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My understanding of the Born approximation is that it valid for scattering with small momentum transfer. In the context of two-particle, isotropic, elastic scattering, I would trust the Born approximation for scattering events where the momentum transfer, $q = 2k|\sin(\theta/2)|$, is much smaller than $k$, the asymptotic relative momentum of the particles. Equivalently, I trust the Born approximation for scattering where the angle of deflection, $\theta$, is small.

I want to know if this understanding is still valid for scattering between two identical particles.

When the particles are indistinguishable (e.g., electrons), we cannot tell the difference between a scattering event with deflection $\theta$ versus one with deflection $\theta+\pi$. The corresponding momentum transfers are $q=2k|\sin\theta|$ versus $q=2k|\cos\theta|$. Accordingly, any small-angle scattering event is paired with an equally valid large-angle one.

I'm torn between two ways of thinking about this:

  1. Compared to the case where the particles are distinguishable, the Born approximation is accurate over an even wider range of $\theta$ because a large-angle deflection can be re-interpreted as a small-angle one.

  2. Compared to the case where the particles are indistinguishable, the Born approximation is accurate over a narrower range of $\theta$ because a small-angle deflection can be re-interpreted as a large-angle one.

This doublethought carries through in practical calculations involving the differential cross-section. For indistinguishable particles, the differential cross-section must be symmetric about $\theta=\pi/2$. This means that in calculating angle-averaged quantities, e.g., the total cross-section $$ \sigma_{\mathrm{tot}}(k) = 2\pi \int_0^\pi \sigma(k,\theta)\sin\theta\,d\theta $$ I am free to integrate either just the small-angle contributions, $0\le\theta\le\frac\pi2$, or just the large-angle contributions, $\frac\pi2\le\theta\le\pi$ (picking up a factor of 2 as well, naturally).

Thus I am led to think that the validity of the Born approximation for identical-particle scattering is more subtle than just requiring small-angle scattering. Is the Born approximation for identical-particle scattering better, worse, or as good compared with distinguishable-particle scattering?

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