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This chapter replies to claims that the pilot-wave theory of de Broglie and Bohm is really a many-worlds theory with a superfluous configuration appended to one of the worlds. Assuming that pilot-wave theory does contain an ontological pilot wave (a complex-valued field in configuration space), the chapter shows that such claims arise from not interpreting pilot-wave theory on its own terms. Specifically, the theory has its own (‘subquantum’) theory of measurement, and in general describes a ‘non-equilibrium’ state that violates the Born rule. Furthermore, in realistic models of the classical limit, one does not obtain localised pieces of an ontological pilot wave following alternative macroscopic trajectories: from a de Broglie–Bohm viewpoint, alternative trajectories are merely mathematical and not ontological. Thus, from the perspective of pilot-wave theory itself, many worlds is an illusion. It is further argued that, even leaving pilot-wave theory aside, the theory of many worlds is rooted in the intrinsically unlikely assumption that quantum measurements should be modelled on classical measurements, and is therefore unlikely to be true.

An unhappy complaint by celebrated Irish physicist John Stuart Bell, who challenged an unchecked quantum orthodoxy, opens Chapter 2. At first his quote seems little more than a disgruntled student blowing off steam. Closer examination reveals much higher stakes. This chapter probes Bell’s frustrations toward his physics training at Queen’s University Belfast in the late 1940s. He complained bitterly about an entrenched quantum orthodoxy supported by canonical narratives that took hold in the early 1930s and continued to dominate the field for decades. The orthodox quantum interpretation eventually became synonymous with the city of Copenhagen and was used widely in the international physics community to filter out unwanted alternate interpretations, shut down interpretational debate, and promote a pragmatically productive culture of scientific consensus.

This chapter presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).

Whatever was going to replace classical physics in the description of radiation and atomic phenomena had to confront the difficult task of somehow reconciling the wave-like and particle-like aspects of light in a single structure. An important clue would come from Einstein’s E=mc2. We tend to want to associate mass (and linear momentum) with material particles. But the Planck–Einstein relation E=hν connects energy with frequency, a determinedly wave-like property. So, here are two very simple yet fundamental equations connecting energy to mass and energy to frequency. Can they be combined? Louis de Broglie thought so, and in 1923 he generalized the discovery made by Einstein in 1905 by extending it to all material particles, and most notably to electrons. The de Broglie relation λ=h/p can be derived quite straightforwardly by comparing the Lorentz transformations for energy and momentum with those for a system of plane waves.

In 1924, de Broglie postulated that particles can behave like waves, thus complementing the observation by Einstein in 1905 that light can behave like particles. This wave–particle duality aspect for both particles and waves had a deep impact on the subsequent development of quantum mechanics. Some highly counterintuitive results, like the Heisenberg uncertainty relation and the Bose–Einstein condensation, that were motivated by wave–particle duality are discussed in this chapter. Following de Broglie’s hypothesis, a wave packet description for a particle is described. An analysis of the Heisenberg microscope is presented, thus motivating the Heisenberg uncertainty relation. The Davisson–Germer experiment that showed that electrons can behave like waves and the Compton effect that provided early conclusive evidence that light can behave like particles are also discussed.

Alongside these social and political readings of the period from the fourth to the seventh century there was also a religious one. The religious reaction to the Ancien Régime and the French Revolution was spearheaded by Chateaubriand, whose novel about the Christian persecutions of the early fourth century Les Martyrs was a source of inspiration for a subsequent generation of writers, about all Ozanam, in a number of works on the early Middle Ages, and Montalembert, in his study of monasticism. Following on from the work of Ozanam, the Prince de Broglie wrote a history of the fourth century which prompted major debates with Guéranger, one of the key figures in the Benedictine revival. The Church of the Age of Persecution was also picked up in England in debates between liberal Anglicans like Kingsley and members of the Oxford Movement, like Newman.

