# $S$-Matrix and Relation to Phase Shifts

Given some potential $$V(x)$$, we can describe the amplitude of incoming and outgoing waves through the scattering matrix $$S$$ whereby $$\begin{pmatrix} B \\ F \end{pmatrix}= \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} A \\ G \end{pmatrix}.$$ I understand that the coefficients of the $$S$$ matrix can give us some information regarding the probabilities of transmission and reflection. I believe however it can also give us some information regarding the phase shift of an incoming and outgoing wave. How may we determine this relationship? Would it be a ratio $$F/A$$ for example, relating the amplitudes of a transmitted incoming wave through a potential?

Indeed, the asymptotic solutions are assumed to be plane waves: $$Ae^{ikx} + Be^{-ikx} \text{ for } x\rightarrow -\infty,\\ Ge^{-ikx} + Fe^{ikx} \text{ for } x\rightarrow +\infty$$ If we now consider a wave incident from the left, we can have only outgoing solution on the right (but we also have a reflected wave on the left): $$e^{ikx} + S_{11}e^{-ikx} \text{ for } x\rightarrow -\infty,\\ S_{21}e^{ikx} \text{ for } x\rightarrow +\infty,$$ where I set $$A=1$$. Thus, the magnitude of $$F$$ characterizes the transmission probability ($$T=|S_{12}|^2$$), while its phase is the phase shift of the transmitted wave. Similarly $$B$$ characterizes the reflection probability ($$R=|S_{11}|^2$$) and the phase of the reflected wave. Similar analysis can be done for the waves incident from the right.
Note that it is the matrix elements that are important here - the absolute values of $$A,B,F,G$$ depend on the normalization (in scattering problems one often uses normalization by the incident flux, which is conserved, rather than by the probability - which is difficult for extended solutions).