Advanced Quantum Mechanics: Materials and Photons

Preface
Contents
To the Students
To the Instructor
1 The Need for Quantum Mechanics
	1.1 Electromagnetic Spectra and Discrete Energy Levels
	1.2 Blackbody Radiation and Planck's Law
	1.3 Blackbody Spectra and Photon Fluxes
	1.4 The Photoelectric Effect
	1.5 Wave-Particle Duality
	1.6 Why Schrödinger's Equation?
	1.7 Interpretation of Schrödinger's Wave Function
	1.8 Problems
2 Self-Adjoint Operators and Eigenfunction Expansions
	2.1 The δ Function and Fourier Transforms
		Sokhotsky–Plemelj Relations
	2.2 Self-Adjoint Operators and Completeness of Eigenstates
	2.3 Problems
3 Simple Model Systems
	3.1 Barriers in Quantum Mechanics
	3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots
		Energy Levels in a Quantum Well
		Energy Levels in a Quantum Wire
		Energy Levels in a Quantum Dot
		Degeneracy of Quantum States
	3.3 The Attractive δ Function Potential
	3.4 Evolution of Free Schrödinger Wave Packets
		The Free Schrödinger Propagator
		Width of Gaussian Wave Packets
		Free Gaussian Wave Packets in Schrödinger Theory
	3.5 Problems
4 Notions from Linear Algebra and Bra-Ket Notation
	4.1 Notions from Linear Algebra
		Tensor Products
		Dual Bases
		Decomposition of the Identity
		An Application of Dual Bases in Solid State Physics: The Laue Conditions for Elastic Scattering off a Crystal
		Bra-ket Notation in Linear Algebra
	4.2 Bra-ket Notation in Quantum Mechanics
	4.3 The Adjoint Schrödinger Equation and the Virial Theorem
	4.4 Problems
5 Formal Developments
	5.1 Uncertainty Relations
	5.2 Frequency Representation of States
	5.3 Dimensions of States
	5.4 Gradients and Laplace Operators in General CoordinateSystems
	5.5 Separation of Differential Equations
	5.6 Problems
6 Harmonic Oscillators and Coherent States
	6.1 Basic Aspects of Harmonic Oscillators
	6.2 Solution of the Harmonic Oscillator by the Operator Method
	6.3 Construction of the x-Representation of the Eigenstates
		Oscillator Eigenstates in k Space and Bilinear Relations for Hermite Polynomials
	6.4 Lemmata for Exponentials of Operators
	6.5 Coherent States
		Scalar Products and Overcompleteness of Coherent States
		Squeezed States
	6.6 Problems
7 Central Forces in Quantum Mechanics
	7.1 Separation of Center of Mass Motion and Relative Motion
	7.2 The Concept of Symmetry Groups
	7.3 Operators for Kinetic Energy and Angular Momentum
	7.4 Matrix Representations of the Rotation Group
		The Defining Representation of the Three-Dimensional Rotation Group
		The General Matrix Representations of the Rotation Group
	7.5 Construction of the Spherical Harmonic Functions
	7.6 Basic Features of Motion in Central Potentials
	7.7 Free Spherical Waves: The Free Particle with Sharp Mz, M2
		Asymptotically Free Angular Momentum Eigenstates
	7.8 Bound Energy Eigenstates of the Hydrogen Atom
	7.9 Spherical Coulomb Waves
	7.10 Problems
8 Spin and Addition of Angular Momentum Type Operators
	8.1 Spin and Magnetic Dipole Interactions
	8.2 Transformation of Scalar, Spinor, and Vector Wave Functions Under Rotations
	8.3 Addition of Angular Momentum Like Quantities
	8.4 Problems
9 Stationary Perturbations in Quantum Mechanics
	9.1 Time-Independent Perturbation Theory Without Degeneracies
		First Order Corrections to the Energy Levels and Eigenstates
		Recursive Solution of Eq.(9.12) for n≥1
		Second Order Corrections to the Energy Levels and Eigenstates
		Summary of Non-degenerate Perturbation Theory in Second Order
	9.2 Time-Independent Perturbation Theory With Degenerate Energy Levels
		First Order Corrections to the Energy Levels
		First Order Corrections to the Energy Eigenstates
		Recursive Solution of Eq.(9.31) for n≥1
		Summary of First Order Shifts of the Level Ei(0) if the Perturbation Lifts the Degeneracy of the Level
