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Quantum Theory Lecture Notes Quantum Theory by Prof. Maximilian Kreuzer Institute for Theoretical Physics Vienna University of Technology covering the contents of 136.019 Quantentheorie I and 136.027 Quantentheorie II Edition 09/10 — Version July 15, 2009 Links The current version of the notes, as well as information on lectures and exams, is available at http://hep.itp.tuwien.ac.at/~kreuzer/QT.html Reports of typos and errors and suggestions for improvements are appreciated, e.g. by e-mail to [email protected] (if possible after cross-checking with the current version). Preface The structure of these lecture notes is mainly motivated by the curricula of the bachelor’s and master’s programs of the faculty of physics at the Vienna University of Technology, which requires a division of quantum mechanics into two parts. The first part Quantum Theory I: chapters 1 – 7 should make available the prerequisites for the subsequent lecture on atomic physics and has to be covered in 45 units of 45 minutes each. After historic recollections in the introduction the principles of quantum theory are first illustrated for one-dimensional examples in chapter 2 and then presented in the proper formalism in chapter 3. In chapters 4 and 5 we solve the Schr¨odinger equation for the spherically symmetric hydrogen atom and treat the quantization and the addition of general angular momenta, respectively. Chapter 6 introduces approximation techniques and chapter 7 initiates relativistic quantum mechanics and derives the Pauli equation and the fine structure corrections in the non-relativistic limit of the Dirac equation. The systematic discussion of symmetries as well as identical particles and many particle theory had to be postponed to part 2, Quantum Theory II: chapters 8 – 11. In chapter 8 we start with 3-dimensional scattering theory. Transformations, symmetries and conservation laws are discussed in chapter 9 and applied to non-relativistic and relativistic contexts. In chapter 10 we discuss many particle systems. The Hartree–Fock approximation is used as a motivation for the introduction of the occupation number representation and the quantization of the radiation field. These three chapters are largely independent so that their order could be permuted with little modifications. In the last chapter we discuss semiclassical methods and the path integral. Acknowledgements A first draft of these lecture notes was created by Katharina Dobes (chap. 1,6,10), Wolfgang Dungel (chap. 3,11), Florian Hinterschuster (chap. 4,5,9) and Daniel Winklehner (2,7,8,9) as a project work. While the text was then largely rewritten by the lecturer, the draft provided many valuable ideas for the structure and the presentation of the contents. My acknowledgements also go to my colleagues at the Institute for Theoretical Physics for sharing their knowledge and ideas, with special thanks to Harald Grosse (Vienna University), Anton Rebhan and Karl Svozil, whose expertise was of great help, and to the late Wolfgang Kummer, from whom I learned quantum mechanics (and quantum field theory) in the first place. In addition to input from many of the books in the references I took advantage of the excellent lecture notes of Profs. Burgd¨orfer, Hafner and Kummer. Often as a first and sometimes as a last resort I used Wikipedia and Google. Last but not least, many thanks to the students who are helping to improve these lecture notes by reporting errors and typos. I Contents 1 Introduction 1 1.1 Historicalnotes................................. ....... 1 1.2 Limitationsofclassicalphysics . ............ 4 1.2.1 Blackbodyradiation . ..... 4 1.2.2 Thephotoelectriceffect . ...... 6 1.2.3 Bohr’s theory of the structure of atoms . ......... 7 1.2.4 TheComptoneffect ............................... 10 1.2.5 Interferencephenomena . ...... 12 2 Wave Mechanics and the Schr¨odinger equation 14 2.1 TheSchr¨odingerequation . ......... 14 2.1.1 Probability density and probability current density ................ 16 2.1.2 Axiomsofquantumtheory . 17 2.1.3 Spreading of free wave packets and uncertainty relation ............. 18 2.2 The time-independent Schr¨odinger equation . .............. 21 2.2.1 One-dimensional square potentials and continuity conditions .......... 23 2.2.2 Bound states and the potential well . ........ 25 2.2.3 Scattering and the tunneling effect . ......... 27 2.2.4 Transfer matrix and scattering matrix . .......... 30 2.3 Theharmonicoscillator . ........ 32 3 Formalism and interpretation 37 3.1 LinearalgebraandDiracnotation . .......... 38 3.2 Operatorcalculus................................ ....... 41 3.3 OperatorsandHilbertspaces . ......... 50 3.3.1 Inequalities .................................. 51 3.3.2 Position and momentum representations . .......... 