Feynman Path Integrals in Quantum Mechanics and Statistical Physics

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Chapter 1: Path Integral Formalism Intuitive Approach
1.1 Probability Amplitude
1.1.1 Double Slit Experiment
1.1.2 Physical State
1.1.3 Probability Amplitude
1.1.4 Revisit Double Slit Experiment
1.1.5 Distinguishability
1.1.6 Superposition Principle
1.1.7 Revisit the Double Slit Experiment/Superposition Principle
1.1.8 Orthogonality
1.1.9 Orthonormality
1.1.10 Change of Basis
1.1.11 Geometrical Interpretation of State Vector
1.1.12 Coordinate Transformation
1.1.13 Projection Operator
1.1.14 Continuous Spectrum
Chapter 2: Matrix Representation of Linear Operators
2.1 Matrix Element
2.2 Linear Self-Adjoint (Hermitian Conjugate) Operators
2.3 Product of Hermitian Operators
2.4 Continuous Spectrum
2.5 Schturm-Liouville Problem: Eigenstates and Eigenvalues
2.6 Revisit Linear Self-Adjoint (Hermitian) Operators
2.7 Unitary Transformation
2.8 Mean (Expectation) Value and Matrix Density
2.9 Degeneracy
2.10 Density Operator
2.11 Commutativity of Operators
Chapter 3: Operators in Phase Space
3.1 Introduction
3.2 Configuration Space
3.3 Position and Wave Function
3.4 Momentum Space
3.5 Classical Action
Chapter 4: Transition Amplitude
4.1 Path Integration in Phase Space
4.1.1 From the Schrödinger Equation to Path Integration
4.1.2 Trotter Product Formula
4.2 Transition Amplitude
4.2.1 Hamiltonian Formulation of Path Integration
4.2.2 Path Integral Subtleties
4.2.2.1 Mid-point Rule
4.2.3 Lagrangian Formulation of Path Integration
4.2.3.1 Complex Gaussian Integral
4.2.4 Transition Amplitude
4.2.5 Law for Consecutive Events
4.2.6 Semigroup Property of the Transition Amplitude
Chapter 5: Stationary and Quasi-Classical Approximations
5.1 Stationary Phase Method/Fourier Integral
5.2 Contribution from Non-Degenerate Stationary Points
5.2.1 Unique Stationary Point
5.3 Quasi-Classical Approximation/Fluctuating Path
5.3.1 Free Particle Classical Action and Transition Amplitude
5.3.1.1 Free Particle Classical Action
5.3.1.2 Free Particle Transition Amplitude
5.3.1.3 From Path Integrals to Quantum Mechanics
5.4 Free and Driven Harmonic Oscillator Classical Action and Transition Amplitude
5.4.1 Free Oscillator Classical Action
5.4.2 Driven or Forced Harmonic Oscillator Classical Action
5.5 Free and Driven Harmonic Oscillator Transition Amplitude
5.6 Fluctuation Contribution to Transition Amplitude
5.6.1 Maslov Correction
Chapter 6: Generalized Feynman Path Integration
6.1 Coordinate Representation
6.2 Free Particle Transition Amplitude
6.3 Gaussian Functional Feynman Path Integrals
6.4 Charged Particle in a Magnetic Field
Chapter 7: From Path Integration to the Schrödinger Equation
7.1 Wave Function
7.2 Schrödinger Equation
7.3 The Schrödinger Equation’s Green’s Function
7.4 Transition Amplitude for a Time-Independent Hamiltonian
7.5 Retarded Green Function
Chapter 8: Quasi-Classical Approximation
8.1 Wentzel-Kramer-Brillouin (WKB) Method
8.1.1 Condition of Applicability of the Quasi-Classical Approximation
8.1.2 Bounded Quasi-Classical Motion
8.1.3 Quasi-Classical Quantization
8.1.4 Path Integral Link
8.2 Potential Well
8.3 Potential Barrier
8.4 Quasi-Classical Derivation of the Propagator
8.5 Reflection and Tunneling via a Barrier
8.6 Transparency of the Quasi-Classical Barrier
8.7 Homogenous Field
8.7.1 Motion in a Central Symmetric Field
8.7.1.1 Polar Equation
8.7.1.2 Radial Equation for a Spherically Symmetric Potential in Three Dimensions
8.7.2 Motion in a Coulombic Field
8.7.2.1 Hydrogen Atom
Chapter 9: Free Particle and Harmonic Oscillator
9.1 Eigenfunction and Eigenvalue
9.1.1 Free Particle
9.1.2 Transition Amplitude for a Particle in a Homogenous Field
9.2 Harmonic Oscillator
9.3 Transition Amplitude Hermiticity
Chapter 10: Matrix Element of a Physical Operator via Functional Integral
10.1 Matrix Representation of the Transition Amplitude of a Forced Harmonic Oscillator
10.1.1 Charged Particle Interaction with Phonons
Chapter 11: Path Integral Perturbation Theory
11.1 Time-Dependent Perturbation
11.2 Transition Probability
11.3 Time-Energy Uncertainty Relation
11.4 Density of Final State
11.4.1 Transition Rate
11.5 Continuous Spectrum due to a Constant Perturbation
11.6 Harmonic Perturbation
Chapter 12: Transition Matrix Element
Chapter 13: Functional Derivative
13.1 Functional Derivative of the Action Functional
13.2 Functional Derivative and Matrix Element
