Quantum mechanics plays the central role in our current understanding of essentially all physical phenomena. The present volume is a textbook of quantum mechanics designed for advanced undergraduate and graduate students of physics, mathematics and chemistry. It provides a pedagogical introduction to the formal apparatus of quantum mechanics, application to various physical problems, and the recent developments in the interpretation of quantum mechanics. Although this book contains considerably more material than can be accomodated in a standard two-semester course, it is hoped that the book will be suitable for a standard course of nonrelativistic quantum mechanics, omitting several sections in accord with the instructor's judgment and preference.
After an introductory Chapter on the mathematical concepts—linear operators, Hilbert space, generalized functions—we develop in Chapter 2 the formal framework of quantum mechanics. Here we introduce the basic postulates, different routes for quantization, uncertainty relations and complimentarity. Chapter 3 is devoted to wave mechanics. The Schrodinger equation and its solution in simple problems are discussed. Chapter 4 to 8 consider the theory of angular momentum and spin. Detailed discussions of rotation matrices, CG coefficients, 3j and 6j symbols and tensor operators are provided. Symmetry and invariance principles are considered in Chapters 9 and 10.

Preface vii

1. Mathematical Preliminaries 1.1

      1.1 Linear Spaces and Linear Functionals 1.1

      1.2 The Dirac Notation 1.5

      1.3 Linear Operators 1.6

      1.4 Generalized Functions 1.12

      (a) Definitions and Simple Properties of Generalized Functions 1.13

      (b) Fourier Transforms of Generalized Functions 1.16

      1.5 The Rigged Hilbert Space 1.19

2. The Basic Concepts 2.1

      2.1 States, Observables, and Operators: The Basic Postulates 2.1

      Postulate I 2.2

      Postulate II 2.2

      Postulate III 2.2

      Postulate IV 2.3

      2.2 Measurements and Observables: Commutability and Compatibility 2.4

      2.3 The Classical Connection 2.8

      (a) The Correspondence Principle 2.8

      (b) The Complementarity Principle 2.9

      (c) The Uncertainty Relations I 2.11

      2.4 The Density Matrix 2.13

      2.5 Unitary Operators and Transformation Theory 2.20

      2.6 Position and Momentum Operators 2.23

      2.7 Canonical Quantization 2.24

      (a) The Canonical Commutation Relations 2.24

      (b) The Uncertainty Relations II 2.29

      (c) Wave Functions 2.31

      2.8 Quantum Dynamics 2.32

      (a) Time Evolution and the Schrodinger Equation 2.32

      (b) Stationary States, Expectation Values 2.34

      (c) The Schrddinger, the Heisenberg and the Interaction Pictures 2.36

      (d)The Energy 一 Time Uncertainty Relations 2.38

      2.9 The Path Integral 2.42

      (a) The Transition Amplitude 2.43

      (b) The Feynman Path Integral 2.45

      (c) The Sum Over Histories 2.49

      (d) Schrodinger's Wave Equation 2.51

      (e) Imaginary Time and Statistical Mechanics 2.52

      (f) Gaussian Integrals 2.54

      (g) Illustrative Examples 2.55

      2.10 Deformation Quantization 2.59

      (a)The Wigner Distribution 2.59

      (b) The Star Product and Quantization 2.62

      (c) Non-commutative Quantum Mechanics 2.66

      Problems 2.67

3. The Schrodinger Equation and Its Solution 3.1

      3.1 The Schrodinger Equation and the Probability Interpretation of the Wave Function 3.1

      3.2 Many-Particle Interactions 3.18

      3.3 Coordinate and Momentum Representations 3.20

      3.4 Stationary States and the Eigenvalue Problem 3.23

      3.5 The Free Particle 3.27

      3.6 The Motion of Wave Packets 3.32

      3.7 The Classical Limit 3.42

      3.8 Illustrative Solutions of the Schrodinger Equation 3.45

      (a) The Potential Step, Reflection and Transmission Coefficients 3.46

      (b) The Infinite Square Well 3.50

      (c) The Finite Square Well 3.52

      (d) The Potential Barrier and the "TUnnel" Effect 3.58

      (e) The Delta Function Potential 3.61

      3.9The Transfer Matrix 3.65

      3.10 The Periodic Potential 3.76

      Problems 3.82

4. Angular Momentum 4.1

      4.1 Commutation Relations 4.1

