As is appropriate in an exposition of the quantum mechanics of vibrators, we started with a quantal calculation and have finished with one; but we have never strayed far from the classical models. We began, indeed, in chapter 13 the task of establishing limits to the validity of classical reasoning, and the very last example of this chapter has been used to demonstrate, somewhat perversely it might be thought, that the method of quantum mechanics can occasionally be applied to systems which physicists and engineers would instinctively regard as classical. So long as the discussions give insight into physical processes and reveal the strengths and weaknesses of the analytical tools available, no apology is needed and no defence is offered except the evidence of the book itself. Even the rather simple systems which provide material for the whole of the volume demonstrate clearly the increased power available to those who can handle both classical and quantal reasoning. The reader who wishes to apply his skill to complex vibratory processes will find his tasks eased if he can use whichever seems advantageous at each stage, and be confident that his understanding of simple processes will enable him to recognize the dangers inherent in both approaches – the danger of carrying classical reasoning too far into the quantum domain, and the danger of forcing over-simplifications on physical systems to make them amenable to the unforgiving methodology of quantum mechanics.
As is appropriate in an exposition of the quantum mechanics of vibrators, we started with a quantal calculation and have finished with one; but we have never strayed far from the classical models. We began, indeed, in chapter 13 the task of establishing limits to the validity of classical reasoning, and the very last example of this chapter has been used to demonstrate, somewhat perversely it might be thought, that the method of quantum mechanics can occasionally be applied to systems which physicists and engineers would instinctively regard as classical. So long as the discussions give insight into physical processes and reveal the strengths and weaknesses of the analytical tools available, no apology is needed and no defence is offered except the evidence of the book itself. Even the rather simple systems which provide material for the whole of the volume demonstrate clearly the increased power available to those who can handle both classical and quantal reasoning. The reader who wishes to apply his skill to complex vibratory processes will find his tasks eased if he can use whichever seems advantageous at each stage, and be confident that his understanding of simple processes will enable him to recognize the dangers inherent in both approaches – the danger of carrying classical reasoning too far into the quantum domain, and the danger of forcing over-simplifications on physical systems to make them amenable to the unforgiving methodology of quantum mechanics.
We begin with some examples of one-dimensional vibration in a nonparabolic potential chosen to permit complete analytical solution of Schrödinger's time-independent equation. These examples are of isochronous vibrators which classically have a frequency independent of amplitude, and which might be expected therefore to have energy levels equally spaced at a separation of ħω 0 . This expectation turns out to be very nearly right and inspires a certain confidence in the semi-classical procedure developed by (among others) Bohr, Wilson and Sommerfeld. We therefore apply this procedure to some non-isochronous systems and find once more rather good agreement with the results of exact quantum mechanics. Periodic systems in fact can often be treated semi-classically with adequate accuracy, and significant economy of effort in comparison with strict quantum-mechanical analysis. This approach pays handsomely when we turn in the next chapter to the quantization of electron cyclotron orbits which, as already discussed in chapter 8, are closely related to harmonic oscillators. Conduction electrons in semi-conductors, and still more in metals, have their behaviour modified by the lattice through which they move, and a complete quantal treatment has never been achieved. It is clear, however, from approximate calculations, often of great complexity, that the semiclassical method describes most of the interesting physical processes correctly and very simply. In chapter 15 we shall describe in outline some of the effects which can be treated quite well enough for most purposes without even writing down Schrödinger's equation.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.
