- #1
kelly0303
- 561
- 33
- 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.