(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 86116, 1906] NotebookOptionsPosition[ 82283, 1790] NotebookOutlinePosition[ 82961, 1814] CellTagsIndexPosition[ 82918, 1811] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Anharmonic Oscillator", "Subtitle"], Cell[CellGroupData[{ Cell["Introduction and the simple harmonic oscillator", "Subsection"], Cell[TextData[{ "In this notebook we study some problems in quantum mechanics using matrix \ methods. We know that we can solve quantum mechanics in any complete set of \ basis functions. If we choose a particular basis, the Hamiltonian will not, \ in general, be diagonal, but the task is to diagonalize it to find the \ eigenvalues (which are the possible results of a measurement of the energy) \ and the eigenvectors. The reference for this material is Kinzel and Reents, \ p. 47-51.\n\nIn many cases this can not be done exactly and some numerical \ approximation is needed. A common approach, which is the basis of a lot of \ quantum chemistry is to take a ", StyleBox["finite ", FontSlant->"Italic"], StyleBox["basis set and diagonalize it numerically. The ground state of this \ reduced basis state will not be the exact ground state, but by increasing \ the size of the basis (up to a point) we can improve the accuracy and see if \ the energy converges as we increase the basis size. We will apply this \ approach here for an anharmonic oscillator. \n\nWe first discuss the exactly \ solvable case of the simple harmonic oscillator. The Hamiltonian is given \ by", FontVariations->{"CompatibilityType"->0}] }], "Text", CellChangeTimes->{{3.482972259807725*^9, 3.482972266549658*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["H", "0"], " ", "=", " ", RowBox[{ FractionBox[ SuperscriptBox["p", "2"], RowBox[{"2", "m"}]], " ", "+", " ", RowBox[{ FractionBox["1", "2"], "m", " ", SuperscriptBox["\[Omega]", "2"], SuperscriptBox["x", "2"]}]}]}]], "DisplayFormula"], Cell["\<\ where p is the momentum, x the position, m the mass and \[Omega] the angular \ frequency of the classical oscillator. This can be written in dimensionless \ form as \ \>", "Text"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["H", "0"], " "}], "\[HBar]\[Omega]"], "=", " ", RowBox[{ RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox[ RowBox[{"(", FractionBox["p", SubscriptBox["p", "0"]], ")"}], "2"]}], "+", " ", RowBox[{ FractionBox["1", "2"], SuperscriptBox[ RowBox[{"(", FractionBox["x", SubscriptBox["x", "0"]], ")"}], "2"]}]}]}]], "DisplayFormula"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"\[HBar]m", " ", "\[Omega]"}]], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"\[HBar]", "/", RowBox[{"(", RowBox[{"m", " ", "\[Omega]"}], ")"}]}]], TraditionalForm]]], ", are the basic momentum and length scales. From now on, we will give the \ energy in units of \[HBar]\[Omega], x in units of ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ", and p in units of ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], ", so the reduced Hamiltonian is " }], "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["H", "0"], " ", "=", " ", RowBox[{ FractionBox[ SuperscriptBox["p", "2"], "2"], " ", "+", " ", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], "2"], "."}]}]}]], "DisplayFormula"], Cell["\<\ In any textbook on quantum mechanics, it is shown that the energy levels are \ given by \ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["E", "n"], "=", " ", RowBox[{"n", " ", "+", " ", FractionBox["1", "2"]}]}], ",", " ", RowBox[{"n", " ", "=", " ", "0"}], ",", " ", "1", ",", " ", "2", ",", " ", RowBox[{"...", ".", " ", "."}]}]], "DisplayFormula"], Cell["and the wavefunctions are given by", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Psi]", "n"], RowBox[{"(", "x", ")"}]}], " ", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["2", "n"], RowBox[{"n", "!"}], " ", SqrtBox["\[Pi]"]}], ")"}], RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], " ", SuperscriptBox["e", RowBox[{ RowBox[{"-", SuperscriptBox["x", "2"]}], "/", "2"}]], " ", SubscriptBox["H", "n"], RowBox[{"(", "x", ")"}]}]}], " ", ","}]], "DisplayFormula"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ SubscriptBox["H", "n"], TraditionalForm]]], "(x) is a Hermite polynomial.