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1 Classical Mechanics1

1.1 Newton's Laws,the Action,and the Hamiltonian1

1.1.1 Newton's Law and Lagrange's Equations1

1.1.2 Hamilton's Principle2

1.1.3 Canonical Momenta and the Hamiltonian Formulation4

1.2 Classical Space-Time Symmetries6

1.2.1 The Space-Time Transformations6

1.2.2 Translations8

1.2.3 Rotations9

1.2.4 Rotation Matrices11

1.2.5 Symmetries and Conservation Laws12

Problems14

2 Fundamentals of Quantum Mechanics19

2.1 The Superposition Principle19

2.1.1 The Double-Slit Experiment19

2.1.2 The Stern-Gerlach Experiment22

2.2 The Mathematical Language of Quantum Mechanics24

2.2.1 Vector Spaces24

2.2.2 The Probability Interpretation27

2.2.3 Linear Operators27

2.2.4 Observables30

2.2.5 Examples31

2.3 Continuous Eigenvalues35

2.3.1 The Dirac Delta Function35

2.3.2 Continuous Observables36

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

2.4 Canonical Commutators and the Schr？dinger Equation38

2.4.1 The Correspondence Principle38

2.4.2 The Canonical Commutation Relations42

2.4.3 Planck's Constant43

2.5 Quantum Dynamics44

2.5.1 The Time-Translation Operator44

2.5.2 The Heisenberg Picture45

2.6 The Uncertainty Principle47

2.7 Wave Functions49

2.7.1 Wave Functions in Coordinate Space49

2.7.2 Momentum and Translations49

2.7.3 Schr？dinger's Wave Equation52

2.7.4 Time-Dependent Free Particle Wave Functions53

Problems55

3 Stationary States62

3.1 Elementary Examples62

3.1.1 States with Definite Energy62

3.1.2 A Two-State System63

3.1.3 One-Dimensional Potential Problems66

3.2 The Harmonic Oscillator68

3.2.1 The Spectrum69

3.2.2 Matrix Elements71

3.2.3 The Ground-State Energy72

3.2.4 Wave Functions73

3.3 Spherically Symmetric Potentials and Angular Momentum74

3.3.1 Spherical Symmetry74

3.3.2 Orbital Angular Momentum as a Difierential Operator75

3.3.3 The Angular Momentum Commutator Algebra76

3.3.4 Classification of the States80

3.4 Spherically Symmetric Potentials:Wave Functions80

3.4.1 Spherical Coordinates and Spherical Harmonics80

3.4.2 The Radial Wave Equation82

3.5 Hydrogenlike Atoms84

3.5.1 The Symmetries84

3.5.2 The Energy Spectrum86

3.5.3 The Radial Wave Functions88

Problems91

4 Symmetry Transformations on States102

4.1 Introduction102

4.1.1 Symmetries and Transformations102

4.1.2 Groups of Transformations103

4.1.3 Classical and Quantum Symmetries105

4.2 The Rotation Group and Algebra105

4.2.1 Representations of Groups105

4.2.2 Representations of the Generators of Rotations106

4.2.3 Generators in an Arbitrary Direction107

4.2.4 Commutators of the Generators108

4.2.5 Explicit Form of the Finite Dimensional Representations111

4.2.6 Summary112

4.3 Spin and Rotations in Quantum Mechanics113

4.3.1 Rotations and Spinless Particles113

4.3.2 Spin114

4.3.3 The Spin-Zero Representation116

4.3.4 The Spin-Half Representation116

4.3.5 Euler Angles117

4.3.6 The Spin-One Representation119

4.3.7 Arbitrary j119

4.4 Addition of Angular Momenta120

4.4.1 Spin and Orbital Angular Momentum120

4.4.2 Two Simple Examples121

4.5 Clebsch-Gordan Coefficients122

4.5.1 Definition of the Coefficients122

4.5.2 Spin Half+Spin Half123

4.5.3 Spin Half+Angular Momentum One125

4.5.4 Spin Half+Angular Momentum l126

4.5.5 The General Rule127

4.5.6 Recursion Relation for the Coefficients129

4.5.7 The Clebsch-Gordan Series129

Problems130

5 Symmetry Transformations on Operators138

5.1 Vector Observables138

5.1.1 Symmetries,Lifetimes,and Selection Rules138

5.1.2 Vector Operators Under Rotations140

5.1.3 Spherical Components of Vector Observables141

5.1.4 Selection Rules for Matrix Elements ofVectors142

5.2 Tensor Observables144

5.2.1 Cartesian Tensor Operators144

5.2.2 Spherical Tensor Components146

5.2.3 Higher Rank Spherical Tensors147

5.2.4 Selection Rules and the Wigner-Eckart Theorem148

5.3 Discrete Symmetries151

5.3.1 Reflections and Parity151

5.3.2 Reversal of the Direction of Motion153