Chapter 14: Quantum Statistical Mechanics Functional Integral Approach
14.1 Introduction
14.2 Density Matrix
14.2.1 Partition Function
14.3 Expectation Value of a Physical Observable
14.4 Density Matrix
14.5 Density Matrix in the Energy Representation
Chapter 15: Partition Function and Density Matrix Path Integral Representation
15.1 Density Matrix Path Integral Representation
15.1.1 Density Matrix Operator Average Value in Phase Space
15.1.1.1 Generalized Gaussian Functional Path Integral in Phase Space
15.1.2 Density Matrix via Transition Amplitude
15.2 Partition Function in the Path integral Representation
15.3 Particle Interaction with a Driven or Forced Harmonic Oscillator: Partition Function
15.4 Free Particle Density Matrix and Partition Function
15.5 Quantum Harmonic Oscillator Density Matrix and Partition Function
Chapter 16: Quasi-Classical Approximation in Quantum Statistical Mechanics
16.1 Centroid Effective Potential
16.2 Expectation Value
Chapter 17: Feynman Variational Method
Chapter 18: Polaron Theory
18.1 Introduction
18.2 Polaron Energy and Effective Mass
18.3 Functional Influence Phase
18.3.1 Polaron Model Lagrangian
18.3.2 Polaron Partition Function
18.4 Influence Phase via Feynman Functional Integral in The Density Matrix Representation
18.4.1 Expectation Value of a Physical Quantity
18.4.1.1 Density matrix
18.5 Full System Polaron Partition Function in a 3D Structure
18.6 Model System Polaron Partition Function in a 3D Structure
18.7 Feynman Inequality and Generating Functional
18.8 Polaron Characteristics in a 3D Structure
18.8.1 Polaron Asymptotic Characteristics
18.9 Polaron Characteristics in a Quasi-1D Quantum Wire
18.9.1 Hamiltonian of the Electron in a Quasi 1D Quantum Wire
18.9.1.1 Lagrangian of the Electron in a Quasi-1D Quantum Wire
18.9.1.2 Partition function of the Electron in a Quasi-1D Quantum Wire
18.10 Polaron Generating Function
18.11 Polaron Asymptotic Characteristics
18.12 Strong Coupling Regime Polaron Characteristics
18.13 Bipolaron Characteristics in a Quasi-1D Quantum Wire
18.13.1 Introduction
18.13.2 Bipolaron Diagrammatic Representation
18.13.3 Bipolaron Lagrangian
18.13.4 Bipolaron Equation of Motion
18.13.5 Transformation into Normal Coordinates
18.13.5.1 Diagonalization of the Lagrangian
18.13.6 Bipolaron Partition Function
18.13.7 Bipolaron Generating Function
18.13.8 Bipolaron Asymptotic Characteristics
18.14 Polaron Characteristics in a Quasi-0D Spherical Quantum Dot
18.14.1 Introduction
18.14.2 Polaron Lagrangian
18.14.3 Normal Modes
18.14.4 Lagrangian Diagonalization
18.14.4.1 Transformation to Normal Coordinates
18.14.5 Polaron Partition Function
18.14.6 Generating Function
18.15 Bipolaron Characteristics in a Quasi-0D Spherical Quantum Dot
18.15.1 Introduction
18.15.2 Model Lagrangian
18.15.3 Model Lagrangian
18.15.3.1 Equation of Motion and Normal Modes
18.15.4 Diagonalization of the Lagrangian
18.15.5 Partition Function
18.15.6 Full System Influence Phase
18.16 Bipolaron Energy
18.16.1 Generating Function
18.16.2 Bipolaron Characteristics
18.17 Polaron Characteristics in a Cylindrical Quantum Dot
18.17.1 System Hamiltonian
18.17.2 Transformation to Normal Coordinates
18.17.2.1 Lagrangian Diagonalization
18.17.3 Polaron Energy/Partition Function
18.17.4 Polaron Generating Function
18.17.5 Polaron Energy
18.18 Bipolaron Characteristics in a Cylindrical Quantum Dot
18.18.1 System Hamiltonian
18.18.1.1 Model System Action Functional
18.18.1.2 Equation of Motion / Normal Modes
18.18.1.3 Lagrangian Diagonalization
18.18.1.4 Bipolaron Partition Function
18.18.1.5 Bipolaron Generating Function
18.18.1.6 Bipolaron Energy
18.19 Polaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic Potential
18.20 Polaron Energy
18.21 Bipolaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic Potential
18.22 Polaron in a Magnetic Field
Chapter 19: Multiphoton Absorption by Polarons in a Spherical Quantum Dot
19.1 Theory of Multiphoton Absorption by Polarons
19.2 Basic Approximations
19.3 Absorption Coefficient
Chapter 20: Polaronic Kinetics in a Spherical Quantum Dot
Chapter 21: Kinetic Theory of Gases
21.1 Distribution Function
21.2 Principle of Detailed Equilibrium
21.3 Transport Phenomenon and Boltzmann-Lorentz Kinetic Equation
21.4 Transport Relaxation Time
21.5 Boltzmann H-Theorem
21.6 Thermal Conductivity
21.7 Diffusion
21.8 Electron–Phonon System Equation of Motion
References
Index
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