(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 76923, 2435]*) (*NotebookOutlinePosition[ 121101, 3898]*) (* CellTagsIndexPosition[ 121019, 3892]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Harmonic Oscillator Matrix Representation James Feagin\ \>", "Subsubtitle", FontFamily->"Comic Sans MS", FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell["\<\ We collect into square arrays the matrix elements of the HO raising and \ lowering operators with respect to HO eigenstates and use the results to \ construct a matrix representation of the HO hamiltonian and related \ operators.\ \>", "Text"], Cell[TextData[StyleBox["", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Text"], Cell[CellGroupData[{ Cell["Kronecker delta function", "Subsubsection", FontFamily->"Comic Sans MS"], Cell["\<\ We start by defining the Kronecker delta function. We will use it to input \ the matrix elements of the raising and lowering operators.\ \>", "Text"], Cell[BoxData[{\(\[Delta][n_, n_] := 1\), "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Delta]", "[", RowBox[{ RowBox[{"n_", "?", StyleBox["IntegerQ", FontColor->RGBColor[0, 0, 1]]}], ",", RowBox[{"m_", "?", StyleBox["IntegerQ", FontColor->RGBColor[0, 0, 1]]}]}], "]"}], ":=", "0"}]}], "Input"], Cell[TextData[{ "The tests with ", StyleBox["IntegerQ", FontColor->RGBColor[0, 0, 1]], " leave \[Delta][m, n] unevaluated when m and n remain unspecified for \ symbolic computing:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\[Delta][1, 1], \ \[Delta][2, 3], \[Delta][m, n]}\)], "Input"], Cell[BoxData[ \({1, 0, \[Delta][m, n]}\)], "Output"] }, Open ]], Cell["We can then substitute for m and n later on:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(% /. m \[Rule] n\)], "Input"], Cell[BoxData[ \({1, 0, 1}\)], "Output"] }, Open ]], Cell[TextData[{ "This is just a compact version of the built-in function ", StyleBox["KroneckerDelta.", FontColor->RGBColor[0, 0, 1]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Identity matrix", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "As an example, we construct the identity matrix. We thus set up a 2D \ matrix of ", StyleBox["\[Delta][m,n] ", FontColor->RGBColor[1, 0, 1]], "values as a ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["Table", FontColor->RGBColor[0, 0, 1]], " summed over rows ", StyleBox["m", FontColor->RGBColor[0, 0, 1]], " and columns ", StyleBox["n", FontColor->RGBColor[0, 0, 1]], ". On the computer, we have to work with truncated representations, i.e. \ finite dimensional matrices. Here, we take ", StyleBox["m, n \[LessSlantEqual] nmax \[Congruent] 5.", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{\(nmax\ = \ 5;\), "\[IndentingNewLine]", RowBox[{ RowBox[{"I\[Delta]", "=", RowBox[{ StyleBox["Table", FontColor->RGBColor[0, 0, 1]], "[", RowBox[{ StyleBox[\(\[Delta][m, n]\), FontColor->RGBColor[1, 0, 1]], ",", RowBox[{ StyleBox["{", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["m", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["0", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["nmax", FontColor->RGBColor[1, 0, 0]]}], StyleBox["}", FontColor->RGBColor[0, 0, 1]]}], StyleBox[",", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["{", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["n", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["0", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["nmax", FontColor->RGBColor[1, 0, 0]]}], StyleBox["}", FontColor->RGBColor[0, 0, 1]]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", \(I\[Delta] // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Note the result is a ", StyleBox["6x6", FontColor->RGBColor[0, 0, 1]], " matrix since by convention the HO ground state is labelled ", StyleBox["n = 0", FontColor->RGBColor[0, 0, 1]], ". We can also do the same thing with a built-in function:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"IdentityMatrix", "[", StyleBox["6", "Subsubsection", FontColor->RGBColor[0, 0, 1]], "]"}], "//", "MatrixForm"}]], "Input",\ CellTags->"KroneckerDelta"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellTags->"KroneckerDelta"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Raising and lowering operators", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "In terms of the \[Delta] function, the matrix elements of the ", StyleBox["raising", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["lowering", FontColor->RGBColor[1, 0, 0]], " operators can be entered directly:" }], "Text"], Cell[BoxData[{ StyleBox[\(\(a\^\[Dagger]\)[m_, n_]\ := \ \(\@\(n + 1\)\) \[Delta][m, n + 1]\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", RowBox[{ StyleBox[ RowBox[{" ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]]}]], StyleBox[\(a[m_, n_]\ := \ \ \ \ \ \ \ \(\@n\) \[Delta][m, n - 1]\), FontColor->RGBColor[1, 0, 0]]}]}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Notation ", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "We now load a package to declare certain input symbols to be single, block \ expressions for all computations. Here, we need symbols ", Cell[BoxData[ \(TraditionalForm\`A\^\[Dagger]\)]], " and A for the raising and lowering matrices. ", StyleBox["(We'll use capital letters to label matrices.) ", FontColor->RGBColor[0, 0, 1]], "With just the palette input ", Cell[BoxData[ \(TraditionalForm\`\[Placeholder]\^\[Dagger]\)]], ", Mathematica will sometimes dump the contents of A into ", Cell[BoxData[ \(TraditionalForm\`A\^\[Dagger]\)]], " literally, which will usually break the computation." }], "Text"], Cell[BoxData[ StyleBox[\(<< Utilities`Notation`\), FontColor->RGBColor[0, 0, 1]]], "Input"], Cell[TextData[{ "Now we can use the pop-up palette which has appeared along with the usual \ 'BasicTypesetting' palette with ", Cell[BoxData[ \(TraditionalForm\`\[Placeholder]\^\[Dagger]\)]], " to declare ", Cell[BoxData[ \(TraditionalForm\`A\^\[Dagger]\)]], " a block symbol. ", StyleBox["Note, you have to use the palette. Typing the words and symbols, \ or using even cut and paste, won't work.", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(A\^\[Dagger]\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell["\<\ The matrices of the raising and lowering operators can now be constructed in \ the same way we set up the identity matrix:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{\(nmax\ = \ 5;\), "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox[\(A\^\[Dagger]\), FontColor->RGBColor[0, 0, 1]], " ", "=", " ", RowBox[{"Table", "[", RowBox[{ StyleBox[\(\(a\^\[Dagger]\)[m, n]\), FontColor->RGBColor[0, 0, 1]], ",", \({m, 0, nmax}\), ",", \({n, 0, nmax}\)}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ StyleBox[\(A\^\[Dagger]\), FontColor->RGBColor[0, 0, 1]], "//", "MatrixForm"}]}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"1", "0", "0", "0", "0", "0"}, {"0", \(\@2\), "0", "0", "0", "0"}, {"0", "0", \(\@3\), "0", "0", "0"}, {"0", "0", "0", "2", "0", "0"}, {"0", "0", "0", "0", \(\@5\), "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ StyleBox["A", FontColor->RGBColor[1, 0, 0]], " ", "=", " ", RowBox[{"Table", "[", RowBox[{ StyleBox[\(a[m, n]\), FontColor->RGBColor[1, 0, 0]], ",", \({m, 0, nmax}\), ",", \({n, 0, nmax}\)}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ StyleBox["A", FontColor->RGBColor[1, 0, 0]], "//", "MatrixForm"}]}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0", "0", "0"}, {"0", "0", \(\@2\), "0", "0", "0"}, {"0", "0", "0", \(\@3\), "0", "0"}, {"0", "0", "0", "0", "2", "0"}, {"0", "0", "0", "0", "0", \(\@5\)}, {"0", "0", "0", "0", "0", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ These matrices are real-valued and the transpose of each other, i.e.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Transpose", "[", StyleBox["A", FontColor->RGBColor[1, 0, 0]], "]"}], " ", "\[Equal]", StyleBox[\(A\^\[Dagger]\), FontColor->RGBColor[0, 0, 1]]}]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The matrix ", "\[EGrave]\.beZ"], Cell[BoxData[ \(A\^\[Dagger]\)], FontColor->RGBColor[0, 0, 1]], "is said to be the '", StyleBox["adjoint", FontColor->RGBColor[0, 0, 1]], "' of ", Cell[BoxData[ \(A\)], FontColor->RGBColor[1, 0, 0]], " (hence the dagger notation), and we see how the adjoint property of the \ raising and lowering operators maps into their matrix representation. We'll \ come back to this point below and examine the self-adjoint form of other \ hermitian observables, like the momentum which is also complex valued." