Linked Questions

3 votes
1 answer
4k views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\...
user avatar
29 votes
2 answers
8k views

How to interpret correlation functions in QFT?

I'm fairly new to the subject of quantum field theory (QFT), and I'm having trouble intuitively grasping what a n-point correlation function physically describes. For example, consider the 2-point ...
user avatar
  • 2,827
22 votes
1 answer
5k views

Propagators, Green’s functions, path integrals and transition amplitudes in quantum mechanics and quantum field theory

I’m trying to make a simple conceptual map regarding the things in the title, and I'm finding that I’m a little perplexed about a couple of items. Let me summarize a few things I regard as being true, ...
user avatar
7 votes
3 answers
8k views

Susceptibilities and response functions

It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very ...
user avatar
  • 4,235
13 votes
2 answers
2k views

Differential equation (Greens function) satisfied by the kernel using path integrals

I'm reading Feynman and Hibbs, Quantum Mechanics and Path Integrals. How do I show that the kernel $$\tag{2-25} K(x_2 ,t_2;x_1, t_1)=\int_{x=x_1}^{x=x_2}\mathcal{D}x~ e^{\frac{i}{\hbar}S[2,1]}$$ ...
user avatar
14 votes
1 answer
2k views

Dirac Delta in definition of Green function

For a inhomogeneous differential equation of the following form $$\hat{L}u(x) = \rho(x) ,$$ the general solution may be written in terms of the Green function, $$u(x) = \int dx' G(x;x')\rho(x'),$$ ...
user avatar
  • 1,950
24 votes
1 answer
12k views

How exactly is the propagator a Green's function for the Schrodinger equation

Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...
user avatar
  • 1,872
7 votes
3 answers
4k views

What's the difference between "boundary value problems" and "initial value problems"?

Mathematically speaking, is there any essential difference between initial value problems and boundary value problems? The specification of the values of a function $f$ and the "velocities" $\frac{\...
user avatar
  • 3,482
8 votes
1 answer
3k views

What is different between resolvent and green function

I bumped into a book, where Resolvent $R^{\pm}(E)$ is defined as $e^{\mp iHt/\hbar}=\pm\frac{i}{2\pi}\int_{-\infty}^{\infty}dER^{\pm}(E)e^{\mp iEt/\hbar}$ and $R^{\pm}(E)=\frac{1}{\pm i\hbar}\int_0^{...
user avatar
  • 1,025
11 votes
2 answers
2k views

Where is the Feynman Green's function in quantum mechanics?

In quantum field theory, the Feynman/time ordered Green's function takes the form $$D_F(p) \sim \frac{1}{p^2 - m^2 + i \epsilon}$$ and the $i \epsilon$ reflects the fact that the Green's function is ...
user avatar
  • 95.5k
6 votes
1 answer
2k views

Causal propagator and Feynman propagator

I have some questions about the Green’s function of the Klein-Gordon operator and the Feynman propagator. The first is about retarded Green’s function: \begin{eqnarray} \int_{-\infty}^\infty\frac{d^...
user avatar
  • 477
2 votes
2 answers
2k views

Green functions, Propagators and probability amplitude that a particle propagates

(The original post has been copiously edited to make it more clear but it still precisely corresponds to what I had intended to ask) My QFT knowledge has very much rusted and i got confused by these ...
user avatar
  • 639
8 votes
1 answer
908 views

Are propagators in QFT really Green functions?

In many textbooks the notions Green function and propagator are used interchangeably. But are they really the same thing? This popular answer argues that a retarded propagator function $D_R(x,t,x',t')...
user avatar
  • 9,301
19 votes
0 answers
356 views

Definition of vacua in QFT in generic spacetimes

I have been learning QFT in curved spaces from various sources (Birrell/Davies, Tom/Parker, some papers), and one thing that confuses me the most is the choice of vacua in various spacetimes, and the ...
user avatar
  • 1,433
3 votes
1 answer
704 views

Single-particle Green's function

Define a single-particle Green's function as \begin{equation} i\hbar G(xt;x't') = \langle x| e^{-iH(t-t')/\hbar} | x'\rangle. \end{equation} By inserting the completeness relation, we have \begin{...
user avatar

15 30 50 per page