Green's function Definition and 185 Threads

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...

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...

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...

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}...

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...

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...

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...

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...

The derivative of the Green's function is: i \dfrac{dG_{A,B}(t)}{dt} =\delta(t) \left< {[A,B]}\right>+G_{[A,H],B}(t) the Fourier transform is: \omega G_{A,B}(t)=\left< {[A,B]}\right>+G_{[A,H],B}(\omega) but this would require that the Green's function is 0 for t->inf. Why is that the case...

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...

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...

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...

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...

I'm trying to understand the derivation for methods of Greens functions for PDEs but I can't get my head around some parts. I'm starting to feel comfortable with the method itself but I want to understand why it works. The thing I have problem with is quite crucial and it is the following: I...

Hello everyone, I would like to know if there is a known, rigorous way to regularize a Green's function in coordinate space. In particular, it is known that the Green's function for a circle of radius R and source located at \vec x_0 is given by: $$ G(\vec x,\vec...

Hello everyone! Does anyone know if there is a know expression for the Green's function for Poisson's equation that vanishes on an ellipse in 2 dimensions? I'm essentially looking for a solution to: $$ \nabla^2G(\vec x-\vec x_0)=\delta^2(\vec x-\vec x_0) $$ in 2 dimensions where $$G(\vec x-\vec...

Hello guys, here's my question is how the book managed to solve this boundary value problem?? can anyone explain it to me in detail? thanks in advance.

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...

Its usually said that the propagator ## K(\mathbf x'',t;\mathbf x',t_0) ## that appears as an integral kernel in integrals in the path integral formulation of QM, is actually the Green's function for the Schrodinger equation and satisfies the equation below: ## \left[ -\frac{\hbar^2}{2m}...

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...

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\...

Homework Statement Consider an object subject to a linear repulsive force, ##F = kx##. Show that the Green’s function for this object is given by: ##G(t-t^\prime)=\frac{1}{m\gamma}sinh(\gamma(t-t^\prime))## where ##\gamma=\sqrt{\frac{k}{m}}## Homework Equations ##sinhx=\frac{e^x+e^{-x}}{2}##...

Hello! I'm having a problem with the Green's function solution of the simplest case of Poisson's equation, namely a single test charge ##q## located at ##\boldsymbol r = \boldsymbol r'##. I've read the related posts on Poisson's equation via Green's function formalism, but they do not answer my...

Dear all, In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modeled as a mass connected to neighbouring atoms through springs). He gives the green...

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...

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...

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## ...

Homework Statement An infinite plane at z=0 is divided into two: the right half of it (y>0) is held at potential zero and the left half of it is held at potential \phi_0 . Over this surface lies a point charge q at (0,y_0,z_0) . Use Green's function to calculate the potential at z>0. The...

So we have derived that for the differential equation: ##x(t)''+x(t)=\delta(t)## The solution is given by ##x=sin(t)H(t)## where ##H## is the Heaviside function. To find this we assumed that the system was in rest before ##t=0## and that position and velocity are continious. QUESTION: I am...

Hello, I'm trying to understand the application of Green's function to find the potential better. I apologize in advance if I start mixing things up a little. From what I understood and seen, we use this method (Green and method of images) in known symmetries (cylindrical/spherical/planar) and...

Edit: I have substantially edited this post from its original form, as I realize that it might have fallen under the label of "textbook-style questions". Really, the heart of my issue here is that, anywhere I look, I can't seem to find a clear description anywhere of the limitations of the...

Homework Statement [/B] I'm trying to show that, \phi(x')=b\frac{\sin kx'}{k}+a\cos kx'+\int_{0}^{x'}\frac{\sin k(x-x')}{k} f(x)dx is the solution of, \frac{d^{2}}{dx'^{2}}\phi(x')+k^{2}\phi(x')=f(x')dx where 0 \leq x'<\infty. 2. Homework Equations N/A The Attempt at a Solution [/B]...

