Spring 2008 10 Misc

Compute the Fourier transform of the three-dimensional Yukawa potential

(1)
\begin{align} V(\bf{x}) = \frac{e^{-m \left |x\right|}}{\left|\bf{x}\right|} \end{align}

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Answer

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Work it in polar coordinates.

(2)
\begin{align} \hat{V}(\bf{k}}) = \int d^3 x \frac{e^{-m \left |x\right|}}{\left|\bf{x}\right|}} e^{-\imath \bf{k} \cdot\bf{x}} \end{align}
(3)
\begin{align} \hat{V}(\bf{k}) = \int_0^{\infty} d r r^2 \int_0^{\pi}d \theta \sin{\theta} \int_0^{2 \pi} d\phi \frac{e^{-(mr+\imath k r \cos{\theta})}}{r} \end{align}
(4)
\begin{align} = \frac{2\pi}{\imath k} \left[ \frac{1}{m- \imath k}-\frac{1}{m+\imath k}\right] \end{align}
(5)
\begin{align} = \frac{4 \pi}{m^2+\bf{k}^2} \end{align}
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