(i) A particle of momentum $\hbar k$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters off a sphericallysymmetric target in three dimensions. Define the corresponding scattering amplitude $f$ as a function of the scattering angle $\theta$. Expand the scattering amplitude in partial waves of definite angular momentum $l$, and determine the coefficients of this expansion in terms of the phase shifts $\delta_{l}(k)$ appearing in the following asymptotic form of the wavefunction, valid at large distance from the target,

$$\psi(\mathbf{r}) \sim \sum_{l=0}^{\infty} \frac{2 l+1}{2 i k}\left[e^{2 i \delta_{l}} \frac{e^{i k r}}{r}-(-1)^{l} \frac{e^{-i k r}}{r}\right] P_{l}(\cos \theta) .$$

Here, $r=|\mathbf{r}|$ is the distance from the target and $P_{l}$ are the Legendre polynomials.

[You may use without derivation the following approximate relation between plane and spherical waves (valid asymptotically for large $r$ ):

$$\exp (i k z) \sim \sum_{l=0}^{\infty}(2 l+1) i^{l} \frac{\sin \left(k r-\frac{1}{2} l \pi\right)}{k r} P_{l}(\cos \theta) .$$

(ii) Suppose that the potential energy takes the form $V(r)=\lambda U(r)$ where $\lambda \ll 1$ is a dimensionless coupling. By expanding the wavefunction in a power series in $\lambda$, derive the Born Approximation to the scattering amplitude in the form

$$f(\theta)=-\frac{2 m \lambda}{\hbar^{2}} \int_{0}^{\infty} U(r) \frac{\sin q r}{q} r d r$$

up to corrections of order $\lambda^{2}$, where $q=2 k \sin (\theta / 2)$. [You may quote any results you need for the Green's function for the differential operator $\nabla^{2}+k^{2}$ provided they are stated clearly.]

(iii) Derive the corresponding order $\lambda$ contribution to the phase shift $\delta_{l}(k)$ of angular momentum $l$.

[You may use the orthogonality relations

$$\int_{-1}^{+1} P_{l}(w) P_{m}(w) d w=\frac{2}{(2 l+1)} \delta_{l m}$$

and the integral formula

$$\int_{0}^{1} P_{l}\left(1-2 x^{2}\right) \sin (a x) d x=\frac{a}{2}\left[j_{l}\left(\frac{a}{2}\right)\right]^{2}$$

where $j_{l}(z)$ is a spherical Bessel function.]