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Integral of radial solution (hydrogen atom) is not evaluated

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91443] Integral of radial solution (hydrogen atom) is not evaluated
  • From: "Gehricht at googlemail.com" <Gehricht at googlemail.com>
  • Date: Fri, 22 Aug 2008 03:10:57 -0400 (EDT)

Hi!

I want to integrate the radial solution of the hydrogen atom from zero
to infinity. The following code (for the corresponding cell
expression, see below) just returns an unevaluated integral:

In:: R=r^l*Exp[-(r/n)]*(2/n)^l*2/n^2*Sqrt[(n-l-1)!/(n+l)!]*LaguerreL[n-
l-1,2*l+1,(2*r)/n]
Assuming[{Element[n,Integers],Element[l,Integers],n>0,n>l>=0},integrand=FullSimplify[(R*r)^2];Simplify[Integrate[integrand,
{r,0,\[Infinity]}]]]

Out::
\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Infinity]\)]\(
FractionBox[\(
\*SuperscriptBox[\(4\), \(1 + l\)]\
\*SuperscriptBox[\(E\), \(-
\*FractionBox[\(2\ r\), \(n\)]\)]\
\*SuperscriptBox[\(n\), \(\(-2\)\ \((2 + l)\)\)]\
\*SuperscriptBox[\(r\), \(2 + 2\ l\)]\ Gamma[\(-l\) + n]\
\*SuperscriptBox[\(LaguerreL[\(-1\) - l + n, 1 + 2\ l,
\*FractionBox[\(2\ r\), \(n\)]]\), \(2\)]\), \(\((l +
      n)\)!\)] \[DifferentialD]r\)\)

I do not know, why the integral is left unevaluated and what I am
doing wrong respectively. Any help appreciated.
With thanks
Yours Wolfgang
---
Cell[BoxData[{
 RowBox[{"R", "=",
  RowBox[{
   SuperscriptBox["r", "l"], "*",
   RowBox[{"Exp", "[",
    RowBox[{"-",
     FractionBox["r", "n"]}], "]"}], "*",
   SuperscriptBox[
    RowBox[{"(",
     FractionBox["2", "n"], ")"}], "l"], "*",
   FractionBox["2",
    SuperscriptBox["n", "2"]], "*",
   SqrtBox[
    FractionBox[
     RowBox[{
      RowBox[{"(",
       RowBox[{"n", "-", "l", "-", "1"}], ")"}], "!"}],
     RowBox[{
      RowBox[{"(",
       RowBox[{"n", "+", "l"}], ")"}], "!"}]]], "*",
   RowBox[{"LaguerreL", "[",
    RowBox[{
     RowBox[{"n", "-", "l", "-", "1"}], ",",
     RowBox[{
      RowBox[{"2", "*", "l"}], "+", "1"}], ",",
     FractionBox[
      RowBox[{"2", "*", "r"}], "n"]}],
    "]"}]}]}], "\[IndentingNewLine]",
 RowBox[{"Assuming", "[",
  RowBox[{
   RowBox[{"{",
    RowBox[{
     RowBox[{"Element", "[",
      RowBox[{"n", ",", "Integers"}], "]"}], ",",
     RowBox[{"Element", "[",
      RowBox[{"l", ",", "Integers"}], "]"}], ",",
     RowBox[{"n", ">", "0"}], ",",
     RowBox[{"n", ">", "l", "\[GreaterEqual]", "0"}]}], "}"}], ",",
   RowBox[{
    RowBox[{"integrand", "=",
     RowBox[{"FullSimplify", "[",
      SuperscriptBox[
       RowBox[{"(",
        RowBox[{"R", "*", "r"}], ")"}], "2"], "]"}]}], ";",
    RowBox[{"Simplify", "[",
     RowBox[{"Integrate", "[",
      RowBox[{"integrand", ",",
       RowBox[{"{",
        RowBox[{"r", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}],
     "]"}]}]}], "]"}]}], "Input",
 CellChangeTimes->{{3.427632357819639*^9, 3.427632377400957*^9}, {
   3.427632644205412*^9, 3.427632664272697*^9},
   3.4282990139911003`*^9, {3.428323376643766*^9,
   3.428323506586858*^9}, {3.42832353786147*^9,
   3.428323548325508*^9}, {3.428324884676128*^9,
   3.428324891782604*^9}}]



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