	9.3 Problems
10 Quantum Aspects of Materials I
	10.1 Bloch's Theorem
		Orthogonality of the Periodic Bloch Factors
	10.2 Wannier States
	10.3 Time-Dependent Wannier States
	10.4 The Kronig-Penney Model
	10.5 kp Perturbation Theory and Effective Mass
	10.6 Problems
11 Scattering Off Potentials
	11.1 The Free Energy-Dependent Green's Function
	11.2 Potential Scattering in the Born Approximation
		The Optical Theorem
		Scattering Phase Shifts
	11.3 Scattering Off a Hard Sphere
	11.4 Rutherford Scattering
		Form Factors
		Mott-Gordon States Revisited
	11.5 Problems
12 The Density of States
	12.1 Counting of Oscillation Modes
		The Reasoning with Periodic Boundary Conditions in a Finite Volume
		The Reasoning Based on the Completeness of Plane Wave States
	12.2 The Continuum Limit
		Another Reasoning for the Continuum Limit
		Different Forms of the Density of States in a Homogeneous Medium
	12.3 The Density of States in the Energy Scale
	12.4 Density of States for Free Non-relativistic Particles and for Radiation
	12.5 The Density of States for Other Quantum Systems
	12.6 Problems
13 Time-Dependent Perturbations in Quantum Mechanics
	13.1 Pictures of Quantum Dynamics
		Time Evolution in the Schrödinger Picture
		The Time Evolution Operator for the Harmonic Oscillator
		The Heisenberg Picture
	13.2 The Dirac Picture
		Dirac Picture for Constant H0
	13.3 Transitions Between Discrete States
		Møller Operators
		First Order Transition Probability Between Discrete Energy Eigenstates
	13.4 Transitions from Discrete States into Continuous States: Ionization or Decay Rates
		Ionization probabilities for hydrogen
		The Golden Rule for Transitions from Discrete States into a Continuum of States
		Time-Dependent Perturbation Theory in Second Order and the Golden Rule #1
	13.5 Transitions from Continuous States into Discrete States: Capture Cross Sections
		Calculation of the Capture Cross Section
	13.6 Transitions Between Continuous States: Scattering
		Cross Section for Scattering Off a Periodic Perturbation
		Scattering Theory in Second Order
	13.7 Expansion of the Scattering Matrix to Higher Orders
	13.8 Energy-Time Uncertainty
	13.9 Problems
14 Path Integrals in Quantum Mechanics
	14.1 Correlation and Green's Functions for Free Particles
	14.2 Time Evolution in the Path Integral Formulation
	14.3 Path Integrals in Scattering Theory
	14.4 Problems
15 Coupling to Electromagnetic Fields
	15.1 Electromagnetic Couplings
		Multipole Moments
		Semiclassical Treatment of the Matter-Radiation System in the Dipole Approximation
		Dipole Selection Rules
	15.2 Stark Effect and Static Polarizability Tensors
		Linear Stark Effect
		Quadratic Stark Effect and the Static Polarizability Tensor
	15.3 Dynamical Polarizability Tensors
		Oscillator Strength
		Thomas-Reiche-Kuhn Sum Rule (f-Sum Rule) for the Oscillator Strength
		Tensorial Oscillator Strengths and Sum Rules
	15.4 Problems
16 Principles of Lagrangian Field Theory
	16.1 Lagrangian Field Theory
		The Lagrange Density for the Schrödinger Field
	16.2 Symmetries and Conservation Laws
		Energy-Momentum Tensors
	16.3 Applications to Schrödinger Field Theory
		Probability and Charge Conservation from Invariance Under Phase Rotations
	16.4 Problems
17 Non-relativistic Quantum Field Theory
	17.1 Quantization of the Schrödinger Field
		Time Evolution of the Field Operators
		k-Space Representation of Quantized Schrödinger Theory
		Field Operators in the Schrödinger Picture and the Fock Space for the Schrödinger Field
		Time-Dependence of H0
	17.2 Time Evolution for Time-Dependent Hamiltonians
	17.3 The Connection Between First and Second Quantized Theory
		General 1-Particle States and Corresponding Annihilation and Creation Operators in Second Quantized Theory
