Quantum physics theory is based on a set of fundamental principles suggested by experimental facts, and an associated formalism to describe the physical world. In this sense, states belong to an appropriate mathematical (Hilbert) space; physical quantities that can be measured (observables) are represented by Hermitian operators; and a law of evolution in spacetime (a dynamics) is postulated as the Schrödinger equation. This chapter is organised in this order. The last section describes composite systems, those defined not by a single physical object but by several, and how to work with them. There we give some basic but important elements of tensor calculus.

This chapter covers quantum error correction, essential for preserving quantum information in the presence of noise. It introduces the bit-flip and phase-flip codes as foundational error-correction methods, building toward Shor’s code, which corrects general single-qubit errors. Logical qubits are formed by encoding physical qubits to maintain stability. Stabilizer codes are presented as a systematic framework for error correction, enabling fault-tolerant quantum computing. These principles are crucial for creating scalable quantum systems that can perform reliable computations, even in noisy environments, addressing a central challenge in quantum computing’s practical implementation.

This is the fifth edition of Making Sense of Mass Education. It offers a nuanced discussion of emerging problems in an ever-changing world. Changes to the field of education have not slowed since the publication of the fourth edition. Of course, this edition offers an updated contemporary assessment of all the topics addressed in the book, but it also provides an extensive discussion of the important and rapidly changing areas that impact mass education and the professional lives of teachers.

Alien abduction reports often follow a strikingly familiar pattern: lost time, immobilization, floating, bright lights, and invasive procedures. These memories are emotionally intense and vividly detailed—even when the events themselves can’t be verified. This chapter explores how neuroscience might explain why such experiences feel real, even when they may not reflect objective reality. Topics include memory formation and reconsolidation, the vulnerability of memory to suggestion, and the ways cultural narratives can shape the content of extraordinary experiences. It also touches on hypnosis, dissociation, and why some individuals may be more prone to magical thinking or altered states of consciousness. Through this lens, alien encounters are reframed as meaningful phenomena rooted in the brain’s powerful (and sometimes flawed) storytelling machinery—offering insight into how belief systems form around experiences that defy conventional explanation.

In this chapter we show that angular momentum quantisation is a general property of any rotating physical system, whether electrically charged or not, massless or not, of microscopic dimensions (e.g., diatomic molecules) or macroscopic size (e.g., superfluids or superconductors). Therefore, we return later to the magnetic properties of electrically charged objects. We first analyse the simple case of a rigid rotor plane (a one-dimensional angular system) to show how the quantisation of the angular momentum, and thus of the rotational energy, follows from the periodicity condition. The generalisation to three-dimensional systems offers the possibility to search for common eigenstates of the modulus and only one component of the angular momentum using ladder operators. The first applications concern molecular rotations and their associated spectroscopies. We then show how quantum superfluids in rotation reveal the quantisation of the circulation through quantum vortices, and how superconductors similarly reveal the quantisation of the magnetic flux. Finally, the quantisation of the angular momentum of light and the associated light vortices are also presented.

This chapter generalises the study of the free and bound states to multiple space variables. Quantum confinement in several dimensions is used in many applications of solid state or atomic physics, especially in nanophysics. The search for stationary states is simple if the different spatial coordinates are separable. We therefore consider the confinement of a set of identical, indistinguishable physical objects in a finite volume. We assume that their mutual interactions are negligible and that we are therefore dealing with an ideal gas. We study the quantum behaviour of an object, focusing on the behaviour of the ensemble, using quantum statistics for fermions or bosons, and classical statistics when the classical limit is reached. As examples we explore a Fermi gas of electrons, a condensate of bosonic atoms, the quantum Hall effect, the spectacular properties of graphene, the quantisation of conductance and the discretisation of energy levels in a quantum dot. We also reconsider the Planck law as a three-dimensional confined photon gas. At the end of the chapter, we extend the analysis of scattering effects (reflection, transmission) to two dimensions.

