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GATE 2025 PH – Question 14

Quantum Mechanics · elementary scattering theory, Born approximation. · 1 mark · Multiple choice

A particle is scattered from a potential $V(\mathbf r)=g\delta^3(\mathbf r)$, where $g$ is a positive constant. Using the first Born approximation, the angular $(\theta,\phi)$ dependence of differential scattering cross section $d\sigma/d\Omega$ is

  1. Independent of $\theta$ but dependent on $\phi$
  2. Dependent on $\theta$ but independent of $\phi$
  3. Dependent on both $\theta$ and $\phi$
  4. Independent of both $\theta$ and $\phi$

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Correct answer: (D) Independent of both $\theta$ and $\phi$

Explanation

In the **first Born approximation** the scattering amplitude is proportional to the Fourier transform of the potential:
$$f(\theta,\phi)=-\frac{m}{2\pi\hbar^2}\int V(\mathbf r)\,e^{-i\mathbf q\cdot\mathbf r}\,d^3r,\qquad\mathbf q=\mathbf k^\prime-\mathbf k .$$

**For $V=g\delta^3(\mathbf r)$:**
$$f=-\frac{m}{2\pi\hbar^2}\,g\int\delta^3(\mathbf r)e^{-i\mathbf q\cdot\mathbf r}d^3r=-\frac{mg}{2\pi\hbar^2},$$
a **constant**, independent of the momentum transfer $\mathbf q$ and so of the angles.

The differential cross section $\dfrac{d\sigma}{d\Omega}=|f|^2$ is therefore **independent of both $\theta$ and $\phi$** (isotropic scattering; option D).