52 3.3.3 Convergence, norms and spectra of Hilbert space operators ........... 54 3.3.4 Self-adjoint operators and spectral representation . ................ 57 3.4 Schr¨odinger, Heisenberg and interaction picture . .................. 60 II 3.5 Ehrenfest theorem and uncertainty relations . ............... 63 3.6 Harmonic oscillator and ladder operators . .............. 66 3.6.1 Coherentstates................................ 69 3.7 Axioms and interpretation of quantum mechanics . .............. 71 3.7.1 Mixed states and the density matrix . ........ 71 3.7.2 Measurements and interpretation . ......... 72 3.7.3 Schr¨odinger’s cat and the Einstein-Podolsky-Rosen argument . 74 4 Orbital angular momentum and the hydrogen atom 77 4.1 The orbital angular momentum . ........ 77 4.1.1 Commutationrelations. ...... 78 4.1.2 Angular momentum and spherical harmonics . ......... 79 4.2 Thehydrogenatom................................. ..... 82 4.2.1 Thetwoparticleproblem . ..... 82 4.2.2 Thehydrogenatom ............................... 83 4.3 Summary ......................................... 88 5 Angular Momentum and Spin 89 5.1 Quantization of angular momenta . .......... 92 5.2 ElectronspinandthePauliequation . ........... 95 5.2.1 Magnetic fields: Pauli equation and spin-orbit coupling ............. 97 5.3 Addition of Angular Momenta . ........ 98 5.3.1 Clebsch-Gordancoefficients . 100 5.3.2 Singlet,tripletandEPRcorrelations . ........... 102 6 Methods of Approximation 104 6.1 Rayleigh–Schr¨odinger perturbation theory . ................ 104 6.1.1 Degenerate time independent perturbation theory . ............. 107 6.2 The fine structure of the hydrogen atom . .......... 107 6.3 External fields: Zeeman effect and Stark effect . ............ 111 6.4 Thevariationalmethod(Riesz) . .......... 115 6.4.1 Ground state energy of the helium atom . ........ 117 III 6.4.2 Applying the variational method and the virial theorem ............. 119 6.5 Timedependentperturbationtheory . ........... 120 6.5.1 Absorption and emission of electromagnetic radiation .............. 124 7 Relativistic Quantum Mechanics 127 7.1 TheDirac-equation................................ 128 7.2 Nonrelativistic limit and the Pauli-equation . ............... 132 8 Scattering Theory 135 8.1 Thecentralpotential. .. .. .. .. .. .. .. ........ 135 8.1.1 Differential cross section and frames of reference . .............. 136 8.1.2 Asymptotic expansion and scattering amplitude . ............. 137 8.2 Partialwaveexpansion. ........ 139 8.2.1 Expansion of a plane wave in spherical harmonics . ............ 141 8.2.2 Scattering amplitude and phase shift . .......... 143 8.2.3 Example: Scatteringbyasquarewell . ........ 145 8.2.4 Interpretation of the phase shift . ......... 147 8.3 The Lippmann-Schwinger equation . ......... 149 8.4 TheBornseries................................... 152 8.4.1 Application: Coulomb scattering and the Yukawa potential ........... 154 8.5 Wave operator, transition operator and S-matrix ..................... 155 9 Symmetries and transformation groups 159 9.1 Transformationgroups . ........ 160 9.2 Noethertheoremandquantization . .......... 163 9.3 Rotationofspins ................................. 167 9.3.1 Tensor operators and the Wigner Eckhart theorem . ........... 171 9.4 Symmetries of relativistic quantum mechanics . ............... 173 9.4.1 Lorentz covariance of the Dirac-equation . ........... 174 9.4.2 Spinandhelicity ............................... 175 9.4.3 Dirac conjugation and Lorentz tensors . .......... 176 9.5 Parity, time reversal and charge-conjugation . ............... 178 IV 9.5.1 Discrete symmetries of the Dirac equation . ........... 180 9.6 Gauge invariance and the Aharonov–Bohm effect . ............ 181 10 Many–particle systems 184 10.1 Identical particles and (anti)symmetrization . .................. 185 10.2 Electron-electronscattering . ............ 189 10.3 Selfconsistent fields and Hartree-Fock . ............. 190 10.4 Occupation number representation . ............ 195 10.4.1 Quantization of the radiation field . .......... 197 10.4.2 Interaction of matter and radiation . ........... 199 10.4.3 Phononsandquasiparticles . ........ 201 11 WKB and the path integral 202 11.1 WKBapproximation............................... 203 11.1.1 Bound states, tunneling, scattering and EKB . ............ 206 11.2Thepathintegral................................ ....... 208 References 211 Chapter 1 Introduction 1.1 Historical notes In the nineteenth century the profession of a specialized scientist was created and the main scientific activity moved to university-like institutions. As a result scientific research flourished. One of the major and at the same time one of the oldest branches of physics was mechanics. Its foundation dates back to 1687, when Isaac Newton
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