dark_matter_is_neat
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- TL;DR
- I'm trying to calculate the scattering cross section between a charged particle and a particle with a magnetic moment using quantum mechanics. I believe the interaction term in this case would be ##\mu \cdot (\vec{v} \times \frac{e\vec{r}}{4 \pi r^{3}})##
I'm trying to calculate the scattering cross section between a charged particle and a particle with a magnetic moment. I believe the interaction term in this case would be ##\mu \cdot (\vec{v} \times \frac{e\vec{r}}{4 \pi r^{3}})## although I'm a bit confused how to properly treat it in this calculation. Would ##\vec{v}## and ##\vec{r}## refer to the relative velocity and position between the two particles, and wouldn't I need to treat ##\vec{v}## as an operator because in principle ##\vec{v} = \vec{p}/m##? If I need to treat the velocity as an operator would I treat ##\vec{p}## as acting on just the wavefunction or should I treat it as acting on the wavefunction and the r terms in the interaction term. In this case the particles are both treated as plane waves so what I'm interested in calculating ##\int d^{3}r_{1}d^{3}r_{2}e^{-i \vec{p'_{1}} \cdot \vec{r_{1}}}e^{-i \vec{p'_{2}} \cdot \vec{r_{2}}}H_{int}e^{i \vec{p_{1}} \cdot \vec{r_{1}}}e^{i \vec{p_{2}} \cdot \vec{r_{2}}}##.