WHAT TO LOOK FOR

In this chapter we introduce the subject of Lagrangian and Hamiltonian formulations, following descriptions that focus on action functionals and their stationary variation to arrive at the relevant equations of motion. Initially we deal with one degree of freedom, and then with finitely many degrees of freedom. These studies are standard and introductory to the case of an infinite number of degrees of freedom, namely the case of fields. In the classical theory, the infinite number of variables case can be approached as a limit of the finite number of variables cases as the number of those variables tends to infinity. This seems completely reasonable, and, more or less, may be taken as an unwritten rule of how to treat classical systems with an infinite number of variables. Unfortunately, as we shall learn much later, this unwritten rule will not always carry over to the quantum theory, at least in any obvious and straightforward fashion.

Lagrangian classical mechanics

Without undue exaggeration, the goal of classical dynamics may be said to be the introduction of dynamical equations of motion for point particles, or for entities that may be idealized as point-like particles, and the analysis of properties of the solution to such equations, i.e., the time-dependent path x(t) describing the classical motion.

WHAT TO LOOK FOR

Although the classical theory developed in Chapter 2 can be used to describe a great many phenomena in the real world, there is little doubt that the proper description of the world is quantum. To a large measure we perceive the world classically and we must find ways to uncover the underlying quantum theory. The process of “quantization” normally consists of turning a classical theory into a corresponding quantum theory. In turn, the “classical limit” is the process by which a quantum theory is brought to its associated classical theory. The parameter ħ = 1.0545 × 10−27 (≃ 10−27) erg-seconds (cgs units), referred to as Planck's constant, sets the scale of quantum phenomena. In many applications it will prove useful to use “natural” units in which ħ = 1 just so that formulas are less cumbersome. For most typical formulations of quantum mechanics the classical limit means the limit that ħ → 0, and in this text we shall have many occasions to use this definition as well. In addition, we shall also learn to speak about the classical and quantum theories simultaneously coexisting – as they do in the real world – without the need to take the limit ħ → 0.

There are three generally accepted “royal” routes to quantization, due principally to Heisenberg, Schrödinger, and Feynman, respectively, and we shall exploit them all. In each case we begin with a single degree of freedom (N = 1).

WHAT TO LOOK FOR

In its simplest characterization, the quantum theory of scalar fields is nothing but the quantum mechanics of N canonical degrees of freedom in the limit that N → ∞. Of course, things are not quite that simple since sequences do not always have the virtue of converging, and even if they do converge they do not always converge to acceptable limits. However, in this chapter we take a simple, pragmatic point of view and assume that any needed limits converge and, in addition, that the resultant answers are acceptable. Partway through this chapter we introduce units in which ħ = 1.

Classical scalar fields

For purposes of the present section we let g(x) denote a real scalar field defined for x ∈ ℝn, where n ≥ 1 denotes the dimension of space-time. If n = 1 then that single dimension is the time dimension and so there is no space dimension – hence no space – and thus one is really back in the case of classical mechanics. We shall have essentially no occasion to study the case n = 1 in this chapter. For n ≥ 2 the number of space dimensions is s ≡ n − 1 ≥ 1. (We note that the symbol n has several uses in this chapter, often as a dummy index of summation, but the context is generally clear in each case.)

This book is intended as a text for a first-year physical-chemistry or chemical-physics graduate course in quantum mechanics. Emphasis is placed on a rigorous mathematical presentation of the principles of quantum mechanics with applications serving as illustrations of the basic theory. The material is normally covered in the first semester of a two-term sequence and is based on the graduate course that I have taught from time to time at the University of Pennsylvania. The book may also be used for independent study and as a reference throughout and beyond the student's academic program.

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrödinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions.

Chapter 3 is the heart of the book. It presents the postulates of quantum mechanics and the mathematics required for understanding and applying the postulates.

The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences.

Permutations of identical particles

Particles are identical if they cannot be distinguished one from another by any intrinsic property, such as mass, charge, or spin. There does not exist, in fact and in principle, any experimental procedure which can identify any one of the particles. In classical mechanics, even though all particles in the system may have the same intrinsic properties, each may be identified, at least in principle, by its precise trajectory as governed by Newton's laws of motion. This identification is not possible in quantum theory because each particle does not possess a trajectory; instead, the wave function gives the probability density for finding the particle at each point in space. When a particle is found to be in some small region, there is no way of determining either theoretically or experimentally which particle it is.

A molecule is composed of positively charged nuclei surrounded by electrons. The stability of a molecule is due to a balance among the mutual repulsions of nuclear pairs, attractions of nuclear–electron pairs, and repulsions of electron pairs as modified by the interactions of their spins. Both the nuclei and the electrons are in constant motion relative to the center of mass of the molecule. However, the nuclear masses are much greater than the electronic mass and, as a result, the nuclei move much more slowly than the electrons. Thus, the basic molecular structure is a stable framework of nuclei undergoing rotational and vibrational motions surrounded by a cloud of electrons described by the electronic probability density.

In this chapter we present in detail the separation of the nuclear and electronic motions, the nuclear motion within a molecule, and the coupling between nuclear and electronic motion.

Nuclear structure and motion

We consider a molecule with Ω nuclei, each with atomic number Zα and mass Mα(α = 1, 2, …, Ω), and N electrons, each of mass me. We denote by Q the set of all nuclear coordinates and by r the set of all electronic coordinates. The positions of the nuclei and electrons are specified relative to an external set of coordinate axes which are fixed in space.