(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 26474, 719] NotebookOptionsPosition[ 26032, 700] NotebookOutlinePosition[ 26375, 715] CellTagsIndexPosition[ 26332, 712] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Spin three halves operator question (Sakurai 3.33). Implement operators \ for ", Cell[BoxData[ RowBox[{ SubscriptBox["S", "+"], ",", " ", SubscriptBox["S", "-"], ",", " ", RowBox[{"and", " ", SubscriptBox["S", "z"]}]}]], CellChangeTimes->{{3.6591761462362566`*^9, 3.6591763065036855`*^9}, { 3.6591774238525934`*^9, 3.6591774434264517`*^9}, {3.659177500734326*^9, 3.6591775632792015`*^9}, {3.659177593737369*^9, 3.6591777062917204`*^9}, { 3.6591777595780773`*^9, 3.6591780194550138`*^9}, {3.659178074708047*^9, 3.659178554250741*^9}, {3.6591786306107516`*^9, 3.659178835426964*^9}, { 3.6591788732543907`*^9, 3.6591789328791437`*^9}, {3.6591790334314375`*^9, 3.6591795575940332`*^9}, {3.659179592032323*^9, 3.6591796088404636`*^9}, { 3.659179648100952*^9, 3.6591798983097506`*^9}}], " that act on kets |\[PlusMinus]3/2>,|\[PlusMinus]1/2>. These operators act \ on only a single basis element, but are used to construct the matrix \ representations for these operators (which are more general). Use those \ matrices to compute the representation of the Hamiltonian for the problem and \ compute its energy eigenvalues." }], "Text", CellChangeTimes->{{3.6591761039778214`*^9, 3.6591761166280994`*^9}, { 3.6591799559278245`*^9, 3.6591801048675003`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"ClearAll", "[", RowBox[{ "a", ",", "b", ",", " ", "hbar", ",", " ", "display", ",", " ", "basisVectors", ",", " ", "mValues", ",", " ", "m", ",", " ", "vec", ",", " ", "raiseOrLower", ",", " ", "raise", ",", " ", "lower", ",", " ", "splus", ",", " ", "sminus", ",", " ", "sz", ",", " ", "szOp", ",", " ", "sSquared", ",", " ", "h", ",", " ", "sopts"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"On", "[", "Assert", "]"}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"$Assumptions", "=", RowBox[{"(", RowBox[{"And", "@@", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"#", ">", "0"}], "&"}], ")"}], "/@", RowBox[{"{", RowBox[{"hbar", ",", " ", "a", ",", "b"}], "}"}]}]}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"display", " ", ":=", " ", RowBox[{"{", RowBox[{ RowBox[{"hbar", " ", "\[Rule]", " ", "\[HBar]"}], ",", " ", RowBox[{"a", "\[Rule]", " ", "\"\\""}], ",", " ", RowBox[{"b", "\[Rule]", "\"\\""}]}], "}"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"basisVectors", " ", "=", " ", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"values", " ", "for", " ", "spin", " ", RowBox[{"3", "/", "2"}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"mValues", " ", ":=", " ", RowBox[{"{", RowBox[{ RowBox[{"3", "/", "2"}], ",", RowBox[{"1", "/", "2"}], ",", RowBox[{ RowBox[{"-", "1"}], "/", "2"}], ",", RowBox[{ RowBox[{"-", "3"}], "/", "2"}]}], "}"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ RowBox[{"Map", " ", "the", " ", "values", " ", "m"}], ",", " ", RowBox[{"in", "|", RowBox[{"m", ">", " ", RowBox[{ "to", " ", "the", " ", "standard", " ", "basis", " ", "for", " ", SuperscriptBox["R", "4"]}]}]}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"vec", "[", RowBox[{"3", "/", "2"}], "]"}], " ", ":=", " ", RowBox[{"basisVectors", "[", RowBox[{"[", RowBox[{"1", ",", "All"}], "]"}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"vec", "[", RowBox[{"1", "/", "2"}], "]"}], " ", ":=", " ", RowBox[{"basisVectors", "[", RowBox[{"[", RowBox[{"2", ",", "All"}], "]"}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"vec", "[", RowBox[{ RowBox[{"-", "1"}], "/", "2"}], "]"}], " ", ":=", " ", RowBox[{"basisVectors", "[", RowBox[{"[", RowBox[{"3", ",", "All"}], "]"}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"vec", "[", RowBox[{ RowBox[{"-", "3"}], "/", "2"}], "]"}], " ", ":=", " ", RowBox[{"basisVectors", "[", RowBox[{"[", RowBox[{"4", ",", "All"}], "]"}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Any", " ", "invalid", " ", "value", " ", "is", " ", "mapped", " ", "to", " ", "a", " ", "zero", " ", "vector", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"i", ".", "e", ".", " ", SubscriptBox["S", "+"]}], "|", RowBox[{ RowBox[{"3", "/", "2"}], ">"}]}], ",", " ", RowBox[{ RowBox[{"or", " ", SubscriptBox["S", "-"]}], "|", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "/", "2"}], ">"}]}]}]}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"vec", "[", "n_", "]"}], " ", ":=", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"For", " ", "a", " ", "basis", " ", "vector"}], " ", "|", RowBox[{"m", ">", " ", RowBox[{ RowBox[{"(", RowBox[{"not", " ", "a", " ", "linear", " ", "combination"}], ")"}], " ", "return", " ", "the", " ", "numeric", " ", "value", " ", "m"}]}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", "[", "v_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"v", ".", RowBox[{"DiagonalMatrix", "[", "mValues", "]"}]}], " ", "//", " ", "Tr"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"s", " ", "=", " ", RowBox[{"1", " ", "for", " ", "raise"}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"raiseOrLower", "[", RowBox[{"v_", ",", " ", "s_Integer"}], "]"}], " ", ":=", " ", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "mv", "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Assert", "[", RowBox[{ RowBox[{"s", " ", "\[Equal]", " ", "1"}], " ", "||", " ", RowBox[{"s", " ", "\[Equal]", " ", RowBox[{"-", "1"}]}]}], "]"}], " ", ";", "\[IndentingNewLine]", RowBox[{"mv", " ", "=", " ", RowBox[{"m", "[", "v", "]"}]}], " ", ";", "\[IndentingNewLine]", RowBox[{"hbar", " ", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"3", "/", "2"}], " ", "-", RowBox[{"s", " ", "mv"}]}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"5", "/", "2"}], " ", "+", " ", RowBox[{"s", " ", "mv"}]}], ")"}]}], "]"}], " ", RowBox[{"vec", "[", RowBox[{"mv", "+", " ", "s"}], " ", "]"}]}]}]}], "\[IndentingNewLine]", "]"}]}], " ", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"raise", "[", "v_", "]"}], " ", ":=", " ", RowBox[{"raiseOrLower", "[", RowBox[{"v", ",", "1"}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"lower", "[", "v_", "]"}], " ", ":=", " ", RowBox[{"raiseOrLower", "[", RowBox[{"v", ",", RowBox[{"-", "1"}]}], "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"szOp", "[", "v_", "]"}], " ", ":=", " ", RowBox[{"hbar", " ", RowBox[{"m", "[", "v", "]"}], " ", "v"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Form", " ", "the", " ", "matrix", " ", "for", " ", "an", " ", "operator", " ", "that", " ", "acts", " ", "on", " ", RowBox[{"vec", "[", "m", "]"}], " ", RowBox[{"values", "."}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"operatorToMatrix", "[", "o_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"o", "[", RowBox[{"vec", "[", "#", "]"}], "]"}], " ", "&"}], "/@", "mValues"}], ")"}], "//", " ", "Transpose"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Matrixes", " ", "for", " ", SubscriptBox["S", "+"]}], ",", " ", SubscriptBox["S", "-"], ",", " ", RowBox[{"and", " ", SubscriptBox["S", "z"]}], ",", " ", SuperscriptBox["S", "2"]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"splus", " ", "=", " ", RowBox[{"operatorToMatrix", "[", "raise", "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sminus", " ", "=", " ", RowBox[{"operatorToMatrix", "[", "lower", "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sz", " ", ":=", " ", RowBox[{"operatorToMatrix", "[", "szOp", "]"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"sSquared", " ", ":=", " ", RowBox[{ RowBox[{"(", RowBox[{"3", "/", "2"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"3", "/", "2"}], "+", "1"}], ")"}], " ", RowBox[{"hbar", "^", "2"}], " ", RowBox[{"IdentityMatrix", "[", "4", "]"}]}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"The", " ", "Hamiltonian", " ", "for", " ", "the", " ", RowBox[{"problem", "."}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"h", " ", "=", " ", RowBox[{ RowBox[{"a", " ", RowBox[{"(", " ", RowBox[{ RowBox[{"3", " ", RowBox[{"sz", ".", "sz"}]}], " ", "-", " ", "sSquared"}], ")"}]}], " ", "+", " ", RowBox[{"b", " ", RowBox[{"(", " ", RowBox[{ RowBox[{"splus", ".", "splus"}], " ", "+", " ", RowBox[{"sminus", ".", "sminus"}]}], ")"}]}]}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Display", " ", "the", " ", "Hamiltonian", " ", "and", " ", "the", " ", "energy", " ", RowBox[{"eigenvalues", "."