(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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[ 190805, 9504]*) (*NotebookOutlinePosition[ 226542, 10729]*) (* CellTagsIndexPosition[ 226498, 10725]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "Associated Legendre Functions and Spherical Harmonics"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell["R.G. Palmer", "Subsubtitle", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Background and Copyright", "Subsection"], Cell[TextData[{ "This ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook was written by Richard G. Palmer (Physics Department, Duke \ University) for use in a course he taught. As of 1999, it has been made \ available for general non-profit use under the following copyright \ provision." }], "Text"], Cell[TextData[{ StyleBox["This Mathematica Notebook is Copyright Richard G. Palmer, 1996", FontWeight->"Bold"], ". It may be freely used by individuals, and by classes at academic \ institutions, provided:\n1. Credit is given to Richard Palmer as the original \ author; and\n2. It is not bought or sold or exchanged for profit, or \ incorporated into material that is bought or sold or exchanged for profit.\n\ Any other use requires the written permission of Richard Palmer, Dept. of \ Physics, Box 90305, Duke University, Durham, NC 27708, USA. ", "See ", StyleBox["http://www.phy.duke.edu/~palmer", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for the email address." }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Version: 4.\nDate: 7/21/00." }], "Text"], Cell[TextData[{ "See ", StyleBox["http://www.phy.duke.edu/~palmer/notebooks/", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for other ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks by Richard Palmer." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Preface"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This Notebook concerns the associated Legendre functions P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) and the spherical harmonics Y", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(", StyleBox["\[Theta],\[Phi]", FontFamily->"Symbol"], "). Ordinary Legendre functions are ", StyleBox["not", FontSlant->"Italic"], " covered here; they're in the ", StyleBox["Legendre.nb", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " notebook. Note that I start off using \"n\", not \"l\" for the subscript \ index, just as in the ", StyleBox["Legendre.nb", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " notebook, because \"l\" looks too much like \"1\". But then I switch \ about half way through:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["RALPH WALDO EMERSON ", FontFamily->"Times"], StyleBox[ "1803\[CapitalEth]1882\nA foolish consistency is the hobgoblin of little \ minds, adored\nby little statesmen and philosophers and divines. With \ consistency\na great soul has simply nothing to do.", FontFamily->"Times", FontWeight->"Plain"] }], "Input", AspectRatioFixed->True], Cell[TextData[ "This Notebook is focused more on the Math than on Mathematica. (Do I hear a \ sigh of relief?) But don't forget everything you've learned."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "There are 4 problems embedded in the Notebook. The first three are \ easy..."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Associated Legendre Functions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Introduction"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The associated Legendre functions are the extension of the Legendre \ functions that we use in physics and engineering when there ", StyleBox["is", FontSlant->"Italic"], " some ", StyleBox["\[Phi]", FontFamily->"Symbol"], " (azimuthal) dependence. When we use spherical coordinates to separate" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["1. Laplace's equation, or"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "2. the Helmholtz equation (which comes from the wave equation or the \ heat/diffusion equation), or"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["3. Schrodinger's equation with a central potential,"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["we get the simple ODE"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " d", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[CapitalPhi]", FontFamily->"Symbol"], "\n ---- = - m", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], " ", StyleBox["\[CapitalPhi]", FontFamily->"Symbol"], "\n d ", StyleBox["\[Phi]", FontFamily->"Symbol"], StyleBox["2", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "for the azimuthal part ", StyleBox["\[CapitalPhi](\[Phi])", FontFamily->"Symbol"], ", with solutions" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " i m ", StyleBox["\[Phi]", FontFamily->"Symbol"], " -i m ", StyleBox["\[Phi]", FontFamily->"Symbol"], "\ne ", StyleBox["or", FontFamily->"Times", FontSize->14], " e ", StyleBox["or", FontFamily->"Times", FontSize->14], " cos(m ", StyleBox["\[Phi]", FontFamily->"Symbol"], ") ", StyleBox["or", FontFamily->"Times", FontSize->14], " sin(m ", StyleBox["\[Phi]", FontFamily->"Symbol"], ")" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "or linear combinations of these. Continuity when ", StyleBox["\[Phi]", FontFamily->"Symbol"], " is increased by 2", StyleBox["\[Pi]", FontFamily->"Symbol"], " forces us to make m an integer." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The ", StyleBox["\[Theta]", FontFamily->"Symbol"], " part then gives the Associated Legendre Equation, in which m appears as a \ parameter:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " 2 \n 2 d y d y \ m\n(1 - x ) ---- - 2x --- + [ n(n+1) - ------ ] y == 0\n \ 2 d x 2\n d x \ 1 - x"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "where x = cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ".\n\nWhen m = 0 this is the ordinary Legendre equation, as we already \ know---the Legendre polynomials P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], "(cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ") are the regular solutions (well-behaved at ", StyleBox["\[PlusMinus]", FontFamily->"Symbol"], "1) that go with ", StyleBox["no ", FontSlant->"Italic"], StyleBox["\[Phi]", FontFamily->"Symbol"], " dependence.\n\nWhen m is non-zero, the regular solutions (well-behaved \ at ", StyleBox["\[PlusMinus]", FontFamily->"Symbol"], "1) are the Associated Legendre Functions P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], "). Mathematica uses the function ", StyleBox["LegendreP[n, m, x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for them, just like the ordinary Legendre functions ", StyleBox["LegendreP[n, x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " but with an extra argument." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We'll follow the same sequence as we did for the Legendre Polynomials in \ the ", StyleBox["Legendre.nb", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " notebook, but more briefly. We won't bother with the generating function \ (though there is one), or recurrence relations (though there are lots, in \ both n and m)." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Explicit Form"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "0 ", StyleBox["\[LessEqual]", FontFamily->"Symbol", FontWeight->"Plain"], " m ", StyleBox["\[LessEqual]", FontFamily->"Symbol", FontWeight->"Plain"], " n" }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Let's begin by asking Mathematica to show us some explicit P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x)'s. We'll choose", StyleBox[" n=4 ", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "and try various ", StyleBox["m", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "'s." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["LegendreP[4, 1, x]", "Input", AspectRatioFixed->True], Cell["LegendreP[4, 2, x]", "Input", AspectRatioFixed->True], Cell["LegendreP[4, 3, x]", "Input", AspectRatioFixed->True], Cell["LegendreP[4, 4, x]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Can you see the pattern? For odd m there's a non-polynomial factor (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], ", times a polynomial of order n-m. For the even m case we just seem to \ get a polynomial of order n, but it also factors to give a (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " term. For example:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Factor[%]", "Input", AspectRatioFixed->True], Cell[TextData[{ "which is just 105 (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ". So in general (trust me!):" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " m 2 m/2\nP (x) = (1 - x ) (even/odd polynomial of order n-m in \ x)\n n"], "Print", CellMargins->{{24, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "By \"even/odd\" I mean that it has only even or only odd powers (like the \ plain Legendre polynomials), depending on whether n-m is even or odd."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "If we set x = cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ", as in most physics applications, then (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["1/2", FontVariations->{"CompatibilityType"->"Superscript"}], " is sin ", StyleBox["\[Theta]", FontFamily->"Symbol"], ", so we have the general form:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " m m\nP (cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ") = sin ", StyleBox["\[Theta]", FontFamily->"Symbol"], " (even/odd poly of order n-m in cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ")\n n" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "It's not too easy to get Mathematica to display this form directly, but \ the following works. Look carefully: it cancels out the (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " factor and multiplies by sin(", StyleBox["\[Theta]", FontFamily->"Symbol"], ")", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ assleg[n_, m_, theta_] := Module[{x}, \t(Cancel[LegendreP[n,m,x] / (1-x^2)^(m/2)] /. \t x-> Cos[theta]) * Sin[theta]^m ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "I stuck a ", StyleBox["Module[{x}, ...]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " around this to make ", StyleBox["x", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " a local variable. I'll do that sort of thing with little or no comment \ from now on. We're getting sophisticated, right?" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Cancel[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " is another useful function for simplification; it tries to find and \ cancel out common factors in the numerator and denominator of ", StyleBox["expr", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], ". OK, OK, OK, I know: there are too many different functions like ", StyleBox["Cancel[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", ", StyleBox["Expand[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", ", StyleBox["ExpandAll[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", ", StyleBox["Together[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", ", StyleBox["Simplify[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", etc. (how many more can ", StyleBox["you", FontSlant->"Italic"], " name?). I seem to pull a new one out of the hat in almost every \ notebook. How's anyone to know what to use, and where? I wish I had an easy \ answer, but all I can really say is \"experience\". What a cop out! But if \ it's any consolation, I usually have to play around quite a bit myself before \ finding the best approach. C'est la vie." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["For example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["assleg[4,1,\[Theta]]"], "Input", AspectRatioFixed->True], Cell[TextData["assleg[4,2,\[Theta]]"], "Input", AspectRatioFixed->True], Cell[TextData["And here are all of them for n=5:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Table[assleg[5,m,\[Theta]], {m,0,5}] //ColumnForm"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Notice that the last one, with n = m, is necessarily just a power of \ Sin[", StyleBox["\[Theta]", FontFamily->"Symbol"], StyleBox["]", FontSize->12], "; the polynomial part is of order 0 -- just a constant. You'll figure out \ that constant in a problem later." