Green's function Definition and 213 Threads

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...

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 differential...

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...

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 My question comes from problem 2 of this homework set, but is dependent on problem 1 of this same homework set. In problem 1 I used the method of images to find the potential everywhere in two dimensions due to an infinite uniform line charge located some distance from a...

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 =...

I am studying scattering from these notes. There I came across Green's function in one dimension which is computed as \langle x|G_o|x'\rangle = -\frac{iM}{\hbar ^2k}\exp(ik|x-x'|) I understand Green's function as a sort of propagator from x' to x. There are two observations that can be made...

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 I'm using the book "Principles of nano-optics" by Novotny and Hecht. I'm stuck a bit at understanding the derivation of the point spread function. It's just given as \mathbf{G} = \frac{\exp(i k_1 r)}{4 \pi r} \exp[-i k_1(x_0 x / r + y_0 y / r + z_0 z / r)]\\ \qquad\times...

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 I'm trying to show that the Green's function for the Laplace operator $-\nabla^2$ is badly behaved at infinity. I.e. $$\int d^dx|G(x,y)|^2=\infty$$ for d=1,2,3. What happens when d>4? I know the 1D Green's function is given by $$G(x,y)=-\frac{|x-y|}{2}$$ Homework...

Homework Statement L[y] = \frac{d^2y}{dx^2} Show that the Green's function for the boundary value problem with y(-1) = 0 and y(1) = 0 is given by G(x,y) = \frac{1}{2}(1-x)(1+y) for -1\leq y \leq x \leq 1\ G(x,y) = \frac{1}{2}(1+x)(1-y) for -1\leq x \leq y \leq...

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...

I am trying to find a Green's function for a third order ODE. (\lambda - d3/dx3 - \mu d/dx)u = f(x) with boundary conditions u(0) = uxx(0) = ux(L) = 0 (L is finite). This has solution u(x) = c1em1x + c2em2x + c3em3x where m1, m2 and m3 are the roots of the characteristic equation from...

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 Obtain the Green's function for BVP (and use it to express the solution for the given data): -y''(x) = f(x), 0 < x < 1, y'(0) = a, y(1) = b Homework Equations The Attempt at a Solution I have found 2 solutions to the homogeneous equation y1(x) = ax...

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...

In the x-y plane, we have the equation \nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0}) with \Psi = 0 at the rectangular boundaries, of size L. A paper I'm looking at says that for R^{2} = (x-x_{0})^{2} + (y-y_{0})^{2} << L^{2} , that is, for points...