Dyson's equation and Green's functions

  • A
  • Thread starter Yiheng Xu
  • Start date
  • Tags
    Functions
In summary, the Dyson's equation is basis-independent and does not require a specific basis set. However, the Green's functions are typically defined in the position basis and completeness relations must be used to convert them to other basis sets. If the basis functions are not orthogonal, additional steps must be taken to ensure proper use of the completeness relations.
  • #1
Yiheng Xu
1
0
Hi,
Is the Dyson's equation basis independent (for instance, I construct the basis set where the elements are atomic orbitals and those orbitals are non-orthogonal) ?

What is the unperturbed retarded Green's function for one-particle case in matrix notation if the basis functions are not orthogonal?

Thank you!
 
Physics news on Phys.org
  • #2
I'm pretty sure the general form of Dyson's equation is basis-independent (there is no reference to a basis).
$$G(\mathbf{k},\omega) = G_0(\mathbf{k},\omega)+G_0(\mathbf{k},\omega)\Sigma(\mathbf{k},\omega)G(\mathbf{k},\omega)$$
But the Green's functions are typically defined in the position basis, so you have to use completeness relations to go to the atomic orbital basis.

Edit: So if you have
$$i\hbar G^+ (\mathbf{x}_2,t_2;\mathbf{x}_1,t_1) = \theta(t_2-t_1)\langle\mathbf{x}_2|U(t_2,t_1)|\mathbf{x}_1\rangle$$
and you want to work in the energy eigenbasis ##\{\phi_n\}##, you need to insert the completeness relations in the appropriate places:
$$i\hbar G^+ (\mathbf{x}_2,t_2;\mathbf{x}_1,t_1) = \theta(t_2-t_1)\sum_{m,n}\langle\mathbf{x}_2|\phi_m\rangle\langle \phi_m|U(t_2,t_1)|\phi_n\rangle\langle \phi_n|\mathbf{x}_1\rangle$$
and since ##\langle\mathbf{x}_2|\phi_m\rangle = \phi_m(\mathbf{x}_2)##, we can rewrite this as:
$$i\hbar G^+ (\mathbf{x}_2,t_2;\mathbf{x}_1,t_1) = \theta(t_2-t_1)\sum_{m,n}\phi_m(\mathbf{x}_2)\phi_n^*(\mathbf{x}_1)\langle \phi_m|U(t_2,t_1)|\phi_n\rangle$$

Further edit (based on post #6 below): you don't have to use an orthogonal basis if you don't want to, but you can't use the completeness relations as I illustrated above. You would have to orthogonalize the non-orthogonal basis for use in the completeness relations.
 
Last edited:

1. What is Dyson's equation?

Dyson's equation is a mathematical formula that describes the relationship between a particle's propagation in a medium and its interactions with that medium. It is commonly used in quantum mechanics to calculate the behavior of particles in a many-body system.

2. How is Dyson's equation related to Green's functions?

Dyson's equation is a type of integral equation, where the solution is expressed in terms of an operator acting on a Green's function. The Green's function represents the propagation of a particle in a medium, and the operator represents the interactions with that medium. Therefore, Dyson's equation can be thought of as a way to solve for the Green's function in a given system.

3. What is the significance of Green's functions in physics?

Green's functions are important in physics because they allow us to solve complex problems involving interactions between particles in a medium. They provide a convenient way to calculate the response of a system to an external perturbation, such as an applied force or potential.

4. How is Dyson's equation used in practice?

In practice, Dyson's equation is used to calculate the behavior of particles in a many-body system, such as atoms in a solid or molecules in a liquid. It is often used in conjunction with other techniques, such as perturbation theory or numerical methods, to solve for the Green's function in a given system.

5. Are there any limitations to using Dyson's equation and Green's functions?

Like any mathematical model, Dyson's equation and Green's functions have their limitations. They are most useful for studying non-relativistic systems, where the particles are moving at speeds much slower than the speed of light. Additionally, they may not be accurate in extreme conditions, such as at very high energies or densities.

Similar threads

  • Atomic and Condensed Matter
Replies
1
Views
911
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Calculus
Replies
8
Views
813
Replies
7
Views
751
  • Beyond the Standard Models
Replies
26
Views
428
  • Atomic and Condensed Matter
Replies
0
Views
247
Replies
3
Views
1K
  • Quantum Physics
Replies
1
Views
581
Replies
9
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
1K
Back
Top