Florian Scheck. Quantum Physics. With 76 Figures, 102 Exercises, Hints and Solutions

Transcription

1 Quantum Physics

2 Florian Scheck Quantum Physics With 76 Figures, 102 Exercises, Hints and Solutions 1 3

3 Professor Dr. Florian Scheck Universität Mainz Institut für Physik, Theoretische Elementarteilchenphysik Staudinger Weg Mainz, Germany scheck@thep.physik.uni-mainz.de ISBN Springer Berlin Heidelberg New York Cataloging-in-Publication Data Library of Congress Control Number: 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 /3100/YL Printed on acid-free paper

4 To the memory of my father, Gustav O. Scheck ( ), who was a great musician and an exceptional personality

5 Preface VII 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 1 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 Hamilton- Jacobi 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.

6 VIII Preface Chapter 2 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 3 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 4 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 5 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 Poincaré groups is taken up from the perspective of particle properties, and some of their unitary representations are worked out. Chapter 7 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 8 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 9 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-

7 Preface IX 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 1 I will keep track of possible errata on an internet page attached to my home page. The latter can be accessed via I will be grateful for hints to misprints or errors.

8 Table of Contents XI PART ONE From the Uncertainty Relation to Many-Body Systems 1. Quantum Mechanics of Point Particles 1.1 Limitations of Classical Physics Heisenberg s Uncertainty Relation for Position and Momentum Uncertainties of Observables Quantum Mechanical Uncertainties of Canonically Conjugate Variables Examples for Heisenberg s Uncertainty Relation The Particle-Wave Dualism The Wave Function and its Interpretation A First Link to Classical Mechanics Gaussian Wave Packet Electron in External Electromagnetic Fields Schrödinger Equation and Born s Interpretation of the Wave Function Expectation Values and Observables Observables as Self-Adjoint Operators on L 2 (R 3 ) Ehrenfest s Theorem A Discrete Spectrum: Harmonic Oscillator in one Dimension Orthogonal Polynomials in One Real Variable Observables and Expectation Values Observables With Nondegenerate Spectrum An Example: Coherent States Observables with Degenerate, Discrete Spectrum Observables with Purely Continuous Spectrum Central Forces and the Schrödinger Equation The Orbital Angular Momentum: Eigenvalues and Eigenfunctions Radial Momentum and Kinetic Energy Force Free Motion with Sharp Angular Momentum The Spherical Oscillator Mixed Spectrum: The Hydrogen Atom Scattering of Particles by Potentials 2.1 Macroscopic and Microscopic Scales Scattering on a Central Potential Partial Wave Analysis How to Calculate Scattering Phases Potentials with Infinite Range: Coulomb Potential Born Series and Born Approximation First Born Approximation Form Factors in Elastic Scattering *Analytical Properties of Partial Wave Amplitudes

9 XII Table of Contents Jost Functions Dynamic and Kinematic Cuts Partial Wave Amplitudes as Analytic Functions Resonances Scattering Length and Effective Range Inelastic Scattering and Partial Wave Analysis The Principles of Quantum Theory 3.1 Representation Theory Dirac s Bracket Notation Transformations Relating Different Representations The Concept of Hilbert Space Definition of Hilbert Spaces Subspaces of Hilbert Spaces Dual Space of a Hilbert Space and Dirac s Notation Linear Operators on Hilbert Spaces Self-Adjoint Operators Projection Operators Spectral Theory of Observables Unitary Operators Time Evolution of Quantum Systems Quantum States Preparation of States Statistical Operator and Density Matrix Dependence of a State on Its History Examples for Preparation of States A First Summary Schrödinger and Heisenberg Pictures Path Integrals The Action in Classical Mechanics The Action in Quantum Mechanics Classical and Quantum Paths Space-Time Symmetries in Quantum Physics 4.1 The Rotation Group (Part 1) Generators of the Rotation Group Representations of the Rotation Group The Rotation Matrices D Examples and Some Formulae for D-Matrices Spin and Magnetic Moment of Particles with j = 1/ Clebsch-Gordan Series and Coupling of Angular Momenta Spin and Orbital Wave Functions Pure and Mixed States for Spin 1/ Space Reflection and Time Reversal in Quantum Mechanics Space Reflection and Parity Reversal of Motion and of Time Concluding Remarks on T and Π Symmetry and Antisymmetry of Identical Particles Two Distinct Particles in Interaction Identical Particles with the Example N = Extension to N Identical Particles Connection between Spin and Statistics