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Number-operator matrix", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[StyleBox["We can now begin to construct a matrix \ representation, albeit truncated, of all HO observables. We'll demonstrate \ along the way that the operator algebra fully maps into the matrix \ representation.", "\[EGrave]\.beZ"]], "Text"], Cell[TextData[{ "The number operator ", Cell[BoxData[ \(TraditionalForm\`N\)]], " = ", Cell[BoxData[ \(TraditionalForm\`a\^\[Dagger]\)]], Cell[BoxData[ \(TraditionalForm\`a\)]], " has the HO states ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_n\)]], " as eigenstates with eigenvalues n, the excitation index of the nth energy \ level. Here, the dot product ", Cell[BoxData[ StyleBox[\(A\^\[Dagger] . A\), FontColor->RGBColor[0, 0, 1]]], FontColor->RGBColor[0, 0, 1]], " here ensures matrix multiplication (and thus that ", Cell[BoxData[ \(TraditionalForm\`a\^\[Dagger]\)]], "operates on ", Cell[BoxData[ \(TraditionalForm\`a\)]], "). " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalN]", "=", StyleBox[\(A\^\[Dagger] . A\), FontColor->RGBColor[0, 0, 1]]}], ";"}], "\[IndentingNewLine]", \(\[DoubleStruckCapitalN] // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "2", "0", "0", "0"}, {"0", "0", "0", "3", "0", "0"}, {"0", "0", "0", "0", "4", "0"}, {"0", "0", "0", "0", "0", "5"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["HO hamiltonian matrix", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "Note, the identity matrix ", StyleBox["I\[Delta]", FontColor->RGBColor[0, 0, 1]], " is needed to add in ", "\[HBar] \[Omega]/2", ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", " ", "=", " ", RowBox[{"\[HBar]", " ", "\[Omega]", " ", RowBox[{"(", RowBox[{"\[DoubleStruckCapitalN]", " ", "+", " ", RowBox[{\(1\/2\), StyleBox["I\[Delta]", FontColor->RGBColor[0, 0, 1]]}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", \(H // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0", "0"}, {"0", \(\(3\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0"}, {"0", "0", \(\(5\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0"}, {"0", "0", "0", \(\(7\ \[Omega]\ \[HBar]\)\/2\), "0", "0"}, {"0", "0", "0", "0", \(\(9\ \[Omega]\ \[HBar]\)\/2\), "0"}, {"0", "0", "0", "0", "0", \(\(11\ \[Omega]\ \[HBar]\)\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Of course, with respect to the HO basis, the diagonal matrix elements of the \ hamiltonian, i.e. its expectation values in the HO basis, are just the energy \ eigenvalues while the off-diagonal elements all vanish due to orthogonality \ of the basis states.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"H", " ", "\[Equal]", " ", RowBox[{"Table", "[", RowBox[{ StyleBox[\(\[HBar]\ \[Omega] \((n + 1/2)\) \[Delta][m, n]\), FontColor->RGBColor[0, 0, 1]], ",", \({m, 0, nmax}\), ",", \({n, 0, nmax}\)}], "]"}]}]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Position X and momentum P matrices", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(X\ = \ \(\@\(\[HBar]\/\(2 m\ \[Omega]\)\)\) \((A\^\[Dagger] + A)\);\)\), "\[IndentingNewLine]", \(X // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", \(\@\(\[HBar]\/\(m\ \[Omega]\)\)\/\@2\), "0", "0", "0", "0"}, {\(\@\(\[HBar]\/\(m\ \[Omega]\)\)\/\@2\), "0", \(\@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0", "0", "0"}, {"0", \(\@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0", \(\@\(3\/2\)\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0", "0"}, {"0", "0", \(\@\(3\/2\)\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0", \(\@2\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0"}, {"0", "0", "0", \(\@2\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0", \(\@\(5\/2\)\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\)}, {"0", "0", "0", "0", \(\@\(5\/2\)\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\), "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(P\ = \ \(\@\(\(m\ \[HBar]\ \[Omega]\)\/2\)\) \(A - \ A\^\[Dagger]\)\/I;\)\), "\[IndentingNewLine]", \(P // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", \(-\(\(\[ImaginaryI]\ \@\(m\ \[Omega]\ \[HBar]\)\)\/\@2\)\), "0", "0", "0", "0"}, {\(\(\[ImaginaryI]\ \@\(m\ \[Omega]\ \[HBar]\)\)\/\@2\), "0", \(\(-\[ImaginaryI]\)\ \@\(m\ \[Omega]\ \[HBar]\)\), "0", "0", "0"}, {"0", \(\[ImaginaryI]\ \@\(m\ \[Omega]\ \[HBar]\)\), "0", \(\(-\[ImaginaryI]\)\ \@\(3\/2\)\ \@\(m\ \[Omega]\ \[HBar]\ \)\), "0", "0"}, {"0", "0", \(\[ImaginaryI]\ \@\(3\/2\)\ \@\(m\ \[Omega]\ \[HBar]\)\), "0", \(\(-\[ImaginaryI]\)\ \@2\ \@\(m\ \[Omega]\ \[HBar]\)\), "0"}, {"0", "0", "0", \(\[ImaginaryI]\ \@2\ \@\(m\ \[Omega]\ \[HBar]\)\), "0", \(\(-\[ImaginaryI]\)\ \@\(5\/2\)\ \@\(m\ \[Omega]\ \[HBar]\ \)\)}, {"0", "0", "0", "0", \(\[ImaginaryI]\ \@\(5\/2\)\ \@\(m\ \[Omega]\ \[HBar]\)\), "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 1.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Verify our matrix results by working Griffith's problem 3.50 by \ hand. You can start with the matrix elements of ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(A\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(A\^\[Dagger]\)], FontColor->RGBColor[0, 0, 1]], StyleBox["to verify eq. 3.155. I don't mind if you simply show that \ \[DoubleStruckCapitalN] is diagonal and from there obtain H.", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ "We can check that these observables are hermitian, i.e. their matrices are \ self adjoint. Since some observables like the momentum are also complex \ valued, we extend our usual complex conjugation rule to include matrix \ transpose and thus define the ", StyleBox["adjoint ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(M\^\[Dagger]\)], FontColor->RGBColor[0, 0, 1]], "to a matrix ", Cell[BoxData[ \(M\)]], "." }], "Text"], Cell[BoxData[{\(\(\[Psi]_\^*\) := \ \[Psi] /. \ Complex[x_, y_] \[Rule] \ x - I\ y\), "\[IndentingNewLine]", StyleBox[\(M_\^\[Dagger] := \ Transpose[\(M\^*\)]\), FontColor->RGBColor[0, 0, 1]]}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \({X\^\[Dagger] \[Equal] X, P\^\[Dagger] \[Equal] P}\)], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Potential V and kinetic K energy matrices", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"V", "=", RowBox[{\(1\/2\), "m", " ", \(\[Omega]\^\(\(2\)\(\ \)\)\), StyleBox[\(X . X\), FontColor->RGBColor[0, 0, 1]]}]}], ";"}], "\[IndentingNewLine]", \(V // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[Omega]\ \[HBar]\)\/4\), "0", \(\(\[Omega]\ \[HBar]\)\/\(2\ \@2\)\), "0", "0", "0"}, {"0", \(\(3\ \[Omega]\ \[HBar]\)\/4\), "0", \(1\/2\ \@\(3\/2\)\ \[Omega]\ \[HBar]\), "0", "0"}, {\(\(\[Omega]\ \[HBar]\)\/\(2\ \@2\)\), "0", \(\(5\ \[Omega]\ \[HBar]\)\/4\), "0", \(1\/2\ \@3\ \[Omega]\ \[HBar]\), "0"}, {"0", \(1\/2\ \@\(3\/2\)\ \[Omega]\ \[HBar]\), "0", \(\(7\ \[Omega]\ \[HBar]\)\/4\), "0", \(1\/2\ \@5\ \[Omega]\ \[HBar]\)}, {"0", "0", \(1\/2\ \@3\ \[Omega]\ \[HBar]\), "0", \(\(9\ \[Omega]\ \[HBar]\)\/4\), "0"}, {"0", "0", "0", \(1\/2\ \@5\ \[Omega]\ \[HBar]\), "0", \(\(5\ \[Omega]\ \[HBar]\)\/4\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Notice the truncation error here in the last element ", StyleBox["V[6,6]", FontColor->RGBColor[0, 0, 1]], ". The matrix multiply ", StyleBox["X.X", FontColor->RGBColor[0, 0, 1]], " for any given element needs elements of ", StyleBox["X", FontColor->RGBColor[0, 0, 1]], " from its neighboring rows and columns. Here, we've truncated our matrices \ to 6 x 6." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(K = P . P\/\(2 m\);\)\), "\[IndentingNewLine]", \(K // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[Omega]\ \[HBar]\)\/4\), "0", \(-\(\(\[Omega]\ \[HBar]\)\/\(2\ \@2\)\)\), "0", "0", "0"}, {"0", \(\(3\ \[Omega]\ \[HBar]\)\/4\), "0", \(\(-\(1\/2\)\)\ \@\(3\/2\)\ \[Omega]\ \[HBar]\), "0", "0"}, {\(-\(\(\[Omega]\ \[HBar]\)\/\(2\ \@2\)\)\), "0", \(\(5\ \[Omega]\ \[HBar]\)\/4\), "0", \(\(-\(1\/2\)\)\ \@3\ \[Omega]\ \[HBar]\), "0"}, {"0", \(\(-\(1\/2\)\)\ \@\(3\/2\)\ \[Omega]\ \[HBar]\), "0", \(\(7\ \[Omega]\ \[HBar]\)\/4\), "0", \(\(-\(1\/2\)\)\ \@5\ \[Omega]\ \[HBar]\)}, {"0", "0", \(\(-\(1\/2\)\)\ \@3\ \[Omega]\ \[HBar]\), "0", \(\(9\ \[Omega]\ \[HBar]\)\/4\), "0"}, {"0", "0", "0", \(\(-\(1\/2\)\)\ \@5\ \[Omega]\ \[HBar]\), "0", \(\(5\ \[Omega]\ \[HBar]\)\/4\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Hamiltonian matrix", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(K + V // MatrixForm\)\)\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0", "0"}, {"0", \(\(3\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0"}, {"0", "0", \(\(5\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0"}, {"0", "0", "0", \(\(7\ \[Omega]\ \[HBar]\)\/2\), "0", "0"}, {"0", "0", "0", "0", \(\(9\ \[Omega]\ \[HBar]\)\/2\), "0"}, {"0", "0", "0", "0", "0", \(\(5\ \[Omega]\ \[HBar]\)\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Subtracting this result from our previous hamiltonian defined using the \ number operator \[DoubleStruckCapitalN] makes the truncation error clear. \ (Can you see why the truncation errors don't appear in \ \[DoubleStruckCapitalN]?)