Question

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:

• A. Independent of \( \theta \) but dependent on \( \phi \)
• B. Dependent on \( \theta \) but independent of \( \phi \)
• C. Dependent on both \( \theta \) and \( \phi \)
• D. Independent of both \( \theta \) and \( \phi \)

Hint:

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.

Answer 1

The Correct Option is D

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.