This chapter is devoted to a technique for solving the Schrödinger equation for a physical object in the presence of a one-dimensional field of potential energy. The linearity of this equation strongly suggests the use of matrices. The transfer matrix method makes it possible to treat any form of potential energy, to systematise it and to simplify it by avoiding tedious calculations. We study the one-dimensional propagation of a probability wave, describing the potential energy profile as a sequence of steps. We introduce two elementary matrices: the refraction–reflection matrix, which describes the interaction with each potential energy step, and the translation matrix, which describes the wave propagation in a homogeneous medium between two successive steps. An important example is the case of a periodic potential energy, which is particularly useful for the study of solids. The transfer matrix method is of great interest for efficient numerical calculations. This technique is reminiscent of that used in ray optics for a series of spherical dioptres, or in electronics for a series of filters. More generally, this approach can be applied to any linear problem in physics.

The principle of superposition of states applied to a composite system allows for the existence of entangled states, with no equivalence in classical physics. In this chapter we examine this concept, introduced by Schrödinger in 1935 and popularised by the thought experiment of the ’Schrödinger cat’. In the same year, Einstein, Podolsky and Rosen presented an epistemological argument that claimed to reveal the incompleteness of current quantum theory. We examine how this gave rise to a lot of theoretical and experimental work that provided scientific answers to the philosophical debate that had arisen. The result was not only an intellectual breakthrough, but also the opening up of new applications centred on the transmission and processing of information, such as cryptography, teleportation and quantum computing. Since most physical systems do not exist in pure states but in mixtures of states, we show how to adapt the formalism by introducing the statistical (or density) operator. This allows us to analyse a measurement process (i.e., the way of extracting information) on a microscopic physical object, knowing that the measurement apparatus and the observer are of macroscopic size.

In this final chapter we see how special relativity is taken into account in quantum physics. We find significant changes in the atomic energy levels, especially for the innermost electronic orbitals. We show that the fine structure is a relativistic effect, through the relationship between linear momentum and energy, and due to spin–orbit coupling. We give a theoretical framework to unify quantum physics and special relativity with the Schrödinger–Klein–Gordon equation for spinless physical objects. A second approach is proposed using the Dirac equation for electrons. It is based on the introduction of a new Hamiltonian in which the spatial derivatives are of the first order, like those with respect to time. However, due to some difficulties and shortcomings, Dirac’s theory has been replaced by a new construction that we now call quantum electrodynamics and which is an example of what we more generally call a quantum field theory. We give some elements of quantum field theory that we think are the most important for a physicist. In particular, we discuss the Feynman variational principle of quantum physics and we explain why it provides a unification of quantum and classical physics.

This chapter introduces seminal quantum algorithms that illustrate quantum computation’s efficiency over classical methods. The Deutsch and Deutsch–Jozsa algorithms showcase quantum parallelism, offering solutions to specific problems with fewer computational steps. The quantum Fourier transform (QFT) is introduced, underpinning period-finding algorithms as well as Shor’s algorithm for integer factorization, which has major implications for cryptography. Grover’s algorithm demonstrates a quadratic speedup for unstructured search problems. By using superposition, entanglement, and phase manipulation, these algorithms highlight the computational power of quantum mechanics and its potential to outperform classical techniques, particularly for complex or classically intractable tasks.

This chapter argues that even though we all have a pretty good idea of what is meant by the term ‘social class’, it is far from being a straightforward matter. After all, there is only tenuous agreement about exactly what it is, how prevalent it is, how it organises the life opportunities of our citizens and how best to study it. To make it more difficult still, this is a subject that many feel uncomfortable discussing, let alone applying to themselves or anyone else.

This chapter serves as an introduction, a synthesis and a reference point to which the reader can always return when working with the other chapters of the book. In order to situate quantum physics, we recall the foundations of classical physics and explain the criterion of ’quantumness’ – when to use quantum physics or just classical physics as an approximation. A list of the most important fundamental constants is given. This chapter is useful to show that physics is a process of unification and simplification. An overview of the history of quantum physics is also provided, where we give the main stages of its development. Finally, we summarise the foundations of quantum physics, the novelty and enrichment it represents in comparison with classical physics, and, in a final section, we mention some important applications.

It is likely that you have experienced the impact of place on your education without even thinking about it. Maybe you’ve had a class on a boiling hot day, with bad lighting and no aircon. Maybe you’ve had to sit in traffic on the way to class, and thought ‘Wow, I wish I didn’t have to be at school by 8 am!’. Maybe you’ve accessed your education online, and felt the differences (good and bad), between in-person and online learning. Or perhaps you’ve sat under a lovely tree after class and chatted with your friends. Maybe you’ve experienced traditional ways of learning on Country, and connectedness to the environment around you. Whatever it may be, you get the drift – if you’ve had an education, it’s happened somewhere.

Beginning with the eerie history of Edinburgh’s South Bridge vaults, Chapter 3 investigates how supernatural encounters are often reported in places associated with death, decay, and sensory uncertainty. Here, we explore the connection between electromagnetic fluctuations, ambiguous sensory experiences, and supernatural perceptions. The chapter explores the human tendency to assign meaning to ambiguous stimuli and introduces key concepts in measurement science, such as reliability and validity. It also addresses the limited evidence for human sensitivity to EMF changes. Disruptions in spatial and body awareness in the brain can lead to experiences like feeling a presence or seeing a shadow figure. Together, these ideas offer plausible brain-based explanations for some ghostly encounters and demonstrate how the brain strives to make sense of the unknown when sensory information is unclear.

