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

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.