QuantumMechanics Franz Schwabl Quantum Mechanics FourthEdition WithFigures,Tables, NumerousWorkedExamplesandProblems 123 ProfessorDr.FranzSchwabl Physik-Department TechnischeUniversitätMünchen James-Franck-Strasse Garching,Germany E-mail:[email protected] Thefirstedition,,wastranslatedbyDr.RonaldKates TitleoftheoriginalGermanedition:Quantenmechanikthedition (Springer-Lehrbuch)ISBN---- ©Springer-VerlagBerlinHeidelberg LibraryofCongressControlNumber: ISBN ---- thed.SpringerBerlinHeidelbergNewYork ISBN---- rded.SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember, ,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg,,, Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantpro- tectivelawsandregulationsandthereforefreeforgeneraluse. Typesettingandproduction:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Coverdesign:eStudioCalamarS.L.,F.Steinen-Broo,Pau/Girona,Spain SPIN: //YL       Printedonacid-freepaper Preface to the Fourth Edition In this latest edition new material has been added, which includes many additional clarifying remarks to some of the more advanced chapters. The design of many figures has been reworked to enhance the didactic appeal of the book. However,in the course of these changes, I have attempted to keep intact the underlying compact nature of the book. I am grateful to many colleagues for their help with this substantial re- vision. Special thanks go to Uwe Ta¨uber and Roger Hilton for discussions, comments and many constructive suggestions on this new edition. Some of the figures which were of a purely qualitative nature have been improved by Robert Seyrkammer in now being computer-generated. I am very obliged to AndrejVilfanforredoingandcheckingthecomputationofsomeofthescien- tificallymoredemandingfigures.IamalsoverygratefultoMsUlrikeOllinger who undertook the graphical design of the diagrams. It is my pleasure to thank Dr. Thorsten Schneider and Mrs Jacqueline Lenz of Springer for their excellent co-operation, as well as the LE-TEX setting team for their careful incorporationoftheamendmentsforthisnewedition.Finally,Ishouldliketo thankallcolleaguesandstudentswho,overtheyears,havemadesuggestions to improve the usefulness of this book. Munich, August 2007 F. Schwabl Preface to the First Edition This is a textbook on quantum mechanics. In an introductory chapter, the basic postulates are established, beginning with the historical development, by the analysisofaninterferenceexperiment.Fromthenonthe organization is purely deductive. In addition to the basic ideas and numerous applica- tions, new aspects of quantum mechanics and their experimental tests are presented. In the text, emphasis is placed on a concise, yet self-contained, presentation. The comprehensibility is guaranteed by giving all mathemati- cal steps and by carrying out the intermediate calculations completely and thoroughly. Thebooktreatsnonrelativisticquantummechanicswithoutsecondquan- tization, except for an elementary treatment of the quantization of the radi- ation field in the context of optical transitions. Aside from the essential core of quantum mechanics, within which scattering theory, time-dependent phe- nomena, andthe density matrix are thoroughlydiscussed, the book presents the theory ofmeasurementand the Bell inequality. The penultimate chapter is devotedto supersymmetric quantum mechanics,a topic whichto date has only been accessible in the researchliterature. For didactic reasons,we begin with wave mechanics; from Chap. 8 on we introduce the Dirac notation. Intermediate calculations and remarks not es- sentialforcomprehensionarepresentedinsmallprint.Onlyinthesomewhat more advanced sections are references given, which even there, are not in- tended to be complete, but rather to stimulate further reading. Problems at the end of the chapters are intended to consolidate the student’s knowledge. The book is recommended to students of physics and related areas with some knowledge of mechanics and classical electrodynamics, and we hope it will augment teaching material already available. This book came about as the result of lectures on quantum mechanics given by the author since 1973 at the University of Linz and the Technical University of Munich. Some parts of the original rough draft, figures, and tables were completed with the help of R. Alkofer, E. Frey and H.-T. Janka. Careful reading of the proofs by Chr. Baumga¨rtel, R. Eckl, N. Knoblauch, J. Krumrey and W. Rossmann-Bloeck ensured the factual accuracy of the translation. W. Gasser read the entire manuscript and made useful sugges- tionsaboutmanyofthechaptersofthebook.Here,Iwouldliketoexpressmy sinceregratitudetothem,andtoallmyothercolleagueswhogaveimportant assistance in producing this book, as well as to the publisher. Munich, June 1991 F. Schwabl Table of Contents 1. Historical and Experimental Foundations ................. 