Integral computation in scattering theory

[jouvelot]

Hello everyone,

Still working on my reading of Weinberg's Lecture notes on QM book. At one point, the following integral $$\int d^3 xd^3 x' V(\vec{x~}) V(\vec{x'}) / |\vec{x~} - \vec{x'}|^2$$ has to be computed in the case where $V(r) = -e^{-r/R}/r$. This reminds me of retarded potentials, but I don't have my other books with me, and cannot find a way to compute this integral. I've been trying with some residues, but this is unwieldy. The result should be $8\pi^2 R^2$.
Thanks for any hints that would help me compute it.
Bye,
Pierre

 

[blue_leaf77]

I think you can employ the generating function of the Legendre polynomials. Using this, you can write
$$
\frac{1}{\sqrt{|\mathbf{x}-\mathbf{x}'|}} = \sum_{l=0}^\infty \sum_{m=-l}^{m=l} \frac{4\pi}{2l+1} \frac{(x_<)^l}{(x_>)^{l+1}} Y_{lm}^*(\theta,\phi) Y_{lm}(\theta',\phi')
$$
using the orthonormality of the spherical harmonics, you can simplify the sums.
I haven't tried myself though, nor know whether the resulting integral will be solvable but at least I know where this is going to.

 

[jouvelot]

Hi blue_leaf,

Thanks a lot for the hint. I'll look into this approach :)

Bye,

Pierre

 

[jouvelot]

Hi blue_leaf,

Indeed, this seems to (almost) work, using a squared Legendre expansion of $1/|\vec{x}-\vec{x'}|$ in the orthogonal $P_l(cos (\gamma))$ polynomials (no need to go to harmonics). I used the online version of Wolfram Alpha (great tool, and free) to compute the sum and integrate the resulting function; I do get something in $\pi^2R^2$, although the coefficient is off by a factor of 2 -- although I can almost make up a story why just half of the result is valid, due to convergence constraints. But since they are other minor details that elude me (limit behavior of Exponential integral Ei and tanh${}^{-1}$, among others), I'll feel happy for now :)

Thanks for your help, which had me dig more into these issues.

Pierre