Green's function Definition and 213 Threads

Homework Statement I am trying to find the Green's function in one space dimension. The Green's function is G(x,y) = \Phi(x-y) - \phi(x,y) where \phi(x,y) is the solution to the Laplace problem (x fixed): \Deltay\phi = 0 in \Omega with \phi(x,\sigma) = \Phi(x-\sigma) for \sigma on...

http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf Consider the Feynman rules for Green's Functions given at the top of p79 in these notes. Now let us consider the diagram given in the example on p78. Take for example the 2nd diagram in the sum i.e. the cross one where x1 is joined to x4...

I have this problem: Consider the differential equation y'' + P(x) y' + Q(x) y = 0 on the interval a ≤ x ≤ b. Suppose we know two solutions y1(x), y2(x) such that y1(a) = 0, y1(b) ≠ 0 y2(a) ≠ 0, y2(b) = 0 Give the solution of the equation y'' + P(x) y' + Q(x) y = f(x) which...

Homework Statement Find a green's function G(x,t) for the BVP y'' + y' = f(x), y(0) = 0, y'(1) = 0. Homework Equations The Attempt at a Solution I solved the homogeneous equation, looking for 2 linearly independent solutions, and found A (constant) and exp(-x). I am struggling...

Homework Statement Hello. I'm taking a course on Mathematical Physics, based on Eugene Butkov's book. I'm having trouble solving a DE with variable coefficients to find Green's Function. The problem asks to find Green's Function through direct construction. Homework Equations...

This appears on the bottom of p.279 of this book. The author begins with Green's second identity: \int_V \alpha \nabla^2 \beta - \beta \nabla^2 \alpha \ dV = \int_C \left( \alpha \frac{\partial \beta}{\partial n} - \beta \frac{\partial \alpha}{\partial n} \right) \ ds Here, C is a...

Show, from it's definition, \psi(x,t) = \int dx' G(x,t;x',t_0) \psi(x',t_0) G(x,t;x',t_0)= \langle x | U(t,t_0) | x' \rangle that the propagator G(x,t;x',t') is the Green Function of the Time-Dependent Schrodinger Equation, \left ( H_x - i \hbar \frac{\partial}{\partial t} \right )...

How does Green's function work in electromagnetism?

This isn't so much a problem as a step in some maths that I don't understand: (I'm trying to follow a very badly written help sheet) Here's how it goes: Given Newtons equation m \ddot{x} = F The Green's function for this equation is given by \ddot{G}(t,t^\prime)=\delta(t-t^\prime) (1)...

Homework Statement The Green function for the three dimensional wave equation is defined by, \left ( \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right ) G(\vec r, t) = \delta(\vec r) \delta(t) The solution is, G(\vec r, t) = -\frac{1}{4 \pi r} \delta\left ( t - \frac{r}{c}...

Use the method of images to find a Green's function for the problem in the attached image. Demonstrate the functions satisfies the homogenous boundary condition.

Homework Statement Use a Green's function to solve: u" + 2u' + u = e-x with u(0) = 0 and u(1) = 1 on 0\leqx\leq1 Homework Equations This from the lecture notes in my course: The Attempt at a Solution Solving for the homogeneous equation first: u" + 2u' + u = 0...

Homework Statement Find the Green's function for the Dirichlet boundary conditions for the interior of an infinite cylinder of radius a. Homework Equations \nabla^2 G(x,x') = -4 \pi \delta(x-x') and in general, Green's functions are of the form G(x,x') =...

In QFT expressions such as these hold: real scalar: \Delta_F(x-x')\propto\langle 0| T\phi(x)\phi(x')|0\rangle 4-spinor S_F(x-x')]\propto\langle 0| T\psi(x)\bar{\psi}(x')|0\rangle where T is the time-ordering operation and the proportionality depends on the choice of normalization...

This is not homework. This is actually a subset of proofing G(\vec{x},\vec{x_0}) = G(\vec{x_0},\vec{x}) where G is the Green's function. I don't want to present the whole thing, just the part I have question. Let D be an open solid region with surface S. Let P \;=\; G(\vec{x},\vec{a})...

For circular region, why is \frac{\partial}{\partial n}G(r,\theta,r_0,\phi)= \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) ? Where \; \hat{n} \: is the outward unit normal of C_R. Let circular region D_R with radius R \hbox { and possitive oriented boundary }\; C_R. Let u(r_0,\theta) be...

