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Quantum theory: a wide spectrum

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Main Author:	Manoukian, E. B. (Author)
Format:	Book
Language:	English
Published:	Dordrecht Springer Netherland 2006
Subjects:	Quantentheorie Lehrbuch
Online Access:	Inhaltstext Inhaltsverzeichnis
Physical Description:	XIX, 1011 S. graph. Darst.
ISBN:	1402041896 9781402041891

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adam_text	Contents Acknowledgments ............................................ XV Preface ..................................................... XVII 1 Fundamentals ............................................. 1 1.1 Selective Measurements ................................. 2 1.2 А, В, С 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 Hubert Space; Rigged Hubert 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 (xlp) ....................... 98 2.5 Quantum Dynamics and Construction of Hamiltonians ...... 100 2.5.1 The Time Evolution: Schrödinger 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 Prom Below ___151 3.3 Boundedness of Hamiltonians Prom 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 ___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 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 v = 1 ............................192 4.2.5 Sufficiency Conditions for the Existence of a Bound-State for v = 2 ............................194 4.2.6 Sufficiency Conditions for the Existence of a Bound-State for v = 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 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 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 ^-Factor ................436 8.3.1 Landau Levels ...................................436 8.3.2 Expression for the ^-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 Ramsey Oscillatory Fields Method and Spin Flip; 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 -Н 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 p° < 0, λ < 0............... 594 9.9.3 The Positive Spectrum p° > 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 Functional ......................................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; Sudden Approximation and the Adiabatic 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 Appendix В to §13.1: The TF Density Actually Gives the Smallest Value for the Energy Density Functional in (13.1.6).........752 13.2 Correction due to Electrons Bound Near the Nucleus ........753 XII Contents 13.3 The Exchange Term .................................... 756 13.4 Quantum Correction ....................................759 13.5 Adding Up the Various Contributions: Estimation of E{Z) ... 762 Problems .................................................. 762 14 Quantum Physics and the Stability of Matter .............765 14.1 Lower Bound to the Multi-Particle Repulsive Coulomb Potential Energy .......................................767 Appendix to §14.1: A Thomas-Fermi-Like Energy Functional and No Binding ............................................769 14.2 Lower and Upper Bounds for the Ground-State Energy and the Stability of Matter ..................................774 14.2.1 A Lower Bound ..................................774 14.2.2 Upper Bounds ...................................777 14.3 Investigation of the High-Density Limit for Matter and Its Stability ...............................................780 14.3.1 Upper Bound of the Average Kinetic Energy of Electrons in Matter ...............................780 14.3.2 Inflation of Matter ................................781 14.4 The Collapse of Bosonic Matter .........,................783 14.4.1 A Lower Bound .................................. 784 14.4.2 An Upper Bound ................................. 786 Appendix to §14.4: Upper Bounds for <HX) in (14.4.47).......... 793 Problems .................................................. 796 15 Quantum Scattering ......................................799 15.1 Interacting States and Asymptotic Boundary Conditions .... 800 15.2 Particle Detection and Connection between Configuration and Momentum Spaces in Scattering ......................807 Appendix to §15.2: Some Properties of F (u, v) .................812 15.3 Differential Cross Sections ...............................814 15.3.1 Expression for the Differential Cross Section .........814 15.3.2 Sufficiency Conditions for the Validity of the Born Expansion .......................................816 15.3.3 Two-Particle Scattering ...........................818 15.4 The Optical Theorem and Its Interpretation; Phase Shifts ... 821 15.4.1 The Optical Theorem .............................821 15.4.2 Phase Shifts Analysis .............................825 15.5 Coulomb Scattering .....................................830 15.5.1 Asymptotically Free Coulomb Green Functions .....830 15.5.2 Asymptotic Time Development of a Charged Particle State ...........................................832 15.5.3 The Full Green Function G+ Near the Energy Shell ... 833 15.5.4 The Scattering Amplitude via Evolution Operators ... 834 Contents XIII 15.6 Functional Treatment of Scattering Theory ................838 15.7 Scattering at Small Deflection Angles at High Energies: Eikonal Approximation ..................................842 15.7.1 Eikonal Approximation ...........................842 15.7.2 Determination of Asymptotic Free Green Function of the Coulomb Interaction ........................845 15.8 Multi-Channel Scatterings of Clusters and Bound Systems . .. 846 15.8.1 Channels and Channel Hamiltonians ................847 15.8.2 Interacting States Corresponding to Preparatory Channels ........................................851 15.8.3 Transition Probabilities and the Optical Theorem .... 