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.

Historical remarks

The quantum mechanical formalism discovered by Heisenberg [Heis 25] an Schrödinger [Schrö 26] in 1925 was first interpreted in a statistical sense by Born [Born 26]. The formal expressions p(φ,ai) = |(φ,φai)|2, i ∈ N, were interpreted as the probabilities that a quantum system S with preparation φ possesses the value ai that belongs to the state φai. This original Born interpretation, which was formulated for scattering processes, was, however, not tenable in the general case. The probabilities must not be related to the system S in state φ, since in the preparation φ the value ai of an observable A is in general not subjectively unknown but objectively undecided. Instead, one has to interpret the formal expressions p(ai, ai) as the probabilities of finding the value ai after measurement of the observable A of the system S with preparation φ. In this improved version, the statistical or Born interpretation is used in the present-day literature.

On the one hand, the statistical (Born) interpretation of quantum mechanics is usually taken for granted, and the formalism of quantum mechanics is considered as a theory that provides statistical predictions referring to a sufficiently large ensemble of identically prepared systems S(φ) after the measurement of the observable in question. On the other hand, the meaning of the same formal terms p(φ,ai) for an individual system is highly problematic.

In chapter 5 we saw how in quantum mechanics electrons are bound to nuclei so as to form atoms. We now give a rather abbreviated account of how atoms bind together to form molecules. There is more than one type of molecular binding. We shall confine our discussion to the type known as covalent binding. The possibility of this type of binding relies on an effect that is peculiar to quantum mechanics, the tunnel effect, which we have already encountered in chapter 3.

The ionised hydrogen molecule

The simplest molecule is the ionised hydrogen molecule, which consists of two protons and one electron. The Coulomb force between the two protons tends to push them apart; we investigate how the presence of the electron overcomes this repulsion and holds the molecule together.

An exact calculation is difficult, but we can discuss the general features of the bonding by making suitable approximations. As the protons are much heavier than the electron, we may neglect their motion compared with that of the electron, and so regard them as fixed. We show that the expectation value of the energy, considered as a function of the proton separation R, has a minimum for a certain value of R, so that there is a stable equilibrium configuration.

Suppose first that R is so large that in the vicinity of each of the protons the Coulomb field of the other is completely negligible.