How Does the Born Series Expansion Translate to Spatial Representation?

[PineApple2]

Hello. I read about the born series in scattering,
<br /> |\psi&gt; = (1+G_0V+\ldots)|\psi_0&gt; <br />
Now when I want to move to spatial representation, the textbook asserts I should get
<br /> \psi(\vec{r})=\psi_0(\vec{r}) + \int dV&#039; G_0(\vec{r},\vec{r&#039;}) V(\vec{r&#039;})\psi_0(\vec{r&#039;})+\ldots<br />
by operating with &lt;\vec{r}| from the left. However I don't know how to get the 2nd term (the integral). I tried to insert a complete basis like this:
<br /> &lt;\vec{r}|G_0V|\psi_0&gt; = \int d^3r&#039;&lt;\vec{r}|G_0|\vec{r&#039;}&gt;&lt;\vec{r&#039;}|V|\psi_0&gt;<br />
however I don't know how to get V(\vec{r&#039;}) from the second bracketed term. Any help?

By the way: is there a "nicer" way to write 'bra' and 'ket' in this forum?

 

[vanhees71]

Introduce another unit operator in terms of the completeness relation for the position-eigenbasis. Then you use

\langle \vec{x}&#039; |V(\hat{\vec{x}}\vec{x}&#039;&#039; \rangle = V(\vec{x}&#039;&#039;) \langle \vec{x}&#039;| \vec{x}&#039;&#039; \rangle=V(\vec{x}&#039;) \delta^{(3)}(\vec{x}&#039;-\vec{x}&#039;&#039;).

Then one of the integrals from the completeness relations can be used to get rid of the \delta distribution, and you arrive at Born's series in the position representation as given by your textbook.

 

[PineApple2]

I see. And then V(\vec{x&#039;}) can be taken out as |\vec{x&#039;}&gt; are its eigenstates and it is taken out as a scalar.
Thanks!