5.3.3 Identical Particles157

5.4 Internal Symmetries:Isospin159

Problems164

6 Interlude171

6.1 External Magnetic Fields171

6.1.1 Natural Units171

6.1.2 Gauge Invariance172

6.1.3 Constant Magnetic Fields and Landau Levels173

6.1.4 Magnetic Moment176

6.1.5 The Hydrogen Atom in a Magnetic Field177

6.2 The Density Matrix178

6.2.1 Definition178

6.2.2 Example:Thermodynamic Equilibrium180

6.2.3 Example:Spin-Half Systems181

6.2.4 Spin Magnetic Resonance182

6.3 Neutrino Interference185

6.3.1 Neutrinos185

6.3.2 Neutrino Mixing186

6.3.3 Neutrino Oscillations and the Mass Splitting187

6.3.4 Solar Neutrinos189

6.3.5 Neutrino Oscillations in Matter189

6.4 Measurements in Quantum Mechanics191

6.4.1 Wave-Function Collapse191

6.4.2 The EPR Paradox191

6.4.3 Bell's Inequality193

Problems195

7 Approximation Methods for Bound States202

7.1 Bound-State Perturbation Theory202

7.1.1 The Perturbation Expansion202

7.1.2 Example:Harmonic Oscillator205

7.2 Static External Electric Fields206

7.2.1 Perturbation of the First Excited Level207

7.2.2 Polarizabilitv of the Ground State208

7.3 Fine Structure of the Hydrogen Atom212

7.3.1 The Spin-Orbit Coupling212

7.3.2 Correction to Energy Levels214

7.3.3 The Relativistic Kinetic Energy Correction215

7.3.4 The Fine Structure of the Hydrogen Atom216

7.3.5 External Magnetic Field Again217

7.3.6 The Hyperfine Structure of the Hydrogen Atom218

7.4 Other Atoms220

7.4.1 The Ground State of Helium220

7.4.2 The Perturbation Method for the Helium Atom222

7.5 The Variational Method223

7.5.1 The General Method223

7.5.2 The Helium Atom225

7.5.3 The Eigenvalue-Variational Scheme228

7.6 Molecules229

7.6.1 The Born-Oppenheimer Approximation229

7.6.2 The Hydrogen Molecular Ion233

7.7 The WKB Method237

7.7.1 Turning Points and Connection Formulas238

7.7.2 The Linear Approximation240

7.7.3 Bound States243

7.7.4 Tunneling through a Barrier247

Problems248

8 Potential Scattering264

8.1 Introduction264

8.1.1 Kinematics of Scattering264

8.1.2 Scattering and Wave Functions265

8.2 The Scattering Amplitude269

8.2.1 Equation for the Scattering Amplitude269

8.2.2 The Born Series270

8.2.3 Spherically Symmetric Potentials270

8.2.4 The Optical Theorem272

8.2.5 The Refractive Index273

8.3 Partial Waves275

8.3.1 Expansion of a Plane Wave in a Legendre Series276

8.3.2 Partial Wave Expansion of？(r)278

8.3.3 Calculation of the Phase Shift280

8.4 The Radial Wave Function281

8.4.1 The Integral Equation281

8.4.2 Partial Wave Green's Functions281

8.4.3 Scattering by an Impenetrable Sphere283

Problems284

9 Transitions288

9.1 Transitions in an External Field288

9.1.1 Time-Dependent Perturbations288

9.1.2 The Semiclassical Method288

9.2 The Transition Matrix291

9.2.1 The Transition Matrix292

9.2.2 The Lippmann-Schwinger Equation295

9.2.3 Relation to the Scattering Amplitude297

9.3 Scattering and Cross Sections298

9.3.1 The Scattering Matrix298

9.3.2 The Transition Probability299

9.3.3 Cross Sections300

9.3.4 Scattering of Electrons by Atoms302

9.3.5 Scattering with Recoil303

9.3.6 Identical Particle Scattering305

9.4 Decays of Excited States307

9.4.1 Lowest-Order Transition Rates308

9.4.2 Time Dependence of the Initial State310

9.4.3 Distribution of the Final States314

Problems316

10 Further Topics in Quantum Dynamics324

10.1 Path Integration324

10.1.1 The Propagator as an Integral over Paths324

10.1.2 The Free Particle Propagator327

10.1.3 The Harmonic Oscillator328

10.1.4 The Euclidean Formalism331

10.1.5 The Ground-State Energy332

10.2 Path Integration:Some Applications334

10.2.1 The Born Series334

10.2.2 External Fields and Gauge Invariance337

10.2.3 The Aharonov-Bohm Effect338

10.3 Berry's Phase341

10.3.1 Origin of the Phase341

10.3.2 Example:Electron in a Precessing Magnetic Field343

10.3.3 The General Formula344

10.3.4 Two States near a Degeneracy347

10.3.5 Fast and Slow Coordinates348

10.3.6 The Aharonov-Bohm Effect Again351

