Green function Definition and 85 Threads

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is the linear differential operator, then

the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f ), where G is the Green's function.Through the superposition principle, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.

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  1. I

    A Circuit ODE with multiple short voltage impulses

    I am trying to understand some introductory material on applications of Green's functions and the book which I am following uses the example of an electric circuit subjected to multiple voltage impulses. This is the Google drive viewing link to the few pages I am referring to: . I had to post...
  2. physicsxanime

    Calculating exciton Green's function on a tight binding chain

    Honestly, I have no real idea. I know for sure the equation connects the initial state ##|\Psi^N_0>##to a final state ##|\Psi^{N+1}_0>##, ##E## is the energy and ##E^N_0## etc are the energy of the initial state and final state. I also know that these energy are related to conduction band and...
  3. S

    I Toy problem to motivate the idea of advanced Green's functions

    The Wikipedia article on Green's functions says, What would be a really simple physical toy problem that would motivate and explain the concept of advanced Green's functions?
  4. P

    Studying Should I study relativistic QFT to get non-relativistic QFT?

    First time in PF, I am sorry if I did not choose the right category. I have been doing theory in condensed matter (mostly numerics) as a PhD but I never got to learn proper quantum field theory (QFT). Aside from a few introductory courses at university, I never learned what is a many-body...
  5. forever_physicist

    A From two-body Green's function to one-body in perturbation theory

    Hi guys! I am trying to derive the Hedin's equations used for GW on my own, and I found this equation, but I cannot really derive it, nor find a source where they explain how this can be derived (they link to other papers that in the end don't show where this is coming from). The equation I am...
  6. kakaho345

    Finding free electron gas Green function in Fourier space

    As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...
  7. D

    A Calculating nonequilibrium Green's Functions

    Hey :) Firstly I want to thank everyone who takes their time to read through this post and who tries to help me. So the issue is the following: I wrote a python code that creates a Lindbladian, and I wanted to try to calculate the Greens function using the Lehmann representation. For the...
  8. A

    A Green's function for tunneling electrons between quantum dots

    Good afternoon! I am writing with such a problem, I hope to find someone who could help me. I'm almost desperate! So, there is such a thing as the Braess paradox, this is a classic paradox for roads and power grids, and there is also such an article...
  9. MahdiI84

    I Obtain the Density of State using the Green function

    Hi everyone. Using the Green function, I want to obtain the density of states of a one-dimensional (linear) lattice. Depending on the problem conditions, we will have an iterative loop with 4,000 data for the energy component and a iteration loop with 2,000 data for the wave number component. In...
  10. H

    Method of image with Green function

    A conducting sphere of radius a is made up of 3 different shells, upper part a/2 z -a/2, the middle part -a/2 z a, the lower part -a z -a/2 where the center of the sphere is at the origin. The upper part V and the lower part has -V potentials, while the mid part is grounded. Find the...
  11. Rzbs

    A How Does the Two-Particle Green Function Relate to Hartree-Fock Theory?

    could anyone explain why in the page of book this figure is related to hartree-fock? I mean why if t1>t2 we have these possibilities? and why not particle propagate from x2t2 to x3t3 instead x3t3+?
  12. Mounice

    Solving Potential of a Charge Outside a Sphere with Green Functions

    I was wondering if there is a way to deduce the solution of the potential of a charge outside a sphere given by the image method, though Green functions. Because of a Dirichlet condition (GD(R,r')=0), I know that a solution can be written as GD=Go+L, where ∇2L=0. But in order to approach this...
  13. filip97

    A Huygens Principle - how to explain this with classical language?

    I was read this article(https://engineering.purdue.edu/wcchew/ece604f19/Lecture%20Notes/Lect31.pdf). I was read this paper about Huygens' principle(https://engineering.purdue.edu/wcchew/ece604f19/Lecture%20Notes/Lect31.pdf) Main idea of Huygens' principle is how wave function ##ψ(r)##...
  14. L

    How to Apply Fourier Transform to Green's Functions?

    In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...
  15. M

    Show that the given Green Function is the propagator of a certain Lagrangian

    My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...
  16. H

    Green's function and the resistance across a Hypercube

    Homework Statement: I do know how to solve the resistance network problem in two dimensions. However, in this problem they want it in 3 dimensions and higher and I don't know how to do that Homework Equations: - In the picture you can see the solution to the two dimensional version
  17. sergiokapone

    1D Green function for a charged layer

    I came across an example of a solution to finding the potential of a charged layer using the Green function (here, pdf). The standard algorithm for finding the Green function by boundary conditions for many problems is understandable: \begin{align*} G_\mathrm{Left} = Ax+ B \\ G_\mathrm{Right} =...
  18. Ramil

    Quantum Monte-Carlo calculation of Green's function

    Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program. i - index enumerating the nodes along t-coordinate, j - along the r-coordinate, k - along the theta-coordinate, l - along the phi-coordinate. N_t - the number of nodes along t-coordinate. N_r...
  19. J

    A Which operator corresponds to the Green function in QFT?

