How can I use spherical coordinates to simplify the Fourier transform equation?

[kelly0303]

Homework Statement

(This is part of a longer problem) Write the following potential in the momentum space:
$$V(r_1-r_2)=(\vec{\sigma_1}\cdot\vec{\nabla_1})(\vec{\sigma_2}\cdot\vec{\nabla_2})\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|}$$ where $\sigma$ is the Pauli matrix.

Relevant Equations

$$\phi(k)=\frac{1}{(\sqrt{2\pi})^3}\int{\psi(r)e^{-ik\cdot r}}d^3r$$

By applying the Fourier transform equation, and expanding the dot product, I get a sum of terms of the form: $$V(k)=\sigma_1^x\nabla_1^x\sigma_2^y\nabla_2^y\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|}e^{-ik(r_2-r_1)} = \sigma_1^x\sigma_2^y\nabla_1^x\nabla_2^y\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|} e^{-ik(r_2-r_1)} $$ But I don't really know what to do with this. Integration by parts doesn't seem to help too much.

 

[Charles Link]

There are a couple of things that I think you are needing for this problem:
1) https://math.stackexchange.com/ques...awa-potential-fourier-representation-integral
2) Follow these simple steps if you can:
$\psi(r)=\frac{1}{2 \pi} \int d^3k \, \hat{\psi}(k) e^{ik \cdot r}$.
$\nabla \psi(r)=\frac{1}{2 \pi} \int d^3k \, ik \hat{\psi}(k) e^{ik \cdot r}$.
This last equation is the result that $ik \hat{\psi}(k)$ is necessarily the F.T. of $\nabla \psi$.
Some additional work might be in order to complete this problem, and I'm not even sure I would know how to finish it up, but perhaps what I gave you will help.

 

[Abhishek11235]

Try using spherical coordinates. In these coordinates $|r_{1} - r_{2}|^2= |r_{1}|^2 +|r_{2}|^2-2r_{1}r_{2}cos\theta$. Also look up gradient in spherical coordinate.