Editing S-matrix (section)

==== Transfer matrix ====
The ''transfer matrix'' <math>M</math> relates the plane waves <math>C e^{ikx}</math> and <math>D e^{-ikx}</math> on the ''right'' side of scattering potential to the plane waves <math>A e^{ikx}</math> and <math>B e^{-ikx}</math> on the ''left'' side:<ref name="cas.cz">{{cite web |url=https://gemma.ujf.cas.cz/~krejcirik/AAMP13/slides/Mostafazadeh.pdf |website=gemma.ujf.cas.cz |title=Transfer Matrix Formulation of Scattering Theory in Arbitrary Dimensions |access-date=29 October 2022}}</ref>  

<math display="block">\begin{pmatrix}C \\ D \end{pmatrix} = \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}\begin{pmatrix} A \\ B \end{pmatrix}</math> and its components can be derived from the components of the S-matrix via:<ref name="ucr.edu">{{cite web |url=https://intra.ece.ucr.edu/~korotkov/courses/EE201/EE201-slides/EE201-Lec-6.pdf |title=EE201/MSE207 Lecture 6 |website=intra.ece.ucr.edu |access-date=29 October 2022}}</ref> <math>M_{11}=1/S_{12}^*= 1/S_{21} ^* {,}\  M_{22}= M_{11}^*</math> and <math>M_{12}=-S_{11}^*/S_{12}^* = S_{22}/S_{12} {,}\ M_{21} = M_{12}^*</math>, whereby time-reversal symmetry is assumed.

In the case of time-reversal symmetry, the transfer matrix <math>\mathbf{M}</math> can be expressed by three real parameters:

<math display="block">M = \frac{1}{\sqrt{1-r^2}} \begin{pmatrix} e^{i\varphi} & -r\cdot e^{-i\delta} \\ -r\cdot e^{i\delta} & e^{-i\varphi} \end{pmatrix}</math> 
with <math>\delta,\varphi \in [0;2\pi]</math> and <math>r\in [0;1]</math> (in case {{math|1=''r'' = 1}} there would be no connection between the left and the right side)