# Completeness relations

1. Aug 12, 2007

### 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

2. Aug 12, 2007

### 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.

3. Aug 12, 2007

### ehrenfest

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

4. Aug 12, 2007

### 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 $x^{\prime}$."