Quantum physics f scheck

Quantum Physics Florian Scheck Quantum Physics With 76 Figures, 102 Exercises, Hints and Solutions Professor Dr Florian Scheck Universität Mainz Institut für Physik, Theoretische Elementarteilchenphysik Staudinger Weg 55099 Mainz, Germany e-mail: scheck@thep.physik.uni-mainz.de ISBN 978-3-540-25645-8 Springer Berlin Heidelberg New York Cataloging-in-Publication Data Library of Congress Control Number: 2007921674 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover Design: WMXDesign GmbH, Heidelberg SPIN 11000198 56/3100/YL - Printed on acid-free paper To the memory of my father, Gustav O Scheck (1901–1984), who was a great musician and an exceptional personality VII Preface This book is divided into two parts: Part One deals with nonrelativistic quantum mechanics, from bound states of a single particle (harmonic oscillator, hydrogen atom) to fermionic many-body systems Part Two is devoted to the theory of quantized fields and ranges from canonical quantization to quantum electrodynamics and some elements of electroweak interactions Quantum mechanics provides both the conceptual and the practical basis for almost all branches of modern physics, atomic and molecular physics, condensed matter physics, nuclear and elementary particle physics By itself it is a fascinating, though difficult, part of theoretical physics whose physical interpretation gives rise, still today, to surprises in novel applications, and to controversies regarding its foundations The mathematical framework, in principle, ranges from ordinary and partial differential equations to the theory of Lie groups, of Hilbert spaces and linear operators, to functional analysis, more generally He or she who wants to learn quantum mechanics and is not familiar with these topics, may introduce much of the necessary mathematics in a heuristic manner, by invoking analogies to linear algebra and to classical mechanics (Although this is not a prerequisite it is certainly very helpful to know a good deal of canonical mechanics!) Quantum field theory deals with quantum systems whith an infinite number of degrees of freedom and generalizes the principles of quantum theory to fields, instead of finitely many point particles As Sergio Doplicher once remarked, quantum field theory is, after all, the real theory of matter and radiation So, in spite of its technical difficulties, every physicist should learn, at least to some extent, concepts and methods of quantum field theory Chapter starts with examples for failures of classical mechanics and classical electrodynamics in describing quantum systems and develops what might be called elementary quantum mechanics The particle-wave dualism, together with certain analogies to HamiltonJacobi mechanics are shown to lead to the Schrödinger equation in a rather natural way, leaving open, however, the question of interpretation of the wave function This problem is solved in a convincing way by Born’s statistical interpretation which, in turn, is corroborated by the concept of expectation value and by Ehrenfest’s theorem Having learned how to describe observables of quantum systems one then solves single-particle problems such as the harmonic oscillator in one dimension, the spherical oscillator in three dimensions, and the hydrogen atom VIII Preface Chapter develops scattering theory for particles scattered on a given potential Partial wave analysis of the scattering amplitude as an example for an exact solution, as well as Born approximation for an approximate description are worked out and are illustrated by examples The chapter also discusses briefly the analytical properties of partial wave amplitudes and the extension of the formalism to inelastic scattering Chapter formalizes the general principles of quantum theory, on the basis of the empirical approach adopted in the first chapter It starts with representation theory for quantum states, moves on to the concept of Hilbert space, and describes classes of linear operators acting on this space With these tools at hand, it then develops the description and preparation of quantum states by means of the density matrix Chapter discusses space-time symmetries in quantum physics, a first tour through the rotation group in nonrelativistic quantum mechanics and its representations, space reflection, and time reversal It also addresses symmetry and antisymmetry of systems of a finite number of identical particles Chapter which concludes Part One, is devoted to important practical applications of quantum mechanics, ranging from quantum information to time independent as well as time dependent perturbation theory, and to the description of many-body systems of identical fermions Chapter 6, the first of Part Two, begins with an extended analysis of symmetries and symmetry groups in quantum physics Wigner’s theorem on the unitary or antiunitary realization of symmetry transformations is in the focus here There follows more material on the rotation group and its use in quantum mechanics, as well as a brief excursion to internal symmetries The analysis of the Lorentz and Poincar´e groups is taken up from the perspective of particle properties, and some of their unitary representations are worked out Chapter describes the principles of canonical quantization of Lorentz covariant field theories and illustrates them by the examples of the real and complex scalar field, and the Maxwell field A section on the interaction of quantum Maxwell fields with nonrelativistic matter illustrates the use of second quantization by a number of physically interesting examples The