PRINCIPLES OF QUANTUM MECHANICS: as Applied to Chemistry and Chemical Physics CAMBRIDGE UNIVERSITY PRESS DONALD D. FITTS PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The ®rst two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to SchroÈdinger's wave mech- anics, the postulates of quantum mechanics are presented along with the essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This advanced text is appropriate for beginning graduate students in chem- istry, chemical physics, molecular physics, and materials science. A native of the state of New Hampshire, Donald Fitts developed an interest in chemistry at the age of eleven. He was awarded an A.B. degree, magna cum laude with highest honors in chemistry, in 1954 from Harvard University and a Ph.D. degree in chemistry in 1957 from Yale University for his theoretical work with John G. Kirkwood. After one-year appointments as a National Science Foundation Postdoctoral Fellow at the Institute for Theoretical Physics, Uni- versity of Amsterdam, and as a Research Fellow at Yale's Chemistry Depart- ment, he joined the faculty of the University of Pennsylvania, rising to the rank of Professor of Chemistry. In Penn's School of Arts and Sciences, Professor Fitts also served as Acting Dean for one year and as Associate Dean and Director of the Graduate Division for ®fteen years. His sabbatical leaves were spent in Britain as a NATO Senior Science Fellow at Imperial College, London, as an Academic Visitor in Physical Chemistry, University of Oxford, and as a Visiting Fellow at Corpus Christi College, Cambridge. He is the author of two other books, Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics DONALD D. FITTS University of Pennsylvania PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia http://www.cambridge.org © D. D. Fitts 1999 This edition © D. D. Fitts 2002 First published in printed format 1999 A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 65124 7 hardback Original ISBN 0 521 65841 1 paperback ISBN 0 511 00763 9 virtual (netLibrary Edition) Contents Preface viii Chapter 1 The wave function 1 1.1 Wave motion 2 1.2 Wave packet 8 1.3 Dispersion of a wave packet 15 1.4 Particles and waves 18 1.5 Heisenberg uncertainty principle 21 1.6 Young's double-slit experiment 23 1.7 Stern±Gerlach experiment 26 1.8 Physical interpretation of the wave function 29 Problems 34 Chapter 2 SchroÈdinger wave mechanics 36 2.1 The SchroÈdinger equation 36 2.2 The wave function 37 2.3 Expectation values of dynamical quantities 41 2.4 Time-independent SchroÈdinger equation 46 2.5 Particle in a one-dimensional box 48 2.6 Tunneling 53 2.7 Particles in three dimensions 57 2.8 Particle in a three-dimensional box 61 Problems 64 Chapter 3 General principles of quantum theory 65 3.1 Linear operators 65 3.2 Eigenfunctions and eigenvalues 67 3.3 Hermitian operators 69 v 3.4 Eigenfunction expansions 75 3.5 Simultaneous eigenfunctions 77 3.6 Hilbert space and Dirac notation 80 3.7 Postulates of quantum mechanics 85 3.8 Parity operator 94 3.9 Hellmann±Feynman theorem 96 3.10 Time dependence of the expectation value 97 3.11 Heisenberg uncertainty principle 99 Problems 104 Chapter 4 Harmonic oscillator 106 4.1 Classical treatment 106 4.2 Quantum treatment 109 4.3 Eigenfunctions 114 4.4 Matrix elements 121 4.5 Heisenberg uncertainty relation 125 4.6 Three-dimensional harmonic oscillator 125 Problems 128 Chapter 5 Angular momentum 130 5.1 Orbital angular momentum 130 5.2 Generalized angular momentum 132 5.3 Application to orbital angular momentum 138 5.4 The rigid rotor 148 5.5 Magnetic moment 151 Problems 155 Chapter 6 The hydrogen atom 156 6.1 Two-particle problem 157 6.2 The hydrogen-like atom 160 6.3 The radial equation 161 6.4 Atomic orbitals 175 6.5 Spectra 187 Problems 192 Chapter 7 Spin 194 7.1 Electron spin 194 7.2 Spin angular momentum 196 7.3 Spin one-half 198 7.4 Spin±orbit interaction 201 Problems 206 vi Contents Chapter 8 Systems of identical particles 208 8.1 Permutations of identical particles 208 8.2 Bosons and fermions 217 8.3 Completeness relation 218 8.4 Non-interacting particles 220 8.5 The free-electron gas 226 8.6 Bose±Einstein condensation 229 Problems 230 Chapter 9 Approximation methods 232 9.1 Variation method 232 9.2 Linear variation functions 237 9.3 Non-degenerate perturbation theory 239 9.4 Perturbed harmonic oscillator 246 9.5 Degenerate perturbation theory 248 9.6 Ground state of the helium atom 256 Problems 260 Chapter 10 Molecular structure 263 10.1 Nuclear structure and motion 263 10.2 Nuclear motion in diatomic molecules 269 Problems 279 Appendix A