Psychedelic substances like ayahuasca, psilocybin, and LSD have been used for thousands of years in spiritual ceremonies, with users often reporting transcendent and life-altering experiences. Chapter 8 traces the arc of psychedelic use from ancient rituals and colonization to the countercultural revolution and modern neuroscience labs. The chapter blends cultural history with psychopharmacology, showing how these compounds mimic serotonin and interact with the brain’s serotonin receptors to create altered states of consciousness. Citing research from neuroscience and psychology, the chapter considers how psychedelics affect the default mode network, ego, and self-referential processing. These effects can lead to feelings of oneness, ego death, and even reductions in depression and anxiety. The chapter asks whether the mystical states brought on by psychedelics are supernatural or simply deeply meaningful expressions of altered neural processing. Regardless, their potential therapeutic value, especially when guided in proper set and setting, positions psychedelics at the intersection of brain, mind, and meaning.

It is argued here that the modern school isn’t just about ‘education’ in some abstract, humanist sort of way; rather, schools have an essential role to play in how we govern our society. It is tempting to think that the process of teaching children has always been pretty much the same, and that mass schooling emerged as a result of greater concern for the wellbeing of the young. The evidence paints a somewhat different picture, wherein mass schooling formed a crucial component of a new form of social regulation based upon an increasing focus on individuality, where the school subtly conforms to the requirements of the state and where the disciplinary management of the population is made possible through continual surveillance and the close regulation of space, time and conduct.

A system can have two types of angular momentum, orbital and spin: an electron in an atom, a proton in a nucleus or a photon in a light beam. There are many other situations in which it is possible to combine two angular momenta, such as two spins or two orbital momenta. A peculiarity relates to quantisation, which has two linked features: one for the square of the angular momentum and the other for a component. This chapter explains what ’combining two angular momenta’ means, how to do it, how to determine the state vectors of a system in terms of the total angular momentum and how the quantisation of angular momenta is affected by the combination. This chapter is rather technical but has many applications since it concerns atomic and molecular, nuclear or optical physics – isolated objects that together form a composite system. We proceed by understanding the meaning of the word ’combination’, looking at the archetypal system of two spin 1/2s and generalising to any system. Some basic considerations are given about the combination of more than two angular momenta.

Symmetry means that there is no change under a transformation (or operation). It can refer to a system – a state or a physical quantity – or to a physical law. The importance of symmetries in physics is that they are most often equivalent to a law of conservation. In this chapter we first consider symmetries related to spacetime, which are geometrical symmetries. These are translations, rotations, inversions of space and time. A second type are internal (or dynamical) symmetries, because they concern the dynamics of the fundamental interaction and/or inherent properties of systems: charge conjugation C, parity P and the time reversal T. We analyse another important symmetry, gauge invariance in electromagnetism. We also distinguish symmetries with a finite number of transformations from continuous symmetries.

In this chapter we consider stationary scattering in three dimensions (interactions that are not explicitly time dependent), and in particular the important case of spherically symmetric interactions whose strength decreases sufficiently rapidly with distance. In this context, we present the partial wave approach and we study resonances (interaction enhancement), which are certainly the most spectacular manifestation of scattering. We limit ourselves to a non-relativistic treatment and to single scattering (a physical object interacts only once with a target). First of all, we propose a unified definition of scattering in terms of information and recall the concept of cross section. We also present the optical model and discuss the treatment of inelastic scattering.

Psychics, mediums, and fortune tellers may seem to possess supernatural insight, but many of their most impressive feats can be explained by the brain’s natural tendencies toward pattern recognition, suggestion, and belief confirmation. This chapter explores the psychological mechanisms behind psychic predictions, including confirmation bias, selective attention, and the Barnum effect. It also examines how experimenter bias and subtle behavioral cues can shape perceived psychic accuracy—even when no one is intentionally deceiving anyone. Using demonstrations from visual neuroscience, the chapter reveals how much information the brain fills in without our awareness. Alongside compelling case studies and historical context, readers are invited to consider how intuition, belief, and cognitive shortcuts can converge to create compelling—yet illusory—experiences. Whether or not psychic powers exist, the feeling of being seen or understood can be profoundly real, and this chapter examines how those feelings might arise from within.

This chapter first recalls important results on the energy and momentum of electromagnetic waves, on superposition and interference, and on first-order coherence. It then presents a synthesis of how quantisation occurs in the exchange of energy and linear momentum. In 1901, Planck proposed an interpretation of the continuous radiation emitted by condensed objects in thermal equilibrium. Four years later, Einstein proposed an explanation of light–matter interactions in terms of ’light’ quanta (i.e., quantum of light energy), explaining in particular the photoelectric effect – the extraction of an electron from a metal irradiated with light. Both contributions assumed a quantised exchange of energy between light and matter. We explain how it is a radical departure from the classical thinking that considers light energy and its variation during the exchange with matter as continuous. The light quanta are then endowed with a linear momentum as was first done theoretically by Einstein in 1917 and then experimentally by Compton in 1923. We focus on the main processes and introduce basic but necessary information about photodetectors.