PRINCIPLES OF QUANTUM MECHANICS: as Applied to Chemistry

361 Pages·2004·1.48 MB·English

PRINCIPLES OF
QUANTUM
MECHANICS:
as Applied to Chemistry
and Chemical Physics
DONALD D. FITTS
CAMBRIDGE UNIVERSITY PRESS
PRINCIPLES OF QUANTUM MECHANICS
as Applied toChemistryand Chemical Physics
This text presents a rigorous mathematical account of the principles of
quantum mechanics, in particular as applied to chemistry and chemical
physics.Applicationsareusedasillustrationsofthebasictheory.
Thefirsttwochaptersserveasanintroductiontoquantumtheory,althoughit
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 Schro¨dinger’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
operatorsareintroducedandusedthroughout.
This advanced text is appropriate for beginning graduate students in chem-
istry,chemicalphysics,molecular physics,andmaterialsscience.
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
laudewith highest honors in chemistry, in 1954from Harvard University and a
Ph.D.degreeinchemistryin1957fromYaleUniversityforhistheoreticalwork
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 joinedthe facultyofthe UniversityofPennsylvania,risingtothe rank
ofProfessorofChemistry.
In Penn’s School of Arts and Sciences, Professor Fitts also served as Acting
DeanforoneyearandasAssociateDeanandDirectoroftheGraduateDivision
for fifteen 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
ChristiCollege,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,theoryofliquids,intermolecularforces,andsurfacephenomena.
PRINCIPLES OF
QUANTUM MECHANICS
as Applied to Chemistry and Chemical Physics
DONALD D. FITTS
UniversityofPennsylvania
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
Chapter1 Thewavefunction 1
1.1 Wavemotion 2
1.2 Wavepacket 8
1.3 Dispersionofawavepacket 15
1.4 Particlesandwaves 18
1.5 Heisenberguncertaintyprinciple 21
1.6 Young’sdouble-slitexperiment 23
1.7 Stern–Gerlachexperiment 26
1.8 Physicalinterpretationofthewavefunction 29
Problems 34
Chapter2 Schro¨dingerwavemechanics 36
2.1 TheSchro¨dingerequation 36
2.2 Thewavefunction 37
2.3 Expectationvaluesofdynamicalquantities 41
2.4 Time-independentSchro¨dingerequation 46
2.5 Particleinaone-dimensionalbox 48
2.6 Tunneling 53
2.7 Particlesinthreedimensions 57
2.8 Particleinathree-dimensionalbox 61
Problems 64
Chapter3 Generalprinciplesofquantumtheory 65
3.1 Linearoperators 65
3.2 Eigenfunctionsandeigenvalues 67
3.3 Hermitianoperators 69
v
vi Contents
3.4 Eigenfunctionexpansions 75
3.5 Simultaneouseigenfunctions 77
3.6 HilbertspaceandDiracnotation 80
3.7 Postulatesofquantummechanics 85
3.8 Parityoperator 94
3.9 Hellmann–Feynmantheorem 96
3.10 Timedependenceoftheexpectationvalue 97
3.11 Heisenberguncertaintyprinciple 99
Problems 104
Chapter4 Harmonicoscillator 106
4.1 Classicaltreatment 106
4.2 Quantumtreatment 109
4.3 Eigenfunctions 114
4.4 Matrixelements 121
4.5 Heisenberguncertaintyrelation 125
4.6 Three-dimensionalharmonicoscillator 125
Problems 128
Chapter5 Angularmomentum 130
5.1 Orbitalangularmomentum 130
5.2 Generalizedangularmomentum 132
5.3 Applicationtoorbitalangularmomentum 138
5.4 Therigidrotor 148
5.5 Magneticmoment 151
Problems 155
Chapter6 Thehydrogenatom 156
6.1 Two-particleproblem 157
6.2 Thehydrogen-likeatom 160
6.3 Theradialequation 161
6.4 Atomicorbitals 175
6.5 Spectra 187
Problems 192
Chapter7 Spin 194
7.1 Electronspin 194
7.2 Spinangularmomentum 196
7.3 Spinone-half 198
7.4 Spin–orbitinteraction 201
Problems 206
Contents vii
Chapter8 Systemsof identicalparticles 208
8.1 Permutationsof identicalparticles 208
8.2 Bosonsandfermions 217
8.3 Completenessrelation 218
8.4 Non-interactingparticles 220
8.5 Thefree-electrongas 226
8.6 Bose–Einsteincondensation 229
Problems 230
Chapter9 Approximationmethods 232
9.1 Variationmethod 232
9.2 Linearvariationfunctions 237
9.3 Non-degenerateperturbationtheory 239
9.4 Perturbedharmonicoscillator 246
9.5 Degenerateperturbationtheory 248
9.6 Groundstateoftheheliumatom 256
Problems 260
Chapter10 Molecularstructure 263
10.1 Nuclearstructureandmotion 263
10.2 Nuclearmotionindiatomicmolecules 269
Problems 279
AppendixA Mathematicalformulas 281
AppendixB FourierseriesandFourierintegral 285
AppendixC Diracdeltafunction 292
AppendixD Hermitepolynomials 296
AppendixE LegendreandassociatedLegendrepolynomials 301
AppendixF LaguerreandassociatedLaguerrepolynomials 310
AppendixG Seriessolutionsofdifferentialequations 318
AppendixH Recurrencerelationforhydrogen-atomexpectationvalues 329
AppendixI Matrices 331
AppendixJ Evaluationofthetwo-electroninteractionintegral 341
Selectedbibliography 344
Index 347
Physicalconstants
Preface
This book is intended as a text for a first-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 first 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
referencethroughoutandbeyondthestudent’sacademicprogram.
The first 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
wavemotionandwavepacketsandthenrelatesparticlemotiontowavemotion.
In Chapter 2 the time-dependent and time-independent Schro¨dinger equations
are introduced along with a discussion of wave functions for particles in a
potential field. Some instructors may wish to omit the first or both of these
chaptersortopresentabbreviatedversions.
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
mechanicsfromanundergraduatecourseishighlydesirable.
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
Preface ix
entialequationbytheseriessolutionmethod.Ladderoperatorsprovidepractice
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
methodispresentedinanappendix.
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
treatsmolecularstructureandnuclearmotion.
The first-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.Theintentistomakethebookusefulasareferenceaswellasatext.
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
UniversityPressfortheireffortsinproducingthisbook.
DonaldD.Fitts
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 superfluids. Applications of quantum behavior give us
transistors, computer chips, lasers, and masers. The relatively new field 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 first half of the twentieth century
throughtheeffortsofmanyscientists.In1926,E.Schro¨dingerinterjectedwave
mechanics into the array of ideas, equations, explanations, and theories that
were prevalent at the time toexplain thegrowing accumulation of observations
of quantum phenomena. His theory introduced the wave function and the
differential wave equation that it obeys. Schro¨dinger’s wave mechanics is now
the backbone of our current conceptional understanding and our mathematical
proceduresforthestudyofquantumphenomena.
Our presentation of the basic principles of quantum mechanics is contained
in the first 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 Schro¨dinger differential wave equation is introduced and the
wave function concept is extended to include particles in an external potential
field. The formal mathematical postulates of quantum theory are presented in
Chapter3.
1