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 1-4020-4189-6 (HB) ISBN-13 978-1-4020-4189-1 (HB) ISBN-10 1-4020-4190-X (e-book) ISBN-13 978-1-4020-4190-7 (e-book) Published by Springer, P.O. Box 17, 3300 AADordrecht, 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 1 Fundamentals............................................. 1 1.1 Selective Measurements ................................. 2 1.2 A, B, C to Probabilities ................................. 8 1.3 Expectation Values and Matrix Representations ............ 10 1.3.1 Probabilities and Expectation Values ............... 10 1.3.2 Representations of Simple Machines ................ 13 1.4 Generation of States, Inner-Product Spaces, Hermitian Operators and the Eigenvalue Problem.................... 15 1.4.1 Generation of States and Vector Spaces ............. 16 1.4.2 Transformation Functions and Wavefunctions in Different Descriptions............................. 18 1.4.3 An Illustration ................................... 19 1.4.4 Generation of Inner Product Spaces ................ 23 1.4.5 Hermitian Operators and the Eigenvalue Problem .... 24 1.5 Pure Ensembles and Mixtures............................ 25 1.6 Polarization of Light: An Interlude ....................... 29 1.7 The Hilbert Space; Rigged Hilbert Space .................. 33 1.8 Self-Adjoint Operators and Their Spectra.................. 39 1.9 Wigner’s Theorem on Symmetry Transformations .......... 55 1.10 Probability, Conditional Probability and Measurement ...... 65 1.10.1 Correlation of a Physical System and an Apparatus... 66 1.10.2 Probability and Conditional Probability............. 68 1.10.3 An Exactly Solvable Model ........................ 70 Problems .................................................. 79 VI Contents 2 Symmetries and Transformations ......................... 81 2.1 Galilean Space-Time Coordinate Transformations........... 81 2.2 Successive Galilean Transformations and the Closed Path.... 86 2.3 Quantum Galilean Transformations and Their Generators ... 89 2.4 The Transformation Function (cid:1)x|p(cid:2)....................... 98 2.5 Quantum Dynamics and Construction of Hamiltonians ...... 100 .. 2.5.1 The Time Evolution: Schrodinger Equation .......... 100 2.5.2 Time as an Operator?............................. 101 2.5.3 Construction of Hamiltonians ...................... 102 2.5.4 Multi-Particle Hamiltonians ....................... 104 2.5.5 Two-Particle Systems and Relative Motion .......... 104 2.5.6 Multi-Electron Atoms with Positions of the Electrons Defined Relative to the Nucleus .................... 105 2.5.7 Decompositions into Clusters of Particles............ 106 Appendix to §2.5: Time-Evolution for Time-Dependent Hamiltonians .......................................... 109 2.6 Discrete Transformations: Parity and Time Reversal ........ 112 2.7 Orbital Angular Momentum and Spin..................... 116 2.8 Spinors and Arbitrary Spins ............................. 121 2.8.1 Spinors and Generation of Arbitrary Spins........... 121 2.8.2 Rotation of a Spinor by 2π Radians................. 129 2.8.3 Time Reversal and Parity Transformation ........... 130 2.8.4 Kramers Degeneracy.............................. 132 Appendix to §2.8: Transformation Rule of a Spinor of Rank One Under a Coordinate Rotation ............................ 133 2.9 Supersymmetry ........................................ 136 Problems .................................................. 139 3 Uncertainties, Localization, Stability and Decay of Quantum Systems ........................................ 143 3.1 Uncertainties, Localization and Stability................... 143 3.1.1 A Basic Inequality................................ 143 3.1.2 Uncertainties .................................... 144 3.1.3 Localization and Stability ......................... 145 3.1.4 Localization, Stability and Multi-Particle Systems .... 148 3.2 Boundedness of the Spectra of Hamiltonians From Below .... 151 3.3 Boundedness of Hamiltonians From Below: General Classes of Interactions ......................................... 152 3.4 Boundedness of Hamiltonian From Below: Multi-Particle Systems ............................................... 163 3.4.1 Multi-Particle Systems with Two-Body Potentials .... 164 3.4.2 Multi-Particle Systems and Other Potentials......... 166 3.4.3 Multi-Particle Systems with Coulomb Interactions.... 167 3.5 Decay of Quantum Systems.............................. 