      4.2 Eigenvalues and Eigenfunctions of the Angular Momentum Operators 4.4

      Problems 4.13

5. Rotations 5.1

      5.1 Rotations 5.1

      5.2 The Euler Angles 5.2

      5.3 Active and Passive Rotations 5.3

      5.4 Rotations as Operators; Infinitesimal Rotations 5.4

      5.5 Rotation matrices 5.8

      5.6 The Spherical harmonics 5.10

      5.7 Group theoretical considerations 5.14

      5.8 The Rotation of a Rigid Body 5.21

      5.9 Schwinger Operator Method 5.23

      Problems 3.28

6.Spin 6.1

      6.1 The Description of a Spin-5 Particle 6.1

      6.2 The Stern-Gerlach Experiment 6.8

      6.3 Spin Precession in a Magnetic Field 6.11

      6.4 The Pauli Equation 6.12

      6.5 The Density Matrix for a Spin-s System 6.14

      6.6 The Helicity Formalism 6.18

      Problems 6.20

7. The Addition of Angular Momenta 7.1

      7.1 Two Simple Examples 7.1

      7.2 Addition of Two Angular Momenta 7.5

      7.3 The Clebsch-Gordan Coefficients 7.8

      7.4 The Recoupling of Three Angular Momenta 7.19

      7.5Four Angular Momenta 7.23

      Problems 7.27

8.Tensor Operators 8.1

      8.1 Irreducible Tensor Operators 8.1

      8.2 Product of Tensor Operators 8.6

      8.3 Matrix Elements of Tensor Operators and the Wigner-Eckart Theorem 8.9

      Problems 8.17

9. Symmetry Transformations 9.1

      9.1 Introduction 9.1

      9.2 Symmetry Transformations and Wigner's Theorem 9.3

      9.3 Antiunitary Operators 9.7

      9.4 Symmetry Groups 9.9

      9.5 Spatial Translations 9.16

      9.6 Time Translation 9.17

      9.7 Space Rotations 9.19

      9.8 Galilei Transformations and Galilei Group 9.19

      9.9 Isospin 9.22

      9.10 Superselection Rules 9.31

      9.11 Dynamical Symmetries 9.32

      Problems 9.40

10.Discrete Symmetries 10.1

      10.1 Lattice Translation 10.1

      10.2 Space Inversion 10.5

      10.3 Time Reversal 10.14

      Problems 10.22

11. Applications 11.1

      11.1 The Rigid Rotator 11.1

      11.2 The Harmonic Oscillator 11.2

      (a) The Linear Harmonic Oscillator 11.3

      (b) The Two-Dimensional Oscillator 11.23

      (c) The Harmonic Oscillator in Three Dimensions 11.28

      11.3 Coherent States 11.33

      (a) Definitions and Properties 11.33

      (b)Squeezed States 11.39

      11.4 Motion in a Central Field of Force 11.40

      11.5 The Hydrogen Atom 11.42

      (a) Spherical Polar Coordinates 11.43

      (b) Parabolic Coordinates 11.48

      (c) Schwinger's Method 11.52

      (d) Wave Functions in Momentum Space 11.54

      11.6 Kaon physics 11.60

      11.7 Motion of a Charged Particle in a Magnetic Field 11.66

      (a) Equation of Motion and the Energy Levels 11.66

      (b) The Aharonov-Bohm Effect 11.73

      11.8 Supersymmetric Quantum Mechanics 11.76

      Problems 11.85

12. Stationary State Perturbation Theory 12.1

      12.1 The Rayleigh-Schrodinger Perturbation Expansion—The Nondegenerate Case 12.1

      12.2 The Rayleigh-Schrodinger Perturbation Expansion—The Degenerate Case 12.6

      12.3 Brillouin-Wigner Perturbation Theory 12.11

      12.4 The Anharmonic Oscillator and Large Orders of Perturbation Theory 12.14

      12.5 Hydrogenic Atoms: Fine Structure and other Corrections 12.20

      12.6 The Hydrogen Atom in a Magnetic Field 12.38

      12.7 The Stark Effect 12.47

      Problems 12.56

13.Nonperturbative Methods 13.1

      13.1 The Variational Method 13.1

      13.2 The Linear Variation Method 13.8

      13.3 The Variation-Perturbation Method 13.9

      13.4 The Hellmann-Feynman and Vinal Theorems 13.10

      13.5 van der Waals' Interaction 13.13

      13.6 The WKB Approximation 13.16

      13.7 The Connection Formulae 13.20

      13.8 Applications 13.25

      Problems 13.51

14. Methods fbr Time-Dependent Problems 14.1

      14.1Time-dependent Perturbation Theory 14.1

      14.2 The Interaction Picture 14.4

      14.3 Transition Probability 14.6

      14.4 Time-Dependent Two-State Problem: Magnetic Resonance 14.8

      14.5 Constant and Harmonic Perturbations 14.13

      14.6 Ionization of a Hydrogen Atom 14.20

      14.7 Sudden and Adiabatic Changes 14.22

      14.8 Geometric Phases 14.34

      Problems 14.44

15. Interaction of Radiation with Matter 15.1

