(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 220563, 4487]*) (*NotebookOutlinePosition[ 221235, 4510]*) (* CellTagsIndexPosition[ 221191, 4506]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Physics 631 Quantum Infinite Square Well", "Subtitle"], Cell[BoxData[ \(To\ use\ this\ notebook, \ read\ and\ execute\ each\ command\ using\ shift - return . \ Play\ with\ the\ entries\ by\ varying\ constants, \ functions, \ etc . \ The\ notebook\ is\ designed\ to\ easily\ play\ with\ a\ few\ \ things\ by\ modifying\ their\ definitions\ at\ the\ very\ top, \ then\ re - executing\ everything\ starting\ with\ \(\(ClearAll\)\(.\)\)\)], "Text",\ CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell["One Particle in an Infinite Square Well", "Subsection"], Cell[BoxData[{ RowBox[{ StyleBox[\(In\ this\ notebook\ we\ will\ explore\ a\ particle\ in\ an\ \ infinite\ square\ well . \ \ The\ potential\ is\ infinite\ for\ x < 0\ and\ for\ x > a . \ The\ eigenstates\ of\ the\ Hamiltonian\ vanish\ outside\ the\ \ box\), FontFamily->"Times"], StyleBox[",", FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], StyleBox[\(in\ the\ box\ \ \(satisfy : \[IndentingNewLine]\ \[IndentingNewLine]\ \ \(-\(1\/\(2\ m\)\)\)\ \ \(d\^2\/dx\^2\) \ \(\[CurlyPhi]\_n\) \((x)\)\) = \[Omega]\_n\ \(\[CurlyPhi]\_n\) \((x)\)\), FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], StyleBox[",", FontFamily->"Times"], StyleBox["\[IndentingNewLine]", FontFamily->"Times"], StyleBox["\[IndentingNewLine]", FontFamily->"Times"], RowBox[{ RowBox[{ StyleBox["and", FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], StyleBox["vanish", FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], StyleBox["at", FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], StyleBox["the", FontFamily->"Times"], StyleBox[" ", FontFamily->"Times"], RowBox[{ StyleBox["boundaries", FontFamily->"Times"], StyleBox[":", FontFamily->"Times"], StyleBox["\[IndentingNewLine]", FontFamily->"Times"], StyleBox["\[IndentingNewLine]", FontFamily->"Times"], \(\(\[CurlyPhi]\_n\) \((0)\)\)}]}], "=", \(\(\[CurlyPhi]\_n\) \((a)\)\ . \[IndentingNewLine]\ \[IndentingNewLine]Mathematica\ can\ solve\ these\ equations\ \ analytically\)}], ",", " ", \(but\ the\ focus\ is\ on\ using\ the\ solutions\ to\ solve\ \ simple\ quantum\ mechanics\ problems . \ The\ solutions\ \(are : \[IndentingNewLine]\[IndentingNewLine]\ \ \(\[CurlyPhi]\_n\) \((x)\)\)\ = \ \@\(2\/a\)\ sin \((\(k\_n\) x)\); \ \ \ \ k\_n\ = \ \(n\ \[Pi]\)\/a; \ n = 1\), ",", "2", ",", "3", ",", \(\(...\)\(;\)\)}], "\[IndentingNewLine]", RowBox[{ "\t ", \(\[Omega]\_n = \(\(k\_n\^2\/\(2\ \ m\)\)\(.\)\(\[IndentingNewLine]\)\)\)}]}], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(ClearAll["\"]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m = 0.5\)], "Input"], Cell[BoxData[ \(0.5`\)], "Output"] }, Open ]], Cell["\<\ This selects the unit of length if we let \[HBar] and c equal 1. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a = 100\)], "Input"], Cell[BoxData[ \(100\)], "Output"] }, Open ]], Cell["This sets the box size.", "Text"], Cell[BoxData[ \(\[CurlyPhi][n_, x_] := \ \@\(2\/a\)\ Sin[\(\(n\ \[Pi]\)\/a\) x]\)], "Input"], Cell["Now all of the eigenstates are defined. 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ooooo`04o`000?ooooooooooo`0000Sooooo00Ko0000oooooooooooo0000ooooool00003ooooo`03 o`000?oooooooooo00;ooooo00?o0000ooooooooool0o?ooool000Sooooo0_l0000:ooooo`;o0000 0oooool3o`0000Gooooo00?o0000ooooooooool0o?ooool00?oooooo8Oooool00?oooooo8Oooool0 0?oooooo8Oooool00?oooooo8Oooool00?oooooo8Oooool00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-13.5686, -0.00220532, \ 0.408522, 0.0001322}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "Let's look at the eigenstate components of some functions. If the function \ is normalized and entirely in the box, vanishing at the edge, we can expand \ it using the energy eigenstates as our basis. Each component is given by ", Cell[BoxData[ \(TraditionalForm\`c\_n\)]], " = <", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_n\)]], "|\[Psi]>, so let's pick a function and look at the first few components." }], "Text"], Cell[BoxData[ \(f[x_] := \(\@\(630\/a\^9\)\) x^2*\((a - x)\)^2\)], "Input"], Cell["\<\ This is a sample function that could be used for a wave function at \ t=0. Check that this is normalized. The scalar products we need are:\ \>", \ "Text"], Cell[BoxData[ \(c[n_] := Integrate[\[CurlyPhi][n, x]\ f[x], {x, 0, a}]\)], "Input"], Cell[TextData[{ "If this does not work because ", StyleBox["Mathematica", FontSlant->"Italic"], " cannot do the integral analytically, just use. USE ONLY ONE DEFINITION \ FOR c[n_] !" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c[1]; c[2]; c[3]; c[4]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[ \(c[n_] := NIntegrate[\[CurlyPhi][n, x]\ f[x], {x, 0, a}]\)], "Input"], Cell[TextData[{ "Compute a few of these and check that \[Psi]=", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] c\_n\)]], Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_n\)]], " should work by using a few terms in the infinite sum as an approximation. \ Change the function and find one that cannot be approximated by the first few \ terms." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[c[1]]\)], "Input"], Cell[BoxData[ \(0.9884533022688091`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[c[2]]\)], "Input"], Cell[BoxData[ \(\(-3.469446951953614`*^-17\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[c[3]]\)], "Input"], Cell[BoxData[ \(\(-0.14668994097899674`\)\)], "Output"] }, Open 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