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Cell[CellGroupData[{
Cell["Schwinger pair production", "Subtitle",
  Background->RGBColor[1, 1, 0]],

Cell[CellGroupData[{

Cell["Introduction", "Section",
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Consider the homogeneous static electric field E, which produces the force \
F= (-e)E on the electron, which is positive along the ",
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  " "
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Cell["\<\
The electron charge is called here (-e), presuming that  e>0. Describe this \
force by the potential energy U\
\>", "Text"],

Cell[BoxData[
    \(U = \(\(-F\)\(\ \)\(x\)\(\ \ \ \)\)\)], "NumberedEquation",
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Cell["Electron energy and momentum", "Text"],

Cell[BoxData[
    \(\((\[CurlyEpsilon] - U)\)\^2 - p\^2 = m\^2\)], "NumberedEquation",
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Cell[BoxData[
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Cell["Classical motion is allowed in the region", "Text"],

Cell[BoxData[
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Cell["which splits in two, either", "Text"],

Cell[BoxData[
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Cell["or altenatively", "Text"],

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Cell["\<\
In the first region, when \[CurlyEpsilon] + e E x \[LessEqual] - m, the \
electron lives in the Dirac see, i.e. it is in the vacuum. To see this more \
clearly think of the limit E\[Rule]0, in which \[CurlyEpsilon] \[LessEqual] - \
m.

In the second region, where \[CurlyEpsilon]+e E x \[GreaterEqual]  m, the \
electron occupies the conventional electron state in the upper continuum, \
compare \[CurlyEpsilon]\[GreaterEqual] m for E\[Rule]0.\
\>", "Text"],

Cell["The intermediate region", "Text"],

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Cell["is classically forbidden", "Text"],

Cell["\<\
A transition of the electron from the Dirac see (first region) into the \
conventional upper continuum (second region) describes the pair creation. In \
order to fulfill this transition the electron must cross the forbidden \
region.\
\>", "Text"]
}, Open  ]],

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Cell["Semiclassical approximation", "Section",
  Background->RGBColor[1, 1, 0]],

Cell["\<\
When the variation of the potential on the wavelenght is small, smaller then \
the energy itself,  the situation is lose to the classical picture. In this \
case the quantum effects can be described by the semiclassical approximation, \
called also the WKB-approximation. In this approx the wave function is \
written as\
\>", "Text"],

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A central role plays the calssicalmomentum, which defines the main \
exponential factor, and also the pre-exponential coefficient\
\>", "Text"],

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(Though this coefficient would not be considered below in detail, beein \
approximated by a constant A(x)\[Tilde]A.)\
\>", "Text"]
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Cell["Transition over the forbidden region", "Section",
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Cell[TextData[{
  "In order to estimate the descreas of the wave function when transition \
from the lower to the upper continuum takes place considered consider the \
semiclassical wave function, in which ",
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upper continuum (region two). It suffices to bring both these points to the \
boundaries of each region"
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Cell["\<\
 Then with the exponential accuracy the amplitude of the transition is \
described by\
\>", "Text"],

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Cell["Thus", "Text"],

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Cell["Amplitude is", "Text"],

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Cell["Probability of the pair creation is", "Text"],

Cell[BoxData[
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Cell["\<\
More accurately, the rate of the pair production per unit volume can be shown \
to be\
\>", "Text"],

Cell[BoxData[
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\) exp \((\(-\[Pi]\) 
              E\_QED\/E)\) \[Congruent] \
\[IndentingNewLine]\[IndentingNewLine]\(1\/\(4  \[Pi]\^3\)\) \
\((mc\^2\/\:f7d8)\) \(\((mc\/\:f7d8)\)\^3\) \(\((E\/E\_QED)\)\^2\) 
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Cell["It is valid provided", "Text"],

Cell[BoxData[
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