Mathematical formulas 281 Appendix B Fourier series and Fourier integral 285 Appendix C Dirac delta function 292 Appendix D Hermite polynomials 296 Appendix E Legendre and associated Legendre polynomials 301 Appendix F Laguerre and associated Laguerre polynomials 310 Appendix G Series solutions of differential equations 318 Appendix H Recurrence relation for hydrogen-atom expectation values 329 Appendix I Matrices 331 Appendix J Evaluation of the two-electron interaction integral 341 Selected bibliography 344 Index 347 Physical constants Contents vii Preface This book is intended as a text for a ®rst-year physical-chemistry or chemical- physics graduate course in quantum mechanics. Emphasis is placed on a rigorous mathematical presentation of the principles of quantum mechanics with applications serving as illustrations of the basic theory. The material is normally covered in the ®rst semester of a two-term sequence and is based on the graduate course that I have taught from time to time at the University of Pennsylvania. The book may also be used for independent study and as a reference throughout and beyond the student's academic program. The ®rst two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent SchroÈdinger equations are introduced along with a discussion of wave functions for particles in a potential ®eld. Some instructors may wish to omit the ®rst or both of these chapters or to present abbreviated versions. Chapter 3 is the heart of the book. It presents the postulates of quantum mechanics and the mathematics required for understanding and applying the postulates. This chapter stands on its own and does not require the student to have read Chapters 1 and 2, although some previous knowledge of quantum mechanics from an undergraduate course is highly desirable. Chapters 4, 5, and 6 discuss basic applications of importance to chemists. In all cases the eigenfunctions and eigenvalues are obtained by means of raising and lowering operators. There are several advantages to using this ladder operator technique over the older procedure of solving a second-order differ- viii ential equation by the series solution method. Ladder operators provide practice for the student in operations that are used in more advanced quantum theory and in advanced statistical mechanics. Moreover, they yield the eigenvalues and eigenfunctions more simply and more directly without the need to introduce generating functions and recursion relations and to consider asymp- totic behavior and convergence. Although there is no need to invoke Hermite, Legendre, and Laguerre polynomials when using ladder operators, these func- tions are nevertheless introduced in the body of the chapters and their proper- ties are discussed in the appendices. For traditionalists, the series-solution method is presented in an appendix. Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the free-electron gas and Bose± Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. The ®rst-year graduate course in quantum mechanics is used in many chemistry graduate programs as a vehicle for teaching mathematical analysis. For this reason, this book treats mathematical topics in considerable detail, both in the main text and especially in the appendices. The appendices on Fourier series and the Fourier integral, the Dirac delta function, and matrices discuss these topics independently of their application to quantum mechanics. Moreover, the discussions of Hermite, Legendre, associated Legendre, La- guerre, and associated Laguerre polynomials in Appendices D, E, and F are more comprehensive than the minimum needed for understanding the main text. The intent is to make the book useful as a reference as well as a text. I should like to thank Corpus Christi College, Cambridge for a Visiting Fellowship, during which part of this book was written. I also thank Simon Capelin, Jo Clegg, Miranda Fyfe, and Peter Waterhouse of the Cambridge University Press for their efforts in producing this book. Donald D. Fitts Preface ix 1 The wave function Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon±the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and super¯uids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new ®eld of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and tech- nology. Quantum theory was developed during the ®rst half of the twentieth century through the efforts of many scientists. In 1926, E. SchroÈdinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. SchroÈdinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the ®rst three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the SchroÈdinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential ®eld. The formal mathematical postulates of quantum theory are presented in Chapter 3. 