Homework Statement Green's function G(x_0,y_0,x,y) =v(x_0,y_0,x,y) + h(x_0,y_0,x,y) in a region \Omega \hbox { with boundary } \Gamma. Also v(x_0,y_0,x,y) = -h(x_0,y_0,x,y) on boundary \Gamma and both v(x_0,y_0,x,y) \hbox { and }h(x_0,y_0,x,y) are harmonic function in \Omega...

Green's function G(x_0,y_0,x,y) =v(x_0,y_0,x,y) + h(x_0,y_0,x,y) in a region \Omega \hbox { with boundary } \Gamma. Also v(x_0,y_0,x,y) = -h(x_0,y_0,x,y) on boundary \Gamma and both v(x_0,y_0,x,y) \hbox { and }h(x_0,y_0,x,y) are harmonic function in \Omega v=\frac{1}{2}ln[(x-x_0)^2 +...

The propagator D for a particle is basically the Green's function of the differential operator that describes that particle, e.g. (\partial^2 + m^2) D(x-y) = \delta^4 (x-y). This propagator is supposed to give the probability of the particle propagating from x to y. Why does this make...

Hi, While considering perturbed gravitational potential of incompressible fluid in rectangular configuration, I encountered two dimensional Poisson's equation including the step function. I want to solve this equation \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2}...

Boundary Value Problem + Green's Function Consider the BVP y''+4y=e^x y(0)=0 y'(1)=0 Find the Green's function for this problem. I am completely lost can someone help me out?

I have a serious blind-spot with mathematics (but I keep trying) Can someone help me with this. I have a relation A = \mu_{0}/4\pi\int J/r \ dVol Which (apparently!) can be written \nabla^{2} A = - \mu_{0} J I know that \nabla^{2} A = 1/r \ \delta^{2} ( r A ) / \delta r^{2} which is...

Consider the BVP y''+4y=f(x) (0\leqx\leq1) y(0)=0 y'(1)=0 Find the Green's function (two-sided) for this problem. Working: So firstly, I let y(x)=Asin2x+Bcos2x Then using the boundary conditions, Asin(2.0)+Bcos(2.0)=0 => B=0 y'(x)=2Acos(2x)-2Asin(2x) y'(0)=2A=0...

Hello! Something about N-point Green's function in QFT really troubles me... In the path-integral formalism,why will we introduce the N-point Green's function? I mean is it enough because we have calculated the 2-point green's function. And in the canonical formalism, it seems we can finish...

Hi, I am trying to compute the Green's function for a Cauchy-Euler equidimensional equation, \frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x') If the impulse is located at a location x'\neq x_c then computation of Green's function is not an issue. What happens when x'= x_c ...

Is it possible to define unambiguously retarded and advanced Green's function in spacetime without timelike Killing vector. Most often e.g. retarded Green function G_R(t,\vec{x},t',\vec{x}') is defined to be 0 unless t'<t but maybe one can express this condition using only casual structure...

Is it true that there always exists Green's function for Dirichlet boundary problem. I mean a function G(r,r') which fullfils the following conditions: div (\epsilon grad G(r,r')) =- \delta(r,r') inside volume V and G(r,r') is 0 on boundary of V. If V is whole space there exists obvious...

The differential equation is as follows: [d/dx^2 + k^2 - tau * dirac_delta(x-x') ] * G(x,x') = dirac_delta(x-x') where tau is a complex valued scattering strength, and assuming scattering waves at infinity. The problem asks to derive the solution to this equation. I've looked over...

Homework Statement If a hollow spherical shell of radius a is held at potential \Phi(a, \theta ', \phi '), then the potential at an arbitrary point is given by, \Phi(r, \theta, \phi)=\frac{1}{4 \pi} \oint \Phi(a, \theta ', \phi ') \frac{\delta G(r, r')}{\delta n '}dS' where G(r...

Homework Statement I am being asked to consider a Dirac spinor with two complex components and the following Lagrangian: L = L_{Dirac}-\stackrel{g}{4}{(\psi\bar{\psi})^{2}} I am asked to derive the Feynman rules for this theory which I can do using the standard methods. However, I am...

Is there a physical unit related to the Green's function of the wave equation? In particular, let \nabla^2 P -\frac{1}{c^2}\frac{\partial^2 P}{\partial t^2} = f(t) where P is pressure in Pa. Since the Green's function solves the PDE when f(t) is the delta function, the Green's function G...

Hi there, could anyone help me on this particularly frustrating problem I am having... I have a linear parabolic homogeneous PDE in two variables with a boundary condition that is a piecewise function. I can solve the pde (with a homogeneous BC) however trying to impose the actual BC makes...