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(H\ - \ % // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", \(3\ \[Omega]\ \[HBar]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Commutation", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "We can use the Notation Palette to define a compact and more conventional \ notation for matrix commutation. Again, clicking on the Notation palette and \ selecting ", StyleBox["Notation", FontColor->RGBColor[0, 0, 1]], " (the version with \[DoubleLeftRightArrow]) and typing in the commutator \ ", StyleBox["A_ .B_ -B_ .A_", FontColor->RGBColor[0, 0, 1]], " of two dummy matrices ", StyleBox["A_ ", FontColor->RGBColor[0, 0, 1]], "and", StyleBox[" B_", FontColor->RGBColor[0, 0, 1]], " (using matrix multiplication to combine the matrices), we define an alias \ notation ", Cell[BoxData[ \(\([A_, B_]\)\_C\)], FontColor->RGBColor[1, 0, 0]], " with" }], "Text"], Cell[BoxData[ RowBox[{ StyleBox["Notation", FontColor->RGBColor[0, 0, 1]], "[", RowBox[{ StyleBox[ TagBox[\(\([A_, B_]\)\_C\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], FontColor->RGBColor[1, 0, 0]], " ", "\[DoubleLongLeftRightArrow]", " ", StyleBox[ TagBox[\(A_\ . B_\ - B_\ . A_\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], FontColor->RGBColor[0, 0, 1]]}], "]"}]], "Input"], Cell[TextData[{ "We include the subscript ", StyleBox["C", FontColor->RGBColor[1, 0, 0]], " to distinguish commutator brackets from function-argument brackets, as in \ for example ", StyleBox["\[Delta][m, n]", FontColor->RGBColor[0, 0, 1]], ". " }], "Text"], Cell["Thus, we verify the fundamental commutation relations:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[ SubscriptBox[ RowBox[{"[", StyleBox[\(X, P\), FontColor->RGBColor[0, 0, 1]], "]"}], "C"], FontColor->RGBColor[1, 0, 0]], "//", "PowerExpand"}], "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[ImaginaryI]\ \[HBar]\), "0", "0", "0", "0", "0"}, {"0", \(\[ImaginaryI]\ \[HBar]\), "0", "0", "0", "0"}, {"0", "0", \(\[ImaginaryI]\ \[HBar]\), "0", "0", "0"}, {"0", "0", "0", \(\[ImaginaryI]\ \[HBar]\), "0", "0"}, {"0", "0", "0", "0", \(\[ImaginaryI]\ \[HBar]\), "0"}, {"0", "0", "0", "0", "0", \(\(-5\)\ \[ImaginaryI]\ \[HBar]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[ SubscriptBox[ RowBox[{ StyleBox["[", FontColor->RGBColor[1, 0, 0]], RowBox[{"A", ",", StyleBox[\(A\^\[Dagger]\), FontColor->RGBColor[0, 0, 1]]}], StyleBox["]", FontColor->RGBColor[1, 0, 0]]}], "C"], FontColor->RGBColor[1, 0, 0]], " ", "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", \(-5\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "We also readily verify one of the Ehrenfest results, ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_t\ \[LeftAngleBracket]x\ \[RightAngleBracket]\ = \ \[LeftAngleBracket]p\[RightAngleBracket]/ m\)\(,\)\)\)]], " which gives the following identity for the quantum velocity:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{\(P\/m\), "\[Equal]", " ", FractionBox[ StyleBox[ SubscriptBox[ RowBox[{ StyleBox["[", FontColor->RGBColor[1, 0, 0]], StyleBox[\(X, H\), FontColor->RGBColor[0, 0, 1]], StyleBox["]", FontColor->RGBColor[1, 0, 0]]}], "C"], FontColor->RGBColor[1, 0, 0]], \(I\ \[HBar]\)]}], "//", "PowerExpand"}], "//", "Simplify"}]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 2.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Check the identities in Griffith's problem 3.41. In (a) use say \ X, K and V, in (b) use ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[\(X\^3\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], ", ", StyleBox["and in (c) use V. Note, there may be truncation error.", FontColor->RGBColor[0, 0, 1]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Energy eigenvectors", "Subsubsection", FontFamily->"Comic Sans MS"], Cell["\<\ The HO ground state in our 6 x 6 matrix representation in which the \ hamiltonian is diagonal is a particularly simple 6D unit vector:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Phi][0] = \ {1, 0, 0, 0, 0, 0}\)], "Input"], Cell[BoxData[ \({1, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ We can check it by showing that multiplication by the hamiltonian matrix is \ equivalent to multiplication by the HO ground-state energy:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(H . \[Phi][0] \[Equal] \ \(\(\[HBar]\ \[Omega]\)\/2\) \[Phi][ 0]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Alternatively, 'dotting' the vector ", StyleBox["H.\[Phi][0]", FontColor->RGBColor[0, 0, 1]], " with the ground state gives the energy (hamiltonian) expectation value in \ the ground state:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(\[Phi][0]\), ".", StyleBox["H", FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[Phi][0]\), FontColor->RGBColor[0, 0, 1]]}]], "Input"], Cell[BoxData[ \(\(\[Omega]\ \[HBar]\)\/2\)], "Output"] }, Open ]], Cell[TextData[{ "Evidently, ", StyleBox["\[Delta][", FontColor->RGBColor[0, 0, 1]], StyleBox["n", FontColor->RGBColor[1, 0, 0]], StyleBox[",m]", FontColor->RGBColor[0, 0, 1]], " gives the ", StyleBox["mth", FontColor->RGBColor[0, 0, 1]], " element of the ", StyleBox["nth", FontColor->RGBColor[1, 0, 0]], " eigenvector, so that all ", StyleBox["nmax + 1", FontColor->RGBColor[1, 0, 1]], " eigenstates can be constructed as:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "[", StyleBox["n_", FontColor->RGBColor[1, 0, 0]], "]"}], ":=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], StyleBox["[", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["n", FontColor->RGBColor[1, 0, 0]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["m", FontColor->RGBColor[0, 0, 1]]}], StyleBox["]", FontColor->RGBColor[0, 0, 1]]}], ",", RowBox[{"{", RowBox[{ StyleBox["m", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["0", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox["nmax", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}]], "Input"], Cell["These vectors are orthonormal,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", " ", RowBox[{ StyleBox[\(\[Phi][m] . \[Phi][n]\), FontColor->RGBColor[0, 0, 1]], ",", \({m, 0, nmax}\), ",", \({n, 0, nmax}\)}], "]"}], "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ and thus we readily check they're also eigenvectors of the hamiltonian by \ recomputing the full hamiltonian matrix:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", " ", RowBox[{ StyleBox[\(\[Phi][m] . H . \[Phi][n]\), FontColor->RGBColor[0, 0, 1]], ",", \({m, 0, nmax}\), ",", \({n, 0, nmax}\)}], "]"}], "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0", "0"}, {"0", \(\(3\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0", "0"}, {"0", "0", \(\(5\ \[Omega]\ \[HBar]\)\/2\), "0", "0", "0"}, {"0", "0", "0", \(\(7\ \[Omega]\ \[HBar]\)\/2\), "0", "0"}, {"0", "0", "0", "0", \(\(9\ \[Omega]\ \[HBar]\)\/2\), "0"}, {"0", "0", "0", "0", "0", \(\(11\ \[Omega]\ \[HBar]\)\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ The eigenvectors also form a complete set in the 6D truncated subspace. We \ can thus expand arbitrary initial wavepackets, as for example,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], "=", \(\(1\/\@50\) \((3 \[Phi][0] + 4\ \[Phi][1] + 5\ \[Phi][3])\)\)}]], "Input"], Cell[BoxData[ \({3\/\(5\ \@2\), \(2\ \@2\)\/5, 0, 1\/\@2, 0, 0}\)], "Output"] }, Open ]], Cell["and check normalization and compute its energy:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox[\(\[Psi]\_0 . \[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], ",", RowBox[{ StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], ".", "H", ".", StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]]}]}], "}"}]], "Input"], Cell[BoxData[ \({1, \(58\ \[Omega]\ \[HBar]\)\/25}\)], "Output"] }, Open ]], Cell[TextData[{ "Due to the simplicity of eigenvectors in a diagonal matrix representation, \ we see that the elements of ", Cell[BoxData[ RowBox[{ StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], " "}]]], "are just the expansion coefficients of the eigenvectors ", StyleBox["\[Phi][n] ", FontColor->RGBColor[0, 0, 1]], "in the superposition, in particular" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", " ", RowBox[{ RowBox[{ StyleBox[\(\[Phi][n]\), FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]]}], ",", \({n, 0, nmax}\)}], "]"}]], "Input"], Cell[BoxData[ \({3\/\(5\ \@2\), \(2\ \@2\)\/5, 0, 1\/\@2, 0, 0}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Time