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 ixAbout the Companion Website xi1 Basics 11.1 Wave Mechanics of de Broglie and Schrödinger 11.2 Klein-Gordon Equation 21.3 Non-Relativistic Approximation 21.4 Free-Particle Probability Current 31.5 Expectation Values 41.6 Particle in a Static, Conservative Force Field 61.7 Ehrenfest Theorem 61.8 Schrödinger Equation in Momentum Space 81.9 Spread in Time of a Free-Particle Wave Packet 81.10 The Nature of Solutions to the Schrödinger Equation 91.11 A Bound-State Problem: Linear Potential 101.12 Sturm-Liouville Eigenvalue Problem 111.13 Linear Operators on Functions 131.14 Eigenvalue Problem for a Hermitian Operator 141.15 Variational Methods for Energy Eigenvalues 141.16 Rayleigh-Ritz Method 16Problems 182 Reformulation 212.1 Stern-Gerlach Experiment 222.2 Linear Vector Spaces 222.3 Linear Operators 252.4 Unitary Transformations of Operators 272.5 Generalized Uncertainty Relation for Self-Adjoint Operators 272.6 Infinite-Dimensional Vector Spaces - Hilbert Space 282.7 Assumptions of Quantum Mechanics 292.8 Mixtures and the Density Matrix 302.9 Measurement 322.10 Classical vs. Quantum Probabilities 332.11 Capsule Review of Classical Mechanics and Conservation Laws 342.12 Translation Invariance and Momentum Conservation 372.13 Dirac’s p’s and q’s 382.14 Time Development of the State Vector 412.15 Schrödinger and Heisenberg Pictures 422.16 Simple Harmonic Oscillator 46Problems 513 Wentzel-Kramers-Brillouin (WKB) Method 553.1 Semi-classical Approximation 553.2 Solution in One Dimension 563.3 Schrödinger Equation for the Linear Potential 583.4 Connection Formulae for the WKB Method 633.5 WKB Formula for Bound States 653.6 Example of WKB with a Power Law Potential 673.7 Normalization of WKB Bound State Wave Functions 683.8 Bohr’s Correspondence Principle and Classical Motion 683.9 Power of WKB 723.10 Barrier Penetration with the WKB Method 733.11 Symmetrical Double-Well Potential 753.12 Application of the WKB Method to Ammonia Molecule 79Problems 804 Rotations, Angular Momentum, and Central Force Motion 854.1 Infinitesimal Rotations 854.2 Construction of Irreducible Representations 884.3 Coordinate Representation of Angular Momentum Eigenvectors 914.4 Observation of Sign Change for Rotation by 2π 924.5 Euler Angles, Wigner d-functions 954.6 Application to Nuclear Magnetic Resonance 984.7 Addition of Angular Momenta 1044.8 Integration Over the Rotation Group 1064.9 Gaunt Integral 1084.10 Tensor Operators 1094.11 Wigner-Eckart Theorem 1124.12 Applications of the Wigner-Eckart Theorem 1144.13 Two-Body Central Force Motion 1184.14 The Coulomb Problem 1214.15 Patterns of Bound States 1254.16 Hellmann-Feynman Theorem 127Problems 1285 Time-Independent Perturbation Theory 1355.1 Time-Independent Perturbation Expansion 1355.2 Interlude: Spectra and History 1375.3 Fine Structure of Hydrogen 1395.4 Stark Effect in Ground-State Hydrogen 1415.5 Perturbation Theory with Degeneracy 1435.6 Linear Stark Effect in Hydrogen 1455.7 Perturbation Theory with Near Degeneracy 1465.8 Zeeman and Paschen-Back Effects in Hydrogen 149Problems 1496 Atomic Structure 1516.1 Parity 1516.2 Identical Particles and the Pauli Exclusion Principle 1536.3 Atoms 1586.4 Helium Atom 1596.5 Periodic Table 1646.6 Multiplet Structure, Russell-Saunders Coupling 1656.7 Spin-Orbit Interaction 1726.8 Intermediate Coupling 1766.9 Thomas-Fermi Atom 1806.10 Hartree-Fock Approximation 185Problems 1897 Time-Dependent Perturbation Theory and Scattering 1977.1 Time-dependent Perturbation Theory 1977.2 Fermi’s Golden Rule 2027.3 Scattering Amplitude 2047.4 Born Approximation 2057.5 Scattering Theory from Fermi’s Golden Rule 2077.6 Inelastic Scattering 2117.7 Optical Theorem 2147.8 Validity Criterion for the First Born Approximation 2167.9 Eikonal Approximation 2167.10 Method of Partial Waves 2237.11 Behavior of the Cross Section and the Argand