853 15.8.4 Basic Processes ..................................854 15.8.5 Born Approximation, Connectedness and Faddeev Equations .......................................858 15.8.6 Phase Shifts Analysis .............................865 15.9 Passage of Particles through Media; Neutron Interferometer .. 867 15.9.1 Passage of Charged Particles through Hydrogen ......867 15.9.2 Neutron Interferometer ............................871 Problems ..................................................877 16 Quantum Description of Relativistic Particles ............881 16.1 The Dirac Equation and Pauli s Fundamental Theorem ......884 Appendix to §16.1: Pauli s Fundamental Theorem ...............889 16.2 Lorentz Covariance, Boosts and Spatial Rotations ..........892 16.2.1 Lorentz Transformations ..........................892 16.2.2 Lorentz Covariance, Boosts and Spatial Rotations .... 894 16.2.3 Lorentz Invariant Scalar Products of Spinors, Lorentz Scalars and Lorentz Vectors .......................898 16.3 Spin, Helicity and V, C, T Transformations ................900 16.3.1 Spin к Helicity.................................. 900 16.3.2 V, C, T Transformations ..........................902 16.4 General Solution of the Dirac Equation ....................903 16.5 Massless Dirac Particles .................................912 16.6 Physical Interpretation, Localization and Particle Content ... 916 16.6.1 Probability, Probability Current and the Initial Value Problem .........................................917 16.6.2 Diagonalization of the Hamiltonian and Definitions of Position Operators .............................919 16.6.3 Origin of Relativistic Corrections in the Hydrogen Atom ...........................................926 16.6.4 The Positron and Emergence of a Many-Particle Theory ..........................................931 Appendix to §16.6: Exact Treatment of the Dirac Equation in the Bound Coulomb Problem ................................933 XIV Contents 16.7 The Klein-Gordon Equation .............................9.47 16.7.1 Setting Up Spin 0 Equations .......................937 16.7.2 A Continuity Equation ............................9 Ш 16.7.3 General Solution of the Free Feshbach- Villars Equation ........................................9 J1 16.7.4 Diagonalization of the Hamiltonian and Definit ion tit Position Operators ...............................912 16.7.5 The External Field Problem .......................911 16.8 Relativistic Wave Equations for Any Mass and Any Spin .... 917 16.8.1 M > 0:..........................................917 16.8.2 M = 0:.......................................... 9Г)() 16.9 Spin & Statistics .......................................953 16.9.1 Quantum Fields ..................................951 16.9.2 Lagrangian for Spin 0 Particles ..................... 9ПГ) 16.9.3 Lagrangian for Spin 1/2 Particles ...................957 16.9.4 Schwinger s Constructive Approach .................958 16.9.5 The Spin and Statistics Connection .................962 Appendix to §16.9: The Action Integral ........................965 Problems ..................................................968 Mathematical Appendices ....................................971 I Variations of the Baker-Campbell-Hausdorff Formula .....973 1. Integral Expression for the Product of the Exponentials of Operators .............................................973 2. Derivative of the Exponential of Operator-Valued Functions . 973 3. The Classic Baker-Campbell-Hausdorff Formula ............975 4. A Modification of the Baker-Campbell-Hausdorff Formula . .. 975 II Convexity and Basic Inequalities .........................977 1. General Convexity Theorem ............................. 977 2. Minkowski s Inequality for Integrals ....................... 978 3. Holder s Inequality for Integrals .......................... 979 4. Young s Inequality for Integrals .......................... 980 III The Poisson Equation in 4D ..............................981 1. The Poisson Equation ...................................982 2. Generating Function ....................................983 3. Expansion Theorem ....................................984 4. Generalized Orthogonality Relation .......................985 References . 987 Index .........................................................999
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institution	BVB
isbn	1402041896 9781402041891
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-020132158
oclc_num	255854455
open_access_boolean
owner	DE-11 DE-355 DE-BY-UBR
owner_facet	DE-11 DE-355 DE-BY-UBR
physical	XIX, 1011 S. graph. Darst.
publishDate	2006
publishDateSearch	2006
publishDateSort	2006
publisher	Springer Netherland
record_format	marc
spelling	Manoukian, E. B. Verfasser aut Quantum theory a wide spectrum E. B. Manoukian Dordrecht Springer Netherland 2006 XIX, 1011 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Quantentheorie (DE-588)4047992-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantentheorie (DE-588)4047992-4 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2743519&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020132158&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Manoukian, E. B. Quantum theory a wide spectrum Quantentheorie (DE-588)4047992-4 gnd
subject_GND	(DE-588)4047992-4 (DE-588)4123623-3
title	Quantum theory a wide spectrum
title_auth	Quantum theory a wide spectrum
title_exact_search	Quantum theory a wide spectrum
title_full	Quantum theory a wide spectrum E. B. Manoukian
title_fullStr	Quantum theory a wide spectrum E. B. Manoukian
title_full_unstemmed	Quantum theory a wide spectrum E. B. Manoukian
title_short	Quantum theory
title_sort	quantum theory a wide spectrum
title_sub	a wide spectrum
topic	Quantentheorie (DE-588)4047992-4 gnd
topic_facet	Quantentheorie Lehrbuch
url	http://deposit.dnb.de/cgi-bin/dokserv?id=2743519&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020132158&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT manoukianeb quantumtheoryawidespectrum

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