(************** 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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With this notation, we may write the \ wave function for a state (j,m) obtained by combining orbital angular moment \ 1 and spin angular momentum 1/2 as" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPsi][j_, m_]\ = Sum[ClebschGordan[{1, m - \[Mu]}, {1/2, \[Mu]}, {j, m}]\ \ \[Chi][\[Mu]]\ \ Y[1, m - \[Mu], \[Theta], \[Phi]], {\[Mu], \(-j\), j}]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(\[Sum]\+\(\[Mu] = \(-j\)\)\%j\((\(-1\))\)\^\(1\/2 + m\)\ \@\(1 + 2\ j\)\ \ ThreeJSymbol[{1, m - \[Mu]}, {1\/2, \[Mu]}, {j, \(-m\)}]\ Y[1, m - \[Mu], \[Theta], \[Phi]]\ \[Chi][\[Mu]]\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["\<\ There are two possibilities, j = 1/2 and j = 3/2. For the first \ case (j=1/2) the two substates are:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\[CapitalPsi][1/2, m], {m, \(-1\)/2, 1/2}]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \({\(Y[1, 0, \[Theta], \[Phi]]\ \[Chi][\(-\(1\/2\)\)]\)\/\@3 - \@\(2\/3\)\ \ Y[1, \(-1\), \[Theta], \[Phi]]\ \[Chi][1\/2], \@\(2\/3\)\ Y[1, 1, \[Theta], \[Phi]]\ \[Chi][\(-\(1\/2\)\)] - \(Y[1, 0, \[Theta], \ \[Phi]]\ \[Chi][1\/2]\)\/\@3}\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell["For the second case (j=3/2) the 4 substates are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\[CapitalPsi][3/2, m], {m, \(-3\)/2, 3/2}]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \({Y[1, \(-1\), \[Theta], \[Phi]]\ \[Chi][\(-\(1\/2\)\)], \@\(2\/3\)\ Y[ 1, 0, \[Theta], \[Phi]]\ \[Chi][\(-\(1\/2\)\)] + \(Y[1, \(-1\), \ \[Theta], \[Phi]]\ \[Chi][1\/2]\)\/\@3, \(Y[1, 1, \[Theta], \[Phi]]\ \ \[Chi][\(-\(1\/2\)\)]\)\/\@3 + \@\(2\/3\)\ Y[1, 0, \[Theta], \[Phi]]\ \[Chi][ 1\/2], Y[1, 1, \[Theta], \[Phi]]\ \[Chi][1\/2]}\)], "Output", CellLabel->"Out[3]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Prob 2", "Subsection"], Cell["\<\ Combining nuclear spin state \[CapitalTheta][m] ( I= 9/2) and \ atomic angular momentum state \[Psi][\[Mu]] ( J = 1/2), we obtain two possible values of F (4 & 5) with wavefunctions \ \[CapitalPsi][F,M]. The degeneracies of these states are 9 & 11, respectively. The corresponding wave functions are:\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPsi][F_, M_]\ = Sum[ClebschGordan[{9/2, M - \[Mu]}, {1/2, \[Mu]}, {F, M}]\ \ \[Psi][\[Mu]]\ \ \[CapitalTheta][ M - \[Mu]], {\[Mu], \(-F\) - 1/2, F + 1/2}]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\[Sum]\+\(\[Mu] = \(-\(1\/2\)\) - F\)\%\(1\/2 + F\)\((\(-1\))\)\^\(4 + \ M\)\ \@\(1 + 2\ F\)\ ThreeJSymbol[{9\/2, M - \[Mu]}, {1\/2, \[Mu]}, {F, \(-M\)}]\ \[CapitalTheta][ M - \[Mu]]\ \[Psi][\[Mu]]\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\[CapitalPsi][4, m], {m, \(-4\), 4}]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \({\(\[CapitalTheta][\(-\(7\/2\)\)]\ \[Psi][\(-\(1\/2\)\)]\)\/\@10 - \(3\ \ \[CapitalTheta][\(-\(9\/2\)\)]\ \[Psi][1\/2]\)\/\@10, \ \(\[CapitalTheta][\(-\(5\/2\)\)]\ \[Psi][\(-\(1\/2\)\)]\)\/\@5 - \(2\ \ \[CapitalTheta][\(-\(7\/2\)\)]\ \[Psi][1\/2]\)\/\@5, \@\(3\/10\)\ \ \[CapitalTheta][\(-\(3\/2\)\)]\ \[Psi][\(-\(1\/2\)\)] - \@\(7\/10\)\ \ \[CapitalTheta][\(-\(5\/2\)\)]\ \[Psi][ 1\/2], \@\(2\/5\)\ \[CapitalTheta][\(-\(1\/2\)\)]\ \ \[Psi][\(-\(1\/2\)\)] - \@\(3\/5\)\ \[CapitalTheta][\(-\(3\/2\)\)]\ \[Psi][ 1\/2], \(\[CapitalTheta][1\/2]\ \[Psi][\(-\(1\/2\)\)]\)\/\@2 - \(\ \[CapitalTheta][\(-\(1\/2\)\)]\ \[Psi][1\/2]\)\/\@2, \@\(3\/5\)\ \ \[CapitalTheta][3\/2]\ \[Psi][\(-\(1\/2\)\)] - \@\(2\/5\)\ \[CapitalTheta][ 1\/2]\ \[Psi][1\/2], \@\(7\/10\)\ \[CapitalTheta][ 5\/2]\ \[Psi][\(-\(1\/2\)\)] - \@\(3\/10\)\ \[CapitalTheta][ 3\/2]\ \[Psi][ 1\/2], \(2\ \[CapitalTheta][7\/2]\ \[Psi][\(-\(1\/2\)\)]\)\/\@5 - \ \(\[CapitalTheta][5\/2]\ \[Psi][1\/2]\)\/\@5, \(3\ \[CapitalTheta][9\/2]\ \ \[Psi][\(-\(1\/2\)\)]\)\/\@10 - \(\[CapitalTheta][7\/2]\ \ \[Psi][1\/2]\)\/\@10}\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(Table[\[CapitalPsi][5, m], {m, \(-5\), 5}]\)\)\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \({\[CapitalTheta][\(-\(9\/2\)\)]\ \[Psi][\(-\(1\/2\)\)], \(3\ \ \[CapitalTheta][\(-\(7\/2\)\)]\ \[Psi][\(-\(1\/2\)\)]\)\/\@10 + \(\ \[CapitalTheta][\(-\(9\/2\)\)]\ \[Psi][1\/2]\)\/\@10, \(2\ \ \[CapitalTheta][\(-\(5\/2\)\)]\ \[Psi][\(-\(1\/2\)\)]\)\/\@5 + \(\ \[CapitalTheta][\(-\(7\/2\)\)]\ \[Psi][1\/2]\)\/\@5, \@\(7\/10\)\ \ \[CapitalTheta][\(-\(3\/2\)\)]\ \[Psi][\(-\(1\/2\)\)] + \@\(3\/10\)\ \ \[CapitalTheta][\(-\(5\/2\)\)]\ \[Psi][ 1\/2], \@\(3\/5\)\ \[CapitalTheta][\(-\(1\/2\)\)]\ \ \[Psi][\(-\(1\/2\)\)] + \@\(2\/5\)\ \[CapitalTheta][\(-\(3\/2\)\)]\ \[Psi][ 1\/2], \(\[CapitalTheta][1\/2]\ \[Psi][\(-\(1\/2\)\)]\)\/\@2 + \(\ \[CapitalTheta][\(-\(1\/2\)\)]\ \[Psi][1\/2]\)\/\@2, \@\(2\/5\)\ \ \[CapitalTheta][3\/2]\ \[Psi][\(-\(1\/2\)\)] + \@\(3\/5\)\ \[CapitalTheta][ 1\/2]\ \[Psi][1\/2], \@\(3\/10\)\ \[CapitalTheta][ 5\/2]\ \[Psi][\(-\(1\/2\)\)] + \@\(7\/10\)\ \[CapitalTheta][ 3\/2]\ \[Psi][ 1\/2], \(\[CapitalTheta][7\/2]\ \[Psi][\(-\(1\/2\)\)]\)\/\@5 + \ \(2\ \[CapitalTheta][5\/2]\ \[Psi][1\/2]\)\/\@5, \(\[CapitalTheta][9\/2]\ \ \[Psi][\(-\(1\/2\)\)]\)\/\@10 + \(3\ \[CapitalTheta][7\/2]\ \ \[Psi][1\/2]\)\/\@10, \[CapitalTheta][9\/2]\ \[Psi][1\/2]}\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\[CapitalPsi][4, 4]\ \[CapitalPsi][5, 4]]\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(3\/10\ \[CapitalTheta][9\/2]\^2\ \[Psi][\(-\(1\/2\)\)]\^2 + 4\/5\ \[CapitalTheta][7\/2]\ \[CapitalTheta][ 9\/2]\ \[Psi][\(-\(1\/2\)\)]\ \[Psi][1\/2] - 3\/10\ \[CapitalTheta][7\/2]\^2\ \[Psi][1\/2]\^2\)], "Output", CellLabel->"Out[28]="] }, Open ]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{686, 613}, WindowMargins->{{Automatic, 217}, {Automatic, 109}}, 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, 44, 0, 37, "Subtitle"], Cell[CellGroupData[{ Cell[1845, 57, 28, 0, 54, "Subsection"], Cell[1876, 59, 439, 10, 44, "Text"], Cell[CellGroupData[{ Cell[2340, 73, 248, 5, 58, "Input"], Cell[2591, 80, 250, 4, 81, "Output"] }, Open ]], Cell[2856, 87, 125, 3, 26, "Text"], Cell[CellGroupData[{ Cell[3006, 94, 105, 2, 42, "Input"], Cell[3114, 98, 306, 5, 119, "Output"] }, Open ]], Cell[3435, 106, 64, 0, 26, "Text"], Cell[CellGroupData[{ Cell[3524, 110, 105, 2, 42, "Input"], Cell[3632, 114, 415, 6, 119, "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[4096, 126, 28, 0, 32, "Subsection"], Cell[4127, 128, 331, 7, 62, "Text"], Cell[CellGroupData[{ Cell[4483, 139, 256, 5, 58, "Input"], Cell[4742, 146, 279, 5, 97, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5058, 156, 100, 2, 42, "Input"], Cell[5161, 160, 1202, 19, 259, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6400, 184, 114, 2, 42, "Input"], Cell[6517, 188, 1296, 20, 264, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7850, 213, 106, 2, 42, "Input"], Cell[7959, 217, 282, 5, 60, "Output"] }, Open ]] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)