(************** 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). 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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[ 167306, 4428]*) (*NotebookOutlinePosition[ 168491, 4465]*) (* CellTagsIndexPosition[ 168447, 4461]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ \(The\ Shooting\ Method\), "\[IndentingNewLine]", \(by\ James\ Feagin\)}], "Input"], Cell[BoxData[ \(Method\ Shooting\ The\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(James\\)\\\" is similar to existing symbol \ \\\"\\!\\(Names\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(by\ Feagin\ James\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Homework Exercise: ", FontColor->RGBColor[0, 0, 1]], StyleBox["Redo this notebook for the 1D harmonic oscillator and produce a \ plot of the first four eigenstates stacked inside the potential.", FontColor->RGBColor[1, 0, 1]] }], "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Approx numerical integration and the '", StyleBox["Shooting method' for estimating eigenvalues", FontColor->RGBColor[0, 0, 1]], ".\nG", StyleBox["uess an energy eigenvalue \[Epsilon] and numerically integrate \ the differential equation starting with the left (or right) boundary \ condition and integrating across to the right (or left) boundary condition. \ Adjust \[Epsilon] and repeat until your 'shot' at the far boundary condition \ scores a hit.", FontSize->12] }], "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["H-atom radial equation 0 < r < \[Infinity]", FontColor->RGBColor[1, 0, 0]], "\nTo simply, let's first switch to reduced radial function u[r] = r R[r]. \ This also allows us the convenient boundary condition u[0] = 0. We'll shoot \ at the right asymptotic boundary condition u \[Rule] 0 for large r." }], "Subsubsection"], Cell[BoxData[ RowBox[{"<<", "Calculus`VectorAnalysis`", " ", StyleBox[\( (*load\ a\ package\ not\ already\ in\ the\ mma\ kernal\ *) \ \), FontColor->RGBColor[0, 0, 1]]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(SetCoordinates[Spherical[r, \[Theta], \[Phi]]]\), " ", StyleBox[\( (*\ define\ the\ coordinate\ system\ *) \), FontColor->RGBColor[0, 0, 1]]}]], "Input"], Cell[BoxData[ \(Spherical[r, \[Theta], \[Phi]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Laplacian[u[r]\/r]\)], "Input"], Cell[BoxData[ FractionBox[ RowBox[{\(Csc[\[Theta]]\), " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "r", " ", \(Sin[\[Theta]]\), " ", RowBox[{"(", RowBox[{\(-\(u[r]\/r\^2\)\), "+", FractionBox[ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "r", "]"}], "r"]}], ")"}]}], "+", RowBox[{\(r\^2\), " ", \(Sin[\[Theta]]\), " ", RowBox[{"(", 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Note, the initial slope of the shot ", FontColor->RGBColor[0, 0, 1]], "u'[0]\[Equal]1", StyleBox[" is arbitrary and will be ultimately fixed by overall \ wavefunction norm.", FontColor->RGBColor[0, 0, 1]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Ground state ", StyleBox["n = 1\n", FontColor->RGBColor[1, 0, 0]], "Result of NDSolve is a replacement rule for the computed eigenfunction. \ The interpolated result means we can plot, differentiate and integrate the \ result like a regular continuous function. \nTurns out, the bigger n is, the \ longer the tail of the wavefunction. Hence, we scale \[Rho]max with n." }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ StyleBox[\(n\ = \ 1\), FontColor->RGBColor[1, 0, 0]], ";", " ", StyleBox[\(\[Epsilon]\ = 1\/n\^2\), FontColor->RGBColor[1, 0, 0]], ";", " ", StyleBox[\(\[Rho]min = 0.000001\), FontColor->RGBColor[0, 0, 1]], ";", " ", RowBox[{"\[Rho]max", " ", "=", " ", RowBox[{ StyleBox["n", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "10."}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"radialEq", "[", StyleBox["\[Epsilon]", FontColor->RGBColor[1, 0, 0]], "]"}], ",", StyleBox[\(u[\[Rho]min] \[Equal] 0\), FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\(u'\)[\[Rho]min] \[Equal] 1\), FontColor->RGBColor[0, 0, 1]]}], "}"}], ",", "u", ",", \({\[Rho], \[Rho]min, \[Rho]max}\)}], "]"}]}], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"u", "\[Rule]", TagBox[\(InterpolatingFunction[{{1.`*^-6, 10.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Let's define the ground state by replacing ", StyleBox["u[\[Rho]] ", FontColor->RGBColor[0, 0, 1]], "with our numerical solution. 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