specific problems related to quantized Maxwell theory are analyzed and solved in its covariant quantization and in an investigation of the state space of quantum electrodynamics Chapter takes up scattering theory in a more general framework by defining the S-matrix and by deriving its properties The optical theorem is proved for the general case of elastic and inelastic final states and formulae for cross sections and decay widths are worked out in terms of the scattering matrix Chapter deals exclusively with the Dirac equation and with quantized fields describing spin-1/2 particles After the construction of the quantized Dirac field and a first analysis of its interactions we also ex- Preface plore the question to which extent the Dirac equation may be useful as an approximate single-particle theory Chapter 10 describes covariant perturbation theory and develops the technique of Feynman diagrams and their translation to analytic amplitudes A number of physically relevant tree processes of quantum electrodynamics are worked out in detail Higher order terms and the specific problems they raise serve to introduce and to motivate the concepts of regularization and of renormalization in a heuristic manner Some prominent examples of radiative corrections serve to illustrate their relevance for atomic and particle physics as well as their physical interpretation The chapter concludes with a short excursion into weak interactions, placing these in the framework of electroweak interactions The book covers material (more than) sufficient for two full courses and, thus, may serve as accompanying textbook for courses on quantum mechanics and introductory quantum field theory However, as the main text is largely self-contained and contains a considerable number of worked-out examples, it may also be useful for independent individual study The choice of topics and their presentation closely follows a two-volume German text well established at German speaking universities Much of the material was tested and fine-tuned in lectures I gave at Johannes Gutenberg University in Mainz The book contains many exercises for some of which I included complete solutions or gave some hints In addition, there are a number of appendices collecting or explaining more technical aspects Finally, I included some historical remarks about the people who pioneered quantum mechanics and quantum field theory, or helped to shape our present understanding of quantum theory.1 I am grateful to the students who followed my courses and to my collaborators in research for their questions and critical comments some of which helped to clarify matters and to improve the presentation Among the many colleagues and friends from whom I learnt a lot about the quantum world I owe special thanks to Martin Reuter who also read large parts of the original German manuscript, to Wolfgang Bulla who made constructive remarks on formal aspects of quantum mechanics, and to Othmar Steinmann from whom I learnt a good deal of quantum field theory during my years at ETH and PSI in Zurich The excellent cooperation with the people at Springer-Verlag, notably Dr Thorsten Schneider and his crew, is gratefully acknowledged Mainz, December 2006 Florian Scheck I will keep track of possible errata on an internet page attached to my home page The latter can be accessed via http://wwwthep.uni-mainz.de/staff.html I will be grateful for hints to misprints or errors IX XI Table of Contents PART ONE From the Uncertainty Relation to Many-Body Systems Quantum Mechanics of Point Particles 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Limitations of Classical Physics Heisenberg’s Uncertainty Relation for Position and Momentum 1.2.1 Uncertainties of Observables 1.2.2 Quantum Mechanical Uncertainties of Canonically Conjugate Variables 1.2.3 Examples for Heisenberg’s Uncertainty Relation The Particle-Wave Dualism 1.3.1 The Wave Function and its Interpretation 1.3.2 A First Link to Classical Mechanics 1.3.3 Gaussian Wave Packet 1.3.4 Electron in External Electromagnetic Fields Schrödinger Equation and Born’s Interpretation of the Wave Function Expectation Values and Observables 1.5.1 Observables as Self-Adjoint Operators on L2 (Ê ) 1.5.2 Ehrenfest’s Theorem A Discrete Spectrum: Harmonic Oscillator in one Dimension Orthogonal Polynomials in One Real Variable Observables and Expectation Values 1.8.1 Observables With Nondegenerate Spectrum 1.8.2 An Example: Coherent States 1.8.3 Observables with Degenerate, Discrete Spectrum 1.8.4 Observables with Purely Continuous Spectrum Central Forces and the Schrödinger Equation 1.9.1 The Orbital Angular Momentum: Eigenvalues and Eigenfunctions 1.9.2 Radial Momentum and Kinetic Energy 1.9.3 Force Free Motion with Sharp Angular Momentum 1.9.4 The Spherical Oscillator 1.9.5 Mixed Spectrum: The Hydrogen Atom 16 17 20 24 26 28 31 32 35 39 45 47 50 53 65 72 72 77 81 86 91 91 101 104 111 118 Scattering of Particles by Potentials 2.1 2.2 2.3 2.4 2.5 Macroscopic and Microscopic Scales Scattering on a Central Potential Partial Wave Analysis 2.3.1 How to Calculate Scattering Phases 2.3.2 Potentials with Infinite Range: Coulomb Potential Born Series and Born Approximation 2.4.1 First Born Approximation 2.4.2 Form Factors in Elastic Scattering *Analytical Properties of Partial Wave Amplitudes 129 131 136 140 144 147 150 152 156 XII Table of Contents 2.6 2.5.1 Jost Functions 2.5.2 Dynamic and Kinematic Cuts 2.5.3 Partial Wave Amplitudes as Analytic Functions 2.5.4 Resonances 2.5.5 Scattering Length and Effective Range Inelastic Scattering and Partial Wave Analysis 157 158 161 161 164 