\n\nThe books also show that it is easier to \ determine the energy levels using operator methods rather than the Schr\ \[ODoubleDot]dinger equation, and that is the approach that we will take \ here. In this approach, one introduces so-called raising and lowering \ operators, ", Cell[BoxData[ FormBox[ SuperscriptBox["a", "\[Dagger]"], TraditionalForm]]], "and a, related to x and p by" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"x", " ", "=", " ", RowBox[{ FractionBox["1", SqrtBox["2"]], RowBox[{"(", RowBox[{ SuperscriptBox["a", "\[Dagger]"], "+", " ", "a"}], ")"}]}]}], ",", " ", RowBox[{"p", " ", "=", " ", RowBox[{ FractionBox["i", SqrtBox["2"]], RowBox[{"(", RowBox[{ SuperscriptBox["a", "\[Dagger]"], " ", "-", " ", "a"}], ")"}]}]}], ","}]], "DisplayFormula"], Cell["which have the commutation relations", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"[", " ", RowBox[{"a", ",", SuperscriptBox["a", "\[Dagger]"]}], "]"}], " ", "=", " ", RowBox[{"1", " ", "."}]}]], "DisplayFormula"], Cell["and in terms of which the Hamiltonian is written", "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["H", "0"], " ", "=", " ", RowBox[{ RowBox[{ SuperscriptBox["a", "\[Dagger]"], "a"}], " ", "+", " ", RowBox[{ FractionBox["1", "2"], "."}]}]}]], "DisplayFormula"], Cell["\<\ Denoting eigenstates of the Hamiltonian by |n\[RightAngleBracket], then one \ finds\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["a", "\[Dagger]"], "|", "n"}], "\[RightAngleBracket]"}], " ", "=", " ", RowBox[{ SqrtBox[ RowBox[{"n", "+", "1"}]], "|", RowBox[{"n", "+", "1"}]}]}], "\[RightAngleBracket]"}], ",", " ", RowBox[{"a", "|", "n"}]}], "\[RightAngleBracket]"}], " ", "=", " ", RowBox[{ SqrtBox["n"], "|", RowBox[{"n", "-", "1"}]}]}], "\[RightAngleBracket]"}], " "}]], "DisplayFormula"], Cell["and so", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["a", "\[Dagger]"], "a"}], "|", "n"}], "\[RightAngleBracket]"}], " ", "=", " ", RowBox[{"n", "|", "n"}]}], "\[RightAngleBracket]"}], ",", " "}]], "DisplayFormula"], Cell[TextData[{ "from which it follows that the energy ", Cell[BoxData[ FormBox[ SubscriptBox["E", "n"], TraditionalForm]]], "is equal to (n + 1/2) as stated above.\n\nThe matrix elements of x in the \ basis |n", "\[RightAngleBracket]", " are given by" }], "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["X", "nm"], "=", " ", RowBox[{ RowBox[{"\[LeftAngleBracket]", RowBox[{"n", "|", "x", "|", "m"}], "\[RightAngleBracket]"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", SqrtBox["2"]], SqrtBox[ RowBox[{"m", "+", "1"}]], SubscriptBox["\[Delta]", RowBox[{"n", ",", RowBox[{"m", "+", "1"}]}]]}], "+", " ", RowBox[{ FractionBox["1", SqrtBox["2"]], SqrtBox["m"], SubscriptBox["\[Delta]", RowBox[{"n", ",", RowBox[{"m", "-", "1"}]}]]}]}], " ", "=", " ", RowBox[{ FractionBox["1", "2"], SqrtBox[ RowBox[{"n", "+", "m", "+", "1"}]], RowBox[{ SubscriptBox["\[Delta]", RowBox[{ RowBox[{"|", RowBox[{"n", "-", "m"}], "|"}], ",", "1"}]], "."}]}]}]}]}]], "DisplayFormula"], Cell["\<\ Hence we can conveniently define the matrix elements for x as follows:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "\"\\"", "]"}], ";", " ", RowBox[{"Off", "[", RowBox[{"General", "::", "spell1"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"x", "[", RowBox[{"n_", ",", " ", "m_"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"(", RowBox[{"n", "+", "m", "+", "1"}], ")"}], "]"}], "/", "2"}], " ", "/;", " ", RowBox[{ RowBox[{"Abs", "[", RowBox[{"n", "-", "m"}], "]"}], " ", "==", " ", "1"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"x", "[", RowBox[{"n_", ",", " ", "m_"}], "]"}], " ", ":=", " ", RowBox[{"0", " ", "/;", " ", RowBox[{ RowBox[{"Abs", "[", RowBox[{"n", "-", "m"}], "]"}], " ", "!