One-dimensional S-matrix

For a reflection-invariant localized potential, order incoming amplitudes from the left and right and outgoing amplitudes toward the left and right. Thenwhere $r$ and $t$ are the reflection and transmission amplitudes. Probability-current conservation makes $S$ unitary for a real potential.

For a parity-symmetric one-dimensional potential, the even and odd parity states diagonalize the One-dimensional S-matrix:For a real potential, unitarity further gives $|S_{++}|=|S_{--}|=1$ at real positive energy.

If the left and right travelling-wave coefficients are related bythen