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1 Classical Mechanics 1

1.1 Newton's Laws,the Action,and the Hamiltonian 1

1.1.1 Newton's Law and Lagrange's Equations 1

1.1.2 Hamilton's Principle 2

1.1.3 Canonical Momenta and the Hamiltonian Formulation 4

1.2 Classical Space-Time Symmetries 6

1.2.1 The Space-Time Transformations 6

1.2.2 Translations 8

1.2.3 Rotations 9

1.2.4 Rotation Matrices 11

1.2.5 Symmetries and Conservation Laws 12

Problems 14

2 Fundamentals of Quantum Mechanics 19

2.1 The Superposition Principle 19

2.1.1 The Double-Slit Experiment 19

2.1.2 The Stern-Gerlach Experiment 22

2.2 The Mathematical Language of Quantum Mechanics 24

2.2.1 Vector Spaces 24

2.2.2 The Probability Interpretation 27

2.2.3 Linear Operators 27

2.2.4 Observables 30

2.2.5 Examples 31

2.3 Continuous Eigenvalues 35

2.3.1 The Dirac Delta Function 35

2.3.2 Continuous Observables 36

2.3.3 Fourier's Theorem and Representations of δ(x) 37

2.4 Canonical Commutators and the Schr？dinger Equation 38

2.4.1 The Correspondence Principle 38

2.4.2 The Canonical Commutation Relations 42

2.4.3 Planck's Constant 43

2.5 Quantum Dynamics 44

2.5.1 The Time-Translation Operator 44

2.5.2 The Heisenberg Picture 45

2.6 The Uncertainty Principle 47

2.7 Wave Functions 49

2.7.1 Wave Functions in Coordinate Space 49

2.7.2 Momentum and Translations 49

2.7.3 Schr？dinger's Wave Equation 52

2.7.4 Time-Dependent Free Particle Wave Functions 53

Problems 55

3 Stationary States 62

3.1 Elementary Examples 62

3.1.1 States with Definite Energy 62

3.1.2 A Two-State System 63

3.1.3 One-Dimensional Potential Problems 66

3.2 The Harmonic Oscillator 68

3.2.1 The Spectrum 69

3.2.2 Matrix Elements 71

3.2.3 The Ground-State Energy 72

3.2.4 Wave Functions 73

3.3 Spherically Symmetric Potentials and Angular Momentum 74

3.3.1 Spherical Symmetry 74

3.3.2 Orbital Angular Momentum as a Difierential Operator 75

3.3.3 The Angular Momentum Commutator Algebra 76

3.3.4 Classification of the States 80

3.4 Spherically Symmetric Potentials:Wave Functions 80

3.4.1 Spherical Coordinates and Spherical Harmonics 80

3.4.2 The Radial Wave Equation 82

3.5 Hydrogenlike Atoms 84

3.5.1 The Symmetries 84

3.5.2 The Energy Spectrum 86

3.5.3 The Radial Wave Functions 88

Problems 91

4 Symmetry Transformations on States 102

4.1 Introduction 102

4.1.1 Symmetries and Transformations 102

4.1.2 Groups of Transformations 103

4.1.3 Classical and Quantum Symmetries 105

4.2 The Rotation Group and Algebra 105

4.2.1 Representations of Groups 105

4.2.2 Representations of the Generators of Rotations 106

4.2.3 Generators in an Arbitrary Direction 107

4.2.4 Commutators of the Generators 108

4.2.5 Explicit Form of the Finite Dimensional Representations 111

4.2.6 Summary 112

4.3 Spin and Rotations in Quantum Mechanics 113

4.3.1 Rotations and Spinless Particles 113

4.3.2 Spin 114

4.3.3 The Spin-Zero Representation 116

4.3.4 The Spin-Half Representation 116

4.3.5 Euler Angles 117

4.3.6 The Spin-One Representation 119

4.3.7 Arbitrary j 119

