Green's function Definition and 185 Threads

Homework Statement A dynamical system has a response, y(t), to a driving force, f(t), that satisfies a differential equation involving a third time derivative: \frac{d^{3}y}{dt^{3}} = f(t) Obtain the solution to the homogeneous equation, and use this to derive the causal Green's function...

Homework Statement A parallel plate waveguide has perfectly conducting plates at y = 0 and y = b for 0 ≤ x < ∞ and -∞ < z < ∞. Inside that bound, the waveguide is filled with a dielectric with k as a propagation constant. The Green's function to be satisfied is \nabla^2G + k^2G =...

Hi everyone, I'm going through some lecture notes on Quantum Field Theory and I came across a derivation of an explicit form of the Pauli Jordan Green's function for the Klein-Gordon field. The equations used in my lecture notes are equivalent to the ones in...

Consider ##\nabla^2 u(x,y)=f(x,y)## in rectangular region bounded by (0,0),(0,b),(a,b)(a,0). And ##u(x,y)=0## on the boundary. Find Green's function ##G(x,y,x_0,y_0)##. For Poisson's eq, let...

Again, from the Peskin and Schroeder's book, I can't quite see how this computation goes: See file attached The thing I don't get is how the term with (\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle vanishes, and also why they only get a \langle 0 | [\pi(x),\phi(y)] | 0 \rangle...

The normal form of Green's function is ##\oint_c\vec F\cdot \hat n dl'=\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy## I want to get to \oint _cMdy-Ndx=\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy Let ##\vec...

Homework Statement A force Fext(t) = F0[ 1−e(−αt) ] acts, for time t > 0, on an oscillator which is at rest at x=0 at time 0. The mass is m; the spring constant is k; and the damping force is −b x′. The parameters satisfy these relations: b = m q , k = 4 m q2 where q is a constant...

Green's function?? Physical interpretation?? Hi friends.. Can anyone help me to understand the physical interpretation of the green's function with help of some physical application example such as that from electrostatic?? I am unable to understand what is meant by linear operator in green...

I'm looking at scattering theory and eventually the Born approximation... In the notes I am reading it says we want to solve the Schrodinger equation written in the form: ##\left(\nabla ^2+k^2\right)\psi =V \psi## Of which there are two solutions, the homogeneous solution which tends to...

Hello, I was wondering what the use in the Green's function for the Klein-Gordon equation was, I have listed it below: \int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}e^{ip\cdot(x-x')} We find this gives an infinite result when the Klein gordon equation is applied to it and if x=x', what...

Homework Statement Hello guys. I've been stuck on a problem when searching for the Green function. Here is the problem: Find the solution of x^2 y''-2y=x for 1 \leq x < \infty with the boundary conditions y(1)=y(\infty ) =0, using the appropriate Green function.Homework Equations The general...

The problem is showing (□+m^2)<0| T(∅(x)∅(y)) |0> = -δ^4 (x-y) I know that it is relavent to Green's function, but the problem is that it should be alternatively solved without any information of Green's function, and using equal time commutation relations. Does Anyone know that?

I'm trying to derive the x-space result for the Green's function for the Klein-Gordon equation, but my complex analysis skills seems to be insufficient. The result should be: \begin{eqnarray} G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int...

Hi, It's about green's function in the book Messiah - Quantum Mechanics II - Chapter 16.3.2 (see http://books.google.de/books?id=OJ1XQ5hnINwC&pg=PA200&lpg=PA202&ots=NWr6A89Mkt&dq=messiah+quantenmechanik+kapitel+16.3&hl=de). The book actually is in german, but I guess that doesn't matter...

Graphene -- Green's function technique Hi, I am looking for a comprehensive review about using Matsubara Green's function technique for graphene (or at least some hints in the following problem). I have already learned some finite temperature Green's function technique, but only the basics...

Hi, I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: \nabla^2 D = f, in 3D the solution is: D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}', and in 2D the solution is: D(\mathbf...

