The one-particle-at-a-time double-slit and many other early twentieth-century experiments convince us that a photon or electron is associated with a “wavefunction.” This function follows the normal mathematics of waves (including constructive and destructive interference), and probabilistically guides the position and other properties of the particle.

We solve the radial differential equation to find the quantized energies and the radial wave functions of the bound states of the hydrogen atom. We present the energy spectrum of hydrogen and the electron probability densities of the energy eigenstates and of superposition states.

Chapter 3 presented a series of experiments that demanded serious changes to classical physics. These experiments pointed to particle-like behavior – quantization – of electromagnetic radiation.

A brief history of smallness. • Seventeenth and eighteenth centuries: Scientists understand that matter is made of particles too small to see, and they devise a host of ingenious experiments to determine the nature and properties of these particles. In hindsight, we would describe much of this work as groping toward a clear distinction between atoms and molecules.

We present a table of useful physical constants and conversion factors.

According to the theory of relativity, Newton’s laws only work for objects traveling much slower than the speed of light. This does not mean that we need one set of laws for fast objects and a different set for slow objects; the equations of relativity work at all speeds. But Einstein’s equations and Newton’s laws make essentially the same predictions as each other for slow objects, and diverge significantly for fast ones. Early twentieth-century physicists were able to measure objects moving close to the speed of light, and such objects followed Einstein’s laws – not Newton’s.

This appendix collects a series of analytical methods that are needed in various parts of the book. All the tools are oriented toward the diagonalization of Hamiltonian, including some cases that allow a complete analytical diagonalization (e.g., coupled resonators) as well as perturbation theory methods for cases where the fully interacting model is too complex for an exact solution.

We introduce the notion of qubit as unit of quantum information, illustrating how this notion can be implemented in nonlinear superconducting circuits via the charge and current degrees of freedom. Within these two types of qubits, we discuss the charge qubit, the transmon, and the flux qubit, illustrating the nature of the states that implement the qubit subspace and how they can be controlled and measured. We discuss how qubits can interact with each other directly or through mediators, illustrating different limits of interaction, introducing the notion of dipolar electric and magnetic moments, and demonstrating the tunability of interactions by different means. The chapter closes with a brief study of qubit coherence along the history of this field, with an outlook to potential near-term improvements.

Almost all superconducting quantum technologies are built using a combination of qubits and microwave resonators. In this chapter, we develop the theory to study coherent qubit–photon interaction in such devices. We start with the equivalent of an atom in free space, studying a qubit in an open waveguide. We develop the spin-boson Hamiltonian, with specific methods to solve its dynamics in the limits of few excitations. Using these tools, we can study how an excited qubit can relax to the ground state, producing a photon, and how a propagating photon can interact with a qubit. We then move to closed environments where the photons are confined in cavities or resonators, developing the theory of cavity-QED. Using this theory, we study the Purcell enhancement of interactions, the Jaynes–Cummings model, Rabi oscillations, and vacuum Rabi splitting. We close the chapter illustrating some limits in which cavities can be used to control and measure qubits.

This chapter studies linear circuits built from capacitors, inductors, and waveguides. It shows how the excitations of these circuits are quantized and can be described as collections of quantum harmonic oscillators. It discusses the quantum states and quantum operations that are accessible by means of these circuits and external microwave drives. We show how to create coherent states, how microwave resonators decay and decohere, how to amplify and measure the quantum state of a resonator, and what states (e.g., Fock states, individual photons) require other, non-Gaussian means to be produced and detected.