This chapter relates the first steps of X-ray crystallography in 1913. It starts with a brief account of the flock of experiments in England and Germany, prompted by Bragg’s experiment with mica. It then tells how Wulff, Laue himself, Ewald, and Friedel interpreted Laue’s relations in terms of the reflection by a set of lattice planes. Laue admitted his early misconceptions and gave a correct interpretation of Friedrich and Knipping’s experiment. The chapter proceeds on with Terada’s and Nishikawa’s early experiments in Japan, and de Broglie’s experiments in France. In England, W. H. Bragg developed the ionization spectrometer, and W. L. Bragg made the first crystal structure determinations. Moseley determined the high-frequency spectra of the elements and established its relations with the atomic numbers. In Zürich, Debye derived the influence of thermal agitation on the intensity of diffracted intensities. In France, de Broglie introduced the rotating crystal method, and Friedel related X-ray diffraction and crystal symmetry. In the United States, the first X-ray spectrometer was built in 1914.

This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.

This chapter presents the properties of thermal neutrons. Their wavelength (from the de Broglie equation) is well suited to the investigation of condensed matter, i.e., to the study of liquids, glasses (amorphous materials), and crystalline materials with varying degrees of order. That the neutrons carry magnetic moment also makes them well suited to the study of magnetic ordering. The theory of nuclear and magnetic scattering from individual atoms and from assemblies of atoms is presented, this leading to the definition of neutron scattering length and to the concepts of coherent and incoherent scattering. The focus then shifts to the direction and intensity of diffraction from crystalline materials (Bragg's law, structure factors), and to the description of this scattering when samples are presented in polycrystalline or powder form (Debye-Scherrer cones).

A common understanding of both the many-worlds theory and the original version of the GRW theory holds that they are ontologically monistic, postulating only the existence of the wavefunction and nothing else. Sometimes an appeal to Occam's razor is used to promote this as a boon for these theories. But without more detailed argumentation, it is hard to see how such an austere ontology can make comprehensible contact with the experimental facts that inspired the development of quantum theory in the first place, or indeed with our whole pre-theoretical picture of the physical world.

Experiments on Bell’s theorem, or Bell’s inequality, were carried out by John Clauser and Stuart Freeman, whose results agreed with quantum theory, and by Richard Holt and Frank Pipkin, whose results disagreed with quantum theory but agreed with local causality. However, it came to be accepted that quantum theory was right: local causality was not obeyed. The locality loophole, the detector loophole, and the freedom of choice loophole were introduced. Bell encouraged the plans of Alain Aspect, who wanted to perform experiments where information about the directions of the polarizer in one wing of the experiment would not have time to reach the other wing. Bell refined his ideas on quantum theory, introducing the ‘beable’, in contrast to the usual ‘observable’, and commenting on many-worlds theories and the de Broglie–Bohm interpretation. He made important contributions to elementary particle physics through his studies of shadow scattering, and models of quarks.

It was discovered in the early decades of the twentieth century that light and matter exhibit both wave-like and particle-like behaviour and this led to the development of quantum theory. Thomas Young demonstrated the wave-like behaviour of light 200 years ago. Thomas Young had wide-ranging interests and played an important role in deciphering hieroglyphic inscriptions. In 1905 Einstein showed that light also behaves like a stream of particles. Louis De Broglie suggested that particles such as electrons might also show wave-like behaviour and his prediction was soon verified in the laboratory. This led to the development of quantum mechanics, as well as important technological applications, such as the electron microscope.

Electromagnetic optical phenomena are generally understood via Maxwell’s equations for the electric and magnetic vector fields. Neutron optical phenomena are based upon de Broglie matter waves and the Schrödinger wave equation which involves a scalar wave function. Nevertheless, a description of interference effects requires the solution of a Helmholtz scalar wave equation in both cases. Gravitational, inertial, and motional effects lead to rather large phase shifts in neutron interferometry. This chapter describes how the quantum phase shift is calculated, usually by evaluating the action as an integral over the Lagrangian. A connection to relativity theory is given when the Compton frequency depending on the mass of the neutron is taken as a physical quantity. It concludes with a list of the approximately 40 neutron interferometry experiments carried out in various laboratories worldwide, which form the subject of this book.