Problems352

11 The Quantized Electromagnetic Field356

11.1 The Classical Electromagnetic Field Hamiltonian356

11.1.1 Maxwell's Equations and the Transverse Gauge Condition356

11.1.2 The Independent Modes358

11.1.3 The Classical Hamiltonian360

11.1.4 The Canonical Coordinates361

11.2 The Quantized Radiation Field362

11.2.1 The Heisenberg Picture362

11.2.2 Canonical Quantization362

11.2.3 Photons364

11.3 Properties of the Quantum Electromagnetic Field365

11.3.1 The Momentum of the Field365

11.3.2 The Angular Momentum of the Field366

11.3.3 The Photon Spin367

11.4 Electromagnetic Decays of Excited States369

11.4.1 The Unperturbed Hamiltonian369

11.4.2 The Vector Potential Interaction369

11.4.3 The Spin Interaction370

11.4.4 The Rate for Photon Emission370

11.4.5 Multipole Matrix Elements372

11.5 Examples373

11.5.1 Decay of the 2P State of Atomic Hydrogen373

11.5.2 Hyperfine Emission374

11.6 Absorption and Stimulated Emission of Radiation378

11.6.1 Periodic Boundary Conditions378

11.6.2 Absorption379

11.6.3 Stimulated Emission381

11.6.4 The Blackbody Formula382

11.7 Scattering of Photons by Atoms383

11.7.1 The Photoelectric Effect383

11.7.2 Elastic Scattering of Photons385

11.7.3 Scattering by a Free Electron391

11.8 The Casimir Effect394

11.8.1 The Ground-State Energy of the Electromagnetic Field394

11.8.2 The Casimir Force with an Elementary Cutoff397

11.8.3 The General Calculation400

Problems402

12 Relativistic Wave Equations407

12.1 Lorentz Transformations407

12.1.1 Four-Vectors and Tensors407

12.1.2 Lorentz Transformations408

12.1.3 Spin411

12.2 Vector and Scalar Fields413

12.2.1 The Electromagnetic Field413

12.2.2 The Klein-Gordon Equation414

12.3 Relativistic Spin-Half Equations417

12.3.1 Two Component Spin-Half Equations417

12.3.2 The Dirac Equation420

12.3.3 Free Particle Solutions421

12.3.4 Probability Current and Hole Theory423

12.4 Dirac Electron in an Electromagnetic Field423

12.4.1 Second-Order Form of the Dirac Equation423

12.4.2 The Gyromagnetic Ratio424

12.4.3 The Nonrelativistic Limit and the Fine Structure425

12.5 The Dirac Hydrogen Atom428

12.5.1 Second-Order Equation428

12.5.2 Spherically Symmetric Potentials431

12.5.3 Radial Equations432

12.5.4 The Hydrogen Atom433

Problems435

13 Identical Particles438

13.1 Nonrelativistic Identical-Particle Systems438

13.1.1 Creation and Annihilation Operators for Bosons438

13.1.2 Creation and Annihilation Operators for Fermions443

13.2 Elementary Applications444

13.2.1 Ideal Gas Distributions444

13.2.2 Ideal Electron Gas446

13.2.3 Collapsed Stars448

13.3 Relativistic Spinless Particles453

13.3.1 The Neutral Scalar Field453

13.3.2 The Classical Theory453

13.3.3 The Quantum Theory455

13.3.4 Charged Particles456

13.4 The Quantized Dirac Field458

13.4.1 The Dirac Action458

13.4.2 The Plane Wave Expansion459

13.5 Interacting Relativistic Fields462

13.5.1 Normal Ordering462

13.5.2 Example:The φ4 interaction463

Problems465

APPENDICES470

A Mathematical Tools470

A.1 Miscellaneous Tools470

A.1.1 The Dirac Delta Function470

A.1.2 The Levi-Civita Symbol472

A.1.3 Some Integrals472

A.1.4 The Trapezoidal Approximation Series474

A.2 Special Functions476

A.2.1 Gamma Function476

A.2.2 Legendre Polynomials479

A.2.3 Solutions to the Free Radial Equation484

A.2.4 Hermite Polynomials487

A.2.5 Bessel Functions489

A.3 Orthogonal Curvilinear Coordinates491

A.3.1 Vector Calculus in Orthogonal Curvilinear Coordinates491

A.3.2 Hydrogen Atom in Parabolic Coordinates495

A.3.3 Elliptic Coordinates498

B Rotation Matrices500

B.1 Rotation Matrices—Ⅰ500

B.1.1 Rotation Matrices and Spherical Harmonics500

B.1.2 The Explicit Form of the Rotation Matrices502

B.1.3 The Projection Theorem507

B.2 Rotation Matrices—Ⅱ508

B.2.1 Averages over Products of Rotation Matrices508

B.2.2 The Wigner-Eckart Theorem Again510

C SU(3)512

C.1 The Group and Algebra512

C.2 Some Representations513

D References516

Index519