# How Does the Two-Particle Green Function Relate to Hartree-Fock Theory?

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• Rzbs
In summary, this figure is related to Hartree-Fock because if you write ##G_{12} = G_1G_2## you can get HF equations.
Rzbs
TL;DR Summary
relation between two-particle green function and Hartree-Fock
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+?

Because you can show that if you write ##G_{12} = G_1G_2## you can get HF equations. It's not difficult but it's not easy as well. I can look up my old notes if you are interested in a formal proof.

Rzbs
dRic2 said:
Because you can show that if you write ##G_{12} = G_1G_2## you can get HF equations. It's not difficult but it's not easy as well. I can look up my old notes if you are interested in a formal proof.
I really want to why these spliting of G2 result in HF and I want to match the graphs to relation 6.62 but I think the right-up graph does'n match to any of G1s.
I will be grateful if you check your notes or introduce a book with similar notation of this book: Inkson, manybody theory of solids.

fateme said:
but I think the right-up graph does'n match to any of G1s.
It looks like an error to me too, but don't take my world without checking for yourself.

Anyway I'll be back this afternoon or tomorrow because I need some time to find and make sense of my notes again

Thanks again for replying.

Last edited:
Thanks a lot but I think it's not the thing I'm looking for. Because I am new in many-body physics I want a simple approach and according to Inkson's book there are two approaches "equation of motion method ...more physically understandable than the use of wick's theorem" . And many of books I think use wick theorem, whereas I need another method.
So I think I need a proof for HF like its proof for Hartree potential.

The article I linked says exactly the same things as the one you posted. What is not clear to you ?

Actually I didn't follow the proof and it's notations. I can't understand the relation between the HF formula that was in the first chapter and the HF formula in terms of green function; It's not clear to me. I want to start with 6.66 eq. and reach to 1.30 or 1.31 eq. It's not clear for me at all how can I do this.

## 1. What is a Two-particle Green function?

A Two-particle Green function, also known as a two-point correlation function, is a mathematical tool used in quantum mechanics to describe the behavior of two interacting particles. It is a function that describes the probability of finding two particles at specific positions at a given time.

## 2. How is the Two-particle Green function used in physics?

The Two-particle Green function is used to calculate the correlation between two particles in a quantum system. It is used to study the behavior of particles in many-body systems, such as solids, liquids, and gases. It is also used in the study of quantum field theory and statistical mechanics.

## 3. What information can be obtained from the Two-particle Green function?

The Two-particle Green function provides information about the correlation between two particles in a system. It can be used to calculate the energy levels, momentum distribution, and other properties of the system. It can also be used to study the interactions between particles and the effects of external forces.

## 4. How is the Two-particle Green function calculated?

The Two-particle Green function is calculated using mathematical techniques such as perturbation theory and Feynman diagrams. It involves solving complex equations and integrals to obtain the correlation function between the two particles. Advanced computational methods are often used to calculate the Green function for complex systems.

## 5. What are some applications of the Two-particle Green function?

The Two-particle Green function has many applications in physics, including the study of electronic and magnetic properties of materials, the behavior of particles in high-energy collisions, and the dynamics of quantum systems. It is also used in the development of new technologies, such as quantum computing and nanotechnology.

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