(* 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[ 172789, 3471] NotebookOptionsPosition[ 167569, 3312] NotebookOutlinePosition[ 168260, 3336] CellTagsIndexPosition[ 168217, 3333] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ The Shooting Method (application to energy levels of the simple harmonic \ oscillator and other problems of a particle in a potential well)\ \>", "Subtitle", CellChangeTimes->{{3.421509332481022*^9, 3.42150936566607*^9}}], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "In the previous handout we found the eigenvalues of a quantum particle in a \ potential well where the potential vanishes for |x| greater than some value, \ which has the advantage that we know the wavefunction exactly in this large \ |x| region. The same method can be used for problems where the potential, \ while not exactly zero at large |x|, is sufficiently close to zero that the \ error in assuming it vanishes is negligible.\n\nHere we give a generalization \ of this approach to problems where the potential does ", StyleBox["not", FontSlant->"Italic"], " tend to zero at large |x|. This more general approach is often called the \ ", StyleBox["shooting method", FontSlant->"Italic"], " . This handout is very similar to the earlier one except for the way it \ handles the boundary conditions at large |x|." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Setting up the Problem of the Simple Harmonic Oscillator", "Section"], Cell[TextData[{ "As an illustration,we take the ", StyleBox["simple harmonic oscillator", FontWeight->"Bold"], " (SHO) potential with \[HBar]=\[Omega]=m=1,for which ", StyleBox["there is an analytic solution", FontWeight->"Bold"], ", discussed in all books on quantum mechanics. The energy levels are" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["E", RowBox[{"n", " "}]], "=", " ", RowBox[{"n", " ", "+", " ", FractionBox["1", "2"]}]}], " ", ",", " ", RowBox[{"(", RowBox[{ RowBox[{"n", "=", "0"}], ",", " ", "1", ",", " ", "2", ",", " ", "..."}], " ", ")"}]}], TraditionalForm]], "DisplayFormula"], Cell["First we set up the potential and plot it.", "Text", CellChangeTimes->{3.4215093761962214`*^9, 3.45311717987337*^9}], Cell[BoxData[ RowBox[{"Clear", " ", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"v0", " ", "=", " ", "1"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"v", "[", "x_", "]"}], " ", ":=", RowBox[{"v0", " ", RowBox[{ RowBox[{"x", "^", "2"}], " ", "/", "2"}]}]}]], "Input", CellChangeTimes->{{3.453117199283065*^9, 3.453117210113427*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"v", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "4"}], ",", " ", 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Cell["\<\ We define the Schrodinger equation using a delayed assignment, \":=\", since \ we will only use it later:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"eqn", "[", "en_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{"u", "''"}], "[", "x", "]"}], " ", "+", " ", RowBox[{"2", RowBox[{"(", RowBox[{"en", " ", "-", " ", RowBox[{"v", "[", "x", "]"}]}], ")"}], RowBox[{"u", "[", "x", "]"}]}]}]}]], "Input"], Cell["\<\ (we call the wavefunction u(x) here.) We will solve the equation in the range \ -L \[LessEqual] x \[LessEqual] 0, and choose L such that V(x) >> E at x=-L, \ i.e. x = -L is well to the left of the \"turning point\" where V(x) = E. We \ take L = 4, which is fine for the lowest level. However we will need to \ increase L to get the higher levels accurately.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"L", " ", "=", " ", "4"}], ";"}]], "Input"], Cell[TextData[{ "At x = -L, we take, arbitrarily, u(-L) = 0, u'(-L) = 1. This corresponds \ to a ", StyleBox["linear superposition", FontSlant->"Italic"], " of the solution which decays exponentially to the left and the one which \ increases exponentially. We just want the solution which decreases \ exponentially to the left. However, if L is deep inside the region where V(x) \ > E the error will be negligible since we integrate to the right (not the \ left) and so the unwanted solution will be exponentially ", StyleBox["surpressed", FontSlant->"Italic"], " as we integrate to the right towards the negative-x turning point." }], "Text", CellChangeTimes->{{3.421509406691531*^9, 3.42150941596169*^9}, { 3.482972492703491*^9, 3.4829724949096107`*^9}}], Cell["\<\ We set up the calculation of the wavefunction in the region between -L and 0, \ matching the function and its derivative to the specified values at x = -L.