How Are Completeness Relations and Green's Functions Related?

[ehrenfest]

I am confused about completeness relations. I thought a completeness relation was something like:

I = \sum_{i = 1}^n |i><i| = \sum_{i=1}^n P_i [

where P_i is the projection operator onto i. Now I saw this called a completeness relation as well:

\delta(x - x') = \sum_{n=0}^\infty \Psi_n(x) \Psi_n(x')

How is that the same as my first equation? What is the difference between x and x'? The second equation can be found at http://en.wikipedia.org/wiki/Green's_function

 

[jostpuur]

As set of functions \psi_n being complete means that you can write down arbitrary function (of some kind, so not really arbitrary) as

<br /> f(x)=\sum_{n=0}^{\infty} c_n \psi_n(x)<br />

where the coefficients are given by an inner product

<br /> c_n = \int_{-\infty}^{\infty} dx&#039;\; \psi^*_n(x&#039;)f(x&#039;).<br />

But you can rewrite this as

<br /> f(x) = \sum_{n=0}^{\infty} \Big(\int_{-\infty}^{\infty} dx&#039;\; \psi^*_n(x&#039;)f(x&#039;)\Big) \psi_n(x) = \int_{-\infty}^{\infty} dx&#039;\; \Big(\sum_{n=0}^{\infty} \psi^*_n(x&#039;)\psi_n(x)\Big) f(x&#039;)<br />

so there you see that it is pretty much the same as the sum being a delta function.

 

[ehrenfest]

I see how it works mathematically thanks. Still a little confused about "how to think about" x and x'...

 

[jpr0]

In the completeness relation there is really nothing to think about, as far as I'm aware... For the Green's function there is an interpretation which is something like "the wavefunction at x resulting from a unit excitation applied at x^{\prime}."