The separation of variables procedure permits us to simplify a partial differential equation by separating out the dependence on the different independent variables and creating multiple ordinary differential equations. To illustrate the method, we apply a six-step process to the classical wave equation to show how the time dependence of the wave function can be found through a separate ordinary differential equation.

Welearn the key aspect of quantum mechanics – how to predict the future with Schrödinger’s equation. We learn the general recipe for solving time-dependent problems by diagonalizing the Hamiltonian to find the energy eigenvalues and eigenvectors.

You probably learned in school that matter comes in three phases: solid, liquid, and gas. (A fourth phase called “plasma” only tends to occur in extreme environments like the center of the Sun or physics laboratories, so your teachers can be forgiven if they left it out.) Gases can flow and conform their shapes to their containers, and can also compress or expand; liquids can also flow and conform shape, but they cannot compress or expand; solids can’t really flow, conform, compress, or expand.

We present some of the basic definitions and properties of matrices necessary to implement the matrix formulation of quantum mechanics.

We learn time-dependent perturbation theory, where we focus on finding the probability that an applied perturbation causes a transition between energy levels of the unperturbed Hamiltonian. We calculate the probability amplitude for a transition from an initial state to a final state subject to a time-dependent perturbation. We learn that an excited state in an atom has a finite lifetime due to spontaneous emission. We learn that electric dipole transitions obey selection rules.

The story of atoms so far, in three parts: 1. 1911: Rutherford describes an atom as being made of small negatively charged electrons orbiting a large positively charged nucleus, all very analogous to planets orbiting the Sun. 2. 1913: Bohr addresses both of those problems by proposing that the angular momentum of an orbiting electron can only take on certain discrete values, and can jump discontinuously between those values. Like Planck’s resolution of the ultraviolet catastrophe and Einstein’s explanation of the photoelectric effect, this fits the data but does not provide any fundamental principles. 3. 1926: Schrödinger publishes his wave equation. Eventually, all the ad hoc hypotheses of the old quantum theory are seen to be consequences of Schrödinger’s wave mechanics.

We present a few of the gedanken- and real experiments that demonstrate the spookiness of quantum mechanics. We discuss the Einstein,Podolsky, and Rosengedankenexperiment that invokes hidden variables to create a paradox. We analyze Bell’s analysis of the paradox, which shows that the predictions of quantum mechanics are inconsistent with local hidden variable theories. We discuss the Schrödinger cat paradox, and the Copenhagen interpretation of quantum mechanics.

Complex numbers are a critical component of the mathematics of quantum mechanics, so we provide a brief review. Topics include imaginary numbers, Euler’s formula, modulus, phase, and complex conjugate.

For a proper quantum mechanical description of multiple-particle systems, we must account for the indistinguishability of fundamental particles. The symmetrization postulate requires that the quantum state vector of a system of identical particles be either symmetric or antisymmetric with respect to exchange of any pair of identical particles within the system. Nature dictates that integer spin particles – bosons – have symmetric states, while half-integer spin particles – fermions – have antisymmetric states. The best-known manifestation of this is the Pauli exclusion principle, which limits the number of electrons in given atomic levels and leads to the structure of the periodic table.

Chapter 8 talked about atoms in isolation. But most of the atoms around you are joined together, a fact that can dramatically change their physical and chemical properties. In this chapter we explore atoms that are joined together in molecules. Chapter 11 will describe solids, large collections of atoms or molecules bound into a macroscopic size.

We spend the last chapter using the learned quantum mechanical tool set to examine two current research topics that are extensions of some of the examples of quantum mechanics studied in the text. We examine quantum mechanical forces on atoms and quantum information processing, which both have important connections to Stern-Gerlach spin-1/2 experiments and to resonant atom-light interactions

We extend the mathematical description of quantum mechanics by using operators to represent physical observables. The only possible results of measurements are the eigenvalues of operators. The eigenvectors of the operator are the basis states corresponding to each possible eigenvalue. We find the eigenvalues and eigenvectors by diagonalizing the matrix representing the operator, which allows us to predict the results of measurements. We characterize quantum mechanical measurements of an observable A by the expectation value and the uncertainty. We quantify the disturbance that measurement inflicts on quantum systems through the quantum mechanical uncertainty principle. We also introduce the projection postulate, which states how the quantum state vector is changed after a measurement.

We learn the language of the wave function, which is the representation of the quantum state vector in position space. We introduce the position and momentum operators and learn the rules for translating bra-ket formulae to wave function formulae. We use these new tools to solve the infinite square potential energy well problem and the finite square well problem.

We discuss the basic concepts of waves, including phase velocity, dispersion, group velocity. We show how to use the Fourier principle to construct any general wave from the harmonic waves.