What is Green's function: Definition and 213 Discussions
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is the linear differential operator, then
the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f ), where G is the Green's function.Through the superposition principle, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.
Let's say we want to calculate the two-point Green's function for a fermion to a given order for a two particle interaction of the form ##U(x,y)=U(y,x)##. For the first order calculation we have to do all contractions related to...
I am trying to understand Green's functions in many-body theory for condensed matter. After much struggle, I managed to calculate my first diagrammatic expansion. However I am perplexed by getting more of the usual results.
The Hartree–Fock energy result I know from second quantization can be...
we know that, using the Green's identity ##\iiint\limits_V (\varphi \Delta\psi -\psi \Delta\varphi)\ dV =\iint_{\partial V} (\varphi \frac {\partial \psi}{\partial n}-\psi \frac {\partial\varphi}{\partial n})\ da## and substituting ##\varphi=\phi## and ##\psi=G## here, we can write the potential...
From the table of Green functions on Wikipedia we can get the generic 2-D Green's function for the Laplacian operator. But how would one apply boundary conditions like u = 0 along a rectangular boundary? Would we visualize a sort of rectangle-based, tilted pyramid, with logarithmically changing...
Doing some revision and getting confused. It's under GR but may as well be under electromagnetism or calculus because that is where the problem is. Taking a shell of mass ##\rho = M\delta(r-R)/(4\pi R^2)## and four velocity corresponding to rotation about ##z## axis i.e. ##U = (1, -\omega y...
Here is the conclusion of the derivation in question:
where ##\phi_n## are eigenfunctions of the Hamiltonian.
I don't see how at the very end the ##\sum ...## becomes ##\delta (x-y)##. What do I miss?
I started from eq(3.113) and (3.114) of Peskin and merge them with upper relation for $S_F$, as following:
\begin{align}
S_F(x-y) &=
\theta(x^0-y^0)(i \partial_x +m) D(x-y) -\theta(y^0-x^0)(i \partial_x -m) D(y-x) \\
&= \theta(x^0-y^0)(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >...
Hi all,
Consider the following Green's function:
where ##\Theta(t)## is the Heaviside step function and ##\tilde{\Theta}(t)## is defined as
I want to understand the following calculation:
More specifically, the ##\text{Im}(G(\textbf{k},t)G(\textbf{k},-t))## from the first line to the second...
I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers.
##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also...
I'm reading about fundamental solutions to differential operators in Ordinary Differential Equations by Andersson and Böiers. There is a remark that succeeds a theorem that I struggle with verifying. First, the theorem:
If the leading coefficient in ##(1)## is not ##1## but ##a_n(t)##, then...
I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
Hi wizards,
I'm working through Jackson's book on E&M (3rd edition) and got stuck in section 3.12 on expansions of Green functions. I have three questions regarding section 3.12:
First, why is Jackson trying to find a Green function that satisfies equation 3.156? To my beginner mind, it...
Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...
The heat conduction equation for a semi-infinite slab with a boundary condition of the first kind is as follows:
The problem is delta is a very small number, so the first exponential will tend to infinity. I am programming this in Fortran and it can accommodate values up to magnitude of 310...
So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force.
$$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$
$$\nabla \cdot \textbf{u}=0$$
We can apply the...
Hi PF!
Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function?
This is the...
Hi PF!
I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this
NIntegrate[-(1/((-1.` + x)^2 (1.` + x)^2 (1.` + y)^2))
3.9787262092516675`*^14 (3.9999999999999907` +
x (-14.99999999999903` +
x (20.00000000000097` -...
Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is:
$$
u(x, t) = \int \phi(x-y, t)f(y)dy
$$
where ##\phi(x, t)## is the heat kernel.
The integral looks a lot similar to using Green's function to solve differential...
Good afternoon!
I am writing with such a problem, I hope to find someone who could help me. I'm almost desperate! So, there is such a thing as the Braess paradox, this is a classic paradox for roads and power grids, and there is also such an article...
This is from Evans page 37. I seem to be missing a basic but perhaps subtle point.
Definition. Green's function for the half-space ##\mathbb{R}^n_+,## is
\begin{gather*}
G(x,y) = \Phi(y-x) - \Phi(y-\tilde{x}) \qquad x,y \in \mathbb{R}^n_+, \quad x \neq y.
\end{gather*}
What's the proper way to...
I am studying the 'toy' Lagrangian (Quantum Field Theory In a Nutshell by A.Zee).
$$\mathcal{L} = - \frac{1}{4} F_{\mu \nu}F^{\mu \nu} + \frac{m^2}{2}A_{\mu}A^{\mu}$$
Which assumes a massive photon (which is of course not what it is experimentally observed; photons are massless).
The...
First off let me say I am a bit confused by this question.
Searching for some references I found the following related to the KG propagator, given by (P&S, chapter 2 pages 29, 30)
Then they Fourier-transformed the KG propagator
Is this what is aimed with this exercise? If yes, could you...
I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...
Let's say you have a tensor u with the following components:
$$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$
Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it...
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
Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and...
Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program.
i - index enumerating the nodes along t-coordinate,
j - along the r-coordinate,
k - along the theta-coordinate,
l - along the phi-coordinate.
