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.
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.
The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.
The early twentieth century saw two major revolutions in the way physicists understand the world. One was quantum mechanics itself and the other was the theory of relativity. Important results also emerged when these two ideas were brought together: some of these have been referred to in earlier chapters-in particular the fact that fundamental particles such as electrons have intrinsic angular momentum (spin) was stated to be a relativistic effect. This chapter explores the relationship between relativity and quantum mechanics more deeply A full understanding of relativistic quantum mechanics is well outside the scope of this book, but many of the important results can be understood at this level and these will be discussed in this chapter. After a short summary of the main results of special relativity, we show how combining this with the time-dependent Schro¨dinger equation leads to a new wave equation known as the Dirac equation. We show how the Dirac equation requires particles such as electrons to have intrinsic angular momentum (spin) and we explore some of its other consequences. The chapter concludes with an outline of some more advanced ideas known as quantum field theory.
Special relativity is a fascinating and challenging branch of physics. It describes the physics of the high velocity/high energy regime, frequently turning up phenomena that appear paradoxical in view of our everyday experience. In this book we will be quite selective in our presentation of the theory of special relativity: we will concentrate on those features that we consider necessary for the later applications to relativistic quantum chemistry. We do this in good conscience, knowing that there is a vast literature on the subject, catering to a wide range of audiences—from the quite elementary to the very sophisticated. A few examples are listed in the reference list, but a visit to any nearby physics library will provide an ample selection of reading material for those wishing to delve deeper into the matter. In the present chapter we adopt a minimalist approach. We develop some of the basic concepts and formulas of special relativity, building on a rather elementary level of basic physics. The aim is to provide a sufficient foundation for those who want to proceed as quickly as possible to the more quantum chemical parts of the text. In later chapters we will introduce more advanced tools of physics and revisit some of the subjects treated here. The theory of special relativity deals with the description of physical phenomena in frames that move at constant velocity relative to each other. The classroom is one such frame, the car passing at constant speed outside the classroom is another. The trajectory of a ball being thrown up vertically in the car will look quite different whether we describe it relative to the interior of the car or relative to the interior of the classroom. In particular we will be concerned with inertial frames. We define an inertial frame as a frame where spatial relations are Euclidean and where there is a universal time such that free particles move with constant velocities. In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S’ are expressed in terms of the Galilean transformations.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are signicant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benets of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantum-mechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic eects become observable, for which non-relativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.33) must be used. There are, however, more subtle eects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.