}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"sopts", " ", "=", " ", RowBox[{"{", RowBox[{"14", ",", " ", "\"\\""}], "}"}]}], " ", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Style", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"#1", ",", "\"\< = \>\"", ",", " ", RowBox[{"#2", " ", "/.", " ", "display"}]}], " ", "}"}], " ", "&"}], "@@@", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(S\), \(+\)]\)\>\"", ",", RowBox[{"splus", " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(S\), \(-\)]\)\>\"", ",", RowBox[{"sminus", " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(S\), \(x\)]\)\>\"", ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"splus", " ", "+", " ", "sminus"}], ")"}], "/", RowBox[{"(", "2", ")"}]}], " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(S\), \(y\)]\)\>\"", ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"splus", " ", "-", " ", "sminus"}], ")"}], "/", RowBox[{"(", RowBox[{"2", "I"}], ")"}]}], " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(S\), \(z\)]\)\>\"", ",", RowBox[{"sz", " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\\"", ",", RowBox[{"h", " ", "//", " ", "MatrixForm"}]}], " ", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\\"", ",", RowBox[{ RowBox[{"Eigenvalues", "[", "h", "]"}], " ", "//", " ", "Column"}]}], "}"}]}], "\[IndentingNewLine]", "}"}]}], " ", "//", " ", "Grid"}], " ", ",", "sopts"}], "]"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.6591761462362566`*^9, 3.6591763065036855`*^9}, { 3.6591774238525934`*^9, 3.6591774434264517`*^9}, {3.659177500734326*^9, 3.6591775632792015`*^9}, {3.659177593737369*^9, 3.6591777062917204`*^9}, { 3.6591777595780773`*^9, 3.6591780194550138`*^9}, {3.659178074708047*^9, 3.659178554250741*^9}, {3.6591786306107516`*^9, 3.659178835426964*^9}, { 3.6591788732543907`*^9, 3.6591789328791437`*^9}, {3.6591790334314375`*^9, 3.6591795575940332`*^9}, {3.659179592032323*^9, 3.6591796088404636`*^9}, { 3.659179648100952*^9, 3.6591798983097506`*^9}}], Cell[BoxData[ StyleBox[ TagBox[GridBox[{ {"\<\"\\!\\(\\*SubscriptBox[\\(S\\), \\(+\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "0", "0"}, {"0", "0", RowBox[{"2", " ", "\[HBar]"}], "0"}, {"0", "0", "0", RowBox[{ SqrtBox["3"], " ", "\[HBar]"}]}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SubscriptBox[\\(S\\), \\(-\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, { RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "0", "0", "0"}, {"0", RowBox[{"2", " ", "\[HBar]"}], "0", "0"}, {"0", "0", RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SubscriptBox[\\(S\\), \\(x\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", FractionBox[ RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "2"], "0", "0"}, { FractionBox[ RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "2"], "0", "\[HBar]", "0"}, {"0", "\[HBar]", "0", FractionBox[ RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "2"]}, {"0", "0", FractionBox[ RowBox[{ SqrtBox["3"], " ", "\[HBar]"}], "2"], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SubscriptBox[\\(S\\), \\(y\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", "\[ImaginaryI]", " ", SqrtBox["3"], " ", "\[HBar]"}], "0", "0"}, { RowBox[{ FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SqrtBox["3"], " ", "\[HBar]"}], "0", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "\[HBar]"}], "0"}, {"0", RowBox[{"\[ImaginaryI]", " ", "\[HBar]"}], "0", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", "\[ImaginaryI]", " ", SqrtBox["3"], " ", "\[HBar]"}]}, {"0", "0", RowBox[{ FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SqrtBox["3"], " ", "\[HBar]"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SubscriptBox[\\(S\\), \\(z\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { FractionBox[ RowBox[{"3", " ", "\[HBar]"}], "2"], "0", "0", "0"}, {"0", FractionBox["\[HBar]", "2"], "0", "0"}, {"0", "0", RowBox[{"-", FractionBox["\[HBar]", "2"]}], "0"}, {"0", "0", "0", RowBox[{"-", FractionBox[ RowBox[{"3", " ", "\[HBar]"}], "2"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"H\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"3", " ", "\<\"A\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0", RowBox[{"2", " ", SqrtBox["3"], " ", "\<\"B\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0"}, {"0", RowBox[{ RowBox[{"-", "3"}], " ", "\<\"A\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0", RowBox[{"2", " ", SqrtBox["3"], " ", "\<\"B\"\>", " ", SuperscriptBox["\[HBar]", "2"]}]}, { RowBox[{"2", " ", SqrtBox["3"], " ", "\<\"B\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0", RowBox[{ RowBox[{"-", "3"}], " ", "\<\"A\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0"}, {"0", RowBox[{"2", " ", SqrtBox["3"], " ", "\<\"B\"\>", " ", SuperscriptBox["\[HBar]", "2"]}], "0", RowBox[{"3", " ", "\<\"A\"\>", " ", SuperscriptBox["\[HBar]", "2"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}, {"\<\"eigenvalues\"\>", "\<\" = \"\>", TagBox[GridBox[{ { RowBox[{ RowBox[{"-", SqrtBox["3"]}], " ", SqrtBox[ RowBox[{ RowBox[{"3", " ", SuperscriptBox["\<\"A\"\>", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["\<\"B\"\>", "2"]}]}]], " ", SuperscriptBox["\[HBar]", "2"]}]}, { RowBox[{ RowBox[{"-", SqrtBox["3"]}], " ", SqrtBox[ RowBox[{ RowBox[{"3", " ", SuperscriptBox["\<\"A\"\>", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["\<\"B\"\>", "2"]}]}]], " ", SuperscriptBox["\[HBar]", "2"]}]}, { RowBox[{ SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"3", " ", SuperscriptBox["\<\"A\"\>", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["\<\"B\"\>", "2"]}]}]], " ", SuperscriptBox["\[HBar]", "2"]}]}, { RowBox[{ SqrtBox["3"], " ", SqrtBox[ RowBox[{ RowBox[{"3", " ", SuperscriptBox["\<\"A\"\>", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["\<\"B\"\>", "2"]}]}]], " ", SuperscriptBox["\[HBar]", "2"]}]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]} }, AutoDelete->False, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], "Text", StripOnInput->False, FontSize->14]], "Output", CellChangeTimes->{{3.659176268035035*^9, 3.6591762773328876`*^9}, 3.6591763080485153`*^9, {3.659177433923667*^9, 3.65917744415814*^9}, 3.659177501681103*^9, {3.6591776884823494`*^9, 3.6591777069042344`*^9}, 3.6591777680800896`*^9, {3.6591779837159786`*^9, 3.6591780206070747`*^9}, { 3.6591781166527634`*^9, 3.6591781341787252`*^9}, {3.659178167862095*^9, 3.659178221247567*^9}, 3.659178255524681*^9, {3.6591785099933033`*^9, 3.659178555019436*^9}, 3.6591786730256443`*^9, {3.6591788293080907`*^9, 3.6591788362216673`*^9}, {3.659178877607311*^9, 3.6591788900368853`*^9}, 3.6591795615627446`*^9, {3.6591796010904837`*^9, 3.6591796106510983`*^9}, 3.6591798194549694`*^9, 3.65917986043865*^9, 3.6591799002655134`*^9, 3.6591818276513753`*^9}] }, Open ]], Cell["Scratch expressions.", "Text", CellChangeTimes->{{3.6591801295543437`*^9, 3.65918013507231*^9}}], Cell[BoxData[{ RowBox[{"m", "[", RowBox[{"vec", "[", RowBox[{"3", "/", "2"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"vec", "[", "#", "]"}], " ", "//", " ", "MatrixForm"}], ")"}], "&"}], "/@", " ", "mValues"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"raise", "[", RowBox[{"vec", "[", "#", "]"}], "]"}], " ", "//", " ", "MatrixForm"}], ")"}], "&"}], "/@", " ", "mValues"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"lower", "[", RowBox[{"vec", "[", "#", "]"}], "]"}], " ", "//", " ", "MatrixForm"}], ")"}], "&"}], "/@", " ", "mValues"}], "\[IndentingNewLine]", RowBox[{"splus", " ", "//", " ", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sminus", " ", "//", " ", "MatrixForm"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"splus", " ", ".", " ", RowBox[{"vec", "[", "#", "]"}]}], " ", "//", " ", "MatrixForm"}], ")"}], "&"}], "/@", " ", "mValues"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"sminus", " ", ".", " ", RowBox[{"vec", "[", "#", "]"}]}], " ", "//", " ", "MatrixForm"}], ")"}], "&"}], "/@", " ", "mValues"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"splus", ".", "splus"}], " ", "//", " ", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sminus", ".", "sminus"}], " ", "//", " ", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sz", ".", "sz"}], " ", "//", " ", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{"sSquared", " ", "//", " ", "MatrixForm"}]}], "Input", CellChangeTimes->{{3.6591789874190083`*^9, 3.6591790222153363`*^9}}] }, WindowSize->{1102, 671}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"9.0 for Microsoft Windows (64-bit) (January 25, 2013)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[557, 20, 1318, 24, 119, "Text"], Cell[CellGroupData[{ Cell[1900, 48, 12091, 328, 1575, "Input"], Cell[13994, 378, 9983, 258, 1013, "Output"] }, Open ]], Cell[23992, 639, 103, 1, 40, "Text"], Cell[24098, 642, 1930, 56, 391, "Input"] } ] *) (* End of internal cache information *)