(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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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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[ 20009, 694]*) (*NotebookOutlinePosition[ 21182, 730]*) (* CellTagsIndexPosition[ 21138, 726]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem Set 1", "Subtitle"], Cell[CellGroupData[{ Cell["Prob 1", "Subsection"], Cell["\<\ Start with a general function of \[Theta] and \[Phi], g =\[Psi](\ \[Theta],\[Phi])\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g\ = \ \[Psi][\[Theta], \[Phi]]\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(\[Psi][\[Theta], \[Phi]]\)], "Output", CellLabel->"Out[1]="] }, Open ]], Cell[TextData[{ "Evaluate lsq = ", Cell[BoxData[ \(TraditionalForm\`L\^2\)]], "\[Psi](\[Theta],\[Phi])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lsq\ = Simplify[\ \(-\ \((Csc[\[Theta]] D[Sin[\[Theta]] D[g, \[Theta]], \[Theta]] + \ \((Csc[\[Theta]])\)^2 D[D[g, \[Phi]], \[Phi]])\)\)]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ RowBox[{ RowBox[{\(-Csc[\[Theta]]\^2\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "-", RowBox[{\(Cot[\[Theta]]\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "-", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}]], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ "Evaluate lz = ", Cell[BoxData[ \(TraditionalForm\`L\_z\)]], " \[Psi](\[Theta],\[Phi]) and lz2 = ", Cell[BoxData[ \(TraditionalForm\`L\_z\%2\)]], " \[Psi](\[Theta],\[Phi])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lz\ = \ I\ D[g, \[Phi]]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}]], "Output", CellLabel->"Out[2]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(lz2\ = \ I\ D[lz, \[Phi]]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ RowBox[{"-", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}]], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[TextData[{ "Introduce the function lpg = ", Cell[BoxData[ \(TraditionalForm\`L\_\(\(+\)\(\ \)\)\)]], "\[Psi](\[Theta],\[Phi])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(lpg\ = \ \ E^\((I\ \[Phi])\)\ \((D[g, \[Theta]] + \ I\ Cot[\[Theta]]\ D[g, \[Phi]])\)\)\(\[IndentingNewLine]\) \)\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[ImaginaryI]", " ", \(Cot[\[Theta]]\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "+", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], ")"}]}]], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "Now, evaluate lmp = ", Cell[BoxData[ \(TraditionalForm\`\(L\_-\)\)]], "[lpg] = ", Cell[BoxData[ \(TraditionalForm\`\(L\_-\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(L\_+\)\)]], " \[Psi](\[Theta],\[Phi])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lmp\ = \ Simplify[\ E^\((\(-I\)\ \[Phi])\)\ \((\(-D[lpg, \[Theta]]\) + \ I\ Cot[\[Theta]]\ D[lpg, \[Phi]])\)]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ RowBox[{ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "-", RowBox[{\(Cot[\[Theta]]\^2\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "-", RowBox[{\(Cot[\[Theta]]\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}], "-", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(\[Theta], \[Phi]\), "]"}]}]], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[TextData[{ "With the aid of this, we can determine form1 = (", Cell[BoxData[ \(TraditionalForm\`\(L\_-\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(L\_+\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(L\_z\%2\)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`L\_z\)]], ") \[Psi](\[Theta],\[Phi])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(form1\ = \ Simplify[lmp\ + \ lz2\ - 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Note: The factor \ (-1)^(3m+Abs[m]) reduces to (-1)^m for m<0 but reduces to (-1)^(2m) == +1 for m>0. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tab1\ = Table[\((\(-1\))\)^\((\((3\ m + Abs[m])\)/2)\)\ Sqrt[ 1/\((2\ Pi)\)]\ E^\((I\ m\ \[Phi])\)\ FullSimplify[ posf[\([Abs[m] + 1]\)]], {m, \(-l\), l}]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({3\/16\ \[ExponentialE]\^\(\(-4\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\(2\ \ \[Pi]\)\)\ Sin[\[Theta]]\^4, 3\/8\ \[ExponentialE]\^\(\(-3\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\[Pi]\ \)\ Cos[\[Theta]]\ Sin[\[Theta]]\^3, 3\/16\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(5\/\(2\ \ \[Pi]\)\)\ \((5 + 7\ Cos[2\ \[Theta]])\)\ Sin[\[Theta]]\^2, 3\/32\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \@\(5\/\[Pi]\)\ \ \((9\ Cos[\[Theta]] + 7\ Cos[3\ \[Theta]])\)\ Sin[\[Theta]], \(3\ \((9 + 20\ Cos[2\ \ \[Theta]] + 35\ Cos[4\ \[Theta]])\)\)\/\(128\ \@\[Pi]\), \(-\(3\/32\)\)\ \ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ \@\(5\/\[Pi]\)\ \((9\ \ Cos[\[Theta]] + 7\ Cos[3\ \[Theta]])\)\ Sin[\[Theta]], 3\/16\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \@\(5\/\(2\ \[Pi]\ \)\)\ \((5 + 7\ Cos[2\ \[Theta]])\)\ Sin[\[Theta]]\^2, \(-\(3\/8\)\)\ \ \[ExponentialE]\^\(3\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\[Pi]\)\ Cos[\[Theta]]\ \ Sin[\[Theta]]\^3, 3\/16\ \[ExponentialE]\^\(4\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\(2\ \ \[Pi]\)\)\ Sin[\[Theta]]\^4}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["Make a table of spherical harmonics generated by mathematica", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tab2\ = \ Table[SphericalHarmonicY[l, m, \[Theta], \[Phi]], {m, \(-l\), l}]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \({3\/16\ \[ExponentialE]\^\(\(-4\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\(2\ \ \[Pi]\)\)\ Sin[\[Theta]]\^4, 3\/8\ \[ExponentialE]\^\(\(-3\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\[Pi]\ \)\ Cos[\[Theta]]\ Sin[\[Theta]]\^3, 3\/8\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ \[Phi]\)\ \@\(5\/\(2\ \ \[Pi]\)\)\ \((\(-1\) + 7\ Cos[\[Theta]]\^2)\)\ Sin[\[Theta]]\^2, 3\/8\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \@\(5\/\[Pi]\)\ \ Cos[\[Theta]]\ \((\(-3\) + 7\ Cos[\[Theta]]\^2)\)\ Sin[\[Theta]], \(3\ \((3 - 30\ Cos[\ \[Theta]]\^2 + 35\ Cos[\[Theta]]\^4)\)\)\/\(16\ \@\[Pi]\), \(-\(3\/8\)\)\ \ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ \@\(5\/\[Pi]\)\ Cos[\[Theta]]\ \ \((\(-3\) + 7\ Cos[\[Theta]]\^2)\)\ Sin[\[Theta]], 3\/8\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \@\(5\/\(2\ \ \[Pi]\)\)\ \((\(-1\) + 7\ Cos[\[Theta]]\^2)\)\ Sin[\[Theta]]\^2, \(-\(3\/8\)\)\ \ \[ExponentialE]\^\(3\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\[Pi]\)\ Cos[\[Theta]]\ \ Sin[\[Theta]]\^3, 3\/16\ \[ExponentialE]\^\(4\ \[ImaginaryI]\ \[Phi]\)\ \@\(35\/\(2\ \ \[Pi]\)\)\ Sin[\[Theta]]\^4}\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell["\<\ Some of these formulas look the same but others are different. Are \ they equivalent? Let us expand both and test for equality.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ TrigExpand[tab1[\([m]\)]] === TrigExpand[tab2[\([m]\)]], {m, 1, 2 l + 1}]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \({True, True, True, True, True, True, True, True, True}\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["Check orthonormality", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Integrate[ Sin[\[Theta]]\ Integrate[\ Conjugate[\ tab1[\([i]\)]]\ tab1[\([j]\)], {\[Phi], 0, 2\ Pi}], {\[Theta], 0, Pi}], {i, 1, 2\ l + 1}, {j, \ 1, \ 2\ l + 1}]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \({{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}}\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[%]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "1", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[8]//MatrixForm="] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{800, 884}, WindowMargins->{{184, Automatic}, {6, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PrintingMargins"->{{54, 54}, {72, 72}}, "PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PrintCellBrackets"->False, "PrintRegistrationMarks"->True, "PrintMultipleHorizontalPages"->False, "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "afs", "nd.edu", \ "users", "johnson", "Class", "Class05F"}, "ProblemSet1.nb.ps", \ CharacterEncoding -> "iso8859-1"], "Magnification"->1}, CellLabelAutoDelete->False, Magnification->1, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* Cached data follows. 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