    The Feynman propagator: $$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$ is the Green's function of the operator (except maybe for a constant): $$ (\Box + m^2)$$ In other words: $$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$ My question is: Which is the operator that...
  20. L

    The OVGF (outer valence Green Function) method

    Is there anyone here that know and understand the OVGF method who can help me? I have some doubts about it, and there is almost nothing about it in literature.
  21. H

    A Green's function for the wave function

    We want to solve the equation. $$H\Psi = i\hbar\frac{\partial \Psi}{\partial t} $$ (1) If we solve the following equation for G $$(H-i\hbar\frac{\partial }{\partial t})G(t,t_{0}) \Psi(t_{0}) = -i\hbar\delta(t-t_{0})$$ (2) The final solution for our wave function is, $$\Psi(t) =...
  22. Phys pilot

    Transform Dirichlet condition into mixed boundary condition

    Hello, If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating...
  23. C

    Order of multi-variable integration of infinite range

    Homework Statement If the Green's function of the electric field in a system is G(x,x')=e^{-i(x-x')^2} I want to calculate the phase of the electric field at x if the source is uniformly distributed at x'=-\infty to x'=\infty Homework EquationsThe Attempt at a Solution Then, the phase of...
  24. T

    Can not find correct green's function

    Homework Statement We have long wire with constant charge density that is put inside a grounded metal housing with a shape of cylindrical section (a ≤ r ≤ b and 0 ≤ ϕ ≤ α). We need to find potential inside the box. 2. Homework Equations Δf=-(μ/ε0)*∂^2(r), where μ is linear charge density...
  25. W

    A Is Normalizing a 4x4 Matrix Possible Using Multiple Methods?

    I am trying to normalize 4x4 matrix (g and f are functions): \begin{equation} G=\begin{matrix} (1-g^2) &0& 0& 0&\\ 0& (1+f^2)& (-g^2-f^2)& 0 \\ 0 &(-g^2-f^2)& (1+f^2)& 0 &\\ 0& 0& 0& (1-g^2) \end{matrix} \end{equation} It's a matrix that's in a research paper (which I don't have) which gives...
  26. W

    Mathematica Is the integral for current correct?

    Hi, I am using Mathematica to calculate density of states and current of the Green's function times self energy in most simple form. I am not sure if I am getting current integral over energy implemented correctly. Shouldnt first current plot be a line with a slope? Below is my code...
  27. Twigg

    A Green function issue: can bound. cond. be applied mode-wise?

    Hello, Before I begin, a lot of the math I try to describe in this post is stuff I worked out myself and have scribbled down. If something looks fishy or doesn't make sense, it could very well be totally wrong! Apologies for any confusions in advance. Consider a slab of linear isotropic...
  28. L

    Semi-infinite slab, surface heating on a radius r=a; T=?

    Homework Statement [/B] We are heating a semi-infinite slab with a laser (radius of a stream is ##a##), which presents us with a steady surface heating (at ##z=0##), everywhere else on the surface the slab is isolated. How does the temperature change with time? Look at the limit cases: at ##t...
  29. Summer95

    I Confused about Finding the Green Function

    Suppose we have a differential equation with initial conditions ##y_{0}=y^{\prime}_{0}=0## and we need to solve it using a Green Function. Then we set up our differential equation with the right side "forcing function" as ##\delta(t^{\prime}-t)## (or with ##t^{\prime}## and ##t## switched I'm a...
  30. L

    Green function in electrostatics

    Hello, I'm taking a course in electrostatics and electrodynamics. We learned about finding a potentional using unique Green functions that are dependent of the geometry of the problem. Specificly on a Dirichlet problem we get the solution: Φ(x)=∫ρ(x')G(x,x')d3x' -...
  31. hideelo

    Q about Poisson eqn w/ Neumann boundary conditions as in Jackson

    I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is $$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4...
  32. P

    Jackson Problem 3.12/3.18 -- Electric potential near two plates

    Homework Statement I need to solve a problem like Jackson 3.18. I need to find potential due to the same configuration but the position of two plates is opposite i.e. Plate at Z=0 contains disc with potential V and plate at Z=0 is grounded. Homework EquationsThe Attempt at a Solution I think...
  33. F

    Good values for gravitational potential

    In the context of a project, I had to solve numerically Poisson equation with cylindrical coordinates. I put here results for z = 0 on a 3D mesh 256x256x256. Below 1 figure representing the final solution (in absolute value) in the case of a galaxy; I use the CGS units for the potential. I...
  34. Z

    MATLAB Is FFT the Proper Method for Transforming Green Function in MATLAB?

