Introduction While it might appear that the nineteenth-century physics presented in this chapter has no place in a topic as quantum mechanically oriented as atomic physics, this is simply not the case. The purpose of this first chapter is to try to convince the readers that the material learned in their early years of studying physics is not disjoint from modern topics, but in fact can provide a foundation for their further understanding. Other examples will occur later. The reader may be a bit surprised to see that the results of the classical description turn out to be so close to the quantum mechanical results. Where there are significant disagreements between classical and quantum descriptions, it is necessary to explain why these occur. The correspondence principle requires that in the limit of large quantum numbers the complete quantum mechanical description must agree with the classical one. It is always better to need fewer of these explanations and to be able to treat physics as a coherent whole instead of a collection of topics. The consequences of classical physics are intuitively familiar, and this is often most helpful in gaining deeper understanding. In fact, the approach taken in very many of the chapters in this book is to start with a purely classical description of a topic, and then transport it directly into the Schrödinger equation. Radiation from an accelerated charge, as treated here in the harmonic case, provides an ideal setting for such pedagogy. Even though classical mechanics is used in place of quantum mechanics, for the case of harmonic motion, the results are quite compatible. The early history of atomic physics contains many topics that are well worth reading about, especially the experimental facilities. For example, there were no electric motors, wire was not readily available, and making even a crude vacuum was an experimental tour de force. Very many textbooks have discussions of these topics, and so they will not be presented here. But the reader is warned that certain oft-repeated modern descriptions are simply wrong. Perhaps the most important notion to be gained from this chapter is that the classical description of a harmonically bound electron provides quite accurate results about the interaction between atoms and light. Of course, the utility of this description is limited because the classical harmonic oscillator has no internal structure, is infinitely deep, and has a continuous energy spectrum.
Introduction While it might appear that the nineteenth-century physics presented in this chapter has no place in a topic as quantum mechanically oriented as atomic physics, this is simply not the case. The purpose of this first chapter is to try to convince the readers that the material learned in their early years of studying physics is not disjoint from modern topics, but in fact can provide a foundation for their further understanding. Other examples will occur later. The reader may be a bit surprised to see that the results of the classical description turn out to be so close to the quantum mechanical results. Where there are significant disagreements between classical and quantum descriptions, it is necessary to explain why these occur. The correspondence principle requires that in the limit of large quantum numbers the complete quantum mechanical description must agree with the classical one. It is always better to need fewer of these explanations and to be able to treat physics as a coherent whole instead of a collection of topics. The consequences of classical physics are intuitively familiar, and this is often most helpful in gaining deeper understanding. In fact, the approach taken in very many of the chapters in this book is to start with a purely classical description of a topic, and then transport it directly into the Schrödinger equation. Radiation from an accelerated charge, as treated here in the harmonic case, provides an ideal setting for such pedagogy. Even though classical mechanics is used in place of quantum mechanics, for the case of harmonic motion, the results are quite compatible. The early history of atomic physics contains many topics that are well worth reading about, especially the experimental facilities. For example, there were no electric motors, wire was not readily available, and making even a crude vacuum was an experimental tour de force. Very many textbooks have discussions of these topics, and so they will not be presented here. But the reader is warned that certain oft-repeated modern descriptions are simply wrong. Perhaps the most important notion to be gained from this chapter is that the classical description of a harmonically bound electron provides quite accurate results about the interaction between atoms and light. Of course, the utility of this description is limited because the classical harmonic oscillator has no internal structure, is infinitely deep, and has a continuous energy spectrum.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.
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.
I conclude the first part of this book by carrying the story up to 1928, to the lightning flash in Eddington's mind produced by Dirac's paper on the wave equation of the electron. Setting out the context and Eddington's thinking about it is a very different matter for quantum mechanics and relativity. Both theories are commonly expressed in austere mathematical language. In the case of relativity the basic ideas behind the mathematics are now well understood and can be set out with little complication. These ideas were mostly already clear to Eddington. But for quantum mechanics the mathematics, yoked to the wealth of experimental results, drove the development of the theory at a breakneck speed, mostly without anyone pausing for deeper understanding. It is indeed only in the last decade that it has become generally accepted that quantum mechanics still lacks any coherent interpretation. To explain Eddington's context it is therefore necessary to say something about the mathematical formalism. It is true, as will become clear, that Eddington was a little more sceptical than most, but by and large the mathematics drove him as it did those directly working in the theory. I begin, then, with a description of that part of the early (pre-1925) history of quantum mechanics that was in Eddington's mind in 1923. Then I explain what happened in 1925–6 and Eddington's reaction to it. The chapter concludes with Dirac's relativistic wave equation of 1928 and Eddington's further reaction to that. The old quantum theory What is now called the old quantum theory arose at the turn of the century. Two related experimental contradictions with classical physics played important roles in this.
We tried to introduce the idea of quantization in Chapter 9 as a completion of a one semester course. However, we skipped over some really interesting events in the history of Science between 1900 and 1913 when Bohr derived the quantized energy of the H atom. First we want to carry out the 1901 Planck derivation of the formula for blackbody radiation [1]. Many texts just show the curve, write e¼ hn, and move on. As a student, this author found that limited explanation very frustrating since energy quantization is a fundamental concept. Even among graduate texts in quantum mechanics, we are aware of only one that does the complete treatment which we will draw upon for the mathematics [2] but supplement with a narrative that we have found helpful to students over the years. Then in 1905, Albert Einstein (1879-1955), one of the most influential scientists of all time, gave an explanation of the photoelectric effect [3] for which he received the Nobel Prize in 1921. Even reducing our list to essential topics, we need to discuss the Davisson-Germer experiment [4]. The photoelectric effect introduces the idea that light waves can act as particles while the Davisson-Germer experiment showed that particles can act like waves and confirmed the De Broglie equation [5]. However, you can be assured that it will not be as difficult as you might have anticipated and if you can absorb the meaning of just these three key experiments you should be able to begin thinking in terms of quantum mechanics! Despite our slow historical development, this is 2010 and we have to get to the twenty-first century somehow!
Introduction Quantum mechanics is based on the intuition that the essential difference between atomic physics and classical physics is the occurrence of discontinuities (see in particular [1,4,58–63]). Quantum mechanics should thus be considered a direct continuation of the quantum theory founded by Planck, Einstein and Bohr. Bohr in particular stressed repeatedly, already before the birth of quantum mechanics, that the discontinuities must lead to the introduction of new kinematical and mechanical concepts, so that indeed classical mechanics and its corresponding conceptual scheme should be abandoned [1,4]. Quantum mechanics tries to introduce the new concepts through a precise analysis of what is ‘observable in principle’. In fact, this does not mean setting up the principle that a sharp division between ‘observable’ and ‘unobservable’ quantities is possible and necessary. As soon as a conceptual scheme is given, one can infer from the observations to other facts that are actually not observable directly, and the boundary between ‘observable’ and ‘unobservable’ quantities becomes altogether indeterminate. But if the conceptual scheme itself is still unknown, it will be expedient to enquire only about the observations themselves, without drawing conclusions from them, because otherwise wrong concepts and prejudices taken over from before will block the way to recognising the physical relationships [Zusammenhänge].