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Other values of m"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We've been assuming that m lies between 0 and n. Of course the associated \ Legendre differential equation only contains m", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ", so we'd expect the same solutions to work for ", StyleBox["\[PlusMinus]", FontFamily->"Symbol"], "m. In fact P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) and P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["-m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) are both defined (as we'll discuss later), and are just proportional \ to one another. For example:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["LegendreP[4,1,x]", "Input", AspectRatioFixed->True], Cell["LegendreP[4,-1,x]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Of course our description above, in terms of (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " etc, must be modified to say |m| instead of just m. It also needs \ restriction to m an integer. Thus a better definition is:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Clear[assleg]; assleg[n_, m_Integer, theta_] := Module[{x}, \t(Cancel[LegendreP[n,m,x] / (1-x^2)^(Abs[m]/2)] /. \t x-> Cos[theta]) * Sin[theta]^Abs[m] ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData["giving"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["assleg[4,1,\[Theta]]"], "Input", AspectRatioFixed->True], Cell[TextData["assleg[4,-1,\[Theta]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Finally, if |m| > n, we get zero (still assuming that n and m are \ integers):"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["LegendreP[4,5,x]", "Input", AspectRatioFixed->True], Cell["LegendreP[6,-10,x]", "Input", AspectRatioFixed->True], Cell[TextData[{ "The latter ought to be 0 according to the usual definition, and indeed was \ in the previous version of ", StyleBox["Mathematica", FontSlant->"Italic"], ". I'm not sure whether ", StyleBox["Mathematica", FontSlant->"Italic"], " is using a different definition, or has a bug. The functions it gives do \ satisfy the differential equation though, so I suspect that it's a different \ definition. But we won't worry about it further." }], "Text"], Cell[TextData[{ "Our ", StyleBox["assleg[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " is based on ", StyleBox["LegendreP[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", so it'll give 0 too if that does." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Plots"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Here's a picture of P", StyleBox["4", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) against x for n = 0, 1, ... 4. We'll stick with the conventional \ domain, from -1 to 1:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot[Evaluate[Table[LegendreP[4,m,x], {m, 0, 4}]], \t {x, -1, 1}, PlotRange->All]\ \>", "Input", AspectRatioFixed->True], Cell[TextData["Or, if we focus in closer for smaller y's:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" Show[%, PlotRange->{-5,5}]", "Input", AspectRatioFixed->True], Cell[TextData[{ "These aren't very enlightening (but if you feel enlightened I'm very happy \ for you). Let's note just the following points:\n\n1. They all go to 0 at +1 \ and -1 except for the m = 0 ones (the plain Legendre polynomials), because of \ the (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " factor." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "2. They're alternately even and odd functions. In fact the parity relation \ is"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " m n+m m\n P (-x) = (-1) P (x)\n n \ n"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "3. They get ", StyleBox["BIG", FontSize->24], "; they are ", StyleBox["not", FontSlant->"Italic"], " constrained to a -1 to 1 range except for m = 0." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Definitions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Up to now we've been using the associated Legendre functions without really \ seeing a definition. As with the plain Legendre polynomials, there are \ actually many different ways to define them."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Two of the most useful are:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "1. As m", StyleBox["th", FontVariations->{"CompatibilityType"->"Superscript"}], " derivatives of the ordinary Legendre functions, with a prefactor:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " d", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "\nP", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) = (-1)", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], " (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " --", StyleBox["-", FontWeight->"Bold"], " P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], "(x)", StyleBox["\n", FontWeight->"Bold"], " dx", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Obviously this only works for ", StyleBox["m", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " a positive integer or 0. The (-1)", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], " factor is not included by everyone, but I put it in to agree with \ Mathematica. Arfken does ", StyleBox["not", FontSlant->"Italic"], " include it; see his equation 12.81 and the footnote on the next page \ (724)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Here's a Mathematica definition. Note the condition on ", StyleBox["m", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and the use of a local variable ", StyleBox["t", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for the calculation, followed by substitution with ", StyleBox["x", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ";. If we'd used ", StyleBox["x", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " all the way through it wouldn't work unless ", StyleBox["x", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " were an undefined symbol (Mathematica doesn't take kindly to being asked \ to differentiate with respect to 1, for example). " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ associated[n_Integer, m_Integer /; m>=0, x_] := Module[{t}, \tCancel[ \t\t(-1)^m (1-t^2)^(m/2) D[LegendreP[n,t], {t,m}] \t] /. t->x ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "The ", StyleBox["Cancel[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " makes Mathematica cancel out common numerical factors in the numerator \ and denominator. For example:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["associated[4,1,x]", "Input", AspectRatioFixed->True], Cell[TextData["should be equivalent to:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["LegendreP[4,1,x]", "Input", AspectRatioFixed->True], Cell[TextData["\n2. From an extension of Rodrigues' formula:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " (-1)", StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], " (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ")", StyleBox["m/2", FontVariations->{"CompatibilityType"->"Superscript"}], " d", StyleBox["n+m", FontVariations->{"CompatibilityType"->"Superscript"}], "\nP", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) = --------------- --", StyleBox["-", FontWeight->"Bold"], " (x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], " - 1)", StyleBox["n", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\n", FontWeight->"Bold"], " 2", StyleBox["n", FontVariations->{"CompatibilityType"->"Superscript"}], " n! dx", StyleBox["n+m", FontVariations->{"CompatibilityType"->"Superscript"}] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "It's easy to see how this comes from the first definition and the original \ Rodrigues' formula. The crucial thing to notice is that it's an (n+m)th \ derivative. Its big advantage is that it's valid for negative m too (for |m| \ ", StyleBox["\[LessEqual]", FontFamily->"Symbol"], " n). " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In Mathematica, putting the n+m", StyleBox["\[GreaterEqual]", FontFamily->"Symbol"], "0 condition at the end:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ rod[n_Integer, m_Integer, x_] := Module[{t}, \tCancel[ \t (-1)^m (1-t^2)^(m/2) D[(t^2-1)^n, {t,n+m}]/(2^n n!) \t] /. t->x ] /; n+m >= 0\ \>", "Input", AspectRatioFixed->True], Cell[TextData["For example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["rod[4,1,x]", "Input", AspectRatioFixed->True], Cell["rod[4,5,x]", "Input", AspectRatioFixed->True], Cell["rod[4,-5,x] (* not defined *)", "Input", AspectRatioFixed->True], Cell["rod[4,-1,x]", "Input", AspectRatioFixed->True], Cell[TextData[{ "The last result doesn't look right; we were expecting a simple multiple of \ P", StyleBox["4", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["1", FontVariations->{"CompatibilityType"->"Superscript"}], "(x). But it is OK; there's a factor of (1-x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ") in the numerator:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Factor[%]", "Input", AspectRatioFixed->True], Cell[TextData[{ "So, dealing with ", StyleBox["Sqrt[1-x^2]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " in the usual way:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["PowerExpand[% /. 1-x^2 -> (1-x)(1+x)]", "Input", AspectRatioFixed->True], Cell[TextData[ "Well, that's close enough. Mathematica can be very frustrating when you \ want it to put things into a particular form."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Problem 1"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Figure out a formula from the above extension of Rodrigues' formula for \ the constant A", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], " such that" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " n n\n P (cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ") = A sin ", StyleBox["\[Theta]", FontFamily->"Symbol"], " \n n n" }], "Info", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[{ "and check it against the values given by ", StyleBox["assleg[n, n, theta]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". The neatest way to write the result is in terms of a double factorial. \ This is really more of a \"paper problem\" than a Mathematica problem. " }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Differential Equation"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We've already seen the associated Legendre differential equation:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ alde[y_, x_, n_, m_] := (1-x^2) y''[x] - 2x y'[x] + (n(n+1)-m^2/(1-x^2)) y[x] == 0\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "So let's just check that ", StyleBox["LegendreP[n, m, x]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " satisfies the equation, say for n=4, m=1:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" alde[LegendreP[4,1,#]&, x, 4, 1]", "Input", AspectRatioFixed->True], Cell[" Simplify[%]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Of course we could ", StyleBox["define", FontSlant->"Italic"], " the associated Legendre functions to be the solutions of the associated \ Legendre differential equation which:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["a. Remain finite at x = +1 and x = -1; and"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "b. Are suitably normalized. The normalization is not easy to express; \ it's no good doing it at ", StyleBox["\[PlusMinus]", FontFamily->"Symbol"], "1, since the associated Legendre functions vanish there (for nonzero m); \ or at 0, since at least some (those with n+m odd) vanish there too. So in \ practice the extended Rodrigues formula provides a more useful definition." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Orthogonality"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The associated Legendre differential equation can readily be put into \ Sturm-Liouville form:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "D[(1-x^2) y'[x], x] - m^2/(1-x^2) y[x] == -n(n+1) y[x]"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The boundary conditions that define the associated Legendre polynomials ", StyleBox["are", FontSlant->"Italic"], " of suitable form, so again we have a Hermitian operator and can expect" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["a. Real eigenvalues."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["b. Orthogonal eigenfunctions."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["c. Completeness and closure."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Note, however, that m is now part of the left hand side of the equation, \ and so we only expect the above properties for ", StyleBox["fixed ", FontSlant->"Italic"], " m. The eigenvalues (in terms of n) are" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" n = m, m+1, m+2, ...."], "Info", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "since we must have m ", StyleBox["\[LessEqual]", FontFamily->"Symbol"], " n." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We'll omit discussing completeness and closure. But orthogonality is very \ important. From the Sturm-Liouville form we expect that"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ assint[n1_, n2_, m_] := Module[{x}, \tIntegrate[ \t\tLegendreP[n1,m,x] LegendreP[n2,m,x], \t\t{x, -1, 1} \t] ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "should be zero if n1 ", StyleBox["\[NotEqual]", FontFamily->"Symbol"], " n2. Note that we put the ", StyleBox["same", FontSlant->"Italic"], " value of m in both P's -- otherwise it will ", StyleBox["not", FontSlant->"Italic"], " work. Let's try it:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" assint[2,3,1]", "Input", AspectRatioFixed->True], Cell[TextData["Good. What about n1 = n2? For example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" assint[4,4,1]", "Input", AspectRatioFixed->True], Cell[TextData[{ "It's ", StyleBox["not", FontSlant->"Italic"], " 1; the associated Legendre functions are orthogonal, but not orthonormal. \ the general result is" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " 2 (n + m)!