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Green's functions and retarded potentials

Mauro Fabrizio and Morro Angelo

in Electromagnetism of Continuous Media: Mathematical Modelling and Applications

This chapter gives a concise and self-contained presentation of the use of Green’s functions to solve linear problems with a given point source. Specific functions are determined for the harmonic ... More

This chapter gives a concise and self-contained presentation of the use of Green’s functions to solve linear problems with a given point source. Specific functions are determined for the harmonic oscillator, the wave equation, and the lossy wave equation. The solution to the inhomogeneous wave equation is derived through a retarded potential and by the Kirchhoff procedure. The field generated by a moving charge is determined through the Lienard-Wiechert potential. A Cauchy problem and a boundary-value problem are investigated for the wave equation and the telegraph equation.Less

Scattering by a Buried Object

Peter Monk

in Finite Element Methods for Maxwell's Equations

This chapter presents a model problem for scattering by a buried object. The ground is modeled by a two-layered medium (the air and the earth) with a planar interface. For the two-layered medium, it ... More

This chapter presents a model problem for scattering by a buried object. The ground is modeled by a two-layered medium (the air and the earth) with a planar interface. For the two-layered medium, it is possible to derive the dyadic Green’s function using Hertz potentials as in Sommerfeld’s book. This Green’s function can then be used to implement a finite element method for the scattering problem using the method of Hazard and Lenoir. In particular, the scatterer is surrounded by an artificial boundary (not necessarily a sphere). Inside the artificial boundary, finite elements are used to represent the solution. Outside the scatterer, the Stratton-Chu formula provides a representation in terms of unknown fields on the surface of the scatterer. This representation is then used to provide a boundary condition on the artificial boundary. The resulting method has great flexibility in the choice of the artificial boundary. The applications of this method to a bounded scatterer in half-space with perfectly conducting boundary, and to scattering by a bounded scatterer in an infinite inhomogeneous background are discussed. Error estimates are proven.Less

DYNAMICS OF IMPERFECT LATTICES

A.M. Stoneham

in Theory of Defects in Solids: Electronic Structure of Defects in Insulators and Semiconductors

Defects and impurities affect the vibrations of a solid, and hence properties such as infra-red spectra and thermal conductivity. There can be new, high-frequency, local modes; there can be new, ... More

Defects and impurities affect the vibrations of a solid, and hence properties such as infra-red spectra and thermal conductivity. There can be new, high-frequency, local modes; there can be new, low-frequency, resonances. This chapter analyses key results systematically, including some convenient simple limits. It also goes into detail concerning classical Green's functions, thermodynamic Green's functions, response functions, isotopic impurity, and asymptotic expansions. Local and resonance modes, Rayleigh scattering, and the peak theorem are also considered.Less

Green’s functions and the dynamical action principle

JAGDISH MEHRA and KIMBALL A. MILTON

in Climbing the Mountain: The Scientific Biography of Julian Schwinger

In a remarkable lecture Julian Schwinger delivered at the University of Nottingham on July 14, 1993, on the occasion of his receiving an honorary degree, entitled ‘The Greening of quantum field ... More

In a remarkable lecture Julian Schwinger delivered at the University of Nottingham on July 14, 1993, on the occasion of his receiving an honorary degree, entitled ‘The Greening of quantum field theory: George and I’, he summarised the central role Green's function played throughout his career. Schwinger then went on to recount his experience at the Massachusetts Institute of Technology's Radiation Laboratory during World War II and traced the influences of George Green on his own works. This chapter chronicles Schwinger's research in relation to Green's function, his first trip to Europe, and his work on the gauge invariance and vacuum polarization, the quantum action principle, electrodynamic displacements of energy levels, quantum field theory, and condensed matter physics.Less

Causality and Green’s functions

Sandip Tiwari

in Semiconductor Physics: Principles, Theory and Nanoscale

The chapter elucidates causality to prevent its erroneous use. It then develops Green’s functions to analyze and predict systems under stimulation in the presence of cause and chance. Causality is ... More

The chapter elucidates causality to prevent its erroneous use. It then develops Green’s functions to analyze and predict systems under stimulation in the presence of cause and chance. Causality is the tracing of an observed phenomenon entirely to an identified cause. Cause and chance both lead to observations in the natural world. Chance is tackled through probabilities. A toy quantum model emphasizing uncertainty is used to illustrate cause and chance. Several examples are discussed to illustrate the development of Green’s functions—examining both advancing Green’s functions and retarding Green’s functions—for classical and quantum evolution. The S-matrix is also discussed.Less