    Hi everybody. I have a (1×N) Green function in MATLAB. I want to use the FFt function for Green function to transform the time domain. Does the FFT command work correctly for Green function using: A=fft(Green function, nfft). Here nfft is the number of transformed points. Is it necessary that...
  35. Z

    MATLAB FFT for one 1*N Green function in MATLAB

    I have a 1×N Green function in energy domain. I want to use the FFT (fast Fourier transform) for this Green function in MATLAB. But this function is non-periodic. Could you help me about this? How can I change the energy interval to convert Green function as a periodic function?
  36. L

    Retarded Green's Function for D'Alembertian

    Hey All, I recently posted this in another area but was suggested to put it here instead. Here is my original post:
  37. L

    Retarded Green's Function for D'Alembertian

    Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...
  38. F

    Poisson equation/zero padding and duplicating Green function

    Hello, I need to solve the Poisson equation in gravitational case (for galaxy dynamics) with Green's function by applying Fast Fourier Transform. I don't understand the method used for an isolated system from (Hockney & Eastwood 1981); it says : I have 2 questions: * Why we duplicate the...
  39. P

    Solve 2D Green Function: Star Equation

    Hello, i don't understand how i integrate the star equation at the figure. Fot example \int d x_1 \int d x_2 \frac{d^2G}{d x_1} = \int d x_1 \frac{d^2G}{d x_1} \int d x_2 but \int d x_2 x_2|^\epsilon_{-\epsilon}=0 ...sure this is a error, but i don't understand how i find the answer...
  40. J

    Calculation and application of dyadic Green's function

    Hello everyone: I'm confusing with the construction and application of dyadic green's function. If we are in the ideal resonant system where only certain resonant mode is supported in this space (such as cavity), the Green's function can be constructed by the mode expansion that is: Gij(r,r')...
  41. J

    Transverse and longitudinal electric Green function

    Hello everyone: This is what I read in a paper, the spontaneous emission rate written by Fermi's Golden Rule is just related to the local transverse electric field. Dose anyone can explain to me what's the meaning of "transverse mode" here? Why the emission is not related to longitudinal...
  42. J

    Green function for forced harmonic oscillator

    Homework Statement The problem requires to solve the integration to find ## G(t) ## after ##G(\omega)## is found via Fourier transform. We have G(\omega)= \frac{1}{2\pi}\frac{1}{\omega _{0}^2 - \omega ^2} Homework Equations As mentioned previously, the question asks to find ##G(t)## The...
  43. A

    Image charge method to find Green's function

    hi guys, my professor told me in the class that when we would like to determine green function there are two general method i.e using image charge and using orthonormal eigen function. However I don't understand what are the specific differences between them. Anybody can help me? Moreover in the...
  44. A

    Green's Function Using Image Charge

    Homework Statement Write an expression for the Dirichlet Green's function of the part of the space bounded by two infinite conducting plates parallel each other and separated by distance of d. Use Image charge method Homework Equations G (at z=0) =0, G (at z=d) =0 I guess The Attempt at a...
  45. C

    Green Function SHO - Reading Materials & Math Background

    hi, I need some reading materials on green function for SHO. my instructor provided a GF frequency and wanted us to find the deformation of poles , boundary conditions for the function. I need to know which mathematical background should I have to solve this. any useful material suggestion will...
  46. A

    Meaning of Dyadic Green Function in Electromagnetism

    I know that in the vector potential formulation one can use a scalar Green function (to find the said potential and from then on the electric and magnetic fields), and that this works because the components of the potential are in the same direction as those of the source - i.e. a current in the...
  47. F

    Particle Mesh Method - FFT with Green Function

    Hello, For generating initial velocities on a N-Body code (modeling galaxy dynamics), I need to solve the Poisson equation with Green's function by applying Fast Fourier Transform. The Fourier analysis being more straightforwardly performed in a periodic system, I use the "zero padding trick"...
  48. Einj

    Dimension of n-point Green function

    Hi everyone. I have a very quick question. Can someone tell me how to compute the energy dimensions of an n-point Green function. Consider for example a \lambda\phi^4 scalar theory. I know that the dimensions of an n-pt Green function are 4-n (or something like that). How do I prove it? Thanks
  49. R

    Green function, Delta and Heaviside

    Homework Statement Show that the explicitly covariant expression: GR(x-y) = θ(x0-y0)δ((\vec{x}-\vec{y})2)/2\pi agrees with the retarded Green function: δ(x0-y0-|\vec{x}-\vec{y}|) / (4\pi|\vec{x}-\vec{y}|) Homework Equations N/A The Attempt at a Solution I know that the...
  50. B

    Green function for the particle hopping on a lattice, meaning?

    I am having trouble understanding something that I am sure is very basic. Let's say I have a particle that is hopping on a 1d lattice with a hard wall at x=0 in the presence of some potential - anything, say linear ##H_0=F*i## or Coulomb ##H_0=C/i## where i is the label of the site the particle...
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