(************** 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). 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[ 286771, 12268]*) (*NotebookOutlinePosition[ 287476, 12292]*) (* CellTagsIndexPosition[ 287432, 12288]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Visualizing Particle-in-a-Box Wavefunctions ", "Title", TextAlignment->Center], Cell["\<\ By Edmund L. Tisko Department of Chemistry University of Nebraska at Omaha Omaha, NE 68182 etisko@unomaha.edu Translated and modified from the MathCAD document by Alex Grushow 2004\ \>", \ "Text", TextAlignment->Center], Cell["\<\ \[Copyright] Copyright 2003, 2005 by the Division of Chemical \ Education, Inc., American Chemical Society. All rights reserved. For \ classroom use by teachers, one copy per student in the class may be made free \ of charge. Write to JCE Online, jceonline@chem.wisc.edu, for permission to \ place a document, free of charge, on a class Intranet. \ \>", "Text"], Cell[CellGroupData[{ Cell["Prerequisites", "Subsection"], Cell[TextData[{ "This document was constructed for use in a physical chemistry \ computational modeling laboratory using ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.1. To use the worksheet, students need a minimal amount of experience \ with the program. Instructions have been written as to allow the student to \ construct the worksheet keystroke by keystroke. Students should have a \ rudimentary understanding of the particle-in-a-box problem as covered in \ texts such as:\n Physical Chemistry, 3rd ed., Keith J. Laidler and \ John H. Meiser, Houghton Mifflin Co., (1999).\n Physical Chemistry, \ 6th ed., Peter W. Atkins, W. H. Freeman and Co., (1998).\n Physical \ Chemistry, 3rd ed., Robert J. Silbey and Robert A. Alberty, John Wiley and \ Sons, (2001).\t \n Physical Chemistry, 5th ed., Ira \ N. Levine, McGraw-Hill, (2002)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Learning Goals", "Subsection"], Cell["\<\ \t1) Seeing a variety of particle-in-a-box wavefunctions with \ different potential functions. \t2) Seeing how the application of boundary conditions forces the energy \ levels of the \t particle-in-a-box to be quantized. \t3) Verifying the relationship between quantum number and energy for a \ simple particle-in-a-box. \t4) Verifying the relationship between the length of the box and energy for \ a simple particle-in-a-box. \t5) Seeing how the requirement for a finite wavefunction forces tunneling \ into a potential barrier. \t6) Seeing how increasing the potential energy decreases the kinetic energy \ and thus decreases \t\t \t the curvature of the wavefunction. \t7) Gaining an intuitive feel for an iterative numerical process.\ \>", \ "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Learning Objectives", "Subsection"], Cell["\<\ At the end of this exercise you should be able to: \t1) Compare and contrast the wavefunctions for a particle in a box when V=0 \ throughout the box to the situations where there is a step potential in the box \ or a barrier potential in the box. \t2) Explain how the boundary conditions forces the energy levels of the \ particle-in-a-box to be quantized. \t3) Verify the relationship between quantum number and energy for a \ simple-particle-in-a-box. \t4) Verify the relationship between the length of the box and energy for a \ simple particle-in-a-box. \t5) Describe tunneling with respect to the barrier and step potential in a \ box. Explain how the wavefunctions behave under these conditions. \t6) Correlate curvature of a wavefunction with the energy associated with \ the wavefunction and the potential and kinetic energy components of the total \ energy.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "The particle-in-a-box is usually the first or second potential energy \ function considered when quantum chemistry students begin solving the Schr\ \[ODoubleDot]dinger equation. The potential function consists of putting a \ particle with no potential energy between two walls that have infinite \ potential energy. A general solution or wavefunction for the particle inside \ the box is\n", Cell[BoxData[ \(\[CapitalPsi][x] = A\ Sin[\(\@\(\(8 \( \[Pi]\^2\) m\ E\)\/h\^2\)\) x] + B\ Cos[\(\@\(\(8 \( \[Pi]\^2\) m\ E\)\/h\^2\)\) x]\)]], "\nwhere n is the quantum number, m is the particle mass and h is Planck's \ constant.\n\nThis general wavefunction is the same wavefunction as for a \ particle-in-free-space, that is, a particle no potential energy anywhere. \ However, the particle-in-a-box wavefunction becomes different than the \ particle-in-free-space wavefunction when the boundary requirements of the \ wavefunction are forced upon it.\n\nThe wavefunction must thus meet the \ following requirements if the wavefunction is to contain meaningful \ information.\n\t1. The wavefunction must be finite everywhere.\n\t2. The \ first derivative of the wavefunction must be finite everywhere.\n\t3. The \ wavefunction must be continuous everywhere.\n\t4. The first derivative of \ the wavefunction must be continuous everywhere.\n\t5. The wavefunction must \ be normalizable. (The integral of the square of the wavefunction over all \ space must be finite.)\n\t\nThe most relevant requirements for our purposes \ are that the wavefunction must be finite everywhere and that the wavefunction \ must be continuous everywhere. These requirements, sometimes referred to as \ boundary conditions, force the value of the wavefunction at the walls of the \ box to be zero, that is\n\[CapitalPsi][0]=0 and \[CapitalPsi][L]=0, where \ L is the length of the box.\nWhen these conditions are applied, the \ wavefunction becomes\n", Cell[BoxData[ \(\[CapitalPsi][x] = A\ Sin[\(\@\(\(8 \( \[Pi]\^2\) m\ E\)\/h\^2\)\) x]\)]], " \n\nand ", Cell[BoxData[ \(\(\(\ \ \)\(E = \(\(n\^2\) h\^2\)\/\(8\ m\ a\^2\)\)\)\)]], "\n\nThus the energy, E, becomes quantized. Only specific values of E will \ yield a wavefunction that satisfies the \nboundary conditions.\n\nLet us \ consider the energy for a particle-in-a-box. \nThe Hamiltonian for the \ particle inside the box is the kinetic energy operator.\n", Cell[BoxData[ \(\(\[ScriptCapitalH]\&^\) = \(\(-\[HBar]\^2\)\/\(2\ m\)\) d\^2\/dx\^2\)]], "\nApplying the Hamiltonian within the Schr\[ODoubleDot]dinger equation \ implies that we find the energy for the particle-in-a-box by taking the \ second derivative of the wavefunction and multiplying by some constants.\n\n\ Recall from calculus that the second derivative is a measure of the curvature \ of a function. When the wavefunction has the correct energy, i.e., the \ correct curvature, the boundary conditions are met. Thus looking at the \ curvature of a particle-in-a-box wavefunction can help in finding correct \ energy eigenvalues.\n\nAlso, since the kinetic energy operator is \ proportional to the second derivative, the curvature of a particle's \ wavefunction is a measure of the particle's kinetic energy.\n\nIn this \ laboratory, we will be examining the effects of the potential energy function \ on the energies and wavefunctions of a particle trapped in a box. For each \ investigation, four steps will be followed.\n\t1. We will learn how to \ construct the potential function.\n\t2. We will construct a simple algorithm \ that will allow us to use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s differential equation\n \t solvers to solve the Schr\ \[ODoubleDot]dinger equation.\n\t3. We will learn how to plot the solution \ of the Schr\[ODoubleDot]dinger equation so that we can see the results.\n\t4. \ We will vary the energy of the particle so that solution meets the correct \ boundary conditions." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Laboratory Procedure", "Subsection"], Cell[CellGroupData[{ Cell["Construction of the Potential energy", "Subsubsection"], Cell[TextData[{ "Construction of the potential function is done using the Which function of \ ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell[BoxData[ \(V[x_] := Which[x < 0, 400, x > 4, 400, True, 0]\)], "Input"], Cell["\<\ This function will generate a value of 400 when x is greater than 4 \ and less than 0. Otherwise V[x]=0. \ \>", "Text"], Cell[TextData[{ "Below we have plotted the potential function. Note that the \"walls\" of \ the function are not infinite because ", StyleBox["Mathematica", FontSlant->"Italic"], " will not evaluate this value in a function. 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.23497 .01212 L .23516 .01212 L .23534 .01212 L .23554 .01212 L .23575 .01212 L .23593 .01212 L .23613 .01212 L .23634 .01212 L .23646 .01212 L .23656 .01212 L .23677 .01212 L Mistroke .23697 .01212 L .23714 .01212 L .23733 .01212 L .23753 .01212 L .23771 .01212 L .23788 .01212 L .23803 .01212 L .23821 .01212 L .23839 .01212 L .23857 .01212 L .23877 .01212 L .23895 .01212 L .23912 .01212 L .23932 .01212 L .23943 .01212 L .23953 .01212 L .23972 .01212 L .23992 .01212 L .24012 .01212 L .24033 .01212 L .24053 .01212 L .24072 .01212 L .2409 .01212 L .24107 .01212 L .24126 .01212 L .24144 .01212 L .24164 .01212 L .24185 .01212 L .24203 .01212 L .24223 .01212 L .24244 .01212 L .24256 .01212 L .24267 .01212 L .24288 .01212 L .24308 .01212 L .24325 .01212 L .24344 .01212 L .24363 .01212 L .24381 .01212 L .24399 .01212 L .24414 .01212 L .24432 .01212 L .24449 .01212 L .24468 .01212 L .24487 .01212 L .24506 .01212 L .24523 .01212 L .24543 .01212 L .24554 .01212 L .24564 .01212 L Mistroke .24583 .01212 L .24603 .01212 L .24623 .01212 L .24634 .01212 L .24644 .01212 L .24663 .01212 L .24683 .01212 L .24701 .01212 L .24717 .01212 L .24736 .01212 L .24756 .01212 L .24775 .01212 L .24787 .01212 L .24797 .01212 L .24818 .01212 L .2483 .01212 L .24841 .01212 L .2486 .01212 L .24881 .01212 L .24901 .01212 L .24919 .01212 L .24936 .01212 L .24955 .01212 L .24972 .01212 L .24988 .01212 L .25006 .01212 L .25026 .01212 L .25045 .01212 L .25063 .01212 L .25083 .01212 L .25101 .01212 L .25118 .01212 L .25136 .01212 L .25156 .01212 L .25177 .01212 L .25195 .01212 L .25215 .01212 L .25236 .01212 L .25255 .01212 L .25265 .01212 L .25276 .01212 L .25288 .01212 L .25299 .01212 L .25319 .01212 L .25338 .01212 L .25358 .01212 L .25379 .01212 L .25397 .01212 L .25416 .01212 L .25435 .01212 L Mistroke .25451 .01212 L .2547 .01212 L .25491 .01212 L .25511 .01212 L .25529 .01212 L .25549 .01212 L .2557 .01212 L .25589 .01212 L .25607 .01212 L .25628 .01212 L .2564 .01212 L .25651 .01212 L .25672 .01212 L .25682 .01212 L .25693 .01212 L .25704 .01212 L .25716 .01212 L .25736 .01212 L .25748 .01212 L .25758 .01212 L .2577 .01212 L .25782 .01212 L .25793 .01212 L .25805 .01212 L .25816 .01212 L .25826 .01212 L .25846 .01212 L .25866 .01212 L .25886 .01212 L .25907 .01212 L .25927 .01212 L .25946 .01212 L .25964 .01212 L .25981 .01212 L .26 .01212 L .26018 .01212 L .26038 .01212 L .26059 .01212 L .26077 .01212 L .26097 .01212 L .26118 .01212 L .2613 .01212 L .2614 .01212 L .26161 .01212 L .26181 .01212 L .26198 .01212 L .26217 .01212 L .26236 .01212 L .26255 .01212 L .26272 .01212 L Mistroke .26287 .01212 L .26304 .01212 L .26323 .01212 L .26341 .01212 L .26361 .01212 L .26379 .01212 L .26396 .01212 L .26416 .01212 L .26427 .01212 L .26437 .01212 L .26456 .01212 L .26476 .01212 L .26496 .01212 L .26517 .01212 L .26537 .01212 L .26556 .01212 L .26574 .01212 L .26591 .01212 L .2661 .01212 L .26628 .01212 L .26648 .01212 L .26669 .01212 L .26687 .01212 L .26707 .01212 L .26728 .01212 L .2674 .01212 L .26751 .01212 L .26772 .01212 L .26792 .01212 L .26809 .01212 L .26827 .01212 L .26847 .01212 L .26865 .01212 L .26882 .01212 L .26898 .01212 L .26916 .01212 L .26933 .01212 L .26951 .01212 L .26971 .01212 L .2699 .01212 L .27006 .01212 L .27026 .01212 L .27037 .01212 L .27048 .01212 L .27067 .01212 L .27087 .01212 L .27106 .01212 L .27117 .01212 L .27128 .01212 L .27146 .01212 L Mistroke .27166 .01212 L .27184 .01212 L .27201 .01212 L .27219 .01212 L .27239 .01212 L .27259 .01212 L .2727 .01212 L .27281 .01212 L .27302 .01212 L .27314 .01212 L .27324 .01212 L .27344 .01212 L .27365 .01212 L .27384 .01212 L .27403 .01212 L .2742 .01212 L .27438 .01212 L .27455 .01212 L .27471 .01212 L .27489 .01212 L .27509 .01212 L .27528 .01212 L .27546 .01212 L .27566 .01212 L .27585 .01212 L .27601 .01212 L .2762 .01212 L .27639 .01212 L .2766 .01212 L .27678 .01212 L .27698 .01212 L .27719 .01212 L .27738 .01212 L .27749 .01212 L .2776 .01212 L .27772 .01212 L .27783 .01212 L .27803 .01212 L .27821 .01212 L .27841 .01212 L .27863 .01212 L .27881 .01212 L .279 .01212 L .27918 .01212 L .27934 .01212 L .27954 .01212 L .27975 .01212 L .27994 .01212 L .28013 .01212 L .28032 .01212 L Mistroke .28053 .01212 L .28073 .01212 L .28091 .01212 L .28112 .01212 L .28124 .01212 L .28135 .01212 L .28155 .01212 L .28165 .01212 L .28176 .01212 L .28187 .01212 L .28199 .01212 L .28219 .01212 L .28231 .01212 L .28241 .01212 L .28253 .01212 L .28266 .01212 L .28277 .01212 L .28288 .01212 L .28299 .01212 L .2831 .01212 L .28329 .01212 L .2835 .01212 L .28369 .01212 L .2839 .01212 L .2841 .01212 L .28429 .01212 L .28447 .01212 L .28464 .01212 L .28483 .01212 L .28501 .01212 L .28521 .01212 L .28542 .01212 L .2856 .01212 L .2858 .01212 L .28601 .01212 L .28613 .01212 L .28623 .01212 L .28644 .01212 L .28664 .01212 L .28681 .01212 L .287 .01212 L .28719 .01212 L .28738 .01212 L .28755 .01212 L .2877 .01212 L .28787 .01212 L .28806 .01212 L .28824 .01212 L .28844 .01212 L .28862 .01212 L Mistroke .28879 .01212 L .28899 .01212 L .2891 .01212 L .2892 .01212 L .28939 .01212 L .28959 .01212 L .28979 .01212 L .29 .01212 L .2902 .01212 L .29039 .01212 L .29057 .01212 L .29074 .01212 L .29093 .01212 L .29111 .01212 L .29131 .01212 L .29152 .01212 L .2917 .01212 L .2919 .01212 L .29211 .01212 L .29223 .01212 L .29234 .01212 L .29255 .01212 L .29274 .01212 L .29291 .01212 L .2931 .01212 L .2933 .01212 L .29348 .01212 L .29365 .01212 L .29381 .01212 L .29398 .01212 L .29416 .01212 L .29434 .01212 L .29454 .01212 L .29472 .01212 L .29489 .01212 L .29509 .01212 L .2952 .01212 L .29531 .01212 L .29549 .01212 L .29569 .01212 L .29589 .01212 L .296 .01212 L .2961 .01212 L .29629 .01212 L .29649 .01212 L .29667 .01212 L .29684 .01212 L .29702 .01212 L .29722 .01212 L .29742 .01212 L Mistroke .29753 .01212 L .29763 .01212 L .29785 .01212 L .29796 .01212 L .29807 .01212 L .29826 .01212 L .29847 .01212 L .29867 .01212 L .29885 .01212 L .29902 .01212 L .29921 .01212 L .29938 .01212 L .29954 .01212 L .29972 .01212 L .29992 .01212 L .30011 .01212 L .30029 .01212 L .30049 .01212 L .30067 .01212 L .30084 .01212 L .30102 .01212 L .30122 .01212 L .30143 .01212 L .30161 .01212 L .30181 .01212 L .30202 .01212 L .30221 .01212 L .30231 .01212 L .30242 .01212 L .30254 .01212 L .30265 .01212 L .30285 .01212 L .30304 .01212 L .30324 .01212 L .30345 .01212 L .30363 .01212 L .30382 .01212 L .304 .01212 L .30417 .01212 L .30436 .01212 L .30457 .01212 L .30477 .01212 L .30495 .01212 L .30515 .01212 L .30535 .01212 L .30555 .01212 L .30573 .01212 L .30594 .01212 L .30606 .01212 L .30617 .01212 L Mistroke .30637 .01212 L .30649 .01212 L .30659 .01212 L .3067 .01212 L .30681 .01212 L .30702 .01212 L .30713 .01212 L .30724 .01212 L .30736 .01212 L .30748 .01212 L .30759 .01212 L .30771 .01212 L .30782 .01212 L .30792 .01212 L .30811 .01212 L .30832 .01212 L .30851 .01212 L .30873 .01212 L .30893 .01212 L .30911 .01212 L .3093 .01212 L .30946 .01212 L .30965 .01212 L .30984 .01212 L .31003 .01212 L .31024 .01212 L .31042 .01212 L .31062 .01212 L .31083 .01212 L .31095 .01212 L .31106 .01212 L .31127 .01212 L .31147 .01212 L .31164 .01212 L .31182 .01212 L .31202 .01212 L .3122 .01212 L .31237 .01212 L .31253 .01212 L .3127 .01212 L .31288 .01212 L .31306 .01212 L .31326 .01212 L .31344 .01212 L .31361 .01212 L .31381 .01212 L .31392 .01212 L .31402 .01212 L .31421 .01212 L .31441 .01212 L Mistroke .31461 .01212 L .31482 .01212 L .31502 .01212 L .31521 .01212 L .31539 .01212 L .31556 .01212 L .31575 .01212 L .31593 .01212 L .31613 .01212 L .31634 .01212 L .31652 .01212 L .31672 .01212 L .31693 .01212 L .31705 .01212 L .31716 .01212 L .31737 .01212 L .31757 .01212 L .31773 .01212 L .31792 .01212 L .31812 .01212 L .3183 .01212 L .31847 .01212 L .31863 .01212 L .3188 .01212 L .31898 .01212 L .31916 .01212 L .31936 .01212 L .31954 .01212 L .31971 .01212 L .31991 .01212 L .32002 .01212 L .32013 .01212 L .32031 .01212 L .32051 .01212 L .32071 .01212 L .32082 .01212 L .32092 .01212 L .32111 .01212 L .32131 .01212 L .32149 .01212 L .32166 .01212 L .32184 .01212 L .32204 .01212 L .32224 .01212 L .32235 .01212 L .32245 .01212 L .32266 .01212 L .32278 .01212 L .32289 .01212 L .32308 .01212 L Mistroke .32329 .01212 L .32349 .01212 L .32367 .01212 L .32384 .01212 L .32402 .01212 L .3242 .01212 L .32435 .01212 L .32454 .01212 L .32474 .01212 L .32493 .01212 L .32511 .01212 L .3253 .01212 L .32549 .01212 L .32566 .01212 L .32584 .01212 L .32604 .01212 L .32625 .01212 L .32643 .01212 L .32662 .01212 L .32683 .01212 L .32703 .01212 L .32713 .01212 L .32724 .01212 L .32736 .01212 L .32747 .01212 L .32767 .01212 L .32785 .01212 L .32806 .01212 L .32827 .01212 L .32845 .01212 L .32864 .01212 L .32882 .01212 L .32898 .01212 L .32918 .01212 L .32939 .01212 L .32959 .01212 L .32977 .01212 L .32996 .01212 L .33017 .01212 L .33037 .01212 L .33055 .01212 L .33076 .01212 L .33088 .01212 L .33099 .01212 L .33119 .01212 L .3313 .01212 L .3314 .01212 L .33151 .01212 L .33163 .01212 L .33183 .01212 L Mistroke .33195 .01212 L .33206 .01212 L .33217 .01212 L .3323 .01212 L .33241 .01212 L .33252 .01212 L .33263 .01212 L .33274 .01212 L .33293 .01212 L .33314 .01212 L .33333 .01212 L .33354 .01212 L .33374 .01212 L .33393 .01212 L .33411 .01212 L .33428 .01212 L .33447 .01212 L .33465 .01212 L .33485 .01212 L .33506 .01212 L .33524 .01212 L .33543 .01212 L .33565 .01212 L .33576 .01212 L .33587 .01212 L .33608 .01212 L .33628 .01212 L .33645 .01212 L .33664 .01212 L .33683 .01212 L .33701 .01212 L .33718 .01212 L .33734 .01212 L .33751 .01212 L .33769 .01212 L .33788 .01212 L .33807 .01212 L .33826 .01212 L .33843 .01212 L .33862 .01212 L .33874 .01212 L .33884 .01212 L .33903 .01212 L .33923 .01212 L .33942 .01212 L .33963 .01212 L .33984 .01212 L .34002 .01212 L .34021 .01212 L .34037 .01212 L Mistroke .34056 .01212 L .34075 .01212 L .34094 .01212 L .34115 .01212 L .34133 .01212 L .34153 .01212 L .34174 .01212 L .34186 .01212 L .34197 .01212 L .34218 .01212 L .34238 .01212 L .34255 .01212 L .34273 .01212 L .34293 .01212 L .34311 .01212 L .34328 .01212 L .34344 .01212 L .34361 .01212 L .34379 .01212 L .34398 .01212 L .34417 .01212 L .34436 .01212 L .34453 .01212 L .34472 .01212 L .34483 .01212 L .34494 .01212 L .34513 .01212 L .34533 .01212 L .34552 .01212 L .34574 .01212 L .34594 .01212 L .34612 .01212 L .34631 .01212 L .34647 .01212 L .34667 .01212 L .34685 .01212 L .34705 .01212 L .34726 .01212 L .34744 .01212 L .34764 .01212 L .34785 .01212 L .34797 .01212 L .34807 .01212 L .34828 .01212 L .34848 .01212 L .34865 .01212 L .34884 .01212 L .34904 .01212 L .34922 .01212 L .34939 .01212 L Mistroke .34955 .01212 L .34973 .01212 L .3499 .01212 L .35008 .01212 L .35028 .01212 L .35046 .01212 L .35063 .01212 L .35083 .01212 L .35094 .01212 L .35105 .01212 L .35123 .01212 L .35143 .01212 L .35163 .01212 L .35174 .01212 L .35184 .01212 L .35203 .01212 L .35223 .01212 L .35241 .01212 L .35258 .01212 L .35276 .01212 L .35296 .01212 L .35316 .01212 L .35327 .01212 L .35338 .01212 L .35359 .01212 L .35371 .01212 L .35381 .01212 L .35401 .01212 L .35422 .01212 L .35442 .01212 L .3546 .01212 L .35477 .01212 L .35495 .01212 L .35512 .01212 L .35528 .01212 L .35546 .01212 L .35567 .01212 L .35586 .01212 L .35603 .01212 L .35623 .01212 L .35642 .01212 L .35659 .01212 L .35677 .01212 L .35697 .01212 L .35718 .01212 L .35735 .01212 L .35755 .01212 L .35776 .01212 L .35796 .01212 L .35806 .01212 L Mistroke .35817 .01212 L .35829 .01212 L .3584 .01212 L .3586 .01212 L .35878 .01212 L .35899 .01212 L .3592 .01212 L .35938 .01212 L .35957 .01212 L .35975 .01212 L .35992 .01212 L .36011 .01212 L .36032 .01212 L .36052 .01212 L .3607 .01212 L .36089 .01212 L .3611 .01212 L .3613 .01212 L .36148 .01212 L .36169 .01212 L .36181 .01212 L .36192 .01212 L .36212 .01212 L .36222 .01212 L .36234 .01212 L .36245 .01212 L .36256 .01212 L .36277 .01212 L .36288 .01212 L .36299 .01212 L .36311 .01212 L .36323 .01212 L .36334 .01212 L .36346 .01212 L .36357 .01212 L .36367 .01212 L .36386 .01212 L .36407 .01212 L .36426 .01212 L .36448 .01212 L .36468 .01212 L .36486 .01212 L .36505 .01212 L .36521 .01212 L .36541 .01212 L .36559 .01212 L .36579 .01212 L .366 .01212 L .36617 .01212 L .36637 .01212 L Mistroke .36658 .01212 L .3667 .01212 L .36681 .01212 L .36702 .01212 L .36722 .01212 L .36739 .01212 L .36757 .01212 L .36777 .01212 L .36795 .01212 L .36812 .01212 L .36828 .01212 L .36845 .01212 L .36863 .01212 L .36882 .01212 L .36901 .01212 L .3692 .01212 L .36936 .01212 L .36956 .01212 L .36967 .01212 L .36978 .01212 L .36997 .01212 L .37017 .01212 L .37036 .01212 L .37058 .01212 L .37078 .01212 L .37096 .01212 L .37115 .01212 L .37131 .01212 L .37151 .01212 L .37169 .01212 L .37189 .01212 L .3721 .01212 L .37228 .01212 L .37247 .01212 L .37269 .01212 L .3728 .01212 L .37291 .01212 L .37312 .01212 L .37332 .01212 L .37349 .01212 L .37368 .01212 L .37387 .01212 L .37406 .01212 L .37423 .01212 L .37438 .01212 L .37457 .01212 L .37474 .01212 L .37492 .01212 L .37512 .01212 L .3753 .01212 L Mistroke .37547 .01212 L .37567 .01212 L .37578 .01212 L .37588 .01212 L .37607 .01212 L .37627 .01212 L .37647 .01212 L .37658 .01212 L .37668 .01212 L .37687 .01212 L .37707 .01212 L .37725 .01212 L .37742 .01212 L .3776 .01212 L .3778 .01212 L .378 .01212 L .37811 .01212 L .37821 .01212 L .37843 .01212 L .37854 .01212 L .37865 .01212 L .37884 .01212 L .37905 .01212 L .37925 .01212 L .37943 .01212 L .3796 .01212 L .37979 .01212 L .37996 .01212 L .38012 .01212 L .3803 .01212 L .3805 .01212 L .38069 .01212 L .38087 .01212 L .38107 .01212 L .38125 .01212 L .38142 .01212 L .3816 .01212 L .3818 .01212 L .38201 .01212 L .38219 .01212 L .38239 .01212 L .3826 .01212 L .38279 .01212 L .38289 .01212 L .38301 .01212 L .38312 .01212 L .38323 .01212 L .38343 .01212 L .38362 .01212 L .38382 .01212 L Mistroke .38403 .01212 L .38421 .01212 L .38441 .01212 L .38459 .01212 L .38475 .01212 L .38494 .01212 L .38515 .01212 L .38535 .01212 L .38553 .01212 L .38573 .01212 L .38594 .01212 L .38613 .01212 L .38631 .01212 L .38652 .01212 L .38664 .01212 L .38675 .01212 L .38696 .01212 L .38706 .01212 L .38717 .01212 L .38728 .01212 L .3874 .01212 L .3876 .01212 L .38772 .01212 L .38782 .01212 L .38794 .01212 L .38806 .01212 L .38817 .01212 L .38829 .01212 L .3884 .01212 L .3885 .01212 L .3887 .01212 L .3889 .01212 L .3891 .01212 L .38931 .01212 L .38951 .01212 L .3897 .01212 L .38988 .01212 L .39005 .01212 L .39024 .01212 L .39042 .01212 L .39062 .01212 L .39083 .01212 L .39101 .01212 L .3912 .01212 L .39142 .01212 L .39153 .01212 L .39164 .01212 L .39185 .01212 L .39205 .01212 L .39222 .01212 L Mistroke .39241 .01212 L .3926 .01212 L .39278 .01212 L .39296 .01212 L .39311 .01212 L .39328 .01212 L .39346 .01212 L .39365 .01212 L .39384 .01212 L .39403 .01212 L .3942 .01212 L .3944 .01212 L .39451 .01212 L .39461 .01212 L .3948 .01212 L .395 .01212 L .39519 .01212 L .39541 .01212 L .39561 .01212 L .39579 .01212 L .39598 .01212 L .39615 .01212 L .39634 .01212 L .39652 .01212 L .39672 .01212 L .39693 .01212 L .39711 .01212 L .3973 .01212 L .39752 .01212 L .39764 .01212 L .39774 .01212 L .39795 .01212 L .39815 .01212 L .39832 .01212 L .39851 .01212 L .3987 .01212 L .39889 .01212 L .39906 .01212 L .39921 .01212 L .39939 .01212 L .39957 .01212 L .39975 .01212 L .39995 .01212 L .40013 .01212 L .4003 .01212 L .4005 .01212 L .40061 .01212 L .40071 .01212 L .4009 .01212 L .4011 .01212 L Mistroke .4013 .01212 L .40141 .01212 L .40151 .01212 L .4017 .01212 L .4019 .01212 L .40208 .01212 L .40225 .01212 L .40243 .01212 L .40263 .01212 L .40283 .01212 L .40294 .01212 L .40304 .01212 L .40325 .01212 L .40337 .01212 L .40348 .01212 L .40367 .01212 L .40388 .01212 L .40408 .01212 L .40426 .01212 L .40443 .01212 L .40462 .01212 L .40479 .01212 L .40495 .01212 L .40513 .01212 L .40533 .01212 L .40552 .01212 L .4057 .01212 L .4059 .01212 L .40608 .01212 L .40625 .01212 L .40643 .01212 L .40663 .01212 L .40684 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.01212 L .41743 .01212 L .41761 .01212 L .41778 .01212 L .41794 .01212 L .41811 .01212 L Mistroke .41829 .01212 L .41847 .01212 L .41867 .01212 L .41885 .01212 L .41902 .01212 L .41922 .01212 L .41933 .01212 L .41943 .01212 L .41962 .01212 L .41982 .01212 L .42002 .01212 L .42023 .01212 L .42043 .01212 L .42062 .01212 L .4208 .01212 L .42097 .01212 L .42116 .01212 L .42134 .01212 L .42154 .01212 L .42175 .01212 L .42193 .01212 L .42213 .01212 L .42234 .01212 L .42246 .01212 L .42257 .01212 L .42278 .01212 L .42298 .01212 L .42315 .01212 L .42333 .01212 L .42353 .01212 L .42371 .01212 L .42388 .01212 L .42404 .01212 L .42421 .01212 L .42439 .01212 L .42457 .01212 L .42477 .01212 L .42496 .01212 L .42512 .01212 L .42532 .01212 L .42543 .01212 L .42554 .01212 L .42573 .01212 L .42593 .01212 L .42612 .01212 L .42623 .01212 L .42633 .01212 L .42652 .01212 L .42672 .01212 L .4269 .01212 L Mistroke .42707 .01212 L .42725 .01212 L .42745 .01212 L .42765 .01212 L .42776 .01212 L .42786 .01212 L .42808 .01212 L .42819 .01212 L .4283 .01212 L .42849 .01212 L .4287 .01212 L .4289 .01212 L .42908 .01212 L .42925 .01212 L .42944 .01212 L .42961 .01212 L .42977 .01212 L .42995 .01212 L .43015 .01212 L .43034 .01212 L .43052 .01212 L .43072 .01212 L .4309 .01212 L .43107 .01212 L .43125 .01212 L .43145 .01212 L .43166 .01212 L .43184 .01212 L .43204 .01212 L .43225 .01212 L .43244 .01212 L .43254 .01212 L .43265 .01212 L .43277 .01212 L .43288 .01212 L .43308 .01212 L .43327 .01212 L .43347 .01212 L .43368 .01212 L .43386 .01212 L .43405 .01212 L .43423 .01212 L .4344 .01212 L .43459 .01212 L .4348 .01212 L .435 .01212 L .43518 .01212 L .43537 .01212 L .43558 .01212 L .43578 .01212 L Mistroke .43596 .01212 L .43617 .01212 L .43629 .01212 L .4364 .01212 L .4366 .01212 L .43671 .01212 L .43682 .01212 L .43693 .01212 L .43704 .01212 L .43725 .01212 L .43736 .01212 L .43747 .01212 L .43758 .01212 L .43771 .01212 L .43782 .01212 L .43793 .01212 L .43805 .01212 L .43815 .01212 L .43834 .01212 L .43855 .01212 L .43874 .01212 L .43895 .01212 L .43915 .01212 L .43934 .01212 L .43952 .01212 L .43969 .01212 L .43988 .01212 L .44006 .01212 L .44026 .01212 L .44047 .01212 L .44065 .01212 L .44085 .01212 L .44106 .01212 L .44118 .01212 L .44128 .01212 L .44149 .01212 L .44169 .01212 L .44186 .01212 L .44205 .01212 L .44224 .01212 L .44243 .01212 L .4426 .01212 L .44275 .01212 L .44292 .01212 L .4431 .01212 L .44329 .01212 L .44348 .01212 L .44367 .01212 L .44384 .01212 L .44404 .01212 L Mistroke .44415 .01212 L .44425 .01212 L .44444 .01212 L .44464 .01212 L .44483 .01212 L .44505 .01212 L .44525 .01212 L .44543 .01212 L .44562 .01212 L .44578 .01212 L .44598 .01212 L .44616 .01212 L .44636 .01212 L .44657 .01212 L .44675 .01212 L .44694 .01212 L .44715 .01212 L .44727 .01212 L .44738 .01212 L .44759 .01212 L .44779 .01212 L .44796 .01212 L .44815 .01212 L .44834 .01212 L .44852 .01212 L .4487 .01212 L .44885 .01212 L .44902 .01212 L .4492 .01212 L .44939 .01212 L .44958 .01212 L .44977 .01212 L .44994 .01212 L .45014 .01212 L .45025 .01212 L .45035 .01212 L .45054 .01212 L .45074 .01212 L .45093 .01212 L .45115 .01212 L .45135 .01212 L .45154 .01212 L .4517 .01212 L .45189 .01212 L .45208 .01212 L .45226 .01212 L .45246 .01212 L .45267 .01212 L .45285 .01212 L .45305 .01212 L Mistroke .45326 .01212 L .45338 .01212 L .45349 .01212 L .4537 .01212 L .4539 .01212 L .45407 .01212 L .45425 .01212 L .45445 .01212 L .45463 .01212 L .4548 .01212 L .45496 .01212 L .45514 .01212 L .45531 .01212 L .4555 .01212 L .45569 .01212 L .45588 .01212 L .45605 .01212 L .45624 .01212 L .45636 .01212 L .45646 .01212 L .45665 .01212 L .45685 .01212 L .45704 .01212 L .45715 .01212 L .45726 .01212 L .45745 .01212 L .45765 .01212 L .45782 .01212 L .45799 .01212 L .45818 .01212 L .45837 .01212 L .45857 .01212 L .45869 .01212 L .45879 .01212 L .459 .01212 L .45912 .01212 L .45923 .01212 L .45942 .01212 L .45963 .01212 L .45983 .01212 L .46001 .01212 L .46018 .01212 L .46036 .01212 L .46054 .01212 L .46069 .01212 L .46088 .01212 L .46108 .01212 L .46127 .01212 L .46145 .01212 L .46164 .01212 L Mistroke .46183 .01212 L .462 .01212 L .46218 .01212 L .46238 .01212 L .46259 .01212 L .46277 .01212 L .46296 .01212 L .46317 .01212 L .46337 .01212 L .46347 .01212 L .46358 .01212 L .4637 .01212 L .46381 .01212 L .46401 .01212 L .4642 .01212 L .4644 .01212 L .46461 .01212 L .46479 .01212 L .46498 .01212 L .46516 .01212 L .46533 .01212 L .46552 .01212 L .46573 .01212 L .46593 .01212 L .46611 .01212 L .46631 .01212 L .46652 .01212 L .46671 .01212 L .46689 .01212 L .4671 .01212 L .46722 .01212 L .46733 .01212 L .46754 .01212 L .46764 .01212 L .46775 .01212 L .46786 .01212 L .46798 .01212 L .46818 .01212 L .4683 .01212 L .4684 .01212 L .46852 .01212 L .46864 .01212 L .46875 .01212 L .46887 .01212 L .46898 .01212 L .46908 .01212 L .46928 .01212 L .46948 .01212 L .46968 .01212 L .46989 .01212 L Mistroke .47009 .01212 L .47028 .01212 L .47046 .01212 L .47063 .01212 L .47082 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.48199 .01212 L .4821 .01212 L .48229 .01212 L .48249 .01212 L .48266 .01212 L .48283 .01212 L .48302 .01212 L .48321 .01212 L .48341 .01212 L .48352 .01212 L .48363 .01212 L .48384 .01212 L .48396 .01212 L .48407 .01212 L .48426 .01212 L .48447 .01212 L .48467 .01212 L .48485 .01212 L .48502 .01212 L .4852 .01212 L .48537 .01212 L .48553 .01212 L .48572 .01212 L .48592 .01212 L .48611 .01212 L .48628 .01212 L .48648 .01212 L .48667 .01212 L .48684 .01212 L .48702 .01212 L .48722 .01212 L .48743 .01212 L .48761 .01212 L Mistroke .4878 .01212 L .48801 .01212 L .48821 .01212 L .48831 .01212 L .48842 .01212 L .48854 .01212 L .48865 .01212 L .48885 .01212 L .48903 .01212 L .48924 .01212 L .48945 .01212 L .48963 .01212 L .48982 .01212 L .49 .01212 L .49017 .01212 L .49036 .01212 L .49057 .01212 L .49077 .01212 L .49095 .01212 L .49114 .01212 L .49135 .01212 L .49155 .01212 L .49173 .01212 L .49194 .01212 L .49206 .01212 L .49217 .01212 L .49237 .01212 L .49247 .01212 L .49259 .01212 L .4927 .01212 L .49281 .01212 L .49302 .01212 L .49313 .01212 L .49324 .01212 L .49336 .01212 L .49348 .01212 L .49359 .01212 L .49371 .01212 L .49382 .01212 L .49392 .01212 L .49411 .01212 L .49432 .01212 L .49451 .01212 L .49473 .01212 L .49493 .01212 L .49511 .01212 L .4953 .01212 L .49546 .01212 L .49566 .01212 L .49584 .01212 L Mistroke .49603 .01212 L .49624 .01212 L .49642 .01212 L .49662 .01212 L .49683 .01212 L .49695 .01212 L .49706 .01212 L .49727 .01212 L .49747 .01212 L .49764 .01212 L .49782 .01212 L .49802 .01212 L .4982 .01212 L .49837 .01212 L .49853 .01212 L .4987 .01212 L .49888 .01212 L .49906 .01212 L .49926 .01212 L .49944 .01212 L .49961 .01212 L .49981 .01212 L .49992 .01212 L .50003 .01212 L .50022 .01212 L .50042 .01212 L .50061 .01212 L .50082 .01212 L .50102 .01212 L .50121 .01212 L .50139 .01212 L .50156 .01212 L .50175 .01212 L .50194 .01212 L .50214 .01212 L .50235 .01212 L .50252 .01212 L .50272 .01212 L .50293 .01212 L .50305 .01212 L .50316 .01212 L .50337 .01212 L .50357 .01212 L .50374 .01212 L .50393 .01212 L .50412 .01212 L .5043 .01212 L .50448 .01212 L .50463 .01212 L .50481 .01212 L Mistroke .50498 .01212 L .50517 .01212 L .50536 .01212 L .50555 .01212 L .50572 .01212 L .50592 .01212 L .50603 .01212 L .50613 .01212 L .50632 .01212 L .50652 .01212 L .50672 .01212 L .50683 .01212 L .50693 .01212 L .50712 .01212 L .50732 .01212 L .5075 .01212 L .50766 .01212 L .50785 .01212 L .50804 .01212 L .50824 .01212 L .50836 .01212 L .50846 .01212 L .50867 .01212 L .50879 .01212 L .5089 .01212 L .50909 .01212 L .5093 .01212 L .5095 .01212 L .50968 .01212 L .50985 .01212 L .51003 .01212 L .5102 .01212 L .51036 .01212 L .51055 .01212 L .51075 .01212 L .51094 .01212 L .51111 .01212 L .51131 .01212 L .5115 .01212 L .51167 .01212 L .51185 .01212 L .51205 .01212 L .51226 .01212 L .51244 .01212 L .51263 .01212 L .51284 .01212 L .51304 .01212 L .51314 .01212 L .51325 .01212 L .51337 .01212 L Mistroke .51348 .01212 L .51368 .01212 L .51386 .01212 L .51407 .01212 L .51428 .01212 L .51446 .01212 L .51465 .01212 L .51483 .01212 L .51499 .01212 L .51519 .01212 L .5154 .01212 L .5156 .01212 L .51578 .01212 L .51597 .01212 L .51618 .01212 L .51638 .01212 L .51656 .01212 L .51677 .01212 L .51689 .01212 L .517 .01212 L .5172 .01212 L .51731 .01212 L .51742 .01212 L .51752 .01212 L .51764 .01212 L .51784 .01212 L .51796 .01212 L .51807 .01212 L .51818 .01212 L .51831 .01212 L .51842 .01212 L .51853 .01212 L .51865 .01212 L .51875 .01212 L .51894 .01212 L .51915 .01212 L .51934 .01212 L .51955 .01212 L .51975 .01212 L .51994 .01212 L .52012 .01212 L .52029 .01212 L .52048 .01212 L .52066 .01212 L .52086 .01212 L .52107 .01212 L .52125 .01212 L .52145 .01212 L .52166 .01212 L .52178 .01212 L Mistroke .52188 .01212 L .52209 .01212 L .52229 .01212 L .52246 .01212 L .52265 .01212 L .52284 .01212 L .52303 .01212 L .5232 .01212 L .52335 .01212 L .52353 .01212 L .52371 .01212 L .52389 .01212 L .52409 .01212 L .52427 .01212 L .52444 .01212 L .52464 .01212 L .52475 .01212 L .52485 .01212 L .52504 .01212 L .52524 .01212 L .52544 .01212 L .52565 .01212 L .52585 .01212 L .52604 .01212 L .52622 .01212 L .52639 .01212 L .52658 .01212 L .52676 .01212 L .52696 .01212 L .52717 .01212 L .52735 .01212 L .52755 .01212 L .52776 .01212 L .52788 .01212 L .52798 .01212 L .5282 .01212 L .52839 .01212 L .52856 .01212 L .52875 .01212 L .52895 .01212 L .52913 .01212 L .5293 .01212 L .52946 .01212 L .52963 .01212 L .52981 .01212 L .52999 .01212 L .53019 .01212 L .53037 .01212 L .53054 .01212 L .53074 .01212 L Mistroke .53085 .01212 L .53095 .01212 L .53114 .01212 L .53134 .01212 L .53154 .01212 L .53165 .01212 L .53175 .01212 L .53194 .01212 L .53214 .01212 L .53232 .01212 L .53249 .01212 L .53267 .01212 L .53287 .01212 L .53307 .01212 L .53318 .01212 L .53328 .01212 L .53349 .01212 L .53361 .01212 L .53372 .01212 L .53391 .01212 L .53412 .01212 L .53432 .01212 L .5345 .01212 L .53467 .01212 L .53485 .01212 L .53503 .01212 L .53518 .01212 L .53537 .01212 L .53557 .01212 L .53576 .01212 L .53594 .01212 L .53613 .01212 L .53632 .01212 L .53649 .01212 L .53667 .01212 L .53687 .01212 L .53708 .01212 L .53726 .01212 L .53745 .01212 L .53766 .01212 L .53786 .01212 L .53796 .01212 L .53807 .01212 L .53819 .01212 L .5383 .01212 L .5385 .01212 L .53868 .01212 L .53889 .01212 L .5391 .01212 L .53928 .01212 L Mistroke .53947 .01212 L .53965 .01212 L .53982 .01212 L .54001 .01212 L .54022 .01212 L .54042 .01212 L .5406 .01212 L .54079 .01212 L .541 .01212 L .5412 .01212 L .54138 .01212 L .54159 .01212 L .54171 .01212 L .54182 .01212 L .54202 .01212 L .54213 .01212 L .54224 .01212 L .54234 .01212 L .54246 .01212 L .54266 .01212 L .54278 .01212 L .54289 .01212 L .543 .01212 L .54313 .01212 L .54324 .01212 L .54335 .01212 L .54347 .01212 L .54357 .01212 L .54376 .01212 L .54397 .01212 L .54416 .01212 L .54437 .01212 L .54457 .01212 L .54476 .01212 L .54494 .01212 L .54511 .01212 L .5453 .01212 L .54548 .01212 L .54568 .01212 L .54589 .01212 L .54607 .01212 L .54627 .01212 L .54648 .01212 L .5466 .01212 L .5467 .01212 L .54691 .01212 L .54711 .01212 L .54728 .01212 L .54747 .01212 L .54766 .01212 L Mistroke .54785 .01212 L .54802 .01212 L .54817 .01212 L .54834 .01212 L .54852 .01212 L .54871 .01212 L .5489 .01212 L .54909 .01212 L .54926 .01212 L .54946 .01212 L .54957 .01212 L .54967 .01212 L .54986 .01212 L .55006 .01212 L .55025 .01212 L .55047 .01212 L .55067 .01212 L .55085 .01212 L .55104 .01212 L .5512 .01212 L .5514 .01212 L .55158 .01212 L .55178 .01212 L .55199 .01212 L .55217 .01212 L .55236 .01212 L .55257 .01212 L .55269 .01212 L .5528 .01212 L .55301 .01212 L .55321 .01212 L .55338 .01212 L .55357 .01212 L .55376 .01212 L .55395 .01212 L .55412 .01212 L .55427 .01212 L .55444 .01212 L .55462 .01212 L .55481 .01212 L .555 .01212 L .55519 .01212 L .55536 .01212 L .55556 .01212 L .55567 .01212 L .55577 .01212 L .55596 .01212 L .55616 .01212 L .55636 .01212 L .55657 .01212 L Mistroke .55677 .01212 L .55696 .01212 L .55712 .01212 L .55731 .01212 L .5575 .01212 L .55768 .01212 L .55788 .01212 L .55809 .01212 L .55827 .01212 L .55847 .01212 L .55868 .01212 L .5588 .01212 L .55891 .01212 L .55912 .01212 L .55932 .01212 L .55949 .01212 L .55967 .01212 L .55987 .01212 L .56005 .01212 L .56022 .01212 L .56038 .01212 L .56056 .01212 L .56073 .01212 L .56092 .01212 L .56111 .01212 L .5613 .01212 L .56147 .01212 L .56167 .01212 L .56178 .01212 L .56188 .01212 L .56207 .01212 L .56227 .01212 L .56246 .01212 L .56258 .01212 L .56268 .01212 L .56287 .01212 L .56307 .01212 L .56325 .01212 L .56341 .01212 L .5636 .01212 L .5638 .01212 L .56399 .01212 L .56411 .01212 L .56421 .01212 L .56442 .01212 L .56454 .01212 L .56465 .01212 L .56484 .01212 L .56505 .01212 L .56525 .01212 L Mistroke .56543 .01212 L .5656 .01212 L .56578 .01212 L .56596 .01212 L .56612 .01212 L .5663 .01212 L .5665 .01212 L .56669 .01212 L .56687 .01212 L .56707 .01212 L .56725 .01212 L .56742 .01212 L .5676 .01212 L .5678 .01212 L .56801 .01212 L .56819 .01212 L .56839 .01212 L .5686 .01212 L .56879 .01212 L .56889 .01212 L .56901 .01212 L .56912 .01212 L .56923 .01212 L .56943 .01212 L .56962 .01212 L .56982 .01212 L .57003 .01212 L .57021 .01212 L .57041 .01212 L .57059 .01212 L .57075 .01212 L .57094 .01212 L .57115 .01212 L .57135 .01212 L .57153 .01212 L .57173 .01212 L .57194 .01212 L .57214 .01212 L .57232 .01212 L .57253 .01212 L .57265 .01212 L .57276 .01212 L .57296 .01212 L .57306 .01212 L .57317 .01212 L .57328 .01212 L .5734 .01212 L .5736 .01212 L .57372 .01212 L .57382 .01212 L Mistroke .57394 .01212 L .57407 .01212 L .57417 .01212 L .57429 .01212 L .5744 .01212 L .57451 .01212 L .5747 .01212 L .57491 .01212 L .5751 .01212 L .57531 .01212 L .57551 .01212 L .5757 .01212 L .57588 .01212 L .57605 .01212 L .57624 .01212 L .57642 .01212 L .57662 .01212 L .57683 .01212 L .57701 .01212 L .57721 .01212 L .57742 .01212 L .57754 .01212 L .57765 .01212 L .57786 .01212 L .57806 .01212 L .57822 .01212 L .57841 .01212 L .57861 .01212 L .57879 .01212 L .57896 .01212 L .57912 .01212 L .57929 .01212 L .57947 .01212 L .57965 .01212 L .57985 .01212 L .58003 .01212 L .5802 .01212 L .5804 .01212 L .58051 .01212 L .58061 .01212 L .5808 .01212 L .581 .01212 L .5812 .01212 L .58141 .01212 L .58161 .01212 L .5818 .01212 L .58198 .01212 L .58215 .01212 L .58234 .01212 L .58253 .01212 L Mistroke .58272 .01212 L .58294 .01212 L .58311 .01212 L .58331 .01212 L .58352 .01212 L .58364 .01212 L .58375 .01212 L .58396 .01212 L .58416 .01212 L .58433 .01212 L .58452 .01212 L .58471 .01212 L .58489 .01212 L .58507 .01212 L .58522 .01212 L .5854 .01212 L .58557 .01212 L .58576 .01212 L .58595 .01212 L .58614 .01212 L .58631 .01212 L .58651 .01212 L .58662 .01212 L .58672 .01212 L .58691 .01212 L .58711 .01212 L .58731 .01212 L .58742 .01212 L .58752 .01212 L .58771 .01212 L .58791 .01212 L .58809 .01212 L .58825 .01212 L .58844 .01212 L .58864 .01212 L .58883 .01212 L .58895 .01212 L .58905 .01212 L .58926 .01212 L .58938 .01212 L .58949 .01212 L .58968 .01212 L .58989 .01212 L .59009 .01212 L .59027 .01212 L .59044 .01212 L .59063 .01212 L .5908 .01212 L .59096 .01212 L .59114 .01212 L Mistroke .59134 .01212 L .59153 .01212 L .59171 .01212 L .59191 .01212 L .59209 .01212 L .59226 .01212 L .59244 .01212 L .59264 .01212 L .59285 .01212 L .59303 .01212 L .59323 .01212 L .59344 .01212 L .59363 .01212 L .59373 .01212 L .59384 .01212 L .59396 .01212 L .59407 .01212 L .59427 .01212 L .59446 .01212 L .59466 .01212 L .59487 .01212 L .59505 .01212 L .59524 .01212 L .59543 .01212 L .59559 .01212 L .59578 .01212 L .59599 .01212 L .59619 .01212 L .59637 .01212 L .59657 .01212 L .59678 .01212 L .59697 .01212 L .59715 .01212 L .59736 .01212 L .59748 .01212 L .59759 .01212 L .5978 .01212 L .5979 .01212 L .59801 .01212 L .59812 .01212 L .59823 .01212 L .59844 .01212 L .59856 .01212 L .59866 .01212 L .59878 .01212 L .5989 .01212 L .59901 .01212 L .59913 .01212 L .59924 .01212 L .59934 .01212 L Mistroke .59954 .01212 L .59974 .01212 L .59994 .01212 L .60015 .01212 L .60035 .01212 L .60054 .01212 L .60072 .01212 L .60089 .01212 L .60108 .01212 L .60126 .01212 L .60146 .01212 L .60167 .01212 L .60185 .01212 L .60204 .01212 L .60226 .01212 L .60237 .01212 L .60248 .01212 L .60269 .01212 L .60289 .01212 L .60306 .01212 L .60325 .01212 L .60344 .01212 L .60363 .01212 L .6038 .01212 L .60395 .01212 L .60412 .01212 L .6043 .01212 L .60449 .01212 L .60468 .01212 L .60487 .01212 L .60504 .01212 L .60524 .01212 L .60535 .01212 L .60545 .01212 L .60564 .01212 L .60584 .01212 L .60603 .01212 L .60625 .01212 L .60645 .01212 L .60663 .01212 L .60682 .01212 L .60699 .01212 L .60718 .01212 L .60736 .01212 L .60756 .01212 L .60777 .01212 L .60795 .01212 L .60815 .01212 L .60836 .01212 L .60848 .01212 L Mistroke .60858 .01212 L .60879 .01212 L .60899 .01212 L .60916 .01212 L .60935 .01212 L .60955 .01212 L .60973 .01212 L .6099 .01212 L .61006 .01212 L .61024 .01212 L .61041 .01212 L .61059 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L .62161 .01212 L .6218 .01212 L .62198 .01212 L .62219 .01212 L .62231 .01212 L .62242 .01212 L .62263 .01212 L .62274 .01212 L .62284 .01212 L .62295 .01212 L .62306 .01212 L .62327 .01212 L .62338 .01212 L .62349 .01212 L .62361 .01212 L .62373 .01212 L .62384 .01212 L .62396 .01212 L .62407 .01212 L .62417 .01212 L .62437 .01212 L .62457 .01212 L .62477 .01212 L .62498 .01212 L .62518 .01212 L .62536 .01212 L Mistroke .62555 .01212 L .62572 .01212 L .62591 .01212 L .62609 .01212 L .62629 .01212 L .6265 .01212 L .62668 .01212 L .62687 .01212 L .62708 .01212 L .6272 .01212 L .62731 .01212 L .62752 .01212 L .62772 .01212 L .62789 .01212 L .62808 .01212 L .62827 .01212 L .62845 .01212 L .62862 .01212 L .62878 .01212 L .62895 .01212 L .62913 .01212 L .62932 .01212 L .62951 .01212 L .6297 .01212 L .62987 .01212 L .63006 .01212 L .63017 .01212 L .63028 .01212 L .63047 .01212 L .63067 .01212 L .63086 .01212 L .63107 .01212 L .63128 .01212 L .63146 .01212 L .63165 .01212 L .63181 .01212 L .63201 .01212 L .63219 .01212 L .63239 .01212 L .6326 .01212 L .63278 .01212 L .63297 .01212 L .63318 .01212 L .6333 .01212 L .63341 .01212 L .63362 .01212 L .63382 .01212 L .63399 .01212 L .63418 .01212 L .63437 .01212 L Mistroke .63456 .01212 L .63473 .01212 L .63488 .01212 L .63505 .01212 L .63523 .01212 L .63542 .01212 L .63561 .01212 L .6358 .01212 L .63597 .01212 L .63617 .01212 L .63628 .01212 L .63638 .01212 L .63657 .01212 L .63677 .01212 L .63697 .01212 L .63708 .01212 L .63718 .01212 L .63737 .01212 L .63757 .01212 L .63775 .01212 L .63791 .01212 L .6381 .01212 L .63829 .01212 L .63849 .01212 L .6386 .01212 L .63871 .01212 L .63892 .01212 L .63904 .01212 L .63915 .01212 L .63934 .01212 L .63955 .01212 L .63975 .01212 L .63993 .01212 L .6401 .01212 L .64028 .01212 L .64045 .01212 L .64061 .01212 L .6408 .01212 L .641 .01212 L .64119 .01212 L .64136 .01212 L .64156 .01212 L .64175 .01212 L .64192 .01212 L .6421 .01212 L .6423 .01212 L .64251 .01212 L .64268 .01212 L .64288 .01212 L .64309 .01212 L Mistroke .64329 .01212 L .64339 .01212 L .6435 .01212 L .64362 .01212 L .64373 .01212 L .64393 .01212 L .64411 .01212 L .64431 .01212 L .64453 .01212 L .64471 .01212 L .6449 .01212 L .64508 .01212 L .64524 .01212 L .64544 .01212 L .64565 .01212 L .64584 .01212 L .64603 .01212 L .64622 .01212 L .64643 .01212 L .64663 .01212 L .64681 .01212 L .64702 .01212 L .64714 .01212 L .64725 .01212 L .64745 .01212 L .64756 .01212 L .64766 .01212 L .64777 .01212 L .64789 .01212 L .64809 .01212 L .64821 .01212 L .64831 .01212 L .64843 .01212 L .64856 .01212 L .64866 .01212 L .64878 .01212 L .64889 .01212 L .649 .01212 L .64919 .01212 L .6494 .01212 L .64959 .01212 L .6498 .01212 L .65 .01212 L .65019 .01212 L .65037 .01212 L .65054 .01212 L .65073 .01212 L .65091 .01212 L .65111 .01212 L .65132 .01212 L Mistroke .6515 .01212 L .65169 .01212 L .6519 .01212 L .65202 .01212 L .65213 .01212 L .65234 .01212 L .65254 .01212 L .65271 .01212 L .6529 .01212 L .65309 .01212 L .65327 .01212 L .65344 .01212 L .6536 .01212 L .65377 .01212 L .65395 .01212 L .65414 .01212 L .65433 .01212 L .65452 .01212 L .65469 .01212 L .65488 .01212 L .65499 .01212 L .6551 .01212 L .65529 .01212 L .65549 .01212 L .65568 .01212 L .65589 .01212 L .6561 .01212 L .65628 .01212 L .65647 .01212 L .65663 .01212 L .65683 .01212 L .65701 .01212 L .65721 .01212 L .65742 .01212 L .65759 .01212 L .65779 .01212 L .658 .01212 L .65812 .01212 L .65823 .01212 L .65844 .01212 L .65864 .01212 L .65881 .01212 L .659 .01212 L .65919 .01212 L .65937 .01212 L .65954 .01212 L .6597 .01212 L .65987 .01212 L .66005 .01212 L .66024 .01212 L Mistroke .66043 .01212 L .66062 .01212 L .66079 .01212 L .66098 .01212 L .6611 .01212 L .6612 .01212 L .66139 .01212 L .66159 .01212 L .66178 .01212 L .662 .01212 L .6622 .01212 L .66238 .01212 L .66255 .01212 L .66274 .01212 L .66293 .01212 L .66311 .01212 L .66331 .01212 L .66352 .01212 L .6637 .01212 L .6639 .01212 L .66411 .01212 L .66423 .01212 L .66434 .01212 L .66455 .01212 L .66475 .01212 L .66492 .01212 L .6651 .01212 L .6653 .01212 L .66548 .01212 L .66565 .01212 L .66581 .01212 L .66599 .01212 L .66616 .01212 L .66635 .01212 L .66654 .01212 L .66673 .01212 L .6669 .01212 L .66709 .01212 L .66721 .01212 L .66731 .01212 L .6675 .01212 L .6677 .01212 L .66789 .01212 L .668 .01212 L .66811 .01212 L .6683 .01212 L .6685 .01212 L .66867 .01212 L .66884 .01212 L .66903 .01212 L Mistroke .66922 .01212 L .66942 .01212 L .66954 .01212 L .66964 .01212 L .66985 .01212 L .66997 .01212 L .67008 .01212 L .67027 .01212 L .67048 .01212 L .67068 .01212 L .67086 .01212 L .67103 .01212 L .67121 .01212 L .67139 .01212 L .67154 .01212 L .67173 .01212 L .67193 .01212 L .67212 .01212 L .6723 .01212 L .67249 .01212 L .67268 .01212 L .67285 .01212 L .67303 .01212 L .67323 .01212 L .67344 .01212 L .67362 .01212 L .67381 .01212 L .67403 .01212 L .67422 .01212 L .67432 .01212 L .67443 .01212 L .67455 .01212 L .67466 .01212 L .67486 .01212 L .67505 .01212 L .67526 .01212 L .67546 .01212 L .67564 .01212 L .67583 .01212 L .67602 .01212 L .67618 .01212 L .67637 .01212 L .67658 .01212 L .67678 .01212 L .67696 .01212 L .67716 .01212 L .67737 .01212 L .67757 .01212 L .67775 .01212 L .67796 .01212 L Mistroke .67808 .01212 L .67819 .01212 L .67839 .01212 L .67849 .01212 L .6786 .01212 L .67871 .01212 L .67883 .01212 L .67903 .01212 L .67915 .01212 L .67925 .01212 L .67937 .01212 L .6795 .01212 L .6796 .01212 L .67972 .01212 L .67983 .01212 L .67994 .01212 L .68013 .01212 L .68034 .01212 L .68053 .01212 L .68074 .01212 L .68094 .01212 L .68113 .01212 L .68131 .01212 L .68148 .01212 L .68167 .01212 L .68185 .01212 L .68205 .01212 L .68226 .01212 L .68244 .01212 L .68264 .01212 L .68285 .01212 L .68297 .01212 L .68308 .01212 L .68329 .01212 L .68349 .01212 L .68365 .01212 L .68384 .01212 L .68404 .01212 L .68422 .01212 L .68439 .01212 L .68455 .01212 L .68472 .01212 L .6849 .01212 L .68508 .01212 L .68528 .01212 L .68546 .01212 L .68563 .01212 L .68583 .01212 L .68594 .01212 L .68604 .01212 L Mistroke .68623 .01212 L .68643 .01212 L .68663 .01212 L .68684 .01212 L .68704 .01212 L .68723 .01212 L .68741 .01212 L .68758 .01212 L .68777 .01212 L .68796 .01212 L .68816 .01212 L .68837 .01212 L .68855 .01212 L .68874 .01212 L .68895 .01212 L .68907 .01212 L .68918 .01212 L .68939 .01212 L .68959 .01212 L .68976 .01212 L .68995 .01212 L .69014 .01212 L .69033 .01212 L .6905 .01212 L .69065 .01212 L .69083 .01212 L .691 .01212 L .69119 .01212 L .69139 .01212 L .69157 .01212 L .69174 .01212 L .69194 .01212 L .69205 .01212 L .69215 .01212 L .69234 .01212 L .69254 .01212 L .69274 .01212 L .69285 .01212 L .69295 .01212 L .69314 .01212 L .69334 .01212 L .69352 .01212 L .69369 .01212 L .69387 .01212 L .69407 .01212 L .69427 .01212 L .69438 .01212 L .69448 .01212 L .69469 .01212 L .69481 .01212 L Mistroke .69492 .01212 L .69511 .01212 L .69532 .01212 L .69552 .01212 L .6957 .01212 L .69587 .01212 L .69606 .01212 L .69623 .01212 L .69639 .01212 L .69657 .01212 L .69677 .01212 L .69696 .01212 L .69714 .01212 L .69734 .01212 L .69752 .01212 L .69769 .01212 L .69787 .01212 L .69807 .01212 L .69828 .01212 L .69846 .01212 L .69866 .01212 L .69887 .01212 L .69906 .01212 L .69916 .01212 L .69928 .01212 L .69939 .01212 L .6995 .01212 L .6997 .01212 L .69989 .01212 L .70009 .01212 L .7003 .01212 L .70048 .01212 L .70068 .01212 L .70086 .01212 L .70102 .01212 L .70121 .01212 L .70142 .01212 L .70162 .01212 L .7018 .01212 L .702 .01212 L .70221 .01212 L .7024 .01212 L .70258 .01212 L .7028 .01212 L .70291 .01212 L .70303 .01212 L .70323 .01212 L .70333 .01212 L .70344 .01212 L .70355 .01212 L Mistroke .70367 .01212 L .70387 .01212 L .70399 .01212 L .70409 .01212 L .70421 .01212 L .70434 .01212 L .70444 .01212 L .70456 .01212 L .70467 .01212 L .70478 .01212 L .70497 .01212 L .70518 .01212 L .70537 .01212 L .70558 .01212 L .70578 .01212 L .70597 .01212 L .70615 .01212 L .70632 .01212 L .70651 .01212 L .70669 .01212 L .70689 .01212 L .7071 .01212 L .70728 .01212 L .70748 .01212 L .70769 .01212 L .70781 .01212 L .70791 .01212 L .70812 .01212 L .70832 .01212 L .70849 .01212 L .70868 .01212 L .70887 .01212 L .70906 .01212 L .70923 .01212 L .70938 .01212 L .70956 .01212 L .70974 .01212 L .70992 .01212 L .71012 .01212 L .7103 .01212 L .71047 .01212 L .71067 .01212 L .71078 .01212 L .71088 .01212 L .71107 .01212 L .71127 .01212 L .71147 .01212 L .71168 .01212 L .71188 .01212 L .71207 .01212 L Mistroke .71225 .01212 L .71242 .01212 L .71261 .01212 L .71279 .01212 L .71299 .01212 L .7132 .01212 L .71338 .01212 L .71358 .01212 L .71379 .01212 L .71391 .01212 L .71402 .01212 L .71423 .01212 L .71443 .01212 L .7146 .01212 L .71478 .01212 L .71498 .01212 L .71516 .01212 L .71533 .01212 L .71549 .01212 L .71567 .01212 L .71584 .01212 L .71602 .01212 L .71622 .01212 L .71641 .01212 L .71657 .01212 L .71677 .01212 L .71688 .01212 L .71699 .01212 L .71718 .01212 L .71738 .01212 L .71757 .01212 L .71768 .01212 L .71779 .01212 L .71797 .01212 L .71817 .01212 L .71835 .01212 L .71852 .01212 L .7187 .01212 L .7189 .01212 L .7191 .01212 L .71921 .01212 L .71932 .01212 L .71953 .01212 L .71965 .01212 L .71976 .01212 L .71995 .01212 L .72016 .01212 L .72036 .01212 L .72054 .01212 L .72071 .01212 L Mistroke .72089 .01212 L .72106 .01212 L .72122 .01212 L .7214 .01212 L .72161 .01212 L .7218 .01212 L .72197 .01212 L .72217 .01212 L .72236 .01212 L .72253 .01212 L .72271 .01212 L .72291 .01212 L .72311 .01212 L .72329 .01212 L .72349 .01212 L .7237 .01212 L .7239 .01212 L .724 .01212 L .72411 .01212 L .72423 .01212 L .72434 .01212 L .72454 .01212 L .72472 .01212 L .72492 .01212 L .72514 .01212 L .72532 .01212 L .72551 .01212 L .72569 .01212 L .72585 .01212 L .72605 .01212 L .72626 .01212 L .72645 .01212 L .72664 .01212 L .72683 .01212 L .72704 .01212 L .72724 .01212 L .72742 .01212 L .72763 .01212 L .72775 .01212 L .72786 .01212 L .72806 .01212 L .72816 .01212 L .72827 .01212 L .72838 .01212 L .7285 .01212 L .7287 .01212 L 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Before this can be done we need to pick a trial \ Energy for the system. This energy will be varied to make sure the boundary conditions discussed in \ the introduction are met. The first argument ( \[CapitalPsi] ''[x]==2(V[x]-Energy)\[CapitalPsi][x] ) in \ the NDSolve is the second order differential equation we are solving. The \ next two are boundary conditions. The first makes sure that the value of the \ wavefunction is zero at the left edge of the potential barrier. The second \ is a dummy condition. The next argument indicates the function to be solved, \ \[CapitalPsi]. This is followed by the independent variable and the interval \ in which to solve the differential equation. 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