The Green’s function in linear elasticity

Adrian P. Sutton

in Physics of Elasticity and Crystal Defects

The elastostatic Green’s tensor function is the solution of a differential equation for the displacement field created by a unit point force in an infinite continuum. Its symmetry is derived using ... More

The elastostatic Green’s tensor function is the solution of a differential equation for the displacement field created by a unit point force in an infinite continuum. Its symmetry is derived using Maxwell’s reciprocity theorem. A general integral expression is derived for the Green’s function in anisotropic media. The Green’s function in isotropic elasticity is derived in closed form. The relation between the elastic Green’s function in a continuum and in a harmonic crystal lattice is shown. The application of the Green’s function to solving displacement fields of point defects exerting defect forces on neighbouring atoms leads to dipole, quadrupole, octupole, etc. tensors for point defects. Eshelby’s ellipsoidal inclusion problem is solved in isotropic elasticity. Using perturbation theory analytic expressions for the Green’s function in a weakly anisotropic cubic crystal are obtained in problem 3 of set 4. The derivation of the elastodynamic Green’s function in isotropic elasticity is outlined.Less

Retarded Green’s Functions

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, ... More

Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, which represent the time development of one (or a few) state(s) that may be understood as localized but are in interaction with a continuum of states, are discussed and applied to chemisorption. Introductions are also made to the Dyson integral equation, T-matrix and the Dirac delta-function potential, with the latter applied to random impurity scattering. The retarded Green’s function in the presence of random impurity scattering is exhibited in the Born and self-consistent Born approximations, with application to Ando’s semi-elliptic density of states for the 2D Landau-quantized electron-impurity system. Important retarded Green’s functions and their methods of derivation are discussed. These include Green’s functions for electrons in magnetic fields in both three dimensions and two dimensions, also a Hamilton equation-of-motion method for the determination of Green’s functions with application to a 2D saddle potential in a time-dependent electric field. Moreover, separable Hamiltonians and their product Green’s functions are discussed with application to a one-dimensional superlattice in axial electric and magnetic fields. Green’s function matching/joining techniques are introduced and applied to spatially varying mass (heterostructures) and non-local electrostatics (surface plasmons).Less

Tunneling Matrix Elements

C. Julian Chen

in Introduction to Scanning Tunneling Microscopy: Second Edition

This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of ... More

This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample, roughly in the middle of the tunneling gap. By expanding the tip wavefunction in terms of spherical harmonics and spherical modified Bessel functions, very simple analytic expressions for the tunneling matrix elements are derived: the tunneling matrix elements are proportional to the amplitudes or the corresponding x-, y-, or z-derivatives of the sample wavefunction at the center of the tip. Two proofs are presented. The first proof is based on the Green's function of the Schrödinger's equation in vacuum. The second proof is based on a power-series expansion of the tip wavefunctions.Less

Electric Fields and Currents in Biological Tissue

Paul L. Nunez and Ramesh Srinivasan

in Electric Fields of the Brain: The neurophysics of EEG

This chapter continues the physical principles of Chapter 3 in the context of biological tissue. Electrophysiology spans about five orders of magnitude of spatial scale, ranging from the ... More

This chapter continues the physical principles of Chapter 3 in the context of biological tissue. Electrophysiology spans about five orders of magnitude of spatial scale, ranging from the microelectrode recordings of transmembrane potentials to millimeter-scale intracranial recordings to centimeter-scale scalp potentials. The classic membrane diffusion equation (core conductor model) is derived from basic principles without reference to any “equivalent circuit.” Tissue electrical properties at several scales are considered with emphasis on the cortical and skull tissues. A volume (current) microsource function and a millimeter scale (current) source function P(r,t) are defined based on fundamental physical and physiological principles. A low-pass filtering effect on cortical potentials is predicted based on reduced (pyramidal cell) microsource/sink separations in cortex at frequencies of perhaps 50 to 100 Hz. The relationship of P(r,t) to scalp potentials is discussed in the context of a Green's function for the head volume conductor, providing equivalent “electrical distances” between sources and scalp electrodes.Less