4.4 Addition of Angular Momenta 120

4.4.1 Spin and Orbital Angular Momentum 120

4.4.2 Two Simple Examples 121

4.5 Clebsch-Gordan Coefficients 122

4.5.1 Definition of the Coefficients 122

4.5.2 Spin Half+Spin Half 123

4.5.3 Spin Half+Angular Momentum One 125

4.5.4 Spin Half+Angular Momentum l 126

4.5.5 The General Rule 127

4.5.6 Recursion Relation for the Coefficients 129

4.5.7 The Clebsch-Gordan Series 129

Problems 130

5 Symmetry Transformations on Operators 138

5.1 Vector Observables 138

5.1.1 Symmetries,Lifetimes,and Selection Rules 138

5.1.2 Vector Operators Under Rotations 140

5.1.3 Spherical Components of Vector Observables 141

5.1.4 Selection Rules for Matrix Elements ofVectors 142

5.2 Tensor Observables 144

5.2.1 Cartesian Tensor Operators 144

5.2.2 Spherical Tensor Components 146

5.2.3 Higher Rank Spherical Tensors 147

5.2.4 Selection Rules and the Wigner-Eckart Theorem 148

5.3 Discrete Symmetries 151

5.3.1 Reflections and Parity 151

5.3.2 Reversal of the Direction of Motion 153

5.3.3 Identical Particles 157

5.4 Internal Symmetries:Isospin 159

Problems 164

6 Interlude 171

6.1 External Magnetic Fields 171

6.1.1 Natural Units 171

6.1.2 Gauge Invariance 172

6.1.3 Constant Magnetic Fields and Landau Levels 173

6.1.4 Magnetic Moment 176

6.1.5 The Hydrogen Atom in a Magnetic Field 177

6.2 The Density Matrix 178

6.2.1 Definition 178

6.2.2 Example:Thermodynamic Equilibrium 180

6.2.3 Example:Spin-Half Systems 181

6.2.4 Spin Magnetic Resonance 182

6.3 Neutrino Interference 185

6.3.1 Neutrinos 185

6.3.2 Neutrino Mixing 186

6.3.3 Neutrino Oscillations and the Mass Splitting 187

6.3.4 Solar Neutrinos 189

6.3.5 Neutrino Oscillations in Matter 189

6.4 Measurements in Quantum Mechanics 191

6.4.1 Wave-Function Collapse 191

6.4.2 The EPR Paradox 191

6.4.3 Bell's Inequality 193

Problems 195

7 Approximation Methods for Bound States 202

7.1 Bound-State Perturbation Theory 202

7.1.1 The Perturbation Expansion 202

7.1.2 Example:Harmonic Oscillator 205

7.2 Static External Electric Fields 206

7.2.1 Perturbation of the First Excited Level 207

7.2.2 Polarizabilitv of the Ground State 208

7.3 Fine Structure of the Hydrogen Atom 212

7.3.1 The Spin-Orbit Coupling 212

7.3.2 Correction to Energy Levels 214

7.3.3 The Relativistic Kinetic Energy Correction 215

7.3.4 The Fine Structure of the Hydrogen Atom 216

7.3.5 External Magnetic Field Again 217

7.3.6 The Hyperfine Structure of the Hydrogen Atom 218

7.4 Other Atoms 220

7.4.1 The Ground State of Helium 220

7.4.2 The Perturbation Method for the Helium Atom 222

7.5 The Variational Method 223

7.5.1 The General Method 223

7.5.2 The Helium Atom 225

7.5.3 The Eigenvalue-Variational Scheme 228

7.6 Molecules 229

7.6.1 The Born-Oppenheimer Approximation 229

7.6.2 The Hydrogen Molecular Ion 233

7.7 The WKB Method 237

7.7.1 Turning Points and Connection Formulas 238

7.7.2 The Linear Approximation 240

7.7.3 Bound States 243

7.7.4 Tunneling through a Barrier 247

Problems 248

8 Potential Scattering 264

8.1 Introduction 264

