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?
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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).
See also: Deriving Unitarity of S-matrix in 1D Quantum Mechanics
