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MathGroup Archive 2001

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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

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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[
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\)\/\(1 + x\^2\) + \(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"],

Cell[BoxData[
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                12\ \((1 + 5\ x\^2 + 4\ x\^4)\)\ Cos[
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                          2\ x]])\)\)\/\(35\ \[Pi]\ \((1 + 5\ x\^2 + 4\ x\^4)\
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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