8.1.1 Kinematics of Scattering 264

8.1.2 Scattering and Wave Functions 265

8.2 The Scattering Amplitude 269

8.2.1 Equation for the Scattering Amplitude 269

8.2.2 The Born Series 270

8.2.3 Spherically Symmetric Potentials 270

8.2.4 The Optical Theorem 272

8.2.5 The Refractive Index 273

8.3 Partial Waves 275

8.3.1 Expansion of a Plane Wave in a Legendre Series 276

8.3.2 Partial Wave Expansion of？(r) 278

8.3.3 Calculation of the Phase Shift 280

8.4 The Radial Wave Function 281

8.4.1 The Integral Equation 281

8.4.2 Partial Wave Green's Functions 281

8.4.3 Scattering by an Impenetrable Sphere 283

Problems 284

9 Transitions 288

9.1 Transitions in an External Field 288

9.1.1 Time-Dependent Perturbations 288

9.1.2 The Semiclassical Method 288

9.2 The Transition Matrix 291

9.2.1 The Transition Matrix 292

9.2.2 The Lippmann-Schwinger Equation 295

9.2.3 Relation to the Scattering Amplitude 297

9.3 Scattering and Cross Sections 298

9.3.1 The Scattering Matrix 298

9.3.2 The Transition Probability 299

9.3.3 Cross Sections 300

9.3.4 Scattering of Electrons by Atoms 302

9.3.5 Scattering with Recoil 303

9.3.6 Identical Particle Scattering 305

9.4 Decays of Excited States 307

9.4.1 Lowest-Order Transition Rates 308

9.4.2 Time Dependence of the Initial State 310

9.4.3 Distribution of the Final States 314

Problems 316

10 Further Topics in Quantum Dynamics 324

10.1 Path Integration 324

10.1.1 The Propagator as an Integral over Paths 324

10.1.2 The Free Particle Propagator 327

10.1.3 The Harmonic Oscillator 328

10.1.4 The Euclidean Formalism 331

10.1.5 The Ground-State Energy 332

10.2 Path Integration:Some Applications 334

10.2.1 The Born Series 334

10.2.2 External Fields and Gauge Invariance 337

10.2.3 The Aharonov-Bohm Effect 338

10.3 Berry's Phase 341

10.3.1 Origin of the Phase 341

10.3.2 Example:Electron in a Precessing Magnetic Field 343

10.3.3 The General Formula 344

10.3.4 Two States near a Degeneracy 347

10.3.5 Fast and Slow Coordinates 348

10.3.6 The Aharonov-Bohm Effect Again 351

Problems 352

11 The Quantized Electromagnetic Field 356

11.1 The Classical Electromagnetic Field Hamiltonian 356

11.1.1 Maxwell's Equations and the Transverse Gauge Condition 356

11.1.2 The Independent Modes 358

11.1.3 The Classical Hamiltonian 360

11.1.4 The Canonical Coordinates 361

11.2 The Quantized Radiation Field 362

11.2.1 The Heisenberg Picture 362

11.2.2 Canonical Quantization 362

11.2.3 Photons 364

11.3 Properties of the Quantum Electromagnetic Field 365

11.3.1 The Momentum of the Field 365

11.3.2 The Angular Momentum of the Field 366

11.3.3 The Photon Spin 367

11.4 Electromagnetic Decays of Excited States 369

11.4.1 The Unperturbed Hamiltonian 369

11.4.2 The Vector Potential Interaction 369

11.4.3 The Spin Interaction 370

11.4.4 The Rate for Photon Emission 370

11.4.5 Multipole Matrix Elements 372

11.5 Examples 373

11.5.1 Decay of the 2P State of Atomic Hydrogen 373

11.5.2 Hyperfine Emission 374

11.6 Absorption and Stimulated Emission of Radiation 378

11.6.1 Periodic Boundary Conditions 378

