Introduction

Quantum theory predicts the striking and paradoxical result that when a system is continuously watched, it does not evolve! Although this effect was noticed much earlier by a number of people [146], [147], [148], [149], [150], it was first formally stated by Misra and Sudarshan [151] and given the appellation ‘Zeno's paradox’ because it evokes the famous paradox of Zeno denying the possibility of motion to a flying arrow. It is as startling as a pot of water on a heater that refuses to boil when continuously watched. This is why it is also called the ‘watched pot effect’. The effect is the result of repeated, frequent measurements on the system, each measurement projecting the system back to its initial state. In other words, the wave function of the system must repeatedly collapse. It is also necessary that the time interval between successive measurements must be much shorter than the critical time of coherent evolution of the system, called the Zeno time [152]. For decays this Zeno time is the time of coherent evolution before the irreversible exponential decay sets in, and is governed by the reciprocal of the range of energies accessible to the decay products. For most decays this is extremely short and hard to detect [153]. In the case of non-exponential time evolution, the relevant Zeno time can be much longer.

I always thought I would write a book and this is it. In the end, though, I hardly wrote it at all, it evolved from my research notes, from essays I wrote for postgraduates starting work with me, and from lecture handouts I distribute to students taking the relativistic quantum mechanics option in the Physics department at Keele University. Therefore the early chapters of this book discuss pure relativistic quantum mechanics and the later chapters discuss applications of relevance in condensed matter physics. This book, then, is written with an audience ranging from advanced students to professional researchers in mind. I wrote it because anyone aiming to do research in relativistic quantum theory applied to condensed matter has to pull together information from a wide range of sources using different conventions, notation and units, which can lead to a lot of confusion (I speak from experience). Most relativistic quantum mechanics books, it seems to me, are directed towards quantum field theory and particle physics, not condensed matter physics, and many start off at too advanced a level for present day physics graduates from a British university. Therefore, I have tried to start at a sufficiently elementary level, and have used the SI system of units throughout.

In the previous chapter we discussed the direct relativistic generalization of the Schrödinger theory of quantum mechanics. It was shown that this leads to the Klein—Gordon equation from which some fundamental physics follows. Spin does not appear in either the Schrödinger or the Klein—Gordon theory. Indeed, the Klein—Gordon equation is only appropriate for particles with spin zero. Most of the particles encountered in everyday life (not that one actually encounters many fundamental particles in everyday life) are not spin zero. The most common ones, the neutron, proton and electron, all have spin 1/2. In this chapter we find the equation to describe particles with spin 1/2 and explore its properties. From the title of this chapter you will not be surprised to learn that it is known as the Dirac equation. The Dirac equation is more general than anything that has gone before, and therefore cannot really be derived. Much of the rest of this book involves looking for solutions of the Dirac equation under various circumstances.

Although the Dirac equation cannot really be derived from anything learnt up to the present level, some plausibility arguments for its existence can be given and a couple of these are presented in the first section.

Perhaps a natural reaction to the title of this chapter is that it should be rather short. In condensed matter physics we are interested in the particles that make up the world. They are the protons, neutrons and electrons, of course, and they all have spin 1/2. A few particles, such as the pi-meson, which we come across in particle physics do have spin zero, but they only exist for a very brief time before decaying, so how much can be said that is relevant to condensed matter physics?

This point can actually be answered rather easily. The generalization of quantum mechanics to include relativity is, to say the least, a nontrivial problem. As we shall see, spin-1/2 particles are well described by the Dirac equation. That equation describes both the relativistic nature of the particles and their spin (although the two are not really divisible). Treating spin-zero particles first means we can understand many aspects of the relativistic nature of quantum theory without the added complication of spin. Furthermore, formulae derived in this chapter can be compared with those in later chapters to give added insight into the nature of spin. This is particularly true when we look at the properties of a mythical spin-zero electron in a central Coulomb potential.

One of the principal applications of quantum theory in physics is in the interpretation of spectroscopies. Spectroscopy is one of the key tools for learning about condensed matter on the microscopic scale. In this chapter we are going to consider the theory of spectroscopy on a quantum mechanical level with particular emphasis on effects that are intrinsically relativistic. This boils down to a study of the interaction of the electrons in the material with incident photons.

A bit of quantum field theory is unavoidable in this chapter. The field theory here is as elementary as it gets. Furthermore, it is conceptually easy and is introduced as a natural extension of the relativistic quantum theory already covered in this book, so it should present no difficulty.

If you scan through this chapter the mathematics appears pretty daunting, (so what's new). Don't worry, it could be worse, at least we make plenty of use of the Dirac notation introduced in chapter 2. Many of the equations in this chapter would be horrendous without it. In the first sections we discuss some properties of the photon and quantization of the electromagnetic field. Then we come to the backbone of the chapter, time-dependent perturbation theory, from which the Golden Rule for transition rates is derived to second order.

In this chapter we are going to make a very detailed study of the one-electron atom using relativistic quantum mechanics. The one-electron atom is the simplest bound system that occurs in nature, and plays a central role in both classical and quantum theory. There are two reasons for this. Firstly, it is a model that can be solved exactly at many levels of theory — the classical Bohr atom, the non-relativistic quantum mechanical one-electron atom, and the relativistic one-electron atom with spin zero and spin 1/2. Secondly, most of our understanding of multielectron atoms, molecules and solids is based on this model.

Of course, a good description of the hydrogen atom is provided by non-relativistic Schrödinger theory. However, once the relativistic quantum theory had been discovered it was of great importance to validate the theory by describing hydrogen at least as well as Schrödinger theory. As we shall see, the Dirac theory was exceedingly successful in this, and furthermore, at several points in the theory the non-relativistic limit can be taken and the effect of relativity in one-electron atoms can be seen explicitly. This connection with the simpler theory will be made here wherever insight may be gained from doing so.

One of the key advances brought about by the advent of quantum theory was that it defined eigenfunctions as well as the eigenvalues predicted by the Bohr theory.

The scattering of fast particles is an important tool in many fields of physics. In particular, virtually all that is known about elementary particles is a result of the interpretation of scattering experiments. In condensed matter physics as well, the bulk of our understanding of materials on a microscopic level comes from the scattering of neutrons, photons and electrons. Neutrons are used to determine crystal structures and to probe the dynamical properties of solids; photons are used in a plethora of spectroscopies to elucidate the details of the electronic and magnetic structure. Some of these will be discussed in chapter 12. Electrons can be used to determine the behaviour of surface plasmons and also to look at electronic transitions.

For these and many other reasons, an understanding of the quantum theory of scattering is of key importance for a theoretical physicist. Therefore in this chapter we develop relativistic scattering theory from scratch. This chapter doesn't assume any knowledge of non-relativistic scattering theory although such knowledge will aid your understanding.

In this chapter it has been necessary to include some mathematical preliminaries. Therefore the first three sections are an introduction to Green's functions and their uses. Later in the chapter we have some followup sections on free-particle and scattering Green's functions.

In this chapter we are going to discuss some rather esoteric topics. Despite this nature they are of fundamental significance in relativistic quantum theory and have profound consequences. Clearly we could discuss a lot of topics that routinely occur in non-relativistic quantum theory under such a chapter heading. We will not do this, but only consider topics of specific importance in relativistic quantum theory.

We will start this chapter by introducing a new type of operator known as a projection operator. This is an operator that acts on some wave-function and projects out the part of the wavefunction corresponding to particular properties. In particular there are energy projection operators which can project out the positive or negative energy part of a wavefunction and spin projection operators which (surprise surprise!) project out the part of the wavefunction corresponding to a particular spin direction. Such operators form an essential part of the theory of high energy scattering. We will not be using them much in our discussion of scattering because our aim there is towards solid state applications. For further discussion of projection operators, the books by Rose (1961), Bjorken and Drell (1964) and Greiner (1990) are useful.

Secondly, we will look at some symmetries that occur in the Dirac equation.

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.