This chapter introduces quantum mechanics through the experimental discoveries that revealed the limits of classical physics. Beginning with the historical debate over whether light is a wave or a particle, it examines Young’s double-slit experiment and its modern single-particle versions, which demonstrate that quantum objects can exist in superpositions of mutually exclusive states. The chapter explains how interference arises when alternative possibilities are not measured and how interaction with the environment suppresses these effects through decoherence, giving rise to classical behavior at macroscopic scales. A sequence of thought experiments involving interferometers and Stern–Gerlach devices highlights the counterintuitive structure of quantum measurement and the incompatibility of certain physical properties, as captured by the uncertainty principle. The chapter emphasizes that quantum mechanics does not describe particles taking multiple paths simultaneously in a classical sense but rather requires a new framework for probability based on amplitudes and interference. Finally, the discussion connects these foundational ideas to emerging technologies such as quantum computing, showing how superposition and entanglement can be harnessed to perform computations that have no classical analogue.

In this chapter, the Lieb–Mattis model for antiferromagnets is introduced in order to analyse the fate of superposed symmetry-breaking states. An explicit calculation is achieved for the spin-half case.

Can de Broglie’s hypothesis be generalised to any physical object interacting with its environment? Schrödinger answered this question by introducing a complex-valued wave-function that fully characterises the state of an object as informational content. If its spatial extent is limited, it accounts for the information localisation of a classical body. We also show that physical quantities (energy, linear momentum or position) are represented by operators, and how their measurements are made in quantum physics. In particular, stationary states are eigenstates of the Hamiltonian with determined energy values. The evolution of the wave-function is governed by the Schrödinger equation, which is the fundamental equation of quantum physics. Examples are taken for a stationary (i.e., time-independent) one-dimensional interaction, when the considered physical object is in a free (or scattering) state – not classically constrained to remain in a spatially bounded domain. We consider potential energy steps that model localised interactions on which a physical object is scattered with determined probabilities of reflection and transmission.

This chapter introduces quantum computation by comparing classical and quantum computers. Core concepts including qubits, superposition, and entanglement are introduced, setting the stage for deeper exploration. Various quantum computing models are summarized, with a focus on the circuit and topological models. The chapter explains why quantum computing is necessary, especially for tasks beyond classical computing’s limits. It discusses existing quantum platforms and provides an overview of their capabilities and limitations. The chapter also offers a brief historical perspective, touches on computational energy efficiency, and forecasts a quantum future where quantum and classical computing work in tandem. This groundwork provides essential insights into quantum computation’s potential and upcoming chapters’ explorations of algorithmic and theoretical principles.

Voltage and current sources, both independent and dependent, are introduced, along with resistors and their equivalent circuit laws. The Thevenin and Norton theorems are presented. Several examples of resistor applications are given. Various techniques for solving circuit problems are discussed, including Kirchhoff’s laws, the mesh loop method, superposition, and source transformation. Input resistance of measuring instruments is discussed and the various types of AC signals are presented.

Rivers and their valleys have long been a source of contemplation and wonder. They are not only key geomorphic agents, but they are also economically important, acting as transportation arteries, sources of irrigation water and food, and as generators of hydropower. We also use rivers for drinking, waste disposal, and for a variety of recreational activities. Many geomorphologists consider running water to be the most dominant and important geomorphic process – shaping landscapes everywhere. Even in deserts, running water is often the most important and widespread geomorphic agent.

Most valleys have a stream or a river at their bottom. In ancient days, it was thought that water simply “found” its way into preexisting valleys, forming rivers there. Geomorphologists now know that most valleys were formed by the rivers currently within them, which moved sediment out and carved the valley over time.

The difference between quantum and classical mechanics does not involve just a small tweak. Instead it is a root and branch overhaul of the entire framework. In this chapter we introduce the key concept that underlies this new framework: the quantum state, as manifested in the wavefunction.