		Time Evolution of 1-Particle States in Second Quantized Theory
	17.4 The Dirac Picture in Quantum Field Theory
	17.5 Inclusion of Spin
	17.6 Two-Particle Interaction Potentials and Equations of Motion
		Equation of Motion
		Relation to Other Equations of Motion
	17.7 Expectation Values and Exchange Terms
	17.8 From Many Particle Theory to Second Quantization
	17.9 Problems
18 Quantization of the Maxwell Field: Photons
	18.1 Lagrange Density and Mode Expansion for the Maxwell Field
		Energy-Momentum Tensor for the Free Maxwell Field
	18.2 Photons
	18.3 Coherent States of the Electromagnetic Field
	18.4 Photon Coupling to Relative Motion
	18.5 Energy-Momentum Densities and Time Evolution in Quantum Optics
	18.6 Photon Emission Rates
		Evaluation of the Transition Matrix Element in the Dipole Approximation
		Energy-Time Uncertainty for Photons
	18.7 Photon Absorption
		Photon Absorption into Discrete States
		Photon Absorption into Continuous States
		Photon Absorption Coefficients
	18.8 Stimulated Emission of Photons
	18.9 Photon Scattering
		Thomson Cross Section
		Rayleigh Scattering
	18.10 Problems
19 Epistemic and Ontic Quantum States
	19.1 Stern-Gerlach Experiments
	19.2 Non-locality from Entanglement?
	19.3 Quantum Jumps and the Continuous Evolution of Quantum States
	19.4 Photon Emission Revisited
	19.5 Particle Location
	19.6 Problems
20 Quantum Aspects of Materials II
	20.1 The Born-Oppenheimer Approximation
	20.2 Covalent Bonding: The Dihydrogen Cation
	20.3 Bloch and Wannier Operators
	20.4 The Hubbard Model
	20.5 Vibrations in Molecules and Lattices
		Normal Coordinates and Normal Oscillations
		Eigenmodes of Three Masses
		The Diatomic Linear Chain
		Quantization of N-particle Oscillations
	20.6 Quantized Lattice Vibrations: Phonons
	20.7 Electron-Phonon Interactions
	20.8 Problems
21 Dimensional Effects in Low-Dimensional Systems
	21.1 Quantum Mechanics in d Dimensions
	21.2 Inter-Dimensional Effects in Interfaces and Thin Layers
		Two-Dimensional Behavior from a Thin Quantum Well
	21.3 Problems
22 Relativistic Quantum Fields
	22.1 The Klein-Gordon Equation
		Mode Expansion and Quantization of the Klein-Gordon Field
		The Charge Operator of the Klein-Gordon Field
		Hamiltonian and Momentum Operators for the Klein-Gordon Field
		Non-relativistic Limit of the Klein-Gordon Field
	22.2 Klein's Paradox
	22.3 The Dirac Equation
		Solutions of the Free Dirac Equation
		Charge Operators and Quantization of the Dirac Field
	22.4 The Energy-Momentum Tensor for Quantum Electrodynamics
		Energy and Momentum in QED in Coulomb Gauge
	22.5 The Non-relativistic Limit of the Dirac Equation
		Higher Order Terms and Spin-Orbit Coupling
	22.6 Covariant Quantization of the Maxwell Field
	22.7 Problems
23 Applications of Spinor QED
	23.1 Two-Particle Scattering Cross Sections
		Measures for Final States with Two Identical Particles
	23.2 Electron Scattering off an Atomic Nucleus
	23.3 Photon Scattering by Free Electrons
	23.4 Møller Scattering
	23.5 Problems
A Lagrangian Mechanics
	Derivation of the Lagrange Equations for the Generalized Coordinates qa from d'Alembert's Principle
	Symmetries and Conservation Laws in Classical Mechanics
B The Covariant Formulation of Electrodynamics
	Lorentz Transformations
	The Manifestly Covariant Formulation of Electrodynamics
	Relativistic Mechanics
	Classical Electromagnetic Hamiltonian in Coulomb Gauge
	Classical Electromagnetic Hamiltonian in Lorentz Gauge
	Relativistic Center of Mass Frame
C Completeness of Sturm–Liouville Eigenfunctions
	Sturm–Liouville Problems
	Liouville's Normal Form of Sturm's Equation
	Nodes of Sturm–Liouville Eigenfunctions
	Sturm's Comparison Theorem and Estimates for the Locations of the Nodes yn(λ)
	Eigenvalue Estimates for the Sturm–Liouville Problem
	Completeness of Sturm–Liouville Eigenstates
D Properties of Hermite Polynomials
E The Baker–Campbell–Hausdorff Formula
F The Logarithm of a Matrix
G Dirac γ Matrices
	γ-Matrices in d Dimensions
	Proof that in Irreducible Representations 0,1,…d-11 for Odd Spacetime Dimension d
	Recursive Construction of γ-Matrices in Different Dimensions
	Proof That Every Set of γ-Matrices is Equivalent to a Set Which Satisfies Eq.(G.23)
	Uniqueness Theorem for γ Matrices
	Contraction and Trace Theorems for γ Matrices
H Spinor Representations of the Lorentz Group
	Generators of Proper Orthochronous Lorentz Transformations in the Vector and Spinor Representations
	Verification of the Lorentz Commutation Relations for the Spinor Representations
	Scalar Products of Spinors and the Lagrangian for the Dirac Equation
	The Spinor Representation in the Weyl and Dirac Bases of γ-Matrices
	Construction of the Vector Representation from the Spinor Representation
	Construction of the Free Dirac Spinors from Spinors at Rest
	Lorentz Covariance of Charge Conjugation
I Transformation of Fields Under Reflections
J Green's Functions in d Dimensions
	Green's Functions for the Schrödinger Equation
	Polar Coordinates in d Dimensions
	The Time Evolution Operator in Various Representations
	Relativistic Green's Functions in d Spatial Dimensions
	Retarded Relativistic Green's Functions in (x,t) Representation
	Green's Functions for Dirac Operators in d Dimensions
	Green's Functions in Covariant Notation
	Green's Functions as Reproducing Kernels
	Liénard–Wiechert Potentials in Low Dimensions
References
Index