We solve the quantum mechanical harmonic oscillator problem using an operator approach. We define the lowering and raising operators. We use the quantum mechanical harmonic oscillator to review the fundamental ideas of quantum mechanics.We study some examples of time dependence in the harmonic oscillator including the coherent state. We apply the quantum mechanical harmonic oscillator to the study of the vibrations of the nuclei of molecules.

Through the Stern-Gerlach experiment, we demonstrate several key concepts about quantum mechanics: quantum mechanics is probabilistic; spin measurements are quantized; quantum measurements disturb the system. We show how to describe the state of a quantum mechanical system mathematically using a ket, which represents all the information we can know about that state.

We learn about unbound states and find that the energies are no longer quantized. We learn about momentum eigenstates and superposing momentum eigenstates in a wave packet. We apply unbound states to the problem of scattering from potential wells and barriers in one dimension.

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.

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.

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.

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.