Chapter 1 begins by re-examining the textbook quantum postulates. It concludes with the realization that some of them are inconsistent with quantum mathematics, but also that they may not have to be postulated. Indeed, in the following two chapters it is shown that their consequences follow from the other, consistent postulates. This simplification of the quantum foundations provides a consistent, convenient, and solid starting point. The emergence of the classical from the quantum substrate is based on this foundation of “core quantum postulates”—the “quantum credo”. Discussion of the postulates is accompanied by a brief summary of their implications for the interpretation of quantum theory. This discussion touches on questions of interpretation that are implicit throughout the book, but will be addressed more fully in Chapter 9. Chapter 1 ends with a “decoherence primer” that provides a quick introduction to decoherence (discussed in detail in Part II). Its aim is to provide the reader with an overview of the process that will play an important role throughout the book, and to motivate Chapters 2 and 3 that lay the foundations for the physics of decoherence (Part II) as well as for quantum Darwinism, the subject of Chapters 7 and 8.

Spoken language exhibits not only grammatical constructions but also prosodic constructions. While the latter are also form–function mappings, there are differences: Prosodic constructions involve temporal configurations of diverse prosodic features, their functions are primarily pragmatics-related and interactional, they can be present to greater or lesser degrees, and they are frequently superimposed and aligned in complex ways with other prosodic constructions and with grammatical constructions. This chapter illustrates these properties with examples from American English.

One-dimensional traveling wave pulses are defined and generalized to sinusoidal traveling waves. The mathematical description of sinusoidal waves is considered in depth. It is shown that any traveling wave can be considered the sum of standing waves, and vice versa. Longitudinal polarization is introduced and contrasted with the previous examples involving transverse polarization. The effects of dispersion, where the wave speed may depend on frequency, on wave propagation are briefly discussed.

We learn the language of the wave function, which is the representation of the quantum state vector in position space. We introduce the position and momentum operators and learn the rules for translating bra-ket formulae to wave function formulae. We use these new tools to solve the infinite square potential energy well problem and the finite square well problem.

We introduce the idea of orbital angular momentum and illustrate its importance in solving the three-dimensional differential equation that is the energy eigenvalue equation for the hydrogen atom. By separating variables in the eigenvalue equation, we isolate the differential equations for the angular variables from the differential equation for the radial variable. We solve the angular equations to discover the spherical harmonics and the angular momentum quantum numbers.

We solve the radial differential equation to find the quantized energies and the radial wave functions of the bound states of the hydrogen atom. We present the energy spectrum of hydrogen and the electron probability densities of the energy eigenstates and of superposition states.

The Introduction to this chapter raises the problem of how to distinguish between tolerable and intolerable inconsistencies in linguistic theorising. In , Rescher and Brandom’s () logic will be introduced, which differentiates between paraconsistency, allowing for a special kind of inconsistency without ex contradictione quodlibet, and strong inconsistency, which is exposed to logical chaos. The workability of this distinction in theoretical linguistics will be exemplified by two case studies in Sections 4.3 and 4.4, respectively. They will illustrate, first, that tolerable inconsistency can be represented as paraconsistency, while the unavailability of a paraconsistent resolution may leave us with an intolerable strong inconsistency. Second, they will also show that in the cases examined the emergence of inconsistencies is closely related to the data handling techniques applied.summarises the solution to the problem raised in this chapter and highlights that there is a need to re-evaluate the way data are treated in theoretical linguistics; but tackling the question of how this should be done will be postponed to later chapters.

This appendix explains quantum effects: uncertainty, entanglement, and superposition; and explains how these effects form the basis of quantum sensing, computing and communication. This appendix summarizes the history and debates of wave mechanics, which was developed at the start of the Twentieth Century. Examples are given of macro-level quantum effects that the reader can observe in an attempt to start building an intuitive sense of quantum effects. These macro-level quantum phenomena are the dual-slit experiment, black-body radiation, and the characteristics of polarized light. Much attention is given to the characteristics of light, both because light provides examples of quantum effects but also because photonic emitters and sensors play a key role in quantum sensing, computing, and communication.