John David Jackson

John David Jackson (1925-2016) was a revered physics professor at the University of California, Berkeley, a faculty Senior Scientist at Lawrence Berkeley National Laboratory, and a member of the National Academy of Sciences. A theoretical physicist, he is well known for numerous publications and summer-school lectures in nuclear and particle physics, as well as for his definitive text, Classical Electrodynamics. Robert N. Cahn is Senior Scientist, emeritus, at the Lawrence Berkeley National Laboratory. He has conducted research in theoretical and experimental particle physics and cosmology. The co-author, with Gerson Goldhaber, of the text Experimental Foundations of Particle Physics, he has taught physics at both the undergraduate and graduate levels.

Preface ix

About the Companion Website xi

1 Basics 1

1.1 Wave Mechanics of de Broglie and Schrödinger 1

1.2 Klein-Gordon Equation 2

1.3 Non-Relativistic Approximation 2

1.4 Free-Particle Probability Current 3

1.5 Expectation Values 4

1.6 Particle in a Static, Conservative Force Field 6

1.7 Ehrenfest Theorem 6

1.8 Schrödinger Equation in Momentum Space 8

1.9 Spread in Time of a Free-Particle Wave Packet 8

1.10 The Nature of Solutions to the Schrödinger Equation 9

1.11 A Bound-State Problem: Linear Potential 10

1.12 Sturm-Liouville Eigenvalue Problem 11

1.13 Linear Operators on Functions 13

1.14 Eigenvalue Problem for a Hermitian Operator 14

1.15 Variational Methods for Energy Eigenvalues 14

1.16 Rayleigh-Ritz Method 16

Problems 18

2 Reformulation 21

2.1 Stern-Gerlach Experiment 22

2.2 Linear Vector Spaces 22

2.3 Linear Operators 25

2.4 Unitary Transformations of Operators 27

2.5 Generalized Uncertainty Relation for Self-Adjoint Operators 27

2.6 Infinite-Dimensional Vector Spaces - Hilbert Space 28

2.7 Assumptions of Quantum Mechanics 29

2.8 Mixtures and the Density Matrix 30

2.9 Measurement 32

2.10 Classical vs. Quantum Probabilities 33

2.11 Capsule Review of Classical Mechanics and Conservation Laws 34

2.12 Translation Invariance and Momentum Conservation 37

2.13 Dirac’s p’s and q’s 38

2.14 Time Development of the State Vector 41

2.15 Schrödinger and Heisenberg Pictures 42

2.16 Simple Harmonic Oscillator 46

Problems 51

3 Wentzel-Kramers-Brillouin (WKB) Method 55

3.1 Semi-classical Approximation 55

3.2 Solution in One Dimension 56

3.3 Schrödinger Equation for the Linear Potential 58

3.4 Connection Formulae for the WKB Method 63

3.5 WKB Formula for Bound States 65

3.6 Example of WKB with a Power Law Potential 67

3.7 Normalization of WKB Bound State Wave Functions 68

3.8 Bohr’s Correspondence Principle and Classical Motion 68

3.9 Power of WKB 72

3.10 Barrier Penetration with the WKB Method 73

3.11 Symmetrical Double-Well Potential 75

3.12 Application of the WKB Method to Ammonia Molecule 79

Problems 80

4 Rotations, Angular Momentum, and Central Force Motion 85

4.1 Infinitesimal Rotations 85

4.2 Construction of Irreducible Representations 88

4.3 Coordinate Representation of Angular Momentum Eigenvectors 91

4.4 Observation of Sign Change for Rotation by 2π 92

4.5 Euler Angles, Wigner d-functions 95

4.6 Application to Nuclear Magnetic Resonance 98

4.7 Addition of Angular Momenta 104

4.8 Integration Over the Rotation Group 106

4.9 Gaunt Integral 108

4.10 Tensor Operators 109

4.11 Wigner-Eckart Theorem 112

4.12 Applications of the Wigner-Eckart Theorem 114