N_t - the number of nodes along t-coordinate.
N_r...
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...
Very often, the term "Green's function" is used more than "correlations" in QFT. For example, the notation:
$$<\Omega|T\{...\}|\Omega> =: <...>$$
appears in Schwartz's QFT book. And it seems very natural, basically because the path integral definition of those terms "looks like" the...
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?
$$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
Homework Statement
We have two semi-infinite coplanar planes defined by z=0, one corresponding to x<0 set at potential zero, and one corresponding to x> set to potential ##V_0##.
a) Find the Green function for the potential in this region
b) Find the potential ##\Phi(r)## for all points in...
Homework Statement
I am trying to fill in the gaps of a calculation (computing the deflection potential ##\psi##) in this paper:
http://adsabs.harvard.edu/abs/1994A%26A...284..285K
We have the Poisson equation:
##\frac{1}{x}\frac{\partial}{\partial x} \left( x \frac{\partial \psi}{\partial...
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,
$$\Psi(t) =...
Homework Statement
I try to integral as picture 1.
The result that is found by me, it doesn't satisfy Green's function for boundary value problem.
Homework EquationsThe Attempt at a Solution
show in picture 2 & picture 3.
Dear all,
Need your suggestions as to how I can arrive at the expression for the Dyadic Green's function.
The scalar case is simple:
Consider the standard equation of motion in Fourier space: ## \omega^2 \hat{x}(\omega) = \omega_0^2 \hat{x}(\omega) - i \delta \omega \hat{x}(\omega)+ F(\omega)...
First, is it suitable to solve a Green's function by one-order self-energy, since it only consider partial high order perturbation, so it's unclear that this calculation corresponding to which order perturbation. In other word, if one wants to use self-energy to get Green's function, he should...
Homework Statement
Find the Green's function for
$$f''(x) + \cos^2 a f(x) = 0;\\
\pm f'(x) + \cos a \cot a f(x)|_{x=x_0(a)}=0$$
where ##a## is a parameter and ##x_0## is defined as
$$x_0(a) = \sec a\arcsin(\cos a)$$.
Homework Equations
Standard variation of parameters
The Attempt at a...
Find Green's function of $$K(f(x)) = (1-x^2)f''(x)-2xf'(x)+\left(2-\frac{1}{1-x^2}\right)f(x):x\in[cos(\alpha),1]$$
subject to boundary conditions: $$f|_{x=1} < \infty\\
f|_{x=\cos(\alpha)} = 0.$$
Two fundamental solutions are associated Legendre polynomials (after all, this is Legendre's...
Homework Statement
We have long wire with constant charge density that is put inside a grounded metal housing with a shape of cylindrical section (a ≤ r ≤ b and 0 ≤ ϕ ≤ α). We need to find potential inside the box.
2. Homework Equations
Δf=-(μ/ε0)*∂^2(r), where μ is linear charge density...
I am solving the Laplace equation in 3D:
\nabla^{2}V=0
I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are:
V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
Homework Statement
Find Green's function of $$K(\phi(s)) = \phi''(s)+\cot(s)\phi'(s)+\left(2-\frac{1}{\sin(s)^2}\right)\phi(s):s\in[0,\alpha]$$
subject to boundary conditions: $$\phi|_{s=0} < \infty\\
\phi|_{s=\alpha} = 0.$$
Homework Equations
Green's function ##G## is found via variation of...
Hi PF!
Given operator ##B## defined as $$ B[u(s)] = c u(s) - u''(s) - \frac{1}{2 s_0}\int_{-s_0}^{s_0}(c u(s) - u''(s))\, ds$$ I'm trying to find it's inverse operator ##B^{-1}##. The journal I'm reading states ##B^{-1}## is an integral operator $$B^{-1}(u(s)) =...
Homework Statement [/B]
Determine the Green's functions for the two-point boundary value problem u''(x) = f(x) on 0 < x < 1 with a Neumann boundary condition at x = 0 and a Dirichlet condition at x = 1, i.e, find the function G(x; x) solving
u''(x) = delta(x - xbar) (the Dirac delta...
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
Find Green's function of ##u''+u=f##.
Homework Equations
What we all know.
The Attempt at a Solution
Let Greens function be ##G##. Then ##G''+G=\delta(x-x_0)##. This admits solutions superimposed of sine and cosine. Let's split the function at ##x=x_0##. Then we require...
I have this BVP $$u''+u' =f(x)-\lambda |u(x)| $$, ##x\in [0,1]## we BC ## u(0)=u(1)=0##.
Following an ''algorithm'' for calculating the green's function I got something like $$g(x,t)=\Theta(x-t)(1+e^{t-x}) + \frac{e^{t}-e}{e-1} +\frac{e-e^{t}}{e-1}e^{-x}$$. At some point there is this integral...
Homework Statement
Hi there,
I was reading a book discussing on the topic of compact Green's function in 2D. However,I have been stuck for a while on some mathematical manipulations depicted below.
Homework EquationsThe Attempt at a Solution
In 2nd box,I guess the author was trying to pull out...