11.6.2 Absorption 379

11.6.3 Stimulated Emission 381

11.6.4 The Blackbody Formula 382

11.7 Scattering of Photons by Atoms 383

11.7.1 The Photoelectric Effect 383

11.7.2 Elastic Scattering of Photons 385

11.7.3 Scattering by a Free Electron 391

11.8 The Casimir Effect 394

11.8.1 The Ground-State Energy of the Electromagnetic Field 394

11.8.2 The Casimir Force with an Elementary Cutoff 397

11.8.3 The General Calculation 400

Problems 402

12 Relativistic Wave Equations 407

12.1 Lorentz Transformations 407

12.1.1 Four-Vectors and Tensors 407

12.1.2 Lorentz Transformations 408

12.1.3 Spin 411

12.2 Vector and Scalar Fields 413

12.2.1 The Electromagnetic Field 413

12.2.2 The Klein-Gordon Equation 414

12.3 Relativistic Spin-Half Equations 417

12.3.1 Two Component Spin-Half Equations 417

12.3.2 The Dirac Equation 420

12.3.3 Free Particle Solutions 421

12.3.4 Probability Current and Hole Theory 423

12.4 Dirac Electron in an Electromagnetic Field 423

12.4.1 Second-Order Form of the Dirac Equation 423

12.4.2 The Gyromagnetic Ratio 424

12.4.3 The Nonrelativistic Limit and the Fine Structure 425

12.5 The Dirac Hydrogen Atom 428

12.5.1 Second-Order Equation 428

12.5.2 Spherically Symmetric Potentials 431

12.5.3 Radial Equations 432

12.5.4 The Hydrogen Atom 433

Problems 435

13 Identical Particles 438

13.1 Nonrelativistic Identical-Particle Systems 438

13.1.1 Creation and Annihilation Operators for Bosons 438

13.1.2 Creation and Annihilation Operators for Fermions 443

13.2 Elementary Applications 444

13.2.1 Ideal Gas Distributions 444

13.2.2 Ideal Electron Gas 446

13.2.3 Collapsed Stars 448

13.3 Relativistic Spinless Particles 453

13.3.1 The Neutral Scalar Field 453

13.3.2 The Classical Theory 453

13.3.3 The Quantum Theory 455

13.3.4 Charged Particles 456

13.4 The Quantized Dirac Field 458

13.4.1 The Dirac Action 458

13.4.2 The Plane Wave Expansion 459

13.5 Interacting Relativistic Fields 462

13.5.1 Normal Ordering 462

13.5.2 Example:The φ4 interaction 463

Problems 465

APPENDICES 470

A Mathematical Tools 470

A.1 Miscellaneous Tools 470

A.1.1 The Dirac Delta Function 470

A.1.2 The Levi-Civita Symbol 472

A.1.3 Some Integrals 472

A.1.4 The Trapezoidal Approximation Series 474

A.2 Special Functions 476

A.2.1 Gamma Function 476

A.2.2 Legendre Polynomials 479

A.2.3 Solutions to the Free Radial Equation 484

A.2.4 Hermite Polynomials 487

A.2.5 Bessel Functions 489

A.3 Orthogonal Curvilinear Coordinates 491

A.3.1 Vector Calculus in Orthogonal Curvilinear Coordinates 491

A.3.2 Hydrogen Atom in Parabolic Coordinates 495

A.3.3 Elliptic Coordinates 498

B Rotation Matrices 500

B.1 Rotation Matrices—Ⅰ 500

B.1.1 Rotation Matrices and Spherical Harmonics 500

B.1.2 The Explicit Form of the Rotation Matrices 502

B.1.3 The Projection Theorem 507

B.2 Rotation Matrices—Ⅱ 508

B.2.1 Averages over Products of Rotation Matrices 508

B.2.2 The Wigner-Eckart Theorem Again 510

C SU(3) 512

C.1 The Group and Algebra 512

C.2 Some Representations 513

D References 516

Index 519