This chapter reviews the role of time as a parameter in quantum mechanics, rather than a dynamical variable. To emphasize this role, the chapter discusses the relationship between the Schrödinger picture and the Heisenberg picture in quantum mechanics. The chapter discusses the issues with the superluminal speed of de Broglie waves in quantum wave mechanics. A review of quantum uncertainty relations is given followed by an application to time and energy. The chapter discusses Pauli’s theorem and comment on the need to interpret the time–energy uncertainty relation contextually, taking into account the fact that time is not a dynamical variable. The chapter finishes with a discussion of a potential clash between relativity and quantum mechanics when the quantum propagator for a relativistic point particle is calculated via a Feynman path integral.

This chapter discusses the electron wavelength, relativistic effects, ray diagram for a typical electron microscope, simple electron optics theory and approximations for ideal lenses. The constant field approximation, projector lenses, objective lenses, lens design, aberrations and the pre-field are also covered.

The concept of wavefunction was introduced in the first 1926 paper by Erwin Schrödinger as the central object of the atomic world and the cornerstone of quantum mechanics. It is a mathematical representation of de Broglie’s postulate that the electron is a material wave. It was defined as everywhere real, single-valued, finite, and continuously differentiable up to the second order. Nevertheless, for many decades, wavefunction has not been characterized as an observable. First, it is too small. The typical size is a small fraction of a nanometer. Second, it is too fragile. The typical bonding energy of a wavefunction is a few electron volts. The advancement of STM and AFM has made wavefunctions observable. The accuracy of position measurement is in picometers. Both STM and AFM measurements are non-destructive, which leaves the wavefunctions under observation undisturbed. Finally, the meaning of direct experimental7 observation and mapping of wavefunctions is discussed.

The arrangement of electrons around the nucleus of an atom is known as its electronic structure. Since electrons determine all the chemical and most physical properties of an atomic system, it is important to understand the electronic structure. Much of our understanding has come from spectroscopy, the analysis of the light absorbed or emitted by a substance. Electromagnetic radiation is a form of energy; light is the most familiar type of electromagnetic radiation. But radio waves, microwaves, X-rays, and many other similar phenomena are also types of electromagnetic radiation. All these exhibit wavelike properties, and all travel through a vacuum at the speed of light. The wavelike propagation of electromagnetic radiation can be described by its frequency (ν), wavelength (λ), and speed (c). Wavelength (lambda, λ): The wavelength of a wave is the distance between two successive peaks or troughs. Frequency (nu, ν): The frequency of a wave is the number of waves (or cycles) that pass a given point in space in one second. The unit is expressed as the reciprocal of seconds (s−1) or as hertz (Hz). A hertz is one cycle per second (1 Hz = 1 s−1). Speed of light (c): The speed of light in a vacuum is one of the fundamental constants of nature, and does not vary with the wavelength. It has a numerical value of 2.9979 × 108 m/s, but for convenience we use 3.0 × 108 m/s. These measurements are related by the equation: Speed of light =Wavelength×Frequency c = λν This expression can be rearranged to give: λ = c/v, or ν = c/λ Wave number (⊽): The wave number is a characteristic of a wave that is proportional to energy. It is defined as the number of wavelengths per unit of length (usually in centimeter, cm).Wave number may be expressed as ⊽ =1/λ While electromagnetic radiation behaves like a wave, with characteristic frequency and wavelength, experiment has shown that electromagnetic radiation also behaves as a continuous stream of particles or energy packets.

The value of recognising the status of ‘information’ as an abstract noun is illustrated by way of application to the analysis of quantum teleportation. It is argued that when one notes that ‘the information’ does not refer to a concrete particular or a sort of pseudo-substance, any puzzles thought to surround the process are quickly dispelled. The central moral is that one should not be seeking, in an information-theoretic protocol, for some particular ‘the information’, whose path one is to follow, but rather concentrating on the physical processes by which the end result of the protocol is brought about. When this is borne in mind for teleportation, it is seen that the only remaining source for dispute over the protocol is the straightforward one regarding what interpretation of quantum mechanics one wishes to adopt. How the teleportation protocol looks within a number of familiar interpretations is then described.

In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.