168 Appendix to §3.5: The Paley-Wiener Theorem .................. 174 Contents VII Problems .................................................. 178 4 Spectra of Hamiltonians .................................. 181 4.1 Hamiltonians with Potentials Vanishing at Infinity.......... 182 4.2 On Bound-States ....................................... 187 4.2.1 A Potential Well ................................. 187 4.2.2 Limit of the Potential Well ........................ 190 4.2.3 The Dirac Delta Potential ......................... 190 4.2.4 Sufficiency Conditions for the Existence of a Bound-State for ν =1 ............................ 192 4.2.5 Sufficiency Conditions for the Existence of a Bound-State for ν =2 ............................ 194 4.2.6 Sufficiency Conditions for the Existence of a Bound-State for ν =3 ............................ 195 4.2.7 No-Binding Theorems............................. 197 4.3 Hamiltonians with Potentials Approaching Finite Constants at Infinity ............................................. 199 4.4 Hamiltonians with Potentials Increasing with No Bound at Infinity................................................ 200 4.5 Counting the Number of Eigenvalues...................... 203 4.5.1 General Treatment of the Problem.................. 203 4.5.2 Counting the Number of Eigenvalues................ 206 4.5.3 The Sum of the Negative Eigenvalues ............... 216 Appendix to §4.5: Evaluation of Certain Integrals ............... 219 4.6 Lower Bounds to the Expectation Value of the Kinetic Energy: An Application of Counting Eigenvalues ........... 220 4.6.1 One-Particle Systems ............................. 220 4.6.2 Multi-Particle States: Fermions .................... 222 4.6.3 Multi-Particle States: Bosons ...................... 224 4.7 The Eigenvalue Problem and Supersymmetry .............. 224 4.7.1 General Aspects.................................. 224 4.7.2 Construction of Supersymmetric Hamiltonians ....... 226 4.7.3 The Eigenvalue Problem .......................... 230 Problems .................................................. 244 5 Angular Momentum Gymnastics.......................... 249 5.1 The Eigenvalue Problem ................................ 251 5.2 Matrix Elements of Finite Rotations ...................... 254 5.3 Orbital Angular Momentum ............................. 258 5.3.1 Transformation Theory ........................... 258 5.3.2 Half-Odd Integral Values? ......................... 259 5.3.3 The Spherical Harmonics.......................... 262 5.3.4 Addition Theorem of Spherical Harmonics........... 267 5.4 Spin .................................................. 269 5.4.1 General Structure ................................ 269 VIII Contents 5.4.2 Spin 1/2 ........................................ 270 5.4.3 Spin 1 .......................................... 272 5.4.4 Arbitrary Spins .................................. 274 5.5 Addition of Angular Momenta ........................... 275 5.6 Explicit Expression for the Clebsch-Gordan Coefficients ..... 284 5.7 Vector Operators ....................................... 290 5.8 Tensor Operators....................................... 296 5.9 Combining Several Angular Momenta: 6-j and 9-j Symbols .. 304 5.10 Particle States and Angular Momentum; Helicity States ..... 307 5.10.1 Single Particle States ............................. 307 5.10.2 Two Particle States............................... 317 Problems .................................................. 324 6 Intricacies of Harmonic Oscillators........................ 329 6.1 The Harmonic Oscillator ................................ 329 6.2 Transition to and Between Excited States in the Presence of a Time-Dependent Disturbance .......................... 335 6.3 The Harmonic Oscillator in the Presence of a Disturbance at Finite Temperature .................................. 340 6.4 The Fermi Oscillator.................................... 343 6.5 Bose-Fermi Oscillators and Supersymmetric Bose-Fermi Transformations........................................ 346 6.6 Coherent State of the Harmonic Oscillator................. 349 Problems .................................................. 356 7 Intricacies of the Hydrogen Atom......................... 359 7.1 Stability of the Hydrogen Atom .......................... 360 7.2 The Eigenvalue Problem ................................ 363 7.3 The Eigenstates ........................................ 