Editing S-matrix (section)

=== Definition ===
Consider a localized one dimensional [[potential barrier]] {{math|''V''(''x'')}}, subjected to a beam of quantum particles with energy {{math|''E''}}. These particles are incident on the potential barrier from left to right.

The solutions of the [[Schrödinger equation]] outside the potential barrier are [[plane waves]] given by
<math display="block">\psi_{\rm L}(x)= A e^{ikx} + B e^{-ikx}</math>
for the region to the left of the potential barrier, and
<math display="block">\psi_{\rm R}(x)= C e^{ikx} + D e^{-ikx}</math>
for the region to the right to the potential barrier, where
<math display="block">k=\sqrt{2m E/\hbar^{2}}</math>
is the [[wave vector]]. The time dependence is not needed in our overview and is hence omitted. The term with coefficient {{math|''A''}} represents the incoming wave, whereas term with coefficient {{math|''C''}} represents the outgoing wave. {{math|''B''}} stands for the reflecting wave. Since we set the incoming wave moving in the positive direction (coming from the left), {{math|''D''}} is zero and can be omitted.

The "scattering amplitude", i.e., the transition overlap of the outgoing waves with the incoming waves is a linear relation defining the ''S''-matrix,
<math display="block">\begin{pmatrix} B \\ C \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} A \\ D \end{pmatrix}.</math>

The above relation can be written as
<math display="block">\Psi_{\rm out}=S \Psi_{\rm in}</math>
where
<math display="block">\Psi_{\rm out}=\begin{pmatrix}B \\ C \end{pmatrix}, \quad \Psi_{\rm in}=\begin{pmatrix}A \\ D \end{pmatrix}, \qquad S=\begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}.</math>
The elements of {{math|''S''}} completely characterize the scattering properties of the potential barrier {{math|''V''(''x'')}}.