QUANTUM MECHANICS www.com QUANTUM MECHANICS A Conceptual Approach HENDRIK F. HAMEKA A John Wiley & Sons, Inc.com Copyright # 2004 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
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Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Hameka, Hendrik F. Quantum mechanics : a conceptual approach / Hendrik F. : acid-free paper) 1.12–dc22 2004000645 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 www.com To Charlotte www.com CONTENTS Preface xi 1 The Discovery of Quantum Mechanics 1 I Introduction, 1 II Planck and Quantization, 3 III Bohr and the Hydrogen Atom, 7 IV Matrix Mechanics, 11 V The Uncertainty Relations, 13 VI Wave Mechanics, 14 VII The Final Touches of Quantum Mechanics, 20 VIII Concluding Remarks, 22 2 The Mathematics of Quantum Mechanics 23 I Introduction, 23 II Differential Equations, 24 III Kummer’s Function, 25 IV Matrices, 27 V Permutations, 30 VI Determinants, 31 vii www.com viii CONTENTS VII Properties of Determinants, 32 VIII Linear Equations and Eigenvalues, 35 IX Problems, 37 3 Classical Mechanics 39 I Introduction, 39 II Vectors and Vector Fields, 40 III Hamiltonian Mechanics, 43 IV The Classical Harmonic Oscillator, 44 V Angular Momentum, 45 VI Polar Coordinates, 49 VII Problems, 51 4 Wave Mechanics of a Free Particle 52 I Introduction, 52 II The Mathematics of Plane Waves, 53 III The Schrödinger Equation of a Free Particle, 54 IV The Interpretation of the Wave Function, 56 V Wave Packets, 58 VI Concluding Remarks, 62 VII Problems, 63 5 The Schrödinger Equation 64 I Introduction, 64 II Operators, 66 III The Particle in a Box, 68 IV Concluding Remarks, 71 V Problems, 72 6 Applications 73 I Introduction, 73 II A Particle in a Finite Box, 74 www.com CONTENTS ix III Tunneling, 78 IV The Harmonic Oscillator, 81 V Problems, 87 7 Angular Momentum 88 I Introduction, 88 II Commuting Operators, 89 III Commutation Relations of the Angular Momentum, 90 IV The Rigid Rotor, 91 V Eigenfunctions of the Angular Momentum, 93 VI Concluding Remarks, 96 VII Problems, 96 8 The Hydrogen Atom 98 I Introduction, 98 II Solving the Schrödinger Equation, 99 III Deriving the Energy Eigenvalues, 101 IV The Behavior of the Eigenfunctions, 103 V Problems, 106 9 Approximate Methods 108 I Introduction, 108 II The Variational Principle, 109 III Applications of the Variational Principle, 111 IV Perturbation Theory for a Nondegenerate State, 113 V The Stark Effect of the Hydrogen Atom, 116 VI Perturbation Theory for Degenerate States, 119 VII Concluding Remarks, 120 VIII Problems, 120 10 The Helium Atom 122 I Introduction, 122 www.com x CONTENTS II Experimental Developments, 123 III Pauli’s Exclusion Principle, 126 IV The Discovery of the Electron Spin, 127 V The Mathematical Description of the Electron Spin, 129 VI The Exclusion Principle Revisited, 132 VII Two-Electron Systems, 133 VIII The Helium Atom, 135 IX The Helium Atom Orbitals, 138 X Concluding Remarks, 139 XI Problems, 140 11 Atomic Structure 142 I Introduction, 142 II Atomic and Molecular Wave Function, 145 III The Hartree-Fock Method, 146 IV Slater Orbitals, 152 V Multiplet Theory, 154 VI Concluding Remarks, 158 VII Problems, 158 12 Molecular Structure 160 I Introduction, 160 II The Born-Oppenheimer Approximation, 161 III Nuclear Motion of Diatomic Molecules, 164 IV The Hydrogen Molecular Ion, 169 V The Hydrogen Molecule, 173 VI The Chemical Bond, 176 VII The Structures of Some Simple Polyatomic Molecules, 179 VIII The Hückel Molecular Orbital Method, 183 IX Problems, 189 Index 191 www.com PREFACE The physical laws and mathematical structure that constitute the basis of quantum mechanics were derived by physicists, but subsequent applications became of inter- est not just to the physicists but also to chemists, biologists, medical scientists, engineers, and philosophers.
Quantum mechanical descriptions of atomic and mole- cular structure are now taught in freshman chemistry and even in some high school chemistry courses. Sophisticated computer programs are routinely used for predict- ing the structures and geometries of large organic molecules or for the indentifica- tion and evaluation of new medicinal drugs. Engineers have incorporated the quantum mechanical tunneling effect into the design of new electronic devices, and philosophers have studied the consequences of some of the novel concepts of quantum mechanics. They have also compared the relative merits of different axiomatic approaches to the subject.
