(************** 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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Determine its eigenvalue." }], "Text"], Cell[CellGroupData[{ Cell["Solution", "Subtitle"], Cell[TextData[{ "Strategy.\n", Cell[BoxData[ FormBox[ StyleBox[\(H\&^\), FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["for the 3D rigid rotor is shown in Engel, eq. 7.17 (p. 112). The \ operator contains two terms, but the second term can be ignored because it is \ a multiple of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(d\^2\/\(d\ \[Phi]\^2\)\), FontWeight->"Plain"], TraditionalForm]], FontWeight->"Plain"], StyleBox["and ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\(\(d\^2\) \[Psi]\)\/\(d\ \[Phi]\^2\) = \ 0\), FontWeight->"Plain"], TraditionalForm]], FontWeight->"Plain"], StyleBox[". Thus, for our purposes, ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ OverscriptBox[ StyleBox["H", FontWeight->"Plain"], "^"], FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["is given by\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[\(H\&^\), FontWeight->"Plain"], " ", "=", " ", \(\(\(-\ hbar\^2\)\/\(2\ I\)\)\ [ 1\/\(sin\ \[Theta]\)\ \(d\/\(d\ \[Theta]\)\) \((\ sin\ \[Theta]\ d\/\(d\ \[Theta]\))\)]\)}], "\[IndentingNewLine]"}], TraditionalForm]], FontWeight->"Plain"], StyleBox["\n", FontWeight->"Plain"], "Execution.", StyleBox["\nTo simplify things, I will leave off the normalization factor. \ It does not affect the outcome of the eigenfunction test or the determination \ of the eigenvalue. I will conduct the eigenfunction test by evaluating ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\(H\&^\) \[Psi]\), FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["step-by-step.", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Psi]\ = \ 3\ Cos[\[Theta]]\^2\ - \ 1\), "\[IndentingNewLine]", \(w\ = \ D[\[Psi], \[Theta]]\)}], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \(\(-1\) + 3\ Cos[\[Theta]]\^2\)], "Output", CellLabel->"Out[37]="], Cell[BoxData[ \(\(-6\)\ Cos[\[Theta]]\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[38]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x\ = \ w\ Sin[\[Theta]]\)], "Input", CellLabel->"In[39]:="], Cell[BoxData[ \(\(-6\)\ Cos[\[Theta]]\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[39]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y\ = \ D[x, \[Theta]]\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \(\(-12\)\ Cos[\[Theta]]\^2\ Sin[\[Theta]] + 6\ Sin[\[Theta]]\^3\)], "Output", CellLabel->"Out[40]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(z\ = \ y/Sin[\[Theta]]\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(Csc[\[Theta]]\ \((\(-12\)\ Cos[\[Theta]]\^2\ Sin[\[Theta]] + 6\ Sin[\[Theta]]\^3)\)\)], "Output", CellLabel->"Out[41]="] }, Open ]], Cell["\<\ This result looks nothing like \[Psi]. However, we can divide it by \[Psi] \ and simplify to see whether it is a constant multiple of \[Psi].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[z/\[Psi]]\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ \(\(-6\)\)], "Output", CellLabel->"Out[47]="] }, Open ]], Cell[TextData[{ "Recalling that we still need to multiply this value by ", Cell[BoxData[ \(TraditionalForm\`\(-\ hbar\^2\)\/\(2\ I\)\)]], ", we can see that ", Cell[BoxData[ \(TraditionalForm\`\(H\&^\) \[Psi]\ = \ 6\ \(hbar\^2\/\(2\ I\)\) \[Psi]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`E\ = \ 6\ hbar\^2\/\(2\ I\)\)]], ".\n\n", StyleBox["Comment.", FontWeight->"Bold"], " Checking Engel eq. 7.33 (p. 116) shows that we have been working with ", StyleBox["Y", FontSlant->"Italic"], "(", StyleBox["l", FontSlant->"Italic"], " = 2, ", StyleBox["m", FontSlant->"Italic"], " = 0). The energy of this wave function can be calculated using Engel eq. \ 7.26 (p. 113) as ", Cell[BoxData[ \(TraditionalForm\`\(hbar\^2\/\(2\ I\)\) \((2)\) \((2 + 1)\)\ = \ 6\ hbar\^2\/\(2\ I\)\)]], " which is exactly what we found." }], "Text"] }, 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, 361, 9, 64, "Text"], Cell[CellGroupData[{ Cell[2202, 68, 28, 0, 56, "Subtitle"], Cell[2233, 70, 1983, 61, 323, "Text"], Cell[CellGroupData[{ Cell[4241, 135, 161, 3, 91, "Input"], Cell[4405, 140, 87, 2, 65, "Output"], Cell[4495, 144, 95, 2, 63, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4627, 151, 82, 2, 64, "Input"], Cell[4712, 155, 98, 2, 65, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4847, 162, 80, 2, 64, "Input"], Cell[4930, 166, 128, 3, 65, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5095, 174, 81, 2, 64, "Input"], Cell[5179, 178, 153, 3, 65, "Output"] }, Open ]], Cell[5347, 184, 165, 3, 64, "Text"], Cell[CellGroupData[{ Cell[5537, 191, 76, 2, 64, "Input"], Cell[5616, 195, 65, 2, 63, "Output"] }, Open ]], Cell[5696, 200, 920, 29, 201, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)