=", " ", "1"}]}]}]], "Input"], Cell["\<\ and hence generate a matrix for x, if we keep a basis of \"basissize\" \ states:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"x", "[", "basissize_", "]"}], " ", ":=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"x", "[", RowBox[{"n", ",", " ", "m"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", " ", "0", ",", " ", RowBox[{"basissize", "-", "1"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"m", ",", " ", "0", ",", " ", RowBox[{"basissize", "-", "1"}]}], "}"}]}], " ", "]"}]}]], "Input"], Cell["Lets check that this is correct for basissize = 4", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", "[", "4", "]"}], " ", "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", FractionBox["1", SqrtBox["2"]], "0", "0"}, { FractionBox["1", SqrtBox["2"]], "0", "1", "0"}, {"0", "1", "0", SqrtBox[ FractionBox["3", "2"]]}, {"0", "0", SqrtBox[ FractionBox["3", "2"]], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], 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We will \ take it to be proportional to ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]]], ", i.e." }], "Text"], Cell[BoxData[ RowBox[{"H", " ", "=", " ", RowBox[{ SubscriptBox["H", RowBox[{"0", " "}]], "+", " ", RowBox[{"\[Lambda]", " ", SuperscriptBox["x", "4"]}]}]}]], "DisplayFormula"], Cell[TextData[{ "It is easy to generate the matrix for H using the matrix obtained above for \ x and the convenient \"dot\" notation in ", StyleBox["Mathematica", FontSlant->"Italic"], " for performing matrix products:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"h", "[", RowBox[{"basissize_", ",", " ", "\[Lambda]_"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"h0", "[", "basissize", "]"}], " ", "+", " ", RowBox[{"\[Lambda]", " ", RowBox[{ RowBox[{"x", "[", "basissize", "]"}], " ", ".", " ", RowBox[{"x", "[", "basissize", "]"}], " ", ".", " ", RowBox[{"x", "[", "basissize", "]"}], " ", ".", " ", RowBox[{"x", "[", "basissize", "]"}]}]}]}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"h", "[", RowBox[{"4", ",", " ", "\[Lambda]"}], "]"}], " ", "//", " ", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ FractionBox["1", "2"], "+", FractionBox[ RowBox[{"3", " ", "\[Lambda]"}], "4"]}], "0", FractionBox[ RowBox[{"3", " ", "\[Lambda]"}], SqrtBox["2"]], "0"}, {"0", RowBox[{ FractionBox["3", "2"], "+", FractionBox[ RowBox[{"15", " ", "\[Lambda]"}], "4"]}], "0", RowBox[{"3", " ", SqrtBox[ FractionBox["3", "2"]], " ", "\[Lambda]"}]}, { FractionBox[ RowBox[{"3", " ", "\[Lambda]"}], SqrtBox["2"]], "0", RowBox[{ FractionBox["5", "2"], "+", FractionBox[ RowBox[{"27", " ", "\[Lambda]"}], "4"]}], "0"}, {"0", RowBox[{"3", " ", SqrtBox[ FractionBox["3", "2"]], " ", "\[Lambda]"}], "0", RowBox[{ FractionBox["7", "2"], "+", FractionBox[ RowBox[{"15", " ", "\[Lambda]"}], "4"]}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.453061990645669*^9, 3.4530632975027733`*^9, 3.453135012671556*^9, 3.482972225627348*^9}] }, Open ]], Cell["\<\ The eigenvalues can also be obtained numerically and then sorted. Here we \ give a function (with delayed assignment) for doing this:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"evals", "[", RowBox[{"basissize_", ",", " ", "\[Lambda]_"}], "]"}], " ", ":=", " ", RowBox[{"Sort", " ", "[", " ", RowBox[{"Eigenvalues", " ", "[", " ", RowBox[{"N", "[", " ", RowBox[{"h", "[", RowBox[{"basissize", ",", " ", "\[Lambda]"}], "]"}], " ", "]"}], " ", "]"}], " ", "]"}]}]], "Input"], Cell["\<\ Now we get some numbers. 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This is not surprising because the highest of the plotted \ levels are some of the highest possible for the size-10 basis, and so the \ effects of the truncation of the basis are naturally large in that region. \ \>", "Text", CellChangeTimes->{3.453063520438367*^9}], Cell[TextData[{ "To be more systematic, we focus on the ground state, which is given by the \ first element of ", StyleBox["evals", FontWeight->"Bold"], ". 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Here are some \ results:\ \>", "Text", CellChangeTimes->{3.45306364272682*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "/", "basissize"}], ",", " ", RowBox[{ RowBox[{"evals", "[", RowBox[{"basissize", ",", " ", "0.2"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"basissize", ",", " ", "9", ",", " ", "30"}], "}"}]}], "]"}], ",", " ", RowBox[{"Frame", " ", "\[Rule]", " ", "True"}], ",", " ", RowBox[{"Axes", " ", "\[Rule]", " ", "False"}], ",", " ", RowBox[{"PlotStyle", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.02", "]"}], ",", " ", RowBox[{"Hue", "[", "0", "]"}]}], " ", "}"}]}], ",", " ", RowBox[{"Epilog", "\[Rule]", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "/", "30"}], ",", " ", "0.6024051508367307`"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"1", "/", "8"}], ",", "0.6024051508367307`"}], "}"}]}], "}"}], "]"}]}], ",", " ", RowBox[{"PlotRange", " ", "\[Rule]", " ", "All"}], ",", " ", RowBox[{"FrameLabel", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ "\"\<1/(basis size)\>\"", ",", " ", "\"\<\!