QUANTUM THEORY Quantum Theory A Wide Spectrum by E.B MANOUKIAN Suranaree University of Technology, Nakhon Ratchasima, Thailand A C.I.P Catalogue record for this book is available from the Library of Congress ISBN-10 ISBN-13 ISBN-10 ISBN-13 1-4020-4189-6 (HB) 978-1-4020-4189-1 (HB) 1-4020-4190-X (e-book) 978-1-4020-4190-7 (e-book) Published by Springer, P.O Box 17, 3300 AA Dordrecht, The Netherlands www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Contents Acknowledgments XV Preface XVII Fundamentals 1.1 Selective Measurements 1.2 A, B, C to Probabilities 1.3 Expectation Values and Matrix Representations 1.3.1 Probabilities and Expectation Values 1.3.2 Representations of Simple Machines 1.4 Generation of States, Inner-Product Spaces, Hermitian Operators and the Eigenvalue Problem 1.4.1 Generation of States and Vector Spaces 1.4.2 Transformation Functions and Wavefunctions in Different Descriptions 1.4.3 An Illustration 1.4.4 Generation of Inner Product Spaces 1.4.5 Hermitian Operators and the Eigenvalue Problem 1.5 Pure Ensembles and Mixtures 1.6 Polarization of Light: An Interlude 1.7 The Hilbert Space; Rigged Hilbert Space 1.8 Self-Adjoint Operators and Their Spectra 1.9 Wigner’s Theorem on Symmetry Transformations 1.10 Probability, Conditional Probability and Measurement 1.10.1 Correlation of a Physical System and an Apparatus 1.10.2 Probability and Conditional Probability 1.10.3 An Exactly Solvable Model Problems 10 10 13 15 16 18 19 23 24 25 29 33 39 55 65 66 68 70 79 VI Contents Symmetries and Transformations 2.1 Galilean Space-Time Coordinate Transformations 2.2 Successive Galilean Transformations and the Closed Path 2.3 Quantum Galilean Transformations and Their Generators 2.4 The Transformation Function x|p 2.5 Quantum Dynamics and Construction of Hamiltonians 2.5.1 The Time Evolution: Schrodinger Equation 2.5.2 Time as an Operator? 2.5.3 Construction of Hamiltonians 2.5.4 Multi-Particle Hamiltonians 2.5.5 Two-Particle Systems and Relative Motion 2.5.6 Multi-Electron Atoms with Positions of the Electrons Defined Relative to the Nucleus 2.5.7 Decompositions into Clusters of Particles Appendix to §2.5: Time-Evolution for Time-Dependent Hamiltonians 2.6 Discrete Transformations: Parity and Time Reversal 2.7 Orbital Angular Momentum and Spin 2.8 Spinors and Arbitrary Spins 2.8.1 Spinors and Generation of Arbitrary Spins 2.8.2 Rotation of a Spinor by 2π Radians 2.8.3 Time Reversal and Parity Transformation 2.8.4 Kramers Degeneracy Appendix to §2.8: Transformation Rule of a Spinor of Rank One Under a Coordinate Rotation 2.9 Supersymmetry Problems Uncertainties, Localization, Stability and Decay of Quantum Systems 3.1 Uncertainties, Localization and Stability 3.1.1 A Basic Inequality 3.1.2 Uncertainties 3.1.3 Localization and Stability 3.1.4 Localization, Stability and Multi-Particle Systems 3.2 Boundedness of the Spectra of Hamiltonians From Below 3.3 Boundedness of Hamiltonians From Below: General Classes of Interactions 3.4 Boundedness of Hamiltonian From Below: Multi-Particle Systems 3.4.1 Multi-Particle Systems with Two-Body Potentials 3.4.2 Multi-Particle Systems and Other Potentials 3.4.3 Multi-Particle Systems with Coulomb Interactions 3.5 Decay of Quantum Systems Appendix to §3.5: The Paley-Wiener Theorem 81 81 86 89 98 100 100 101 102 104 104 105 106 109 112 116 121 121 129 130 132 133 136 139 143 143 143 144 145 148 151 152 163 164 166 167 168 174 Contents VII Problems 178 Spectra of Hamiltonians 4.1 Hamiltonians with Potentials Vanishing at Infinity 4.2 On Bound-States 4.2.1 A Potential Well 4.2.2 Limit of the Potential Well 4.2.3 The Dirac Delta Potential 4.2.4 Sufficiency Conditions for the Existence of a Bound-State for ν = 4.2.5 Sufficiency Conditions for the Existence of a Bound-State for ν = 4.2.6 Sufficiency Conditions for the Existence of a Bound-State for ν = 4.2.7 No-Binding Theorems 4.3 Hamiltonians with Potentials Approaching Finite Constants at Infinity 4.4 Hamiltonians with Potentials Increasing with No Bound at Infinity 4.5 Counting the Number