# Validity of the Born approximation for identical-particle scattering

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.

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).