Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 1 / 4 / Solution

Put $\beta =m\alpha /{\hslash }^2$. The delta interaction reduces the integral equation to ${\psi }_s=e^{ikx}+i\beta e^{ik|x|}{\psi }_s(0)/k$. At zero, ${\psi }_s(0)=(1-i\beta /k{)}^{-1}$. HenceTheir sum is one, as required for a real potential. The explicit wave is the indicated incident-plus-reflected form for $x<0$ and $t{e}^{ikx}$ for $x>0$; it is continuous and has derivative jump $-2\beta \psi (0)$.