(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that for a spherically symmetric potential

[itex]\int _{all space} V(\vec{r})exp(i\vec{k}\cdot\vec{r})d\tau = \frac{4\pi}{r}\int_{0}^{\infty} V(r) sin(\kappa r)dr[/itex]

3. The attempt at a solution

Given that the potential is spherically symmetric we have azimuthal symmetry and zenithal symmetry, so that the integral reduces to

[itex]\int _{all space} V(\vec{r})exp(i\vec{k}\cdot\vec{r})d\tau = 4\pi \int _{0}^{\infty}V(r)r^{2}exp(i\vec{k}\cdot\vec{r})dr[/itex]

From here, I am not sure how to work with the exponential portion. I've thought that perhaps since this is spherically symmetric we can reduce the dot product into [itex]\vec{k}\cdot\vec{r}=kr cos(\theta)[/itex] or something of that nature, but I really don't see how this helps me. If you have any suggestions or references that would help, please let me know. Thanks in advance.

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# Spherically symmetric potential

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