      15. 1The Electromagnetic Field and the Interaction Hamiltonian 15.1

      15.2 Absorption and Induced Emission 15.6

      15.3 Quantization of the Electromagnetic Field 15.9

      15.4 Einstein's A and B Coefficients 15.15

      15.5 Photons and Atoms: Emission and Absorption 15.16

      15.6 Electric Dipole Transitions 15.19

      15.7 Magnetic Dipole and Electric Quadrupole Transitions, Higher Multipole Transitions 15.24

      15.8 Lifetimes, Line Intensities,Sum Rules and Line Breadths 15.30

      15.9 The Photoeffect 15.37

      15.10 Rayleigh Scattering, Thomson Scattering and the Raman Effect. Resonance Fluorescence 15.42

      15.11 The Casimir Effect 15.48

      Problems 15.53

16. I dentical Particles 16.1

      16.1 Permutations and Symmetry 16.1

      16.2 Indistinguishability of Particles 16.3

      16.3 The Symmetrization Postulate 16.4

      16.4 The Permutation Group 16.6

      16.5 Symmetries of Many-Particle Wave Functions and Young Tableaux 16.12

      16.6 The SU(3) Symmetry 16.16

      Problem 16.19

17. Many-Particle Systems 17.1

      17.1 The Occupation Number Representation 17.1

      17.2 Creation and Annihilation Operators 17.3

      (a) Fermions 17.3

      (b) Bosons 17.6

      17.3 One-Particle and Two-Particle Operators 17.7

      17.4 Particle-Hole Formalism for Fermions 17.12

      17.5 Field Operators 17.14

      17.6 Wick's Theorem 17.16

      (a) Time Ordered Product (Wick T-product) 17.16

      (b) Normal Product (Wick N-product) 17.16

      (c) Contraction 17.17

      (d) A Convention 17.18

      (e) For Operators A, B, .... at Equal Times 17.18

      17.7 The Hartree Approximation 17.22

      17.8 The Hartree-Fock Approximation 17.24

      17.9 The Free-Electron Gas 17.29

      17.10 Correlations 17.35

      17.11 Intensity-Fluctuation Correlations (The Hanbury Brown and Twiss Effect) 17.38

      17.12 Two Electron Systems 17.40

      Problems 17.50

18. The Description of Scattering Processes 18.1

      18.1 The Wave Packet Description of Scattering Processes 18.1

      18.2 The Scattering Cross Section 18.7

      18.3 The General Description of Scattering Processes 18.8

      Problems 18.9

19. Formal Scattering Theory 19.1

      19.1 The Lippmann-Schwinger Equation 19.1

      19.2 In-States and Out-States 19.6

      19.3 The S-operator and the Wave Operators 19.8

      19.4 Symmetry Considerations 19.19

      19.5 Scattering from Two Potentials 19.22

      Problems 19.23

20. Potential Scattering: Methods of Solution 20.1

      20.1 The Integral Form of the Schrodinger Equation 20.1

      20.2 The Optical Theorem 20.3

      20.3 Partial Waves and Phase Shifts 20.4

      20.4 The Bom Approximation and the Fredholm Method 20.17

      20.5 The Eikonal Approximation 20.29

      20.6 The Distorted-Wave Bom Approximation (DWBA) 20.34

      20.7 The Jost Function 20.36

      20.8 Low Energy Scattering and Bound States 20.38

      20.9 Resonance Scattering 20.43

      20.10 Variable Phase Method 20.50

      20.11 Analytic Properties of the S-matrix 20.52

      20.12 Dispersion Relations 20.54

      20.13 Coulomb Scattering 20.57

      20.14 Inelastic Electron-Atom Scattering 20.64

      20.15 Regge Poles 20.69

      Problems 20.77

21. Scattering of Identical Particles and Polarized Particles 21.1

      21.1 Scattering of Identical Particles 21.1

      21.2 Scattering of Polarized Particles 21.5

      Problems 21.12

      22. Iterpretation 22.1

      22.1Quantum Entanglement 22.2

      22.2 Hidden Variables 22.3

      22.3 Nonlocality and the Bell Inequalities 22.5

      (a)The Critique of Einstein, Podolsky, and Rosen and the Incompleteness of Quantum Theory 22.5

      (b) The Bell Inequalities 22.7

      (c) The Clauser-Home Inequality 22.8

      22.4The Problem of Measurement 22.12

      (a) Quantum measurement scheme 22.12

      (b) Reduction of the State Vector 22.13

      (c) The Watched Pot 22.14

      (d) The No-Cloning Theorem 22.15

      22.5Other Views 22.16

      (a)Consistent—Histories Interpretation 22.16

      (b)The "Many Worlds", Interpretation 22.17

      (c) Decoherence 22.18

      22.6 Epilogue 22.19

Bibliography B.l

Index I.I

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