(a) A classical particle of mass $m$ scatters on a central potential $V(r)$ with energy $E$, impact parameter $b$, and scattering angle $\theta$. Define the corresponding differential cross-section.

For particle trajectories in the Coulomb potential,

$$V_{C}(r)=\frac{e^{2}}{4 \pi \epsilon_{0} r}$$

the impact parameter is given by

$$b=\frac{e^{2}}{8 \pi \epsilon_{0} E} \cot \left(\frac{\theta}{2}\right)$$

Find the differential cross-section as a function of $E$ and $\theta$.

(b) A quantum particle of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters in a localised potential $V(\mathbf{r})$. With reference to the asymptotic form of the wavefunction at large $|\mathbf{r}|$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ as a function of the incident and outgoing wavevectors $\mathbf{k}$ and $\mathbf{k}^{\prime}$ (where $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|=k$ ). Define the differential cross-section for this process and express it in terms of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$.

Now consider a potential of the form $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless coupling and $U$ does not depend on $\lambda$. You may assume that the Schrödinger equation for the wavefunction $\psi(\mathbf{k} ; \mathbf{r})$ of a scattering state with incident wavevector $\mathbf{k}$ may be written as the integral equation

$$\psi(\mathbf{k} ; \mathbf{r})=\exp (i \mathbf{k} \cdot \mathbf{r})+\frac{2 m \lambda}{\hbar^{2}} \int d^{3} r^{\prime} \mathcal{G}_{0}^{(+)}\left(k ; \mathbf{r}-\mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$$

where

$$\mathcal{G}_{0}^{(+)}(k ; \mathbf{r})=-\frac{1}{4 \pi} \frac{\exp (i k|\mathbf{r}|)}{|\mathbf{r}|}$$

Show that the corresponding scattering amplitude is given by

$$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m \lambda}{2 \pi \hbar^{2}} \int d^{3} r^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$$

By expanding the wavefunction in powers of $\lambda$ and keeping only the leading term, calculate the leading-order contribution to the differential cross-section, and evaluate it for the case of the Yukawa potential

$$V(\mathbf{r})=\lambda \frac{\exp (-\mu r)}{r}$$

By taking a suitable limit, obtain the differential cross-section for quantum scattering in the Coulomb potential $V_{C}(r)$ defined in Part (a) above, correct to leading order in an expansion in powers of the constant $\tilde{\alpha}=e^{2} / 4 \pi \epsilon_{0}$. Express your answer as a function of the particle energy $E$ and scattering angle $\theta$, and compare it to the corresponding classical cross-section calculated in Part (a).

Part II, $2014 \quad$ List of Questions

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