This chapter begins Part II of the book, in which we build on the basic definition of a category and think about particular types of structure that might be of interest in any given category. This chapter is about how category theory provides a more nuanced approach to sameness, called isomorphism. We define inverses and isomorphisms. We give a sense in which a category treats isomorphic objects as the same. We then study isomorphisms of sets and show that the categorical definition corresponds to the elementary notion of bijection (where “elementary” means “defined with reference to elements”). We then look at isomorphisms of monoids, groups, and partially ordered sets, showing that these are just structure-preserving maps that also happen to be a bijection, and we discuss how these exhibit things with the same structure. We show that the situation is different for topological spaces, as not every bijective continuous map has a continuous inverse. We briefly touch on the idea of isomorphisms of categories, explaining that this is not the best level of sameness of categories. We finish by mentioning further topics: groupoids, categorical uniqueness, and categorification.

The field of quantum research is currently undergoing a revolution. A variety of tools and platforms for controlling individual quantum particles have emerged, which can be utilized to develop entirely new technologies for computation, communication, and sensing. In particular, these technologies will enable applications of quantum information science that can fundamentally change the way we store, process, and transmit information. Exciting theoretical predictions exist for quantum computers, with some proof-of-principle experiments, to perform calculations that would overwhelm the world’s best conventional supercomputers. Quantum research is rapidly developing, and the race is intensifying for quantum technology development, involving some of the high-tech giants. In this chapter we will introduce some key concepts in the materials and devices behind these technological developments. Becoming familiar with these concepts in this first chapter should provide the reader with concrete goals and motivations for studying the quantum methods and tools described in subsequent chapters.

Quantum mechanics is currently the most fundamental theory in use in many disciplines of science and engineering. It is particularly important when one is dealing with nanoscale and atomic-scale systems. However, many phenomena and properties that occur at atomic scales are strange and nonintuitive. There are a number of concepts that simply do not exist in the macroscopic world where we live. Wave–particle duality is one of them. In this chapter, we examine how and when classical particles start behaving as quantum mechanical waves, derive the most important wave equation that quantum particles obey, Schrödinger’s equation, and solve it for the elementary problems of electron waves in given potential energy landscapes. We will also learn how to calculate the expectation values of observables when the wavefunction is known. Schrödinger’s equation will be extensively used throughout the rest of this textbook. More complicated potential energy problems, particularly those relevant to materials and devices, will be dealt with in Chapters 5 and 7, building upon the formulations developed in this chapter.

The band theory of solids provides a general framework with which to understand properties of materials. It not only explains the fundamental differences in electronic structure between insulators, semiconductors, and metals but also provides guidelines for finding optimum materials for specific device applications. For example, a semiconductor with a light effective mass is suited for high-electron-mobility transistors (HEMTs) because the mobility ${\mu }_{\mathrm{e}}$ is inversely proportional to the effective mass, ${\mu }_{\mathrm{e}}=e\tau /m^{*}$, where τ is the scattering time. For developing LEDs and laser diodes, a direct band gap material – i.e., a material in which the conduction-band bottom and the valence-band top occur at the same k – is necessary for momentum conservation since the momentum of photons is negligibly small compared with crystal momenta. In this chapter, after reviewing the basic concepts of atomic and molecular orbitals, bonds and bands, crystal lattices and reciprocal lattices, we provide an overview of the band structure of technologically important materials, including both traditional and emerging materials.

The theory of the interaction of radiation with matter is fundamentally important for describing how modern semiconductor devices generate, detect, and modulate light. These devices, known as optoelectronic devices, are behind today’s technology in diverse areas, including communications, imaging, spectroscopy, sensing, and energy harvesting. They may also become essential components in future quantum technology based on photons. In this chapter we will learn the basic theoretical formalism for describing light–matter interaction phenomena, starting from microscopic processes such as absorption, spontaneous emission, and stimulated emission and ending with the conditions for achieving gain, which is a fundamental requirement for a laser.

The current–voltage characteristics of modern electronic devices consisting of semiconductor heterostructures, such as resonant tunneling diodes, quantum cascade lasers, and tandem solar cells, are determined by the dynamics of electrons propagating through quantum-engineered 1D potential landscapes. In this chapter, we will develop a general formalism with which to describe transmission probabilities for electron waves propagating through arbitrary potentials, which can be used for analyzing electron motion in semiconductor devices. Furthermore, we will extend our formalism to 1D electrons moving in a general spatially periodic potential, based on which we will describe the basic concepts of the band theory of solids. The central theorem in band theory is the Bloch theorem, which we will derive and then use for discussing the dynamics of electrons in crystalline solids (or Bloch electrons).

The field of quantum computing is developing at a rapid pace, and one can expect paradigm-shifting advances in coming years. The goal of this chapter is for the reader to understand fully the language and basic concepts of quantum information science needed to engage in research and development in this very exciting field in the future. We will apply the mathematical machinery we have acquired so far to develop the quantum counterparts to the classical notions of bits, logic gates, circuits, and algorithms. We will also review some of the promising examples of quantum hardware for physically realizing quantum information processing.

A regrettable amount of mathematical machinery goes into a good understanding of quantum mechanics. This could be avoided if a good intuitive understanding of many quantum systems was possible, but as intuition is generally derived from daily experience (which is governed by classical laws), we cannot expect this to be the case in general. Here, we present an in-depth introduction to the mathematical foundations of quantum mechanics, accompanied, wherever appropriate, by detailed explanations of relevant quantum concepts such as superposition, wavefunction collapse, and the uncertainty principle. As an additional benefit, the language developed in this chapter will be especially useful for describing quantum information science in .