A particle is scattered from a potential \( V_{\vec{r}} = g \delta^3(\vec{r}) \), where \( g \) is a positive constant. Using the first Born approximation, the angular \((\theta, \phi)\) dependence of the differential scattering cross section \( \frac{d\sigma}{d\Omega} \) is:
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For a spherically symmetric potential, the differential cross section is independent of both the scattering angle \( \theta \) and the azimuthal angle \( \phi \) in the first Born approximation.
Independent of \( \theta \) but dependent on \( \phi \)
Dependent on \( \theta \) but independent of \( \phi \)
Dependent on both \( \theta \) and \( \phi \)
Independent of both \( \theta \) and \( \phi \)
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The Correct Option isD
Solution and Explanation
In the first Born approximation for a delta-function potential \( V_{\vec{r}} = g \delta^3(\vec{r}) \), the scattering amplitude is proportional to the Fourier transform of the potential. For a spherically symmetric potential such as \( \delta^3(\vec{r}) \), the differential cross section \( \frac{d\sigma}{d\Omega} \) is independent of both the scattering angles \( \theta \) and \( \phi \), since the delta-function potential is rotationally invariant. This results in an isotropic scattering cross section.
