QUANTUM MECHANICS QUANTUM MECHANICS A Conceptual Approach HENDRIK F. HAMEKA A John Wiley & Sons, Inc. Publication Copyright#2004byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,978-750-8400, fax978-646-8600,oronthewebatwww.copyright.com.RequeststothePublisherforpermission shouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet, Hoboken,NJ07030,(201)748-6011,fax(201)748-6008. 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LibraryofCongressCataloging-in-PublicationData: Hameka,HendrikF. Quantummechanics:aconceptualapproach/HendrikF.Hameka. p. cm. Includesindex. ISBN0-471-64965-1(pbk.:acid-freepaper) 1.Quantumtheory. I.Title. QC174.12.H3532004 530.12–dc22 2004000645 PrintedintheUnitedStatesofAmerica 10987654321 To Charlotte 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 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 Schro¨dinger 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 Schro¨dinger 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 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 Schro¨dinger 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 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 Hu¨ckel Molecular Orbital Method, 183 IX Problems, 189 Index 191 PREFACE The physical laws and mathematical structure that constitute the basis of quantum mechanicswerederivedbyphysicists,butsubsequentapplicationsbecameofinter- est not just to the physicists but also to chemists, biologists, medical scientists, engineers,andphilosophers.Quantummechanicaldescriptionsofatomicandmole- cularstructurearenowtaughtinfreshmanchemistryandeveninsomehighschool chemistrycourses.Sophisticatedcomputerprogramsareroutinelyusedforpredict- ingthe structuresand geometriesoflargeorganicmolecules orfor 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, butalmostallofthesetextbooksassumethattheirreadershaveanextensiveback- ground in physics and mathematics; very few of these books make an effort to explain the subject in simple non-mathematical terms. Inthisbookwetrytopresentthefundamentalsandsomesimpleapplicationsof quantum mechanics byemphasizing thebasicconceptsandby keeping the mathe- maticsassimpleaspossible.Wedoassumethatthereaderisfamiliarwithelemen- tary calculus; it is after all not possible to explain the Scho¨dinger equation to someonewhodoesnotknowwhataderivativeoranintegralis.Someofthemathe- maticaltechniquesthatareessentialforunderstandingquantummechanics,suchas matrices and determinants, differential equations, Fourier analysis, and so on are xi xii PREFACE described in a simple manner. We also present some applications to atomic and molecularstructurethatconstitutethebasisofthevariousmolecularstructurecom- puter programs, but we do not attempt to describe the computation techniques in detail. Manyauthorspresentquantummechanicsbymeansoftheaxiomaticapproach, which leads to a rigorous mathematical representation of the subject. However, in some instances it is not easy for an average reader to evenunderstand the axioms, letalonethetheoremsthatarederivedfromthem.Ihavealwayslookeduponquan- 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- standingand appreciation ofthese concepts if thereader is familiar with the back- ground and the personalities of the scientists who conceived them and with the reasoningandargumentsthatledtotheirconception.Ourapproachtothepresenta- tionofquantummechanicsmaythenbecalledhistoricorconceptualbutisperhaps best described as pragmatic. Also, the inclusion of some historical background makes the book more readable. I did not givea detailed description of thevarious 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 mostreliablesourcesaretheoriginalpublicationswherethenewquantummechan- icalideaswerefirstproposed.Thesearereadilyavailableinthescientificliterature, and I was intrigued in reading some of the original papers. I also read various biographiesandautobiographies.IfoundMoore’sbiographyofSchro¨edinger,Con- stance Reid’s biographies of Hilbert and Courant, Abraham Pais’ reminiscences, and the autobiographies of Elsasser and Casimir particularly interesting. I should mentionthatKramerswastheprofessoroftheoreticalphysicswhenIwasastudent atLeidenUniversity.HediedbeforeIfinishedmystudiesandIneverworkedunder his supervision, but I did learn quantum mechanics by reading his book and by attending his lectures. FinallyIwishtoexpressmythankstoMrs.AliceChenforhervaluablehelpin typing and preparing the manuscript. HENDRIK F. HAMEKA