The Green function is a very genius way to find the particular solution for an in-homogeneous equation.

$L \psi(x) = f(x)$

$L$  is called the operator of the equation. A Green function $G(x,x')$ is a 2D function, such that

$L G(x,x') = \delta ( x-x')$

Many beginner will feel uncomfortable and wonder why the Green function depends on 2 variables rather 1. Just relax, the operator only affect the x part, leave x’ unchanged, so, beginner can always treat x’ as a constant, a parameter, like the slope in linear equation. For those who touched multi-variable function, x’ is just another dimension. anyway, if we integrate it with the function f(x),

$\int { L G(x,x') f(x') dx'} = \int {\delta(x-x') f(x') dx' } = f(x) = L \psi(x)$

since the operator only act on the x part, not x’, so $L$ can be pulled out, then,

$\psi(x) = \int { G(x,x') f(x') dx' }$

which is our solution! But sadly, we have to find the green function rather then the solution! Ok, lets find the Green function.

to find the Green function, we have to solve the eigen equation of the operator.

$L \alpha_n(x) = \lambda_n \alpha_n(x)$

the eigen functions must be orthonormal ( the normality of the eigen function can be done by divided a constant) :

$\int { \alpha_n^* (x) \alpha_m(x) dx } = \delta_{nm}$

which this, the eigen function space must span every function. thus, the Green function can be spanned by:

$G(x,x') = \sum { \alpha_n(x) c_n }$

by using the orthonormal properties. the coefficient can be found and the Green function is:

$G(x,x') = \sum { \alpha_n^*(x') \frac {1} {\lambda_n} \alpha_n(x) }$

thus the particular solution is

$\psi(x) = \sum { \left( \int {\alpha_n^*(x') f(x') dx'} \right) \frac {1} {\lambda_n} \alpha_n(x) }$

If we use the eigenfunction expansion on the solution, this is the same result. However, the Green Function provided an “once and for all ” method to solve any function. and this often give a theoretical insight. For example, in solving static electric potential.

when we sub the eigen function expansion  of the Green function back into the operator. we will have 1 more discovery.

$L G(x,x') = \sum { \alpha_n^*(x') \alpha_n(x) } = \delta(x-x')$

the delta function is equal to the sum of the eigenfunctions!

OK. the stuff on above is very common and textbook or wiki. now i am going to show how this Green function stated in Dirac notation of ket and bra.

for an operator A, which is hermitian, so that it has complete eigenket.

$A \left| \psi \right> = \left| f \right>$

and the eigenkets are

$A \left| \alpha_n \right> = \left| \alpha_n \right> \lambda_n$

thus, the solution is

$\left| \psi \right> = \sum{ \left| \alpha_n \right> \frac { 1} { \lambda_n} \left< \alpha_n | f \right> }$

we can compare with the above formalism, and see the Green function in Dirac notation is:

$G = \sum{ \left| \alpha_n \right> \frac { 1} { \lambda_n} \left< \alpha_n \right| }$

which is rather an operator then a function, so, we call it Green Operator. if we notices that the operator A is related to the eigenket by

$A = \sum{ \left| \alpha_n \right> \lambda_n \left< \alpha_n \right| }$

Thus, the Green operator is just the inverse of the operator A!

By using the Dirac notation, we have a more understanding on the Green function. and more easy approach! Dirac rocks again by great invention of notation!