IGNOU MPH 8 2026 - Using the Born Approximation, calculate the differential cross-section

Question

Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: $V(r) = V_0 \frac{a}{r} \exp\left( -\frac{r}{a} \right)$ . You may use:
$$\int_0^\infty e^{-\alpha x} \sin(\beta x) dx = \frac{\beta}{\alpha^2 + \beta^2} \text{ for } \alpha > 0$ $

19 Jan 2026

Answer :

Word Count : 242

The Born approximation gives the scattering amplitude as: [ f(\theta) = -\frac{2m}{\hbar^2} \frac{1}{4\pi} \int e^{i \mathbf{q} \cdot \mathbf{r}} V(r) , d^3r ] where (\mathbf{q} = \mathbf{k}' - \mathbf{k}) is the momentum transfer, with magnitude (q = 2k \sin(\theta/2)). Given the potential: [ V(r) = V_0 \frac{a}{r} e^{-r/a} ] The scattering amplitude in Born approximation simplifies (spherically symmetric potential) to: [ f(\theta) = -\frac{2m}{\hbar^2} \frac{1}{q} \int_0^\infty r V(r) \sin(qr) , dr ] Substitute (V(r)): [ f(\theta) = -\frac{2m}{\hbar^2}

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