The question posed by de Broglie in 1923 concerned the possible manifestation of wave behaviour for any physical object with well-defined energy and linear momentum. The idea of a union between the wave and particle concepts for any physical object was strongly suggested by Einstein’s theory of special relativity. This chapter focuses on the de Broglie hypothesis and its immediate consequences. How can the behaviour of waves and particles be reconciled? Heisenberg answered this question in 1925 with his famous inequalities. What is the nature of the de Broglie wave? Born proposed a probabilistic interpretation in 1926. This chapter presents these points in detail in the case of free objects, that is, one that does not interact with its environment. The end of the chapter is devoted to phase effects. We see that the phase of a de Broglian wave can be changed by a spacetime transformation or by an electromagnetic field. Many modern applications (e.g., gyrometers) are based on phase effects, which are also fundamental to understanding the foundations of quantum physics; in particular, they allow us to see interference as a question of topology.

Throughout this work we have emphasised the informational content of a physical system and its accessible part. We have argued and reasoned by considering that the state vector (in general, a ray), and particularly the wave-function in spacetime, represents this knowable information. Schrödinger’s equation then governs the dynamics of the informational content within an accessible domain. Resulting or equivalent equations, for example Heisenberg’s equation relative to a physical quantity (an operator or its expected value), also describe the evolution of the accessible information. Finally, Feynman’s variational approach highlights the spacetime field of accessible information, stating that we are most likely to access information where the paths between initial and final events in this field interfere constructively, which corresponds to the loci in spacetime where classical action is stationary.

This chapter explores the neuroscience of courage and the brain’s ability to override fear. Building on the fear circuits introduced in Chapter 1, it dives into the neural pathways that allow organisms to detect threats and either flee or take action. Chapter 2 examines how fear responses can be overridden through higher-order processing in the cortex. Readers are introduced to LeDoux’s “low road” and “high road” pathways to the amygdala, and the role of cortical regions like the anterior cingulate cortex and medial prefrontal cortex in fear regulation. The chapter also explores how basic synaptic mechanisms contribute to courage and social decision-making, and ends with a discussion of altruism and the neuroscience of self-sacrifice. Together, these systems suggest that bravery is not the absence of fear, but a coordinated biological response that can transcend it.

In this chapter we ask about the physical evolution of a system, that is, for a non-stationary state vector. From an initial state, how can we determine the state and physical quantities of the system at any time? The answer is provided by the Schrödinger equation that describes the dynamics of the system under the Hamiltonian associated with the total energy. We consider here the special, yet important, case of a Hamiltonian that does not explicitly depend on time. We say that such systems are conservative. We show how the law of conservation of the total energy is fulfilled in quantum physics. To do this, we consider the evolution of various systems already described in previous chapters, namely the quantum beats between two quantum states corresponding to different energies, different positions or different angular momenta; the propagation of a wave packet; and the coherent oscillations of a harmonic oscillator. Several applications illustrate the importance of such quantum phenomena, including quantum beat spectroscopy, superconducting quantum interference devices or the generation of coherent states of any oscillator.

This chapter argues that the relationship between the online world and the classroom remains a contentious issue. Popular culture, and the increasing use of social media by young people and children has seen many traditionalists lament how our culture has declined, and worry about how educationally corrupted our schools have become. Its absence has been used to suggest that our schools are out of touch with their primary constituency – children and young people. The keen-eyed among you might note that this chapter is full of false binaries... perhaps this tells us something about the nature of the topic. This is not a simple issue to address; even the notion of ‘culture’ itself is subject to considerable disagreement. This is not even a simple chapter to write; the references will likely be outdated by the time I finish writing this sentence. So read on with a little grace, and a little humor.

This chapter presents key quantum mechanics principles essential for understanding quantum computation. The postulates of quantum mechanics, mixed states, and density matrices are introduced, along with the Stern–Gerlach experiment’s role in illustrating quantum behavior. Topics such as quantum coherence, entanglement, and the EPR paradox are covered to clarify the fundamental distinctions between classical and quantum systems. Measurement is explored with an emphasis on positive operator-valued measures (POVM), a key concept in understanding quantum state collapse. These principles provide a foundation for studying quantum computation and are essential for understanding qubit behavior, quantum information processing, and subsequent algorithmic structures.

In this chapter we turn to much more recent aspects of photons. Specifically, we discuss single-photon optical experiments to capture the fundamental nature of the photon. In particular, we argue that the old concepts of particle and wave are simplistic, and that we prefer the unifying term of physical object that is ubiquitous throughout the book. These experiments raise the question of photon counting, or photon statistics. In particular, we will see that different light sources can be characterised by the way in which photons are distributed in the emitted light flux. Historically, the question of the distribution of photons in a light flux arose after the Hanbury Brown and Twiss (HBT) interference experiments. They are a cornerstone in the history of optics, stimulating research into the structure of light beams and quantum coherence. Here we look at how HBT interference can be interpreted in terms of photons.

