Chapter 15 is devoted to systems of identical particles and the associated quantum statistics. It is shown how the spin (whether integer or half-integer) determines fundamental features of the system’s behavior that cannot be obtained by approximate methods, in particular the exchange effects characteristic of states of two or more identical particles. The spin-symmetry relation is discussed in detail for both the Bose–Einstein and the Fermi–Dirac statistics. Quantum collective phenomena that are a defining feature of condensed matter, including phonons, superconductivity, Bose–Einstein condensates and the quantum Hall effect, are briefly introduced. Since most real quantum systems consist of subsystems in different quantum states that are mixtures of pure states, the chapter includes an introduction to the density matrix, its basic properties and its application to quantum statistics. It concludes with a critical discussion of decoherence and the significance of the collapse of the wave function.

Chapter 14 is entirely devoted to the electron spin and an introduction to quantum entanglement. The first part deals with the groundbreaking discovery of spin, its introduction into quantum formalism, and some of its most important effects on atomic spectra, notably the anomalous Zeeman effect. Historically, spin has been considered as an angular momentum that particles can have by the mere fact of their existence, which is called "intrinsic" and does not require any explanation. To address this shortcoming, Section 4 presents a possible explanation for the origin of electron spin as a result of its interaction with the vacuum field. Section 5 introduces the entangled system of two particles with spin, which provides an opportunity to discuss, necessarily schematically, the Schrödinger cat and the Einstein-Podolsky-Rosen thought experiment, as well as the Bell inequalities.

Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.

Chapter 1 begins by introducing the quantum world as part of the physical world, not in opposition to or separate from the classical world. It then introduces the basic tools needed to work with quantum mechanics, and describes the scope and purpose of this textbook. It discusses how despite its successes, current quantum mechanics lacks adequate explanations for important observed atomic properties, suggesting an incompleteness of the theory. After a brief overview of its main interpretative issues , the chapter concludes with an introduction of the vacuum or zero-point field as the physical element that restores causality and serves to complete the quantum picture by providing a causal explanation of characteristic quantum phenomena.

Chapter 16 begins by showing how quantum mechanics explains the structure and basic properties of multielectron atoms and their position in the periodic table of elements. It introduces the main approximation methods that have been developed to study such systems, explains the atomic shell structure, and discusses the helium atom in detail. It then looks at the different types of forces that bind atoms together to form molecules, focusing on diatomic molecules and the hydrogen molecule in particular. It discusses the long-range intermolecular forces that are responsible for many of the chemical and structural properties of matter, and identifies the zero-point or vacuum field as being responsible for the van der Waals and Casimir forces. In the final section, the properties of atomic nuclei are shown to reveal an internal structure with periodicities reminiscent of those of atoms and to disclose the effect of the nuclear spin–orbit interaction.

Chapter 18 is entirely devoted to the quantum theory of scattering, which is normally covered in more detail in a course on quantum field theory. We introduce the main concepts, including scattering amplitude, differential and total cross sections, and form factors, and derive the main formulas for elastic scattering, in a nonrelativistic framework; this allows us to appreciate how the information obtained from scattering experiments is used to explore the intricacies of quantum particles that are otherwise inaccessible. The Born approximation is studied and applied to scattering by a periodic potential, a weak potential, and Rutherford scattering. The partial wave expansion is derived and applied to obtain the optical theorem. The chapter concludes with a complementary section on resonant scattering.

Chapter 4 shows that, just as Heisenberg’s matrix mechanics represented a break with classical physics and the beginning of a new physics, Schrödinger’s wave mechanics represented another, no less radical, break with classical physics and at the same time opened up another path to the quantum world. Schrödinger’s theory is shown to make substantial use of de Broglie’s waves and related phenomena. The Schrödinger equation, being wave-like, constitutes mathematically an eigenvalue problem for bounded systems, and thus naturally leads to quantization, as is illustrated with a couple of simple examples. The chapter ends with Dirac’s abstract formulation, which has the advantage of being suitable for any description, be it Heisenberg’s, Schrödinger’s or any other.

Chapter 8 is devoted to the study of the dynamics of quantum systems, based on the formalism presented in Chapter 7 and previous chapters. The operator notation and its matrix representation are used without distinction, moving freely from one to the other as convenient. This allows us to study the evolution of the system from different perspectives, depending on the description adopted. The chapter includes a discussion of the statistical information contained in the quantum formalism, a formal derivation of the Heisenberg inequalities and a clarification of their physical meaning. A section is dedicated to quantum canonical and related transformations. More advanced topics, which provide a valuable complement, include Noether’s theorem on the relationship between the symmetry properties and the conserved dynamical variables of the system.

Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.

Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.

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.