With the exception of the latter half of chapter 6 we have, up to this point, been discussing single-particle quantum mechanics. This is easy (although you may not think so). One can certainly gain a lot of insight and understanding from a single-particle theory. However, in real life there are very few (no) situations in physics in which a single-particle theory is able to paint the whole picture. Any real physical process involves the interaction of many particles. In fact even that is a vast simplification. Really any physical process involves the interaction of all particles. Even the gravitational attraction due to an electron at the other end of the universe is felt by an electron on earth.

To describe all particles in a calculation is, of course, absurd. You would have to include the particles of the paper you are writing the calculation down on and the particles in your brain thinking about the many-body problem. However, many-body theory on a more limited scale is feasible. In this chapter we discuss two ways of going beyond the one-electron approximation. Actually, we are not going very far beyond the one-electron approximation and you will see what is meant by that soon.

This chapter is essentially divided into two halves.

I believe that physicists gain much of their physics intuition from solving simple model problems explicitly. Apart from the hydrogen atom, this is something that is not often done in relativistic quantum theory books (except the book by Greiner (1990)). In this chapter we set up and solve five simple models that have exact analytical solutions. Furthermore, they are all related to well-known non-relativistic counterparts, and most find application in many areas of physics, particularly solid state physics. These relationships will be pointed out and where appropriate we will also mention the applications and consequences of the models.

There are very few models in quantum mechanics that yield exact solutions, hence any that do are of fundamental interest, the hydrogen atom being, perhaps, the most famous example. The hydrogen atom with the potential V(r) = —Ze2/4πε0r is unique in that it can be solved analytically classically and in both non-relativistic and relativistic quantum theory, with and without spin. This has been discussed in detail in chapters 3 and 8.

The five examples we choose to consider here are the following. Firstly we will solve the Dirac equation for an electron in a one-dimensional well, a relativistic generalization of the non-relativistic particle in a box problem.

Spin is well known to be an intrinsically relativistic property of particles. Nonetheless its effects are seen in many physical situations which are not obviously relativistic, perhaps the most obvious examples being magnets and the quantum mechanically permitted electronic configurations of the elements in the periodic table. The view of spin adopted in these and other problems is that the electron has a quantized spin (s = 1/2) and that is a fundamental tenet of the theory, rather than something that has to be explained. In this chapter we are going to discuss the behaviour of spin-1/2 particles (electrons, protons and neutrons for example) without much direct reference to relativistic quantum theory and the origins of spin. This chapter should be instructive in its own right and as a guide to understanding spin when we come to discuss it in a fully relativistic context at various stages throughout this book. Unless otherwise specified, we will refer to electrons, but it should be borne in mind that the theory is equally applicable to any spin-1/2 particle.

Students of quantum theory cannot delve very deeply into the subject without coming across the quantization of angular momentum. The orbital angular momentum usually surfaces in the theory leading up to the quantum description of the hydrogen atom.

Relativistic quantum mechanics is the unification into a consistent theory of Einstein's theory of relativity and the quantum mechanics of physicists such as Bohr, Schrödinger, and Heisenberg. Evidently, to appreciate relativistic quantum theory it is necessary to have a good understanding of these component theories. Apart from this chapter we assume the reader has this understanding. However, here we are going to recall some of the important points of the classical theory of special relativity. There is good reason for doing this. As you will discover all too soon, relativistic quantum mechanics is a very mathematical subject and my experience has been that the complexity of the mathematics often obscures the physics being described. To facilitate the interpretation of the mathematics here, appropriate limits are taken wherever possible, to obtain expressions with which the reader should be familiar. Clearly, when this is done it is useful to have the limiting expressions handy. Presenting them in this chapter means they can be referred to easily.

Taking the above argument to its logical conclusion means we should include a chapter on non-relativistic quantum mechanics as well. However, that is too vast a subject to include in a single chapter. Furthermore, there already exists a plethora of good books on the subject.

As in non-relativistic quantum theory, the simplest problem to solve in relativistic quantum theory is that of describing a free particle. Much can be learned from this case which will be of use in interpreting the topics covered in later chapters. Furthermore, some of the most profound features of relativistic quantum theory are well illustrated by the free particle, so it is a very instructive problem to consider in detail. Another advantage of the free-particle problem is that the mathematics involved in solving it is not nearly as involved as that necessary for solving problems involving particles under the influence of potentials.

Firstly we shall look briefly at the formulae for the current and probability density, then we shall go on to examine the solutions of the Dirac equation and investigate their behaviour. This leads us to a discussion of spin, the Pauli limit, and the relativistic spin operator. Next we consider the negative energy solutions and show how relativistic quantum theory predicts the existence of antiparticles. Some of the dilemmas this concept introduces and their resolution are discussed. At the end of the chapter we go back to the Klein paradox, and examine it for an incident spin-1/2 particle. We find that the Klein paradox exists for Dirac particles in exactly the same way as it existed for Klein—Gordon particles and has the same resolution and interpretation.

