• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Jackson p244,Green function for wave equation

Jackson electrodynamics 3rd. p244

I understood that
G=\frac{e^{ikR}}{R}
is a spetial solution for
( \nabla ^2 + k^2 )G =0 (R>0) .

but,why G=\frac{e^{ikR}}/{R} satisfy
( \nabla ^2 + k^2 )G =-4\pi \delta (\mathbf{R}) ?

How to normalize the Green function?
( \nabla ^2 + k^2 )\frac{e^{ikR}}{R}=...calculate...=0.

I can't understand.Please help me...
 
255
0
Jackson electrodynamics 3rd. p244

I understood that
G=\frac{e^{ikR}}{R}
is a spetial solution for
( \nabla ^2 + k^2 )G =0 (R>0) .
Firstly
[tex]
G=\frac{e^{ikR}}{R}
[/tex]
is not strictly a solution of homogenous equation (i.e. equation without delta function
on RHS) because G is not well behaved for R=0 and because of it one has to consider all
derivatives near the point R=0 in distributional sense. It is true that for all points
[tex]R \neq 0 [/tex] we have [tex]( \nabla ^2 + k^2 )G =0 [/tex] but it's not true for
R=0 as G blows up there.

but,why G=\frac{e^{ikR}}/{R} satisfy
( \nabla ^2 + k^2 )G =-4\pi \delta (\mathbf{R}) ?
There is a nice trick which shows why it is the case. Let's integrate the expression
[tex] ( \nabla ^2 + k^2 )G [/tex] over small ball of radious r in the limit r goes to 0.
Notice that:
[tex]
\lim_{r \rightarrow 0} \int_{K(0,r)} d^3 x ( \nabla ^2 + k^2 )G =
\lim_{r \rightarrow 0} \int_{K(0,r)} d^3 x ( \nabla ^2)G
[/tex]
Gauss theorem tells
us that we can change integration over ball K(0,r) for integration over sphere:
[tex]
\int_{K(0,r)} d^3 x \nabla ^2 G = \int_{K(0,r)} d^3 x ~ \textrm{div} ~\textrm{grad} G
= \int_{S(0,r)} dS \vec{n} ~\textrm{grad} G
[/tex]
So we have:
[tex]
\lim_{r \rightarrow 0} \int_{K(0,r)} d^3 x ( \nabla ^2 + k^2 )G = \lim_{r \rightarrow 0}
\int_{S(0,r)} dS \left( \frac{(ik R -1)\exp(ikR)}{R^2} \right) = -4\pi
[/tex]
Because integration of [tex] ( \nabla ^2 + k^2 )G [/tex] gives [tex]-4\pi[/tex] and
[tex] ( \nabla ^2 + k^2 )G [/tex] is zero everywher beside R=0 it has
to be equal to [tex] -4\pi \delta(\vec{R}) [/tex].

How to normalize the Green function?
( \nabla ^2 + k^2 )\frac{e^{ikR}}{R}=...calculate...=0.
Because [tex]( \nabla ^2 + k^2 )G =-4\pi \delta (\vec{R}) [/tex] is not a homogenous
equation if you multiply G by a constant it won't be a solution to this eqaution.
 
Oh...It's a very nice trick.
Thank you for your help!
 

Related Threads for: Jackson p244,Green function for wave equation

  • Posted
Replies
6
Views
3K
Replies
1
Views
2K
Replies
2
Views
893
Replies
4
Views
6K
Replies
8
Views
2K
Replies
3
Views
440
  • Posted
Replies
5
Views
2K
  • Posted
Replies
12
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top