(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 21335, 613] NotebookOptionsPosition[ 19071, 546] NotebookOutlinePosition[ 19473, 563] CellTagsIndexPosition[ 19430, 560] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lecture 10", "Section", CellChangeTimes->{{3.426947592244849*^9, 3.4269475961628494`*^9}, { 3.4315423019150314`*^9, 3.4315423352420316`*^9}, {3.4327427093176003`*^9, 3.4327427098915997`*^9}, {3.432826830388052*^9, 3.432826830967052*^9}, { 3.4333500665675044`*^9, 3.4333500670195045`*^9}, {3.4340534150565147`*^9, 3.4340534158645144`*^9}, {3.4345732876377974`*^9, 3.4345732888757973`*^9}, { 3.434897875502897*^9, 3.434897876013897*^9}}], Cell["\<\ In the Lecture we discuss the general structure of group theory and the \ associated algebra and generators. Here we will use Mathematica to look at \ explicit representations of the elements of the related groups SU(2) and \ SO(3). We begin by the defining explicit representations of the generators. \ For SU(2) we have the Pauli matrices \ \>", "Text", CellChangeTimes->{{3.4269476003598495`*^9, 3.4269476357228494`*^9}, 3.4315423175140314`*^9, {3.431542352626032*^9, 3.431542424331032*^9}, { 3.4327427140746*^9, 3.4327429812265997`*^9}, 3.4327511315046*^9, 3.4327511653416*^9, {3.432826858714052*^9, 3.432827111073052*^9}, { 3.432827229514052*^9, 3.4328272577850523`*^9}, {3.432827310610052*^9, 3.432827312426052*^9}, 3.432827345393052*^9, {3.4328274568730516`*^9, 3.432827466097052*^9}, {3.4328340733070517`*^9, 3.432834109282052*^9}, { 3.4333500834845047`*^9, 3.4333502558835044`*^9}, {3.4333503504285045`*^9, 3.4333503966995044`*^9}, {3.4333514295795045`*^9, 3.4333514323355045`*^9}, {3.4340534436005144`*^9, 3.434053562807514*^9}, { 3.4341070329600143`*^9, 3.434107035436014*^9}, {3.4345732986842976`*^9, 3.4345733013132973`*^9}, {3.4345734090392976`*^9, 3.4345734322927976`*^9}, {3.4345735243652973`*^9, 3.434573548324298*^9}, { 3.4345735826372976`*^9, 3.4345736734922976`*^9}, 3.434653478878298*^9, { 3.4348978938068967`*^9, 3.4348980800508966`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[Sigma]", "1"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["\[Sigma]", "2"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "\[ImaginaryI]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[ImaginaryI]", ",", "0"}], "}"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["\[Sigma]", "3"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}]}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.434898084994897*^9, 3.434898159278897*^9}}], Cell["which are traceless", "Text", CellChangeTimes->{{3.4348981824918966`*^9, 3.434898192619897*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["\[Sigma]", "1"], "]"}]], "Input", CellChangeTimes->{{3.4348982005318966`*^9, 3.4348982085718966`*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["\[Sigma]", "2"], "]"}]], "Input", CellChangeTimes->{{3.4348982005318966`*^9, 3.4348982237868967`*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["\[Sigma]", "3"], "]"}]], "Input", CellChangeTimes->{{3.4348982005318966`*^9, 3.434898228810897*^9}}], Cell[TextData[{ "and satisfy the commutator (algebra) [", Cell[BoxData[ RowBox[{ SubscriptBox["\[Sigma]", "k"], "/", "2"}]], CellChangeTimes->{{3.4348982005318966`*^9, 3.434898228810897*^9}}], Cell[BoxData[ RowBox[{",", RowBox[{ SubscriptBox["\[Sigma]", "l"], "/", "2"}]}]], CellChangeTimes->{{3.4348982005318966`*^9, 3.434898228810897*^9}}], "] = \[ImaginaryI] ", Cell[BoxData[ FormBox[ SubscriptBox["\[Epsilon]", "klm"], TraditionalForm]]], Cell[BoxData[ SubscriptBox["\[Sigma]", "m"]], CellChangeTimes->{{3.4348982005318966`*^9, 3.434898228810897*^9}}], "/2" }], "Text", CellChangeTimes->{{3.4348982317938967`*^9, 3.4348983365828967`*^9}, { 3.434898443086897*^9, 