Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 33C / a / i / Solution

With the incoming basis ${\mathcal{I}}_{\pm }$ and outgoing basis ${\mathcal{O}}_{\pm }$ as printed, expand any scattering solution as $\psi \sim a_{+}{\mathcal{I}}_{+}+a_{-}{\mathcal{I}}_{-}+b_{+}{\mathcal{O}}_{+}+b_{-}{\mathcal{O}}_{-}$. Linearity definesThe column indexed by a channel records outgoing amplitudes for a unit incoming wave in that channel. The extra minus sign in ${\mathcal{O}}_{-}$ is a channel phase convention; it affects the sign of the free odd-channel amplitude but not flux probabilities. For a real potential and no absorbing or additional channels, probability-current conservation makes the flux-normalized matrix unitary.