Hi,everyone. Recently, I am studying green's function in many body physics and suffer from trouble.Following are my problems.
(1) What is the origin of the definition of green's function in many body physics?
(2) What is the physical meaning of self energy ? It seems like it is the correction...
I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get
g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0}
But I can't seem to get that to work out.
Using the Cauchy integral formula above, I take one pole at -w_0 and get
\frac{ie^{i\omega_0(t-t')}}{2\omega_0}
and...
Why can't G and its derivative be continuous in the relation below?
$$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
I am solving the Laplace equation in 3D:
\nabla^{2}V=0
I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are:
V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
Homework Statement
Find out the Green's function, ##G(\vec{r}, \vec{r}')##, for the following partial differential equation:
$$\left(-2\frac{\partial ^2}{\partial t \partial x} + \frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \right) F(\vec{r}) = g(\vec{r})$$
Here ##\vec{r}...
Homework Statement
Hi all,
I came across these steps in my notes, relating to a step whereby,
$$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$
and performing the following operation on ##\hat{G}## gives the following expression,
$$[\frac{\partial}{\partial t}...
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...
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
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...
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...
Dear All,
I am trying to understand what operators actually mean when deriving the definition of green's function. Is this integral representation of an operator in the ##x-basis## correct ?
## D = <x|\int dx|D|x>##
I am asking this because the identity operator for non-denumerable or...
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...
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)$$ (where ##a## is an arbitrary constant shift.)
My question is, given such a translationally invariant system, how does one...
Concerning green's function with the same time and spatial argument(i.e. ##G_0(x,t;x,t)##, mostly in QFT), I have the following question
1. Is green's function well defined at this point?
2. if green's function is well defined at this point, is it continuous here?
3. In quantum many body...
Regarding interacting green's function, I found two different description:
1. usually in QFT:
<\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>}
2. usually in quantum many body systems...
Homework Statement
This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is:
-- The problem is to show that
##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x')...
Homework Statement
This comes up in the context of Poisson's equation
Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \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
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...