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Cambridge University Press eBooks

Introduction

Michel Le Bellac

Published Mar 23, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511616471.003

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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.
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.
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Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
Chapter 2 explains how in constructing quantum mechanics, old ideas had to be discarded and well-established principles had to be modified or even abandoned. First it was necessary to be convinced of the physical reality of the atomic structure of matter; then to show that Newtonian mechanics is not directly applicable to the study of the atom; later to show that Maxwell’s electrodynamics alone does not describe all the elementary processes of interaction between an atom and other quantum particles. The chapter takes us from Planck’s energy quantum to Einstein’s quantization as a universal phenomenon, from Bohr’s quantum atom model, and the early quantization rules, to de Broglie’s waves associated with corpuscles in motion. It ends with a corollary on the electrodynamic nature of quantum mechanics.
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!
This chapter is an introduction to the study of the quantized nature of matter. It begins with an introduction to the line spectrum produced by energized gases and the Bohr model that was created to explain the hydrogen spectrum. Emphasis is put on the energy analysis of the Bohr model to connect back to the theme of this volume of the text. The wave nature of matter is introduced as a way to explain the assumptions that Bohr made in his model and how these concepts led to the development of quantum mechanics. This chapter concludes with an introduction to quantum mechanics as an expression of the energy of a matter wave and as a way to explain atomic orbitals.