1 1.1 Introduction and Overview ............................. 1 1.2 Historically Fundamental Experiments and Insights........ 3 1.2.1 Particle Properties of Electromagnetic Waves...... 3 1.2.2 Wave Properties of Particles, Diffraction of Matter Waves..................... 7 1.2.3 Discrete States ................................ 8 2. The Wave Function and the Schro¨dinger Equation........ 13 2.1 The Wave Function and Its Probability Interpretation ..... 13 2.2 The Schro¨dinger Equation for Free Particles .............. 15 2.3 Superposition of Plane Waves........................... 16 2.4 The Probability Distribution for a Measurement of Momentum ........................................ 19 2.4.1 Illustration of the Uncertainty Principle .......... 21 2.4.2 Momentum in Coordinate Space ................. 22 2.4.3 Operators and the Scalar Product................ 23 2.5 The Correspondence Principle and the Schr¨odinger Equation 26 2.5.1 The Correspondence Principle ................... 26 2.5.2 The Postulates of Quantum Theory .............. 27 2.5.3 Many-Particle Systems ......................... 28 2.6 The Ehrenfest Theorem................................ 28 2.7 The Continuity Equation for the Probability Density ...... 31 2.8 Stationary Solutions of the Schro¨dinger Equation, Eigenvalue Equations .................................. 32 2.8.1 Stationary States .............................. 32 2.8.2 Eigenvalue Equations........................... 33 2.8.3 Expansion in Stationary States .................. 35 2.9 The Physical Significance of the Eigenvalues of an Operator 36 2.9.1 Some Concepts from Probability Theory.......... 36 2.9.2 Application to Operators with Discrete Eigenvalues 37 2.9.3 Application to Operators with a Continuous Spectrum .................... 38 2.9.4 Axioms of Quantum Theory..................... 40 X Table of Contents 2.10 Additional Points ..................................... 41 2.10.1 The General Wave Packet....................... 41 2.10.2 Remark on the Normalizability of the Continuum States........................ 43 Problems .................................................. 44 3. One-Dimensional Problems ............................... 47 3.1 The Harmonic Oscillator ............................... 47 3.1.1 The Algebraic Method.......................... 48 3.1.2 The Hermite Polynomials ....................... 52 3.1.3 The Zero-PointEnergy ......................... 54 3.1.4 Coherent States ............................... 56 3.2 Potential Steps ....................................... 58 3.2.1 Continuity of ψ(x) and ψ(cid:2)(x) for a Piecewise Continuous Potential ............. 58 3.2.2 The Potential Step............................. 59 3.3 The Tunneling Effect, the Potential Barrier............... 64 3.3.1 The Potential Barrier .......................... 64 3.3.2 The Continuous Potential Barrier................ 67 3.3.3 Example of Application: α-decay................. 68 3.4 The Potential Well .................................... 71 3.4.1 Even Symmetry ............................... 72 3.4.2 Odd Symmetry ................................ 73 3.5 Symmetry Properties .................................. 76 3.5.1 Parity ........................................ 76 3.5.2 Conjugation................................... 77 3.6 General Discussion of the One-Dimensional Schr¨odinger Equation ............ 77 3.7 The Potential Well, Resonances ......................... 81 3.7.1 Analytic Properties of the Transmission Coefficient. 83 3.7.2 The Motion of a Wave Packet Near a Resonance... 87 Problems .................................................. 92 4. The Uncertainty Relation................................. 97 4.1 The Heisenberg Uncertainty Relation .................... 97 4.1.1 The Schwarz Inequality......................... 97 4.1.2 The General Uncertainty Relation ............... 97 4.2 Energy–Time Uncertainty .............................. 99 4.2.1 Passage Time and Energy Uncertainty............ 100 4.2.2 Duration of an Energy Measurement and Energy Uncertainty ........................ 100 4.2.3 Lifetime and Energy Uncertainty ................ 101 4.3 Common Eigenfunctions of Commuting Operators......... 102 Problems .................................................. 106 Table of Contents XI 5. Angular Momentum ...................................... 107 5.1 Commutation Relations, Rotations ...................... 107 5.2 Eigenvalues of Angular Momentum Operators ............ 110 5.3 Orbital Angular Momentum in Polar Coordinates ......... 