1.5 Relativity and Electromagnetism

1.6 The Compton Effect

1.7 Problems

2 Aspects of Angular Momentum

2.1 Various Angular Momenta

2.2 Angular Momentum and Rotations

2.3 Operators and Eigenvectors for Spin 1/2

2.4 Operators for Higher Spins

2.5 Orbital Magnetic Moments

2.6 Spin Without Relativity

2.7 Thomas Precession

2.8 The Pauli Equation in a Central Potential

2.9 Dirac Notation

2.10 Clebsch-Gordan and Racah Coefficients

2.11 Relativistic Quantum Numbers and Spin-Angular Functions

2.12 Energy Levels of the One-Electron Atom

2.13 Plane Wave Expansions

2.14 Problems

3 Particles of Spin Zero

3.1 The Klein-Gordon Equation

3.2 Relativistic Wavefunctions, Probabilities and Currents

3.3 The Fine Structure Constant

3.4 The Two-Component Klein-Gordon Equation

3.5 Free Klein-Gordon Particles/Antiparticles

3.6 The Klein Paradox

3.7 The Radial Klein-Gordon Equation

3.8 The Spinless Electron Atom

3.9 Problems

4 The Dirac Equation

4.1 The Origin of the Dirac Equation

4.2 The Dirac Matrices

4.3 Lorentz Invariance of the Dirac Equation

4.4 The Non-Relativistic Limit of the Dirac Equation

4.5 An Alternative Formulation of the Dirac Equation

4.6 Probabilities and Currents

4.7 Gordon Decomposition

4.8 Forces and Fields

4.9 Gauge Invariance and the Dirac Equation

4.10 Problems

5 Free PaNicles/Antiparticles

5.1 Wavefunctions, Densities and Currents

5.2 Free-Particle Solutions

5.3 Free-Particle Spin

Rotations and Spinors

A Generalized Spin Operator

5.4 Negative Energy States, Antiparticles

5.5 Classical Negative Energy Particles?

5.6 The Klein Paradox Revisited

5.7 Lorentz Transformation of the Free-Particle Wavefunction

5.8 Problems

6 Symmetries and Operators

6.1 Non-Relativistic Spin Projection Operators

6.2 Relativistic Energy and Spin Projection Operators

6.3 Charge Conjugation

6.5 Parity

6.6

6.7 Angular Momentum Again

6.8 Non-Relativistic Limits Again

6.9 Second Quantization

6.10 Field Operators

6.11 Second Quantization in Relativistic Quantum Mechanics

6.12 Problems

7 Separating Particles from Antiparticles

7.1 The Foldy-Wouthuysen Transformation for a Free Particle

7.2 Foldy-Wouthuysen Transformation of Operators

7.3 Zitterbewegung

7.4 Foldy-Wouthuysen Transformation of the Wavefunction

7.5 The F-W Transformation in an Electromagnetic Field

7.6 Problems

8 One-Electron Atoms

8.1 The Radial Dirac Equation

8.2 Free-Electron Solutions

8.3 One-Electron Atoms, Eigenvectors and Eigenvalues

8.4 Behaviour of the Radial Functions

8.5 The Zeeman Effect

8.6 Magnetic Dichroism

8.7 Problems

9 Potential Problems

9.1 A Particle in a One-Dimensional Well

9.2 The Dirac Oscillator

The Non-Relativistic Limit

Solution of the Dirac Oscillator

Expectation Values and the Uncertainty Principle

9.3 Bloch's Theorem

9.4 The Relativistic Kronig-Penney Model

A One-Dimensional Time-Independent Dirac Equation

A Potential Step

A One-Dimensional Solid

9.5 An Electron in Crossed Electric and Magnetic Fields

An Electron in a Constant Magnetic Field

An Electron in a Field for which

9.6 Non-Linear Dirac Equations, the Dirac Soliton

9.7 Problems

10 More Than One Electron

10.1 The Breit Interaction

10.2 Two Electrons

10.3 Many-Electron Wavefunctions

10.4 The Many-Electron Hamiltonian

10.5 Dirac-Hartree-Fock Integrals

Single-Particle Integrals

Two-Particle Integrals

The Direct Coulomb Integral

The Exchange Integral

10.6 The Dirac-Hartree-Fock Equations

The One-Electron Atom

The Many-Electron Atom

10.7 Koopmans' Theorem

10.8 Implementation of the Dirac-Hartree-Fock Method

10.9 Introduction to Density Functional Theory

10.10 Non-Relativistic Density Functional Theory

10.11 The Variational Principle and the Kohn-Sham Equation

10.12 Density Functional Theory and Magnetism

Density Functional Theory in a Weak Magnetic Field

Density Functional Theory in a Strong Magnetic Field

10.13 The Exchange-Correlation Energy

10.14 Relativistic Density Functional Theory (RDFT)

RDFT with an External Scalar Potential

RDFT with an External Vector Potential

The Dirac-Kohn-Sham Equation

10.15 An Approximate Relativistic Density Functional Theory

10.16 Further Development of RDFT

10.17 Relativistic Exchange-Correlation Functionals

10.18 Implementation of RDFT

11 Scattering Theory

11.1 Green's Functions

11.2 Time-Dependent Green's Functions

11.3 The T-Operator

11.4 The Relativistic Free-Particle Green's Function

11.5 The Scattered Particle Wavefunction

11.6 The Scattering Experiment

11.7 Single-Site Scattering in Zero Field

11.8 Radial Dirac Equation in a Magnetic Field

11.9 Single-Site Scattering in a Magnetic Field

11.10 The Single-Site Scattering Green's Function

11.11 Transforming Between Representations

11.12 The Scattering Path Operator

11.13 The Non-Relativistic Free Particle Green's Function

11.14 Multiple Scattering Theory

11.15 The Multiple Scattering Green's Function

11.16 The Average T-Matrix Approximation

11.17 The Calculation of Observables

The Band Structure

The Fermi Surface

The Density of States

The Charge Density

Magnetic Moments

Energetic Quantities

11.18 Magnetic Anisotropy

The Non-Relativistic Limit, the RKKY Interaction

The Orion of Anisotropy

12 Electrons and Photons

12.1 Photon Polarization and Angular Momentum

12.2 Quantizing the Electromagnetic Field

12.3 Time-Dependent Perturbation Theory

12.4 Photon Absorption and Emission in Condensed Matter

12.5 Magneto-Optical Effects

12.6 Photon Scattering Theory

12.7 Thomson Scattering

12.8 Rayleigh Scattering

12.9 Compton Scattering

12.10 Magnetic Scattering of X-Rays

12.11 Resonant Scattering of X-Rays

13 Superconductivity

13.1 Do Electrons Find Each Other Attractive?

13.2 Superconductivity, the Hamiltonian

13.3 The Dirac——Bogolubov-de Gennes Equation

13.4 Solution of the Dirac-Bogolubov-de Gennes Equations

13.5 Observable Properties of Superconductors

13.6 Elcctrodynamics of Superconductors

Appendix A The Uncertainty Principle

Appendix B The Confluent Hypergeometrie Function

B.1 Relations to Other Functions

Appendix C Spherical Harmonics

Appendix D Unit Systems

Appendix E Fundamental Constants

References

Index