Read about green's function | 26 Discussions | Page 1

  1. J

    A Studying Green's function in many body physics

    Hi,everyone. Recently, I am studying green's function in many body physics and suffer from trouble.Following are my problems. (1) What is the origin of the definition of green's function in many body physics? (2) What is the physical meaning of self energy ? It seems like it is the correction...
  2. TheBigDig

    Green's Function for a harmonic oscillator

    I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0} But I can't seem to get that to work out. Using the Cauchy integral formula above, I take one pole at -w_0 and get \frac{ie^{i\omega_0(t-t')}}{2\omega_0} and...
  3. W

    I Continuity of Green's function

    Why can't G and its derivative be continuous in the relation below? $$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
  4. H

    A Applying boundary conditions on an almost spherical body

    I am solving the Laplace equation in 3D: \nabla^{2}V=0 I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are: V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
  5. arpon

    Green's function of a PDE

    Homework Statement Find out the Green's function, ##G(\vec{r}, \vec{r}')##, for the following partial differential equation: $$\left(-2\frac{\partial ^2}{\partial t \partial x} + \frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \right) F(\vec{r}) = g(\vec{r})$$ Here ##\vec{r}...
  6. W

    Green's functions: Logic behind this step

    Homework Statement Hi all, I came across these steps in my notes, relating to a step whereby, $$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$ and performing the following operation on ##\hat{G}## gives the following expression, $$[\frac{\partial}{\partial t}...
  7. G

    General solution for the heat equation of a 1-D circle

    Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
  8. A

    Classical Please recommend two textbookss about the Poisson equation and Green's function

    Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
  9. redtree

    I Green's function and the evolution operator

    The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator: \begin{equation} \begin{split} \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0) \end{split} \end{equation} However, I have seen the following...
  10. BiGyElLoWhAt

    Another Green's function

    Homework Statement Find the green's function for y'' +4y' +3y = 0 with y(0)=y'(0)=0 and use it to solve y'' +4y' +3' =e^-2x Homework Equations ##y = \int_a^b G*f(z)dz## The Attempt at a Solution ##\lambda^2 + 4\lambda + 3 = 0 \to \lambda = -1,-3## ##G(x,z) = \left\{ \begin{array}{ll} Ae^{-x}...
  11. BiGyElLoWhAt

    Green's Function and integral

    Homework Statement Find the green's function for y'' +2y' +2y = 0 with boundary conditions y(0)=y'(0)=0 and use it to solve y'' + 2y' +2y = e^(-2x) Homework Equations ##y = \int_a^b G(x,z)f(z)dz## The Attempt at a Solution I'm going to rush through the first bit. If you need a specific step...
  12. BiGyElLoWhAt

    I A question about boundary conditions in Green's functions

    I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function. I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...
  13. BiGyElLoWhAt

    I A somewhat conceptual question about Green's functions

    I just did a problem for a final that required us to use a green's function to solve a diff eq. y'' +y/4 = sin(2x) I went through and solved it and got a really nasty looking thing, but I checked it in wolfram and it works out. Now, my question is this: After I got the solution from my greens...
  14. F

    A Physical interpretation of correlator

    Consider the 2-point correlator of a real scalar field ##\hat{\phi}(t,\mathbf{x})##, $$\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rangle$$ How does one interpret this quantity physically? Is it quantifying the probability amplitude for a particle to be created at space-time point...
  15. AwesomeTrains

    Green's function differential equation

    Hello I'm doing some problems in QM scattering regarding the Green's function. Homework Statement Determine the differential equation of G(\vec{r},\vec{r}',\omega) Homework Equations I've been given the Fourier transform for the case where the Hamiltonian is time independent...
  16. 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...
  17. Y

    Potential and charge on a plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  18. Y

    Potential and total charge on plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  19. askhetan

    Understanding operators for Green's function derivation

    Dear All, I am trying to understand what operators actually mean when deriving the definition of green's function. Is this integral representation of an operator in the ##x-basis## correct ? ## D = <x|\int dx|D|x>## I am asking this because the identity operator for non-denumerable or...
  20. RUber

    Differentiating Integral with Green's function

    Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following: ## u(x,y) = \frac{\partial}{\partial y} \int_0^\infty\int_{-\infty}^\infty f(x') K_0( \sqrt{ (x - x')^2 + (y-y')^2 } \, dx' dy' ## ##K_0## is the Modified...
  21. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  22. T

    Green's function with same time and spatial arguments

    Concerning green's function with the same time and spatial argument(i.e. ##G_0(x,t;x,t)##, mostly in QFT), I have the following question 1. Is green's function well defined at this point? 2. if green's function is well defined at this point, is it continuous here? 3. In quantum many body...
  23. T

    Relation between adiabatic approximation and imaginary time

    Regarding interacting green's function, I found two different description: 1. usually in QFT: <\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>} 2. usually in quantum many body systems...
  24. RUber

    2D Green's Function - Bessel function equivalence

    Homework Statement This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is: -- The problem is to show that ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x')...
  25. M

    Inverse Fourier Transform of ##1/k^2## in ##\mathbb{R}^N ##

    Homework Statement This comes up in the context of Poisson's equation Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Homework Equations $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...
  26. genxium

    How is (d^3)r in Green's Function equivalent to volume element?

    Homework Statement This is part of the online tutorial I'm reading: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html I'm so confused about the notation of Dirac Delta. It's said that 3-dimensional delta function is denoted as \delta^3(x, y, z)=\delta(x)\delta(y)\delta(z) in...
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