Past exam of the further quantum theory course of the University of Oxford / 2023 / 3 / a / iii / Solution

Beyond the outer classical turning point, let $\chi (x)=\sqrt{2m[V(x)-E]}$ and $U(x)={\hslash }^{-1}{\int }_{x_2}^x\chi (y)dy$. The bound state condition removes the growing branch at positive infinity, leavingHere $F$ is an arbitrary common amplitude pending the matching and wavefunction normalization. For the confining potential energy in the native diagram, the decaying branch gives the required exterior square-integrable function.