\(\*SubscriptBox[\(E\), \(0\)]\)\>\""}], "}"}]}], " ", ",", " ", RowBox[{"RotateLabel", " ", "\[Rule]", " ", "False"}], ",", " ", RowBox[{"PlotLabel", " ", "\[Rule]", " ", "\"\<\[Lambda] = 0.2\>\""}]}], "]"}]], "Input", CellChangeTimes->{3.453061982927841*^9}], Cell[BoxData[ GraphicsBox[ {Hue[0], PointSize[0.02], PointBox[{{0.1111111111111111, 0.6024209543013108}, {0.1, 0.6023775190526743}, {0.09090909090909091, 0.6024183832982735}, { 0.08333333333333333, 0.6024011607654558}, {0.07692307692307693, 0.6024049452074481}, {0.07142857142857142, 0.602405894021111}, { 0.06666666666666667, 0.6024046277198818}, {0.0625, 0.6024053450745795}, { 0.058823529411764705`, 0.6024051371647793}, {0.05555555555555555, 0.6024051377861176}, {0.05263157894736842, 0.6024051875296739}, {0.05, 0.6024051508367516}, {0.047619047619047616`, 0.6024051675730528}, { 0.045454545454545456`, 0.6024051634332768}, {0.043478260869565216`, 0.6024051627967529}, {0.041666666666666664`, 0.6024051643602931}, {0.04, 0.6024051633274045}, {0.038461538461538464`, 0.6024051637875685}, { 0.037037037037037035`, 0.602405163688821}, {0.03571428571428571, 0.6024051636547668}, {0.034482758620689655`, 0.6024051637110869}, { 0.03333333333333333, 0.602405163672897}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], AxesStyle->Thickness[Large], BaseStyle->{15, FontFamily -> "Times", Bold}, Epilog->LineBox[ NCache[{{ Rational[1, 30], 0.6024051508367307}, { Rational[1, 8], 0.6024051508367307}}, {{0.03333333333333333, 0.6024051508367307}, {0.125, 0.6024051508367307}}]], Frame->True, FrameLabel->{ FormBox["\"1/(basis size)\"", TraditionalForm], FormBox[ "\"\\!\\(\\*SubscriptBox[\\(E\\), \\(0\\)]\\)\"", TraditionalForm]}, FrameStyle->Thickness[Large], ImageSize->{460., Automatic}, PlotLabel->FormBox["\"\[Lambda] = 0.2\"", TraditionalForm], PlotRangeClipping->True, RotateLabel->False]], "Output", CellChangeTimes->{3.453061994164588*^9, 3.453063300451911*^9, 3.453135673419445*^9, 3.48297223666746*^9}] }, Open ]], Cell["\<\ We see that for basis size greater than about 16, the energy has converged \ well. As you might expect, and you can verify yourself, a larger basis set is \ needed if \[Lambda] is larger. The eigenvectors can also be found, and these give the linear coefficients \ of the Hermite polynomials which make up the coordinate space wavefunction, \ and hence the wavefunction of the anharmonic oscillator can also be obtained \ by this method.\ \>", "Text", CellChangeTimes->{{3.453062929075835*^9, 3.453062930305891*^9}}], Cell["\<\ In another handout we calculated the lowest three energy levels of this \ anharmonic oscillator in a completely different way, from the Schrodinger \ equation. In that approach we solve a differential equation rather than \ diagonalize a matrix. We found the lowest three energy levels to be 0.602405, 1.95054, 3.5363\ \>", "Text", CellChangeTimes->{{3.453062939219185*^9, 3.4530630350421343`*^9}, { 3.4530631513486767`*^9, 3.453063160218576*^9}, {3.453063666159521*^9, 3.453063688247961*^9}, 3.48297218684897*^9}], Cell["\<\ We see that the lowest agrees very well with the ground state determined \ above from matrix methods. To compare all three numbers we now compute the \ lowest three levels from the matrix method, still with \[Lambda] = 0.2:\ \>", "Text", CellChangeTimes->{{3.4530630398861027`*^9, 3.4530630709152613`*^9}, { 3.453063169050375*^9, 3.453063215803606*^9}, {3.453063701399527*^9, 3.453063707071718*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"evals", "[", RowBox[{"20", ",", " ", "0.2"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", " ", RowBox[{ RowBox[{"evals", "[", RowBox[{"20", ",", " ", "0.2"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{ RowBox[{"evals", "[", RowBox[{"20", ",", " ", "0.2"}], "]"}], "[", RowBox[{"[", "3", "]"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.453063230190669*^9, 3.453063250837049*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.6024051508367516`", ",", "1.9505434140535531`", ",", "3.5363012292982883`"}], "}"}]], "Output", CellChangeTimes->{3.453063251606269*^9, 3.453063300693613*^9, 3.453135769583309*^9, 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