Say we have a 3D function, p(x,y,z) and we define it in terms of another function f(x,y,z) via, \nabla ^2 p = f. I know that if we are working in R^3 space (with no boundaries) we can say that, p= \frac{-1}{4\pi}\iiint \limits_R \frac{f(x',y',z')}{\sqrt{(x-x')^2 +(y-y')^2+(z-z')^2}} dx'...

Matrix "Green's function" Hi. If you have a differential equation \mathcal L y=f where \mathcal L is some linear differential operator, then you can find a particular solution using the Green's function technique. It is then said that the Green's function is kind of the inverse to \mathcal...

I'm seeking help in understanding Kirchoff-Helmholtz Integral. Actually what i am facing the problem here is, i don't understand certain things about Green's 2nd identity which stated that two scalar function can be interchanged, and forming the force F = \phi\nabla\varphi -...

Hi, I have been trying to find the (causal) Green's function of \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + m^2 \phi = 0. What would be a good way to approach this? I have initial values for t=0, so I use...

Hello Forum, given a input=delta located at time t=0, the system will respond generating a function h(t). If the delta is instead located at t=t0 (delayed by tau), the system will respond with a function g(t)=h(t-tau), just a shifted version of the response for the delta a t=0... If...

Here are some pages of Arfken's “Mathematical Methods for Physicists ” I don't How to work out the Green's function! Can anyone explain (9.174)and(9.175) for me ? I'm hoping for your help, Thank you !

Use Green's Functions to solve: \frac{d^{2}y}{dx^{2}} + y = cosec x Subject to the boundary conditions: y\left(0\right) = y\left(\frac{\pi}{2}\right) = 0 Attempt: \frac{d^{2}G\left(x,z\right)}{dx^{2}} + G\left(x,z\right) = \delta\left(x-z\right) For x\neq z the RHS is zero...

By taking the Fourier transform of the fundamental Helmholtz equation (\nabla^2+k^2)G(\vec{x})=-\delta(\vec{x}), one finds that G(\vec{x})=\frac{e^{ikr}}{r} and \tilde{G}(\vec{\xi})=\frac{1}{k^2-\xi^2}. However, I can't figure out how to directly confirm that this Fourier...

This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as: D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k} Where z = x - x' the 4 vector difference, k\cdot z = k_0z_0 -...

I can't figure out how to use the Green's function approach rigorously, i.e., taking into account the fact that the Dirac Delta function is not a function on the reals. Suppose we have Laplace's Equation: \nabla^2 \phi(\vec{x})=f(\vec{x}) The solution, for "well-behaved" f(\vec{x}) is...

Hello I am trying to build a 3D Poisson solver using method of moments. I need to find out the Green's function for the system. My system is a rectangular box and boundary conditions are as follows: On all surfaces BC is neumann. Only on the upper and lower surface, the middle 1/3 region...

Homework Statement I'm asked to calculate Green's function's real and imaginary parts. The expression for the given Green's function is: g00=[1-(1-4t2(z-E0)-2)1/2]/2t2(z-E0)-1 (1) Where, z is the complex variable: z= E+iO+ (2) Homework Equations...

Hey Guys; I'm solving PDE's with the use of Green's function where all the boundary conditions are homogeneous. However, how do you solve ones in which we have non-homogeneous b.c's. In case it helps, the particular PDE I'm looking at is: y'' = -x^2 y(0) + y'(0) = 4, y'(1)= 2...

Homework Statement Consider a potential problem in the half space z>=0 with Dirichlet boundary conditions on the plane z=0. If the potential on the plane z=0 is specified to be V inside a circle of radius a centered at the origin, and Phi=0 outside that circle, show that along the axis of...

Homework Statement \Omega = \left{ \left( x,y,z \right) :,0<z<1 \right} Need to find Green's function using the method of images. Homework Equations none The Attempt at a Solution I can see that I will need an infinite sequence of images at each plane z = k, k = 0, +/- 1, +/- 2,... to...

i am having trouble distinguishing when to use Fourier or laplace transform to solve any linear differential equation (it can be an ODE or PDE). What are the advantages and disadvantages of using each? Also for a green's function (take it to be a function of x, x') when solving for it, is it...

Hello! I have problem with my homework, but what I'm going to ask you is not homework problem so I hope it is OK I'm writing it here :) I need to find Green's function for differential operator L=a\frac{d^2}{dx^2}+b\frac{d}{dx}+c i.e. find solution for differential equation equation...

i find that most books on green's function are burdened with too much formalism i am now reading the book by Rammer, which deals with non-equilibrium physics. The formalism is so lengthy and so confusing. You have to strive hard to remenber the various green's functions, and only to find...