What is Greens function: Definition and 20 Discussions

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. steve1763

    A Green's function for Stokes equation

    So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force. $$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$ $$\nabla \cdot \textbf{u}=0$$ We can apply the...
  2. P

    A Green's function calculation of an infinite lattice with periodicity in 1D

    I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...
  3. peguerosdc

    Understanding Green's second identity and the reciprocity theorem

    This is Jackson's 3rd edition 1.12 problem. So, for both ## \phi ## and ## \phi' ##, I started from Green's second identity: ## \int_V ( \phi \nabla^2 \phi' - \phi' \nabla^2 \phi )dV = \int_S ( \phi \frac {\partial \phi'} {\partial n} - \phi' \frac {\partial \phi} {\partial n} ) dS ## And...
  4. C

    A Help with Correlation/Green's Function of Rotated Variables

    Hello (I'm reposting this from stack exchange, and thought this site may be more appropriate, so if you see it that's why), I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the...
  5. P

    Application of boundary conditions in determining the Green's function

    Homework Statement Find the Green's function $G(t,\tau)$ that satisfies $$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$ under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$ Then...
  6. D

    A little question on the Helmoltz Greens Function

    Homework Statement I was a bit confused when reading my notes.After some derivation it states that G_h = e^(ikr)G_p,where G_h = Helmoltz Green's function G_p = Poisson Green's function = - 1 /(4πr) By definition (D^2)(G_p) = δ^3 (r) (D^2 = Laplacian) Please see the attached (D^2)(G_p) = 0...
  7. ognik

    Fourier Transforms, Green's function, Helmholtz

    Homework Statement I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $$ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) \:is\...
  8. ognik

    MHB Greens Function for Hemmholtz using Fourier

    I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) $ is $ G(\vec{r_1},\vec{r_2})=...
  9. ognik

    MHB Greens function has me blue ....

    Not following this example (PDE for Greens function) in my book: Book states: $ \left( {\nabla}^{2} +{k}^{2}\right)G(r, r_2)=-\delta(r-r_2) = -\int \frac{e^{ip.(r-r_2)}}{\left(2\pi\right)^3} {d}^{3}p$ I recognised this as the Hemlmholtz eqtn, but cannot find where the 3rd term comes from? It...
  10. S

    Finding potential using Greens function

    Homework Statement A potential ##\phi(\rho, \phi ,z)## satisfies ##\nabla^2 \phi=0## in the volume ##V={z\geqslant a}## with boundary condition ##\partial \phi / \partial n =F_{s}(\rho, \phi)## on the surface ##S={z=0}##. a) write the Neumann Green's function ##G_N (x,x')## within V in...
  11. S

    Can we make use of Greens function if there are no charges?

    I can't think of a situation where we can utilize greens function without the presence of a point charge. let's consider the following equation: \Phi=\frac{1}{4\pi \epsilon} \int dv \rho(x')G_{N} (x,x')+ \frac{1}{4\pi} \int da F_{s}(\rho , \phi) G_{N} + <\phi>_S Here we see that a volume...
  12. A

    Greens function path integral representation

    In my book the path integral representation of the green's function is given as that on the attached picture. But how do you go from the usual trace formula for the Green's function 2.6 to this equation?
  13. C

    Klein Gordon ret. Greens function on closed form

    In this note (http://sgovindarajan.wdfiles.com/local--files/serc2009/greenfunction.pdf) the Klein-Gordon retarded green function is derived on the form $$G_{ret}(x − x′) = \theta(t − t') \int \frac{d^3 \vec k}{(2\pi)^3 \omega_k} \sin \omega_k (t − t′) e^{i \vec{k}\cdot (\vec x - \vec x')}$$...
  14. A

    Interpreting the Greens function in frequency space

    I have done a problem with a spin two electron spin system. I have found the Greens function propagator for spin up->spin up, which I have written in a form where the occupancy of spin up electrons is incorparated (attached) and from there I can get the spectral function, which will then be...
  15. G

    Greens function and the conducting sphere

    Can somebody explain to me, when equations 2.48 and 2.50 are applicable and what ##\Phi_s## and ##\Phi## actually are? The thing is, I want to find a general equations that determines the field produced by conducting spherical sphere in an external field and was wondering whether these are the...
  16. C

    Greens Function Solution of Schrodinger Equation

    Homework Statement Using the Greens function technique, reduce the Schrodinger Equation for the following potential: V = V0 , 0<x<a V = infinite, elsewhere. Homework Equations The Attempt at a Solution I have no idea what "reduce" means. The professor did not go over this...
  17. S

    Solving Poisson's equation with the help of Greens function

    Hey all, some weeks ago in a tutorial our TA solved Poissons equation with Greens functions..would be very short, but he also derived the Greens function using a Fourier transform. Two points I really don't get and he could also not explain it. Maybe you can help me? There might be even a short...
  18. M

    How Can Greens Function Be Used to Solve the Heat Equation with Delta Functions?

    Hi, i am trying to find greens function for the heat equation with a=1, i.e du/dt - d2u/dx2 = f(x,t) for 0<x<infinty i also have the conditions, u(x,0)=0 and u(0,t)=0 When i have found my greens function i have to allow f(x,t) = delta(x-1) delta(t-1) and obtain a solution using the greens...
  19. X

    Understanding Greens Function vs. Variation of Parameters

    So I just recently learned about how to use Greens Function to solve a differential equation. The formula was derived and it said the main goal was to find an integral representation to the solution. It seems to me, however that Greens Function is nothing more than variation of parameters with...
  20. S

    Solve Inhomogeneous Problem Using Greens Function: (1-x2)y'' - xy' + y = f(x)

    Homework Statement use the greens function G(x,z) to solve inhomogeneous problem: (1-x2 ) y'' - x y' + y = f(x) y(0) = y(1) = 0 Homework Equations the answer is: G(x,z)= -x for x<z and -z(1-x2 ) 1/2 (1-z2 ) 1/2 The Attempt at a Solution the general solution to the...