Editing S-matrix (section)

=== Optical theorem in one dimension ===
In the case of [[free particle]]s {{math|1=''V''(''x'') = 0}}, the ''S''-matrix is<ref>{{harvnb|Merzbacher|1961}} Ch 6. A more common convention, utilized below, is to have the ''S''-matrix go to the identity in the free particle case.</ref>
<math display="block"> S=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>
Whenever {{math|''V''(''x'')}} is different from zero, however, there is a departure of the ''S''-matrix from the above form, to
<math display="block"> S = \begin{pmatrix} 2ir & 1+2it \\ 1+2it &2ir^* \frac{1+2it}{1-2it^*} \end{pmatrix}.</math>
This departure is parameterized by two [[complex functions]] of energy, {{math|''r''}} and {{math|''t''}}.
From unitarity there also follows a relationship between these two functions,
<math display="block">|r|^2+|t|^2 = \operatorname{Im}(t).</math>

The analogue of this identity in three dimensions is known as the [[optical theorem]].