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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 186630, 3945]*) (*NotebookOutlinePosition[ 187393, 3971]*) (* CellTagsIndexPosition[ 187349, 3967]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem 5.6 (Engel)", "Title"], Cell[TextData[{ "This problem refers to a large amount of material in the book. I will not \ reproduce this material here except when it is needed.\n\nTwo typos that \ should be noted:\n1. ", StyleBox["Part A", FontWeight->"Bold"], " The right-hand side of the equation in the lower left corner should read: \ ", Cell[BoxData[ \(TraditionalForm\`\(\[ImaginaryI]\ k\)\/\[Kappa]\ F\ \[ExponentialE]\^\ \(\(+\[ImaginaryI]\)\ k\ a\)\)]], "\n2. 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\(\(sinh\^2\)(\[Kappa]\ a)\)/4 \((k\ \[Kappa])\)\^2]\)\)]}], FontWeight->"Plain"], TraditionalForm]]], "\n" }], "Text", FontWeight->"Bold"], Cell[TextData[{ StyleBox["Part E.", FontWeight->"Bold"], " Plot the transmission probability ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(F/A\)\( | \^2\)\)\)]], " for an electron as a function of energy for ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], " = 1.6 \[Times] ", Cell[BoxData[ \(TraditionalForm\`10\^\(-19\)\)]], " J and ", StyleBox["a", FontSlant->"Italic"], " = 9.0 \[Times] ", Cell[BoxData[ \(TraditionalForm\`10\^\(-10\)\)]], " m. Follow the transmission probability for energies ranging from 0 to 8 \ \[Times] ", Cell[BoxData[ \(TraditionalForm\`10\^\(-19\)\)]], "J. At what energy is the tunneling (transmission) probability equal to \ 0.1? 0.02?" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Solution to Part E", "Subtitle"], Cell[TextData[{ StyleBox["transmission or tunneling probability = ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"|", \(F\/A\), SuperscriptBox[ StyleBox["|", FontWeight->"Plain"], "2"]}], "=", " ", FractionBox[ StyleBox[\(\(\ \)\(1\)\), FontWeight-> "Plain"], \(1\ + \ \([\(\((k\^2\ + \ \[Kappa]\^2)\)\^2\) \ \(\(sinh\^2\)(\[Kappa]\ a)\)/4 \((k\ \[Kappa])\)\^2]\)\)]}], FontWeight->"Plain"], TraditionalForm]]], "\n\n", StyleBox["where\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(k\ = \ \@\(\(2 m\ E\)\/hbar\^2\)\), FontWeight->"Plain"], TraditionalForm]]], "\t\t", StyleBox["and\t", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\[Kappa]\ = \ \@\(\(2 m\ \((V\_0\ - \ E)\)\)\/hbar\^2\)\ \), FontWeight->"Plain"], TraditionalForm]]], "\n" }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear\ [Energy]\), "\[IndentingNewLine]", \(hbar\ = \ 1.0546\ 10\^\(-34\)\), "\[IndentingNewLine]", \(me\ = \ 9.1094\ 10\^\(-31\)\), "\[IndentingNewLine]", \(V\ = \ 1.6\ 10\^\(-19\)\), "\[IndentingNewLine]", \(a\ = \ 9\ 10\^\(-10\)\), "\[IndentingNewLine]", \(k\ = \ \@\(\(2\ me\ Energy\)\/hbar\^2\)\), "\[IndentingNewLine]", \(\[Kappa]\ = \ \@\(\(2\ me\ \((V\ - \ Energy)\)\)\/hbar\^2\)\), "\ \[IndentingNewLine]", \(TunnelP\ = \ 1\/\(1\ + \ \(\(\((k\^2 + \ \[Kappa]\^2)\)\^2\) Sinh[\[Kappa]\ a]\^2\)\ \/\(4\ \((k\ \[Kappa])\)\^2\)\)\)}], "Input", CellLabel->"In[159]:="], Cell[BoxData[ \(1.0546`*^-34\)], "Output", CellLabel->"Out[160]="], Cell[BoxData[ \(9.109400000000001`*^-31\)], "Output", CellLabel->"Out[161]="], Cell[BoxData[ \(1.6000000000000002`*^-19\)], "Output", CellLabel->"Out[162]="], Cell[BoxData[ \(9\/10000000000\)], "Output", CellLabel->"Out[163]="], Cell[BoxData[ \(1.2798884419092589`*^19\ \@Energy\)], "Output", CellLabel->"Out[164]="], Cell[BoxData[ \(1.2798884419092589`*^19\ \@\(1.6000000000000002`*^-19 - Energy\)\)], \ "Output", CellLabel->"Out[165]="], Cell[BoxData[ \(1/\((1 + \((9.316473221465272`*^-78\ \((1.6381144237329102`*^38\ \ \((1.6000000000000002`*^-19 - Energy)\) + 1.6381144237329102`*^38\ \ Energy)\)\^2\ Sinh[1.151899597718333`*^10\ \@\(1.6000000000000002`*^-19 - \ Energy\)]\^2)\)/\((\((1.6000000000000002`*^-19 - Energy)\)\ Energy)\))\)\)], "Output", CellLabel->"Out[166]="] }, Open ]], Cell[TextData[{ "Plotting tunneling probability over the energy range given in the book \ shows that the probability rises to one once ", Cell[BoxData[ \(TraditionalForm\`E\ > V\_0\)]], ". 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