Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S18 / 1 / c / Solution

Write $L_0\psi =-U\psi$ and use the outgoing inverse of $L_0$. Adding the homogeneous incoming wave yieldsFor the attractive delta potential, let $\eta =m\alpha /(k{\hslash }^2)$. The equation reduces to $\psi (x)=e^{ikx}+i\eta e^{ik|x|}\psi (0)$ and $\psi (0)=1/(1-i\eta )$. Separating the two half-lines givesBoth sides have the same group velocity, soThe same amplitudes satisfy continuity and the exact delta-potential derivative jump ${\psi }^{\prime }(0^{+})-{\psi }^{\prime }(0^{-})=-2m\alpha \psi (0)/{\hslash }^2$, providing an independent local check on the sign.