Development", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "Because the hamiltonian matrix is diagonal, we can readily exponentiate it \ with the built-in function ", StyleBox["MatrixExp", FontColor->RGBColor[0, 0, 1]], " to construct the time-development operator:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"U", " ", "=", RowBox[{ StyleBox["MatrixExp", FontColor->RGBColor[0, 0, 1]], "[", \(\(-I\)\ H\ t/\[HBar]\), "]"}]}], ";"}], "\[IndentingNewLine]", \(U // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ t\ \[Omega]\)\ \), "0", "0", "0", "0", "0"}, { "0", \(\[ExponentialE]\^\(\(-\(3\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\), "0", "0", "0", "0"}, {"0", "0", \(\[ExponentialE]\^\(\(-\(5\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\), "0", "0", "0"}, {"0", "0", "0", \(\[ExponentialE]\^\(\(-\(7\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\), "0", "0"}, {"0", "0", "0", "0", \(\[ExponentialE]\^\(\(-\(9\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\), "0"}, {"0", "0", "0", "0", "0", \(\[ExponentialE]\^\(\(-\(11\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["MatrixExp", FontColor->RGBColor[0, 0, 1]], " effectively sums the Taylor expansion of ", StyleBox["Exp,", FontColor->RGBColor[0, 0, 1]], " and thus operating on diagonal matrices like ", StyleBox["H", FontColor->RGBColor[0, 0, 1]], " just maps ", StyleBox["Exp", FontColor->RGBColor[0, 0, 1]], " onto the diagonal elements. In general, however, the result for arbitrary \ ", StyleBox["symbolic", FontSlant->"Italic"], " matrices is complicated and not very useful, although for purely ", StyleBox["numerical", FontSlant->"Italic"], " matrices ", StyleBox["MatrixExp", FontColor->RGBColor[0, 0, 1]], " is useful and efficient for performing the necessary arithmetic at \ machine precision ." }], "Text"], Cell["\<\ We apply the time-development operator to evolve the initial wavepacket we \ defined above,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(\[Psi]\_t\), "=", RowBox[{"U", ".", StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]]}]}]], "Input"], Cell[BoxData[ \({\(3\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\)\/\(5\ \@2\), 2\/5\ \@2\ \[ExponentialE]\^\(\(-\(3\/2\)\)\ \[ImaginaryI]\ t\ \[Omega]\ \), 0, \[ExponentialE]\^\(\(-\(7\/2\)\)\ \[ImaginaryI]\ t\ \[Omega]\)\/\@2, 0, 0}\)], "Output"] }, Open ]], Cell["and thus obtain a solution of the wave equation:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(I\ \[HBar] \[PartialD]\_t\ \[Psi]\_t \[Equal] H . \[Psi]\_t\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "We also verify that normalization is preserved. Note, however, ", Cell[BoxData[ \(\[Psi]\_t\)], FontColor->RGBColor[0, 0, 1]], " is complex valued and its role as a '", StyleBox["bra", FontColor->RGBColor[0, 0, 1]], "' in scalar products requires complex conjugation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox[\(\[Psi]\_0 . \[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], ",", RowBox[{ StyleBox[\(\[Psi]\_t\^*\), FontColor->RGBColor[0, 0, 1]], ".", \(\[Psi]\_t\)}]}], "}"}]], "Input"], Cell[BoxData[ \({1, 1}\)], "Output"] }, Open ]], Cell[TextData[{ "This important property follows from the fact that U is ", StyleBox["unitary", FontColor->RGBColor[0, 0, 1]], ", namely its adjoint is its inverse, such that" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(U\^\[Dagger] . U \[Equal] U . U\^\[Dagger] \[Equal] I\[Delta]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["and therefore", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[\(\[Psi]\_t\^*\), FontColor->RGBColor[0, 0, 1]], ".", \(\[Psi]\_t\)}], " ", "\[Equal]", RowBox[{ StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], StyleBox[\(U\^\[Dagger]\), FontColor->RGBColor[0, 0, 1]], ".", StyleBox["U", FontColor->GrayLevel[0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]]}]}]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Moreover, since ", Cell[BoxData[ StyleBox[\(\[Psi]\_0\), FontColor->RGBColor[1, 0, 0]]]], "is time independent, we see that U itself is a solution of the wave \ equation, without any reference to a particular initial state:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(I\ \[HBar] \[PartialD]\_t\ U \[Equal] H . U\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 3.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Griffith's problem 3.12. Hint: The elements ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[\(\((M\^T)\)\_mn\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], StyleBox[" of a transposed matrix M are defined ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\((M\^T)\)\_mn\)], FontColor->RGBColor[0, 0, 1]], StyleBox["\[Congruent] ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(M\_nm\)\(.\)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox["Use summation notation on the matrix elements for matrix \ products. These properties essentially follow one after the other from the \ first.", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ StyleBox["EXERCISE 4.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Griffith's problem 3.47, parts (a) and (d). Hint: Note that ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`M\^3\)], FontColor->RGBColor[0, 0, 1]], StyleBox["in part (a) (i) vanishes. In part (a) (ii), note that even powers \ of M are proportional to the identity matrix, while odd powers of M are \ proportional to M. Hence, re-sum the Taylor expansion into Cos and Sin \ components.", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ "From the expectation values of the ", StyleBox["position", FontColor->RGBColor[1, 0, 0]], " and ", StyleBox["momentum", FontColor->RGBColor[0, 0, 1]], ", we see as expected that the wavepacket describes simple harmonic motion: \ " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ StyleBox[\(\(\[Psi]\_t\^*\) . X . \[Psi]\_t\), FontColor->RGBColor[1, 0, 0]], StyleBox[",", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\(\[Psi]\_t\^*\) . P . \[Psi]\_t\), FontColor->RGBColor[0, 0, 1]]}], "}"}], "//", "FullSimplify"}]], "Input"], Cell[BoxData[ \({6\/25\ \@2\ \@\(\[HBar]\/\(m\ \[Omega]\)\)\ Cos[ t\ \[Omega]], \(-\(6\/25\)\)\ \@2\ \@\(m\ \[Omega]\ \[HBar]\)\ Sin[ t\ \[Omega]]}\)], "Output"] }, Open ]], Cell[TextData[{ "Note, the initial velocity vanishes, and therefore the initial \ displacement does not. ", "To have nonvanishing initial velocity, we need to include an initial \ momentum boost, like the plane-wave factor ", Cell[BoxData[ StyleBox[\(\(\ \)\(Exp[I\ ko\ z]\)\), FontColor->RGBColor[0, 0, 1]]]], " we include in setting up a squeezed state, which we consider shortly." }], "Text"], Cell[TextData[{ "Likewise, the ", StyleBox["potential", FontColor->RGBColor[1, 0, 0]], " and ", StyleBox["kinetic", FontColor->RGBColor[0, 0, 1]], " energies vary harmonically while their sum\[LongDash]the total energy\ \[LongDash]remains of course constant:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ StyleBox[\(\(\[Psi]\_t\^*\) . V . \[Psi]\_t\), FontColor->RGBColor[1, 0, 0]], StyleBox[",", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\(\[Psi]\_t\^*\) . K . \[Psi]\_t\), FontColor->RGBColor[0, 0, 1]], ",", \(\(\[Psi]\_t\^*\) . H . \[Psi]\_t\)}], "}"}], "//", "FullSimplify"}]], "Input"], Cell[BoxData[ \({1\/25\ \[Omega]\ \[HBar]\ \((29 + 5\ \@6\ Cos[2\ t\ \[Omega]])\), 1\/25\ \[Omega]\ \[HBar]\ \((29 - 5\ \@6\ Cos[ 2\ t\ \[Omega]])\), \(58\ \[Omega]\ \[HBar]\)\/25}\)], \ "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 5.