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"wavefunc", "[", "en_", "]"}], " ", ":=", " ", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{ RowBox[{"eqn", "[", "en", "]"}], " ", "==", " ", "0"}], ",", " ", RowBox[{ RowBox[{"u", "[", RowBox[{"-", "L"}], "]"}], " ", "==", " ", "0"}], ",", " ", "\n", "\t\t\t\t\t", RowBox[{ RowBox[{ RowBox[{"u", "'"}], "[", RowBox[{"-", "L"}], "]"}], " ", "==", " ", "1"}]}], "}"}], ",", " ", "u", ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "L"}], ",", " ", "0"}], "}"}]}], " ", "]"}]}]], "Input"], Cell[TextData[{ "Note that \"", StyleBox["wavefunc[en_]", FontWeight->"Bold"], "\" will given as a replacement rule in the form \"", StyleBox["{{u\[Rule]InterpolatingFunction[{{-0.5,0.5}},\"<>\"]}}", FontWeight->"Bold"], "\". In order to directly access the wavefunction we define a function, \ called ", StyleBox["sol[x, en]", FontWeight->"Bold"], ", which applies the replacement rule, and removes one of the sets of curly \ brackets by taking the first element of the list. " }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"sol", "[", RowBox[{ RowBox[{"x_", "?", "NumericQ"}], ",", " ", RowBox[{"en_", "?", "NumericQ"}]}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", RowBox[{ RowBox[{"wavefunc", "[", "en", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input"], Cell[TextData[{ "As before we have added the hieroglyphics \n ", StyleBox["?NumericQ", FontWeight->"Bold"], " \nafter the arguments of ", StyleBox["sol", FontWeight->"Bold"], ". This is necessary in ", StyleBox["Mathematica", FontSlant->"Italic"], " version 5 and later when the solution is put into ", StyleBox["FindRoot", FontWeight->"Bold"], " below to determine the energy eigenvalue.\n\nIt is also convenient to \ define a function for the derivative of the wave function (since we will be \ requiring that this is zero at x = 0 for the even parity solutions):" }], "Text", CellChangeTimes->{{3.421509431526924*^9, 3.4215094385712137`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"solprime", "[", RowBox[{ RowBox[{"x_", "?", "NumericQ"}], ",", " ", RowBox[{"en_", "?", "NumericQ"}]}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{"u", "'"}], "[", "x", "]"}], " ", "/.", RowBox[{ RowBox[{"wavefunc", "[", "en", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Even Parity Solution", "Section"], Cell["\<\ We now find an eigenvalue. We use the \"FindRoot\" command to locate the \ eigenvalue and give it two starting values. The boundary condition is that \ the derivative of the wavefunction is zero at x = 0:\ \>", "Text", CellChangeTimes->{3.4215094523709183`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"evalue", " ", "=", " ", RowBox[{"en", " ", "/.", " ", RowBox[{"FindRoot", " ", "[", " ", RowBox[{ RowBox[{"solprime", "[", RowBox[{"0", ",", " ", "en"}], "]"}], " ", ",", RowBox[{"{", RowBox[{"en", ",", " ", "0", ",", " ", "1"}], "}"}]}], " ", "]"}]}]}]], "Input"], Cell[BoxData["0.5000004979498838`"], "Output", CellChangeTimes->{3.4215070732004013`*^9, 3.452625278242136*^9, 3.453061842896327*^9, 3.453062274387948*^9, 3.453062400115399*^9, 3.4530626032405243`*^9, 3.4531173117893467`*^9, 3.4829723125234222`*^9}] }, Open ]], Cell[TextData[{ "This agrees with the known ground state energy of the simple harmonic \ oscillator, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["E", "0"], "=", " ", RowBox[{"1", "/", "2."}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "Now we want the eigenfunction coresponding to our eigenvalue. Since we now \ have the eigenvalue, we do not want to keep recalculating the wavefunction \ so we define a function \"", StyleBox["efunc", FontWeight->"Bold"], "\" with immediate assignment, where we input the eigenvalue for the \ energy:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"efunc", "[", "x_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", RowBox[{ RowBox[{"wavefunc", "[", "evalue", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}]], "Input"], Cell["\<\ We have now obtained the wavefunction in for x < 0. We now also define it for \ x > 0 (remembering that it's even) and collect these functions into a single \ (not yet normalized) function \[Psi]nn[x_], which can then easily be plotted:\ \ \>", "Text", CellChangeTimes->{{3.4215094627130423`*^9, 3.4215094847975492`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"\[Psi]nn", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"efunc", "[", "x", "]"}], " ", "/;", " ", RowBox[{"x", "\[LessEqual]", " ", "0"}]}]}], " ", ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"\[Psi]nn", " ", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"efunc", "[", RowBox[{"-", "x"}], "]"}], " ", "/;", " ", RowBox[{"x", " ", ">", " ", "0"}]}]}], " ", ";"}]], "Input"], Cell["We now normalize the wavefunction,", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"normconst", " ", "=", " ", RowBox[{"Sqrt", "[", " ", RowBox[{"NIntegrate", " ", "[", " ", RowBox[{ RowBox[{ RowBox[{"\[Psi]nn", "[", "x", "]"}], "^", "2"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", 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Cell[CellGroupData[{ Cell["Odd Parity Solution", "Section"], Cell["Now we look at odd-parity solutions.", "Text"], Cell["\<\ We repeat the previous calculation of the eigenvalue, and calculate the \ eigenfunction, which is then normalized and plotted. We give different \ initial guesses for the eigenvalue from what we took for the even parity \ solution and also take a somewhat larger value for L, in order to get an \ accurate answer for this state which has higher energy than the even parity \ solution discussed in the previous section. \ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"L", " ", "=", " ", "5"}], ";"}]], "Input"], Cell["\<\ The boundary condition is now that the wavefunction vanishes at the origin:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"evalue", " ", "=", RowBox[{"en", " ", "/.", " ", RowBox[{"FindRoot", " ", "[", " ", RowBox[{ RowBox[{"sol", "[", RowBox[{"0", ",", " ", "en"}], "]"}], ",", RowBox[{"{", RowBox[{"en", ",", " ", "1", ",", " ", "3"}], "}"}]}], " ", "]"}]}]}]], "Input"], Cell[BoxData["1.4999999993755186`"], "Output", CellChangeTimes->{3.421507073825392*^9, 3.452625279080573*^9, 3.453061843355721*^9, 3.453062274585264*^9, 3.453062400365458*^9, 3.453062603459527*^9, 3.453117312348048*^9, 3.4829723133744173`*^9}] }, Open ]], Cell[TextData[{ "This agrees with the known energy of the first excited state of the simple \ harmonic oscillator, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["E", RowBox[{"1", " "}]], "=", " ", RowBox[{"3", "/", "2."}]}], TraditionalForm]]], "\n\nNext we calculate the eigenfunction, " }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"efunc", "[", "x_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", RowBox[{ RowBox[{"wavefunc", "[", "evalue", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}]], "Input"], Cell["\<\ redefine it for x > 0 (noting that it is now odd rather than even)\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Psi]nn", " ", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"-", RowBox[{"efunc", "[", RowBox[{"-", "x"}], "]"}]}], " ", "/;", " ", RowBox[{"x", " ", ">", " ", "0"}]}]}]], "Input"], Cell["and recompute the normalization constant", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"normconst", " ", "=", " ", RowBox[{"Sqrt", "[", " ", RowBox[{"NIntegrate", " ", "[", " ", RowBox[{ RowBox[{ RowBox[{"\[Psi]nn", "[", "x", "]"}], "^", "2"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "L"}], ",", " ", "L"}], "}"}]}], " ", "]"}], " ", "]"}]}], ";"}]], "Input"], Cell["\<\ Everything else is the same as for the even-parity eigenfunction and used \ delayed assignment. 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The \"shooting method\" described in this handout can be applied to \ essentially any quantum well problem in one dimension with a symmetric \ potential. The main thing is to ensure that L is far enough into the region \ where the solution is exponentially decaying that the boundary conditions \ applied at x = -L do not introduce a noticeable amount of the \"wrong\" \ solution in the x-region of interest. It is also straightforward to \ generalize the method to the case of a non-symmetric potential.\ \>", "Text", CellChangeTimes->{{3.453117273817061*^9, 3.453117298760639*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Anharmonic Oscillator", "Section"], Cell[TextData[{ "Now we consider a problem ", StyleBox["for which there is no analytic solution", FontWeight->"Bold"], "; an oscillator with a ", StyleBox["quartic", FontSlant->"Italic"], " potential, in addition to the quadratic potential:" }], "Text", CellChangeTimes->{{3.453062118709627*^9, 3.453062127409617*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.453062313576016*^9, 3.4530623153838997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"v", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", "/", " ", "2"}], " ", "+", " ", RowBox[{"\[Lambda]", " ", RowBox[{"x", "^", "4"}]}]}]}]], "Input", CellChangeTimes->{{3.453062132696432*^9, 3.453062196884748*^9}, { 3.453062377247509*^9, 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