, it glosses over the multiphase nature of the CGM. The practical design of high-redshift experiments is such that they are much more statistical in nature than the more granular experiments at low redshift. Thus, high-redshift studies are discussed separately.

Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.

Chapter 13 is devoted to two important approximation methods used to solve time-independent problems, namely stationary perturbation theory and the variational method. Perturbation theory is developed for both non-degenerate states and degenerate states, with illustrative examples. The Stark effect is considered separately. The dielectric constant of a transparent medium as a function of frequency is shown to result from the modification of the wave function due to the perturbation of the radiation field. Two complementary topics are presented: the canonical transformation method and the Feynman-Hellman method. Finally, the possibilities of the variational method are illustrated by using it first to determine an upper bound for the energy of the stationary states of a quantum system, and then to study more complicated cases, including the partially shielded Coulomb potential produced by a neutral atom.

This chapter covers the most challenging aspect of quasar absorption line studies – estimating the densities, dynamic conditions, metallicities, ionization conditions, and general cloud properties (masses, sizes, stability) that match the observed data. The techniques have evolved from single-cloud single phase models that were simply constrained by the measure column densities, to kinematically complex, multi-cloud multiphase models that are constrained by absorption profile morphologies on a pixel-by-pixel basis. In this chapter, we cover the modeling methods by describing them in order of complexity and ambition. These methods are the chi-square method, the density-metallicity locus method, and Bayesian approaches, including Markov Chain Monte Carlo (MCMC) methods and profile-based multiphase Bayesian modeling. Methods are discussed and examples are provided, but modeling absorbers is a scientific artform that requires a deep intuition that can only be developed through lots of practice.

In this chapter, we describe how blended multi-component absorption profiles can be modeled. Simple deblending that bypasses radiative transfer and atomic and gas physics can be performed using multi-component Gaussian fitting. We show how further sophistication can be added by tying doublets or multiplets and forcing Gaussian components to match known line spacings. To extract column densities and Doppler broadening parameters for each component, we use Voigt profile fitting. We begin with a general expression for a multi-component absorption profile for which each component has a unique column density and Doppler broadening parameter. We then discuss progressively more complex Voigt profile fitting, starting with multiple components for a single transition, then multiple components for a doublet (two transitions from a single ion), and then generalize to multi-component multi-transition multi-ion absorption systems. We also discuss methods for measuring the turbulent velocity component and approaches to multiphase decomposition for ions of different ionization levels. We conclude by discussing fitters and fitting philosophies. Optimized AOD column densities are also discussed.

Chapter 6 begins with an exposition of the WKB approximation technique developed in 1926 by Wentzel. Kramers and Brillouin. The WKB method is discussed in detail, and it is shown that it is particularly suitable when the particle is in a sufficiently energetic state that its behavior can be considered to be almost classical, for which reason it is called a semiclassical approximation. The method is applied specifically to the study of the nuclear alpha decay and the calculation of the tunneling time delay. The rest of the chapter is devoted to a discussion of the basic electronic properties of solids and some important applications of these properties, which are easily explained using the results obtained in the first part of the chapter. Starting with the free electron gas model, it concludes with a discussion of semiconductors based on the Kronig and Penney model.

Helium is the second most abundant element in the Universe, and, when singly ionized, is hydrogenic. This means HeII has a hydrogen-like absorption spectrum but with transition energies a factor of 4 higher. This places HeII Ly α forest lines deep into the ultraviolet, the consequences of which highly limit the redshift visibility of HeII studies – only favorable quasar sightlines can be used to study HeII Ly α and Ly β absorption. The column density ratio of HeII to HI is highly sensitive to the shape and intensity of the cosmic ultraviolet background (UVB), and thus is a key quantity for constraining the evolution and patchiness of the UVB. An Epoch of HeII Reionization stretching into the Cosmic Noon era provides insights into the appearance of the first quasars in the Universe. In this chapter, we describe the redshift visibility of HeII absorbers, discuss the cosmic impact of HeII absorption, and describe key observational results, including the so-called hardness parameter, the HeII Gunn-Peterson trough, and HeII Ly α spikes.

Studies of the intermediate-ionization metal-line absorbers provide insights into warm/hot lower-density gas that has been processed through stars in galaxies. These absorbers have been studied primarily using doubly and triply ionized carbon and silicon ions (CIII, CIV, SiIII, and SiIV). CIII arises deep within the spectral range of the Ly α forest and is thus mostly visible at low redshifts where the Ly α forest line density is much smaller. SiIII is adjacent to the Ly α line and is also best surveyed at low redshift. The CIV and SiIV lines are well redward of the Ly α line and thus have visibility over a wide range of redshift. UV and IR spectrographs expanded the redshift coverage from z = 0 to z = 7. The population statistics measured include the redshift path density, the equivalent width and column density distributions, the cosmic mass densities, and the kinematics (broadening parameters, velocity splitting distributions, and absorber velocity widths). In this chapter, we discuss multiple observational programs and their reported findings for several of these ions.