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["The initial kinetic energy is nonvanishing because the initial \ rms deviation in the momentum ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_p\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" is, although the initial velocity vanishes. Verify that", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ FormBox[\(\[Sigma]\_x\), "TraditionalForm"], FormBox[\(\[Sigma]\_p\), "TraditionalForm"]}], "/.", \(\(t\)\(\[Rule]\)\(\ \)\(0\)\(\ \)\)}], FontColor->RGBColor[0, 0, 1]]], "Input", Evaluatable->False], Cell[BoxData[ \(2.065915895516429`\ \[HBar]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Time development in the Heisenberg picture", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "In the usual description of the time development, the state vectors like \ ", Cell[BoxData[ StyleBox[\(\[Psi]\_t\), FontColor->RGBColor[0, 0, 1]]]], "carry the time dependence while the operators representing the observables \ remain time independent. This emphasis can be turned around, however, and an \ equivalent description obtained by letting the observables carry the time \ dependence and keeping the state vectors fixed. The usual description with \ time-dependent state vectors and fixed observables is referred to as the ", StyleBox["Schroedinger picture", FontColor->RGBColor[1, 0, 0]], ", while the alternative with fixed state vectors and time-dependent \ observables is referred to as the ", StyleBox["Heisenberg picture", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], ". " }], "Text"], Cell[TextData[{ "The essential requirement we impose is that the expectation values of our \ observables in either picture be equivalent. Noting that ", Cell[BoxData[ \(\[LeftAngleBracket]Q\[RightAngleBracket]\_t\)], "DisplayFormula", FontColor->RGBColor[0, 0, 1]], StyleBox[" = ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\[LeftAngleBracket]\[Psi]\_t | Q | \[Psi]\_t\[RightAngleBracket]\)], "DisplayFormula", FontColor->RGBColor[0, 0, 1]], StyleBox[" = ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\[LeftAngleBracket]\[Psi]\_0 | U\^\[Dagger] . Q . U | \[Psi]\_0\[RightAngleBracket]\)], "DisplayFormula", FontColor->RGBColor[0, 0, 1]], ", we thus define transformed time-dependent observables in the '", StyleBox["Heisenberg picture", FontColor->RGBColor[0, 0, 1]], "' according to ", Cell[BoxData[ \(Q\_t\ \[Congruent] U\^\[Dagger] . Q . U\)], FontColor->RGBColor[0, 0, 1]], ". For example, we define" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(X\_t = U\^\[Dagger] . X . U\)], "Input"], Cell[BoxData[ \({{0, \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\)\)\/\@2, 0, 0, 0, 0}, {\(\[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \@\(\[HBar]\/\ \(m\ \[Omega]\)\)\)\/\@2, 0, \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\), 0, 0, 0}, {0, \[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\), 0, \@\(3\/2\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\), 0, 0}, {0, 0, \@\(3\/2\)\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \@\(\ \[HBar]\/\(m\ \[Omega]\)\), 0, \@2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \@\(\ \[HBar]\/\(m\ \[Omega]\)\), 0}, {0, 0, 0, \@2\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\), 0, \@\(5\/2\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \ \@\(\[HBar]\/\(m\ \[Omega]\)\)}, {0, 0, 0, 0, \@\(5\/2\)\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \@\(\ \[HBar]\/\(m\ \[Omega]\)\), 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\_t = U\^\[Dagger] . P . U\)], "Input"], Cell[BoxData[ \({{0, \(-\(\(\[ImaginaryI]\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\)\)\/\@2\)\), 0, 0, 0, 0}, {\(\[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ \ \@\(m\ \[Omega]\ \[HBar]\)\)\/\@2, 0, \(-\[ImaginaryI]\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0, 0, 0}, {0, \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0, \(-\[ImaginaryI]\)\ \@\(3\/2\)\ \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \@\(m\ \[Omega]\ \ \[HBar]\), 0, 0}, {0, 0, \[ImaginaryI]\ \@\(3\/2\)\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0, \(-\[ImaginaryI]\)\ \@2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0}, {0, 0, 0, \[ImaginaryI]\ \@2\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0, \(-\[ImaginaryI]\)\ \@\(5\/2\)\ \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \@\(m\ \[Omega]\ \ \[HBar]\)}, {0, 0, 0, 0, \[ImaginaryI]\ \@\(5\/2\)\ \[ExponentialE]\^\(\[ImaginaryI]\ t\ \ \[Omega]\)\ \@\(m\ \[Omega]\ \[HBar]\), 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "and verify that their expectation values calculated in either the ", StyleBox["Schroedinger", FontColor->RGBColor[1, 0, 0]], " or ", StyleBox["Heisenberg", FontColor->RGBColor[0, 0, 1]], " pictures are the same:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ StyleBox[\(\(\[Psi]\_t\^*\) . X . \[Psi]\_t\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\[Equal]", StyleBox[\(\[Psi]\_0 . X\_t . \[Psi]\_0\), FontColor->RGBColor[0, 0, 1]]}], ",", " ", RowBox[{ StyleBox[\(\(\[Psi]\_t\^*\) . P . \[Psi]\_t\), FontColor->RGBColor[1, 0, 0]], "\[Equal]", StyleBox[\(\[Psi]\_0 . P\_t . \[Psi]\_0\), FontColor->RGBColor[0, 0, 1]]}]}], "}"}]], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell["\<\ The 'similarity transformation' (along with the unitarity of U) nevertheless \ preserves hermiticity,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({X\_t\^\[Dagger] \[Equal] \ X\_t, \ P\_t\^\[Dagger] \[Equal] \ P\_t}\)], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell["as well as the fundamental commutation relation,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\([X\_t, P\_t]\)\_C\ // PowerExpand\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[ImaginaryI]\ \[HBar]\), "0", "0", "0", "0", "0"}, {"0", \(\[ImaginaryI]\ \[HBar]\), "0", "0", "0", "0"}, {"0", "0", \(\[ImaginaryI]\ \[HBar]\), "0", "0", "0"}, {"0", "0", "0", \(\[ImaginaryI]\ \[HBar]\), "0", "0"}, {"0", "0", "0", "0", \(\[ImaginaryI]\ \[HBar]\), "0"}, {"0", "0", "0", "0", "0", \(\(-5\)\ \[ImaginaryI]\ \[HBar]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Note, however, as a conserved quantity the hamiltonian is unaffected by the \ similarity transformation and remains independent of time:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(U\^\[Dagger] . H . U\ \[Equal] H\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Because the initial wavepacket ", Cell[BoxData[ \(\[Psi]\_0\)], FontColor->RGBColor[0, 0, 1]], "is time independent, we can pull it out of the fundamental result \t\t", Cell[BoxData[ \(d \[LeftAngleBracket]Q\[RightAngleBracket]\_t/dt\)], FontColor->RGBColor[0, 0, 1]], StyleBox["= ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\[LeftAngleBracket]\([Q, H]\)\[RightAngleBracket]\_t/\((I\ \[HBar])\)\ \)], FontColor->RGBColor[0, 0, 1]], "+", Cell[BoxData[ \(\[LeftAngleBracket]\[PartialD]\_t\ Q\[RightAngleBracket]\_t\)], FontColor->RGBColor[0, 0, 1]], "\nrelating expectation values of an observable ", StyleBox["Q", FontColor->RGBColor[0, 0, 1]], " and its commutator with the hamiltonian. We obtain in this way an equally \ fundamental relation for the time derivative of observables ", Cell[BoxData[ \(Q\_t\ \[Congruent] U\^\[Dagger] . Q . U\)], FontColor->RGBColor[0, 0, 1]], " in the Heisenberg picture, namely,", StyleBox[" \n\t", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(dQ\_t/dt\)], FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\([Q\_t, H]\)/\((I\ \[HBar])\)\)], FontColor->RGBColor[0, 0, 1]], " + ", Cell[BoxData[ \(\(\(\((\[PartialD]\_t\ Q)\)\_t\)\(\ \)\)\)], FontColor->RGBColor[0, 0, 1]], " \nNote, for observables with no intrinsic time dependence of interest to \ us presently, ", Cell[BoxData[ \(\[PartialD]\_t\ Q\ \[Congruent] 0\)], FontColor->RGBColor[0, 0, 1]], " and we can ignore the last term." }], "Text"], Cell["\<\ We can thus verify a variety of expectation-value results directly, for \ example the Ehrenfest-theorem identities for the time-development of the \ position and momentum:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\[PartialD]\_t\ X\_t \[Equal] \([X\_t, H]\)\_C\/\(I\ \[HBar]\), \[PartialD]\_t\ P\_t \[Equal] \([P\_t, H]\)\_C\/\(I\ \[HBar]\)} // PowerExpand\)], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell["\<\ The first result above is the quantum analogue of the velocity, so that\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(P\_t\/m \[Equal] \[PartialD]\_t\ X\_t \[Equal] \([X\_t, H]\)\_C\/\(I\ \[HBar]\) // PowerExpand\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "The second result above is the quantum analogue of the force, ", Cell[BoxData[ \(TraditionalForm\`\(\(\(-\[PartialD]\_x V\) = \ \(-m\)\ \(\[Omega]\^2\) x\)\(,\)\)\)]], " and thus" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[PartialD]\_t\ P\_t\ \[Equal] \([P\_t, H]\)\_C\/\(I\ \[HBar]\) \[Equal] \ \(-m\)\ \ \(\[Omega]\^2\) X\_t\)\(//\)\(PowerExpand\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 6.