of Eigenvalues 4.5.1 General Treatment of the Problem 4.5.2 Counting the Number of Eigenvalues 4.5.3 The Sum of the Negative Eigenvalues Appendix to §4.5: Evaluation of Certain Integrals 4.6 Lower Bounds to the Expectation Value of the Kinetic Energy: An Application of Counting Eigenvalues 4.6.1 One-Particle Systems 4.6.2 Multi-Particle States: Fermions 4.6.3 Multi-Particle States: Bosons 4.7 The Eigenvalue Problem and Supersymmetry 4.7.1 General Aspects 4.7.2 Construction of Supersymmetric Hamiltonians 4.7.3 The Eigenvalue Problem Problems 181 182 187 187 190 190 220 220 222 224 224 224 226 230 244 Angular Momentum Gymnastics 5.1 The Eigenvalue Problem 5.2 Matrix Elements of Finite Rotations 5.3 Orbital Angular Momentum 5.3.1 Transformation Theory 5.3.2 Half-Odd Integral Values? 5.3.3 The Spherical Harmonics 5.3.4 Addition Theorem of Spherical Harmonics 5.4 Spin 5.4.1 General Structure 249 251 254 258 258 259 262 267 269 269 192 194 195 197 199 200 203 203 206 216 219 VIII Contents 5.4.2 Spin 1/2 5.4.3 Spin 5.4.4 Arbitrary Spins 5.5 Addition of Angular Momenta 5.6 Explicit Expression for the Clebsch-Gordan Coefficients 5.7 Vector Operators 5.8 Tensor Operators 5.9 Combining Several Angular Momenta: 6-j and 9-j Symbols 5.10 Particle States and Angular Momentum; Helicity States 5.10.1 Single Particle States 5.10.2 Two Particle States Problems 270 272 274 275 284 290 296 304 307 307 317 324 Intricacies of Harmonic Oscillators 6.1 The Harmonic Oscillator 6.2 Transition to and Between Excited States in the Presence of a Time-Dependent Disturbance 6.3 The Harmonic Oscillator in the Presence of a Disturbance at Finite Temperature 6.4 The Fermi Oscillator 6.5 Bose-Fermi Oscillators and Supersymmetric Bose-Fermi Transformations 6.6 Coherent State of the Harmonic Oscillator Problems 329 329 Intricacies of the Hydrogen Atom 7.1 Stability of the Hydrogen Atom 7.2 The Eigenvalue Problem 7.3 The Eigenstates 7.4 The Hydrogen Atom Including Spin and Relativistic Corrections Appendix to §7.4: Normalization of the Wavefunction Including Spin and Relativistic Corrections 7.5 The Fine-Structure of the Hydrogen Atom Appendix to §7.5: Combining Spin and Angular Momentum in the Atom 7.6 The Hyperfine-Structure of the Hydrogen Atom 7.7 The Non-Relativistic Lamb Shift 7.7.1 The Radiation Field 7.7.2 Expression for the Energy Shifts 7.7.3 The Lamb Shift and Renormalization Appendix to §7.7: Counter-Terms and Mass Renormalization 7.8 Decay of Excited States 7.9 The Hydrogen Atom in External Electromagnetic Fields 7.9.1 The Atom in an External Magnetic Field 335 340 343 346 349 356 359 360 363 366 370 378 379 383 384 391 391 394 398 401 403 406 406 Contents IX 7.9.2 The Atom in an External Electric Field 412 Problems 414 Quantum Physics of Spin 1/2 and Two-Level Systems; Quantum Predictions Using Such Systems 8.1 General Properties of Spin 1/2 and Two-Level Systems 8.1.1 General Aspects of Spin 1/2 8.1.2 Spin 1/2 in External Magnetic Fields 8.1.3 Two-Level Systems; Exponential Decay 8.2 The Pauli Hamiltonian; Supersymmetry 8.2.1 The Pauli Hamiltonian 8.2.2 Supersymmetry 8.3 Landau Levels; Expression for the g-Factor 8.3.1 Landau Levels 8.3.2 Expression for the g-Factor 8.4 Spin Precession and Radiation Losses 8.5 Anomalous Magnetic Moment of the Electron 8.5.1 Observational Aspect of the Anomalous Magnetic Moment 8.5.2 Computation of the Anomalous Magnetic Moment 8.6 Density Operators and Spin 8.6.1 Spin in a General Time-Dependent Magnetic Field 8.6.2 Scattering of Spin 1/2 Particle off a Spin Target 8.6.3 Scattering of Spin 1/2 Particles off a Spin 1/2 Target 8.7 Quantum Interference and Measurement; The Role of the Environment 8.7.1 Interaction with an Apparatus and Unitary Evolution Operator 8.7.2 Interaction with a Harmonic Oscillator in a Coherent State 8.7.3 The Role of the Environment 8.8 Ramsey Oscillatory Fields Method and Spin Flip; Monitoring the Spin 8.8.1 Ramsey Apparatus and Interference; Spin Flip 8.8.2 Monitoring the Spin 8.9 Schrödinger’s Cat and Quantum Decoherence 8.10 Bell’s Test 8.10.1 