(* 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[ 8834, 252] NotebookOptionsPosition[ 8482, 235] NotebookOutlinePosition[ 8824, 250] CellTagsIndexPosition[ 8781, 247] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Sakurai. Problem 5.11 (c). Verify hand calculations (diagonalization)\ \>", "Text", CellChangeTimes->{{3.6584495380073385`*^9, 3.6584495926543636`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"ClearAll", "[", " ", RowBox[{ "u", ",", " ", "h", ",", " ", "v", ",", " ", "delta", ",", " ", "lambda", ",", "e", ",", " ", "display", ",", " ", "d"}], "]"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"display", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ "e", " ", "\[Rule]", " ", "\"\<\!\(\*SubscriptBox[\(E\), \(1\)]\)\>\""}], ",", RowBox[{"lambda", " ", "\[Rule]", " ", "\[Lambda]"}], ",", " ", RowBox[{"delta", " ", "\[Rule]", " ", "\[CapitalDelta]"}]}], "}"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"u", " ", "=", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "/", RowBox[{"Sqrt", "[", "2", "]"}]}], ")"}], " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}]}], "}"}]}], "}"}]}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"h", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"e", ",", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "e"}], "}"}]}], "}"}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v", " ", "=", " ", RowBox[{"lambda", " ", "delta", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], "}"}]}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"d", " ", ":=", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"#", " ", "/.", " ", "display"}], " ", "//", " ", "MatrixForm"}], ")"}], " ", "&"}]}], " ", ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"#1", ",", "\"\< = \>\"", ",", RowBox[{"#2", "//", "d"}]}], " ", "}"}], " ", "&"}], "@@@", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"\"\<\!\(\*SubscriptBox[\(H\), \(0\)]\)\>\"", ",", " ", "h"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\\"", ",", " ", "v"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\"\\"", ",", " ", "u"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ "\"\<\!\(\*SuperscriptBox[\(U\), \(\(\[Dagger]\)\(\\\ \ \)\)]\)\!\(\*SubscriptBox[\(H\), \(0\)]\) U\>\"", ",", " ", RowBox[{"u", " ", ".", " ", "h", " ", ".", " ", "u"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ "\"\<\!\(\*SuperscriptBox[\(U\), \(\(\[Dagger]\)\(\\\ \)\)]\)V U\>\"", ",", " ", RowBox[{"u", " ", ".", " ", "v", " ", ".", " ", "u"}]}], "}"}]}], "\[IndentingNewLine]", "}"}]}], " ", "//", " ", "Grid"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.6584490010570745`*^9, 3.65844905206341*^9}, { 3.6584493085363326`*^9, 3.6584494926283236`*^9}, {3.6584495982794504`*^9, 3.6584496126495333`*^9}, {3.6584497263934197`*^9, 3.658449884836358*^9}}], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\\!\\(\\*SubscriptBox[\\(H\\), \\(0\\)]\\)\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"\\!\\(\\*SubscriptBox[\\(E\\), \\(1\\)]\\)\"\>", "0"}, {"0", "\<\"\\!\\(\\*SubscriptBox[\\(E\\), \\(1\\)]\\)\"\>"} }, 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$, MatrixForm[BoxForm`e$]]]}, {"\<\"V\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", RowBox[{"\[CapitalDelta]", " ", "\[Lambda]"}]}, { RowBox[{"\[CapitalDelta]", " ", "\[Lambda]"}], "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$, MatrixForm[BoxForm`e$]]]}, {"\<\"U\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { FractionBox["1", SqrtBox["2"]], FractionBox["1", SqrtBox["2"]]}, { FractionBox["1", SqrtBox["2"]], RowBox[{"-", FractionBox["1", SqrtBox["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$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SuperscriptBox[\\(U\\), \\(\\(\[Dagger]\\)\\(\\\\ \ \\)\\)]\\)\\!\\(\\*SubscriptBox[\\(H\\), \\(0\\)]\\) U\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"\\!\\(\\*SubscriptBox[\\(E\\), \\(1\\)]\\)\"\>", "0"}, {"0", "\<\"\\!\\(\\*SubscriptBox[\\(E\\), \\(1\\)]\\)\"\>"} }, 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$, MatrixForm[BoxForm`e$]]]}, {"\<\"\\!\\(\\*SuperscriptBox[\\(U\\), \\(\\(\[Dagger]\\)\\(\\\\ \ \\)\\)]\\)V U\"\>", "\<\" = \"\>", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"\[CapitalDelta]", " ", "\[Lambda]"}], "0"}, {"0", RowBox[{ RowBox[{"-", "\[CapitalDelta]"}], " ", "\[Lambda]"}]} }, 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$, MatrixForm[BoxForm`e$]]]} }, AutoDelete->False, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"]], "Output", CellChangeTimes->{{3.658449410942911*^9, 3.6584494162667313`*^9}, { 3.658449484655176*^9, 3.6584494932629404`*^9}, 3.658449600968105*^9, { 3.65844987374691*^9, 3.6584498854579153`*^9}}] }, Open ]] }, WindowSize->{924, 722}, 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, 165, 3, 40, "Text"], Cell[CellGroupData[{ Cell[747, 27, 3156, 89, 432, "Input"], Cell[3906, 118, 4560, 114, 283, "Output"] }, Open ]] } ] *) (* End of internal cache information *)