3.434898443469897*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "1"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "2"], "2"]}], "-", RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "2"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "1"], "2"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}}], Cell[TextData[{ "which we recognize as \[ImaginaryI] ", Cell[BoxData[ SubscriptBox["\[Sigma]", "3"]], CellChangeTimes->{{3.4348982005318966`*^9, 3.434898228810897*^9}}], "/2. For the other cases we have" }], "Text", CellChangeTimes->{{3.4348984497018967`*^9, 3.4348984805558968`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "3"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "1"], "2"]}], "-", RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "1"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "3"], "2"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}, { 3.434898511994897*^9, 3.4348985233058968`*^9}}], Cell[TextData[{ "equals \[ImaginaryI] ", Cell[BoxData[ FormBox[ SubscriptBox["\[Sigma]", "2"], TraditionalForm]]], "/2, while " }], "Text", CellChangeTimes->{{3.4348985394588966`*^9, 3.434898561238897*^9}, { 3.4348985957488966`*^9, 3.4348986033008966`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "2"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "3"], "2"]}], "-", RowBox[{ FractionBox[ SubscriptBox["\[Sigma]", "3"], "2"], ".", FractionBox[ SubscriptBox["\[Sigma]", "2"], "2"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}, { 3.4348985646968966`*^9, 3.434898579846897*^9}}], Cell[TextData[{ "equals \[ImaginaryI] ", Cell[BoxData[ FormBox[ SubscriptBox["\[Sigma]", "1"], TraditionalForm]]], "/2. Now define these matrices as a vector of matrices" }], "Text", CellChangeTimes->{{3.4348985394588966`*^9, 3.434898561238897*^9}, { 3.434898606548897*^9, 3.4348986432508965`*^9}}], Cell[BoxData[ RowBox[{"\[Sigma]", "=", RowBox[{"{", RowBox[{ SubscriptBox["\[Sigma]", "1"], ",", SubscriptBox["\[Sigma]", "2"], ",", SubscriptBox["\[Sigma]", "3"]}], "}"}]}]], "Input", CellChangeTimes->{{3.434898653379897*^9, 3.4348986772008967`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"\[Sigma]", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.4348992762818966`*^9, 3.434899289869897*^9}}], Cell["and a corresponding vector of transformation angles", "Text", CellChangeTimes->{{3.4348986903948965`*^9, 3.434898710909897*^9}, { 3.4348987905858965`*^9, 3.4348987910498967`*^9}}], Cell[BoxData[ RowBox[{"\[CapitalTheta]", "=", RowBox[{"{", RowBox[{"\[Theta]1", ",", "\[Theta]2", ",", "\[Theta]3"}], "}"}]}]], "Input", CellChangeTimes->{{3.434898715999897*^9, 3.4348987415338964`*^9}, 3.434899242961897*^9, {3.434899317756897*^9, 3.434899360619897*^9}, { 3.434900261504897*^9, 3.434900272693897*^9}, 3.434900939183897*^9, { 3.4353308116173196`*^9, 3.4353308146113195`*^9}}], Cell[BoxData[ RowBox[{"Element", "[", RowBox[{ RowBox[{"{", RowBox[{"\[Theta]1", ",", "\[Theta]2", ",", "\[Theta]3"}], "}"}], ",", "Reals"}], "]"}]], "Input", CellChangeTimes->{{3.434901357239897*^9, 3.434901368310897*^9}}], Cell[BoxData["\[CapitalTheta]"], "Input", CellChangeTimes->{{3.4353308235213194`*^9, 3.4353308254883194`*^9}}], Cell[BoxData[ RowBox[{"Norm", "[", "\[CapitalTheta]", "]"}]], "Input", CellChangeTimes->{{3.434900956711897*^9, 3.434900963280897*^9}, { 3.4353308363993196`*^9, 3.43533083861932*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"\[CapitalTheta]", ".", "\[Sigma]"}], "]"}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4328273021790524`*^9, 3.432827304691052*^9}, { 3.4328273870430517`*^9, 3.432827397018052*^9}, {3.4333503994135046`*^9, 3.4333504541405044`*^9}, {3.4340535687635145`*^9, 3.4340536150015144`*^9}, { 3.4345736507752976`*^9, 3.4345737054617977`*^9}, {3.434899374330897*^9, 3.434899381897897*^9}, {3.4349008986898966`*^9, 