Finding free electron gas Green function in Fourier space

In summary, the conversation discusses the use of definitions and quotes from a book to explain the concept of a free electron gas. The speaker expresses confusion about the use of exponential terms and step functions in the equations, seeking clarification on how to deal with them. They then ask for help in understanding the concept in relation to the ground state of the system at a temperature of 0 Kelvin.
  • #1
kakaho345
5
0
Homework Statement
See below
Relevant Equations
See below
As in title:
1681350014004.png


Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971:
1681350111348.png

1681350169049.png

1681350186563.png

1681350198430.png

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 here:
1681350443696.png

I understand that ##e^{ik\cdot(x-x')}## is from terms like ##\psi=e^{ikx}c##, however, the term ##e^{-i\omega_k(t-t')}## the sign doesn't look right to me for the two time region should have different signs in the exponential. Also, I don't know how to deal with the exponential sandwiched between the field operator. The step function in time is from the two pieces of time regions, but I am not sure on the step function in k. I may be from the filled Fermi sea.

I understand this is a very simple question. However, I have been sitting whole day dealing with this. Any help will be appreciated.
 
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  • #2
You simply have to think about, how the ground state looks like! Note that at ##T=0## the system is in a pure state of lowest possible energy under the given constraints. First think what is the constraint here!
 

1. What is a free electron gas Green function?

A free electron gas Green function is a mathematical tool used in quantum mechanics to describe the behavior of a system of free electrons. It represents the probability amplitude for an electron to move from one point to another in the system.

2. What is the significance of finding the Green function in Fourier space?

Finding the Green function in Fourier space allows for a more efficient and accurate calculation of the behavior of the free electron gas system. It takes advantage of the properties of Fourier transforms, making it easier to solve complex problems.

3. How is the free electron gas Green function derived?

The free electron gas Green function is derived using the Schrödinger equation and the concept of a propagator, which describes the time evolution of a quantum system. It involves solving a set of differential equations and applying boundary conditions to obtain the final form.

4. What are the applications of the free electron gas Green function?

The free electron gas Green function has various applications in condensed matter physics, such as studying the electronic properties of metals and semiconductors, as well as in quantum field theory. It is also used in the calculation of transport properties, such as conductivity, in materials.

5. Are there any limitations to using the free electron gas Green function?

While the free electron gas Green function is a powerful tool, it is limited to systems of non-interacting electrons. It also assumes that the electrons behave as free particles, which may not always be the case in real materials. Additionally, it does not take into account relativistic effects, so it is not suitable for systems involving high energies.

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