Interacting Electron–Hole–Phonon System

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting ... More

Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting electron-hole-phonon system. The coupled Fermion Green’s function equations involve five interactions (electron-electron, hole-hole, electron-hole, electron-phonon, and hole-phonon). Starting with quantum Hamilton equations of motion for the various electron/hole creation/annihilation operators and their nonequilibrium average/expectation values, variational differentiation with respect to particle sources leads to a chain of coupled Green’s function equations involving differing species of Green’s functions. For example, the 1-electron Green’s function equation is coupled to the 2-electron Green’s function (as earlier), also to the 1-electron/1-hole Green’s function, and to the Green’s function for 1-electron propagation influenced by a nontrivial phonon field. Similar remarks apply to the 1-hole Green’s function equation, and all others. Higher order Green’s function equations are derived by further variational differentiation with respect to sources, yielding additional couplings. Chapter 11 also introduces the 1-phonon Green’s function, emphasizing the role of electron coupling in phonon propagation, leading to dynamic, nonlocal electron screening of the phonon spectrum and hybridization of the ion and electron plasmons, a Bohm-Staver phonon mode, and the Kohn anomaly. Furthermore, the single-electron Green’s function with only phonon coupling can be rewritten, as usual, coupled to the 2-electron Green’s function with an effective time-dependent electron-electron interaction potential mediated by the 1-phonon Green’s function, leading to the polaron as an electron propagating jointly with its induced lattice polarization. An alternative formulation of the coupled Green’s function equations for the electron-hole-phonon model is applied in the development of a generalized shielded potential approximation, analysing its inverse dielectric screening response function and associated hybridized collective modes. A brief discussion of the (theoretical) origin of the exciton-plasmon interaction follows.Less

Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator ... More

Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.Less

TDDFT and many-body theory

Carsten A. Ullrich

in Time-Dependent Density-Functional Theory: Concepts and Applications

This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the ... More

This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the so-called adiabatic connection. This leads to a perturbative expansion of the xc potential, with exact exchange representing the lowest order. The following section introduces nonequilibrium Green's functions along the Keldysh contour. A Keldysh action principle is formulated which yields a formally exact representation of the time-dependent xc potential. As an alternative, the xc potential is expressed through the so-called Sham-Schlüter equation. The third section establishes a connection between the xc kernels of time-dependent density-functional theory in linear response and many-body perturbation theory. A diagrammatic expansion of the xc kernel is given. Finally, an expression for the xc kernel is derived starting from the Bethe-Salpeter equation. This xc kernel gives excellent agreement with experimental optical absorption spectra of insulators, including excitonic features.Less

Thermodynamic Green’s Functions and Spectral Structure

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the ... More

Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.Less

Electrostatics of semiconductor nanostructures

Thomas Ihn

in Semiconductor Nanostructures: Quantum states and electronic transport

Semiconductor nanostructures consist of materials with dielectric properties. Doping introduces charge distributions, and metallic surface gates establish boundary conditions for electrostatic ... More

Semiconductor nanostructures consist of materials with dielectric properties. Doping introduces charge distributions, and metallic surface gates establish boundary conditions for electrostatic potentials. All these ingredients are accounted for in a general theoretical description of the electrostatic potentials arising in semiconductor nanostructures. A formal solution of the electrostatic problem is derived using Green's functions, and the individual contributions to the total electrostatic energy are identified and interpreted. Furthermore, charges induced on gate electrodes are determined. The general concepts are illustrated with the example of a simple model for a split-gate defined structure.Less

Approximate and Numerical Approaches

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko

in Advances in the Casimir Effect

This chapter considers several approximate methods developed to calculate the Casimir energy and force for nontrivial geometries, where the separation of variables is not possible. One of these ... More

This chapter considers several approximate methods developed to calculate the Casimir energy and force for nontrivial geometries, where the separation of variables is not possible. One of these methods is the multiple-reflection expansion. This allows an iterative calculation of the corresponding Green's function. Another method is the semiclassical one. This is based on the idea of the WKB approximation in quantum mechanics, or, equivalently, the eikonal approximation in optics. Another approximate method for the calculation of the Casimir force considered is the numerical world line approach. Inspired by string theory, it uses the Feynman path-integral representation of transition amplitudes. Two other simple approximate methods are the pairwise summation method and the proximity force approximation. These are repeatedly used in the following chapters of the book.Less

Propagators and Green’s functions

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

This chapter introduces Green’s functions and shows how they are related to a propagator, the amplitude that a particle in some spacetime position will be later found at another spacetime position.