3.434900945903897*^9}, { 3.4353308444153194`*^9, 3.4353308466573195`*^9}}], Cell[BoxData[ RowBox[{"g", "=", RowBox[{"MatrixForm", "[", RowBox[{"MatrixExp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{ RowBox[{"\[CapitalTheta]", ".", "\[Sigma]"}], "/", "2"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.4349012068808966`*^9, 3.4349012077928967`*^9}, { 3.4353308704773197`*^9, 3.4353308725613194`*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", "g", "]"}]], "Input", CellChangeTimes->{{3.4349074768713107`*^9, 3.4349074814567695`*^9}}], Cell[BoxData[ RowBox[{"%", "/.", RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", SuperscriptBox["\[Theta]1", "2"]}], "-", SuperscriptBox["\[Theta]2", "2"], "-", SuperscriptBox["\[Theta]3", "2"]}]], "\[Rule]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Norm", "[", "\[CapitalTheta]", "]"}]}]}]}]], "Input", CellChangeTimes->{{3.434901214455897*^9, 3.4349012257768965`*^9}, { 3.434907504595083*^9, 3.4349075056981936`*^9}, {3.43533088446132*^9, 3.4353308866243196`*^9}}], Cell["\<\ This SU(2) transformation (element of the SU(2) group, as specified by the \ parameter set \[Theta] and obtained by exponentiating, is what we found \ analytically in Ex B in HW V (the extra credit problem) - although \ Mathematica has trouble carrying out the full simplification. In particular \ for a rotation of 2\[Pi] about the z-axis we have\ \>", "Text", CellChangeTimes->{{3.4349001948688965`*^9, 3.4349002870598965`*^9}, { 3.434907741980819*^9, 3.4349078961438813`*^9}, 3.4349110307998815`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"g", "/.", RowBox[{"\[Theta]1", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Theta]2", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Theta]3", "\[Rule]", RowBox[{"2", "\[Pi]"}]}]}]], "Input", CellChangeTimes->{{3.434907899118881*^9, 3.4349079309868813`*^9}}], Cell["\<\ As discussed in class, a 2\[Pi] rotation in SU(2) just changes the sign, but \ \ \>", "Text", CellChangeTimes->{{3.4349079723648815`*^9, 3.4349080127608814`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"g", "/.", RowBox[{"\[Theta]1", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Theta]2", "\[Rule]", "0"}]}], "/.", RowBox[{"\[Theta]3", "\[Rule]", RowBox[{"4", "\[Pi]"}]}]}]], "Input", CellChangeTimes->{{3.434907899118881*^9, 3.4349079309868813`*^9}, { 3.4349080218178816`*^9, 3.4349080219778814`*^9}}], Cell["\<\ gets us back to where we started. Next consider the representations of the \ generators of the group SO(3) given by\ \>", "Text", CellChangeTimes->{{3.4349080311708813`*^9, 3.4349080662138815`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["J", "1"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"-", "\[ImaginaryI]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "\[ImaginaryI]", ",", "0"}], "}"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["J", "2"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "\[ImaginaryI]"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], ",", "0", ",", "0"}], "}"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["J", "3"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "\[ImaginaryI]"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"\[ImaginaryI]", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.4349080871648817`*^9, 3.4349082029278812`*^9}}], Cell["Again traceless, since a generator", "Text", CellChangeTimes->{{3.434908228835881*^9, 3.4349082383158817`*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["J", "1"], "]"}]], "Input", CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["J", "2"], "]"}]], "Input", CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}, { 3.4349082513078814`*^9, 3.4349082515158815`*^9}}], Cell[BoxData[ RowBox[{"Tr", "[", SubscriptBox["J", "3"], "]"}]], "Input", CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}, { 3.434908255323881*^9, 3.4349082559318814`*^9}}], Cell["The commutators are (see Eq. (10.6))", "Text", CellChangeTimes->{{3.434573713909298*^9, 3.4345737214257975`*^9}, { 3.434910217434881*^9, 3.4349102271778812`*^9}, {3.4349111153198814`*^9, 3.4349111230318813`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ SubscriptBox["J", "1"], ".", SubscriptBox["J", "2"]}], "-", RowBox[{ SubscriptBox["J", "2"], ".", SubscriptBox["J", "1"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}, { 3.434910264441881*^9, 3.4349102986148815`*^9}}], Cell[TextData[{ "which is \[ImaginaryI] ", Cell[BoxData[ SubscriptBox["J", "3"]], CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}, { 3.434908255323881*^9, 3.4349082559318814`*^9}}] }], "Text", CellChangeTimes->{{3.4349103416968813`*^9, 3.4349103549648814`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ SubscriptBox["J", "3"], ".", SubscriptBox["J", "1"]}], "-", RowBox[{ SubscriptBox["J", "1"], ".", SubscriptBox["J", "3"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}, { 3.434910264441881*^9, 3.4349102986148815`*^9}, {3.4349103662278814`*^9, 3.4349103739868813`*^9}}], Cell[TextData[{ "which is \[ImaginaryI] ", Cell[BoxData[ SubscriptBox["J", "2"]], CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}, { 3.434908255323881*^9, 3.4349082559318814`*^9}}], ", and " }], "Text", CellChangeTimes->{{3.434910382974881*^9, 3.4349104030718813`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{ SubscriptBox["J", "2"], ".", SubscriptBox["J", "3"]}], "-", RowBox[{ SubscriptBox["J", "3"], ".", SubscriptBox["J", "2"]}]}], "]"}]], "Input", CellChangeTimes->{{3.4348983753738966`*^9, 3.434898435122897*^9}, { 3.434910264441881*^9, 3.4349102986148815`*^9}, {3.4349103662278814`*^9, 3.4349103739868813`*^9}, {3.4349104295838814`*^9, 3.434910436974881*^9}}], Cell[TextData[{ "which is \[ImaginaryI] ", Cell[BoxData[ SubscriptBox["J", "1"]], CellChangeTimes->{{3.4349082116238813`*^9, 3.434908220877881*^9}, { 3.434908255323881*^9, 3.4349082559318814`*^9}}], ". The group elements can again be found by exponentiating the generators \ with the same vector of angles " }], "Text", CellChangeTimes->{{3.434910382974881*^9, 3.4349104030718813`*^9}, { 3.4349104484478817`*^9, 3.4349104947148814`*^9}}], Cell[BoxData[ RowBox[{"J", "=", RowBox[{"{", RowBox[{ SubscriptBox["J", "1"], ",", SubscriptBox["J", "2"], ",", SubscriptBox["J", "3"]}], "}"}]}]], "Input", CellChangeTimes->{{3.4349105314288816`*^9, 3.4349105550798817`*^9}}], Cell[BoxData[ RowBox[{"gg", "=", RowBox[{"MatrixForm", "[", RowBox[{"MatrixExp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"\[CapitalTheta]", ".", "J"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.4349012068808966`*^9, 3.4349012077928967`*^9}, 3.434910523066881*^9, {3.4349105619758816`*^9, 3.4349105883908815`*^9}, { 3.4353309650003195`*^9, 3.4353309676833196`*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", "gg", "]"}]], "Input", CellChangeTimes->{{3.4349074768713107`*^9, 3.4349074814567695`*^9}, 3.434910632179881*^9}], Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"%", "/.", RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", SuperscriptBox["\[Theta]1", "2"]}], "-", SuperscriptBox["\[Theta]2", "2"], "-", SuperscriptBox["\[Theta]3", "2"]}]], "\[Rule]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Norm", "[", "\[CapitalTheta]", "]"}]}]}]}], "]"}]], "Input", CellChangeTimes->{{3.434901214455897*^9, 3.4349012257768965`*^9}, { 3.434907504595083*^9, 3.4349075056981936`*^9}, {3.4349109062928815`*^9, 3.4349109127088814`*^9}, {3.4353309838153195`*^9, 3.4353309864513197`*^9}}],\ Cell["\<\ Again fairly messy in the general case. 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