This chapter delves into the quantum circuit model, a primary framework for quantum computation. It begins with the qubit, exploring its representation on the Bloch sphere and its probabilistic measurement outcomes. Quantum gates are introduced as the basic operational units, transforming qubits via unitary operations. The chapter discusses single- and two-qubit gates, building up to universal quantum computation, which enables any quantum function to be constructed through a finite set of gates. This chapter provides an in-depth understanding of information processing in quantum circuits, establishing a practical foundation for executing quantum algorithms and advancing to topics like entanglement-based operations and fault-tolerant design in later chapters.

We turn in this chapter to the quantisation of the electromagnetic field and its consequences. This belongs to the field of quantum electrodynamics, the study of electromagnetic phenomena, no longer in terms of classical fields but in terms of photons. The correspondence between the field and the photons is analysed in terms of energy, linear and angular momentum. We see that the quantised electromagnetic field explains important phenomena that cannot be understood with classical fields. In particular, an essential concept that emerges as a natural consequence of quantisation is the quantum vacuum. This non-classical state has a deep fundamental meaning with many fascinating concrete effects. In particular, its presence is revealed by the spontaneous emission of atoms and the attraction of two sufficiently close conducting plates, it is at the origin of very small but measurable corrections in atomic spectra and, more generally, it is a key component of one-photon optics, for example in confined spaces (cavity quantum electrodynamics). Finally, we are able to answer the question ’What is a photon?’

This chapter is devoted to quantum dynamics with a time-dependent Hamiltonian. The latter is written as the sum of two contributions: one stationary, the other (with the time parameter) describing the perturbation of a system by its environment and responsible for possible transitions. We thus recover the heuristic hypotheses of Bohr or Einstein on light absorption/emission. We analyse these phenomena in much more detail, for example in the case of confinement in a cavity or to describe magnetic resonance, which is at the origin of important applications. Quantum theory is essentially probabilistic, so we begin by defining the concept of transition probability. We make extensive use of the two-state system, but apply it to the situation of an initial state and a possible final state, both coupled by an explicitly time-dependent interaction. We analyse the important cases of a quasi-stationary perturbation, a harmonic perturbation and coupling to a continuum of states. The chapter ends with a more complete presentation of the theory of time-dependent perturbations according to the Heisenberg and Dirac pictures, and also examines the strong field–atom coupling in a cavity.

This chapter explores the neuroscience of fear, examining how our brains detect and respond to threats, both real and imagined. It introduces major theories of emotion and focuses on the role of the limbic system in processing fear-related stimuli. Through research in animals and humans —including lesion studies and the famous case of patient S.M.—the chapter distinguishes between behavioral responses to danger and the subjective experience of fear. It also challenges the idea of a single “fear center,” emphasizing that fear arises from dynamic interactions across multiple brain regions. These concepts are then applied to ambiguous situations, such as sensing a presence in a dark room, where the brain may interpret uncertainty in emotionally charged ways. Finally, this chapter encourages readers to consider how the brain constructs meaning from unclear stimuli, laying the groundwork for a scientific exploration of the supernatural.

This chapter addresses one of the most important areas of philosophy – ethics – and uses it to examine aspects of the role of the law in education. Of all the areas of philosophy, more has probably been written about ethics, and over a longer period, than any other. In addition, all cultures are structured around a fundamental ethical system: the law. However, irrespective of their importance, both subjects are currently notable for their lowly status within the teacher education curriculum.

This chapter concerns the search for the eigenstates of a conservative system (the Hamiltonian has no explicit time dependence). In general, the eigenvalue equation cannot be solved exactly and we are forced to find approximate solutions. In many cases the Hamiltonian is the sum of terms of different importance, where the main part corresponds to a case for which eigenstates/eigenvalues are known, while the smaller part is considered as a perturbation. This approach, called the theory of time-independent perturbations, is widely used and its applications are numerous in atomic, molecular, nuclear or solid-state physics. An important example we explore is magnetism. Sometimes, a perturbative treatment is not possible and other approaches are needed: the variational method, where we use trial wave-functions to find an approximate value of the ground state of a bound system, and semi-classical approximation, which assumes that the spatial dependence of the interaction affects mainly the phase and slightly the amplitude of the wave-function. This chapter offers the opportunity to physically analyse the validity of these different approaches before solving an eigenvalue equation mathematically (or numerically).

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.