This textbook, now in an expanded third edition, emphasizes the importance of advanced quantum mechanics for materials science and all experimental techniques which employ photon absorption, emission, or scattering. Important aspects of introductory quantum mechanics are covered in the first seven chapters to make the subject self-contained and accessible for a wide audience. Advanced Quantum Mechanics: Materials and Photons can therefore be used for advanced undergraduate courses and introductory graduate courses which are targeted towards students with diverse academic backgrounds from the Natural Sciences or Engineering. To enhance this inclusive aspect of making the subject as accessible as possible, introductions to Lagrangian mechanics and the covariant formulation of electrodynamics are provided in appendices.
This third edition includes 60 new exercises, new and improved illustrations, and new material on interpretations of quantum mechanics. Other special features include an introduction to Lagrangian field theory and an integrated discussion of transition amplitudes with discrete or continuous initial or final states. Once students have acquired an understanding of basic quantum mechanics and classical field theory, canonical field quantization is easy. Furthermore, the integrated discussion of transition amplitudes naturally leads to the notions of transition probabilities, decay rates, absorption cross sections and scattering cross sections, which are important for all experimental techniques that use photon probes.
Author(s): Rainer Dick
Series: Graduate Texts in Physics
Edition: 3
Publisher: Springer
Year: 2020
Language: English
Pages: 832
City: Cham
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
