Green's function expansion in a set of eigenfunction

In summary, the conversation discusses the problem of decomposing the Green function into a set of eigenfunctions. The Green function is given as G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|} and the goal is to express it in terms of cylindrical mode eigenfunctions \Psi ( \vec r; k) = H_m ( q r) sin( h z) e^{i m \phi}, where k^2 = q^2 + h^2 and h = \frac { \pi } {2 L}. The conversation mentions that Hankel's function of the first kind,
  • #1
kristobal hunta
7
0
Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function
[tex] G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|} [/tex]
and I need to express it into series of cylindrical mode eigenfunctions
[tex] \Psi ( \vec r; k) = H_m ( q r) sin( h z) e^{i m \phi} [/tex]
[tex] k^2 = q^2 + h^2, h = \frac { \pi } {2 L} [/tex]

here H - Hankel's function of the first kind.
Eigenfunction forms a complete set, with discrete spectrum of eigenvalues q and h.
I know that we can decompose the Green function into set of eigenfunctions, but I have the Green function for spherical representation, and eigenfunctions are from waveguide formed by two infinite plates parallel to each other. I couldn't find anything relevant about expanding the Green function into arbitrary set of eigenfunctions. Would appreciate any opinion or advice on the matter :)
 
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  • #2
Look in Jackson's Electrodynamics book, I believe that the solution can be found by applying either chapter 2 or 3's methods.
 
  • #3
i want notes about quantum dynamics(schrodinger,heisenberg and interaction representation or pictures of quantum mechanics)
 

1. What is the purpose of using Green's function expansion in a set of eigenfunction?

The purpose of using Green's function expansion in a set of eigenfunction is to solve differential equations in a system with a specific set of boundary conditions. It allows for a more efficient and accurate solution by breaking down the problem into a linear combination of simpler problems.

2. How does Green's function expansion relate to eigenfunctions?

Green's function expansion uses a set of eigenfunctions as a basis to represent the solution of a differential equation. The eigenfunctions are used to construct a Green's function which acts as a "building block" for the overall solution.

3. What are the benefits of using Green's function expansion?

Green's function expansion allows for a systematic and efficient way to solve differential equations with a specific set of boundary conditions. It also allows for the superposition of solutions, making it easier to solve complex problems.

4. Are there any limitations to using Green's function expansion?

One limitation of Green's function expansion is that it is only applicable to linear differential equations. It also requires the knowledge of the eigenfunctions and eigenvalues of the system, which may not always be readily available.

5. How is Green's function expansion used in practical applications?

Green's function expansion is used in various fields of science and engineering, such as electromagnetics, fluid dynamics, and quantum mechanics. It is particularly useful in solving boundary value problems, such as finding the electric potential in a region with a specific charge distribution or determining the stress distribution in a material with specific boundary conditions.

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