Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 3

For an interaction of finite range $b$, higher partial waves are suppressed at $kb\ll 1$ by the centrifugal barrier, away from exceptional higher-wave resonances. Outside the potential, the zero-energy radial S-wave scattering equation is $u^{\prime \prime }=0$, where $\psi =u/r$. Its solution is proportional to $r-a$, so normalization of the constant term givesThe scattering length $a$ is the extrapolated zero of $u(r)$. At nonzero small $k$, this form applies in the overlap region $b\ll r\ll k^{-1}$; it is not the far-field limit at fixed $k$. Equivalently, $k\cot {\delta }_0\to -1/a$ and $f_0\to -a$.

Put $d=|{\mathbf{r}}_1-{\mathbf{r}}_2|$, $g=e^{ikd}/d$ and $s_j=e^{i{\mathbf{k}}_i\cdot {\mathbf{r}}_j}$. In the range-to-wavelength overlap region around centre $j$, the outgoing wave's singular and constant terms are $c_j/(k{\rho }_j)+i{c}_j$. Matching their ratio to the Bethe-Peierls boundary condition yieldsDefine $A_j=1+ik{a}_j$ and $D=A_1{A}_2-a_1{a}_2{g}^2$. Solving the coupled system givesThis is the two-centre scattering-length model result, retaining the outgoing-wave $ik$ term and all repeated exchange between the two centres. If terms of order $k{a}_j$ are neglected in the local matching, set $A_1=A_2=1$ in these formulae; if also $kd\ll 1$, set $g\simeq 1/d$ and $s_j\simeq 1$. Away from a vanishing denominator this gives $c_1\simeq -k{a}_1(1-a_2/d)/(1-a_1{a}_2/d^2)$ and its exchanged partner. The isolated-centre limit is $c_j=-k{a}_j{s}_j/(1+ik{a}_j)$. Finite-range corrections require additional effective-range data, not supplied here.

At large $r$, $|\mathbf{r}-{\mathbf{r}}_j|=r-\hat{\mathbf{r}}\cdot {\mathbf{r}}_j+O(r^{-1})$. Retain the displacement in the phase, but replace the denominator by $r$. ThereforeThe differential scattering cross-section is $|f{|}^2$ for elastic scattering with this unit incident plane-wave normalization. interference depends on outgoing direction and centre separation as well as the incoming phases already contained in $c_j$.

Choose the polar axis along ${\mathbf{r}}_1-{\mathbf{r}}_2$. ThenUsing the angular interference integral for two point scatterers, integrating the two self terms and the interference terms givesFor $kd\ll 1$, the amplitudes add coherently and $\sigma \to 4\pi |c_1+c_2{|}^2/k^2$; when $kd\gg 1$ the angle-integrated interference is suppressed. These limits hold for fixed incoming direction; no incoming-orientation average has been silently imposed.