112 Problems .................................................. 118 6. The Central Potential I................................... 119 6.1 Spherical Coordinates.................................. 119 6.2 Bound States in Three Dimensions ...................... 122 6.3 The Coulomb Potential ................................ 124 6.4 The Two-Body Problem ............................... 138 Problems .................................................. 140 7. Motion in an Electromagnetic Field ...................... 143 7.1 The Hamiltonian...................................... 143 7.2 Constant Magnetic Field B ............................ 144 7.3 The Normal Zeeman Effect ............................. 145 7.4 Canonical and Kinetic Momentum, Gauge Transformation.. 147 7.4.1 Canonical and Kinetic Momentum ............... 147 7.4.2 Change of the Wave Function Under a Gauge Transformation .................. 148 7.5 The Aharonov–BohmEffect ............................ 149 7.5.1 The Wave Function in a Region Free of Magnetic Fields......................... 149 7.5.2 The Aharonov–BohmInterference Experiment..... 150 7.6 Flux Quantization in Superconductors ................... 153 7.7 Free Electrons in a Magnetic Field ...................... 154 Problems .................................................. 155 8. Operators, Matrices, State Vectors ....................... 159 8.1 Matrices, Vectors, and Unitary Transformations........... 159 8.2 State Vectors and Dirac Notation ....................... 164 8.3 The Axioms of Quantum Mechanics ..................... 169 8.3.1 Coordinate Representation...................... 170 8.3.2 Momentum Representation...................... 171 8.3.3 Representation in Terms of a Discrete Basis System 172 8.4 Multidimensional Systems and Many-Particle Systems............................. 172 8.5 The Schro¨dinger, Heisenberg and Interaction Representations......................... 173 8.5.1 The Schro¨dinger Representation ................. 173 8.5.2 The Heisenberg Representation .................. 174 8.5.3 The Interaction Picture (or Dirac Representation) . 176 8.6 The Motion of a Free Electron in a Magnetic Field ........ 177 Problems .................................................. 181 XII Table of Contents 9. Spin ...................................................... 183 9.1 The Experimental Discovery of the Internal Angular Momentum...................... 183 9.1.1 The “Normal” Zeeman Effect.................... 183 9.1.2 The Stern–Gerlach Experiment .................. 183 9.2 Mathematical Formulation for Spin-1/2 .................. 185 9.3 Properties of the Pauli Matrices......................... 186 9.4 States, Spinors........................................ 187 9.5 Magnetic Moment..................................... 188 9.6 Spatial Degrees of Freedom and Spin .................... 189 Problems .................................................. 191 10. Addition of Angular Momenta............................ 193 10.1 Posing the Problem ................................... 193 10.2 Addition of Spin-1/2 Operators ......................... 194 10.3 Orbital Angular Momentum and Spin 1/2................ 196 10.4 The General Case ..................................... 198 Problems .................................................. 201 11. Approximation Methods for Stationary States ............ 203 11.1 Time Independent Perturbation Theory (Rayleigh–Schro¨dinger) ................................ 203 11.1.1 Nondegenerate Perturbation Theory.............. 204 11.1.2 Perturbation Theory for Degenerate States........ 206 11.2 The Variational Principle .............................. 207 11.3 The WKB (Wentzel–Kramers–Brillouin)Method.......... 208 11.4 Brillouin–Wigner Perturbation Theory................... 211 Problems .................................................. 212 12. Relativistic Corrections................................... 215 12.1 Relativistic Kinetic Energy ............................. 215 12.2 Spin–Orbit Coupling .................................. 217 12.3 The Darwin Term ..................................... 219 12.4 Further Corrections ................................... 222 12.4.1 The Lamb Shift ............................... 222 12.4.2 Hyperfine Structure ............................ 222 Problems .................................................. 225 13. Several-Electron Atoms................................... 227 13.1 Identical Particles ..................................... 227 13.1.1 Bosons and Fermions........................... 227 13.1.2 Noninteracting Particles ........................ 230 13.2 Helium .............................................. 233 13.2.1 Without the Electron–ElectronInteraction ........ 233