We explore the energy eigenvalues and eigenstates of a periodic series of potential energy wells with the purpose of creating a rudimentary model of a solid. Our model uses an approximate approach that emphasizes the interaction between neighboring atoms. We learn how the eigenstates of the periodic potential can be constructed from the eigenstates of the single elements of the periodic potential. We also learn that the eigenstates of a solid are characterized by a wavelength, and that the energies of those eigenstates form bands centered near the atomic energy eigenvalues. We model electron motion in solids with the use of a wave packet, a superposition of delocalized Bloch states.

We introduce the idea of orbital angular momentum and illustrate its importance in solving the three-dimensional differential equation that is the energy eigenvalue equation for the hydrogen atom. By separating variables in the eigenvalue equation, we isolate the differential equations for the angular variables from the differential equation for the radial variable. We solve the angular equations to discover the spherical harmonics and the angular momentum quantum numbers.

We learn how to use perturbation theory to solve more realistic problems that do not admit exact solutions. We learn degenerate and nondegenerate perturbation theory and apply them to a variety of problems, including spin magnetic moments in magnetic fields and the Stark effect in hydrogen.

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.