Green function Definition and 66 Threads

The lesser Green's function is defined as: $$G^{<}(t,t')=i\langle C_{\nu}^{\dagger}(t')C_{\nu}(t)\rangle=i\bra{n}C_{\nu}^{\dagger}(t')C_{\nu}(t)\ket{n}$$ where ##\ket{n}## is the many particle ground state...

I am trying to understand some introductory material on applications of Green's functions and the book which I am following uses the example of an electric circuit subjected to multiple voltage impulses. This is the Google drive viewing link to the few pages I am referring to: . I had to post...

Honestly, I have no real idea. I know for sure the equation connects the initial state ##|\Psi^N_0>##to a final state ##|\Psi^{N+1}_0>##, ##E## is the energy and ##E^N_0## etc are the energy of the initial state and final state. I also know that these energy are related to conduction band and...

The Wikipedia article on Green's functions says, What would be a really simple physical toy problem that would motivate and explain the concept of advanced Green's functions?

First time in PF, I am sorry if I did not choose the right category. I have been doing theory in condensed matter (mostly numerics) as a PhD but I never got to learn proper quantum field theory (QFT). Aside from a few introductory courses at university, I never learned what is a many-body...

Hi guys! I am trying to derive the Hedin's equations used for GW on my own, and I found this equation, but I cannot really derive it, nor find a source where they explain how this can be derived (they link to other papers that in the end don't show where this is coming from). The equation I am...

As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...

Hey :) Firstly I want to thank everyone who takes their time to read through this post and who tries to help me. So the issue is the following: I wrote a python code that creates a Lindbladian, and I wanted to try to calculate the Greens function using the Lehmann representation. For the...

Hi everyone. Using the Green function, I want to obtain the density of states of a one-dimensional (linear) lattice. Depending on the problem conditions, we will have an iterative loop with 4,000 data for the energy component and a iteration loop with 2,000 data for the wave number component. In...

A conducting sphere of radius a is made up of 3 different shells, upper part a/2 z -a/2, the middle part -a/2 z a, the lower part -a z -a/2 where the center of the sphere is at the origin. The upper part V and the lower part has -V potentials, while the mid part is grounded. Find the...

could anyone explain why in the page of book this figure is related to hartree-fock? I mean why if t1>t2 we have these possibilities? and why not particle propagate from x2t2 to x3t3 instead x3t3+?

I was wondering if there is a way to deduce the solution of the potential of a charge outside a sphere given by the image method, though Green functions. Because of a Dirichlet condition (GD(R,r')=0), I know that a solution can be written as GD=Go+L, where ∇2L=0. But in order to approach this...

I was read this article(https://engineering.purdue.edu/wcchew/ece604f19/Lecture%20Notes/Lect31.pdf). I was read this paper about Huygens' principle(https://engineering.purdue.edu/wcchew/ece604f19/Lecture%20Notes/Lect31.pdf) Main idea of Huygens' principle is how wave function ##ψ(r)##...

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...

Homework Statement: I do know how to solve the resistance network problem in two dimensions. However, in this problem they want it in 3 dimensions and higher and I don't know how to do that Homework Equations: - In the picture you can see the solution to the two dimensional version

I came across an example of a solution to finding the potential of a charged layer using the Green function (here, pdf). The standard algorithm for finding the Green function by boundary conditions for many problems is understandable: \begin{align*} G_\mathrm{Left} = Ax+ B \\ G_\mathrm{Right} =...

The Feynman propagator: $$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$ is the Green's function of the operator (except maybe for a constant): $$ (\Box + m^2)$$ In other words: $$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$ My question is: Which is the operator that...

We want to solve the equation. $$H\Psi = i\hbar\frac{\partial \Psi}{\partial t} $$ (1) If we solve the following equation for G $$(H-i\hbar\frac{\partial }{\partial t})G(t,t_{0}) \Psi(t_{0}) = -i\hbar\delta(t-t_{0})$$ (2) The final solution for our wave function is, $$\Psi(t) =...

Hello, If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating...

Homework Statement We have long wire with constant charge density that is put inside a grounded metal housing with a shape of cylindrical section (a ≤ r ≤ b and 0 ≤ ϕ ≤ α). We need to find potential inside the box. 2. Homework Equations Δf=-(μ/ε0)*∂^2(r), where μ is linear charge density...

I am trying to normalize 4x4 matrix (g and f are functions): \begin{equation} G=\begin{matrix} (1-g^2) &0& 0& 0&\\ 0& (1+f^2)& (-g^2-f^2)& 0 \\ 0 &(-g^2-f^2)& (1+f^2)& 0 &\\ 0& 0& 0& (1-g^2) \end{matrix} \end{equation} It's a matrix that's in a research paper (which I don't have) which gives...

Hello, Before I begin, a lot of the math I try to describe in this post is stuff I worked out myself and have scribbled down. If something looks fishy or doesn't make sense, it could very well be totally wrong! Apologies for any confusions in advance. Consider a slab of linear isotropic...

Homework Statement [/B] We are heating a semi-infinite slab with a laser (radius of a stream is ##a##), which presents us with a steady surface heating (at ##z=0##), everywhere else on the surface the slab is isolated. How does the temperature change with time? Look at the limit cases: at ##t...

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...

