Basic Concepts of Quantum Mechanics

In this chapter we give a brief review of some of the basic concepts of quantum mechanics with emphasis on salient points of this theory relevant to the central theme of the book. We focus particularly on the electron density because it is the basis of the theory of atoms in molecules (AIM), which is discussed in Chapter 6. The Pauli exclusion principle is also given special attention in view of its role in the VSEPR and LCP models (Chapters 4 and 5). We first revisit the perhaps most characteristic feature of quantum mechanics, which differentiates it from classical mechanics its probabilistic character. For that purpose we go back to the origins of quantum mechanics, a theory that has its roots in attempts to explain the nature of light and its interactions with atoms and molecules. References to more complete and more advanced treatments of quantum mechanics are given at the end of the chapter. 

The reader is assumed to be familiar with some of the basic concepts of quantum mechanics. At this point we will therefore just briefly consider a few central concepts, including the time-dependent Schrodinger equation for nuclear dynamics. This equation allows us to focus on the nuclear motion associated with a chemical reaction. 

In 1921, Stern and Gerlach performed an experiment that later turned out to be a milestone in quantum mechanics.1,2 First, it provided an experimental basis for the concept of electron spin, introduced in 1925 by Goudsmit and Uhlenbeck.3,4 Second, it evolved into the quantum mechanical experiment par excellence. From this experiment, we easily learn basic concepts of quantum mechanics such as the additivity of probability amplitudes, basis states, projection operators, and the resolution of the identity.5 The latter concept relates to the fact that a complete set of basis states (i.e., the identity) can be inserted in any quantum mechanical equation without changing the result. 

In previous chapters we have considered systems for which there is an exact solution to the Schrodinger wave equation, but as we begin to look at atoms containing more than one electron we shall find that it is impossible to solve the Schrodinger equation exactly, and various approximations will have to be introduced to make the problem solvable. Before these are considered it will be useful to look at some basic concepts of quantum mechanics in more detail, so that we can obtain the Schrodinger equation for any system that may be of interest. 

The approach is rather different from that adopted in most books on quantum chemistry in that the Schrbdinger wave equation is introduced at a fairly late stage, after students have become familiar with the application of de Broglie-type wavefunctions to free particles and particles in a box. Likewise, the Hamiltonian operator and the concept of eigenfunctions and eigenvalues are not introduced until the last two chapters of the book, where approximate solutions to the wave equation for many-electron atoms and molecules are discussed. In this way, students receive a gradual introduction to the basic concepts of quantum mechanics. 

The world of the nuclear spins is a true paradise for theoretical and experimental physicists. It supplies, for example, most simple test systems for demonstrating the basic concepts of quantum mechanics and quantum statistics, and numerous textbook-like examples have emerged. On the other hand, the ease of hattdling nuclear spin systems predestinates them for testing novel experimental concepts. Indeed the universal procedures of coherent spectroscopy have been developed predominantly within nuclear magnetic resonance (NMR and have found widespread application in a variety of other fields. - Richard R. Ernst (Nobel Prize Lecture, 1992)... 

---