Diagram 2257.12 Hard Sphere Scattering 2277.13 Strongly Attractive Potentials and Resonance 2297.14 Levinson’s Theorem 232Problems 2348 Semi-Classical and Quantum Electromagnetic Field 2418.1 Electromagnetic Hamiltonian and Gauge Invariance 2418.2 Aharonov-Bohm Effect 2428.3 Semi-Classical Radiation Theory 2448.4 Scalar Field Quantization 2468.5 Quantization of the Radiation Field 2478.6 States of the Electromagnetic Field 2528.7 Vacuum Expectation Values of E, E ⋅ E over Finite Volume 2538.8 Classical vs. Quantum Radiation 2548.9 Quasi-Classical Fields and Coherent States 255Problems 2579 Emission and Absorption of Radiation 2599.1 Matrix Elements and Rates 2599.2 Dipole Transitions 2619.3 General Selection Rules 2629.4 Charged Particle in a Central Field 2639.5 Decay Rates with LS Coupling 2649.6 Line Breadth and Level Shift 2679.7 Alteration of Spontaneous Emission from Changed Density of States 272Problems 27710 Relativistic Quantum Mechanics 28110.1 Klein-Gordon Equation 28110.2 Dirac Equation 28310.3 Angular Momentum in Dirac Equation 28510.4 Two-Component Equation and Plane-Wave Solutions 28610.5 Dirac’s Treatment of Negative Energy States 28810.6 Heisenberg Operators and Equations of Motion 28910.7 Hydrogen in the Dirac Equation 29010.8 Foldy-Wouthuysen Transformation 29110.9 Lorentz Covariance 29410.10 Discrete Symmetries 29710.11 Bilinear Covariants 30110.12 Applications to Electromagnetic Form Factors 30210.13 Potential Scattering of a Dirac Particle 30510.14 Neutron-Electron Scattering 30710.15 Compton Scattering 312Problems 322A Dimensions and Units 327B Mathematical Tools 329B. 1 Contour Integration 329B. 2 Green Function for Helmholtz Equation 333B. 3 Wigner 3-j and 6-j Symbols 335C Selected Solutions 339C. 1 Chapter 1 339C. 2 Chapter 2 340C. 3 Chapter 3 343C. 4 Chapter 4 352C. 5 Chapter 5 362C. 6 Chapter 6 366C. 7 Chapter 7 375C. 8 Chapter 8 377C. 9 Chapter 9 380C. 10 Chapter 10 387Bibliography 393Index 395

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 ixAbout the Companion Website xi1 Basics 11.1 Wave Mechanics of de Broglie and Schrödinger 11.2 Klein-Gordon Equation 21.3 Non-Relativistic Approximation 21.4 Free-Particle Probability Current 31.5 Expectation Values 41.6 Particle in a Static, Conservative Force Field 61.7 Ehrenfest Theorem 61.8 Schrödinger Equation in Momentum Space 81.9 Spread in Time of a Free-Particle Wave Packet 81.10 The Nature of Solutions to the Schrödinger Equation 91.11 A Bound-State Problem: Linear Potential 101.12 Sturm-Liouville Eigenvalue Problem 111.13 Linear Operators on Functions 131.14 Eigenvalue Problem for a Hermitian Operator 141.15 Variational Methods for Energy Eigenvalues 141.16 Rayleigh-Ritz Method 16Problems 182 Reformulation 212.1 Stern-Gerlach Experiment 222.2 Linear Vector Spaces 222.3 Linear Operators 252.4 Unitary Transformations of Operators 272.5 Generalized Uncertainty Relation for Self-Adjoint Operators 272.6 Infinite-Dimensional Vector Spaces - Hilbert Space 282.7 Assumptions of Quantum Mechanics 292.8 Mixtures and the Density Matrix 302.9 Measurement 322.10 Classical vs. Quantum Probabilities 332.11 Capsule Review of Classical Mechanics and Conservation Laws 342.12 Translation Invariance and Momentum Conservation 372.13 Dirac’s p’s and q’s 382.14 Time Development of the State Vector 412.15 Schrödinger and Heisenberg Pictures 422.16 Simple Harmonic Oscillator 46Problems 513 Wentzel-Kramers-Brillouin (WKB) Method 553.1 Semi-classical Approximation 553.2 Solution in One Dimension 563.3 Schrödinger Equation for the Linear Potential 583.4 Connection Formulae for the WKB Method 633.5 WKB Formula for Bound States 653.6 Example of WKB with a Power Law Potential 673.7 Normalization of WKB Bound State Wave Functions 683.8 Bohr’s Correspondence Principle and Classical Motion 683.9 Power of WKB 723.10 Barrier Penetration with the WKB Method 733.11 Symmetrical Double-Well