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

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.

An even potential $V(x)=V(-x)$ preserves parity. Its Hamiltonian operator commutes with the parity operator, so an even incoming solution produces only an even outgoing solution, and similarly for an odd one. ThusAn asymmetric potential generically couples the parity channels; an accidental vanishing at one energy does not imply the symmetry for all energies.