366 7.4 The Hydrogen Atom Including Spin and Relativistic Corrections ............................................ 370 Appendix to §7.4: Normalization of the Wavefunction Including Spin and Relativistic Corrections ......................... 378 7.5 The Fine-Structure of the Hydrogen Atom................. 379 Appendix to §7.5: Combining Spin and Angular Momentum in the Atom.............................................. 383 7.6 The Hyperfine-Structure of the Hydrogen Atom ............ 384 7.7 The Non-Relativistic Lamb Shift ......................... 391 7.7.1 The Radiation Field .............................. 391 7.7.2 Expression for the Energy Shifts ................... 394 7.7.3 The Lamb Shift and Renormalization ............... 398 Appendix to §7.7: Counter-Terms and Mass Renormalization ..... 401 7.8 Decay of Excited States ................................. 403 7.9 The Hydrogen Atom in External Electromagnetic Fields..... 406 7.9.1 The Atom in an External Magnetic Field............ 406 Contents IX 7.9.2 The Atom in an External Electric Field ............. 412 Problems .................................................. 414 8 Quantum Physics of Spin 1/2 and Two-Level Systems; Quantum Predictions Using Such Systems ................ 419 8.1 General Properties of Spin 1/2 and Two-Level Systems...... 420 8.1.1 General Aspects of Spin 1/2 ....................... 420 8.1.2 Spin 1/2 in External Magnetic Fields ............... 423 8.1.3 Two-Level Systems; Exponential Decay.............. 427 8.2 The Pauli Hamiltonian; Supersymmetry ................... 432 8.2.1 The Pauli Hamiltonian............................ 432 8.2.2 Supersymmetry .................................. 434 8.3 Landau Levels; Expression for the g-Factor ................ 436 8.3.1 Landau Levels ................................... 436 8.3.2 Expression for the g-Factor ........................ 440 8.4 Spin Precession and Radiation Losses ..................... 441 8.5 Anomalous Magnetic Moment of the Electron .............. 444 8.5.1 Observational Aspect of the Anomalous Magnetic Moment......................................... 445 8.5.2 Computation of the Anomalous Magnetic Moment.... 446 8.6 Density Operators and Spin ............................. 453 8.6.1 Spin in a General Time-Dependent Magnetic Field ... 453 8.6.2 Scattering of Spin 1/2 Particle off a Spin 0 Target .... 454 8.6.3 Scattering of Spin 1/2 Particles off a Spin 1/2 Target . 459 8.7 Quantum Interference and Measurement; The Role of the Environment........................................... 462 8.7.1 Interaction with an Apparatus and Unitary Evolution Operator ........................................ 463 8.7.2 Interaction with a Harmonic Oscillator in a Coherent State ........................................... 467 8.7.3 The Role of the Environment ...................... 469 8.8 RamseyOscillatoryFieldsMethodandSpinFlip;Monitoring the Spin............................................... 473 8.8.1 Ramsey Apparatus and Interference; Spin Flip ....... 473 8.8.2 Monitoring the Spin .............................. 478 8.9 Schrödinger’s Cat and Quantum Decoherence .............. 482 8.10 Bell’s Test............................................. 486 8.10.1 Bell’s Test....................................... 486 8.10.2 Basic Processes .................................. 490 Appendix to §8.10. Entangled States; The C-H Inequality ........ 499 8.11 Quantum Teleportation and Quantum Cryptography........ 501 8.11.1 Quantum Teleportation ........................... 501 8.11.2 Quantum Cryptography........................... 503 X Contents 8.12 Rotation of a Spinor .................................... 508 8.13 Geometric Phases ...................................... 513 8.13.1 The Berry Phase and the Adiabatic Regime ......... 513 8.13.2 Degeneracy...................................... 518 8.13.3 Aharonov-Anandan (AA) Phase.................... 520 8.13.4 Samuel-Bhandari (SB) Phase ...................... 529 8.14 Quantum Dynamics of the Stern-Gerlach Effect ............ 531 8.14.1 The Quantum Dynamics .......................... 531 8.14.2 The Intensity Distribution......................... 535 Appendix to §8.14: Time Evolution and Intensity Distribution .... 540 Problems .................................................. 544 9 Green Functions .......................................... 547 9.1 The Free Green Functions ............................... 