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

• To: mathgroup at smc.vnet.net
• Subject: [mg91475] Re: Integral of radial solution (hydrogen atom) is not evaluated
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Sat, 23 Aug 2008 01:41:18 -0400 (EDT)
• Organization: Uni Leipzig
• References: <g8lp87\$ih2\$1@smc.vnet.net>

```Hi,

even if you would do it by hand, you have to substitute
2*r/n in the arguments to use the orthogonality relation
and Mathematica can't evaluate
Integrate[
LaguerreL[n1, l1, r]*LaguerreL[n2, l2, r]*Exp[-r], {r, 0, Infinity}]

or

Integrate[
LaguerreL[n, l, r]*LaguerreL[n, l, r]*Exp[-r], {r, 0, Infinity}]

so you have to use you head to solve the integral or define the
corresponding integration rules.

Regards
Jens

> 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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