Bell’s Test 8.10.2 Basic Processes Appendix to §8.10 Entangled States; The C-H Inequality 8.11 Quantum Teleportation and Quantum Cryptography 8.11.1 Quantum Teleportation 8.11.2 Quantum Cryptography 419 420 420 423 427 432 432 434 436 436 440 441 444 445 446 453 453 454 459 462 463 467 469 473 473 478 482 486 486 490 499 501 501 503 X Contents 8.12 Rotation of a Spinor 8.13 Geometric Phases 8.13.1 The Berry Phase and the Adiabatic Regime 8.13.2 Degeneracy 8.13.3 Aharonov-Anandan (AA) Phase 8.13.4 Samuel-Bhandari (SB) Phase 8.14 Quantum Dynamics of the Stern-Gerlach Effect 8.14.1 The Quantum Dynamics 8.14.2 The Intensity Distribution Appendix to §8.14: Time Evolution and Intensity Distribution Problems 508 513 513 518 520 529 531 531 535 540 544 Green Functions 9.1 The Free Green Functions 9.2 Linear and Quadratic Potentials 9.3 The Dirac Delta Potential 9.4 Time-Dependent Forced Dynamics 9.5 The Law of Reflection and Reconciliation with the Classical Law 9.6 Two-Dimensional Green Function in Polar Coordinates: Application to the Aharonov-Bohm Effect 9.7 General Properties of the Full Green Functions and Applications 9.7.1 A Matrix Notation 9.7.2 Applications 9.7.3 An Integral Expression for the (Homogeneous) Green Function 9.8 The Thomas-Fermi Approximation and Deviations Thereof 9.9 The Coulomb Green Function: The Full Spectrum 9.9.1 An Integral Equation 9.9.2 The Negative Spectrum p0 < 0, λ < 9.9.3 The Positive Spectrum p0 > Problems 547 548 555 558 561 10 Path Integrals 10.1 The Free Particle 10.2 Particle in a Given Potential 10.3 Charged Particle in External Electromagnetic Fields: Velocity Dependent Potentials 10.4 Constrained Dynamics 10.4.1 Classical Notions 10.4.2 Constrained Path Integrals 10.4.3 Second Class Constraints and the Dirac Bracket 10.5 Bose Excitations 565 570 580 580 582 586 587 590 590 594 596 598 601 602 604 608 614 614 623 627 628 Contents XI 10.6 Grassmann Variables: Fermi Excitations 10.6.1 Real Grassmann Variables 10.6.2 Complex Grassmann Variables 10.6.3 Fermi Excitations Problems 633 633 637 640 645 11 The Quantum Dynamical Principle 11.1 The Quantum Dynamical Principle 11.2 Expressions for Transformations Functions 11.3 Trace Functionals 11.4 From the Quantum Dynamical Principle to Path Integrals 11.5 Bose/Fermi Excitations 11.6 Closed-Time Path and Expectation-Value Formalism Problems 649 650 656 665 669 672 675 681 12 Approximating Quantum Systems 12.1 Non-Degenerate Perturbation Theory 12.2 Degenerate Perturbation Theory 12.3 Variational Methods 12.4 High-Order Perturbations, Divergent Series; Padé Approximants 12.5 WKB Approximation 12.5.1 General Theory 12.5.2 Barrier Penetration 12.5.3 WKB Quantization Rules 12.5.4 The Radial Equation 12.6 Time-Dependence; Sudden Approximation and the Adiabatic Theorem 12.6.1 Weak Perturbations 12.6.2 Sudden Approximation 12.6.3 The Adiabatic Theorem 12.7 Master Equation; Exponential Law, Coupling to the Environment 12.7.1 Master Equation 12.7.2 Exponential Law 12.7.3 Coupling to the Environment Problems 683 684 688 690 13 Multi-Electron Atoms: Beyond the Thomas-Fermi Atom 13.1 The Thomas-Fermi Atom Appendix A To §13.1: The TF Energy Gives the Leading Contribution to E(Z) for Large Z Appendix B to §13.1: The TF Density Actually Gives the Smallest Value for the Energy Density Functional in (13.1.6) 13.2 Correction due to Electrons Bound Near the Nucleus 739 740 695 703 703 709 712 715 716 717 720 724 727 728 733 734 736 746 752 753 ... selective measurements may be considered, for example, as a preparatory stage for a system before undergoing a subsequent analysis By a selective measurement, for example, one may prepare the... that the measurement of a physical quantity A (also called an observable), as a physical attribute of a system, can lead to a certain finite set of discrete real values {a, a , a , } In general,... work accessible to a wider audience, some of the technical details occurring in the presentations have been relegated to appendices A glance at the Contents will reveal that although the book