167 The Principles of Quantum Theory 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Representation Theory 3.1.1 Dirac’s Bracket Notation 3.1.2 Transformations Relating Different Representations The Concept of Hilbert Space 3.2.1 Definition of Hilbert Spaces 3.2.2 Subspaces of Hilbert Spaces 3.2.3 Dual Space of a Hilbert Space and Dirac’s Notation Linear Operators on Hilbert Spaces 3.3.1 Self-Adjoint Operators 3.3.2 Projection Operators 3.3.3 Spectral Theory of Observables 3.3.4 Unitary Operators 3.3.5 Time Evolution of Quantum Systems Quantum States 3.4.1 Preparation of States 3.4.2 Statistical Operator and Density Matrix 3.4.3 Dependence of a State on Its History 3.4.4 Examples for Preparation of States A First Summary Schrödinger and Heisenberg Pictures Path Integrals 3.7.1 The Action in Classical Mechanics 3.7.2 The Action in Quantum Mechanics 3.7.3 Classical and Quantum Paths 171 174 177 180 182 187 188 190 191 194 196 200 202 203 204 207 210 213 214 216 218 219 220 224 Space-Time Symmetries in Quantum Physics 4.1 4.2 4.3 The Rotation Group (Part 1) 4.1.1 Generators of the Rotation Group 4.1.2 Representations of the Rotation Group 4.1.3 The Rotation Matrices D 4.1.4 Examples and Some Formulae for D-Matrices 4.1.5 Spin and Magnetic Moment of Particles with j = 1/2 4.1.6 Clebsch-Gordan Series and Coupling of Angular Momenta 4.1.7 Spin and Orbital Wave Functions 4.1.8 Pure and Mixed States for Spin 1/2 Space Reflection and Time Reversal in Quantum Mechanics 4.2.1 Space Reflection and Parity 4.2.2 Reversal of Motion and of Time 4.2.3 Concluding Remarks on T and Π Symmetry and Antisymmetry of Identical Particles 4.3.1 Two Distinct Particles in Interaction 4.3.2 Identical Particles with the Example N = 4.3.3 Extension to N Identical Particles 4.3.4 Connection between Spin and Statistics 227 227 230 236 238 239 242 245 246 248 248 251 255 258 258 261 265 266 Exercises, Hints, and Selected Solutions 9.8 Let space reflection x = (t, x) → x = (t, −x) be applied to the Dirac equation (9.56) Show that if ψ(x) is a solution of the Dirac equation, then also ψΠ (x) = γ ψ(x ) is a solution The following two exercises give a hint as to when the Dirac equation is applicable and when it fails as a single-particle equation, the failures being due to the admixture of states with “negative energy.” 9.9 A fermion in one dimension is scattered by a step potential U(x) = U0 Θ(x), see Fig Before the scattering it comes in from x < with momentum p and spin orientation m s = +1/2 Write down, in the standard representation, the incoming wave, the reflected wave, and the wave that goes through towards positive x Use continuity of the solution at x = to fix the free parameters in the three waves (up to a common normalization) Calculate the current going through, and the reflected current, in relation to the incoming current Discuss the results for the case (E − U0 )2 < m and for the case (E − U0 )2 > m , U0 > E + m Comment: The paradoxical results that one obtains in the second case, are called Klein’s paradox 9.10 In close analogy to (9.55) one constructs a wave packet ψ(t, x) from the complete system of free solutions with positive and negative frequency Calculate the spatial current J i for this wave packet Estimate the frequencies of the oscillations in the mixed, positive and negative frequency terms Comment: These rapid oscillations are called Zitterbewegung 9.11 We consider bound states of an electron in the Coulomb potential for weakly relativistic motion Show: For κ < 0, that is, for = −κ − 1, we have rgnκ (r) ≈ yn (r), where yn (r) is the radial function of the nonrelativistic hydrogen atom, (cf (1.155)) Remark: The result shows that in the nonrelativistic limit 1 n, κ, j = −κ − = + and 2 1 n, κ = −κ − 1, j = κ − = − 2 have the same radial function Exercises: Chapter 10 10.1 Work out the rule (R9) in detail 10.2 Assume the current density j µ (x) to be self-adjoint and conserved, ∂µ j µ (x) = Then, in the decomposition in terms of covariants, only the form factors F1 (Q ) and F2 (Q ) are different from zero Show that the form factors are real U(x) U0 x Fig 723 724 Exercises, Hints, and Selected Solutions Solution: The current density being self-adjoint we have p| jµ† (0) |q = q| jµ (0) | p ∗ = p| jµ (0) |q (31) The first equal sign is nothing but the definition of the adjoint operator while the second reflects the assumption Inserting the decomposition (10.76) the middle term of (31) yields ∗ i u † (q)γ0 γµ F1 (Q ) − σµν ( p − q)ν F2 (Q ) u( p) 2m i † ∗ † † = u ( p) γ0 γµ F1 (Q ) + γ0 σµν ( p − q)ν F2∗ (Q ) u(q) 2m i † = u( p) γ0 㵆 γ0 F1∗ (Q ) − γ0 σµν γ0 (q − p)ν F2∗ (Q ) u(q) 2m In the last step we inserted γ0 = 1l between u † ( p) and the expression to the right of it From the properties of the γ -matries we have γ0 㵆 γ0 = γµ , † γ0 σµν γ0 = σµν Hence, the second equal sign in (31) yields F1∗ (Q ) = F1 (Q ) , F2∗ (Q ) = F2 (Q ) (32) Both form factors are indeed real 10.3 Prove the integral that we used in calculating the anomalous magnetic moment, d4 v I := v2 − Λ2 + iε =− iπ 2Λ2 Solution: Splitting into integrals over v0 and over v, the latter is formulated in terms of spherical polar coordinates of R3 With r ≡ |v| one has +∞ I= ∞ dv −∞ dr r ∞ = 4π dr r dv0 −∞ v02 − Λ + iε , where Λ := r + Λ2 The denominator may equivalently be written as ⎯• Λ−ιε Fig +∞ 2 v02 − Λ + iε ⎯ −Λ+ιε • dΩ v02 − (Λ − iε) v0 − (Λ − iε) v0 + (Λ − iε) The positions of the singularities of the integrand in the complex v0 plane are as shown in Fig One closes the path of integration by a semi-circle at infinity, as sketched in the figure, and uses Cauchy’s integral theorem in the form f(ζ) dζ = πi f (z) , (ζ − z)3 Exercises, Hints, and Selected Solutions (It follows from the theorem of residua by twofold derivative in z) One obtains ∞ I = −4π i dr r 12 = − π 2i 2Λ) ∞ dr r2 r + Λ2 5/2 The remaining integral is done in an elementary way, by partial integration with u (r) = r/(r + Λ2 )5/2 and v(r) = r The result is ∞ dr r2 5/2 r + Λ2 = , 3Λ2 from which the assertion follows directly 10.4 At order O(α) the differential cross sections in the center-of-mass system for the following processes are symmetric about 90◦ : e− + e− −→ e− + e− e+ + e− −→ γ + γ e+ + e− −→ µ+ + µ− e+ + e− −→ τ + + τ − Why is this so? Why does this not hold for e+ + e− → e+ + e− ? 