After a short review of some essential concepts of classical physics (with a discussion of the so-called `scientific method', which is rarely found in textbooks of this kind), including the classical description of particles and waves, together with a sketch of statistical thermodynamics, the basics of quantum mechanics are introduced. The third chapter then treats systems of identical particles and their statistics, as well as the statistics of discrete energy levels. This is followed by an extensive presentation of the behaviour of quantum particles in periodic potentials with obvious applications to crystalline materials, including metals, insulators, semiconductors and superconductors. The band structure of some special materials, intrinsic carrier concentrations and electrons in metals are further topics of this main chapter. In most quantum mechanics textbooks the subjects of the fifth chapter, quantum wells, harmonic oscillators and the hydrogen atom, are usually introduced before a discussion of periodic potentials, but the author's exposition makes it quite plausible to place them afterwards. The concept of tunnelling through potential barriers and important application examples, ranging from ohmic contacts to scanning tunnelling microscopy and even radioactivity, make up the next chapter. Some approximation methods, i.e. stationary perturbation theory with applications to coupled wells, and the variational method, are treated in the next chapter. The final chapter dealing with quantum mechanics then is concerned with scattering and collision problems, again, of course, with examples from solid state physics. This last chapter, however, looks somehow merely added on, since it introduces only quite briefly the theory of special relativity -almost certainly in order not to leave out this important revolution of modern physics. Three appendices, on Fermi's Golden Rule, Boltzmann's transport theory and quantum interference devices, round off the contents. The book abounds with links to modern technologies that have evolved out of quantum mechanics (with, of course, the exception of the last chapter on special relativity). To the reviewer this constitutes its outstanding value as a textbook on quantum mechanics for engineering students. In addition, about 100 solved numerical examples in the text illustrate very lucidly some of the more abstract concepts, and at the end of the chapters altogether 200 further problems are posed. Although some of the concepts and results of quantum mechanics are introduced in a seemingly ad hoc manner, nevertheless the book as a whole certainly represents a valuable teaching tool at engineering schools; even physics majors may profit from the wealth of technical applications not usually found in their textbooks on quantum mechanics.
This is an enjoyable book by a particle physicist of some distinction, who wrote several books on the theoretical aspects of his subject. He died soon after the book was finished. This book would seem a splendid tribute to his memory. The first five chapters are an excellent introduction to quantum mechanics from the viewpoint of a particle physicist. (A solid state physicist might include some other topics.) The next three chapters give some feeling for the excitement of particle physics and describe some of the zoo of curious creatures discovered in the depths of particle accelerators, and the forces between them: strong, electromagnetic and weak. The final chapter, 'Quantum Fields', is a tour de force. The author has a light touch and a pleasant sense of humour. He does not attempt to explain everything he mentions, but he makes it very clear when he does not. These points are beautifully illustrated by the following extract from the final chapter (p 245). 'It was said above that virtual particles come into play when the real collision ingredients are all close together. In fact, virtual particles are always in play. Even a single real particle, moving along in isolation, can emit and reabsorb virtual particles over and over again. This has the effect of shifting the physical mass of the particle away from the 'bare' value that entered into the Hamiltonian. That shift inevitably turns out to be slightly infinite, and there is a whole technology for isolating and redefining away this and a few other infinities that are characteristic of renormalisable quantum field theories. But we forebear to pursue these delicacies any further here.' I particularly like 'slightly infinite'. The publisher's blurb describes this book as being suitable for popular science readers. It is not. Its level of mathematics would make it very heavy going for anyone who had not taken at least one course in mathematics at university. This is a book which can be strongly recommended as a supplementary text for undergraduates studying quantum mechanics, and also as a book for professional physicists who wish to obtain an up-to-date description and explanation of particle physics. It ought to be in every library. P Borcherds
This book and the accompanying software (PC Diskette) is a toolbox for performing a set of numerical simulations on simple quantum mechanical problems. It is one in a series of similar books (CUPS) covering different fields in physics from classical mechanics to nuclear and particle physics. Each chapter of the book is written by a different author and deals with a different physical problem. The choice of problems is standard, e.g. bound state and scattering problems in one and three dimensions, quantum mechanical time development; the exceptions are, a study of identical particles and electron states in a one dimensional lattice, an example from solid state physics. Although the chapters are self-contained and can be studied independently (at the expense of some minor duplication of material), care was taken with a uniform style of presentation and similar format in numerical simulations. As a result, it is easy to proceed in the study of the different problems having understood the technical possibilities of one simulation. Each chapter starts with a section presenting the basic theoretical concepts to be studied using the numerical simulation. It is followed by a section with a short description of the numerical technique used and an extensive series of suggestions for problems and quantities to experiment with, using the computer codes provided. A final section describes in detail how to run the computer programs. This book is not, and is not intended to be, a textbook for learning quantum mechanics nor a textbook of numerical analysis. Nevertheless, I find the theoretical presentation clear and straightforward enough for its purpose. The extensive and up to date list of references to books and research articles at the end of each chapter is also very helpful. Technically, I found the software easy to install and, after some initial testing, easy to run. The produced plots are clear and the use of units of atomic dimension gives a touch of reality to the simulations. In contrast to other similar books, this is not a `picture' book of quantum mechanics where results of numerical simulations are illustrated; here the reader is given the necessary means to play and discover for himself the properties of quantum mechanical systems. I believe that this book, used by teachers for demonstrations and students for independent study, provides a versatile tool for numerical experimentation in quantum mechanics.