Homework Statement Find Green's function for ##u''(x) + u(x) = f(x)## subject to ##u(0) = A## and ##u(\pi) + u'(\pi) = B##. Homework Equations No set equation. The Attempt at a Solution I begin by recognizing that green's function ##G## satisfies ##G''(x) + G(x) = \delta(x - x_0)## subject to...

Hi, I'm stuck with a question from one of my examples sheets from uni. The question is as follows: If G(x,x') is a greens function for the linear operator L, then what is the corresponding greens function for the linear operator L'=f(x)L, where f(x) =/=0? So I've started by writing...

There is a second order nonhomogeneous equation of motion with nonzero initial condition given at ##t=-\infty##: ##D^2 y(x)=f(x)## with ##y(-\infty)=e^{-i x}## where I have used the shorthand notation ##D^2## for the full differential operator. Also I have the two solutions ##y_1(x)## and...

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')...

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 $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...

Homework Statement Find the solution for: ({\partial{}_t}^2 -D \Delta^2)G(\vec{r},t;\vec{r}_o,t_o)=\delta(\vec{r}-\vec{r}_o)\delta(t-t_o) In two dimensions. Homework EquationsThe Attempt at a Solution Am I supposed to use bessel eqs? I'm kind of stuck in starting the problem :L

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...

Homework Statement I need to calculate, in D=4, the time component of the vector potential, A_{0}, given the equation (below) for A_{\mu} with the Green's function, also given below. The answer is given to be A^{0}=+\frac{q}{4\pi}\frac{1}{\mid\overline{x}\mid} Homework Equations A_{\mu}=\int...

So if I understood well, Normal ordering just comes due to the conmutation relation of a and a⁺? right? Is just a simple and clever simplification. Wick Theorem is analogue to normal ordering because it is related to the a and a⁺ again (so related to normal ordering, indeed). However I do not...

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...

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...

[SIZE="4"]Definition/Summary Green's function G\left(\mathbf{x},\mathbf{\xi}\right) can be defined thus \mathcal{L}G\left(\mathbf{x},\mathbf{\xi}\right) + \delta\left(\mathbf{x} - \mathbf{\xi}\right) = 0\;\;\; \mathbf{x},\mathbf{\xi} \in \mathbb{R}^n Where \mathcal{L} is a linear...

Suppose we have some partial differential equation for a scalar ##f## $$Df = \rho$$ taking values in ##\mathbb{R}^n##, and further suppose that the differential equation is completely independent of the variable ##y:=x^n## so that the differential operator ##D## only contains derivatives with...

Hi Everyone: I think some of you who familiar with quantum-optics know that the local photonic density of state can be calculated by the imaginary part of electromagnetic Green's function. The Green's function can be further presented by the dipole's mode pattern as G =...

I've been reading this article for a prof this summer: http://arxiv.org/pdf/1302.0245v1.pdf I'm having some trouble following the math in Appendix B: Green's Function Of A Homogeneous Cylinder (page 9). Can someone explain to me why there is a factor of \frac{1}{\rho\rho'} in front of the Green...

This is how I learned about Green's functions: For the 1-D problem with the linear operator L and the inner product, (\cdot,\cdot), Lu(x) = f(x) \rightarrow u=(f(x),G(\xi,x)) if the Green's function G is defined such that L^*G(\xi,x) = \delta(\xi-x) I understand how to arrive at this...

Dear users, right now I am struggling with calculations of the displacement in the analytically way. I am trying to accomplish this with help of the Eshelby's work (Eshelby's tensor). Right now I have a problem with Green's function. Displacement is expressed by Green's function, it looks...

Homework Statement Given a linear operator L=\frac{d^3}{dx^3}-1, show that the Fourier transform of the Green's function is \tilde{G}(k)=\frac{i}{k^3-i} and find the three complex poles. Use the Cauchy integral theorem to compute G(x) for x < 0 and x > 0. Homework Equations The...