10.5 Prove the integral formulae (A.11) which are needed in the analysis of the self-energy Hint: Since x = (x )2 − x2 , the nonvanishing integrals factorize in integrals over each coordinate separately In every one-dimensional integral the path of integration can be rotated by 45◦ in the complex plane The integral then becomes a Gauss integral 10.6 Consider the processes (10.125) – (10.131), sketch tree diagrams for these reactions, and decide to which of them the weak charged current contributes, to which the weak neutral current contributes, and to which both of them contribute 10.7 Compute the decay rate of µ-decay taking into account the W ± propagator Show that the rate is modified as shown (10.139) 725 727 Bibliography Mechanics and Electrodynamics Landau, L D., Lifschitz, E M.: The Classical Theory of Fields (Pergamon Press, Oxford 1987) Jackson, J D.: Classical Electrodynamics (John Wiley, New York 1999) For the sake of reference to specific topics in mechanics (further books on mechanics see therein): Scheck, F.: Mechanics: From Newton’s Laws to Deterministic Chaos (Springer-Verlag, Heidelberg 2005) Classics on Quantum Theory Condon, E U., Shortley, E H.: The Theory of Atomic Spectra (Cambridge University Press, London 1957) Dirac, P A M.: The Principles of Quantum Mechanics (Oxford Science Publications, Clarendon Press Oxford 1996) Heisenberg, W.: Physikalische Prinzipien der Quantentheorie (BI, Mannheim 1958) Heitler, W.: The Quantum Theory of Radiation (Oxford University Press, Oxford 1953) Von Neumann, J.: Mathematical Foundations of Quantum Mechanics (Princeton University Press, German original: Springer-Verlag, Berlin 1932) Pauli, W.: General Principles of Quantum Mechanics (Springer-Verlag, Heidelberg 1980); Wigner, E P.: Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra (Academic Press, New York 1959) Selected Textbooks on Quantum Mechanics Basdevant, J.-L., Dalibard, J.: Quantum Mechanics (Springer-Verlag, Berlin, Heidelberg 2002) Basdevant, J.-L., Dalibard, J.: The Quantum Mechanics Solver (SpringerVerlag, Ecole Polytechnique, Heidelberg 2000) Feynman, R P., Hibbs, A R.: Quantum Mechanics and Path Integrals (McGraw Hill 1965) Gottfried, K., Yan, T M.: Quantum Mechanics: Fundamentals (SpringerVerlag, New York, Berlin 2003) Landau, L D., Lifschitz, E M.: Quantum Mechanics (Pergamon 1977) Merzbacher, E.: Quantum Mechanics (John Wiley and Sons, New York 1997) 728 Bibliography Sakurai, J J.: Modern Quantum Mechanics (Addison-Wesley, Reading, Mass 1994) Schiff, L I.: Quantum Mechanics (McGraw-Hill 1968) Schwinger, J.: Quantum Mechanics, Symbolism of Atomic Measurements (Springer-Verlag, New York, Heidelberg 2001) Thirring, W.: A Course in Mathematical Physics, Volume 3: Quantum Mechanics of Atoms and Molecules (Springer, Berlin, Heidelberg 1981) Extensive Monographs on Quantum Mechanics and Fields Most of these might be to extensive for a first tour through quantum mechanics All of them are suitable, however, for further study and as references for specific topics Cohen-Tannoudji, C., Diu, B., Lalo¨e, F.: M´ecanique Quantique I + II (Hermann, Paris 1977) Galindo, A., Pascual, P.: Quantum mechanics I + II (Springer-Verlag, Berlin Heidelberg New York 1990) Itzykson, Cl., Zuber, J.-B.: Quantum Field Theory (McGraw-Hill, New York 1980) Messiah, A.: Quantum Mechanics + (North Holland Publ., Amsterdam 1964) Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford 1994) Problems of Quantum Mechanics on a PC Feagin, J M.: Quantum Methods with Mathematica (Springer, Berlin, Heidelberg 1994) Horbatsch, M.: Quantum Mechanics using Maple (Springer, Berlin, Heidelberg 1995) Selected Books on Relativistic Quantum Theory and Quantum Field Theory Bethe, H A., Jackiw, R W.: Intermediate Quantum Mechanics (Benjamin Cummings, Menlo Park 1986) Bethe, H A., Salpeter, E E.: Quantum Mechanics of One and Two Electron Atoms (Springer-Verlag, Berlin 1957) Bjork´en, J D., Drell, S D.: Relativistic Quantum Mechanics (McGraw Hill, New York 1964) Bjork´en, J D., Drell, S D.: Relativistic Quantum Fields (McGraw Hill, New York 1965) Bogoliubov, N N , Shirkov, D V.: Introduction to the Theory of Quantized Fields (Interscience, New York 1959) Collins, J C.: Renormalization (Cambridge University Press, Cambridge 1998) Bibliography Dittrich, W., Reuter, M.: Classical and Quantum Dynamics (Springer, Heidelberg 2001) Gasiorowicz, S.: Elementary Particle Physics (John Wiley & Sons, New York 1966) Grosche, Ch., Steiner, F.: Handbook of Feynman Path Integrals (Springer, Heidelberg 1998) Jauch, J M., Rohrlich, F.: The Theory of Photons and Electrons: The Relativistic Quantum Field Theory of Charged Particles with Spin one-half (Springer-Verlag, New York 1976) Jost, R.: The General Theory of Quantized Fields (Am Mathematical Soc., Providence 1965) Källen, G.: Quantum Electrodynamics (Springer-Verlag, Berlin 1972) Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics (World Scientific, Singapore 1990) Le Bellac, M.: Quantum and Statistical Field Theory (Clarendon Press, Oxford 1991) Ramond, P.: Field Theory – a Modern Primer (Benjamin/Cummings Publ Co., Reading 1981) Sakurai, J J.: Advanced Quantum Mechanics (Benjamin/Cummings 1984) Salmhofer, M.: Renormalization (Springer-Verlag, Heidelberg 1999) Scheck, F.: Electroweak and Strong Interactions – An Introduction to Theoretical Particle Physics (Springer-Verlag, Heidelberg 1996) Schweber, S S.: An Introduction to Quantum Field Theory (Row, Peterson & Co., Evanston 1961) Streater, R F., Wightman, A S.: PCT, Spin & Statistics, and All That (Benjamin, New York 1964) Weinberg, S.: The Quantum Theory of Fields I + II (Cambridge University Press, Cambridge 1995, 1996) The Rotation Group in Quantum Mechanics Edmonds, A R.: Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton 1957) Fano, U., Racah, G.: Irreducible Tensorial Sets (Academic Press, New York 1959) Rose, M E.: Elementary Theory of Angular Momentum (Wiley, New York 1957) Rotenberg, M., Bivins, R., Metropolis, N., Wooten, J K.: The j- and j-Symbols (Technical Press MIT, Boston 1959) de Shalit, A., Talmi, I.: Nuclear Shell Theory (Academic Press, New York 1963) Quantum Scattering Theory Calogero, F.: Variable Phase Approach to Potential Scattering (Academic Press, New York 1967) 729 730 Bibliography Goldberger, M L., Watson, K W.: Collision Theory (Wiley, New York 1964) Newton, R G.: Scattering Theory of Waves and Particles (McGraw Hill, 1966) Selected Monographs on Quantized Gauge Theories, and Algebraic Quantum Field Theory These references are meant primarily for those who are, or want to become, theoreticians There are many more books on non-Abelian gauge theories whose list would be too long to be included here Bailin, D., Love, A.: Introduction to Gauge Field Theory (Institute of Physics Publishing, 1993) Bogoliubov, N N., Logunov, A A., Oksak, A I., Todorov, I T.: General Principles of Quantum Field Theory (Kluwer, 1990) Cheng, T P., Li, L F.: Gauge Theory of Elementary Particle Physics (Oxford University Press, Oxford 1984) Deligne, P et al (Eds.): Quantum Fields and Strings: A Course for Mathematicians (Am Math Soc., Inst for Advanced Study, Providence 2000) Faddeev, L D., Slavnov, A A.: Gauge Fields: An Introduction to Quantum Theory (Addison-Wesley, Reading 1991) Haag, R.: Local Quantum Physics (Springer-Verlag, Heidelberg 1999) Piguet, O., Sorella, S P.: Algebraic Renormalization (Lecture Notes in Physics, Springer-Verlag 1995) Steinmann, O.: Perturbative Quantum Electrodynamics and Axiomatic Field Theory (Springer-Verlag, Heidelberg 2000) Handbooks, Special Functions Abramowitz, M., Stegun, I A.: Handbook of Mathematical Functions (Dover, New York 1965) Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F G.: Higher Transcendental Functions, The Bateman Manuscript Project (McGrawHill, New York 1953) Gradshteyn, I S., Ryzhik, I M.: Table of Integrals, Series and Products (Academic Press, New York and London 1965) Whittaker, E T., Watson, G N.: A Course of Modern Analysis (Cambridge University Press, London 1958) Selected Books on the Mathematics of Quantum Theory Some of these books are relatively elementary, others go deep into the theory of Lie groups and Lie algebras, and refer to present-day research in theoretical physics de Azc´arraga, J A., Izquierdo, J M.: Lie Groups, Lie Algebras, Cohomology and some Applications in Physics (Cambridge University Press, Cambridge 1995) Bibliography Blanchard, Ph., Brüning, E.: Mathematical Methods in Physics (Birkhäuser, Boston, Basel, Berlin 2003) Bremermann, H.: Distributions, Complex variables, and Fourier transforms (Addison-Wesley, Reading, Mass 1965) Fuchs, J., Schweigert, Ch.: Symmetries, Lie Algebras and Representations (Cambridge University Press, Cambridge 1997) Hamermesh, M.: Group Theory and Its Applications to Physical Problems (Addison-Wesley, Reading, Mass 1962) Lawson, H B., Michelson, M.-L.: Spin geometry (Princeton University Press, Princeton 1989) Mackey, G W.: The theory of group representations (Univ of Chicago Press, 1976) (Chicago lectures in mathematics) O’Raifertaigh, L.: Group Structure of Gauge Theories (Cambridge Monographs on Mathematical Physics, Cambridge 1986) Richtmyer, R D.: Principles of Advanced Mathematical Physics (Springer-Verlag, New York 1981) Sternberg, S.: Group Theory and Physics (Cambridge University Press, Cambridge 1994) About the Interpretation of Quantum Theory Aharonov, Y., Rohrlich, D.: Quantum Paradoxes – Quantum Theory for the Perplexed (Wiley-VCH Verlag, Weinheim 2005) d’Espagnat, B.: Conceptual Foundations of Quantum Mechanics (Addison-Wesley, Redwood City 1989) Omn`es, R.: The Interpretation of Quantum Mechanics (Princeton University Press, Princeton 1994) Selleri, F.: Quantum Paradoxes and Physical Reality (Kluwer, 1990) References pertaining to the Historical Notes Born, M.: Mein Leben (Nymphenburger Verlagshandlung, München 1975) Born, M (Hrsg.) : Albert Einstein – Max Born, Briefwechsel 1916 - 1955 (Nymphenburger Verlagshandlung, München 1969) Cassidy, D C.: Uncertainty: The Life and Science of Werner Heisenberg (Freeman & Co, 1992) Einstein, A., Podolsky, B., Rosen, N.: Phys Rev 47 (1935) 777 Jost, R.: Das Märchen vom elfenbeinernen Turm (Springer-Verlag, Heidelberg 1995) Heisenberg, W.: Der Teil und das Ganze (Piper Verlag, München 1969) Mehra, J.: The Conceptual Completion and the Extensions of Quantum Mechanics: 1932-1941 (Springer-Verlag, New York 2001) Pais, A.: Niels Bohr’s Times (Clarendon Press, Oxford 1991) Pais, A.: “Subtle is the Lord ”, The