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Suppose I have a spherically symmetric potential and I can find its cross section in configuration space (i.e position-space), $d\sigma / d\theta$. Now I need to find its distribution $d^2\sigma / p_rdp_rdp_\theta$ in momentum space. How can I do this?

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By definition, the cross-section of a potential $d \sigma$ is equal to $d N/n$, where $dN$ is the number of particles, which are deflected in some specified direction and $n$ is the particle (2d)density in the beam before deflection. By definition, particles in the beam all have some initial momentum $p_0$.

As in any potential field the energy is conserved, the momenta of all the particles remain the same as before scattering, so $\sigma \sim \delta(p_r-p_0)$. When it comes to $p_\theta$, the distribution is actually exactly described by $\sigma(\theta)$. That is, the particles with a given $\theta$ are exactly the particles, described by $p_\theta$, and hence $\dfrac{d \sigma}{d p_\theta}=\dfrac{d\sigma}{d\theta}$.

Accounting for both dependencies one may write $\dfrac{d \sigma}{d p_\theta}=\dfrac{d\sigma}{d\theta}\delta(p_r-p_0)$.

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