### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.14)**
**1. Quantum Scattering Theory (Born Approximation)**:
- **Scattering Potential**: $V(\vec{r}) = g\delta^3(\vec{r})$, where $g$ is a positive constant.
- **First Born Approximation**: The scattering amplitude $f(\theta, \phi)$ is proportional to the 3-dimensional spatial Fourier transform of $V(\vec{r})$:
$f(\theta, \phi) = -\frac{m}{2\pi\hbar^2} \int V(\vec{r}) e^{-i\vec{q}\cdot\vec{r}} d^3r$
where $\vec{q} = \vec{k}_f - \vec{k}_i$ is the momentum transfer vector.
**2. Analytical Evaluation**:
- Substitute the delta function potential:
$f(\theta, \phi) = -\frac{m}{2\pi\hbar^2} \int g\delta^3(\vec{r}) e^{-i\vec{q}\cdot\vec{r}} d^3r = -\frac{mg}{2\pi\hbar^2} e^{-i\vec{q}\cdot\vec{0}} = -\frac{mg}{2\pi\hbar^2}$
- Notice that $f(\theta, \phi)$ is a pure constant, entirely independent of scattering angles $\theta$ and $\phi$.
- Differential cross section:
$\frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2 = \left(\frac{mg}{2\pi\hbar^2}\right)^2 = \text{constant}$
- Therefore, the differential scattering cross section is **independent of both $\theta$ and $\phi$** (pure isotropic s-wave scattering).
**3. Official Verification**:
The verified official IIT answer is **Option (D)** (`Independent of both š and š`).
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