Science and the Life of Albert Einstein (Clarendon Press, Oxford 1982) Sciaccia, L.: La Scomparsa di Majorana (Einaudi, Torino 1975) Stern, F.: Einstein’s German World (Princeton University Press 1999) 731 732 Bibliography Weart, S R.: Scientists in Power (Harvard University Press, Cambridge 1979) Interesting information on Nobel price winners who contributed to the development of quantum mechanics and quantum field theory (curricula vitae, Nobel lectures, etc.) can also be found on the internet: www.nobel.se/physics/laureates/index.html 733 Subject Index A Absorption 169 Action Addition of angular momenta 351 Addition theorem – for spherical harmonics 98 Algebra – graded 517 Angular momentum – orbital 91 Annihilation operators – for fermions 536 Annihilation operators – for photons 443 Anomalous magnetic moment – Schwinger 622 Anomaly – of g-factor 610 Anticommutator 517 Antihermitean 228 Anti-isomorphism 189 Antiparticle 256, 523 – with spin 428 Antiunitary operators 252 B Baryon number 330 Baryons 380 Base vectors – in Ê 390 Basis – spherical 291, 445 Bessel functions 144 – spherical 106 Bessel’s differential equation 105 Bhabha scattering 585 Bit – classical 281 – quantum 282 Bogoliubov’s method for pairing forces 322 Bohr magneton 240, 680 Bohr radius 7, 680 Born approximation 147 – first 150 Born series 147, 149 Born’s interpretation 39 – of wave function 41 Bose-Einstein condensation 268 Bose-Einstein statistics 268 Bosons 266 Boundary condition – of Schrödinger 44 – of Born 44 de Broglie–wave length 27 C c-number 411 Campbell-Hausdorff formula 222 Canonical Momentum – for Maxwell field 436 Casimir operators 378 Cauchy series 183 Center-of-mass system – in scattering 492 Central forces 91 Channels 167 Characteristic exponent – in DE of Fuchsian type 105 Charge conjugation 256, 333 – Dirac field 527 Charge operator – for fermions 537 Charged current 641 Chiral Fields 553 Circular orbit 700 Clebsch-Gordan coefficients 242, 244 – being real 355 – symmetry relations 357 Clebsch-Gordan series 242 Clifford algebra 516 Coherent state 79, 705 Colliding beam 492 Completeness 73, 89, 90 – asymptotic 485 – of description 329 – of hydrogen functions 127 – of spherical harmonics 97 Completeness of functions 69 Completeness relation 176 Compton-Effekt 585 Condon-Shortley phase convention 101, 236 Confluent hypergeometric function 114, 664 Contact interaction 642 Continuity equation 38 Contraction 318 contragredient 352 Coordinates – parabolic 145 – spherical 228 Correlation function – two-body 274 Correspondence principle Cosets 338 Coulomb interaction – instantaneous 440 Coulomb phase 125, 147 Coulomb potential 132 Coupling – minimal 429 Coupling constant – running 635 Creation operators – for fermions 536 Creation operators – for photons 443 Cross section – differential 133 – integrated elastic 134 – Rutherford 497 – total 140, 167 Crossing 587 Current – charged 557 734 Subject Index Current density – for complex scalar field 428 Cut – dynamic 158 – kinematic 158, 160 – left 159 D D-functions – basis in Eulerian angles 348 – completeness 348 – orthogonality 344 – symmetry relations 343 dagger 191 Decay – π → 2γ 499 – in particles 498 – in particles 500 – of η-meson in lepton pair 557 – of muon 644 – of pions 256 – of the π − 649 Density 65 – one-body 272 – two-body 273 Density matrix 208 – for neutrinos/antineutrinos 544 Deviation – mean square 19 – standard 19 Differential equation – Fuchsian type 96 – of Kummer 113 Diffraction minima 156 Dirac equation – in momentum space 515 – in polar coordinates 560 – in spacetime 525 Dirac’s bracket notation 174 Dirac’s δ-distribution 653, 658 Dirac’s operator 561 Discrete spectrum 53 Dispersion – of wave packet 35 Dispersion relation 29 Distribution – acausal 424 – tempered 87, 653 Distribution – causal for mass m 420 Dual space of H 188, 189 Dualism – particle-wave 26 Dyson series 304, 575 E Effective range 166 Ehrenfest’s theorem 50, 52 Eigenfunction 72 Eigenspace 194 Eigenvalue 72, 194 Eigenvalues – relativistic H-Atom 568 Einstein–Planck relation 26 Electron radius – classical 460 Elementary charge 680 Emission – induced 452 – spontaneous 452 Energ density – for Dirac field 538 Energy-momentum 387 – tensor field 412, 435 Energy-momentum tensor field – for fermions 538 Ensemble – mixed 208 Entangled state 273 Equations of motion – Heisenberg’s 51, 413 Euler angles 236 Evolution – time 202 Exchange interaction 310 Exchange symmetry 266 Exclusion principle 267 Expectation value 45, 46 F Factor group 338 Fermi constant 680 Fermi distribution 141 Fermi’s constant 642 Fermions 266 – identical 306 Field strength tensor field 432 Fields – electric 11 – magnetic 11 Fine structure 241, 711 Fine structure constant 5, 680 Flux factor – in cross section 492 Fock space 309 Form factor 152 – electric 155, 612 – magnetic 613 – properties 153 Functions – complete set of 71 – orthonormal 71 G g-factor 241 Gamma function 661 Gauge – Coulomb 439 – Feynman- 469 – Landau- 469 – Lorenz 437 – transverse 439 Gauge transformation 694 – global 427 – of Maxwell field 435 Gauss integral 33 Gaussian Wave Packet 32 Gell-Mann matrices 378 Generating function 59 Generator – of unitary transformations 202 Generators – of the rotation group 227 Golden Rule 306 Gordon identity 612 Gradient formula 447 Gram-Schmidt procedure 66 Gravitation constant 680 Green function 659 Group – special linear 508 – unimodular 508 Group velocity 29 Gupta-Bleuler method 473 H Haar measure 343 – for SU(2) 712 Hamilton – operator 50 Hamilton density – for complex scalar field 426 Hamilton–Jacobi differential equation 31 Subject Index