10 Table of Contents XIII 5. Applications of Quantum Mechanics 5.1 Correlated States and Quantum Information Nonlocalities, Entanglement, and Correlations Entanglement, More General Considerations Classical and Quantum Bits Stationary Perturbation Theory Perturbation of a Nondegenerate Energy Spectrum Perturbation of a Spectrum with Degeneracy An Example: Stark Effect Two More Examples: Two-State System, Zeeman-Effect of Hyperfine Structure in Muonium Time Dependent Perturbation Theory and Transition Probabilities Perturbative Expansion of Time Dependent Wave Function First Order and Fermi s Golden Rule Stationary States of N Identical Fermions Self Consistency and Hartree s Method The Method of Second Quantization The Hartree-Fock Equations Hartree-Fock Equations and Residual Interactions Particle and Hole States, Normal Product and Wick s Theorem Application to the Hartree-Fock Ground State PART TWO From Symmetries in Quantum Physics to Electroweak Interactions 6. Symmetries and Symmetry Groups in Quantum Physics 6.1 Action of Symmetries and Wigner s Theorem Coherent Subspaces of Hilbert Space and Superselection Rules Wigner s Theorem The Rotation Group (Part 2) Relationship between SU(2) and SO(3) The Irreducible Unitary Representations of SU(2) Addition of Angular Momenta and Clebsch-Gordan Coefficients Calculating Clebsch-Gordan Coefficients; the 3j-Symbols Tensor Operators and Wigner Eckart Theorem *Intertwiner, 6j- and9j-symbols Reduced Matrix Elements in Coupled States Remarks on Compact Lie Groups and Internal Symmetries Lorentz- and Poincaré Groups The Generators of the Lorentz and Poincaré Groups Energy-Momentum, Mass and Spin Physical Representations of the Poincaré Group Massive Single-Particle States and Poincaré Group Quantized Fields and their Interpretation 7.1 The Klein-Gordon Field

11 XIV Table of Contents The Covariant Normalization A Comment on Physical Units Solutions of the Klein-Gordon Equation for Fixed Four-Momentum Quantization of the Real Klein-Gordon Field Normal Modes, Creation and Annihilation Operators Commutator for Different Times, Propagator The Complex Klein-Gordon Field The Quantized Maxwell Field Maxwell s Theory in the Lagrange Formalism Canonical Momenta, Hamilton- and Momentum Densities Lorenz- and Transversal Gauges Quantization of the Maxwell Field Energy, Momentum, and Spin of Photons Helicity and Orbital Angular Momentum of Photons Interaction of the Quantum Maxwell Field with Matter Many-Photon States and Matrix Elements Absorption and Emission of Single Photons Rayleigh- and Thomson Scattering Covariant Quantization of the Maxwell Field Gauge Fixing and Quantization Normal Modes and One-Photon States Lorenz Condition, Energy and Momentum of the Radiation Field *The State Space of Quantum Electrodynamics *Field Operators and Maxwell s Equations *The Method of Gupta and Bleuler Scattering Matrix and Observables in Scattering and Decays 8.1 Nonrelativistic Scattering Theory in an Operator Formalism The Lippmann-Schwinger Equation T-Matrix and Scattering Amplitude Covariant Scattering Theory Assumptions and Conventions S-Matrix and Optical Theorem Cross Sections for two Particles Decay Widths of Unstable Particles Comment on the Scattering of Wave Packets Particles with Spin 1/2 and the Dirac Equation 9.1 Relationship between SL(2,C )andl Representations with Spin 1/ *Dirac Equation in Momentum Space Solutions of the Dirac Equation in Momentum Space Dirac Equation in Spacetime and Lagrange Density Quantization of the Dirac Field Quantization of Majorana Fields Quantization of Dirac Fields Electric Charge, Energy, and Momentum Dirac Fields and Interactions Spin and Spin Density Matrix The Fermion-Antifermion Propagator Traces of Products of γ -Matrices

12 Table of Contents XV Chiral States and their Couplings to Spin-1 Particles When is the Dirac Equation a One-Particle Theory? Separation of the Dirac Equation in Polar Coordinates Hydrogen-like Atoms from the Dirac Equation Elements of Quantum Electrodynamics and Weak Interactions 10.1 S-Matrix and Perturbation Series Tools of Quantum Electrodynamics with Leptons Feynman Rules for Quantum Electrodynamics with Charged Leptons Some Processes in Tree Approximation Radiative Corrections, Regularization, and Renormalization Self-Energy of Electrons to Order O(e 2 ) Renormalization of the Fermion Mass Scattering on an External Potential Vertex Correction and Anomalous Magnetic Moment Vacuum Polarization Epilogue: Quantum Electrodynamics in the Framework of Electroweak Interactions Weak Interactions with Charged Currents Purely Leptonic Processes and Muon Decay Two Simple Semi-leptonic Processes Appendix A.1 Dirac s δ(x) and Tempered Distributions A.1.1 Test Functions and Tempered Distributions A.1.2 Functions as Distributions A.1.3 Support of a Distribution A.1.4 Derivatives of Tempered Distributions A.1.5 Examples of Distributions A.2 Gamma Function and Hypergeometric Functions A.2.1 The Gamma Function A.2.2 Hypergeometric Functions A.3 Self-energy of the Electron A.4 Renormalization of the Fermion Mass A.5 Proof of the Identity (10.86) A.6 Analysis of Vacuum Polarization A.7 Ward-Takahashi Identity A.8 Some Physical Constants and Units Historical Notes Exercises, Hints, and Selected Solutions Bibliography Subject Index