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Derive ", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(I\ \[HBar]\ dQ\_t/dt\)], FontColor->RGBColor[0, 0, 1]], StyleBox["= ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\([Q\_t, H]\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" directly using ", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(I\ \[HBar]\ dU/dt\)], FontColor->RGBColor[0, 0, 1]], StyleBox["= ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(H\ U\)], FontColor->RGBColor[0, 0, 1]], "." }], "Text"], Cell["\<\ Although the transformed potential and kinetic energies vary with time,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(V\_t = U\^\[Dagger] . V . U\)], "Input"], Cell[BoxData[ \({{\(\[Omega]\ \[HBar]\)\/4, 0, \(\[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar]\)\/\(2\ \@2\), 0, 0, 0}, {0, \(3\ \[Omega]\ \[HBar]\)\/4, 0, 1\/2\ \@\(3\/2\)\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\ \[Omega]\ \[HBar], 0, 0}, {\(\[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \[Omega]\)\ \[Omega]\ \ \[HBar]\)\/\(2\ \@2\), 0, \(5\ \[Omega]\ \[HBar]\)\/4, 0, 1\/2\ \@3\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar], 0}, {0, 1\/2\ \@\(3\/2\)\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar], 0, \(7\ \[Omega]\ \[HBar]\)\/4, 0, 1\/2\ \@5\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar]}, {0, 0, 1\/2\ \@3\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar], 0, \(9\ \[Omega]\ \[HBar]\)\/4, 0}, {0, 0, 0, 1\/2\ \@5\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar], 0, \(5\ \[Omega]\ \[HBar]\)\/4}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(K\_t = U\^\[Dagger] . K . U\)], "Input"], Cell[BoxData[ \({{\(\[Omega]\ \[HBar]\)\/4, 0, \(-\(\(\[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar]\)\/\(2\ \@2\)\)\), 0, 0, 0}, {0, \(3\ \[Omega]\ \[HBar]\)\/4, 0, \(-\(1\/2\)\)\ \@\(3\/2\)\ \[ExponentialE]\^\(\(-2\)\ \ \[ImaginaryI]\ t\ \[Omega]\)\ \[Omega]\ \[HBar], 0, 0}, {\(-\(\(\[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \[Omega]\)\ \ \[Omega]\ \[HBar]\)\/\(2\ \@2\)\)\), 0, \(5\ \[Omega]\ \[HBar]\)\/4, 0, \(-\(1\/2\)\)\ \@3\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\ \[Omega]\ \[HBar], 0}, {0, \(-\(1\/2\)\)\ \@\(3\/2\)\ \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ t\ \[Omega]\)\ \[Omega]\ \[HBar], 0, \(7\ \[Omega]\ \[HBar]\)\/4, 0, \(-\(1\/2\)\)\ \@5\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ t\ \ \[Omega]\)\ \[Omega]\ \[HBar]}, {0, 0, \(-\(1\/2\)\)\ \@3\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \ \[Omega]\)\ \[Omega]\ \[HBar], 0, \(9\ \[Omega]\ \[HBar]\)\/4, 0}, {0, 0, 0, \(-\(1\/2\)\)\ \@5\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ t\ \ \[Omega]\)\ \[Omega]\ \[HBar], 0, \(5\ \[Omega]\ \[HBar]\)\/4}}\)], "Output"] }, Open ]], Cell["their total is of course conserved:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_t\ K\_t + \[PartialD]\_t\ V\_t\)], "Input"], Cell[BoxData[ \({{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["EXERCISE 7.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Verify that", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\(\(\ \)\(X\_t\)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(P\_t\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" are", FontColor->RGBColor[0, 0, 1]], StyleBox[" proportional to their second-order time derivatives in analogy \ with their classical counterparts describing simple harmonic motion and \ determine the constants of proportionality. Verify that the raising- and \ lowering-operator matrices ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(A\_t\^\[Dagger]\)], FontColor->RGBColor[0, 0, 1]], StyleBox["and ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(A\_t\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" in the Heisenberg representation are proportional to their \ first-order time derivatives and determine the constants of proportionality. \ ", FontColor->RGBColor[0, 0, 1]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Squeezed state ", "Subsubsection", FontFamily->"Comic Sans MS"], Cell[TextData[{ "Here is the HO initial wavepacket we've studied before defined with \ arbitrary ", StyleBox["width w", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["position zo", FontColor->RGBColor[1, 0, 1]], ", and ", StyleBox["momentum boost ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(ko\)], FontColor->RGBColor[0, 0, 1]], ". For convenience, we will use scaled HO coordinates and effectively set \ \[HBar] = \[Omega] = m \[Congruent] 1. " }], "Text", FontFamily->"Comic Sans MS"], Cell[BoxData[ RowBox[{ RowBox[{\(\[Chi]\_0\), " ", "=", " ", RowBox[{\(1\/\@\(w \@\(\(\ \)\(\[Pi]\)\)\)\), RowBox[{"Exp", "[", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", StyleBox["zo", FontColor->RGBColor[1, 0, 1]]}], ")"}], "2"], RowBox[{"2", SuperscriptBox[ StyleBox["w", FontColor->RGBColor[1, 0, 0]], "2"]}]]}], "]"}], SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"+", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["I", FontColor->RGBColor[0, 0, 1]]}], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["ko", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["z", FontColor->RGBColor[0, 0, 1]]}]]}]}], ";"}]], "Input"], Cell[TextData[{ "We check ", StyleBox["normalization", FontColor->RGBColor[0, 0, 1]], " and compute the wavepacket's ", StyleBox["energy", FontColor->RGBColor[1, 0, 0]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{\(SetOptions[Integrate, GenerateConditions \[Rule] False]; H\_HO := \ \(1\/2\) \((\(-\[PartialD]\_\(z, z\)#\) + \(z\^2\) #)\) &\), "\ \[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ StyleBox[\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \ \(\[Chi]\_0\^*\) \(\[Chi]\_0\) \[DifferentialD]z\), FontColor->RGBColor[0, 0, 1]], ",", " ", StyleBox[ RowBox[{ SubscriptBox[ StyleBox[ RowBox[{"e", StyleBox["\[Chi]", FontColor->RGBColor[1, 0, 0]]}]], "0"], "=", \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \ \(\[Chi]\_0\^*\) H\_HO\ @\ \[Chi]\_0 \[DifferentialD]z\)}], FontColor->RGBColor[1, 0, 0]]}], "}"}], " ", "//", "PowerExpand"}], "//", "FullSimplify"}]}], "Input"], Cell[BoxData[ RowBox[{\(Unique::"usym"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\(\\*FrameBox[\\\"\\\\\\\"{(\[Omega]*\[HBar])/4, 0, -(\[Omega]*\ \[HBar])/(2*Sqrt[2]), 0, 0, 0}\\\\\\\"\\\", Rule[BoxFrame, False], \ Rule[BoxMargins, False]]\\) is not a symbol or a valid symbol name. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Unique::usym\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ RowBox[{\(Unique::"usym"\), \(\(:\)\(\ \)\), \ "\<\"\\!\\(\\*FrameBox[\\\"\\\\\\\"{(\[Omega]*\[HBar])/4, 0, -(\[Omega]*\ \[HBar])/(2*Sqrt[2]), 0, 0, 0}\\\\\\\"\\\", Rule[BoxFrame, False], \ Rule[BoxMargins, False]]\\) is not a symbol or a valid symbol name. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Unique::usym\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({0, 0}\)], "Output"] }, Open ]], Cell["\<\ With the scaled HO eigenstates in the coordinate representation,\ \>", "Text", FontFamily->"Comic Sans MS"], Cell[BoxData[ RowBox[{ StyleBox[\(\[CurlyPhi]\_n_\), FontColor->RGBColor[1, 0, 0]], ":=", " ", \(\(1\/\@\(\(2\^n\) \(n!\) \@\[Pi]\)\) Exp[\(-\(z\^2\/2\)\)]\ HermiteH[n, z]\)}]], "Input"], Cell[TextData[{ "we compute expansion coefficients, i.e. overlap integrals, for expressing \ the initial state as a generalized Fourier series in the ", Cell[BoxData[ StyleBox[\(\[CurlyPhi]\_n\), FontColor->RGBColor[1, 0, 0]]]], ". We use ", StyleBox["numerical", FontColor->RGBColor[1, 0, 1]], " integration for efficiency and speed, and thus enter values for the \ initial wavepacket ", StyleBox["momentum, position, ", FontColor->RGBColor[0, 0, 1]], "and", StyleBox[" width", FontColor->RGBColor[0, 0, 1]], ":" }], "Text"], Cell[BoxData[ RowBox[{\(c[n_]\), ":=", RowBox[{ StyleBox[ RowBox[{ StyleBox["N", FontColor->RGBColor[1, 0, 1]], "Integrate"}]], "[", " ", RowBox[{ RowBox[{"Evaluate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox[\(\[CurlyPhi]\_n\), FontColor->RGBColor[1, 0, 0]], " ", StyleBox[\(\[Chi]\_0\), FontColor->RGBColor[0, 0, 1]]}], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(ko \[Rule] .75\), FontColor->RGBColor[0, 0, 1]]}], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(zo \[Rule] 1. \), FontColor->RGBColor[0, 0, 1]]}], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(w \[Rule] 1. \), FontColor->RGBColor[0, 0, 1]]}], "]"}], ",", \({z, \(-\[Infinity]\), \[Infinity]}\)}], "]"}]}]], "Input"], Cell["\<\ The vector List of these coefficients defines the matrix representation of \ the initial-state vector:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Chi]\&\[RightVector]\_0\ = Table[\ c[n], {n, 0, nmax}]\)], "Input"], Cell[BoxData[ \({\(\(0.6296129510972491`\)\(\[InvisibleSpace]\)\) + 0.24783239009039712`\ \[ImaginaryI], \(\(0.31377061449852994`\)\(\ \[InvisibleSpace]\)\) + 0.5091466540966074`\ \[ImaginaryI], \(-0.03404468791825199`\) + 0.3722373074334796`\ \[ImaginaryI], \(-0.12787261900137684`\) + 0.14154123016291478`\ \[ImaginaryI], \(-0.08274158438653312`\) + 0.016135033323897593`\ \[ImaginaryI], \(-0.02999194554422996`\) - 0.014521544247835612`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[TextData[{ "Squaring the resulting vector ", StyleBox["elements", FontSlant->"Italic"], " gives us an idea of convergence:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Abs[%]\^2\)], "Input"], Cell[BoxData[ \({0.45783336176730577`, 0.35768231390055544`, 0.13971965382077797`, 0.0363853265263025`, 0.007106509087157069`, 