Hamiltonian density – of Klein-Gordon field 411 Hamiltonian Form – of Dirac equation 529 Hankel functions – spherical 111 Harmonic oscillator 24, 53 Hartree approximation 306 Hartree-Fock equations 311, 312 Hartree-Fock method – time-dependent 322 Hartree-Fock operator 312 Heisenberg – equation of motion 51, 217 – uncertainty relations 696 Heisenberg algebra 62 Heisenberg picture 217 Heisenberg’s commutation relations 179 Heisenberg’s uncertainty relation 16, 21 Helicity 257, 392, 522 Helmholtz equation 445 Hermite polynomials 58 Hermitean 49 Hilbert space 180, 182 Homogeneous coordinates 382 Hydrogen atom 118 Hyperfine interval 299 Hyperfine structure 242, 296 Hypergeometric functions 663 I Idempotent 195 Inelasticity 169 Infraparticle 487, 638 Infrared divergence 601 In-state 131 Interaction – direct 310 – exchange 310 Interaction picture 218 Intertwiner 366 Irreducible 232 Isometry 200 Isospin 377 – nuclear 277 J j-symbols 358 – definition 358 – orthogonality 358 – special values 359 – symmetry relations 358 j-symbols 368 – definition 368 – symmetry relations 368 j-symbols – definition 372 – symmetry relations 373 Jacobi coordinates 259 jj-coupling 366 Jost functions 157 Line width 455 Liouville – theorem of 82 Lippmann-Schwinger equation 482 Locality 421 Lorentz force 407 Lorentz group – commutators 382 – generators 381 – spinor representations 386 Lorenz condition 437 Lowering operator 56 s-coupling 366 K Klein’s paradox 723 Klein-Gordon – equation 400 – field 400 Klein-Gordon equation 693 – mass zero 437 Klein-Nishina – cross section 595 Kramer’s theorem 254 Kramers-Heisenberg formula 460 Kummer’s differential equation 113, 567, 665 Kummer’s relation 665 L Laboratory system 492 Ladder operators 98, 232 Lagrange density – for complex scalar field 425 – for Maxwell theory 434 – free Dirac field 525 – of Klein-Gordon field 401 Laguerre polynomials 127 – associated 127 Lamb shift 630 Laplace-Operator – in dimension 400 Legendre functions – associated, of the first kind 97 – of second kind 151 Legendre polynomials 70, 664 Lepton number 330 – family numbers 639 Level density 306 Levi-Civit`a-Symbol – in dimension 387 Li´enard-Wiechert potentials 13 M Magic numbers 268 Magnetic moment 239 Magnetisation density 240 Majorana particle 528 Mass – in Poincaré group 387 Mass shell 400 Matrix – hermitean 75 Matrix mechanics 180, 217 Maxwell – velocity distribution 20 Maxwell’s equations 407 Mesons 380 Method of second quantization 308 Metric – see scalar product 182 Michel spectrum 647 Micro causality – for Maxwell fields 471 Micro-causality 421 Møller operators 483 Møller scattering 585 Momentum density – for the Dirac field 538 Motion – free, angular momentum basis 104 Multipole fields 447 Muonic Helium 693 Muonium 296, 693 N Neumann functions – spherical 110 Neutral current 643 Neutrino 295, 501 735 736 Subject Index Neutrinos 394 Nondegenerate eigenvalues 73 Norm 183 Normal order – of free fields 416 Normal product 318 – of operators 416 Normalization – covariant 397 Nuclear magneton 680 Number operator 85 O Observable 17, 45, 72, 193, 214 One-body density 272 Operator – adjoint 191 – antiunitary 252 – bounded 190 – densely defined 190 – domain of definition 190 – for particle number 85 – Hilbert-Schmidt 191 – linear 190 – range 190 – self-adjoint 47, 192 – statistical 206 – unbounded 190 – unitary 200 Operator-valued 411 Optical theorem 140, 491, 709 Orbital angular momentum 91 Orthogonal 183 Orthogonal complement 187 Orthogonal polynomial 65 Orthogonality 89, 90 Orthogonality relation 88, 97 Orthonormal 61, 73 Oscillations – zero point 25 Oscillator – spherical 25, 84, 111 Out-state 131 P Pair annihilation 585 Pair creation 556 Parity 248 – even 249 – odd 249 Parity operator 249 Parity violation – maximal 641 Partial wave amplitudes 136 Partial wave analysis 136 Partial waves 106 Particle – massive 388 – massless 389 – quasi-stable 498 Particle number 415 Particle-hole excitation 353 Particles – classification 388 – identical 258, 261 Path integral 218 Pauli matrices 193, 202, 245 Pauli principle 241 Pauli-Lubanski vector 387 PCT 333 Perturbation theory – Golden Rule 304, 306 – stationary 285 – time dependent 300 – with degeneracy 289 – without degeneracy 285 Phase convention – Condon-Shortley 354 Phase convention for D-matrices 237 Phase velocity 29 Photon – Spin 444 Photons – longitudinal 467 – scalar 467 Physical sheet 160 Pion – decay constant 650 Pion decay – charged 557 – neutral 557 Planck’s constant 6, 680 Plane wave 87 Pochhammer’s symbol 663 Poincaré group 381 – covariant generators 385 – representations 388 Poisson bracket 51 Polarizability – electric 292 Polarization 248 Positivity condition 168 Positronium 693 Potential – effective 103 Potential field – electromagnetic 433 Potential well 141 Poynting vector field 14 Pre-Hilbert space 183 Preparation measurement 215 Principle of complementarity 28 Processes – purely leptonic 643 – semi-leptonic 649 Product – time-ordered 423 Projectile 492 Projection operator 194, 206 Projection operators – for spin of fermions 541 Propagator – fermion-antifermion 547 – for scalar field 423 Pseudoscalar 393 Q Quadrupole interaction 363 Quadrupole moment – spectroscopic 363 Quantization 215 – axis of 92 – canonical 411 – Dirac fields 533 – of Majorana field 529 Quantization of atomic bound states quantum – of action – Planck’s Quantum field theory 399 Quantum number – principal 5, 120 Quantum numbers – additive 330 Quark-antiquark – creation in e+ e− 600 Quarks – colour 640 – flavour 640 Qubit 282 R R-matrix 486 Subject Index Radial equation 118 Radial function – large component 