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RE: Integral of radial solution (hydrogen atom) is not evaluated

• To: mathgroup at smc.vnet.net
• Subject: [mg91494] RE: [mg91443] Integral of radial solution (hydrogen atom) is not evaluated
• From: "Tony Harker" <a.harker at ucl.ac.uk>
• Date: Sun, 24 Aug 2008 07:04:46 -0400 (EDT)
• References: <200808220710.DAA18261@smc.vnet.net>

 I'm not entirely surprised that Mathematica fails to evaluate the integral,
as it requires a bit of fiddling of recurrence relations to get it into a
standard form. What is a bit surprising, though, is that the standard form
itslef, the fundamental integral Integrate[r^k Exp[-r] LaguerreL[n, k, r]^2,
{r, 0, Infinity}], is not evaluated in Mathematica 6.0.3 (even with
appropriate Assumptions). 

   Tony Harker

]-> -----Original Message-----
]-> From: Gehricht at googlemail.com [mailto:Gehricht at googlemail.com] 
]-> Sent: 22 August 2008 08:11
]-> To: mathgroup at smc.vnet.net
]-> Subject: [mg91443] Integral of radial solution (hydrogen 
]-> atom) is not evaluated
]-> 
]-> 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}}]
]-> 
]-> 
]-> 

• References:
  • Integral of radial solution (hydrogen atom) is not evaluated
    • From: "Gehricht@googlemail.com" <Gehricht@googlemail.com>