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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[ 57757, 1517]*) (*NotebookOutlinePosition[ 58523, 1543]*) (* CellTagsIndexPosition[ 58479, 1539]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem 6.17 (Engel)", "Title"], Cell[TextData[{ "The questions that follow refer to the particle in a box shown in Figure \ 6.5 (", StyleBox["a", FontSlant->"Italic"], " = 5 nm).", StyleBox[" ", FontWeight->"Bold"], "The overarching goal of these questions is to show how the absolute and \ relative uncertainties in momentum (", Cell[BoxData[ \(TraditionalForm\`p\_x\)]], " in the book, but ", StyleBox["p", FontSlant->"Italic"], " here) represented by a ", StyleBox["single peak", FontVariations->{"Underline"->True}], " in the momentum distribution (Figure 6.5) vary with quantum number ", StyleBox["n", FontSlant->"Italic"], ".\n", StyleBox["\nPart A.", FontWeight->"Bold"], " The kinetic energy and momentum of a moving particle are related by ", StyleBox["E", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`p\^2/2\ m\)]], ". 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Show that ", Cell[BoxData[ \(TraditionalForm\`p\ = \ n\ h\ /\ 2\ a\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["Solution to Part A", "Subtitle"], Cell[TextData[{ StyleBox["The first equation can be rewritten as:\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(p\^2\), FontWeight->"Plain"], " ", "=", " ", \(2\ m\ E\)}], TraditionalForm]], FontWeight->"Plain"], StyleBox["\n\n\t", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"=", " ", RowBox[{"2", " ", "m", FractionBox[ RowBox[{\(n\^2\), " ", \(h\^2\), StyleBox[" ", FontWeight->"Plain"]}], \(8\ m\ a\^2\)]}]}], FontWeight->"Plain"], TraditionalForm]]], "\n\t\n\t", Cell[BoxData[ FormBox[ StyleBox[\(\(=\)\(\ \)\(\(\(n\^2\) h\^2\)\/\(4\ a\^2\)\)\), FontWeight->"Plain"], TraditionalForm]]], "\n\n", StyleBox["Taking the square root of both sides gives the desired result:\n\n\ ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(p\ = \(n\ h\)\/\(2\ a\)\), FontWeight->"Plain"], TraditionalForm]]], "\n" }], "Text", FontWeight->"Bold"], Cell[TextData[{ "Part B.", StyleBox[" The wave length of \[Psi] and the length of the box, ", FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", are related. Use this and the result from part A to show ", FontWeight->"Plain"], StyleBox["p = hbar k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell["Solution to Part B", "Subtitle"], Cell[TextData[{ StyleBox["Recall that for a wave, ", FontWeight->"Plain"], StyleBox["k = ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["2 \[Pi]", FontWeight->"Plain"], StyleBox[" / \[Lambda].\n\nThe wave functions, ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\[Psi]\_n\)], FontWeight->"Plain"], StyleBox[", of a particle in a box are shown in Figure 4.2. Visual \ inspection shows that \[Lambda]", FontWeight->"Plain"], StyleBox[" = 2", FontWeight->"Plain"], StyleBox[" a", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" / ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". Comparing this formula with the result of Part A shows that we \ can rewrite the momentum as ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(p\ = \ h\/\[Lambda]\), FontWeight->"Plain"], TraditionalForm]]], StyleBox[", which is the de Broglie relationship. 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", StyleBox["Part A", FontWeight->"Bold"], " showed that ", StyleBox["p", FontSlant->"Italic"], " is proportional to ", StyleBox["n", FontSlant->"Italic"], ". So it stands to reason that \[CapitalDelta]", StyleBox["p", FontSlant->"Italic"], "/", StyleBox["p", FontSlant->"Italic"], " ~ 1/", StyleBox["n", FontSlant->"Italic"], "." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{1014, 870}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.5, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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