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>
- Integral of radial solution (hydrogen atom) is not evaluated