(************** 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). 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[ 9150, 322]*) (*NotebookOutlinePosition[ 9839, 346]*) (* CellTagsIndexPosition[ 9795, 342]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem Set 6", "Title"], Cell["Excercise on transforming from Ls to jj coupling. ", "Subtitle"], Cell[CellGroupData[{ Cell["Prob 1", "Subsection"], Cell[TextData[{ "Write down the 2 by 2 matrix that transforms the jj coupled states |(1", Cell[BoxData[ \(TraditionalForm\`s\_\(1/2\)\)]], " 2", Cell[BoxData[ \(TraditionalForm\`p\_\(1/2\)\)]], ") 1> \nand |(1", Cell[BoxData[ \(TraditionalForm\`s\_\(1/2\)\)]], " 2", Cell[BoxData[ \(TraditionalForm\`p\_\(3/2\)\)]], ") 1> into LS coupled states (1s2p) ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)\(P\_1\)\)]], " and (1s2p) ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)\(P\_1\)\)]], "." }], "Text"], Cell["\<\ As a preliminary step define the 9j symbol nineJ and the LS-jj \ transformation matrix t[LS,jj]. \ \>", "Text"], Cell[BoxData[ \(nineJ[a_, b_, c_, d_, e_, f_, g_, h_, j_]\ := \[IndentingNewLine]Sum[\((\(-1\))\)^\((2\ x)\)\ \((2\ x\ + \ \ 1)\)\ SixJSymbol[{a, b, c}, {f, j, x}] SixJSymbol[{d, e, f}, {b, x, h}]\ SixJSymbol[{g, h, j}, {x, a, d}], {x, Max[Abs[a - j], Abs[b - f], Abs[d - h]], Min[a + j, b + f, d + h]}]\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(t[l1_, s1_, j1_, l2_, s2_, j2_, l_, s_, j_]\ := \[IndentingNewLine]Sqrt[\((2 l + 1)\) \((2 s + 1)\) \((2 j1 + 1)\) \((2 j2 + 1)\)]\ nineJ[l1, s1, j1, l2, s2, j2, l, s, j]\)], "Input", CellLabel->"In[2]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[Phi]LS[0]\ \ = \ Sum[t[0, 1/2, 1/2, 1, 1/2, j2, 1, 0, 1] \[Psi]jj[j2], {j2, 1/2, 3/2}]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(\[Psi]jj[1\/2]\/\@3 + \@\(2\/3\)\ \[Psi]jj[3\/2]\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Phi]LS[1]\ \ = \ Sum[t[0, 1/2, 1/2, 1, 1/2, j2, 1, 1, 1]\ \[Psi]jj[j2], {j2, 1/2, 3/2}]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(\@\(2\/3\)\ \[Psi]jj[1\/2] - \[Psi]jj[3\/2]\/\@3\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(mat1\ = \ {{\(\[Phi]LS[0]\)[\([1, 1]\)], \(\[Phi]LS[0]\)[\([2, 1]\)]}, {\(\[Phi]LS[1]\)[\([1, 1]\)], \(\[Phi]LS[1]\)[\([2, 1]\)] \(\[Phi]LS[1]\)[\([2, 2]\)]}} // MatrixForm\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\) \)\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@3\), \(\@\(2\/3\)\)}, {\(\@\(2\/3\)\), \(-\(1\/\@3\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[5]//MatrixForm="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Prob 2", "Subsection"], Cell[TextData[{ "\nWrite down the 3 by 3 matrix that transforms the jj coupled states \n\ |(2", Cell[BoxData[ \(TraditionalForm\`p\_\(1/2\)\)]], " 3", Cell[BoxData[ \(TraditionalForm\`d\_\(3/2\)\)]], ") 1>, (2", Cell[BoxData[ \(TraditionalForm\`p\_\(3/2\)\)]], " 3", Cell[BoxData[ \(TraditionalForm\`d\_\(3/2\)\)]], ") 1>, and |(2", Cell[BoxData[ \(TraditionalForm\`p\_\(3/2\)\)]], " 3", Cell[BoxData[ \(TraditionalForm\`d\_\(5/2\)\)]], ") 1 > \n into LS coupled states \n(2p3d) ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^1\)\(P\_1\)\)]], ", (2p3d) ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)\(P\_1\)\)]], " , and (2p3d) ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)\(D\_1\)\)]], ".\n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mat2 = Simplify[{{t[1, 1/2, 1/2, 2, 1/2, 3/2, 1, 0, 1], t[1, 1/2, 3/2, 2, 1/2, 3/2, 1, 0, 1], \[IndentingNewLine]t[1, 1/2, 3/2, 2, 1/2, 5/2, 1, 0, 1]}, {t[1, 1/2, 1/2, 2, 1/2, 3/2, 1, 1, 1], t[1, 1/2, 3/2, 2, 1/2, 3/2, 1, 1, 1], \[IndentingNewLine]t[1, 1/2, 3/2, 2, 1/2, 5/2, 1, 1, 1]}, {t[1, 1/2, 1/2, 2, 1/2, 3/2, 2, 1, 1], t[1, 1/2, 3/2, 2, 1/2, 3/2, 2, 1, 1], \[IndentingNewLine]t[1, 1/2, 3/2, 2, 1/2, 5/2, 2, 1, 1]}}]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \({{1\/\@3, 1\/\@15, \@\(3\/5\)}, {1\/\@6, 2\ \@\(2\/15\), \(-\@\(3\/10\)\)}, {1\/\@2, \(-\@\(2\/5\)\), \(-\(1\/\ \@10\)\)}}\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mat2]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@3\), \(1\/\@15\), \(\@\(3\/5\)\)}, {\(1\/\@6\), \(2\ \@\(2\/15\)\), \(-\@\(3\/10\)\)}, {\(1\/\@2\), \(-\@\(2\/5\)\), \(-\(1\/\@10\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[7]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Inverse[mat2]]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@3\), \(1\/\@6\), \(1\/\@2\)}, {\(1\/\@15\), \(2\ \@\(2\/15\)\), \(-\@\(2\/5\)\)}, {\(\@\(3\/5\)\), \(-\@\(3\/10\)\), \(-\(1\/\@10\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[8]//MatrixForm="] }, Open ]], Cell["\<\ Note that the inverse of this matrix is its transpose, as it should \ be for an orthogonal matrix.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Some Other Examples", "Subsection"], Cell[TextData[{ "Look at the case |(1", Cell[BoxData[ \(TraditionalForm\`s\_\(1/2\)\)]], " 2", Cell[BoxData[ \(TraditionalForm\`p\_\(1/2\)\)]], ") 0> coupled to give ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)\(P\_0\)\)]], ". In this case the matrix is 1 x 1. We find" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t[0, 1/2, 1/2, 1, 1/2, 1/2, 1, 1, 0]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(\(-1\)\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[TextData[{ "Here is another 1 x 1 case: |(1", Cell[BoxData[ \(TraditionalForm\`s\_\(1/2\)\)]], " 2", Cell[BoxData[ \(TraditionalForm\`p\_\(3/2\)\)]], ") 2> coupled to give ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\^3\)\(P\_2\)\)]], ". We find" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t[0, 1/2, 1/2, 1, 1/2, 3/2, 1, 1, 2]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["\<\ In these cases of only one intremediate state, the transformation \ is just the identity or a phase!\ \>", "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{942, 746}, WindowMargins->{{Automatic, 10}, {Automatic, 0}}, 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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