Re: Greens functions for Schrodingers equation

From: Igor Khavkine (k_igor_k_at_lycos.com)
Date: 08/17/04 

Date: Tue, 17 Aug 2004 22:11:08 GMT

On Tue, 17 Aug 2004 09:08:28 -0700, mike wrote:

> Hi
>
> Wonder if anyone out there can help me with this. I write Schrodingers
> equation in Dirac notation as
>
> (H - E) |n> = 0
>
> Where H is the Hamiltonian operator E is energy. I now define an operator
> G that is the solution of the equation
>
> (H-E) G = I (eqn 1)
>
> Where G is my Green's operator and I is the identity opeator (Or matrices
> - whatever you prefer). G is the inverse of the operator ( or matrix)
> (H-E)
>
> This is all standard so I believe. Now I have seen the following
> operator representation of G in various books
>
> G = SUM_n |n><n| (eqn 2)
> -------
> E - E_n

> How do you get from the simple definiton of the Green
> operator as an inverse given by my equation (1) to the operator
> representation given in equation (2).

Very simple. In the energy eigenstate basis |n>, the H-E operator
is diag(E_1 - E, E_2 - E, ..., E_n - E, ...). What's the inverse
of a diagonal matrix? It's just another diagonal matrix with the diagonal
entries inverted: (H-E)^(-1) diag(1/(E_1-E), ..., 1/(E_n-E), ...). I think
you can figure out the Dirac notation representation for (H-E)^(-1) now.
It's just the expression you have for G (except that you are off by a
minus sign). So your (eqn 2) states that G=(H-E)^(-1), which is equivalent
to equation (eqn 1) as long as (H-E) is invertable, i.e., E is not an
eigenvalue of H.

Hope this helps.

Igor

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