We introduce the concept of adding or coupling angular momenta. We introduce the angular momentum ladder operators and learn to transform from the uncoupled basis to the coupled basis. We use these new ideas to study the hyperfine structure of the ground state of hydrogen.

We present a small collection of useful integrals.

How would you model the air in your room?

Quantum mechanics is inherently a probabilistic theory, so we present a brief review of some important concepts in probability theory. We distinguish between discrete probabilities, encountered in spin measurements, and continuous probabilities, encountered in position measurements.

In 1869 Dmitri Mendeleev presented to the Russian Chemical Society a periodic table and a set of laws that laid the foundation for modern chemistry. He showed that the elements could be placed in an order, corresponding loosely but not perfectly to their atomic weights, and this order could be used to classify and predict their properties. He was even able to predict the existence and properties of elements (such as gallium and germanium) that had not yet been discovered.

We study further perturbations of the hydrogen atom due to both external and internal magnetic fields. The internal fields give rise to the fine structure of the hydrogen energy levels. The external fields give rise to the Zeeman effect. We also study internal perturbations due to relativistic effects, which are part of the fine structure.

Einstein’s theories have become part of popular culture. The fact that time passes differently for different observers (“time dilation”) is a staple of science fiction, from Planet of the Apes (1968) to Interstellar (2014).

You’ve been through an entire chapter on special relativity and you haven’t seen the equation $E=m{c}^2$ even once. How weird is that?

Much of this book has focused down to ever smaller scales, from atoms to nuclei to fundamental particles. At the other extreme is cosmology, the study of the overall structure and history of the universe. This chapter will introduce the Big Bang model of cosmology, what it does and doesn’t explain about the history of the universe, and some of the evidence for the model.

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.