4.13 Two-Body Central Force Motion 118

4.14 The Coulomb Problem 121

4.15 Patterns of Bound States 125

4.16 Hellmann-Feynman Theorem 127

Problems 128

5 Time-Independent Perturbation Theory 135

5.1 Time-Independent Perturbation Expansion 135

5.2 Interlude: Spectra and History 137

5.3 Fine Structure of Hydrogen 139

5.4 Stark Effect in Ground-State Hydrogen 141

5.5 Perturbation Theory with Degeneracy 143

5.6 Linear Stark Effect in Hydrogen 145

5.7 Perturbation Theory with Near Degeneracy 146

5.8 Zeeman and Paschen-Back Effects in Hydrogen 149

Problems 149

6 Atomic Structure 151

6.1 Parity 151

6.2 Identical Particles and the Pauli Exclusion Principle 153

6.3 Atoms 158

6.4 Helium Atom 159

6.5 Periodic Table 164

6.6 Multiplet Structure, Russell-Saunders Coupling 165

6.7 Spin-Orbit Interaction 172

6.8 Intermediate Coupling 176

6.9 Thomas-Fermi Atom 180

6.10 Hartree-Fock Approximation 185

Problems 189

7 Time-Dependent Perturbation Theory and Scattering 197

7.1 Time-dependent Perturbation Theory 197

7.2 Fermi’s Golden Rule 202

7.3 Scattering Amplitude 204

7.4 Born Approximation 205

7.5 Scattering Theory from Fermi’s Golden Rule 207

7.6 Inelastic Scattering 211

7.7 Optical Theorem 214

7.8 Validity Criterion for the First Born Approximation 216

7.9 Eikonal Approximation 216

7.10 Method of Partial Waves 223

7.11 Behavior of the Cross Section and the Argand Diagram 225

7.12 Hard Sphere Scattering 227

7.13 Strongly Attractive Potentials and Resonance 229

7.14 Levinson’s Theorem 232

Problems 234

8 Semi-Classical and Quantum Electromagnetic Field 241

8.1 Electromagnetic Hamiltonian and Gauge Invariance 241

8.2 Aharonov-Bohm Effect 242

8.3 Semi-Classical Radiation Theory 244

8.4 Scalar Field Quantization 246

8.5 Quantization of the Radiation Field 247

8.6 States of the Electromagnetic Field 252

8.7 Vacuum Expectation Values of E, E ⋅ E over Finite Volume 253

8.8 Classical vs. Quantum Radiation 254

8.9 Quasi-Classical Fields and Coherent States 255

Problems 257

9 Emission and Absorption of Radiation 259

9.1 Matrix Elements and Rates 259

9.2 Dipole Transitions 261

9.3 General Selection Rules 262

9.4 Charged Particle in a Central Field 263

9.5 Decay Rates with LS Coupling 264

9.6 Line Breadth and Level Shift 267

9.7 Alteration of Spontaneous Emission from Changed Density of States 272

Problems 277

10 Relativistic Quantum Mechanics 281

10.1 Klein-Gordon Equation 281

10.2 Dirac Equation 283

10.3 Angular Momentum in Dirac Equation 285

10.4 Two-Component Equation and Plane-Wave Solutions 286

10.5 Dirac’s Treatment of Negative Energy States 288

10.6 Heisenberg Operators and Equations of Motion 289

10.7 Hydrogen in the Dirac Equation 290

10.8 Foldy-Wouthuysen Transformation 291

10.9 Lorentz Covariance 294

10.10 Discrete Symmetries 297

10.11 Bilinear Covariants 301

10.12 Applications to Electromagnetic Form Factors 302

10.13 Potential Scattering of a Dirac Particle 305

10.14 Neutron-Electron Scattering 307

10.15 Compton Scattering 312

Problems 322

A Dimensions and Units 327

B Mathematical Tools 329

B. 1 Contour Integration 329

B. 2 Green Function for Helmholtz Equation 333

B. 3 Wigner 3-j and 6-j Symbols 335

C Selected Solutions 339

C. 1 Chapter 1 339

C. 2 Chapter 2 340

C. 3 Chapter 3 343

C. 4 Chapter 4 352

C. 5 Chapter 5 362

C. 6 Chapter 6 366

C. 7 Chapter 7 375

C. 8 Chapter 8 377

C. 9 Chapter 9 380

C. 10 Chapter 10 387

Bibliography 393

Index 395