564 – small component 564 Radial functions – of Dirac fields 564 Radial momentum 101 Radiation theory – semi-classical 452 Radius – mean square 154 – proton 155 Raising operator 56 Raleigh scattering 457 Range – effective 164, 165 – of potential 160 Range of Potentials – finite 132 Range of potentials – infinite 144 Reaction matrix 486 Recouplings 374 reduced mass Reduced matrix element – in coupled states 373 Regularization 602 Renormalization 457 – of charge 580 – of mass 580 – of wave function 580 Renormalization constant – Z 608 Renormalization point 633 Renormalization scale 626 Representation – high-energy 518 – coordinate space 172 – defining 231 – Majorana 519 – momentum space 172 – natural 518 – position space 172 – projective 334 – spinor 234 – standard 518 – trivial 231 – unitary irreducible 232 Representation coefficients – of the rotation group 236 Representation theorem by Riesz and Fr´echet 189 Representation theory 171 Representations – spin-1/2 509 Residual interaction 314, 321 Resonance fluorescence 462 Resonance scattering 462 Resonances 161 Reversal of the motion 255 Right chiral 553 Rodrigues – formula 97 Rotation group 227, 335 – generators 230 – representations 230 Rotation matrices 236 Rotational spectrum 350 Rules – for traces 549 Rutherford cross section 147 Rydberg constant 297 Rydberg energy 680 S s-channel 587 S-matrix 485 Scalar product 182 – of operators 374 Scalar/pseudoscalar coupling 559 Scattering – inelastic 167 Scattering amplitude 134 – analytical properties 156 Scattering length 164, 166 Scattering matrix 486 Scattering phase 137, 145 Schmidt rank 280 Schmidt weights 280 Schrödinger equation 39 – time-dependent 40 – time-independent 41 Schrödinger picture 217 Schur’s Lemma 232 Secular equation 290 Selection rules 251 Self consistent 307, 312 Self-energy 601 Sesquilinear form 182 Shell model of nuclei 268, 307 SL(2, ) – ε-matrix 510 – definition 508 SO(3) – definition 335 Sommerfeld’s fine structure constant Sommerfeld’s radiation condition 132 Space of L functions 186 Space of test functions 655 Space reflection 248 Spectral family 197, 198 Spectral representation 196 Spectral theory of observables 196 Spectrum – continuous 172 – degenerate 81 – discrete 53, 171 – fully continuous 54 – fully discrete 54 – mixed 54, 86, 90, 118, 126, 172 – of eigenvalues 198 – purely continuous 86 Speed of light 680 Spherical base vectors 228 Spherical harmonics 97 – reduced matrix elements 376 – vector- 445 Spherical wave – incoming 138 – outgoing 138 Spin 239 – in Poincaré group 387 – massless particle 393 – of the photons 394 Spin and statistics 266 Spin density matrices – for fermions 540 Spin-orbit coupling 241 Spin-statistics relationship 216 Spin-statistics theorem 269 square integrable 43 Standard model 573 – of electroweak interactions 639 – particle content 639 Stark effect 290 State – asymptotic 130 – detection of 203 – entangled 273 – hole 317 – mixed 207, 208 – particle 317 – preparation 203, 213 – pure 207, 208 737 738 Subject Index Statistics – Bose–Einstein 418 Step function 423, 659 Strangeness – of K -mesons 432 Structure of H – metric 181 SU(2) – definition 335 – irreducible representations 340 – Lie algebra 346 SU(3) – Clebsch-Gordan series 379 – decuplet 379 – definition 377 – fundamental representation 379 – generators 378 – octet 379 – structure constants 378 Subspace – coherent 332 Subspace of Hilbert space 187 Superposition principle 30, 181 Superselection rule 331 Support of distribution 657 Symmetry 248 – external 327 – inner 327 – internal 377 T t-channel 587 T -matrix 486 Tamm-Dancoff method 322 Target 492 Tempered distribution 655 Tensor operator 360 Test functions 654 Theorem – no-cloning of qubits 283 – of Wigner 332, 333 – Wigner-Eckart 360 Thomson scattering 457, 462 Time evolution 215 Time reversal 248, 251 – for fermions 528 Time-ordered product 303 Top – symmetric 348 Trace techniques – for fermions 547 Traces – with γ5 550 Transformation – antiunitary 249 – orthogonal 177 – unitary 178, 186 Transition probability 306 Translation formula 413, 419 Tree diagrams 585 Triangular relation for angular momenta 357 Two-body density 273 Two-level system 293 U u-channel 588 Ultraviolet divergence 602 Uncertainties of observables 17 Uncertainty – of observables 19 Uncertainty relation – for position and momentum 16 Unit ray 76 Unit vectors – spherical 228 Unitarity – of S-Matrix 485 Units – natural 405, 406 Universal covering group 337 Universality – lepton 578 Unphysical sheet 160 V Vacuum polarization 624 – charge density 631 – observable 627 – potential 631 Variables – Mandelstam 586 Vector operator 376 Vector/axial vector coupling Velocity – of light Vertices 318 Virial theorem 697 556 W Wave function 28 – antisymmetric 265 – symmetric 265 Wave packet – Gaussian 32 Weight function 65 Weyl’s commutation relation 705 Weyl-Dirac equations 530 Wick’s Theorem 317 Wigner rotation 396 WKBJ-method 31 Y Yukawa potential 141, 143, 151, 152, 403 Z Zeeman effect 296 Zero point energy 25 Zitterbewegung 723 ... Elementarteilchenphysik Staudinger Weg 55099 Mainz, Germany e-mail: scheck@ thep.physik.uni-mainz.de ISBN 97 8-3 -5 4 0-2 564 5-8 Springer Berlin Heidelberg New York Cataloging-in-Publication Data Library of Congress Control.. .Quantum Physics Florian Scheck Quantum Physics With 76 Figures, 102 Exercises, Hints and Solutions Professor Dr Florian Scheck Universität Mainz Institut für... of modern physics, atomic and molecular physics, condensed matter physics, nuclear and elementary particle physics By itself it is a fascinating, though difficult, part of theoretical physics whose