\n assint[n, n, m] = -------- \ --------\n (2n + 1) (n - m)!"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Note that the first factor is just the usual Legendre polynomial \ normalization, valid when m = 0."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "It's also possible to construct an orthogonality relation for two P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x)'s with the same n's but different m's. Just consider the differential \ equation with -m^2 as the eigenvalue, instead of -n(n+1); it's still in the \ Sturm-Liouville form. Write down the the orthogonality relation and check it \ directly, as we did with ", StyleBox["assint[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". (You don't need to check the differential equation itself -- just write \ down the appropriate orthogonality relation and check it for some examples.) \ The \"diagonal\" term (when the same n's and m's are involved) isn't too easy \ to derive; actually it's" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " (n + m)!\n ------------\n |m| (n - m)!"], "Print", Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Spherical Harmonics"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Introduction and Definition"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "As discussed earlier, when we use spherical coordinates to separate the \ variables in Laplace's equation, Helmholtz' equation, or a central-potential \ Schrodinger equation, we get"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " i m ", StyleBox["\[Phi]", FontFamily->"Symbol"], " -i m ", StyleBox["\[Phi]", FontFamily->"Symbol"], "\ne ", StyleBox["or", FontFamily->"Times", FontSize->14], " e ", StyleBox["or", FontFamily->"Times", FontSize->14], " cos(m ", StyleBox["\[Phi]", FontFamily->"Symbol"], ") ", StyleBox["or", FontFamily->"Times", FontSize->14], " sin(m ", StyleBox["\[Phi]", FontFamily->"Symbol"], ")" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "(or linear combinations of these) for the ", StyleBox["\[Phi]", FontFamily->"Symbol"], " part, and an associated Legendre function P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ") for the ", StyleBox["\[Theta]", FontFamily->"Symbol"], " part. Continuity when ", StyleBox["\[Phi]", FontFamily->"Symbol"], " is increased by 2", StyleBox["\[Pi]", FontFamily->"Symbol"], " forces us to make m an integer. The m's in the ", StyleBox["\[Theta]", FontFamily->"Symbol"], " and ", StyleBox["\[Phi]", FontFamily->"Symbol"], " parts must of course agree, except that m and -m give the same thing for \ P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(x) besides a numerical factor." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "It's very useful to combine the two parts, conventionally associating the \ first of the above ", StyleBox["\[Phi]", FontFamily->"Symbol"], " forms, e", StyleBox["im", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], ", with P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "(cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ") for both positive and negative m, so for example e", StyleBox["2i", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], " goes with P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], "(cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], "), and e", StyleBox["-2i", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], " goes with P", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["-2", FontVariations->{"CompatibilityType"->"Superscript"}], "(cos\.80", StyleBox["\[Theta]", FontFamily->"Symbol"], "). We also normalize so that the resulting functions are orthonormal, \ giving the set of ", StyleBox["spherical harmonics", FontSlant->"Italic"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Y[n_,m_,theta_,phi_] := \t\t\t\tSqrt[(2n+1)(n-m)!/(4 Pi (n+m)!)] * \t\t\t\tassleg[n,m,theta] Exp[I m phi]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "The first factor, with the ", StyleBox["Sqrt[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", is just the normalization factor. For the associated Legendre part I \ used the ", StyleBox["assleg[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " function we defined earlier, because it replaces the (1 - cos", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ") factors by sin's. Recall that ", StyleBox["assleg[n,m,theta]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " is equivalent to ", StyleBox["LegendreP[n,m,Cos[theta]]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In case you're doing this Notebook in stages, here's the definition of ", StyleBox["assleg[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " again; there's no harm in re-executing this in any case." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Clear[assleg]; assleg[n_, m_Integer, theta_] := Module[{x}, \t(Cancel[LegendreP[n,m,x] / (1-x^2)^(Abs[m]/2)] /. \t x-> Cos[theta]) * Sin[theta]^Abs[m] ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Y[n,m,theta,phi]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " is called a spherical harmonic, usually written" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " m\n Y (", StyleBox["\[Theta]", FontFamily->"Symbol"], ", ", StyleBox["\[Phi]", FontFamily->"Symbol"], ")\n l" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "using l instead of n. I'll use l instead of n from now on. Thank you, \ Emerson."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["For example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Y[3,2,\[Theta],\[Phi]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Actually, Mathematica has spherical harmonics built in, so we can get the \ same result (more efficiently) by writing"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SphericalHarmonicY[3,2,\[Theta],\[Phi]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "For m = 0, of course, we get a simple Legendre polynomial in cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ", with no ", StyleBox["\[Phi]", FontFamily->"Symbol"], " dependence:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SphericalHarmonicY[4,0,\[Theta],\[Phi]]"], "Input", AspectRatioFixed->True], Cell[TextData["Compare with"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["LegendreP[4,Cos[\[Theta]]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nHere's a table of all of them for l=0, 1, 2. I could have done this \ with ", StyleBox["Table[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", but to get a pretty output list it was easier to write the loops myself \ using ", StyleBox["Do[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", and produce the output with ", StyleBox["Print[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". If you don't know about ", StyleBox["Do[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", do look it up, or do at least do ", StyleBox["?Do", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Do[ Print[\" \"];\n\tPrint[\"============= l = \",l,\" ==============\"];\n\t\ Do[\tPrint[\" \"];\n\t\tPrint[\"m = \",m,\" \",\n\t \ SphericalHarmonicY[l,m,\[Theta],\[Phi]]\n\t\t],\n\t\t{m,-l,l}\n\t],\n\t\ {l,0,2}\n]"], "Input", AspectRatioFixed->True], Cell[TextData["Note the relation between the +m and -m values: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " -m m m *\n Y (", StyleBox["\[Theta]", FontFamily->"Symbol"], ", ", StyleBox["\[Phi]", FontFamily->"Symbol"], ") = (-1) Y (", StyleBox["\[Theta]", FontFamily->"Symbol"], ", ", StyleBox["\[Phi]", FontFamily->"Symbol"], ")\n l l" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "where the * means the complex conjugate, to change e", StyleBox["im", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], " to e", StyleBox["-im", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], "." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Parity"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "It's quite important in quantum mechanics to know the parity relation for \ Y. Taking (vector) ", StyleBox["r", FontWeight->"Bold"], " to ", StyleBox["-r", FontWeight->"Bold"], " means:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " ", StyleBox["\[Phi]", FontFamily->"Symbol"], " --> ", StyleBox["\[Phi]", FontFamily->"Symbol"], " + ", StyleBox["\[Pi]", FontFamily->"Symbol"], "\n ", StyleBox["\[Theta]", FontFamily->"Symbol"], " --> ", StyleBox["\[Pi]", FontFamily->"Symbol"], " - ", StyleBox["\[Theta]", FontFamily->"Symbol"] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The ", StyleBox["\[Phi]", FontFamily->"Symbol"], " transformation gives a factor of" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " i m ", StyleBox["\[Pi]", FontFamily->"Symbol"], " m\n e or (-1)" }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "while the ", StyleBox["\[Theta]", FontFamily->"Symbol"], " transformation leaves sin ", StyleBox["\[Theta]", FontFamily->"Symbol"], " unchanged and reverses the sign of cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], ". But Y contains an even or odd polynomial in cos ", StyleBox["\[Theta]", FontFamily->"Symbol"], " of order l - m, giving a factor (-1)", StyleBox["l-m", FontVariations->{"CompatibilityType"->"Superscript"}], ". Thus overall we get a factor of" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " m l-m l\n (-1) (-1) = (-1)"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "So the parity relation in terms of a direction (unit vector) ", StyleBox["r", FontWeight->"Bold"], " is simply" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " m l m\n Y (-", StyleBox["r", FontWeight->"Bold"], ") = (-1) Y (", StyleBox["r", FontWeight->"Bold"], ")\n l l" }], "Print", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Physicists' Plots"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In quantum mechanics we often have a spherical harmonic Y as the angular \ (", StyleBox["\[Theta]", FontFamily->"Symbol"], ", ", StyleBox["\[Phi]", FontFamily->"Symbol"], ") part of a wave function. This is a complex function (because of the e", StyleBox["im", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\[Phi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Superscript"}], "), and so is not easy to plot directly. We can however plot the \ probability, given by |Y|", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ". Note that this removes all ", StyleBox["\[Phi]", FontFamily->"Symbol"], " dependence, but we'll make some 3D plots anyhow, because they're pretty." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To get |Y|", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], " (which I'll call ", StyleBox["YY", FontWeight->"Bold"], ") we'll take" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ YY[l_,m_,theta_,phi_] := \tSphericalHarmonicY[l,m,theta,-phi] * \tSphericalHarmonicY[l,m,theta,phi]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "since replacing ", StyleBox["\[Phi]", FontFamily->"Symbol"], " by - ", StyleBox["\[Phi]", FontFamily->"Symbol"], " is obviously equivalent to complex conjugation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "What we need now is a 3D plot with radius r representing YY as a function \ of ", StyleBox["\[Theta]", FontFamily->"Symbol"], " and ", StyleBox["\[Phi]", FontFamily->"Symbol"], ". Although we could use the built-in function ", StyleBox["ParametricPlot3D[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", and convert (r,", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ") to (x,y,z) ourselves, it's easier to use the ", StyleBox["SphericalPlot3D[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " function that comes with the ", StyleBox["Graphics`ParametricPlot3D`", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " package:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Needs[\"Graphics`ParametricPlot3D`\"]", "Input", AspectRatioFixed->True], Cell[TextData["Check its definition:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["?SphericalPlot3D", "Input", AspectRatioFixed->True], Cell[TextData[{ "-- the first argument can just give r (as a function of ", StyleBox["\[Theta]", FontFamily->"Symbol"], " and ", StyleBox["\[Phi]", FontFamily->"Symbol"], "), whereas for ", StyleBox["ParametricPlot3D[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " we'd have to give {x,y,z}." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "So here we go for l=2, m=1. The ", StyleBox["Evaluate[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " is just to speed it up, and the ", StyleBox["PlotRange->All", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " makes sure it doesn't clip off any of the surface:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SphericalPlot3D[Evaluate[YY[2,1,\[Theta],\[Phi]]],\n\t\t\t\t\ {\[Theta],0,\[Pi]},{\[Phi],0,2\[Pi]},PlotRange->All]"], "Input", AspectRatioFixed->True], Cell[TextData["Way cool. (?)"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Actually, it's even prettier if you use more ", StyleBox["PlotPoints", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", but of course that takes longer to evaluate, and more memory." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Repeat the above for some other values of l and m. Don't bother with m < 0; \ you'll get the same function as for |m|, except for a numerical prefactor \ which won't show on the plots."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Try to get a general feeling for how the pictures change with l and m, \ paying special attention to the number and placement of nodes as ", StyleBox["\[Theta]", FontFamily->"Symbol"], " varies. 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m .36216 .64427 L .35203 .61249 L p .036 .283 .777 r F P s P p .002 w .36216 .64427 m .50598 .77421 L .50676 .77506 L p 0 .107 .619 r F P s P p .002 w .42736 .71363 m .43534 .70444 L p .31432 .65201 L .594 .644 .87 r F P s P p .002 w .50571 .77325 m .35203 .61249 L .35907 .58005 L p .307 .383 .777 r F P s P p .002 w .35203 .61249 m .50571 .77325 L .50598 .77421 L p .036 .283 .777 r F P s P p .002 w .43534 .70444 m .44079 .69816 L p .42281 .6916 L .594 .644 .87 r F P s P p .31432 .65201 m .43534 .70444 L .42281 .6916 L .594 .644 .87 r F P p .002 w .25915 .63188 m .36488 .68327 L p .31432 .65201 L .594 .644 .87 r F P s P p .002 w .36488 .68327 m .42736 .71363 L p .31432 .65201 L .594 .644 .87 r F P s P p .002 w .68157 .79731 m .56935 .76895 L .54541 .77816 L p .637 .871 .985 r F P s P p .002 w .56935 .76895 m .68157 .79731 L .73536 .75836 L p .697 .875 .965 r F P s P p .002 w .68157 .79731 m .65652 .8265 L .70206 .79365 L p 0 0 0 r F P s P p .002 w .70206 .79365 m .73536 .75836 L .68157 .79731 L p 0 0 0 r F P s P p .002 w .63907 .24924 m .53619 .57888 L .54373 .58501 L p .88 .719 .678 r F P s P p .002 w .53619 .57888 m .63907 .24924 L .58733 .22542 L p .814 .642 .669 r F P s P p .002 w .63907 .24924 m .70555 .22287 L .6274 .18631 L p .559 .73 .955 r F P s P p .002 w .6274 .18631 m .58733 .22542 L .63907 .24924 L p .559 .73 .955 r F P s P p .002 w .74635 .60667 m .64685 .66893 L p .59736 .72208 L .759 .794 .881 r F P s P p .002 w .61592 .68355 m .52823 .76265 L p .52932 .76178 L 0 0 0 r F P s P p .002 w .51507 .77465 m .64835 .6588 L .61592 .68355 L p 0 0 0 r F P s P p .002 w .51407 .77542 m .51507 .77465 L p .52932 .76178 L 0 0 0 r F P s P p .002 w .52823 .76265 m .51407 .77542 L p .52932 .76178 L 0 0 0 r F P s P p .002 w .64835 .6588 m .51507 .77465 L .51561 .77373 L p .802 .988 .73 r F P s P p .002 w .51561 .77373 m .66652 .62861 L .64835 .6588 L p .802 .988 .73 r F P s P p .002 w .64685 .66893 m .58836 .70553 L .59201 .71225 L p .759 .794 .881 r F P s P p .002 w .59201 .71225 m .59736 .72208 L p .64685 .66893 L .759 .794 .881 r F P s P p .002 w .51367 .79564 m .52843 .85879 L .45856 .85548 L p .538 .803 .994 r F P s P p .002 w .52843 .85879 m .51367 .79564 L .52511 .79394 L p .586 .837 .992 r F P s P p .002 w .52511 .79394 m .59661 .84886 L .52843 .85879 L p .586 .837 .992 r F P s P p .002 w .76015 .27436 m .65236 .56465 L .62065 .53798 L p .893 .718 .658 r F P s P p .002 w .62065 .53798 m .70555 .22287 L .76015 .27436 L p .893 .718 .658 r F P s P p .002 w .50148 .57334 m .46377 .2183 L .40668 .23591 L p .62 .477 .666 r F P s P p .002 w .40668 .23591 m .49015 .57648 L .50148 .57334 L p .62 .477 .666 r F P s P p .002 w .35237 .20245 m .40668 .23591 L .46377 .2183 L p .659 .823 .962 r F P s P p .002 w .40668 .23591 m .35237 .20245 L .28517 .24726 L p .701 .831 .944 r F P s P p .002 w .50192 .79507 m .45856 .85548 L .39368 .83919 L p .51 .765 .988 r F P s P p .002 w .49111 .79229 m .50192 .79507 L p .49964 .79601 L .51 .765 .988 r F P s P p .002 w .39368 .83919 m .49111 .79229 L p .49964 .79601 L .51 .765 .988 r F P s P p .002 w .31073 .7793 m .34017 .8113 L .39368 .83919 L p .717 .184 0 r F P s P p .002 w .48237 .78759 m .49111 .79229 L p .48794 .79269 L .504 .728 .973 r F P s P p .002 w .34017 .8113 m .48237 .78759 L p .48794 .79269 L .504 .728 .973 r F P s P p .002 w .42832 .74776 m .26746 .73533 L .31073 .7793 L p .494 .685 .952 r F P s P p .002 w .31073 .7793 m .44232 .76274 L .42832 .74776 L p .494 .685 .952 r F P s P p .002 w .34017 .8113 m .31073 .7793 L .26746 .73533 L p .548 .049 0 r F P s P p .002 w .26746 .73533 m .30373 .77428 L .34017 .8113 L p .548 .049 0 r F P s P p .002 w .47667 .78145 m .48237 .78759 L p .47867 .78731 L .517 .698 .951 r F P s P p .002 w .30373 .77428 m .47667 .78145 L p .47867 .78731 L .517 .698 .951 r F P s P p .002 w .50797 .77572 m .46351 .77417 L .48961 .7809 L p .071 .594 .875 r F P s P p .002 w .46351 .77417 m .50797 .77572 L .50676 .77506 L p .137 .571 .926 r F P s P p .002 w .42309 .73079 m .2492 .68447 L .26746 .73533 L p .524 .661 .923 r F P s P p .002 w .26746 .73533 m .42832 .74776 L .42309 .73079 L p .524 .661 .923 r F P s P p .002 w .50676 .77506 m .44232 .76274 L .46351 .77417 L p .137 .571 .926 r F P s P p .002 w .47463 .77455 m .47667 .78145 L p .47282 .78043 L .541 .676 .925 r F P s P p .002 w .28874 .73165 m .47463 .77455 L p .47282 .78043 L .541 .676 .925 r F P s P p .002 w .50598 .77421 m .42832 .74776 L .44232 .76274 L p .257 .561 .934 r F P s P p .002 w .44232 .76274 m .50676 .77506 L .50598 .77421 L p .257 .561 .934 r F P s P p .002 w .47463 .77455 m .4765 .76761 L p .4762 .76946 L 0 0 .211 r F P s P p .002 w .47667 .78145 m .47463 .77455 L p .47495 .7764 L .942 .718 .246 r F P s P p .002 w .47667 .78145 m .30373 .77428 L .28874 .73165 L p .541 .676 .925 r F P s P p .002 w .47664 .78627 m .47667 .78145 L p .47495 .7764 L .942 .718 .246 r F P s P p .002 w .4746 .77937 m .47664 .78627 L p .47632 .78443 L .942 .718 .246 r F P s P p .002 w .47463 .77455 m .4746 .77937 L p .47632 .78443 L .942 .718 .246 r F P s P p .002 w .4746 .77937 m .47463 .77455 L p .4762 .76946 L 0 0 .211 r F P s P p .002 w .42736 .71363 m .25915 .63188 L .2492 .68447 L p .559 .648 .894 r F P s P p .002 w .2492 .68447 m .42309 .73079 L .42736 .71363 L p .559 .648 .894 r F P s P p .002 w .47463 .77455 m .28874 .73165 L .29772 .68781 L p .57 .663 .9 r F P s P p .002 w .50571 .77325 m .42309 .73079 L .42832 .74776 L p .375 .555 .902 r F P s P p .002 w .42832 .74776 m .50598 .77421 L .50571 .77325 L p .375 .555 .902 r F P s P p .002 w .29772 .68781 m .4765 .76761 L .47463 .77455 L p .57 .663 .9 r F P s P p .002 w .47647 .77243 m .4746 .77937 L p .4749 .77752 L 0 0 .211 r F P s P p .002 w .4765 .76761 m .47647 .77243 L p .4749 .77752 L 0 0 .211 r F P s P p .002 w .47647 .77243 m .4765 .76761 L .4821 .76139 L p .175 .04 .445 r F P s P p .002 w .48207 .76622 m .47647 .77243 L p .47735 .7707 L .175 .04 .445 r F P s P p .002 w .4821 .76139 m .48207 .76622 L p .47735 .7707 L .175 .04 .445 r F P s P p .002 w .35907 .58005 m .50598 .77229 L .50571 .77325 L p .307 .383 .777 r F P s P p .002 w .50598 .77229 m .42736 .71363 L .42309 .73079 L p .469 .556 .861 r F P s P p .002 w .42309 .73079 m .50571 .77325 L .50598 .77229 L p .469 .556 .861 r F P s P p .002 w .50598 .77229 m .35907 .58005 L .383 .55057 L p .461 .441 .75 r F P s P p .002 w .383 .55057 m .50676 .77144 L .50598 .77229 L p .461 .441 .75 r F P s P p .002 w .50676 .77144 m .44079 .69816 L .42736 .71363 L p .542 .564 .827 r F P s P p .002 w .42736 .71363 m .50598 .77229 L .50676 .77144 L p .542 .564 .827 r F P s P p .002 w .49081 .76142 m .48207 .76622 L p .48345 .7647 L .431 .234 .511 r F P s P p .002 w .49083 .75658 m .49081 .76142 L p .48345 .7647 L .431 .234 .511 r F P s P p .002 w .50676 .77144 m .383 .55057 L .42153 .52749 L p .56 .486 .729 r F P s P p .002 w .42153 .52749 m .50796 .77078 L .50676 .77144 L p .56 .486 .729 r F P s P p .002 w .44079 .69816 m .50676 .77144 L .50796 .77078 L p .6 .58 .803 r F P s P p .002 w .49044 .61569 m .51069 .77695 L .48139 .61052 L closepath p .878 .544 .415 r F P s P p .002 w .47529 .58859 m .51069 .77695 L .48109 .58176 L closepath p .285 .186 .563 r F P s P p .002 w .50796 .77078 m .42153 .52749 L .47046 .51364 L p .633 .527 .716 r F P s P p .002 w .47046 .51364 m .50946 .77039 L .50796 .77078 L p .633 .527 .716 r F P s P p .002 w .46204 .68614 m .50796 .77078 L .50946 .77039 L p .649 .602 .788 r F P s P p .002 w .50796 .77078 m .46204 .68614 L .44079 .69816 L p .6 .58 .803 r F P s P p .002 w .50946 .77039 m .48877 .67896 L .46204 .68614 L p .649 .602 .788 r F P s P p .002 w .50174 .75856 m .49081 .76142 L p .49253 .76021 L .571 .344 .533 r F P s P p .002 w .50175 .75372 m .50174 .75856 L p .49253 .76021 L .571 .344 .533 r F P s P p .002 w .51384 .61936 m .51069 .77695 L .50166 .61874 L closepath p .7 .405 .472 r F P s P p .002 w .52569 .61749 m .51069 .77695 L .51384 .61936 L closepath p .602 .324 .468 r F P s P p .002 w .50166 .61874 m .51069 .77695 L .49044 .61569 L closepath p .788 .475 .459 r F P s P p .002 w .53596 .61333 m .51069 .77695 L .52569 .61749 L closepath p .468 .212 .442 r F P s P p .002 w .54353 .6073 m .51069 .77695 L .53596 .61333 L closepath p .24 .021 .367 r F P s P p .002 w .53619 .57888 m .51069 .77695 L .54373 .58501 L closepath p .912 .683 .579 r F P s P p .002 w .48109 .58176 m .51069 .77695 L .49015 .57648 L closepath p .478 .32 .587 r F P s P p .002 w .52584 .57461 m .51069 .77695 L .53619 .57888 L closepath p .829 .6 .591 r F P s P p .002 w .49015 .57648 m .51069 .77695 L .50148 .57334 L closepath p .595 .404 .593 r F P s P p .002 w .51383 .5727 m .51069 .77695 L .52584 .57461 L closepath p .755 .532 .594 r F P s P p .002 w .50148 .57334 m .51069 .77695 L .51383 .5727 L closepath p .68 .47 .594 r F P s P p .002 w .58733 .22542 m .52584 .57461 L .53619 .57888 L p .814 .642 .669 r F P s P p .002 w .52584 .57461 m .58733 .22542 L .52649 .21464 L p .752 .581 .664 r F P s P p .002 w .52649 .21464 m .51383 .5727 L .52584 .57461 L p .752 .581 .664 r F P s P p .002 w .54373 .58501 m .51069 .77695 L .54767 .59233 L closepath p .991 .787 .531 r F P s P p .002 w .5111 .77031 m .52415 .51077 L .57625 .51924 L p .747 .616 .713 r F P s P p .002 w .51269 .77055 m .57625 .51924 L .62065 .53798 L p .8 .671 .721 r F P s P p .002 w .57625 .51924 m .51269 .77055 L .5111 .77031 L p .747 .616 .713 r F P s P p .002 w .51269 .77055 m .5464 .68186 L .518 .67748 L p .732 .665 .784 r F P s P p .002 w .518 .67748 m .5111 .77031 L .51269 .77055 L p .732 .665 .784 r F P s P p .002 w .52415 .51077 m .5111 .77031 L .50946 .77039 L p .693 .569 .711 r F P s P p .002 w .51383 .5727 m .52649 .21464 L .46377 .2183 L p .69 .528 .663 r F P s P p .002 w .46377 .2183 m .50148 .57334 L .51383 .5727 L p .69 .528 .663 r F P s P p .002 w .50946 .77039 m .47046 .51364 L .52415 .51077 L p .693 .569 .711 r F P s P p .002 w .48877 .67896 m .50946 .77039 L .5111 .77031 L p .693 .63 .782 r F P s P p .002 w .5111 .77031 m .518 .67748 L .48877 .67896 L p .693 .63 .782 r F P s P p .002 w .51366 .75313 m .51367 .75797 L .50174 .75856 L p .669 .423 .539 r F P s P p .002 w .50174 .75856 m .51367 .75797 L p .51247 .75665 L .562 .82 .994 r F P s P p .002 w .29755 .58348 m .44079 .69816 L .46204 .68614 L p .627 .648 .852 r F P s P p .002 w .28517 .24726 m .383 .55057 L .35907 .58005 L p .365 .306 .658 r F P s P p .002 w .36114 .54517 m .46204 .68614 L .48877 .67896 L p .658 .66 .841 r F P s P p .002 w .383 .55057 m .28517 .24726 L .35237 .20245 L p .515 .396 .655 r F P s P p .002 w .35237 .20245 m .42153 .52749 L .383 .55057 L p .515 .396 .655 r F P s P p .002 w .48877 .67896 m .44299 .52201 L .36114 .54517 L p .658 .66 .841 r F P s P p .002 w .45367 .5969 m .50175 .75372 L .49083 .75658 L p .658 .671 .851 r F P s P p .002 w .46204 .68614 m .36114 .54517 L .29755 .58348 L p .627 .648 .852 r F P s P p .002 w .49083 .75658 m .3846 .61599 L .45367 .5969 L p .658 .671 .851 r F P s P p .002 w .49081 .76142 m .49083 .75658 L .50175 .75372 L p .571 .344 .533 r F P s P p .002 w .53327 .51719 m .518 .67748 L .5464 .68186 L p .711 .7 .837 r F P s P p .002 w .44299 .52201 m .48877 .67896 L .518 .67748 L p .686 .677 .836 r F P s P p .002 w .518 .67748 m .53327 .51719 L .44299 .52201 L p .686 .677 .836 r F P s P p .002 w .52969 .59294 m .51366 .75313 L .50175 .75372 L p .683 .687 .847 r F P s P p .002 w .50175 .75372 m .45367 .5969 L .52969 .59294 L p .683 .687 .847 r F P s P p .002 w .50174 .75856 m .50175 .75372 L .51366 .75313 L p .669 .423 .539 r F P s P p .002 w .45266 .69011 m .50174 .75856 L p .51247 .75665 L .562 .82 .994 r F P s P p .002 w .49081 .76142 m .50174 .75856 L p .4994 .75759 L .615 .848 .988 r F P s P p .002 w .44079 .69816 m .29755 .58348 L .25915 .63188 L p .594 .644 .87 r F P s P p .002 w .33066 .64764 m .4821 .76139 L .4765 .76761 L p .6 .658 .878 r F P s P p .002 w .4765 .76761 m .29772 .68781 L .33066 .64764 L p .6 .658 .878 r F P s P p .002 w .3846 .61599 m .49083 .75658 L .4821 .76139 L p .63 .661 .862 r F P s P p .002 w .4821 .76139 m .33066 .64764 L .3846 .61599 L p .63 .661 .862 r F P s P p .002 w .48207 .76622 m .4821 .76139 L .49083 .75658 L p .431 .234 .511 r F P s P p .002 w .38243 .70895 m .49081 .76142 L p .4994 .75759 L .615 .848 .988 r F P s P p .002 w .48207 .76622 m .49081 .76142 L p .48757 .761 L .669 .86 .972 r F P s P p .002 w .32761 .7402 m .48207 .76622 L p .48757 .761 L .669 .86 .972 r F P s P p .002 w .47647 .77243 m .48207 .76622 L p .4783 .76649 L .711 .855 .95 r F P s P p .002 w .48237 .78759 m .34017 .8113 L .30373 .77428 L p .517 .698 .951 r F P s P p .002 w .29416 .77984 m .47647 .77243 L p .4783 .76649 L .711 .855 .95 r F P s P p .002 w .4746 .77937 m .47647 .77243 L p .47258 .77346 L .737 .835 .925 r F P s P p .002 w .28509 .82308 m .4746 .77937 L p .47258 .77346 L .737 .835 .925 r F P s P p .002 w .48237 .78759 m .47667 .78145 L p .47757 .78318 L .965 .687 .458 r F P s P p .002 w .48235 .7924 m .48237 .78759 L p .47757 .78318 L .965 .687 .458 r F P s P p .002 w .49111 .79229 m .48237 .78759 L p .48375 .78909 L .876 .601 .517 r F P s P p .002 w .49111 .79229 m .39368 .83919 L .34017 .8113 L p .504 .728 .973 r F P s P p .002 w .49109 .7971 m .49111 .79229 L p .48375 .78909 L .876 .601 .517 r F P s P p .002 w .50192 .79507 m .49111 .79229 L p .49281 .79349 L .792 .527 .537 r F P s P p .002 w .45856 .85548 m .50192 .79507 L p .5125 .79691 L .538 .803 .994 r F P s P p .002 w .50192 .79988 m .50192 .79507 L p .49281 .79349 L .792 .527 .537 r F P s P p .002 w .50192 .79507 m .51367 .79564 L p .5125 .79691 L .538 .803 .994 r F P s P p .002 w .59661 .84886 m .52511 .79394 L .53501 .79014 L p .641 .857 .982 r F P s P p .002 w .53501 .79014 m .65652 .8265 L .59661 .84886 L p .641 .857 .982 r F P s P p .002 w .58733 .22542 m .6274 .18631 L .53473 .16968 L p .579 .767 .968 r F P s P p .002 w .53473 .16968 m .52649 .21464 L .58733 .22542 L p .579 .767 .968 r F P s P p .002 w .43905 .17532 m .46377 .2183 L .52649 .21464 L p .615 .801 .971 r F P s P p .002 w .46377 .2183 m .43905 .17532 L .35237 .20245 L p .659 .823 .962 r F P s P p .002 w .52649 .21464 m .53473 .16968 L .43905 .17532 L p .615 .801 .971 r F P s P p .002 w .73536 .75836 m .58717 .75557 L .56935 .76895 L p .697 .875 .965 r F P s P p .002 w .58717 .75557 m .73536 .75836 L .76656 .71034 L p .738 .859 .937 r F P s P p .002 w .73536 .75836 m .70206 .79365 L .72813 .75331 L p .73 .941 .62 r F P s P p .002 w .72813 .75331 m .76656 .71034 L .73536 .75836 L p .73 .941 .62 r F P s P p .002 w .65652 .8265 m .53501 .79014 L .54231 .78465 L p .691 .861 .963 r F P s P p .002 w .54231 .78465 m .70206 .79365 L .65652 .8265 L p .691 .861 .963 r F P s P p .002 w .70555 .22287 m .62065 .53798 L .57625 .51924 L p .821 .636 .653 r F P s P p .002 w .57625 .51924 m .6274 .18631 L .70555 .22287 L p .821 .636 .653 r F P s P p .002 w .50192 .79988 m .45772 .94511 L .52868 .94837 L p .695 .696 .846 r F P s P p .002 w .52868 .94837 m .51368 .80045 L .50192 .79988 L p .695 .696 .846 r F P s P p .002 w .53206 1.01913 m .52868 .94837 L .45772 .94511 L p .721 .584 .702 r F P s P p .002 w .51368 .80045 m .52868 .94837 L .59793 .93859 L p .671 .678 .848 r F P s P p .002 w .52868 .94837 m .53206 1.01913 L .61444 1.00778 L p .664 .541 .703 r F P s P p .002 w .61444 1.00778 m .59793 .93859 L .52868 .94837 L p .664 .541 .703 r F P s P p .002 w .66652 .62861 m .51561 .77373 L .51561 .77276 L p .928 .942 .786 r F P s P p .002 w .51561 .77276 m .66807 .59605 L .66652 .62861 L p .928 .942 .786 r F P s P p .002 w .45772 .94511 m .4477 1.01535 L .53206 1.01913 L p .721 .584 .702 r F P s P p .002 w .51797 .87466 m .53206 1.01913 L .4477 1.01535 L p .699 .689 .837 r F P s P p .002 w .53206 1.01913 m .51797 .87466 L .54595 .8706 L p .671 .669 .839 r F P s P p .002 w .54595 .8706 m .61444 1.00778 L .53206 1.01913 L p .671 .669 .839 r F P s P p .002 w .49109 .7971 m .39181 .92906 L .45772 .94511 L p .717 .719 .851 r F P s P p .002 w .45772 .94511 m .50192 .79988 L .49109 .7971 L p .717 .719 .851 r F P s P p .002 w .4477 1.01535 m .45772 .94511 L .39181 .92906 L p .774 .634 .707 r F P s P p .002 w .39181 .92906 m .36919 .99672 L .4477 1.01535 L p .774 .634 .707 r F P s P p .002 w .48923 .8733 m .4477 1.01535 L .36919 .99672 L p .723 .715 .842 r F P s P p .002 w .4477 1.01535 m .48923 .8733 L .51797 .87466 L p .699 .689 .837 r F P s P p .002 w .30373 .77428 m .26746 .73533 L .2492 .68447 L p 0 .185 .717 r F P s P p .002 w .59793 .93859 m .52512 .79875 L .51368 .80045 L p .671 .678 .848 r F P s P p .002 w .52512 .79875 m .59793 .93859 L .6588 .91656 L p .644 .665 .855 r F P s P p .002 w .59793 .93859 m .61444 1.00778 L .68709 .98218 L p .6 .499 .712 r F P s P p .002 w .68709 .98218 m .6588 .91656 L .59793 .93859 L p .6 .499 .712 r F P s P p .002 w .61444 1.00778 m .54595 .8706 L .57029 .86152 L p .641 .654 .847 r F P s P p .002 w .57029 .86152 m .68709 .98218 L .61444 1.00778 L p .641 .654 .847 r F P s P p .002 w .42153 .52749 m .35237 .20245 L .43905 .17532 L p .612 .46 .651 r F P s P p .002 w .70206 .79365 m .54231 .78465 L .54622 .77804 L p .726 .847 .938 r F P s P p .002 w .54622 .77804 m .72813 .75331 L .70206 .79365 L p .726 .847 .938 r F P s P p .002 w .48235 .7924 m .33744 .90159 L .39181 .92906 L p .734 .745 .861 r F P s P p .002 w .39181 .92906 m .49109 .7971 L .48235 .7924 L p .734 .745 .861 r F P s P p .002 w .36919 .99672 m .39181 .92906 L .33744 .90159 L p .827 .695 .718 r F P s P p .002 w .33744 .90159 m .3041 .96473 L .36919 .99672 L p .827 .695 .718 r F P s P p .002 w .46271 .86666 m .36919 .99672 L .3041 .96473 L p .743 .745 .854 r F P s P p .002 w .36919 .99672 m .46271 .86666 L .48923 .8733 L p .723 .715 .842 r F P s P p .002 w .6274 .18631 m .57625 .51924 L .52415 .51077 L p .754 .572 .649 r F P s P p .002 w .52415 .51077 m .53473 .16968 L .6274 .18631 L p .754 .572 .649 r F P s P p .002 w .50947 .7761 m .48961 .7809 L .51788 .78228 L p .111 .655 .834 r F P s P p .002 w .51111 .77618 m .51788 .78228 L .54541 .77816 L p .26 .768 .849 r F P s P p .002 w .76656 .71034 m .59689 .73938 L .58717 .75557 L p .738 .859 .937 r F P s P p .002 w .59689 .73938 m .76656 .71034 L .77083 .65791 L p .757 .829 .908 r F P s P p .002 w .76656 .71034 m .72813 .75331 L .7312 .70948 L p .938 .957 .753 r F P s P p .002 w .7312 .70948 m .77083 .65791 L .76656 .71034 L p .938 .957 .753 r F P s P p .002 w .48961 .7809 m .50947 .7761 L .50797 .77572 L p .071 .594 .875 r F P s P p .002 w .66807 .59605 m .51561 .77276 L .51507 .77184 L p .905 .831 .763 r F P s P p .002 w .51507 .77184 m .65236 .56465 L .66807 .59605 L p .905 .831 .763 r F P s P p .002 w .54541 .77816 m .5127 .77595 L .51111 .77618 L p .26 .768 .849 r F P s P p .002 w .5127 .77595 m .54541 .77816 L .56935 .76895 L p .481 .896 .904 r F P s P p .002 w .6588 .91656 m .53503 .79495 L .52512 .79875 L p .644 .665 .855 r F P s P p .002 w .53503 .79495 m .6588 .91656 L .70509 .8842 L p .616 .659 .869 r F P s P p .002 w .6588 .91656 m .68709 .98218 L .74273 .94442 L p .517 .456 .727 r F P s P p .002 w .74273 .94442 m .70509 .8842 L .6588 .91656 L p .517 .456 .727 r F P s P p .002 w .47046 .51364 m .43905 .17532 L .53473 .16968 L p .688 .515 .648 r F P s P p .002 w .43905 .17532 m .47046 .51364 L .42153 .52749 L p .612 .46 .651 r F P s P p .002 w .2492 .68447 m .28874 .73165 L .30373 .77428 L p 0 .185 .717 r F P s P p .002 w .68709 .98218 m .57029 .86152 L .5884 .84834 L p .608 .646 .862 r F P s P p .002 w .5884 .84834 m .74273 .94442 L .68709 .98218 L p .608 .646 .862 r F P s P p .002 w .56935 .76895 m .51407 .77542 L .5127 .77595 L p .481 .896 .904 r F P s P p .002 w .51407 .77542 m .56935 .76895 L .58717 .75557 L p .683 .964 .937 r F P s P p .002 w .53473 .16968 m .52415 .51077 L .47046 .51364 L p .688 .515 .648 r F P s P p .002 w .47664 .78627 m .30037 .86511 L .33744 .90159 L p .745 .775 .878 r F P s P p .002 w .33744 .90159 m .48235 .7924 L .47664 .78627 L p .745 .775 .878 r F P s P p .002 w .3041 .96473 m .33744 .90159 L .30037 .86511 L p .881 .773 .738 r F P s P p .002 w .28874 .73165 m .2492 .68447 L .25915 .63188 L p .246 .334 .757 r F P s P p .002 w .58717 .75557 m .51507 .77465 L .51407 .77542 L p .683 .964 .937 r F P s P p .002 w .51507 .77465 m .58717 .75557 L .59689 .73938 L p .796 .947 .921 r F P s P p .002 w .72813 .75331 m .54622 .77804 L .5463 .77102 L p .744 .822 .912 r F P s P p .002 w .5463 .77102 m .7312 .70948 L .72813 .75331 L p .744 .822 .912 r F P s P p .002 w .30037 .86511 m .2593 .92208 L .3041 .96473 L p .881 .773 .738 r F P s P p .002 w .44117 .8554 m .3041 .96473 L .2593 .92208 L p .756 .78 .872 r F P s P p .002 w .3041 .96473 m .44117 .8554 L .46271 .86666 L p .743 .745 .854 r F P s P p .002 w .65236 .56465 m .51507 .77184 L .51407 .77108 L p .853 .741 .738 r F P s P p .002 w .51407 .77108 m .62065 .53798 L .65236 .56465 L p .853 .741 .738 r F P s P p .002 w .70509 .8842 m .54234 .78946 L .53503 .79495 L p .616 .659 .869 r F P s P p .002 w .54234 .78946 m .70509 .8842 L .73162 .84444 L p .586 .659 .888 r F P s P p .002 w .70509 .8842 m .74273 .94442 L .7751 .89782 L p .397 .405 .751 r F P s P p .002 w .7751 .89782 m .73162 .84444 L .70509 .8842 L p .397 .405 .751 r F P s P p .002 w .59689 .73938 m .51561 .77373 L .51507 .77465 L p .796 .947 .921 r F P s P p .002 w .51561 .77373 m .59689 .73938 L .59736 .72208 L p .831 .885 .881 r F P s P p .002 w .77083 .65791 m .59736 .72208 L .59689 .73938 L p .757 .829 .908 r F P s P p .002 w .51368 .80045 m .51367 .79564 L .50192 .79507 L p .712 .46 .543 r F P s P p .002 w .50192 .79507 m .50192 .79988 L .51368 .80045 L p .712 .46 .543 r F P s P p .002 w .5111 .78359 m .51797 .87466 L .48923 .8733 L p .712 .647 .782 r F P s P p .002 w .51367 .79564 m .51368 .80045 L .52512 .79875 L p .625 .39 .541 r F P s P p .002 w .51797 .87466 m .5111 .78359 L .5127 .78335 L p .672 .616 .784 r F P s P p .002 w .52512 .79875 m .52511 .79394 L .51367 .79564 L p .625 .39 .541 r F P s P p .002 w .5127 .78335 m .54595 .8706 L .51797 .87466 L p .672 .616 .784 r F P s P p .002 w .49111 .79229 m .49109 .7971 L .50192 .79988 L p .792 .527 .537 r F P s P p .002 w .50947 .78351 m .48923 .8733 L .46271 .86666 L p .75 .685 .788 r F P s P p .002 w .48923 .8733 m .50947 .78351 L .5111 .78359 L p .712 .647 .782 r F P s P p .002 w .25915 .63188 m .29772 .68781 L .28874 .73165 L p .246 .334 .757 r F P s P p .002 w .59736 .72208 m .77083 .65791 L .74635 .60667 L p .759 .794 .881 r F P s P p .002 w .77083 .65791 m .7312 .70948 L .70995 .66686 L p .922 .843 .748 r F P s P p .002 w .70995 .66686 m .74635 .60667 L .77083 .65791 L p .922 .843 .748 r F P s P p .002 w .64611 .88427 m .5884 .84834 L .59239 .8419 L p .573 .645 .884 r F P s P p .002 w .52511 .79394 m .52512 .79875 L .53503 .79495 L p .514 .302 .53 r F P s P p .002 w .53503 .79495 m .53501 .79014 L .52511 .79394 L p .514 .302 .53 r F P s P p .002 w .53501 .79014 m .53503 .79495 L .54234 .78946 L p .338 .167 .496 r F P s P p .002 w .51561 .78114 m .59829 .83238 L .5884 .84834 L p .508 .559 .843 r F P s P p .002 w .51508 .78205 m .5884 .84834 L .57029 .86152 L p .573 .572 .814 r F P s P p .002 w .5884 .84834 m .51508 .78205 L .51561 .78114 L p .508 .559 .843 r F P s P p .002 w .59829 .83238 m .51561 .78114 L .51561 .78017 L p .425 .554 .882 r F P s P p .002 w .51561 .78017 m .67775 .93606 L .66115 .90621 L p 0 .205 .73 r F P s P p .002 w .66115 .90621 m .62751 .88084 L p .52959 .79236 L .528 0 0 r F P s P p .002 w .54234 .78946 m .54231 .78465 L .53501 .79014 L p .338 .167 .496 r F P s P p .002 w .51561 .78017 m .59877 .81532 L .59829 .83238 L p .425 .554 .882 r F P s P p .002 w .59877 .81532 m .51561 .78017 L .51508 .77925 L p .318 .557 .921 r F P s P p .002 w .66115 .90621 m .51508 .77925 L .51561 .78017 L p 0 .205 .73 r F P s P p .002 w .51508 .77925 m .53578 .79725 L p .52959 .79236 L .528 0 0 r F P s P p .002 w .53578 .79725 m .66115 .90621 L p .52959 .79236 L .528 0 0 r F P s P p .002 w .74273 .94442 m .64611 .88427 L p .59829 .83238 L .573 .645 .884 r F P s P p .002 w .59239 .8419 m .59829 .83238 L p .64611 .88427 L .573 .645 .884 r F P s P p .002 w .59829 .83238 m .7751 .89782 L .74273 .94442 L p .573 .645 .884 r F P s P p .002 w .54595 .8706 m .5127 .78335 L .51407 .78282 L p .626 .591 .794 r F P s P p .002 w .51407 .78282 m .57029 .86152 L .54595 .8706 L p .626 .591 .794 r F P s P p .002 w .62065 .53798 m .51407 .77108 L .51269 .77055 L p .8 .671 .721 r F P s P p .002 w .50797 .78312 m .46271 .86666 L .44117 .8554 L p .784 .73 .803 r F P s P p .002 w .46271 .86666 m .50797 .78312 L .50947 .78351 L p .75 .685 .788 r F P s P p .002 w .4746 .77937 m .28509 .82308 L .30037 .86511 L p .747 .806 .899 r F P s P p .002 w .30037 .86511 m .47664 .78627 L .4746 .77937 L p .747 .806 .899 r F P s P p .002 w .2593 .92208 m .30037 .86511 L .28509 .82308 L p .928 .875 .764 r F P s P p .002 w .48237 .78759 m .48235 .7924 L .49109 .7971 L p .876 .601 .517 r F P s P p .002 w .59736 .72208 m .51561 .77276 L .51561 .77373 L p .831 .885 .881 r F P s P p .002 w .51561 .77276 m .59736 .72208 L .58836 .70553 L p .824 .817 .843 r F P s P p .002 w .57029 .86152 m .51407 .78282 L .51508 .78205 L p .573 .572 .814 r F P s P p .002 w .47667 .78145 m .47664 .78627 L .48235 .7924 L p .965 .687 .458 r F P s P p .002 w .54231 .78465 m .54234 .78946 L .54625 .78286 L p 0 0 .384 r F P s P p .002 w .54625 .78286 m .54622 .77804 L .54231 .78465 L p 0 0 .384 r F P s P p .002 w .73162 .84444 m .54625 .78286 L .54234 .78946 L p .586 .659 .888 r F P s P p .002 w .50676 .78246 m .44117 .8554 L .42693 .84064 L p .813 .786 .827 r F P s P p .002 w .42693 .84064 m .50597 .78161 L .50676 .78246 L p .813 .786 .827 r F P s P p .002 w .42161 .82391 m .5057 .78065 L .50597 .78161 L p .831 .851 .861 r F P s P p .002 w .50597 .78161 m .42693 .84064 L .42161 .82391 L p .831 .851 .861 r F P s P p .002 w .42593 .80699 m .50597 .7797 L .5057 .78065 L p .821 .919 .902 r F P s P p .002 w .5057 .78065 m .42161 .82391 L .42593 .80699 L p .821 .919 .902 r F P s P p .002 w .43959 .79174 m .50675 .77884 L .50597 .7797 L p .751 .964 .934 r F P s P p .002 w .50597 .7797 m .42593 .80699 L .43959 .79174 L p .751 .964 .934 r F P s P p .002 w .50597 .7797 m .50675 .77884 L p .48471 .79874 L .613 .906 .622 r F P s P p .002 w .42693 .84064 m .43542 .84944 L p .32701 .89415 L .756 .78 .872 r F P s P p .002 w .2593 .92208 m .36444 .871 L p .32701 .89415 L .756 .78 .872 r F P s P p .002 w .36444 .871 m .42693 .84064 L p .32701 .89415 L .756 .78 .872 r F P s P p .002 w .48369 .79979 m .50597 .7797 L p .48471 .79874 L .613 .906 .622 r F P s P p .002 w .43542 .84944 m .44117 .8554 L p .42323 .86198 L .756 .78 .872 r F P s P p .002 w .34211 .95166 m .5057 .78065 L .50597 .7797 L p .898 .986 .778 r F P s P p .42323 .86198 m .2593 .92208 L .43542 .84944 L .756 .78 .872 r F P p .002 w .50597 .7797 m .34948 .92086 L .34211 .95166 L p .898 .986 .778 r F P s P p .002 w .50675 .77884 m .37482 .89282 L .34948 .92086 L p .613 .906 .622 r F P s P p .002 w .50675 .77884 m .50796 .77819 L p .49131 .79252 L 0 0 0 r F P s P p .002 w .37482 .89282 m .48994 .79337 L p .49131 .79252 L 0 0 0 r F P s P p .002 w .48994 .79337 m .50675 .77884 L p .49131 .79252 L 0 0 0 r F P s P p .002 w .34948 .92086 m .48369 .79979 L p .48471 .79874 L .613 .906 .622 r F P s P p .002 w .44117 .8554 m .50676 .78246 L .50797 .78312 L p .784 .73 .803 r F P s P p .002 w .50947 .78351 m .47066 1.0398 L .52423 1.04219 L p .721 .592 .711 r F P s P p .002 w .52423 1.04219 m .5111 .78359 L .50947 .78351 L p .721 .592 .711 r F P s P p .002 w .53591 1.40997 m .52423 1.04219 L .47066 1.0398 L p .721 .544 .651 r F P s P p .002 w .5111 .78359 m .52423 1.04219 L .57647 1.03505 L p .664 .548 .713 r F P s P p .002 w .52423 1.04219 m .53591 1.40997 L .63362 1.3984 L p .654 .491 .651 r F P s P p .002 w .63362 1.3984 m .57647 1.03505 L .52423 1.04219 L p .654 .491 .651 r F P s P p .002 w .54622 .77804 m .54625 .78286 L .54633 .77584 L p 0 0 .019 r F P s P p .002 w .54633 .77584 m .5463 .77102 L .54622 .77804 L p 0 0 .019 r F P s P p .002 w .7312 .70948 m .5463 .77102 L .5425 .76435 L p .747 .792 .889 r F P s P p .002 w .5425 .76435 m .70995 .66686 L .7312 .70948 L p .747 .792 .889 r F P s P p .002 w .50797 .78312 m .42103 1.02811 L .47066 1.0398 L p .774 .642 .716 r F P s P p .002 w .47066 1.0398 m .50947 .78351 L .50797 .78312 L p .774 .642 .716 r F P s P p .002 w .43592 1.40612 m .47066 1.0398 L .42103 1.02811 L p .785 .602 .653 r F P s P p .002 w .47066 1.0398 m .43592 1.40612 L .53591 1.40997 L p .721 .544 .651 r F P s P p .002 w .54625 .78286 m .73162 .84444 L .73479 .80122 L p .556 .668 .912 r F P s P p .002 w .73479 .80122 m .54633 .77584 L .54625 .78286 L p .556 .668 .912 r F P s P p .002 w .58836 .70553 m .51507 .77184 L .51561 .77276 L p .824 .817 .843 r F P s P p .002 w .57647 1.03505 m .5127 .78335 L .5111 .78359 L p .664 .548 .713 r F P s P p .002 w .5127 .78335 m .57647 1.03505 L .62217 1.01902 L p .599 .507 .721 r F P s P p .002 w .62217 1.01902 m .51407 .78282 L .5127 .78335 L p .599 .507 .721 r F P s P p .002 w .51507 .77184 m .58836 .70553 L .57075 .69159 L p .8 .757 .813 r F P s P p .002 w .57075 .69159 m .51407 .77108 L .51507 .77184 L p .8 .757 .813 r F P s P p .002 w .28509 .82308 m .24028 .87268 L .2593 .92208 L p .928 .875 .764 r F P s P p .002 w .42693 .84064 m .2593 .92208 L .24028 .87268 L p .76 .816 .897 r F P s P p .002 w .42103 1.02811 m .50797 .78312 L .50676 .78246 L p .827 .704 .729 r F P s P p .002 w .51407 .78282 m .62217 1.01902 L .65663 .99558 L p .516 .465 .738 r F P s P p .002 w .65663 .99558 m .51508 .78205 L .51407 .78282 L p .516 .465 .738 r F P s P p .002 w .58836 .70553 m .74635 .60667 L .6946 .56262 L p .75 .759 .86 r F P s P p .002 w .6946 .56262 m .57075 .69159 L .58836 .70553 L p .75 .759 .86 r F P s P p .002 w .51069 .77695 m .54565 .98053 L .53755 .98622 L closepath p .397 .262 .579 r F P s P p .002 w .51069 .77695 m .54999 .97369 L .54565 .98053 L closepath p .123 .078 .53 r F P s P p .002 w .51069 .77695 m .53782 .9537 L .54588 .9595 L closepath p .922 .578 .368 r F P s P p .002 w .51069 .77695 m .53755 .98622 L .52659 .99016 L closepath p .542 .365 .591 r F P s P p .002 w .51069 .77695 m .52676 .94967 L .53782 .9537 L closepath p .832 .509 .442 r F P s P p .002 w .51069 .77695 m .52659 .99016 L .51393 .99192 L closepath p .64 .438 .594 r F P s P p .002 w .51069 .77695 m .51392 .94786 L .52676 .94967 L closepath p .744 .44 .468 r F P s P p .002 w .51069 .77695 m .50093 .99133 L .48895 .98845 L closepath p .791 .564 .593 r F P s P p .002 w .51069 .77695 m .51393 .99192 L .50093 .99133 L closepath p .718 .501 .595 r F P s P p .002 w .51069 .77695 m .50072 .94848 L .51392 .94786 L closepath p .653 .367 .472 r F P s P p .002 w .51069 .77695 m .48895 .98845 L .47928 .98358 L closepath p .869 .639 .587 r F P s P p .002 w .51069 .77695 m .48861 .95144 L .50072 .94848 L closepath p .542 .274 .458 r F P s P p .002 w .51069 .77695 m .47928 .98358 L .47295 .97723 L closepath p .955 .733 .564 r F P s P p .002 w .51069 .77695 m .47894 .95643 L .48861 .95144 L closepath p .373 .132 .414 r F P s P p .002 w .47068 .97007 m .51069 .77695 L .47521 .96524 L p .993 .827 .459 r F P s P p .002 w .47521 .96524 m .47295 .97723 L .47068 .97007 L p .993 .827 .459 r F P s P p .002 w .34265 1.38711 m .42103 1.02811 L .38035 1.00816 L p .85 .671 .658 r F P s P p .002 w .42103 1.02811 m .34265 1.38711 L .43592 1.40612 L p .785 .602 .653 r F P s P p .002 w .38035 1.00816 m .2649 1.35434 L .34265 1.38711 L p .85 .671 .658 r F P s P p .002 w .30146 1.2539 m .46483 .97632 L p .47068 .97007 L .847 .921 .528 r F P s P p .002 w .47068 .97007 m .47274 .96288 L p .47308 .95848 L 0 0 0 r F P s P p .002 w .51045 .7781 m .47068 .97007 L p .51046 .77809 L 0 0 0 r F P s P p .002 w .46483 .97632 m .47274 .96288 L .47068 .97007 L p .847 .921 .528 r F P s P p .002 w .29259 1.28913 m .47068 .97007 L .47295 .97723 L p .977 .892 .671 r F P s P p .002 w .47295 .97723 m .30725 1.32341 L .29259 1.28913 L p .977 .892 .671 r F P s P p .002 w .47068 .97007 m .29259 1.28913 L .30146 1.2539 L p .847 .921 .528 r F P s P p .002 w .33383 1.22163 m .4723 .96856 L p .47274 .96288 L 0 0 0 r F P s P p .002 w .4723 .96856 m .47894 .95643 L .47274 .96288 L p 0 0 0 r F P s P p .002 w .47274 .96288 m .51046 .77809 L p .47308 .95848 L 0 0 0 r F P s P p .002 w .51046 .77809 m .51069 .77695 L .51045 .7781 L p 0 0 0 r F P s P p .002 w .55008 .96641 m .54999 .97369 L .54763 .96189 L p 0 0 .307 r F P s P p .002 w .54763 .96189 m .51069 .77695 L .55008 .96641 L p 0 0 .307 r F P s P p .002 w .72012 1.37223 m .62217 1.01902 L .57647 1.03505 L p .574 .436 .655 r F P s P p .002 w .62217 1.01902 m .72012 1.37223 L .78691 1.33347 L p .463 .368 .662 r F P s P p .002 w .72404 1.30655 m .54999 .97369 L .55008 .96641 L p 0 .091 .631 r F P s P p .002 w .55008 .96641 m .72699 1.27131 L .72404 1.30655 L p 0 .091 .631 r F P s P p .002 w .55008 .96641 m .54588 .9595 L p .5526 .97106 L .632 .223 0 r F P s P p .002 w .72699 1.27131 m .55008 .96641 L p .5526 .97106 L .632 .223 0 r F P s P p .002 w .54588 .9595 m .55008 .96641 L .51092 .77808 L p .929 .595 .145 r F P s P p .002 w .51092 .77808 m .51069 .77695 L .51091 .77807 L p .929 .595 .145 r F P s P p .002 w .51091 .77807 m .54588 .9595 L p .51092 .77808 L .929 .595 .145 r F P s P p .002 w .51069 .77695 m .47274 .96288 L .47894 .95643 L closepath p .046 0 .286 r F P s P p .002 w .51074 .77773 m .50981 .77778 L p .51253 .7755 L .172 .705 .833 r F P s P p .002 w .50981 .77778 m .50946 .7778 L p .50957 .77771 L .172 .705 .833 r F P s P p .50957 .77771 m .51253 .7755 L .50981 .77778 L .172 .705 .833 r F P p .002 w .51809 .77134 m .5111 .77772 L .51074 .77773 L p .172 .705 .833 r F P s P p .51253 .7755 m .51809 .77134 L .51074 .77773 L .172 .705 .833 r F P p .002 w .38035 1.00816 m .50676 .78246 L .50597 .78161 L p .88 .783 .75 r F P s P p .002 w .50676 .78246 m .38035 1.00816 L .42103 1.02811 L p .827 .704 .729 r F P s P p .002 w .51407 .77108 m .57075 .69159 L .5464 .68186 L p .768 .706 .794 r F P s P p .002 w .5464 .68186 m .51269 .77055 L .51407 .77108 L p .768 .706 .794 r F P s P p .002 w .29772 .68781 m .25915 .63188 L .29755 .58348 L p .434 .414 .737 r F P s P p .002 w .51508 .78205 m .65663 .99558 L .676 .96695 L p .394 .415 .763 r F P s P p .002 w .676 .96695 m .51561 .78114 L .51508 .78205 L p .394 .415 .763 r F P s P p .002 w .35296 .98181 m .50597 .78161 L .5057 .78065 L p .924 .886 .777 r F P s P p .002 w .50597 .78161 m .35296 .98181 L .38035 1.00816 L p .88 .783 .75 r F P s P p .002 w .51561 .78114 m .676 .96695 L .67775 .93606 L p .192 .341 .786 r F P s P p .002 w .67775 .93606 m .51561 .78017 L .51561 .78114 L p .192 .341 .786 r F P s P p .002 w .5057 .78065 m .34211 .95166 L .35296 .98181 L p .924 .886 .777 r F P s P p .002 w .5463 .77102 m .54633 .77584 L .54253 .76918 L p .988 .721 .365 r F P s P p .002 w .54253 .76918 m .5425 .76435 L .5463 .77102 L p .988 .721 .365 r F P s P p .002 w .57647 1.03505 m .63362 1.3984 L .72012 1.37223 L p .574 .436 .655 r F P s P p .002 w .73162 .84444 m .7751 .89782 L .77966 .84687 L p .196 .329 .773 r F P s P p .002 w .77966 .84687 m .73479 .80122 L .73162 .84444 L p .196 .329 .773 r F P s P p .002 w .51508 .77925 m .58963 .79901 L .59877 .81532 L p .318 .557 .921 r F P s P p .002 w .58963 .79901 m .51508 .77925 L .51407 .77849 L p .194 .565 .937 r F P s P p .002 w .62751 .88084 m .51407 .77849 L .51508 .77925 L p .528 0 0 r F P s P p .002 w .46119 .77988 m .50796 .77819 L .50675 .77884 L p .59 .941 .926 r F P s P p .002 w .50675 .77884 m .43959 .79174 L .46119 .77988 L p .59 .941 .926 r F P s P p .002 w .51407 .77849 m .57173 .78526 L .58963 .79901 L p .194 .565 .937 r F P s P p .002 w .57173 .78526 m .51407 .77849 L .51269 .77795 L p .092 .579 .903 r F P s P p .002 w .51407 .77849 m .62751 .88084 L .58032 .863 L p .638 .08 0 r F P s P p .002 w .58032 .863 m .51269 .77795 L .51407 .77849 L p .638 .08 0 r F P s P p .002 w .48837 .7728 m .50946 .7778 L .50796 .77819 L p .365 .834 .874 r F P s P p .002 w .50796 .77819 m .46119 .77988 L .48837 .7728 L p .365 .834 .874 r F P s P p .002 w .41574 .87085 m .50796 .77819 L .50946 .7778 L p .259 0 0 r F P s P p .002 w .50796 .77819 m .41574 .87085 L .37482 .89282 L p 0 0 0 r F P s P p .002 w .54633 .77584 m .73479 .80122 L .71324 .75917 L p .529 .685 .938 r F P s P p .002 w .71324 .75917 m .54253 .76918 L .54633 .77584 L p .529 .685 .938 r F P s P p .002 w .51269 .77795 m .54697 .77566 L .57173 .78526 L p .092 .579 .903 r F P s P p .002 w .54697 .77566 m .51269 .77795 L .5111 .77772 L p .075 .617 .848 r F P s P p .002 w .51269 .77795 m .58032 .863 L .5249 .85492 L p .586 .064 0 r F P s P p .002 w .5249 .85492 m .5111 .77772 L .51269 .77795 L p .586 .064 0 r F P s P p .002 w .50946 .7778 m .48837 .7728 L .51809 .77134 L p .172 .705 .833 r F P s P p .002 w .46778 .85766 m .50946 .7778 L .5111 .77772 L p .464 0 0 r F P s P p .002 w .50946 .7778 m .46778 .85766 L .41574 .87085 L p .259 0 0 r F P s P p .002 w .5111 .77772 m .51809 .77134 L .54697 .77566 L p .075 .617 .848 r F P s P p .002 w .5111 .77772 m .5249 .85492 L .46778 .85766 L p .464 0 0 r F P s P p .002 w .24028 .87268 m .42161 .82391 L .42693 .84064 L p .76 .816 .897 r F P s P p .002 w .47647 .77243 m .29416 .77984 L .28509 .82308 L p .737 .835 .925 r F P s P p .002 w .70995 .66686 m .5425 .76435 L .53522 .75876 L p .74 .761 .869 r F P s P p .002 w .5425 .76435 m .54253 .76918 L .53524 .7636 L p .925 .643 .488 r F P s P p .002 w .53524 .7636 m .53522 .75876 L .5425 .76435 L p .925 .643 .488 r F P s P p .002 w .54253 .76918 m .71324 .75917 L .66845 .72318 L p .511 .712 .962 r F P s P p .002 w .66845 .72318 m .53524 .7636 L .54253 .76918 L p .511 .712 .962 r F P s P p .002 w .57075 .69159 m .6946 .56262 L .62074 .53139 L p .733 .727 .845 r F P s P p .002 w .62074 .53139 m .5464 .68186 L .57075 .69159 L p .733 .727 .845 r F P s P p .002 w .48207 .76622 m .32761 .7402 L .29416 .77984 L p .711 .855 .95 r F P s P p .002 w .53522 .75876 m .66586 .63039 L .70995 .66686 L p .74 .761 .869 r F P s P p .002 w .66586 .63039 m .53522 .75876 L .52525 .75488 L p .726 .732 .856 r F P s P p .002 w .53522 .75876 m .53524 .7636 L .52526 .75972 L p .835 .561 .525 r F P s P p .002 w .52526 .75972 m .52525 .75488 L .53522 .75876 L p .835 .561 .525 r F P s P p .002 w .7751 .89782 m .59829 .83238 L .59877 .81532 L p .537 .654 .911 r F P s P p .002 w .53524 .7636 m .66845 .72318 L .60494 .69774 L p .507 .746 .981 r F P s P p .002 w .60494 .69774 m .52526 .75972 L .53524 .7636 L p .507 .746 .981 r F P s P p .002 w .52525 .75488 m .6034 .60463 L .66586 .63039 L p .726 .732 .856 r F P s P p .002 w .6034 .60463 m .52525 .75488 L .51366 .75313 L p .706 .707 .848 r F P s P p .002 w .52525 .75488 m .52526 .75972 L .51367 .75797 L p .752 .491 .537 r F P s P p .002 w .51367 .75797 m .51366 .75313 L .52525 .75488 L p .752 .491 .537 r F P s P p .002 w .29755 .58348 m .33066 .64764 L .29772 .68781 L p .434 .414 .737 r F P s P p .002 w .49081 .76142 m .38243 .70895 L .32761 .7402 L p .669 .86 .972 r F P s P p .002 w .78691 1.33347 m .65663 .99558 L .62217 1.01902 L p .463 .368 .662 r F P s P p .002 w .5464 .68186 m .62074 .53139 L .53327 .51719 L p .711 .7 .837 r F P s P p .002 w .51366 .75313 m .52969 .59294 L .6034 .60463 L p .706 .707 .848 r F P s P p .002 w .52526 .75972 m .60494 .69774 L .52999 .68619 L p .524 .784 .992 r F P s P p .002 w .52999 .68619 m .51367 .75797 L .52526 .75972 L p .524 .784 .992 r F P s P p .002 w .50174 .75856 m .45266 .69011 L .38243 .70895 L p .615 .848 .988 r F P s P p .002 w .51367 .75797 m .52999 .68619 L .45266 .69011 L p .562 .82 .994 r F P s P p .002 w .74635 .60667 m .70995 .66686 L .66586 .63039 L p .865 .744 .727 r F P s P p .002 w .66586 .63039 m .6946 .56262 L .74635 .60667 L p .865 .744 .727 r F P s P p .002 w .24028 .87268 m .28509 .82308 L .29416 .77984 L p .912 .98 .767 r F P s P p .002 w .59877 .81532 m .77966 .84687 L .7751 .89782 L p .537 .654 .911 r F P s P p .002 w .29416 .77984 m .25041 .82157 L .24028 .87268 L p .912 .98 .767 r F P s P p .002 w .25041 .82157 m .42593 .80699 L .42161 .82391 L p .749 .85 .925 r F P s P p .002 w .42161 .82391 m .24028 .87268 L .25041 .82157 L p .749 .85 .925 r F P s P p .002 w .2649 1.35434 m .38035 1.00816 L .35296 .98181 L p .922 .761 .664 r F P s P p .002 w .77966 .84687 m .59877 .81532 L .58963 .79901 L p .501 .672 .94 r F P s P p .002 w .25041 .82157 m .29416 .77984 L .32761 .7402 L p .637 .912 .614 r F P s P p .002 w .32761 .7402 m .29003 .77445 L .25041 .82157 L p .637 .912 .614 r F P s P p .002 w .29003 .77445 m .43959 .79174 L .42593 .80699 L p .718 .874 .954 r F P s P p .002 w .42593 .80699 m .25041 .82157 L .29003 .77445 L p .718 .874 .954 r F P s P p .002 w .65663 .99558 m .78691 1.33347 L .82643 1.28537 L p .28 .265 .663 r F P s P p .002 w .82643 1.28537 m .676 .96695 L .65663 .99558 L p .28 .265 .663 r F P s P p .002 w .33066 .64764 m .29755 .58348 L .36114 .54517 L p .547 .468 .718 r F P s P p .002 w .73479 .80122 m .77966 .84687 L .75448 .79703 L p 0 .19 .721 r F P s P p .002 w .75448 .79703 m .71324 .75917 L .73479 .80122 L p 0 .19 .721 r F P s P p .002 w .58963 .79901 m .75448 .79703 L .77966 .84687 L p .501 .672 .94 r F P s P p .002 w .75448 .79703 m .58963 .79901 L .57173 .78526 L p .474 .7 .968 r F P s P p .002 w .36114 .54517 m .3846 .61599 L .33066 .64764 L p .547 .468 .718 r F P s P p .002 w .35577 .73714 m .46119 .77988 L .43959 .79174 L p .664 .88 .979 r F P s P p .002 w .43959 .79174 m .29003 .77445 L .35577 .73714 L p .664 .88 .979 r F P s P p .002 w .21076 1.31045 m .35296 .98181 L .34211 .95166 L p .982 .878 .651 r F P s P p .002 w .35296 .98181 m .21076 1.31045 L .2649 1.35434 L p .922 .761 .664 r F P s P p .002 w .52795 1.39142 m .53591 1.40997 L .43592 1.40612 L p .346 .778 .969 r F P s P p .002 w .53591 1.40997 m .52795 1.39142 L .59494 1.38342 L p .421 .833 .972 r F P s P p .002 w .59494 1.38342 m .63362 1.3984 L .53591 1.40997 L p .421 .833 .972 r F P s P p .002 w .57173 .78526 m .70103 .75414 L .75448 .79703 L p .474 .7 .968 r F P s P p .002 w .70103 .75414 m .57173 .78526 L .54697 .77566 L p .463 .738 .987 r F P s P p .002 w .6946 .56262 m .66586 .63039 L .6034 .60463 L p .807 .669 .711 r F P s P p .002 w .6034 .60463 m .62074 .53139 L .6946 .56262 L p .807 .669 .711 r F P s P p .002 w .4593 1.38875 m .43592 1.40612 L .34265 1.38711 L p .318 .734 .976 r F P s P p .002 w .43592 1.40612 m .4593 1.38875 L .52795 1.39142 L p .346 .778 .969 r F P s P p .002 w .44051 .71456 m .48837 .7728 L .46119 .77988 L p .594 .863 .994 r F P s P p .002 w .46119 .77988 m .35577 .73714 L .44051 .71456 L p .594 .863 .994 r F P s P p .002 w .54697 .77566 m .62459 .72371 L .70103 .75414 L p .463 .738 .987 r F P s P p .002 w .62459 .72371 m .54697 .77566 L .51809 .77134 L p .48 .782 .998 r F P s P p .002 w .676 .96695 m .82643 1.28537 L .83298 1.23244 L p 0 .067 .608 r F P s P p .002 w .83298 1.23244 m .67775 .93606 L .676 .96695 L p 0 .067 .608 r F P s P p .002 w .53402 .70986 m .51809 .77134 L .48837 .7728 L p .527 .827 .999 r F P s P p .002 w .48837 .7728 m .44051 .71456 L .53402 .70986 L p .527 .827 .999 r F P s P p .002 w .51809 .77134 m .53402 .70986 L .62459 .72371 L p .48 .782 .998 r F P s P p .002 w .63362 1.3984 m .59494 1.38342 L .65378 1.36542 L p .525 .884 .981 r F P s P p .002 w .65378 1.36542 m .72012 1.37223 L .63362 1.3984 L p .525 .884 .981 r F P s P p .002 w .4593 1.38875 m .50093 .99133 L .51393 .99192 L p .722 .553 .661 r F P s P p .002 w .51393 .99192 m .52795 1.39142 L .4593 1.38875 L p .722 .553 .661 r F P s P p .002 w .52795 1.39142 m .51393 .99192 L .52659 .99016 L p .655 .5 .662 r F P s P p .002 w .39555 1.37564 m .48895 .98845 L .50093 .99133 L p .785 .611 .664 r F P s P p .002 w .50093 .99133 m .4593 1.38875 L .39555 1.37564 L p .785 .611 .664 r F P s P p .002 w .52659 .99016 m .59494 1.38342 L .52795 1.39142 L p .655 .5 .662 r F P s P p .002 w .59494 1.38342 m .52659 .99016 L .53755 .98622 L 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Just the same information, or \ rather more, is obtained by plotting a cross section for any fixed ", StyleBox["\[Phi]", FontFamily->"Symbol"], ". We could use ", StyleBox["PolarPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " from the ", StyleBox["Graphics`Graphics`", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " package to do that, but that's set up for ", StyleBox["\[Theta]", FontFamily->"Symbol"], "=0 to be horizontal rather than vertical, so let's do it ourselves with ", StyleBox["ParametricPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". That needs the first argument to be a list {x, z}, which we can get \ from r and ", StyleBox["\[Theta]", FontFamily->"Symbol"], " with ", StyleBox["r*{Sin[q], Cos[q]}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "; the ", StyleBox["r", FontWeight->"Bold"], " outside multiplies the two terms inside the list." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Thus, for the l=2, m=1 case that we tried first above"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ParametricPlot[\n\t\ Evaluate[YY[2,1,\[Theta],0]*{Sin[\[Theta]],Cos[\[Theta]]}],\n\t{\[Theta],-\ \[Pi],\[Pi]},\n\tAspectRatio->Automatic\n]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "I set ", StyleBox["AspectRatio->Automatic", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " so that it didn't rescale x and z separately." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Compare this picture to the 3D plot we made earlier.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We can even make a graphical display of a bunch of these, using ", StyleBox["GraphicsArray[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". First we make the plots internally, setting ", StyleBox["DisplayFunction->Identity", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " so Mathematica just stores the graphics objects, not yet rendering them \ into Postscript or pictures. I also turned off axes, which would clutter the \ pictures, and added ", StyleBox["//TableForm", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " to show the form of the output (", StyleBox["TableForm", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " won't affect the ", StyleBox["plot", FontSlant->"Italic"], ")." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Table[\n\tParametricPlot[\n\t\tEvaluate[\n\t\t\tYY[l,m,\[Theta],0]*{Sin[\ \[Theta]],Cos[\[Theta]]}\n\t\t],\n\t\t{\[Theta],-\[Pi],\[Pi]},\n\t\t\ AspectRatio->Automatic,\n\t\tAxes->False,\n\t\tDisplayFunction->Identity\n\t\ ],\n\t{l,0,3},\n\t{m,0,l}\n] // TableForm"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The first row is l=0, m=0, the second row is l=1, m=0,1, and so on. Now \ we can render them as a ", StyleBox["GraphicsArray[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[": ", FontSize->12] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Show[GraphicsArray[%]]", "Input", AspectRatioFixed->True], Cell[TextData[ "You might want to enlarge that; click on the picture and then drag one of \ the corner handles. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Of course we could have done just the same thing with the 3D \ spherical plots, but it's pretty time and memory-consuming.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Chemist's plots"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Chemists (and some atomic and solid state physicists) often construct ", StyleBox["real", FontSlant->"Italic"], " wave-functions by taking linear combinations of our Y", StyleBox["l", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], " and Y", StyleBox["l", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["-m", FontVariations->{"CompatibilityType"->"Superscript"}], " to get cos\.80m", StyleBox["\[Phi]", FontFamily->"Symbol"], " or sin m", StyleBox["\[Phi]", FontFamily->"Symbol"], " for the azimuthal dependence. These are sometimes given weird names like \ ", StyleBox["zonal harmonics", FontSlant->"Italic"], ", (for m=0), ", StyleBox["sectoral harmonics", FontSlant->"Italic"], " (for m=n), and ", StyleBox["tesseral harmonics", FontSlant->"Italic"], " (otherwise)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "There is no particular reason to do this for ", StyleBox["isolated", FontSlant->"Italic"], " atoms, for which it's just a different choice among degenerate \ eigenvalues. But in the presence of an external field, the sine and cosine \ functions sometimes form a better basis than the complex exponentials. \ They're particularly useful in discussing the directionality of chemical \ bonds." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "One of the nice features of the sine/cosine representation is that we can \ picture the azimuthal dependence. Remember that this was impossible with the \ complex exponentials, and we ended up just showing |Y|", StyleBox["2", FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], ". Now we can plot |cos\.80m", StyleBox["\[Phi]", FontFamily->"Symbol"], "| or |sin m", StyleBox["\[Phi]", FontFamily->"Symbol"], "| times the ", StyleBox["\[Theta]", FontFamily->"Symbol"], " part; we still need to take the absolute value to avoid handing ", StyleBox["SphericalPlot", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " a negative radius." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Here's a function to plot the general (l,m) case with the cos m", StyleBox["\[Phi]", FontFamily->"Symbol"], " choice:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "chemist[l_,m_] :=\nModule[{t,\[Phi]},\n\tSphericalPlot3D[\n\t\t\ Evaluate[Abs[LegendreP[l,m,Cos[t]]*Cos[m \[Phi]]]],\n\t\t{t,0,\[Pi]}, \ {\[Phi],0,2\[Pi]},\n\t\tPlotRange->All\n\t]\n]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Note that the ", StyleBox["Abs[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " gets rid of the complex exponential entirely." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here's our l=2, m=1 again: (be patient)"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["chemist[2,1]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Isn't that splendid? (Super?) Because this has l=2 it would be called a \ ", StyleBox["d", FontSlant->"Italic"], " orbital by a chemist; l=0,1,2,3 are called s,p,d,f respectively. But \ note two things:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "1. Some of the lobes of the wavefunction are positive, some are negative; \ we don't see that with our ", StyleBox["Abs[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " function (and ", StyleBox["SphericalPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " won't work if you remove it). Orbital diagrams are normally labelled \ with a + or a - by each lobe." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "2. We are ", StyleBox["not", FontSlant->"Italic"], " plotting complete wave-functions, only the angular part. The full \ wavefunction also has an r dependence, which has more and more oscillations \ as you increase the \"principal quantum number\" n. It's not easy to plot a \ complete wavefunction in 3D (you'd need something like a 3D equivalent of ", StyleBox["DensityPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "), though you could plot cross-sections in a fixed-", StyleBox["\[Phi] ", FontFamily->"Symbol"], "plane fairly easily using ", StyleBox["DensityPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " or ", StyleBox["ContourPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". But that's beyond the present call of duty." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Note that I'm talking about ", StyleBox["one-electron", FontSlant->"Italic"], " wave functions; you wouldn't even think about plotting a two-electron \ wave function, would you? It lives in a 6-dimensional space. ", StyleBox[ "You can't think of wave functions as \"objects\" (or fields) in real space \ for more than one particle", FontSlant->"Italic"], ". Physics denies wave functions any such reality. Or are wave functions \ (or field operators) the only reality? Just be thankful I don't teach \ quantum... Actually people who mention words like ", StyleBox["reality", FontSlant->"Italic"], " probably shouldn't be allowed to teach quantum; they're just not \ compatible. Quantum mechanics does ", StyleBox["work", FontSlant->"Italic"], ", and the party line is to accept that as enough, or as all we can know." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "OK, back to Earth. Play with a few other cases. Again the default ", StyleBox["PlotPoints", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " isn't really enough for large l, but increasing it will cost you time and \ memory." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Angular Momentum Operators"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The most central way that the Y's occur in quantum mechanics is as the \ eigenfunctions of angular momentum. In spherical coordinates we can write \ (after some work):"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Lz[f_] := - I D[f, \[Phi]];\nLL[f_] := -(1/Sin[\[Theta]] \ D[Sin[\[Theta]]D[f,\[Theta]], \[Theta]] + 1/Sin[\[Theta]]^2 D[f, \ {\[Phi],2}])"], "Input", AspectRatioFixed->True], Cell[TextData[{ "where f is a function of ", StyleBox["\[Theta]", FontFamily->"Symbol"], " and ", StyleBox["\[Phi]", FontFamily->"Symbol"], ", Lz is the z-component of the angular momentum operator, and LL is the ", StyleBox["L.L", FontWeight->"Bold"], " operator. The claim is that the spherical harmonics are simultaneous \ eigenfunctions of these two operators, with eigenvalues m for ", StyleBox["Lz", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and l(l+1) for ", StyleBox["LL", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". Let's check that for l=3, m=2:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" y32 = SphericalHarmonicY[3,2,\[Theta],\[Phi]]"], "Input", AspectRatioFixed->True], Cell[" Lz[y32]", "Input", AspectRatioFixed->True], Cell[TextData["Good, that's 2 f."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" LL[y32]", "Input", AspectRatioFixed->True], Cell[" Simplify[%] (* be patient *)", "Input", AspectRatioFixed->True], Cell[TextData["Good, that's 12f, and l(l+1) = 12."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We can also define raising and lowering operators for angular momentum by:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " i", StyleBox["\[Phi]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]", FontFamily->"Symbol"], " \nL = L + iL = e ( -- + i cot(", StyleBox["\[Theta]", FontFamily->"Symbol"], ") -- )\n + x y ", StyleBox["\[PartialD]\[Theta]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]\[Phi]", FontFamily->"Symbol"], " \n\n -i", StyleBox["\[Phi]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]", FontFamily->"Symbol"], " \nL = L - iL = - e ( -- - i cot(", StyleBox["\[Theta]", FontFamily->"Symbol"], ") -- )\n - x y ", StyleBox["\[PartialD]\[Theta]", FontFamily->"Symbol"], " ", StyleBox["\[PartialD]\[Phi]", FontFamily->"Symbol"], " \n" }], "Info", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["or"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Lplus[f_] := Exp[I \[Phi]] (D[f,\[Theta]] + I Cot[\[Theta]] D[f,\[Phi]]);\n\ Lminus[f_] := - Exp[-I \[Phi]] (D[f,\[Theta]] - I Cot[\[Theta]] \ D[f,\[Phi]])"], "Input", AspectRatioFixed->True], Cell[TextData[{ "For example, if we apply L", StyleBox["+", FontVariations->{"CompatibilityType"->"Subscript"}], " to Y", StyleBox["2", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["0", FontVariations->{"CompatibilityType"->"Superscript"}], ", it \"raises\" the m index from m=0 to m=1, and gives a multiple of Y", StyleBox["2", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["1", FontVariations->{"CompatibilityType"->"Superscript"}], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Lplus[SphericalHarmonicY[2,0,\[Theta],\[Phi]]]"], "Input", AspectRatioFixed->True], Cell[TextData["SphericalHarmonicY[2,1,\[Theta],\[Phi]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "It worked. The numerical factor should actually be Sqrt[(l-m)(l+m+1)] \ (where m is the original m before raising), which is Sqrt[6]. And it is."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Raising again:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Lplus[SphericalHarmonicY[2,1,\[Theta],\[Phi]]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "which is obviously a multiple of Y", StyleBox["2", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ". If that's not yet obvious, you need to stare at a table of Y", StyleBox["l", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["m", FontVariations->{"CompatibilityType"->"Superscript"}], "'s until you achieve spherical harmony." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Raising one more time (guess what?):"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Lplus[SphericalHarmonicY[2,2,\[Theta],\[Phi]]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Lowering is more of a pain, because you often get something that needs some \ simplifying, but I'll pretend that isn't so and pick a \"typical\" case. \ [Cynic's view: in a scientific paper a \"typical\" case often means one of \ the best the author has.]"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Lminus[SphericalHarmonicY[2,2,\[Theta],\[Phi]]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "That's a multiple of Y", StyleBox["2", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["1", FontVariations->{"CompatibilityType"->"Superscript"}], ", yes?" }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Orthonormality and Laplace Series"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The orthogonality relation is very pretty: if you integrate a product of \ any two Y's over all solid angles, you get zero unless the l's are equal ", StyleBox["and", FontSlant->"Italic"], " the m's are equal, in which case you get 1. Actually there's one \ complication; you must complex conjugate one of the Y's, or (equivalently) \ replace ", StyleBox["\[Phi]", FontFamily->"Symbol"], " by -", StyleBox["\[Phi]", FontFamily->"Symbol"], " in one of them. Thus the integral can be written:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "yint[l1_,m1_,l2_,m2_] :=\nModule[{\[Theta],\[Phi]},\n\tIntegrate[\n\t\t\ Integrate[\n\t\t\tSphericalHarmonicY[l1, m1, \[Theta], -\[Phi]] *\n\t\t\t\t\ SphericalHarmonicY[l2, m2, \[Theta], \[Phi]],\n\t\t\t{\[Phi], 0, 2\[Pi]}\n\t\t\ ] * Sin[\[Theta]],\n\t\t{\[Theta], 0, \[Pi]}\n\t]\n]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Note that the solid angle integral requires sin(", StyleBox["\[Theta]", FontFamily->"Symbol"], ") d", StyleBox["\[Theta]", FontFamily->"Symbol"], " d", StyleBox["\[Phi]", FontFamily->"Symbol"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Let's try it. It's a double integral, and we should be prepared to wait a \ while, but actually it's pretty quick."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" yint[3,1,3,2] (* same l's, different m's *)", "Input", AspectRatioFixed->True], Cell[" yint[2,1,3,1] (* different l's, same m's *)", "Input", AspectRatioFixed->True], Cell[" yint[2,1,3,2] (* different l's, different m's *)", "Input", AspectRatioFixed->True], Cell[" yint[2,1,2,1] (* same l's, same m's *)", "Input", AspectRatioFixed->True], Cell[TextData["Oh joy!"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Given this orthonormality relation (and completeness of the spherical \ harmonics), we can expand any reasonable function f(", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ") (a function \"on the unit sphere\") in a ", StyleBox["Laplace Series", FontSlant->"Italic"], " of spherical harmonics:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " ", StyleBox["\[Infinity]", FontFamily->"Symbol"], " +l m\n f(", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ") = ", StyleBox["\[Sum]", FontFamily->"Symbol"], " ", StyleBox["\[Sum]", FontFamily->"Symbol"], " a Y (", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ")\n l=0 m=-l lm l" }], "Info", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[{ "and find the coefficients a", StyleBox["lm", FontSize->12, FontVariations->{"CompatibilityType"->"Subscript"}], " by use of the orthogonality relation, just as for Fourier series or \ Legendre series:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " m *\n a = ", StyleBox["\[Integral]", FontFamily->"Symbol"], " Y (", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ") f(", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], ") d", StyleBox["\[CapitalOmega]", FontFamily->"Symbol"], "\n lm l" }], "Info", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[{ "The integral is over all solid angles [sin ", StyleBox["\[Theta]", FontFamily->"Symbol"], " d", StyleBox["\[Theta]", FontFamily->"Symbol"], " d", StyleBox["\[Phi]", FontFamily->"Symbol"], "]. Note that we need to complex conjugate the Y, or (equivalently) \ replace its ", StyleBox["\[Phi]", FontFamily->"Symbol"], " by -", StyleBox["\[Phi]", FontFamily->"Symbol"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Problem 4"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here's a \"pincushion\" function to play with:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ pin[theta_,phi_] := 1/ \t(Sqrt[1 - Sin[theta]^2 Cos[phi]^2] + \t Sqrt[1 - Sin[theta]^2 Sin[phi]^2])\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "Let's plot it using ", StyleBox["SphericalPlot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". I've repeated here the ", StyleBox["Needs[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " command to load the appropriate package, in case you're coming back after \ a break; it doesn't hurt to repeat it anyhow. The ", StyleBox["PlotPoints->17", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " gives us 16 increments in each variable, which shows the four-fold \ symmetry nicely." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Needs[\"Graphics`ParametricPlot3D`\"]", "Input", AspectRatioFixed->True], Cell[TextData[ "SphericalPlot3D[pin[\[Theta],\[Phi]],\n \t\ {\[Theta],0,\[Pi]},{\[Phi],0,2\[Pi]},PlotPoints->17]"], "Input", AspectRatioFixed->True], Cell[TextData[ "The \"equator\" is exactly a square, as you could easily show."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Your job is make a Laplace Series for this ", StyleBox["pin[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " function, and plot the result using terms at least up to l=4. But I'm \ going to help a bit. First fill in the blanks in the following \ observations:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "1. Because of the up-down symmetry, pin[", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], "] = pin[", StyleBox["\[Pi]", FontFamily->"Symbol"], "-", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], "], only .... values of l contribute." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "2. Because of the 4-fold symmetry in ", StyleBox["\[Phi]", FontFamily->"Symbol"], ", only m values that are multiples of .... (including 0) contribute." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "3. Because of the symmetry pin[", StyleBox["\[Theta]", FontFamily->"Symbol"], ",", StyleBox["\[Phi]", FontFamily->"Symbol"], "] = pin[", StyleBox["\[Theta]", FontFamily->"Symbol"], ",-", StyleBox["\[Phi]", FontFamily->"Symbol"], "] and the above symmetries, all coefficients are ", StyleBox["real", FontSlant->"Italic"], ", and a", StyleBox["l,-m", FontVariations->{"CompatibilityType"->"Subscript"}], " = a", StyleBox["l,m", FontVariations->{"CompatibilityType"->"Subscript"}], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "To do the integrals to find the non-zero coefficients, use the following \ outline:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "... =\nModule[{\[Theta], \[Phi]},\n\t16 * Re[NIntegrate[\n\t\tEvaluate[...],\ \n\t\t{\[Phi],0,\[Pi]/4}, {\[Theta],0,\[Pi]/2},\n\t\tAccuracyGoal->6\n\t]]\n\ ]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "where you fill in the ...'s. I suggest the save-the-results trick (e.g. ", StyleBox["f[n_] := f[n] = ", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "...). The ", StyleBox["AccuracyGoal->6", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " quickly gets us sufficient accuracy . The limits are set up to integrate \ over 1/16 of the whole, and we multiply by 16 outside (this is OK for the m \ values we need, as in point 2 above, but not in general). The ", StyleBox["Re[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " takes the real part of the result; the imaginary part should be zero if \ we integrated ", StyleBox["{\[Phi],0,\[Pi]/2}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ", but even then might have a tiny non-zero value from numerical \ inaccuracies. The way I've set it up as an explicit double-integral is ", StyleBox["much", FontSlant->"Italic"], " faster than the nested ", StyleBox["NIntegrate[NIntegrate[...],...]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " approach." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "As a check, you should find that a", StyleBox["44", FontVariations->{"CompatibilityType"->"Subscript"}], " is about 0.0776." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "I calculated all the non-zero coefficients up to l=4 in about 10 seconds, \ and all those up to l=8 in under a minute."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Once you've got the coefficients you need, plot the resulting fit, using ", StyleBox["SphericalPlot3D[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " as above (including the ", StyleBox["PlotPoints->17", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "). 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