If youre going to understand one thing in physics then it should be the harmonic oscillator. It is simple system that underlies nearly everything else that we do. This chapter studies the quantum harmonic oscillator, solving it several times in different ways to highlight different features.

Our discussion in early chapters captures the spirit of quantum mechanics but is restricted to particles moving along a line. Thats not very unrealistic. In this chapter we breathe some life into quantum particles and allow them to roam in three-dimensional space. This entails an understanding of angular momentum. We will pay particular attention to the hydrogen atom, whose quantum solution was one of the first great triumphs of quantum mechanics and still underlies all of atomic physics.

The two body problem is the question of how two objects – say the Sun and the Earth – move under their mutual gravitational attraction. The problem is, happily, fully solvable and the purpose of this chapter is to fully solve it. We will understand how Keplers laws of planetary motion arise from the more fundamental Newtonian law of gravity. Because the electrostatic force has exactly the same form as the force of gravity, we can also use our solutions to understand how electrons scatter off atoms, a famous experiment performed by Rutherford that led to an understanding of the structure of matter.

Drop some ink in a glass of water. It will slowly spread through the whole glass, moving in a manner known as diffusion. This process is so common that it gets its own chapter. We will describe the basics of diffusion, as captured by the heat equation, before understanding how diffusion comes about from an underlying randomness. We will see this through the eyes of the Langevin and Fokker-Planck equations.

Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

Take water and push it through a pipe. If the flow is slow, then everything proceeds in a nice, orderly fashion. But as you force the water to move faster and faster, it starts to wobble. And then those wobbles get bigger until, at some point the flow loses all coherence as it tumbles and turn, tripping over itself in an attempt to push forwards. This is turbulent flow.

Understanding turbulence remains one of the great outstanding questions of classical physics. Why does it occur? How does it occur? How should we characterise such turbulent flows? The purpose of this chapter is to take the first tiny steps towards addressing these questions.

The periodic table is one of the most iconic images in science. All elements are classified in groups, ranging from metals on the left that go bang when you drop them in water through to gases on the right that don’t do very much at all. The purpose of this chapter is to start to look at the periodic table from first principles, to understand the structure and patterns that lie there.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

Any education in theoretical physics begins with the laws of classical mechanics. The basics of the subject were laid down long ago by Galileo and Newton and are enshrined in the famous equation $F=ma$ that we all learn in school. But there is much more to the subject and, in the intervening centuries, the laws of classical mechanics were reformulated to emphasise deeper concepts such as energy, symmetry, and action. This textbook describes these different approaches to classical mechanics, starting with Newton’s laws before turning to subsequent developments such as the Lagrangian and Hamiltonian approaches. The book emphasises Noether’s profound insights into symmetries and conservation laws, as well as Einstein’s vision of spacetime, encapsulated in the theory of special relativity. Classical mechanics is not the last word on theoretical physics. But it is the foundation for all that follows. The purpose of this book is to provide this foundation.

Much of classical mechanics treats particles as infinitesimally small. But most of our world is not like this. Planets and cats and tennis balls are not infinitesimally small, but have an extended size and this can be important for many applications. The purpose of this chapter is to understand how to describe the complicated motion of extended objects as they tumble and turn.

The purpose of this chapter is to understand how quantum particles react to magnetic fields. There are a number of reasons to do be interested in this. First, quantum particles do extraordinary things when subjected to magnetic fields, including forming exotic states of matter known as quantum Hall fluids. But, in addition, magnetic fields bring a number of new conceptual ideas to the table. Among other things, this is where we first start to see the richness that comes from combining quantum mechanics with the gauge fields of electromagnetism.