548 9.2 Linear and Quadratic Potentials.......................... 555 9.3 The Dirac Delta Potential ............................... 558 9.4 Time-Dependent Forced Dynamics........................ 561 9.5 The Law of Reflection and Reconciliation with the Classical Law .................................................. 565 9.6 Two-Dimensional Green Function in Polar Coordinates: Application to the Aharonov-Bohm Effect ................. 570 9.7 General Properties of the Full Green Functions and Applications ........................................... 580 9.7.1 A Matrix Notation ............................... 580 9.7.2 Applications ..................................... 582 9.7.3 An Integral Expression for the (Homogeneous) Green Function ........................................ 586 9.8 The Thomas-Fermi Approximation and Deviations Thereof .. 587 9.9 The Coulomb Green Function: The Full Spectrum .......... 590 9.9.1 An Integral Equation ............................. 590 9.9.2 The Negative Spectrum p0 <0,λ<0 ............... 594 9.9.3 The Positive Spectrum p0 >0...................... 596 Problems .................................................. 598 10 Path Integrals ............................................ 601 10.1 The Free Particle....................................... 602 10.2 Particle in a Given Potential............................. 604 10.3 Charged Particle in External Electromagnetic Fields: Velocity Dependent Potentials ........................... 608 10.4 Constrained Dynamics .................................. 614 10.4.1 Classical Notions ................................. 614 10.4.2 Constrained Path Integrals ........................ 623 10.4.3 Second Class Constraints and the Dirac Bracket...... 627 10.5 Bose Excitations ....................................... 628 Contents XI 10.6 Grassmann Variables: Fermi Excitations................... 633 10.6.1 Real Grassmann Variables......................... 633 10.6.2 Complex Grassmann Variables ..................... 637 10.6.3 Fermi Excitations ................................ 640 Problems .................................................. 645 11 The Quantum Dynamical Principle ....................... 649 11.1 The Quantum Dynamical Principle ....................... 650 11.2 Expressions for Transformations Functions................. 656 11.3 Trace Functionals ...................................... 665 11.4 From the Quantum Dynamical Principle to Path Integrals ... 669 11.5 Bose/Fermi Excitations ................................. 672 11.6 Closed-Time Path and Expectation-Value Formalism........ 675 Problems .................................................. 681 12 Approximating Quantum Systems ........................ 683 12.1 Non-Degenerate Perturbation Theory ..................... 684 12.2 Degenerate Perturbation Theory ......................... 688 12.3 Variational Methods .................................... 690 12.4 High-Order Perturbations, Divergent Series; Padé Approximants.......................................... 695 12.5 WKB Approximation ................................... 703 12.5.1 General Theory .................................. 703 12.5.2 Barrier Penetration............................... 709 12.5.3 WKB Quantization Rules ......................... 712 12.5.4 The Radial Equation ............................. 715 12.6 Time-Dependence;SuddenApproximationandtheAdiabatic Theorem .............................................. 716 12.6.1 Weak Perturbations .............................. 717 12.6.2 Sudden Approximation............................ 720 12.6.3 The Adiabatic Theorem ........................... 724 12.7 Master Equation; Exponential Law, Coupling to the Environment........................................... 727 12.7.1 Master Equation ................................. 728 12.7.2 Exponential Law ................................. 733 12.7.3 Coupling to the Environment ...................... 734 Problems .................................................. 736 13 Multi-Electron Atoms: Beyond the Thomas-Fermi Atom.. 739 13.1 The Thomas-Fermi Atom................................ 740 Appendix A To §13.1: The TF Energy Gives the Leading Contribution to E(Z) for Large Z ........................ 746 AppendixBto§13.1:TheTFDensityActuallyGivestheSmallest Value for the Energy Density Functional in (13.1.6)......... 752 13.2 Correction due to Electrons Bound Near the Nucleus........ 753