While I was studying Ch 2.5 of Sakurai, I have a question about Green's function in time dependent schrodinger equation. (Specifically, page 110~111 are relevant to my question) Eq (2.5.7) and Eq (2.5.12) of Sakurai say \psi(x'',t) = \int d^3x' K(x'',t;x',t_0)\psi(x',t_0) and...

Hi, I have the following problem, I have an electric field (which no charge) which satisfies the usual Laplace equation: \frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}+\frac{\partial^{2}V}{\partial z^{2}}=0 in the region \mathbb{R}^{2}\times [\eta ,\infty ]. So...

Homework Statement Same problem as in https://www.physicsforums.com/showthread.php?t=589704 but instead of a spherical shape, consider an infinite line of constant charge density \lambda _0. Homework Equations Given in the link. The Attempt at a Solution I assume Phi will be the...

Homework Statement Arfken & Weber 9.7.2 - Show that \frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|} satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation. Homework Equations The Helmholtz operator is given by \nabla ^{2}A+k^{2}A...

Homework Statement Use the fundamental solution or Green function for the diffusion/heat equation in (-\infty, \infty ) to determine the fundamental solution to \frac{\partial u }{ \partial t } =k^2 \frac{\partial ^2 u }{ \partial x ^2 } in the semi-line (0, \infty ) with initial condition...

Hello, I am looking for a good reference book that has a detailed derivation of the single particle Green's function. I expected this to be in Sakurai but it's not. I couldn't find the spectral representation of this simple function in Ashcroft or any other solid state book either. Jackson...

Homework Statement Find the Green's Function for the Helmholtz Eqn in the cube 0≤x,y,z≤L by solving the equation: \nabla 2 u+k 2 u=δ(x-x') with u=0 on the surface of the cube This is problem 9-4 in Mathews and Walker Mathematical Methods of Physics Homework Equations Sines, they have the...

the definition of a green's function is: LG(x,s)=δ(x-s) the definition of a right inverse of a function f is: h(y)=x,f(x)=y→f°h=y how does it add up?

I was wondering if someone could help me go through a simple example in using Green's Function. Lets say: x' + x = f(t) with an initial condition of x(t=0,t')=0; Step 1 would be to re-write this as: G(t,t') + G(t,t') = \delta(t-t') then do you multiply by f(t')\ointdt' ? which I...

Homework Statement [PLAIN]http://img836.imageshack.us/img836/2479/stepvt.png Homework Equations H'(t) = \delta(t) The Attempt at a Solution So far I've taken the derivatives of G(x,t) with respect to xx and tt and gotten G_{xx}(x,t) = -\frac{θ^{2}}{c} and G_{tt}(x,t) = θ^{2}c...

Homework Statement So I'm trying to get a grip about those Green functions and how to aply them to solve differential equations. I've searched the forums and read the section on green's functions in my course book both once and twice, and I think I start to understand at least som of it...

Homework Statement Consider critically damped harmonic oscillator, driven by a force F(t) Find the green's function G(t,t') such that x(t) = ∫ dt' G(t,t')F(t') from 0 to T solves the equation of motion with x(0) =0 and x(T) =0Homework Equations x(t) = ∫ dt' G(t,t')F(t') from 0 to TThe Attempt...

Hi, i need help to find linear differential operator for the given green's function. please help. regards sunit

Hi all, I know that the electric field generated by a dipole is given by \mathbf{E}= [1-i(\omega/c) r]\frac{3 (\mathbf{p}\cdot\mathbf{r})\mathbf{r}-\mathbf{p} }{r^3}+(\omega/c)^2\frac{\mathbf{p}-(\mathbf{p}\cdot\mathbf{r})\mathbf{r}}{r} e^{i(\omega/c)r} where \mathbf{p} is the dipole's...

In 3 dimensions, how do I solve the following equation using the Green’s function technique? ∇2∇2φ(r) = ρ(r)

Hello everyone, In Fermi Liquid Theory, I'm trying to understand what the relationship is between the Green's function and the average occupancy of levels. In my lecture they gave the relation \left\langle n_k \right\rangle = G(k,\tau\rightarrow 0^+) Anyone know where this comes from...

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 &leq; x &leq; 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...

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...

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}...