Quantum Mechanics

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

General Relativity

This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the equivalence principle, tidal forces in Newtonian gravity and their connection to curved spacetime geometry. This leads to a derivation of the field equation. Tests of general relativity are considered: Mercury’s perihelion advance, gravitational redshift, the deflection of starlight and gravitational lenses. The exterior and interior Schwarzschild solutions are discussed. Eddington–Finkelstein coordinates are used to describe objects falling into non-rotating black holes. The Kerr metric is used to describe rotating black holes and their astrophysical consequences. Gravitational waves are described and used to explain the orbital decay of binary neutron stars. Their recent detection by LIGO and the beginning of a new era of gravitational wave astronomy is discussed. Finally, the gravitational field equations are derived from the Einstein–Hilbert action.

Nuclear Physics

The chapter gives an overview of nuclear physics from the discovery of the neutron to ongoing research topics. General properties of atomic nuclei are considered: the valley of stability, the nuclear potential, the pairing of nucleons and the strong force. The semi-empirical liquid drop model is presented as a description of relatively large atomic nuclei. The nuclear shell model is described, along with its relationship to magic numbers and beta decay, and is then refined to produce the Nilsson model. Gamow tunnelling is used to explain alpha decay and the Geiger–Nuttall law. It is then applied to nuclear fission and used to calculate rates for thermonuclear fusion in stars. ITER and controlled nuclear fusion are also discussed. Production of superheavy nuclei is detailed and the existence of exotic nuclei, such as halo nuclei, is considered. The Yukawa theory of the strong force is discussed, including its relationship to QCD.

Fields—Maxwell’s Equations

Chapter 3 explores the concept of the field, which is necessary to describe forces without resorting to action at a distance, and uses it to describe electromagnetism, as encapsulated by the Maxwell equations. First, scalar fields and the Klein–Gordon equation are discussed. Vector calculus is introduced. The physical meaning of Maxwell’s equations is explained. The equations are then solved for electrostatic fields. Non-uniform charge distributions and dipole moments are discussed. The vector and scalar potentials are introduced. Electromagnetic wave solutions of Maxwell’s equations are found and the Hertz experiment is described. Magnetostatics is discussed briefly. The Lorentz force is described and used to determine the motion of a charged particle in a cyclotron or synchrotron. The action principle for electromagnetism is described. The energy and momentum carried by the electromagnetic field are calculated. The reaction of a charged particle to its own electromagnetic field is considered.

Fundamental Ideas

Chapter 1 offers a simple introduction to the use of variational principles in physics. This approach to physics plays a key role in the book. The chapter starts with a look at how we might minimize a journey by car, even if this means taking a longer route. Soap films are also discussed. It then turns to geometrical optics and uses Fermat’s principle to explain the reflection and refraction of light. There follows a discussion of the significance of variational principles throughout physics. The chapter also covers some introductory mathematical ideas and techniques that will be used in later chapters. These include the mathematical representation of space and time and the use of vectors; partial differentiation, which is necessary to express all the fundamental equations of physics; and Gaussian integrals, which arise in many physical contexts. These mathematical techniques are illustrated by their application to waves and radioactive decay.

Particle Physics

This chapter offers a brief introduction to quantum field theory and an outline of modern particle physics. The fundamental particles of the Standard Model are introduced. The quantization of fields is described, first the electromagnetic, then the Klein–Gordon and Dirac fields, followed by the prediction and discovery of antimatter. The importance of the action in QFT is outlined, along with its relationship to Feynman diagrams, particle interactions and QED. The path from the strong force to quarks, gluons and QCD is presented. The weak force is discussed, along with the subsequent discoveries of the neutrino and parity violation. The unified electroweak theory is described, including the Higgs mechanism. The discoveries of the W and Z bosons and the Higgs boson are discussed. Quark mixing and the CKM matrix are explained. The experimental determination of the number of generations is discussed. Neutrino oscillation experiments and their theoretical explanation are described.

Atoms, Molecules and Solids

Chapter 9 presents an introductory overview of quantum chemistry and solid state physics. First, the Periodic Table is examined in terms of atomic structure, electron orbitals and the shell model. Simple polar and non-polar molecules are considered in terms of the overlap of atomic orbitals which gives rise to covalent bonding between atoms. Hückel theory is used to analyse the electronic structure of benzene and polyene molecules. These ideas are extended to periodic solids. Bloch’s theorem is used to explain their band structure in terms of molecular orbital theory. Band theory provides an explanation of the distinctions between metals, semi-conductors and insulators. Caesium chloride is used to illustrate how the band structure and properties of an ionic compound arise from its atomic structure. Metals are discussed, with emphasis on copper as an illustrative example, and the significance of the Fermi surface is explained. Ferromagnetism is considered in the transition metals.

Curved Space

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

Motions of Bodies—Newton’s Laws

Chapter 2 covers Newtonian dynamics, Newton’s law of gravitation and the motion of mutually gravitating bodies. The principle of least action is used to provide an alternative approach to Newton’s laws. Motion of several bodies is described. By analogy the same results are used to describe the motion of a single body in three dimensions. The equations of motion are solved for a harmonic oscillator potential. The general central potential is considered. The equations are solved for an attractive inverse square law force and shown to agree with Kepler’s laws of planetary motion. The Michell–Cavendish experiment to determine Newton’s gravitational constant is described. The centre of mass is defined and the motion of composite bodies described. The Kepler 2-body problem is solved and applied to binary stars. The positions of the five Lagrangian points are calculated. Energy conservation in mechanical systems is discussed, and friction and dissipation are considered.

Introduction

We live in a fascinating world, bursting with remarkable phenomena on every scale. Our expanding universe is filled with trillions of galaxies, and a supermassive black hole inhabits the heart of each. Exploding stars seed the galaxies with the dust of life, and eight minutes away a blazing nuclear furnace releases the energy that keeps the Earth green, vibrant and full of life. Our marbled, watery globe may be unique, or one of many where sentient beings have evolved. At a smaller scale, all visible matter is made of just a few fundamental particle types, but these combine into more than a hundred different atoms that bond together in countless ways....