In view of the widespread applications of quantum mechanics to these areas there are now many people who want to learn more about the subject. They may, of course, try to read one of the many quantum textbooks that have been written, but almost all of these textbooks assume that their readers have an extensive back- ground in physics and mathematics; very few of these books make an effort to explain the subject in simple non-mathematical terms. In this book we try to present the fundamentals and some simple applications of quantum mechanics by emphasizing the basic concepts and by keeping the mathe- matics as simple as possible. We do assume that the reader is familiar with elemen- tary calculus; it is after all not possible to explain the Schödinger equation to someone who does not know what a derivative or an integral is.
Some of the mathe- matical techniques that are essential for understanding quantum mechanics, such as matrices and determinants, differential equations, Fourier analysis, and so on are xi www.com xii PREFACE described in a simple manner. We also present some applications to atomic and molecular structure that constitute the basis of the various molecular structure com- puter programs, but we do not attempt to describe the computation techniques in detail. Many authors present quantum mechanics by means of the axiomatic approach, which leads to a rigorous mathematical representation of the subject. However, in some instances it is not easy for an average reader to even understand the axioms, let alone the theorems that are derived from them.
I have always looked upon quan- tum mechanics as a conglomerate of revolutionary new concepts rather than as a rigid mathematical discipline. I also feel that the reader might get a better under- standing and appreciation of these concepts if the reader is familiar with the back- ground and the personalities of the scientists who conceived them and with the reasoning and arguments that led to their conception. Our approach to the presenta- tion of quantum mechanics may then be called historic or conceptual but is perhaps best described as pragmatic. Also, the inclusion of some historical background makes the book more readable.
I did not give a detailed description of the various sources I used in writing the historical sections of the book because many of the facts that are presented were derived from multiple sources. Some of the material was derived from personal conversations with many scientists and from articles in various journals. The most reliable sources are the original publications where the new quantum mechan- ical ideas were first proposed. These are readily available in the scientific literature, and I was intrigued in reading some of the original papers.
I also read various biographies and autobiographies. I found Moore’s biography of Schröedinger, Con- stance Reid’s biographies of Hilbert and Courant, Abraham Pais’ reminiscences, and the autobiographies of Elsasser and Casimir particularly interesting. I should mention that Kramers was the professor of theoretical physics when I was a student at Leiden University. He died before I finished my studies and I never worked under his supervision, but I did learn quantum mechanics by reading his book and by attending his lectures.
Finally I wish to express my thanks to Mrs. Alice Chen for her valuable help in typing and preparing the manuscript.com 1 THE DISCOVERY OF QUANTUM MECHANICS I. INTRODUCTION The laws of classical mechanics were summarized in 1686 by Isaac Newton (1642– 1727) in his famous book Philosophiae Naturalis Principia Mathematica. During the following 200 years, they were universally used for the theoretical interpretation of all known phenomena in physics and astronomy.
However, towards the end of the nineteenth century, new discoveries related to the electronic structure of atoms and molecules and to the nature of light could no longer be interpreted by means of the classical Newtonian laws of mechanics. It therefore became necessary to develop a new and different type of mechanics in order to explain these newly discovered phenomena. This new branch of theoretical physics became known as quantum mechanics or wave mechanics. Initially quantum mechanics was studied solely by theoretical physicists or chemists, and the writers of textbooks assumed that their readers had a thorough knowledge of physics and mathematics.
In recent times the applications of quantum mechanics have expanded dramatically. We feel that there is an increasing number of students who would like to learn the general concepts and fundamental features of quantum mechanics without having to invest an excessive amount of time and effort. The present book is intended for this audience. We plan to explain quantum mechanics from a historical perspective rather than by means of the more common axiomatic approach.
Most fundamental con- cepts of quantum mechanics are far from self-evident, and they gained general Quantum Mechanics: A Conceptual Approach, By Hendrik F. Hameka ISBN 0-471-64965-1 Copyright # 2004 John Wiley & Sons, Inc.com 2 THE DISCOVERY OF QUANTUM MECHANICS TABLE 1-1. Pioneers of Quantum Mechanics Niels Henrik David Bohr (1885–1962) Max Born (1882–1970) Louis Victor Pierre Raymond, Duc de Broglie (1892–1989) Pieter Josephus Wilhelmus Debije (1884–1966) Paul Adrien Maurice Dirac (1902–1984) Paul Ehrenfest (1880–1933) Albert Einstein (1879–1955) Samuel Abraham Goudsmit (1902–1978) Werner Karl Heisenberg (1901–1976) David Hilbert (1862–1943) Hendrik Anton Kramers (1894–1952) Wolfgang Ernst Pauli (1900–1958) Max Karl Ernst Ludwig Planck (1858–1947) Erwin Rudolf Josef Alexander Schrödinger (1887–1961) Arnold Johannes Wilhelm Sommerfeld (1868–1951) George Eugene Uhlenbeck (1900–1988) acceptance only because there were no reasonable alternatives for the interpretation of new experimental discoveries. We believe therefore that they may be easier to understand by learning the motivation and the line of reasoning that led to their discovery.
The discovery of quantum mechanics makes an interesting story, and it has been the subject of a number of historical studies. It extended over a period of about 30 years, from 1900 to about 1930. The historians have even defined a specific date, namely, December 14, 1900, as the birth date of quantum mechanics.