This chapter deals with scattering states for a physical system in one dimension. Specifically, tunnelling (or tunnel effect) and scattering, and especially resonant scattering (or the Ramsauer–Townsend effect), occur when an object interacts with a potential energy barrier or well of finite thickness. The first effect affects any incident object of energy E less than the energy height of the barrier (E < U0). While this barrier is impassable in classical physics, we see from experiment that the object can be detected on the other side of the barrier. This effect occurs when E > U0: it has specific features that can only be explained by a quantum approach. These effects are taken into account by solving the one-dimensional time-independent Schrödinger equation. The subject is also concrete, since tunnelling occurs in a large number of important phenomena (e.g., alpha decay, chemical bonding between atoms, inversion of the ammonia molecule, field emission, the Josephson effect) and is the origin of many technological applications: tunnelling microscopes, electronic devices for computer memories and, more recently, single-electron tunnelling devices used for a new intensity standard.

We analyse Einstein’s introduction of the processes of light–atom interactions to explain Planck’s law of thermal emission and, in particular, the mechanism of stimulated (or induced) emission. We define the statistical Einstein coefficients he introduced to describe these processes and analyse their relative importance. We then discuss how to amplify an electromagnetic wave by inversion of atomic populations, and thus how masers and various lasers work, and look at laser cooling and extremely stable standards for frequency (time) measurements.

This chapter explores classical computation fundamentals, starting with Turing machines as a foundation for defining computability. The universal Turing machine is introduced, emphasizing the theoretical basis for all computable functions. Computational complexity is discussed, differentiating between tractable and intractable problems and explaining complexity classes as a framework for problem-solving. The chapter also covers the circuit model, providing a bridge between theoretical constructs and modern computer architecture. Finally, the concept of reversible computation is introduced, which has implications for energy-efficient processing. Through these topics, the chapter delineates classical computation’s limitations, setting up the motivation to transition into quantum approaches in subsequent chapters.

Sleep paralysis is one of the most terrifying experiences a person can have—and it’s surprisingly common. Cultures around the world describe eerily similar episodes: waking up unable to move, a crushing pressure on the chest, and the overwhelming sense that someone—or something—is in the room. This chapter explores how those experiences may arise from the collision of sleep architecture and perceptual ambiguity. It covers the basic neurobiology of REM sleep, explains what happens when paralysis persists into wakefulness, and investigates how hallucinations can emerge in these liminal states. The chapter also examines the role of the temporoparietal junction in out-of-body experiences and the sensation of a nearby presence. Rooted in both science and cultural context, this chapter offers a grounded explanation for a deeply human phenomenon—one that’s haunted people for centuries and continues to blur the line between brain and belief.

If we’re trying to come up with a theory to explain the sound of footsteps behind you, a feeling of a presence, lights that you can’t explain, or the psychic who knows everything about you, we might be tempted to say that supernatural forces are at work. But we also know that each one of these instances can be easily explained with neuroscience and psychology. This is what I’ve attempted to do in this book.

In 1900, long before Schrödinger’s work, Planck introduced the ad hoc hypothesis of energy exchange by quanta between radiation and matter, modelling it as a set of harmonic oscillators. In this chapter we show that this quantisation hypothesis follows naturally from the Schrödinger equation for a quadratic potential energy. The oscillator is of great importance because it concerns the vibrations of nuclei, atoms, molecules and solids, and gives a description of the electromagnetic field in vacuum. Before discussing the quantum aspects, we recall properties of a harmonic oscillator in classical physics to show just how different its quantum behaviour is; this can only be found under the ’classical approximation’. A very effective way of finding solutions to the Schrödinger equation is to introduce ’ladder’ operators. Their physical interpretation as the creation and annihilation of vibrational energy quanta opens a new approach (the second quantisation) to consider energy eigenvalues and eigenstates. As applications, we focus on far-infrared and Raman spectroscopy, which are important techniques for identifying molecules.

The Science of the Supernatural might, at first, feel like an oxymoron. I don’t think most people would immediately see the myriad connections between the paranormal and psychology. I didn’t at first, either. I’ve always loved ghost stories, horror movies, and scary novels. I have a distinct memory of lying in my bed as a kid, trying unsuccessfully to go to sleep. I had just read Stephen King’s short story “The Boogeyman.” I remember staring at my closet door, sure that it was slowly creaking open. Certain that the boogeyman was on the other side, waiting to kill me.