The majority of all knowledge accumulated in physics and chemistry concerning atoms, molecules, and solids has been derived from applications of group theory to quantum systems.

My (T.W.) first encounter with group theory was as an undergraduate in physics, struggling to understand Wigner's Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (1959). I felt there was something magical about the subject. It was amazing to me that it was possible to analyze a physical system knowing only the symmetry and obtain results that were absolute, independent of any particular model. To me it was a miracle that it was possible to find some exact eigenvectors of a Hamiltonian by simply knowing the geometry of the system or the symmetry of the potential.

Many books devote the initial chapters to deriving abstract theorems before discussing any of the applications of group theory. We have taken a different approach. The first chapter of this book is devoted to finding the molecular vibration eigen-values, eigenvectors, and force constants of a molecule. The theorems required to accomplish this task are introduced as needed and discussed, but the proofs of the theorems are given in the appendices. (In later chapters the theorems needed for the analysis are derived within the discussions.) By means of this applications-oriented approach we are able to immediately give a general picture of how group theory is applied to physical systems. The emphasis is on the process of applying group theory.