Poisson Kernels and Green’s Representation Formula

Ali Taheri

in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

This chapter establishes Green’s representation formula and using it to arrive at Greens functions and Poisson kernels. Presenting explicit examples of Greens functions and Poisson kernels in ... More

This chapter establishes Green’s representation formula and using it to arrive at Greens functions and Poisson kernels. Presenting explicit examples of Greens functions and Poisson kernels in specific geometries, e.g., an hp-ball and a half-space. It presents an introductory study of the Newtonian potential.Less

Random Phase Approximation Plasma Phenomenology, Semiclassical and Hydrodynamic Models; Electrodynamics

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Chapter 10 reviews both homogeneous and inhomogeneous quantum plasma dielectric response phenomenology starting with the RPA polarizability ring diagram in terms of thermal Green’s functions, also ... More

Chapter 10 reviews both homogeneous and inhomogeneous quantum plasma dielectric response phenomenology starting with the RPA polarizability ring diagram in terms of thermal Green’s functions, also energy eigenfunctions. The homogeneous dynamic, non-local inverse dielectric screening functions (K) are exhibited for 3D, 2D, and 1D, encompassing the non-local plasmon spectra and static shielding (e.g. Friedel oscillations and Debye-Thomas-Fermi shielding). The role of a quantizing magnetic field in K is reviewed. Analytically simpler models are described: the semiclassical and classical limits and the hydrodynamic model, including surface plasmons. Exchange and correlation energies are discussed. The van der Waals interaction of two neutral polarizable systems (e.g. physisorption) is described by their individual two-particle Green’s functions: It devolves upon the role of the dynamic, non-local plasma image potential due to screening. The inverse dielectric screening function K also plays a central role in energy loss spectroscopy. Chapter 10 introduces electromagnetic dyadic Green’s functions and the inverse dielectric tensor; also the RPA dynamic, non-local conductivity tensor with application to a planar quantum well. Kramers–Krönig relations are discussed. Determination of electromagnetic response of a compound nanostructure system having several nanostructured parts is discussed, with applications to a quantum well in bulk plasma and also to a superlattice, resulting in coupled plasmon spectra and polaritons.Less

MICROSCOPIC DERIVATION OF THE TDGL EQUATION

Anatoly Larkin and Andrei Varlamov

in Theory of Fluctuations in Superconductors

This chapter presents the basic aspects of the microscopic description of fluctuation phenomena in superconductors. The notion of fluctuation propagator as the vertex part of the electron: electron ... More

This chapter presents the basic aspects of the microscopic description of fluctuation phenomena in superconductors. The notion of fluctuation propagator as the vertex part of the electron: electron interaction in the Cooper channel, diagrammatic representation of fluctuation corrections, the method of their averaging over impurities, are introduced. The developed method of Matsubara temperature Green's functions applied to a description of the fluctuations allows the determination of the values of the phenomenological parameters of the GL theory. It also allows the determination of the treatment of fluctuation effects quantitatively, even far from the transition point, and for strong magnetic fields taking into account the contributions of dynamical and short wavelength fluctuations, as well as the quantum effects eluding from the phenomenological consideration.Less

MATERIAL DEFORMATION

Nasr M. Ghoniem and Daniel D. Walgraef

in Instabilities and Self-Organization in Materials: Volume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology

This chapter discusses the main elements of continuum theory, which will be utilized further in later sections dealing with a variety of applications. First, a basic review of vector and tensor ... More

This chapter discusses the main elements of continuum theory, which will be utilized further in later sections dealing with a variety of applications. First, a basic review of vector and tensor operations is presented, followed by descriptions of stress and strain tensors, and their relationships through thermodynamic functions. The special conditions of anisotropic crystals are also discussed. Green's tensor functions are derived for the displacement field of a material in response to a point force, for both isotropic and anisotropic crystals. Then, the methods of Eshelby for inclusions and inhomogeneities are presented, with analytical solutions to special problems of a second phase within a perfect crystals. This is followed by a presentation of the basic elements of crystal plasticity, and its connection with dislocation (or incompatibility) fields. The elastic fields of complex dislocation loop ensembles are finally given for both isotropic and anisotropic crystals.Less