(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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The variance in our observations is given by:\n\n", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\ = \ \(\(<\)\(\((\(\(B\_i\)\(-\)\) < B > )\)\^2\)\(>\)\)\)]], "\n\nwhere ", Cell[BoxData[ \(TraditionalForm\`B\_i\)]], " is the value observed in a single measurement and <", StyleBox["B", FontSlant->"Italic"], "> is the average or mean value observed over all measurements.\n\nGiven \ the usual method for calculating expectation values or averages from wave \ functions, show that ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\ = \ \(\(<\)\(B\^2\)\(>\)\(\ \)\(-\ \ \(\(<\)\(B\)\( > \^2\)\)\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subtitle"], Cell[TextData[{ "Execution. ", StyleBox["(Note: I will leave the ", FontWeight->"Plain"], StyleBox["\"hat\"", FontWeight->"Plain"], StyleBox[" off of ", FontWeight->"Plain"], StyleBox["B", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" to", FontWeight->"Plain"], StyleBox[" make the solution easier to type.)\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Sigma]\^2\), FontWeight->"Plain"], " ", StyleBox["=", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox[\(\(<\)\(\((\(\(B\_i\)\(-\)\) < B > )\)\^2\)\(>\)\), FontWeight->"Plain"]}], TraditionalForm]]], StyleBox["\n\nThe expression for variance contains the average symbol \"< >\ \" twice. We must calculate the average ", FontWeight->"Plain"], StyleBox["of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\((\(\(B\_i\)\(-\)\) < B > )\)\^2\)], FontWeight->"Plain"], StyleBox[", and this has <", FontWeight->"Plain"], StyleBox["B", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["> embedded inside it.\n", FontWeight->"Plain"], StyleBox["\n\t", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\[Integral]\(\(\(\[Psi]\^*\)(\(\(B\_i\ \)\(-\)\) < B > )\)\^2\) \[Psi] \[DifferentialD]x\)\)\)], FontWeight->"Plain"], "\n\t\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\[Integral]\(\(\[Psi]\^*\)( B\_i\^2 - 2\ B\_i < B > \ \(+\ \(\(<\)\(B\)\( > \^2\)\)\))\)\ \[Psi] \(\(\ \[DifferentialD]\)\(x\)\(\ \)\)\)\)\)], FontWeight->"Plain"], StyleBox["\n\nAt this point, it is helpful to recall that <", FontWeight->"Plain"], StyleBox["B", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["> is just a number, a constant that can be pulled outside the \ integrand.\n", FontWeight->"Plain"], StyleBox["\n\t", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\[Integral]\(\(\[Psi]\^*\)( B\_i\^2)\) \[Psi] \[DifferentialD]x - 2\ < B > \[Integral]\(\(\[Psi]\^*\)( B\_i)\) \[Psi] \[DifferentialD]x\ + \ \(\(<\)\(B\)\( > \^2\ \)\(\[Integral]\(\[Psi]\^*\)\ \[Psi] \(\(\[DifferentialD]\)\(x\)\(\ \ \)\)\)\)\)\)\)], FontWeight->"Plain"], "\n\n", StyleBox["The three integrals, starting on the left, correspond to ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\(<\)\(B\^2\)\(>\)\), FontWeight->"Plain"], TraditionalForm]]], ", <", StyleBox["B", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[">, and 1, respectively. This simplifies the expression:\n", FontWeight->"Plain"], "\t\n\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["=", FontWeight->"Plain"], RowBox[{ StyleBox["<", FontWeight->"Plain"], SuperscriptBox[ StyleBox["B", FontWeight->"Plain"], "2"], ">", " ", \(-\ 2\), " ", "<", "B", ">", "<", "B", ">", " ", \(+\ \(\(<\)\(B\)\( > \^2\)\)\)}]}], TraditionalForm]], FontWeight->"Plain"], StyleBox["\n\n\t", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox["=", FontWeight->"Plain"], RowBox[{ StyleBox["<", FontWeight->"Plain"], SuperscriptBox[ StyleBox["B", FontWeight->"Plain"], "2"], ">", " ", \(-\ \(\(<\)\(B\)\( > \^2\)\)\)}]}], TraditionalForm]], FontWeight->"Plain"] }], "Text", FontWeight->"Bold"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{729, 651}, 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. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 37, 0, 129, "Title"], Cell[1816, 55, 1021, 27, 293, "Text"], Cell[CellGroupData[{ Cell[2862, 86, 28, 0, 56, "Subtitle"], Cell[2893, 88, 3767, 116, 605, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)