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Table of Contents
Intro
Preface
Contents
1 Mathematical Background
1.1 Sets and Maps
1.2 Topological Spaces
1.3 Fields
1.4 Linear Spaces
1.4.1 Inner Product Spaces
1.4.2 Linear Combinations and Basis Sets
1.4.3 Operators and Superoperators
1.4.4 Representations of Linear Spaces
1.4.5 Operator Norms and Inner Products
1.4.6 Representing Matrices with Vectors
1.5 Groups and Algebras
1.5.1 Finite, Discrete, and Continuous Groups
1.5.2 Conjugacy Classes and Centres
1.5.3 Group Actions, Orbits, and Stabilisers
1.5.4 Matrix Representations of Groups
1.5.5 Orthogonality Theorems and Characters
1.5.6 Algebras and Lie Algebras
1.5.7 Exponential and Tangent Maps
1.5.8 Ideals, Simple and Semisimple Algebras
1.5.9 Matrix Representations of Lie Algebras
1.5.10 Envelopes, Complexifications, and Covers
1.5.11 Cartan Subalgebras, Roots, and Weights
1.5.12 Killing Form and Casimir Elements
1.6 Building Blocks of Spin Physics
1.6.1 Euclidean and Minkowski Spaces
1.6.2 Special Orthogonal Group in Three Dimensions
1.6.2.1 Parametrisation of Rotations
1.6.2.2 Euler Angles Parametrisation
1.6.2.3 Angle-Axis Parametrisation
1.6.2.4 Irreducible Representations
1.6.3 Special Unitary Group in Two Dimensions
1.6.3.1 Parametrisation
1.6.3.2 Irreducible Representations
1.6.3.3 Normalisation-Commutation Dilemma
1.6.4 Relationship Between SU(2) and SO(3)
1.7 Linear Time-Invariant Systems
1.7.1 Pulse and Frequency Response
1.7.2 Properties of the Fourier Transform
1.7.3 Causality and Hilbert Transform
2 What Exactly Is Spin?
2.1 Time Translation Group
2.2 Full Translation Group
2.3 Rotation Group
2.4 Lorentz Group
2.4.1 Boost Generators
2.4.2 Irreps of Lorentz Group
2.4.3 Irreps of Lorentz Group with Parity
2.4.4 Poincare Group and the Emergence of Spin
2.5 Dirac's Equation and Electron Spin
2.5.1 Dirac's Equation
2.5.2 Total Angular Momentum and Spin
2.5.3 Total Angular Momentum Representation-Numerical
2.5.4 Total Angular Momentum Representation-Analytical
2.5.5 Benefits of the Individual Momentum Representation
2.6 Weakly Relativistic Limit of Dirac's Equation
2.6.1 Zitterbewegung
2.6.2 Negative Energy Subspace Elimination
2.6.3 Zeeman Interactions and Langevin Susceptibility
2.6.4 Spin-Orbit Coupling
2.6.5 Spinning Charge Analogy
2.6.6 Spin as a Magnetic Moment
2.6.7 Spin Hamiltonian Approximation
2.6.8 Energy Derivative Formalism
2.6.9 Thermal Corrections
3 Bestiary of Spin Hamiltonians
3.1 Physical Side
3.1.1 Nuclear Spin and Magnetic Moment
3.1.2 Nuclear Electric Quadrupole Moment
3.1.3 Electronic Structure Derivative Table
3.1.4 Spin-Independent Susceptibility
3.1.5 Hyperfine Coupling
3.1.6 Electron and Nuclear Shielding
3.1.7 Nuclear Shielding by Susceptibility
3.1.8 Inter-nuclear Dipolar Interaction
Preface
Contents
1 Mathematical Background
1.1 Sets and Maps
1.2 Topological Spaces
1.3 Fields
1.4 Linear Spaces
1.4.1 Inner Product Spaces
1.4.2 Linear Combinations and Basis Sets
1.4.3 Operators and Superoperators
1.4.4 Representations of Linear Spaces
1.4.5 Operator Norms and Inner Products
1.4.6 Representing Matrices with Vectors
1.5 Groups and Algebras
1.5.1 Finite, Discrete, and Continuous Groups
1.5.2 Conjugacy Classes and Centres
1.5.3 Group Actions, Orbits, and Stabilisers
1.5.4 Matrix Representations of Groups
1.5.5 Orthogonality Theorems and Characters
1.5.6 Algebras and Lie Algebras
1.5.7 Exponential and Tangent Maps
1.5.8 Ideals, Simple and Semisimple Algebras
1.5.9 Matrix Representations of Lie Algebras
1.5.10 Envelopes, Complexifications, and Covers
1.5.11 Cartan Subalgebras, Roots, and Weights
1.5.12 Killing Form and Casimir Elements
1.6 Building Blocks of Spin Physics
1.6.1 Euclidean and Minkowski Spaces
1.6.2 Special Orthogonal Group in Three Dimensions
1.6.2.1 Parametrisation of Rotations
1.6.2.2 Euler Angles Parametrisation
1.6.2.3 Angle-Axis Parametrisation
1.6.2.4 Irreducible Representations
1.6.3 Special Unitary Group in Two Dimensions
1.6.3.1 Parametrisation
1.6.3.2 Irreducible Representations
1.6.3.3 Normalisation-Commutation Dilemma
1.6.4 Relationship Between SU(2) and SO(3)
1.7 Linear Time-Invariant Systems
1.7.1 Pulse and Frequency Response
1.7.2 Properties of the Fourier Transform
1.7.3 Causality and Hilbert Transform
2 What Exactly Is Spin?
2.1 Time Translation Group
2.2 Full Translation Group
2.3 Rotation Group
2.4 Lorentz Group
2.4.1 Boost Generators
2.4.2 Irreps of Lorentz Group
2.4.3 Irreps of Lorentz Group with Parity
2.4.4 Poincare Group and the Emergence of Spin
2.5 Dirac's Equation and Electron Spin
2.5.1 Dirac's Equation
2.5.2 Total Angular Momentum and Spin
2.5.3 Total Angular Momentum Representation-Numerical
2.5.4 Total Angular Momentum Representation-Analytical
2.5.5 Benefits of the Individual Momentum Representation
2.6 Weakly Relativistic Limit of Dirac's Equation
2.6.1 Zitterbewegung
2.6.2 Negative Energy Subspace Elimination
2.6.3 Zeeman Interactions and Langevin Susceptibility
2.6.4 Spin-Orbit Coupling
2.6.5 Spinning Charge Analogy
2.6.6 Spin as a Magnetic Moment
2.6.7 Spin Hamiltonian Approximation
2.6.8 Energy Derivative Formalism
2.6.9 Thermal Corrections
3 Bestiary of Spin Hamiltonians
3.1 Physical Side
3.1.1 Nuclear Spin and Magnetic Moment
3.1.2 Nuclear Electric Quadrupole Moment
3.1.3 Electronic Structure Derivative Table
3.1.4 Spin-Independent Susceptibility
3.1.5 Hyperfine Coupling
3.1.6 Electron and Nuclear Shielding
3.1.7 Nuclear Shielding by Susceptibility
3.1.8 Inter-nuclear Dipolar Interaction