Hello, I'm taking a course in electrostatics and electrodynamics. We learned about finding a potentional using unique Green functions that are dependent of the geometry of the problem. Specificly on a Dirichlet problem we get the solution: Φ(x)=∫ρ(x')G(x,x')d3x' -...

I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is $$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4...

Homework Statement I need to solve a problem like Jackson 3.18. I need to find potential due to the same configuration but the position of two plates is opposite i.e. Plate at Z=0 contains disc with potential V and plate at Z=0 is grounded. Homework EquationsThe Attempt at a Solution I think...

In the context of a project, I had to solve numerically Poisson equation with cylindrical coordinates. I put here results for z = 0 on a 3D mesh 256x256x256. Below 1 figure representing the final solution (in absolute value) in the case of a galaxy; I use the CGS units for the potential. I...

Hi everybody. I have a (1×N) Green function in MATLAB. I want to use the FFt function for Green function to transform the time domain. Does the FFT command work correctly for Green function using: A=fft(Green function, nfft). Here nfft is the number of transformed points. Is it necessary that...

I have a 1×N Green function in energy domain. I want to use the FFT (fast Fourier transform) for this Green function in MATLAB. But this function is non-periodic. Could you help me about this? How can I change the energy interval to convert Green function as a periodic function?

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## ...

Hello, i don't understand how i integrate the star equation at the figure. Fot example \int d x_1 \int d x_2 \frac{d^2G}{d x_1} = \int d x_1 \frac{d^2G}{d x_1} \int d x_2 but \int d x_2 x_2|^\epsilon_{-\epsilon}=0 ...sure this is a error, but i don't understand how i find the answer...

Hello everyone: This is what I read in a paper, the spontaneous emission rate written by Fermi's Golden Rule is just related to the local transverse electric field. Dose anyone can explain to me what's the meaning of "transverse mode" here? Why the emission is not related to longitudinal...

Homework Statement The problem requires to solve the integration to find ## G(t) ## after ##G(\omega)## is found via Fourier transform. We have G(\omega)= \frac{1}{2\pi}\frac{1}{\omega _{0}^2 - \omega ^2} Homework Equations As mentioned previously, the question asks to find ##G(t)## The...

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...

hi, I need some reading materials on green function for SHO. my instructor provided a GF frequency and wanted us to find the deformation of poles , boundary conditions for the function. I need to know which mathematical background should I have to solve this. any useful material suggestion will...

I know that in the vector potential formulation one can use a scalar Green function (to find the said potential and from then on the electric and magnetic fields), and that this works because the components of the potential are in the same direction as those of the source - i.e. a current in the...

Hi everyone. I have a very quick question. Can someone tell me how to compute the energy dimensions of an n-point Green function. Consider for example a \lambda\phi^4 scalar theory. I know that the dimensions of an n-pt Green function are 4-n (or something like that). How do I prove it? Thanks

Homework Statement The generalized Green function is $$ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. $$ Show G_g satisfies the equation $$ (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x') $$ where \delta(x - x') = \frac{2}{\ell}\sum_{n =...

I had made a post in the past about the same problem and unfortunately I wasn't clear enough so I am trying again. I am studying an article and there I found without any proof that the solution of: Let ##g \in \mathbb{C}## and let ##u:(0,\infty)\to \mathbb{C}## $$ -u''+\lambda^2u=f\,\, on...

hello all ! my teacher told me to do a research on examples of problems that has connection with green function on solving differential equations (with programmed numerical solutions) in my final year project , can you give me such problems to work on as an undergraduate ? , thank you !

Homework Statement I would like to calculate the electric potential in all the space in the following set up: Conductor sphere of radius a whose surface is kept at a zero potential. 1 point charge ##q_1## at distance ##d_1## from the center of the sphere. 1 point charge ##q_2## at a distance...

Hello there, I would like to obtain the Green function for the operator F, F [u(x)] = u '''', and the boundary conditions u(0) = u'(0) = u (1) = u' (1) = 0. I am looking for a function G ( x, s ) such that G'''' (x,s) = delta (x-s) (the apecis referring to differentiation w.r.t. x, and...

Hello, Can anyone explain to me the next settence, found http://relativity.livingreviews.org/open?pubNo=lrr-2011-7&amp;page=articlese16.html in section 1.4: "The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the retarded Green’s function...

Homework Statement The problem is attached in the first picture, the provided solutions are in the second.The Attempt at a Solution I got to where they are, but aren't they missing an additional term of sin(t)*cos(∏)*f(∏) from the second integral in dx/dt ?

Homework Statement I'm trying to self study Green function and I can't follow the very last step of a demonstration in an online notes (that I attach in this post). Page 7 to 8. Basically he says that from G_{tt}(t,t')+\omega G(t,t')=0 for all t>t' with the conditions G(t,t'+\varepsilon)=0...

If we have Green function g(x,s)=exp[-\int^x_s p(z)dz] we want to think about that as distribution so we multiply it with Heaviside step function g(x,s)=H(x-s)exp[-\int^x_s p(z)dz] Why we can just multiply it with step function and tell that the function is the same. Tnx for the answer.

Please teach me this: Why the electromagnetic potential obeys the Callan-Symazik equation in renormalization group theory like propagator functions.By the way,are there any relation between classical potential and interaction Haminton(the product of different field operators). Thank you very...