5 - Systems with a finite number of levels

Summary

In this chapter we examine some simple applications of quantum mechanics in situations where it is possible to model quantum systems accurately by restricting ourselves to a space of states of finite dimension. If each energy level, including degenerate ones, is counted once, the dimension of ℌ is equal to the number of levels, and this is why we use the term system with a finite number of levels. The first two examples (Section 5.1) are taken from quantum chemistry and allow us to study a stationary situation where the Hamiltonian is time-independent. But the most important point in this chapter is the introduction of time dependence, which will be implemented by coupling a two-level system to an external periodic classical field. This will be illustrated by three examples of great practical importance: nuclear magnetic resonance (Section 5.2), the ammonia molecule (Section 5.3), and the two-level atom (Section 5.4).

Elementary quantum chemistry

The ethylene molecule

The ethylene molecule C₂H₄ will serve as an introduction to the subject. The “skeleton” of this molecule is formed by the so-called σ bonds, pairs of σ electrons of opposite spin common to two carbon atoms or to a carbon and a hydrogen atom, thus forming the (C₂H₄)⁺⁺ ion (Fig. 5.1). The remaining two electrons, called π electrons, are mobile – they can jump from one carbon atom to another. It is said that they are delocalized. The separate treatment of the π and σ electrons is, of course, an approximation, but one that plays an important role in the theory of chemical bonding. Let us begin by putting the first π electron in place.