Superconductivity was discovered by Kammerlingh Onnes (1911) (see Gorter 1964). It turned out to be one of the most difficult problems in condensed matter physics of the twentieth century. There were over 40 years between the discovery of the effect and the development of a satisfactory theory (Cooper 1956, Bardeen, Cooper and Schrieffer 1957). The theory was based on the insightful suggestion by Frohlich (1950) that under some circumstances electrons in a lattice could actually attract one another. The theory of superconductivity divides neatly into two parts. Firstly, there is the theory required to describe the mutual attraction of electrons to form Cooper pairs. Secondly, there is the theory that accepts pairing as a fact and then goes on to calculate observables and properties of superconductors. In this chapter we will be principally concerned with the latter aspects of superconductivity theory. We will start from the many-body theory developed in chapter 6 together with a pairing interaction to get to observables such as the superconducting energy gap, critical fields and temperatures, and to describe the electrodynamics of superconductors. On the whole, though, we will not reproduce superconductivity theory that appears in other textbooks. There are several good books on the non-relativistic theory of superconductivity.

Atoms

An atom consists of a positively charged nucleus, together with a number of negatively charged electrons. Inside the nucleus there are protons, each of which carries positive charge e, and neutrons, which have no charge. So the charge on the nucleus is Ze, where Z, the atomic number, is the number of protons. The charge on each electron is -e, so that when the atom has Z electrons it is electrically neutral. If some of the electrons are stripped off, the atom then has net positive charge; it has been ionised.

The electrons are held in the atom by the electrostatic attraction between each electron and the nucleus. There is also an attraction because of the gravitational force, but this is about 10-40 times less strong, and so may be neglected. The protons and neutrons are held together in the nucleus by a different type of force, the nuclear force. The nuclear force is much stronger than the electrical force, and its attraction more than counteracts the electrostatic repulsion between pairs of protons. The nuclear force does not affect electrons. It is a very short-range force, so that it keeps the neutrons and protons very close together; the diameter of a nucleus is of the order of 10-15 m. By contrast, the diameter of the whole atom is about 10-10 m, so that for many purposes one can think of the nucleus as a point charge.

This book is intended as a first course on quantum mechanics and its applications. It is designed to be a first course rather than a complete one, and it is based on lectures given to mathematics and physics students in Cambridge. The book should be suitable also for engineering students.

The first five chapters deal with basic quantum mechanics, and are followed by a revision quiz to test the reader's understanding of them. The remaining chapters concentrate on applications. In most courses on quantum mechanics, the first application is to scattering problems. While recognising the importance of scattering theory, we have chosen rather to describe the application of quantum mechanics to physical phenomena that are of more everyday interest. These include molecular binding, the physics of masers and lasers, simple properties of crystalline solids arising from their electronic band structure, and the operation of junction transistors.

A few problems are included at the end of each chapter. We urge the student to work through all of these, as they form an integral part of the course. Some hints on their solution may be found at the end of the book.

A previous edition of this book was published under the title Simple Quantum Physics in 1979. In this new edition, the main change is the addition of a chapter on the theory of spin, and its application to magnetic resonance imaging. We have also described the WKB approximation and its application to a-decay, and have made a number of other minor changes.

Self-referential consistency and inconsistency

Universality implies self-referentiality

In the preceding chapters, we have mentioned on several occasions that there are good reasons to consider quantum mechanics as universally valid. Indeed, during the last 70 years quantum mechanics has not been disproved by a single experiment. In spite of numerous attempts to discover the limits of applicability and validity of this theory, there is no indication that the theory should be improved, extended, or reformulated. Moreover, the formal structure of quantum mechanics is based on very few assumptions, and these do not leave much room for alternative formulations. The most radical attempt to justify quantum mechanics, operational quantum logic, begins with the most general preconditions of a scientific language of physical objects, and derives from these preconditions the logico-algebraic structure of quantum mechanical propositions [Mit 78,86], [Sta 80]. There are strong indications that from these structures (orthomodular lattices, Baer*-semigroups, orthomodular posets, etc.) the full quantum mechanics in Hilbert space can be obtained. Although simple application of Piron's representation theorem [Pir 76] does not lead to the desired result, [Kel 80], [Gro 90], there are new and very promising results [Sol 95] which indicate that the intended goal may well be achieved within the next few years. Together with the experimental confirmation and verification of quantum mechanics, these quantum logical results strongly support the hypothesis that quantum mechanics is indeed universally valid.