Completeness relations confusion

[ehrenfest]

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

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

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

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

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 [itex]\psi_n[/itex] being complete means that you can write down arbitrary function (of some kind, so not really arbitrary) as

[tex]
f(x)=\sum_{n=0}^{\infty} c_n \psi_n(x)
[/tex]

where the coefficients are given by an inner product

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

But you can rewrite this as

[tex]
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')
[/tex]

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 [itex]x[/itex] resulting from a unit excitation applied at [itex]x^{\prime}[/itex]."