This one is tougher, but important! Suppose you have to solve Poisson's equation, V2f(=g(F), where f and g are functions of position, F. Show that,...

7.

This one is tougher, but important!

Suppose you have to solve Poisson's equation,

∇

2

f

(

~r

) =

g

(

~r

)

,

where

f

and

g

are functions of position,

~r

. Show that, if we introduce a new function,

G

(

~r,~r

′

), satisfying

∇

2

G

(

~r,~r

′

) =

δ

(3)

(

~r

−

~r

′

)

,

then the solution to Poisson's equation is given by

f

(

~r

) =

∫

G

(

~r,~r

′

)

g

(

~r

′

)

d

3

~r

′

.

Determine the

Green's function

,

G

, in this case, and find the integral for

f

(

~r

).

Note

just for fun, your integral can actually be written in the nice notation

f

(

~r

) =

1

∇

2

g

(

~r

)

,

which means the same thing as the integral that you find. This method of solving

linear differential equations is known as the

Green's function

method, and shows up

extensively in physics.

7. This one is tougher, but important! Suppose you have to solve Poisson’s equation, V2f(fl=g(F), where f and g are functions of position, F. Show that, if we introduce a new function,G (fif’), satisfyingv20 (F, F) = 6(3) (F— F) , then the solution to Poisson’s equation is given by fla=f0mmmofa Determine the Green’s function, G, in this case, and find the integral for f (1‘). Notejust for fun, your integral can actually be written in the nice notation f®=%w@, which means the same thing as the integral that you find. This method of solvinglinear differential equations is known as the Green’s function method, and shows upextensively in physics.