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Spherically symmetric potential

  1. Jan 16, 2012 #1
    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.
     
  2. jcsd
  3. Jan 16, 2012 #2

    Redbelly98

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    That can't be right. r is a variable of integration, it can't appear outside the integral.

    But the [itex]\exp{(i\vec{k}\cdot\vec{r})}[/itex] term spoils that symmetry, doesn't it?

    You might try choosing a coordinate system where k lies along the z-axis, and set up the integral that way.
     
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