1 [...]... photon concept of light as required by Einstein's explanation of the photoelectric effect If the monochromatic beam of light consists of a stream of individual photons, then each photon presumably must pass through either slit A or slit B To test this assertion, detectors are placed directly behind slits A and B and both slits are opened The light beam used is of such low intensity that only one photon... medium such as glass Because the phase velocity of each monochromatic plane wave depends on its wavelength, the beam of light is dispersed, or separated onto its component waves, when passed through a glass prism The wave on the surface of water caused by dropping a stone into the water is another example of dispersive wave motion Addition of two plane waves As a speciđc and yet simple example of composite-wave... nodes of B(x, t) nearest to the maximum occur when (x vg t)ka2 equals ặ, i.e., when x is ặ(2ak) from the point of maximum amplitude If we consider the half width of the wave packet between these two nodes as a measure of the uncertainty x in the location of the wave packet and the width (k 2 k 1 ) of the square pulse A(k) as a measure of the uncertainty k in the value of k, then the product of these... force is positive and the atom moves in the positive x-direction For an angle ố between 908 and 1808, the force is negative and the atom moves in the negative x-direction As the silver atoms escape from the oven, their magnetic moments are randomly oriented so that all possible values of the angle ố occur According to classical mechanics, we should expect the beam of silver atoms to form, on the detection... E of a photon is E "ự (1X32) where " is deđned by h 1X054 57 3 1034 J s " 2 Because the photon travels with velocity c, its motion is governed by relativity 1 The history of the development of quantum concepts to explain observed physical phenomena, which occurred mainly in the đrst three decades of the twentieth century, is discussed in introductory texts on physical chemistry and on atomic physics. .. The wave function As time increases from I to 0, the half width of the wave packet jỉó (x, t)j continuously decreases and the maximum amplitude p continuously increases At t 0 the half width attains its lowest value of 2aỏ and the maximum p amplitude attains its highest value of 1a 2, and both values are in agreement with the wave packet in equation (1.20) As time increases from 0 to I, the half width... illustration of a quantum- mechanical effect which is in direct conict with the concepts of classical theory It was the đrst experiment of a non-optical nature to show quantum behavior directly In the SternGerlach experiment, a beam of silver atoms is produced by evaporating silver in a high-temperature oven and allowing the atoms to escape through a small hole The beam is further collimated by passage through... t 2 and increases as jtj increases Thus, the value of the right-hand side when t 0 is the lower bound for the product xk and is in agreement with the uncertainty relation (1.23) 1.4 Particles and waves To explain the photoelectric effect, Einstein (1905) postulated that light, or electromagnetic radiation, consists of a beam of particles, each of which travels at the same velocity c (the speed of light),... absence of an external force, is expressed as the sum of the kinetic and potential energies and is given by 1 2 p2 V (1X35) mv V 2m 2 where m is the mass and v the velocity of the particle, the linear momentum p is E p mv and V is a constant potential energy The force F acting on the particle is given by dV 0 F dx and vanishes because V is constant In classical mechanics the choice of the zero-level of. .. conđned to a narrow band of values Such a composite wave ỉ(x, t) is known as a wave packet and may be expressed as 1 I A(k)ei( kxự t) dk (1X11) ỉ(x, t) p 2 I The weighting factor A(k) for each plane wave of wave number k is negligible except when k lies within a small interval k For mathematical convenience we have included a factor (2)1a2 on the right-hand side of equation (1.11) This factor merely . PRINCIPLES OF QUANTUM MECHANICS: as Applied to Chemistry and Chemical Physics CAMBRIDGE UNIVERSITY PRESS DONALD D. FITTS PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical. thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics DONALD D. . quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry