Different Integration Results
- To: mathgroup at smc.vnet.net
- Subject: [mg30344] Different Integration Results
- From: Harald Grossauer <Harald.Grossauer at uibk.ac.at>
- Date: Sat, 11 Aug 2001 03:40:04 -0400 (EDT)
- Organization: University of Innsbruck, Austria
- Sender: owner-wri-mathgroup at wolfram.com
Hi, I have got a problem with the attached notebook. In the last two lines, if I use Integrate[ ] the result is 99/35, NIntegrate[ ] says it is "1.". Due to the nature of the problem (quantum theory, fourier transform) I would expect the result to be 1 exactly. What could cause this difference? Greetings, Harald --------------D5A2CDAE8C4B1512F9E18669 filename="problem.nb" (*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at 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[ 20275, 615]*) (*NotebookOutlinePosition[ 20913, 638]*) (* CellTagsIndexPosition[ 20869, 634]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(phi[p_] = 18/Sqrt[35]*UnitStep[p]* p*\((Exp[\(-p\)] - \((1/6)\)*Exp[\(-p\)/2])\)\)], "Input"], Cell[BoxData[ \(\(18\ \((\[ExponentialE]\^\(-p\) - \[ExponentialE]\^\(\(-p\)/2\)\/6)\)\ \ p\ UnitStep[p]\)\/\@35\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[phi[p]^2, {p, \(-Infinity\), Infinity}]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(psi[x_] = Simplify[\((1/Sqrt[2*Pi])\)* Integrate[phi[p]*Exp[I*p*x], {p, \(-Infinity\), Infinity}], Element[x, Reals]]\)], "Input"], Cell[BoxData[ \(9\ \@\(2\/\(35\ \[Pi]\)\)\ \((Cos[2\ ArcTan[x]]\/\(1 + x\^2\) - \(2\ \ Cos[2\ ArcTan[2\ x]]\)\/\(3 + 12\ x\^2\) + \[ImaginaryI]\ \((Sin[2\ \ ArcTan[x]]\/\(1 + x\^2\) - \(2\ Sin[2\ ArcTan[2\ x]]\)\/\(3 + 12\ \ x\^2\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(psiconj[x_] = Simplify[\((1/Sqrt[2*Pi])\)* Integrate[phi[p]*Exp[\(-I\)*p*x], {p, \(-Infinity\), Infinity}], Element[x, Reals]]\)], "Input"], Cell[BoxData[ \(9\ \@\(2\/\(35\ \[Pi]\)\)\ \((Cos[2\ ArcTan[x]]\/\(1 + x\^2\) - \(2\ \ Cos[2\ ArcTan[2\ x]]\)\/\(3 + 12\ x\^2\) - \(\[ImaginaryI]\ Sin[2\ ArcTan[x]]\ \)\/\(1 + x\^2\) + \(2\ \[ImaginaryI]\ Sin[2\ ArcTan[2\ x]]\)\/\(3 + 12\ x\^2\ \))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(rho[x_] = Simplify[psi[x]*psiconj[x], Element[x, Reals]]\)], "Input"], 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