Potential 753.12 Application of the WKB Method to Ammonia Molecule 79Problems 804 Rotations, Angular Momentum, and Central Force Motion 854.1 Infinitesimal Rotations 854.2 Construction of Irreducible Representations 884.3 Coordinate Representation of Angular Momentum Eigenvectors 914.4 Observation of Sign Change for Rotation by 2π 924.5 Euler Angles, Wigner d-functions 954.6 Application to Nuclear Magnetic Resonance 984.7 Addition of Angular Momenta 1044.8 Integration Over the Rotation Group 1064.9 Gaunt Integral 1084.10 Tensor Operators 1094.11 Wigner-Eckart Theorem 1124.12 Applications of the Wigner-Eckart Theorem 1144.13 Two-Body Central Force Motion 1184.14 The Coulomb Problem 1214.15 Patterns of Bound States 1254.16 Hellmann-Feynman Theorem 127Problems 1285 Time-Independent Perturbation Theory 1355.1 Time-Independent Perturbation Expansion 1355.2 Interlude: Spectra and History 1375.3 Fine Structure of Hydrogen 1395.4 Stark Effect in Ground-State Hydrogen 1415.5 Perturbation Theory with Degeneracy 1435.6 Linear Stark Effect in Hydrogen 1455.7 Perturbation Theory with Near Degeneracy 1465.8 Zeeman and Paschen-Back Effects in Hydrogen 149Problems 1496 Atomic Structure 1516.1 Parity 1516.2 Identical Particles and the Pauli Exclusion Principle 1536.3 Atoms 1586.4 Helium Atom 1596.5 Periodic Table 1646.6 Multiplet Structure, Russell-Saunders Coupling 1656.7 Spin-Orbit Interaction 1726.8 Intermediate Coupling 1766.9 Thomas-Fermi Atom 1806.10 Hartree-Fock Approximation 185Problems 1897 Time-Dependent Perturbation Theory and Scattering 1977.1 Time-dependent Perturbation Theory 1977.2 Fermi’s Golden Rule 2027.3 Scattering Amplitude 2047.4 Born Approximation 2057.5 Scattering Theory from Fermi’s Golden Rule 2077.6 Inelastic Scattering 2117.7 Optical Theorem 2147.8 Validity Criterion for the First Born Approximation 2167.9 Eikonal Approximation 2167.10 Method of Partial Waves 2237.11 Behavior of the Cross Section and the Argand Diagram 2257.12 Hard Sphere Scattering 2277.13 Strongly Attractive Potentials and Resonance 2297.14 Levinson’s Theorem 232Problems 2348 Semi-Classical and Quantum Electromagnetic Field 2418.1 Electromagnetic Hamiltonian and Gauge Invariance 2418.2 Aharonov-Bohm Effect 2428.3 Semi-Classical Radiation Theory 2448.4 Scalar Field Quantization 2468.5 Quantization of the Radiation Field 2478.6 States of the Electromagnetic Field 2528.7 Vacuum Expectation Values of E, E ⋅ E over Finite Volume 2538.8 Classical vs. Quantum Radiation 2548.9 Quasi-Classical Fields and Coherent States 255Problems 2579 Emission and Absorption of Radiation 2599.1 Matrix Elements and Rates 2599.2 Dipole Transitions 2619.3 General Selection Rules 2629.4 Charged Particle in a Central Field 2639.5 Decay Rates with LS Coupling 2649.6 Line Breadth and Level Shift 2679.7 Alteration of Spontaneous Emission from Changed Density of States 272Problems 27710 Relativistic Quantum Mechanics 28110.1 Klein-Gordon Equation 28110.2 Dirac Equation 28310.3 Angular Momentum in Dirac Equation 28510.4 Two-Component Equation and Plane-Wave Solutions 28610.5 Dirac’s Treatment of Negative Energy States 28810.6 Heisenberg Operators and Equations of Motion 28910.7 Hydrogen in the Dirac Equation 29010.8 Foldy-Wouthuysen Transformation 29110.9 Lorentz Covariance 29410.10 Discrete Symmetries 29710.11 Bilinear Covariants 30110.12 Applications to Electromagnetic Form Factors 30210.13 Potential Scattering of a Dirac Particle 30510.14 Neutron-Electron Scattering 30710.15 Compton Scattering 312Problems 322A Dimensions and Units 327B Mathematical Tools 329B. 1 Contour Integration 329B. 2 Green Function for Helmholtz Equation 333B. 3 Wigner 3-j and 6-j Symbols 335C Selected Solutions 339C. 1 Chapter 1 339C. 2 Chapter 2 340C. 3 Chapter 3 343C. 4 Chapter 4 352C. 5 Chapter 5 362C. 6 Chapter 6 366C. 7 Chapter 7 375C. 8 Chapter 8 377C. 9 Chapter 9 380C. 10 Chapter 10 387Bibliography 393Index 395