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

$$f(x)=\sum_{n=0}^{\infty} c_n \psi_n(x)$$

where the coefficients are given by an inner product

$$c_n = \int_{-\infty}^{\infty} dx'\; \psi^*_n(x')f(x').$$

But you can rewrite this as

$$f(x) = \sum_{n=0}^{\infty} \Big(\int_{-\infty}^{\infty} dx'\; \psi^*_n(x')f(x')\Big) \psi_n(x) = \int_{-\infty}^{\infty} dx'\; \Big(\sum_{n=0}^{\infty} \psi^*_n(x')\psi_n(x)\Big) f(x')$$

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 $x^{\prime}$."