0.0011103920448699027`}\)], "Output"] }, Open ]], Cell["We readily compute the normalization and energy,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\({\(\[Chi]\&\[RightVector]\_0\^*\) . \[Chi]\&\[RightVector]\_0\ , \(\ \[Chi]\&\[RightVector]\_0\^*\) . H . \[Chi]\&\[RightVector]\_0\ } /. {\[HBar] \[Rule] 1, \[Omega] \[Rule] 1} // Chop\) // Chop\)], "Input"], Cell[BoxData[ \({0.9998375571469686`, 1.280174376267481`}\)], "Output"] }, Open ]], Cell["and compare the energy with the above analytical result:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["e", FontColor->GrayLevel[0]], "\[Chi]"}]], "0"], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(ko \[Rule] .75\), FontColor->RGBColor[0, 0, 1]]}], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(zo \[Rule] 1. \), FontColor->RGBColor[0, 0, 1]]}], StyleBox["/.", FontColor->RGBColor[0, 0, 1]], StyleBox[\(w \[Rule] 1. \), FontColor->RGBColor[0, 0, 1]]}], "//", "FullSimplify"}]], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ There is clearly some loss of precision involved using a truncated matrix \ representation and the effective truncated Fourier series expansion of the \ initial wavepacket. \ \>", "Text"], Cell["\<\ We can now evolve the initial wavevector in time by applying the \ time-development matrix,\ \>", "Text"], Cell[BoxData[ \(\(\[Chi]\&\[RightVector]\_t = U . \[Chi]\&\[RightVector]\_0 /. \[Omega] \[Rule] \ 1 // Chop;\)\)], "Input"], Cell["\<\ and check for example conservation of probability and energy:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{\(\(\[Chi]\&\[RightVector]\_t\^*\) . \[Chi]\&\ \[RightVector]\_t\), " ", ",", RowBox[{ StyleBox[ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["e", FontColor->GrayLevel[0]], StyleBox["\[Chi]", FontColor->RGBColor[0, 0, 1]]}]], "t"], FontColor->GrayLevel[0]], "=", \(\(\[Chi]\&\[RightVector]\_t\^*\) . H . \[Chi]\&\[RightVector]\_t\)}]}], " ", "}"}], "/.", \({\[HBar] \[Rule] 1, \[Omega] \[Rule] 1}\)}], "//", "Chop"}], "//", "Chop"}]], "Input"], Cell[BoxData[ \({0.9998375571469686`, 1.280174376267481`}\)], "Output"] }, Open ]], Cell[TextData[{ "We can also compute for example the time-dependent expectation values of \ the ", StyleBox["X", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["P", FontColor->RGBColor[1, 0, 0]], " matrices:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\[Chi]\&\[RightVector]\_t\^*\), ".", StyleBox["X", FontColor->RGBColor[0, 0, 1]], ".", \(\[Chi]\&\[RightVector]\_t\)}], " ", ",", RowBox[{\(\[Chi]\&\[RightVector]\_t\^*\), ".", StyleBox["P", FontColor->RGBColor[1, 0, 0]], ".", \(\[Chi]\&\[RightVector]\_t\)}]}], " ", "}"}], "/.", \({\[HBar] \[Rule] 1, \[Omega] \[Rule] 1, m \[Rule] 1}\)}], "//", "FullSimplify"}], "//", "Chop"}]], "Input"], Cell[BoxData[ \({0.9987271651091203`\ Cos[t] + 0.7490453737611881`\ Sin[t], 0.7490453737611881`\ Cos[t] - 0.9987271651091202`\ Sin[t]}\)], "Output"] }, Open ]], Cell[TextData[{ "Similarly, we can compute the same thing in the Heisenberg picture using \ ", StyleBox["time-dependent operators", FontColor->RGBColor[1, 0, 0]], " and the ", StyleBox["initial-state vector", FontColor->RGBColor[0, 0, 1]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ StyleBox[\(\[Chi]\&\[RightVector]\_0\^*\), FontColor->RGBColor[0, 0, 1]], ".", StyleBox[\(X\_t\), FontColor->RGBColor[1, 0, 0]], ".", StyleBox[\(\[Chi]\&\[RightVector]\_0\), FontColor->RGBColor[0, 0, 1]]}], " ", ",", RowBox[{ StyleBox[\(\[Chi]\&\[RightVector]\_0\^*\), FontColor->RGBColor[0, 0, 1]], ".", StyleBox[\(P\_t\), FontColor->RGBColor[1, 0, 0]], ".", StyleBox[\(\[Chi]\&\[RightVector]\_0\), FontColor->RGBColor[0, 0, 1]]}]}], " ", "}"}], "/.", \({\[HBar] \[Rule] 1, \[Omega] \[Rule] 1, m \[Rule] 1}\)}], "//", "FullSimplify"}], "//", "Chop"}]], "Input"], Cell[BoxData[ \({0.9987271651091203`\ Cos[t] + 0.7490453737611881`\ Sin[t], 0.7490453737611881`\ Cos[t] - 0.9987271651091202`\ Sin[t]}\)], "Output"] }, Open ]], Cell[TextData[{ "These results show to within round-off errors that ", Cell[BoxData[ \(TraditionalForm\`\(\(\[LeftAngleBracket]z\[RightAngleBracket]\_\(t = \ \ 0\) = \(zo\ \ and\ \[LeftAngleBracket]p\_z\[RightAngleBracket]\_\(t = \ 0\) \ = ko\)\)\(,\)\)\)]], " as required. Moreover, we see that the time development is described by a \ closed orbit in the ", Cell[BoxData[ RowBox[{"\[LeftAngleBracket]", StyleBox[\(X\_t\), FontColor->RGBColor[1, 0, 0]], "\[RightAngleBracket]"}]]], ", ", Cell[BoxData[ RowBox[{"\[LeftAngleBracket]", StyleBox[\(P\_t\), FontColor->RGBColor[1, 0, 0]], "\[RightAngleBracket]"}]]], " phase-space plane in analogy with the classical description." }], "Text"], Cell[TextData[{ StyleBox["EXERCISE 8.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Check convergence in this section by doubling or tripling ", FontColor->RGBColor[0, 0, 1]], StyleBox["nmax", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" and recomputing the results. Make ", FontColor->RGBColor[0, 0, 1]], StyleBox["ParametricPlots", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" of the ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\(\(\[LeftAngleBracket]X\_t\[RightAngleBracket]\)\(,\)\)\)], FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\[LeftAngleBracket]P\_t\[RightAngleBracket]\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" phase-space orbit for half and full periods, or, better, animate \ it. Use the option ", FontColor->RGBColor[0, 0, 1]], StyleBox["AspectRatio \[Rule] Automatic", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontColor->RGBColor[0, 0, 1]] }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Macintosh", ScreenRectangle->{{0, 1920}, {0, 1174}}, WindowSize->{884, 875}, WindowMargins->{{20, Automatic}, {Automatic, 0}}, Visible->True, PrintingOptions->{"PrintingMargins"->{{54, 54}, {72, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->True, "PrintMultipleHorizontalPages"->False}, InputAliases->{"notation"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation>"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation<"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "symb"->RowBox[ {"Symbolize", "[", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], "]"}], "infixnotation"->RowBox[ {"InfixNotation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "addia"->RowBox[ {"AddInputAlias", "[", RowBox[ {"\"\[Placeholder]\"", "\[Rule]", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "pattwraper"->TagBox[ "\[Placeholder]", NotationPatternTag, TagStyle -> "NotationPatternWrapperStyle"], "madeboxeswraper"->TagBox[ "\[Placeholder]", NotationMadeBoxesTag, TagStyle -> "NotationMadeBoxesWrapperStyle"]}, Magnification->1.25, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. Make modifications to any definition using commands in the \ Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, CellLabelMargins->{{12, Inherited}, {Inherited, Inherited}}, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, CellLabelMargins->{{24, Inherited}, {Inherited, Inherited}}, ScriptMinSize->12], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}}, ScriptMinSize->8], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, CellLabelMargins->{{2, Inherited}, {Inherited, Inherited}}, ScriptMinSize->5, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 11}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {12, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 7}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubsection", FontFamily->"Times", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, Hyphenation->True, LineSpacing->{1, 3}, CounterIncrements->"Text"], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, DefaultNewInlineCellStyle->"None", Hyphenation->True, LineSpacing->{1, 3}, LanguageCategory->"NaturalLanguage", CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, TextJustification->0.5, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names. \ Some attributes for these styles are actually set in FormatType Styles (in \ the last section of this stylesheet). \ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontSize->11], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontSize->11], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->11, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->7, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FrameLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["FrameLabel", "Presentation"], FontSize->12], Cell[StyleData["FrameLabel", "Condensed"], FontSize->9], Cell[StyleData["FrameLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a cell \ exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "EnhancedPrintout"], StyleMenuListing->None, FontFamily->"Futura", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Outline Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Outline1"], CellMargins->{{12, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 50}, ParagraphIndent->-38, CounterIncrements->"Outline1", FontSize->18, FontWeight->"Bold", CounterBoxOptions->{ CounterFunction:>Utilities`Notation`Private`CapitalRomanNumeral}], Cell[StyleData["Outline1", "Printout"], CounterBoxOptions->{ CounterFunction:>Utilities`Notation`Private`CapitalRomanNumeral}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline2"], CellMargins->{{59, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 60}, ParagraphIndent->-27, CounterIncrements->"Outline2", FontSize->15, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}], Cell[StyleData["Outline2", "Printout"], CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline3"], CellMargins->{{108, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 70}, ParagraphIndent->-21, CounterIncrements->"Outline3", FontSize->12, CounterBoxOptions->{CounterFunction:>Identity}], Cell[StyleData["Outline3", "Printout"], CounterBoxOptions->{CounterFunction:>Identity}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline4"], CellMargins->{{158, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 80}, ParagraphIndent->-18, CounterIncrements->"Outline4", FontSize->10, CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "a", "z"], #]&)}], Cell[StyleData["Outline4", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder objects in \ palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "Condensed"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an inline cell \ within another.\ \>", "Text"], Cell[StyleData["InlineCell"], LanguageCategory->"Formula", ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display of certain \ types of objects in typeset expressions. For example, \"UnmatchedBracket\" \ style defines how unmatched bracket, curly bracket, and parenthesis \ characters are displayed (typically by coloring them to make them stand out).\ \ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Notation Package Styles", "Section", CellTags->"NotationPackage"], Cell["\<\ The cells below define certain styles needed by the Notation package. These \ styles serve to make visible otherwise invisible tagboxes.\ \>", "Text", CellTags->"NotationPackage"], Cell[StyleData["NotationTemplateStyle"], StyleMenuListing->None, Background->RGBColor[1, 1, 0.850004], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"], Cell[StyleData["NotationPatternWrapperStyle"], StyleMenuListing->None, Background->RGBColor[1, 0.900008, 0.979995], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"], Cell[StyleData["NotationMadeBoxesWrapperStyle"], StyleMenuListing->None, Background->RGBColor[0.900008, 0.889998, 1], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"] }, Closed]] }] ] (******************************************************************* 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->{ "KroneckerDelta"->{ Cell[6934, 229, 225, 7, 33, "Input", CellTags->"KroneckerDelta"], Cell[7162, 238, 473, 12, 145, "Output", CellTags->"KroneckerDelta"]} } *) (*CellTagsIndex CellTagsIndex->{ {"KroneckerDelta", 120838, 3883} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 164, 6, 103, "Subsubtitle"], Cell[1943, 61, 253, 5, 55, "Text"], Cell[2199, 68, 90, 2, 36, "Text"], Cell[CellGroupData[{ Cell[2314, 74, 80, 1, 55, "Subsubsection"], Cell[2397, 77, 159, 3, 36, "Text"], Cell[2559, 82, 385, 9, 52, "Input"], Cell[2947, 93, 202, 6, 36, "Text"], Cell[CellGroupData[{ Cell[3174, 103, 83, 1, 33, "Input"], Cell[3260, 106, 56, 1, 33, "Output"] }, Open ]], Cell[3331, 110, 60, 0, 36, "Text"], Cell[CellGroupData[{ Cell[3416, 114, 49, 1, 33, "Input"], Cell[3468, 117, 43, 1, 33, "Output"] }, Open ]], Cell[3526, 121, 153, 4, 36, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[3716, 130, 71, 1, 55, "Subsubsection"], Cell[3790, 133, 671, 21, 74, "Text"], Cell[CellGroupData[{ Cell[4486, 158, 1672, 43, 71, "Input"], Cell[6161, 203, 443, 11, 145, "Output"] }, Open ]], Cell[6619, 217, 290, 8, 55, "Text"], Cell[CellGroupData[{ Cell[6934, 229, 225, 7, 33, "Input", CellTags->"KroneckerDelta"], Cell[7162, 238, 473, 12, 145, "Output", CellTags->"KroneckerDelta"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[7684, 256, 86, 1, 55, "Subsubsection"], Cell[7773, 259, 261, 8, 36, "Text"], Cell[8037, 269, 421, 10, 66, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[8495, 284, 65, 1, 55, "Subsubsection"], Cell[8563, 287, 660, 15, 76, "Text"], Cell[9226, 304, 101, 2, 33, "Input"], Cell[9330, 308, 469, 12, 56, "Text"], Cell[9802, 322, 160, 4, 35, "Input"], Cell[9965, 328, 146, 3, 36, "Text"], Cell[CellGroupData[{ Cell[10136, 335, 539, 12, 73, "Input"], Cell[10678, 349, 455, 11, 163, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11170, 365, 456, 12, 52, "Input"], Cell[11629, 379, 455, 11, 163, "Output"] }, Open ]], Cell[12099, 393, 92, 2, 36, "Text"], Cell[CellGroupData[{ Cell[12216, 399, 231, 6, 35, "Input"], Cell[12450, 407, 38, 1, 33, "Output"] }, Open ]], Cell[12503, 411, 610, 16, 74, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[13150, 432, 78, 1, 55, "Subsubsection"], Cell[13231, 435, 256, 3, 55, "Text"], Cell[13490, 440, 733, 25, 57, "Text"], Cell[CellGroupData[{ Cell[14248, 469, 254, 6, 54, "Input"], Cell[14505, 477, 443, 11, 145, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[14997, 494, 77, 1, 55, "Subsubsection"], Cell[15077, 497, 175, 7, 36, "Text"], Cell[CellGroupData[{ Cell[15277, 508, 403, 9, 78, "Input"], Cell[15683, 519, 609, 11, 205, "Output"] }, Open ]], Cell[16307, 533, 281, 5, 55, "Text"], Cell[CellGroupData[{ Cell[16613, 542, 291, 6, 33, "Input"], Cell[16907, 550, 38, 1, 33, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[16994, 557, 90, 1, 55, "Subsubsection"], Cell[CellGroupData[{ Cell[17109, 562, 172, 3, 103, "Input"], Cell[17284, 567, 911, 17, 345, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18232, 589, 160, 3, 103, "Input"], Cell[18395, 594, 1133, 24, 227, "Output"] }, Open ]], Cell[19543, 621, 648, 19, 55, "Text"], Cell[20194, 642, 476, 14, 74, "Text"], Cell[20673, 658, 223, 3, 54, "Input"], Cell[CellGroupData[{ Cell[20921, 665, 83, 1, 35, "Input"], Cell[21007, 668, 46, 1, 33, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[21102, 675, 97, 1, 55, "Subsubsection"], Cell[CellGroupData[{ Cell[21224, 680, 266, 6, 78, "Input"], Cell[21493, 688, 984, 19, 219, "Output"] }, Open ]], Cell[22492, 710, 412, 12, 55, "Text"], Cell[CellGroupData[{ Cell[22929, 726, 108, 2, 78, "Input"], Cell[23040, 730, 1078, 21, 219, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[24167, 757, 74, 1, 55, "Subsubsection"], Cell[CellGroupData[{ Cell[24266, 762, 66, 1, 33, "Input"], Cell[24335, 765, 608, 11, 205, "Output"] }, Open ]], Cell[24958, 779, 256, 5, 55, "Text"], Cell[CellGroupData[{ Cell[25239, 788, 56, 1, 33, "Input"], Cell[25298, 791, 464, 11, 145, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[25811, 808, 67, 1, 55, "Subsubsection"], Cell[25881, 811, 726, 22, 74, "Text"], Cell[26610, 835, 537, 15, 34, "Input"], Cell[27150, 852, 275, 9, 36, "Text"], Cell[27428, 863, 70, 0, 36, "Text"], Cell[CellGroupData[{ Cell[27523, 867, 305, 9, 34, "Input"], Cell[27831, 878, 589, 11, 145, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[28457, 894, 432, 12, 36, "Input"], Cell[28892, 908, 446, 11, 145, "Output"] }, Open ]], Cell[29353, 922, 330, 7, 36, "Text"], Cell[CellGroupData[{ Cell[29708, 933, 573, 15, 58, "Input"], Cell[30284, 950, 38, 1, 33, "Output"] }, Open ]], Cell[30337, 954, 489, 15, 56, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[30863, 974, 75, 1, 55, "Subsubsection"], Cell[30941, 977, 158, 3, 36, "Text"], Cell[CellGroupData[{ Cell[31124, 984, 65, 1, 33, "Input"], Cell[31192, 987, 52, 1, 33, "Output"] }, Open ]], Cell[31259, 991, 161, 3, 55, "Text"], Cell[CellGroupData[{ Cell[31445, 998, 107, 2, 58, "Input"], Cell[31555, 1002, 38, 1, 33, "Output"] }, Open ]], Cell[31608, 1006, 229, 6, 36, "Text"], Cell[CellGroupData[{ Cell[31862, 1016, 247, 7, 33, "Input"], Cell[32112, 1025, 58, 1, 51, "Output"] }, Open ]], Cell[32185, 1029, 482, 18, 36, "Text"], Cell[32670, 1049, 1142, 32, 33, "Input"], Cell[33815, 1083, 46, 0, 36, "Text"], Cell[CellGroupData[{ Cell[33886, 1087, 264, 6, 33, "Input"], Cell[34153, 1095, 443, 11, 145, "Output"] }, Open ]], Cell[34611, 1109, 141, 3, 36, "Text"], Cell[CellGroupData[{ Cell[34777, 1116, 268, 6, 33, "Input"], Cell[35048, 1124, 609, 11, 205, "Output"] }, Open ]], Cell[35672, 1138, 163, 3, 55, "Text"], Cell[CellGroupData[{ Cell[35860, 1145, 198, 5, 64, "Input"], Cell[36061, 1152, 81, 1, 62, "Output"] }, Open ]], Cell[36157, 1156, 63, 0, 36, "Text"], Cell[CellGroupDa